Simulation of Wireless Communication Systems

Modeling of Wireless Communication Systems using MATLAB Dr. B.-P. Paris Dept. Electrical and Comp. Engineering George Mason University last updated September 23, 2009 ©2009, B.-P. Paris Wireless Communications 1

Approach This course aims to cover both theory and practice of wireless commuication systems, and the simulation of such systems using MATLAB. Both topics are given approximately equal treatment. After a brief introduction to MATLAB, theory and MATLAB simulation are pursued in parallel. This approach allows us to make concepts concrete and/or to visualize relevant signals. In the process, a toolbox of MATLAB functions is constructed. Hopefully, the toolbox will be useful for your own projects. Illustrates good MATLAB practices. ©2009, B.-P. Paris Wireless Communications 2

Outline - Part I: From Theory to Simulation Part I: Learning Objectives Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation ©2009, B.-P. Paris Wireless Communications 4

Outline - Part II: Digital Modulation and Spectrum Part II: Learning Objectives Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation ©2009, B.-P. Paris Wireless Communications 5

Outline - Part III: The Wireless Channel Part III: Learning Objectives Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels ©2009, B.-P. Paris Wireless Communications 6

User Interface Working with Vectors Visualization Prologue: Just Enough MATLAB to ... MATLAB will be used throughout this course to illustrate theory and key concepts. MATLAB is very well suited to model communications systems: Signals are naturally represented in MATLAB, MATLAB has a very large library of functions for processing signals, Visualization of signals is very well supported in MATLAB. MATLAB is used interactively. Eliminates code, compile, run cycle. Great for rapid prototyping and what-if analysis. ©2009, B.-P. Paris Wireless Communications 9

User Interface Working with Vectors Visualization Learning Objectives Getting around in MATLAB The user interface, Getting help. Modeling signals in MATLAB Using vectors to model signals, Creating and manipulating vectors, Visualizing vectors: plotting. ©2009, B.-P. Paris Wireless Communications 11

User Interface Working with Vectors Visualization MATLAB’s Built-in Help System MATLAB has an extensive built-in help system. On-line documentation reader: contains detailed documentation for entire MATLAB system, is invoked by typing doc at command line clicking “Question Mark” in tool bar of main window, via “Help” menu. Command-line help provides access to documentation inside command window. Helpful commands include: help function-name, e.g., help fft. lookfor keyword, e.g., lookfor inverse. We will learn how to tie into the built-in help system. ©2009, B.-P. Paris Wireless Communications 15

User Interface Working with Vectors Visualization Interacting with MATLAB You interact with MATLAB by typing commands at the command prompt (» ) in the command window. MATLAB’s response depends on whether a semicolon is appended after the command or not. If a semicolon is not appended, then MATLAB displays the result of the command. With a semicolon, the result is not displayed. Examples: The command xx = 1:3 produces xx = 1 2 3 The command xx = 1:3; produces no output. The variable xx still stores the result. Do use a semicolon with xx = 1:30000000; ©2009, B.-P. Paris Wireless Communications 16

User Interface Working with Vectors Visualization Signals and Vectors Our objective is to simulate communication systems in MATLAB. This includes the signals that occur in such systems, and processing applied to these signals. In MATLAB (and any other digital system) signals must be represented by samples. Well-founded theory exists regarding sampling (Nyquist’s sampling theorem). Result: Signals are represented as a sequence of numbers. MATLAB is ideally suited to process sequences of numbers. MATLAB’s basic data types: vectors (and matrices). Vectors are just sequence of numbers. ©2009, B.-P. Paris Wireless Communications 18

User Interface Working with Vectors Visualization Illustration: Generating a Sinusoidal Signal The following, simple task illustrates key beneﬁts from MATLAB’s use of vectors. Task: Generate samples of the sinusoidal signal π x (t ) = 3 · cos(2π440t − ) 4 for t ranging from 0 to 10 ms. The sampling rate is 20 KHz. Objective: Compare how this task is accomplished using MATLAB’s vector function, traditional (C-style) for or while loops. ©2009, B.-P. Paris Wireless Communications 19

User Interface Working with Vectors Visualization Illustration: Generating a Sinusoidal Signal For both approaches, we begin by deﬁning a few parameters. This increases readability and makes it easier to change parameters. %% Set Parameters A = 3; % amplitude f = 440; % frequency phi = -pi/4; % phase 7 fs = 20e3; % sampling rate Ts = 0; % start time Te = 10e-3; % end time ©2009, B.-P. Paris Wireless Communications 20

User Interface Working with Vectors Visualization Using Loops The MATLAB code below uses a while loop to generate the samples one by one. The majority of the code is devoted to “book-keeping” tasks. Listing : generateSinusoidLoop.m %initialize loop variables tcur = Ts; kk = 1; 17 while( tcur <= Te) % compute current sample and store in vector tt(kk) = tcur; xx(kk) = A*cos(2*pi*f*tcur + phi); 22 %increment loop variables kk = kk+1; tcur = tcur + 1/fs; end ©2009, B.-P. Paris Wireless Communications 21

User Interface Working with Vectors Visualization Vectorized Code Much more compact code is possible with MATLAB’s vector functions. There is no overhead for managing a program loop. Notice how similar the instruction to generate the samples is to the equation for the signal. The vector-based approach is the key enabler for rapid prototyping. Listing : generateSinusoid.m %% generate sinusoid tt = Ts : 1/fs : Te; % define time-axis xx = A * cos( 2*pi * f * tt + phi ); ©2009, B.-P. Paris Wireless Communications 22

User Interface Working with Vectors Visualization Commands for Creating Vectors The following commands all create useful vectors. [ ]: the sequence of samples is explicitly speciﬁed. Example: xx = [ 1 3 2 ] produces xx = 1 3 2. :(colon operator): creates a vector of equally spaced samples. Example: tt = 0:2:9 produces tt = 0 2 4 6 8. Example: tt = 1:3 produces tt = 1 2 3. Idiom: tt = ts:1/fs:te creates a vector of sampling times between ts and te with sampling period 1/fs (i.e., the sampling rate is fs). ©2009, B.-P. Paris Wireless Communications 23

User Interface Working with Vectors Visualization Creating Vectors of Constants ones(n,m): creates an n × m matrix with all elements equal to 1. Example: xx = ones(1,5) produces xx = 1 1 1 1 1. Example: xx = 4*ones(1,5) produces xx = 4 4 4 4 4. zeros(n,m): creates an n × m matrix with all elements equal to 0. Often used for initializing a vector. Usage identical to ones. Note: throughout we adopt the convention that signals are represented as row vectors. The ﬁrst (column) dimension equals 1. ©2009, B.-P. Paris Wireless Communications 24

User Interface Working with Vectors Visualization Creating Random Vectors We will often need to create vectors of random numbers. E.g., to simulate noise. The following two functions create random vectors. randn(n,m): creates an n × m matrix of independent Gaussian random numbers with mean zero and variance one. Example: xx = randn(1,5) may produce xx = -0.4326 -1.6656 0.1253 0.2877 -1.1465. rand(n,m): creates an n × m matrix of independent uniformly distributed random numbers between zero and one. Example: xx = rand(1,5) may produce xx = 0.1576 0.9706 0.9572 0.4854 0.8003. ©2009, B.-P. Paris Wireless Communications 25

User Interface Working with Vectors Visualization Addition and Subtraction The standard + and - operators are used to add and subtract vectors. One of two conditions must hold for this operation to succeed. Both vectors must have exactly the same size. In this case, corresponding elements in the two vectors are added and the result is another vector of the same size. Example: [1 3 2] + 1:3 produces 2 5 5. A prominent error message indicates when this condition is violated. One of the operands is a scalar, i.e., a 1 × 1 (degenerate) vector. In this case, each element of the vector has the scalar added to it. The result is a vector of the same size as the vector operand. Example: [1 3 2] + 2 produces 3 5 4. ©2009, B.-P. Paris Wireless Communications 26

User Interface Working with Vectors Visualization Element-wise Multiplication and Division The operators .* and ./ operators multiply or divide two vectors element by element. One of two conditions must hold for this operation to succeed. Both vectors must have exactly the same size. In this case, corresponding elements in the two vectors are multiplied and the result is another vector of the same size. Example: [1 3 2] .* 1:3 produces 1 6 6. An error message indicates when this condition is violated. One of the operands is a scalar. In this case, each element of the vector is multiplied by the scalar. The result is a vector of the same size as the vector operand. Example: [1 3 2] .* 2 produces 2 6 4. If one operand is a scalar the ’.’ may be omitted, i.e., [1 3 2] * 2 also produces 2 6 4. ©2009, B.-P. Paris Wireless Communications 27

User Interface Working with Vectors Visualization Inner Product The operator * with two vector arguments computes the inner product (dot product) of the vectors. Recall the inner product of two vectors is defined as N x ·y = ∑ x (n ) · y (n ). n =1 This implies that the result of the operation is a scalar! The inner product is a useful and important signal processing operation. It is very different from element-wise multiplication. The second dimension of the first operand must equal the first dimension of the second operand. MATLAB error message: Inner matrix dimensions must agree. Example: [1 3 2] * (1:3)’ = 13. The single quote (’) transposes a vector. ©2009, B.-P. Paris Wireless Communications 28

User Interface Working with Vectors Visualization Powers To raise a vector to some power use the .^ operator. Example: [1 3 2].^2 yields 1 9 4. The operator ^ exists but is generally not what you need. Example: [1 3 2]^2 is equivalent to [1 3 2] * [1 3 2] which produces an error. Similarly, to use a vector as the exponent for a scalar base use the .^ operator. Example: 2.^[1 3 2] yields 2 8 4. Finally, to raise a vector of bases to a vector of exponents use the .^ operator. Example: [1 3 2].^(1:3) yields 1 9 8. The two vectors must have the same dimensions. The .^ operator is (nearly) always the right operator. ©2009, B.-P. Paris Wireless Communications 29

User Interface Working with Vectors Visualization Complex Arithmetic MATLAB support complex numbers fully and naturally. √ The imaginary unit i = −1 is a built-in constant named i and j. Creating complex vectors: Example: xx = randn(1,5) + j*randn(1,5) creates a vector of complex Gaussian random numbers. A couple of “gotchas” in connection with complex arithmetic: Never use i and j as variables! Example: After invoking j=2, the above command will produce very unexpected results. Transposition operator (’) transposes and forms conjugate complex. That is very often the right thing to do. Transpose only is performed with .’ operator. ©2009, B.-P. Paris Wireless Communications 30

User Interface Working with Vectors Visualization Vector Functions MATLAB has literally hundreds of built-in functions for manipulating vectors and matrices. The following will come up repeatedly: yy=cos(xx), yy=sin(xx), and yy=exp(xx): compute the cosine, sine, and exponential for each element of vector xx, the result yy is a vector of the same size as xx. XX=fft(xx), xx=ifft(XX): Forward and inverse discrete Fourier transform (DFT), computed via an efﬁcient FFT algorithm. Many algebraic functions are available, including log10, sqrt, abs, and round. Try help elfun for a complete list. ©2009, B.-P. Paris Wireless Communications 31

User Interface Working with Vectors Visualization Functions Returning a Scalar Result Many other functions accept a vector as its input and return a scalar value as the result. Examples include min and max, mean and var or std, sum computes the sum of the elements of a vector, norm provides the square root of the sum of the squares of the elements of a vector. The norm of a vector is related to power and energy. Try help datafun for an extensive list. ©2009, B.-P. Paris Wireless Communications 32

User Interface Working with Vectors Visualization Accessing Elements of a Vector Frequently it is necessary to modify or extract a subset of the elements of a vector. Accessing a single element of a vector: Example: Let xx = [1 3 2], change the third element to 4. Solution: xx(3) = 4; produces xx = 1 3 4. Single elements are accessed by providing the index of the element of interest in parentheses. ©2009, B.-P. Paris Wireless Communications 33

User Interface Working with Vectors Visualization Accessing Elements of a Vector Accessing a range of elements of a vector: Example: Let xx = ones(1,10);, change the ﬁrst ﬁve elements to −1. Solution: xx(1:5) = -1*ones(1,5); Note, xx(1:5) = -1 works as well. Example: Change every other element of xx to 2. Solution: xx(2:2:end) = 2;; Note that end may be use to denote the index of a vector’s last element. This is handy if the length of the vector is not known. Example: Change third and seventh element to 3. Solution: xx([3 7]) = 3;; A set of elements of a vector is accessed by providing a vector of indices in parentheses. ©2009, B.-P. Paris Wireless Communications 34

User Interface Working with Vectors Visualization Accessing Elements that Meet a Condition Frequently one needs to access all elements of a vector that meet a given condition. Clearly, that could be accomplished by writing a loop that examines and processes one element at a time. Such loops are easily avoided. Example: “Poor man’s absolute value” Assume vector xx contains both positive and negative numbers. (e.g., xx = randn(1,10);). Objective: Multiply all negative elements of xx by −1; thus compute the absolute value of all elements of xx. Solution: proceeds in two steps isNegative = (xx < 0); xx(isNegative) = -xx(isNegative); The vector isNegative consists of logical (boolean) values; 1’s appear wherever an element of xx is negative. ©2009, B.-P. Paris Wireless Communications 35

User Interface Working with Vectors Visualization Visualization and Graphics MATLAB has powerful, built-in functions for plotting functions in two and three dimensions. Publication quality graphs are easily produced in a variety of standard graphics formats. MATLAB provides ﬁne-grained control over all aspects of the ﬁnal graph. ©2009, B.-P. Paris Wireless Communications 37

User Interface Working with Vectors Visualization A Basic Plot The sinusoidal signal, we generated earlier is easily plotted via the following sequence of commands: Try help plot for more information about the capabilities of the plot command. %% Plot plot(tt, xx, ’r’) % xy-plot, specify red line xlabel( ’Time (s)’ ) % labels for x and y axis ylabel( ’Amplitude’ ) 10 title( ’x(t) = A cos(2pi f t + phi)’) grid % create a grid axis([0 10e-3 -4 4]) % tweak the range for the axes ©2009, B.-P. Paris Wireless Communications 38

User Interface Working with Vectors Visualization Multiple Plots in One Figure MATLAB can either put multiple graphs in the same plot or put multiple plots side by side. The latter is accomplished with the subplot command. subplot(2,1,1) plot(tt, xx ) % xy-plot xlabel( ’Time (s)’ ) % labels for x and y axis ylabel( ’Amplitude’ ) 12 title( ’x(t) = A cos(2pi f t + phi)’) subplot(2,1,2) plot(tt, yy ) % xy-plot xlabel( ’Time (s)’ ) % labels for x and y axis 17 ylabel( ’Amplitude’ ) title( ’x(t) = A sin(2pi f t + phi)’) ©2009, B.-P. Paris Wireless Communications 40

User Interface Working with Vectors Visualization Resulting Plot x(t) = A cos(2π f t + φ) 4 2 Amplitude 0 −2 −4 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time (s) x(t) = A sin(2π f t + φ) 4 2 Amplitude 0 −2 −4 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time (s) ©2009, B.-P. Paris Wireless Communications 41

User Interface Working with Vectors Visualization 3-D Graphics MATLAB provides several functions that create high-quality three-dimensional graphics. The most important are: plot3(x,y,z): plots a function of two variables. mesh(x,y,Z): plots a mesh of the values stored in matrix Z over the plane spanned by vectors x and y. surf(x,y,Z): plots a surface from the values stored in matrix Z over the plane spanned by vectors x and y. A relevant example is shown on the next slide. The path loss in a two-ray propagation environment over a ﬂat, reﬂecting surface is shown as a function of distance and frequency. ©2009, B.-P. Paris Wireless Communications 42

User Interface Working with Vectors Visualization 1.02 1.01 1 0.99 2 10 0.98 0.97 0.96 0.95 1 10 Frequency (GHz) Distance (m) Figure: Path loss over a ﬂat reﬂecting surface. ©2009, B.-P. Paris Wireless Communications 43

User Interface Working with Vectors Visualization Summary We have taken a brief look at the capabilities of MATLAB. Speciﬁcally, we discussed Vectors as the basic data unit used in MATLAB, Arithmetic with vectors, Prominent vector functions, Visualization in MATLAB. We will build on this basis as we continue and apply MATLAB to the simulation of communication systems. To probe further: Read the built-in documentation. Recommended MATLAB book: D. Hanselman and B. Littleﬁeld, Mastering MATLAB, Prentice-Hall. ©2009, B.-P. Paris Wireless Communications 44

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Introduction: From Theory to Simulation Introduction to digital communications and simulation of digital communications systems. A simple digital communication system and its theoretical underpinnings Introduction to digital modulation Baseband and passband signals: complex envelope Noise and Randomness The matched ﬁlter receiver Bit-error rate Example: BPSK over AWGN, simulation in MATLAB ©2009, B.-P. Paris Wireless Communications 46

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Outline Part I: Learning Objectives Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation ©2009, B.-P. Paris Wireless Communications 47

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Learning Objectives Theory of Digital Communications. Principles of Digital modulation. Communications Channel Model: Additive, White Gaussian Noise. The Matched Filter Receiver. Finding the Probability of Error. Modeling a Digital Communications System in MATLAB. Representing Signals and Noise in MATLAB. Simulating a Communications System. Measuring Probability of Error via MATLAB Simulation. ©2009, B.-P. Paris Wireless Communications 48

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Elements of a Digital Communications System Source: produces a sequence of information symbols b. Transmitter: maps bit sequence to analog signal s (t ). Channel: models corruption of transmitted signal s (t ). Receiver: produces reconstructed sequence of information ˆ symbols b from observed signal R (t ). b s (t ) R (t ) ˆ b Source Transmitter Channel Receiver Figure: Block Diagram of a Generic Digital Communications System ©2009, B.-P. Paris Wireless Communications 50

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation The Source The source models the statistical properties of the digital information source. Three main parameters: Source Alphabet: list of the possible information symbols the source produces. Example: A = {0, 1}; symbols are called bits. Alphabet for a source with M (typically, a power of 2) symbols: A = {0, 1, . . . , M − 1} or A = {±1, ±3, . . . , ±(M − 1)}. Alphabet with positive and negative symbols is often more convenient. Symbols may be complex valued; e.g., A = {±1, ±j }. ©2009, B.-P. Paris Wireless Communications 51

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation A priori Probability: relative frequencies with which the source produces each of the symbols. Example: a binary source that produces (on average) equal numbers of 0 and 1 bits has 1 π0 = π1 = 2 . Notation: πn denotes the probability of observing the n-th symbol. Typically, a-priori probabilities are all equal, 1 i.e., πn = M . A source with M symbols is called an M-ary source. binary (M = 2) ternary (M = 3) quaternary (M = 4) ©2009, B.-P. Paris Wireless Communications 52

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Symbol Rate: The number of information symbols the source produces per second. Also called the baud rate R. Closely related: information rate Rb indicates the number of bits the source produces per second. Relationship: Rb = R · log2 (M ). Also, T = 1/R is the symbol period. Bit 1 Bit 2 Symbol 0 0 0 0 1 1 1 0 2 1 1 3 Table: Two bits can be represented in one quaternary symbol. ©2009, B.-P. Paris Wireless Communications 53

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Remarks This view of the source is simplified. We have omitted important functionality normally found in the source, including error correction coding and interleaving, and mapping bits to symbols. This simplified view is sufficient for our initial discussions. Missing functionality will be revisited when needed. ©2009, B.-P. Paris Wireless Communications 54

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Modeling the Source in MATLAB Objective: Write a MATLAB function to be invoked as: Symbols = RandomSymbols( N, Alphabet, Priors); The input parameters are N: number of input symbols to be produced. Alphabet: source alphabet to draw symbols from. Example: Alphabet = [1 -1]; Priors: a priori probabilities for the input symbols. Example: Priors = ones(size(Alphabet))/length(Alphabet); The output Symbols is a vector with N elements, drawn from Alphabet, and the number of times each symbol occurs is (approximately) proportional to the corresponding element in Priors. ©2009, B.-P. Paris Wireless Communications 55

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Reminders MATLAB’s basic data units are vectors and matrices. Vectors are best thought of as lists of numbers; vectors often contain samples of a signal. There are many ways to create vectors, including Explicitly: Alphabet = [1 -1]; Colon operator: nn = 1:10; Via a function: Priors=ones(1,5)/5; This leads to very concise programs; for-loops are rarely needed. MATLAB has a very large number of available functions. Reduces programming to combining existing building blocks. Difﬁculty: ﬁnd out what is available; use built-in help. ©2009, B.-P. Paris Wireless Communications 56

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Writing a MATLAB Function A MATLAB function must begin with a line of the form function [out1,out2] = FunctionName(in1, in2, in3) be stored in a file with the same name as the function name and extension ’.m’. For our symbol generator, the file name must be RandomSymbols.m and the first line must be function Symbols = RandomSymbols(N, Alphabet, Priors) ©2009, B.-P. Paris Wireless Communications 57

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Writing a MATLAB Function A MATLAB function should have a second line of the form %FunctionName - brief description of function This line is called the “H1 header.” have a more detailed description of the function and how to use it on subsequent lines. The detailed description is separated from the H1 header by a line with only a %. Each of these lines must begin with a % to mark it as a comment. These comments become part of the built-in help system. ©2009, B.-P. Paris Wireless Communications 58

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation The Header of Function RandomSymbols function Symbols = RandomSymbols(N, Alphabet, Priors) % RandomSymbols - generate a vector of random information symbols % % A vector of N random information symbols drawn from a given 5 % alphabet and with specified a priori probabilities is produced. % % Inputs: % N - number of symbols to be generated % Alphabet - vector containing permitted symbols 10 % Priors - a priori probabilities for symbols % % Example: % Symbols = RandomSymbols(N, Alphabet, Priors) ©2009, B.-P. Paris Wireless Communications 59

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Algorithm for Generating Random Symbols For each of the symbols to be generated we use the following algorithm: Begin by computing the cumulative sum over the priors. Example: Let Priors = [0.25 0.25 0.5], then the cumulative sum equals CPriors = [0 0.25 0.5 1]. For each symbol, generate a uniform random number between zero and one. The MATLAB function rand does that. Determine between which elements of the cumulative sum the random number falls and select the corresponding symbol from the alphabet. Example: Assume the random number generated is 0.3. This number falls between the second and third element of CPriors. The second symbol from the alphabet is selected. ©2009, B.-P. Paris Wireless Communications 60

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Implementation In MATLAB, the above algorithm can be “vectorized” to work on the entire sequence at once. CPriors = [0 cumsum( Priors )]; rr = rand(1, N); for kk=1:length(Alphabet) 42 Matches = rr > CPriors(kk) & rr <= CPriors(kk+1); Symbols( Matches ) = Alphabet( kk ); end ©2009, B.-P. Paris Wireless Communications 61

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Testing Function RandomSymols We can invoke and test the function RandomSymbols as shown below. A histogram of the generated symbols should reﬂect the speciﬁed a priori probabilities. %% set parameters N = 1000; Alphabet = [-3 -1 1 3]; Priors = [0.1 0.2 0.3 0.4]; 10 %% generate symbols and plot histogram Symbols = RandomSymbols( N, Alphabet, Priors ); hist(Symbols, -4:4 ); grid 15 xlabel(’Symbol Value’) ylabel(’Number of Occurences’) ©2009, B.-P. Paris Wireless Communications 62

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Resulting Histogram 400 350 300 Number of Occurences 250 200 150 100 50 0 −4 −3 −2 −1 0 1 2 3 4 Symbol Value ©2009, B.-P. Paris Wireless Communications 63

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation The Transmitter The transmitter translates the information symbols at its input into signals that are “appropriate” for the channel, e.g., meet bandwidth requirements due to regulatory or propagation considerations, provide good receiver performance in the face of channel impairments: noise, distortion (i.e., undesired linear ﬁltering), interference. A digital communication system transmits only a discrete set of information symbols. Correspondingly, only a discrete set of possible signals is employed by the transmitter. The transmitted signal is an analog (continuous-time, continuous amplitude) signal. ©2009, B.-P. Paris Wireless Communications 64

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Illustrative Example The sources produces symbols from the alphabet A = {0, 1}. The transmitter uses the following rule to map symbols to signals: If the n-th symbol is bn = 0, then the transmitter sends the signal A for (n − 1)T ≤ t < nT s0 (t ) = 0 else. If the n-th symbol is bn = 1, then the transmitter sends the signal for (n − 1)T ≤ t < (n − 1 )T   A 2 s1 (t ) = −A for (n − 1 )T ≤ t < nT 2 0 else.  ©2009, B.-P. Paris Wireless Communications 65

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Symbol Sequence b = {1, 0, 1, 1, 0, 0, 1, 0, 1, 0} 4 2 Amplitude 0 −2 −4 0 2 4 6 8 10 Time/T ©2009, B.-P. Paris Wireless Communications 66

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Code for Example Listing : plot_TxExampleOrth.m b = [ 1 0 1 1 0 0 1 0 1 0]; %symbol sequence fsT = 20; % samples per symbol period A = 3; 6 Signals = A*[ ones(1,fsT); % signals, one per row ones(1,fsT/2) -ones(1,fsT/2)]; tt = 0:1/fsT:length(b)-1/fsT; % time axis for plotting 11 %% generate signal ... TXSignal = []; for kk=1:length(b) TXSignal = [ TXSignal Signals( b(kk)+1, : ) ]; 16 end ©2009, B.-P. Paris Wireless Communications 67

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Code for Example Listing : plot_TxExampleOrth.m %% ... and plot plot(tt, TXSignal) 20 axis([0 length(b) -(A+1) (A+1)]); grid xlabel(’Time/T’) ©2009, B.-P. Paris Wireless Communications 68

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation The Communications Channel The communications channel models the degradation the transmitted signal experiences on its way to the receiver. For wireless communications systems, we are concerned primarily with: Noise: random signal added to received signal. Mainly due to thermal noise from electronic components in the receiver. Can also model interference from other emitters in the vicinity of the receiver. Statistical model is used to describe noise. Distortion: undesired ﬁltering during propagation. Mainly due to multi-path propagation. Both deterministic and statistical models are appropriate depending on time-scale of interest. Nature and dynamics of distortion is a key difference to wired systems. ©2009, B.-P. Paris Wireless Communications 69

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Thermal Noise At temperatures above absolute zero, electrons move randomly in a conducting medium, including the electronic components in the front-end of a receiver. This leads to a random waveform. The power of the random waveform equals PN = kT0 B. k : Boltzmann’s constant (1.38 · 10−23 Ws/K). T0 : temperature in degrees Kelvin (room temperature ≈ 290 K). For bandwidth equal to 1 MHz, PN ≈ 4 · 10−15 W (−114 dBm). Noise power is small, but power of received signal decreases rapidly with distance from transmitter. Noise provides a fundamental limit to the range and/or rate at which communication is possible. ©2009, B.-P. Paris Wireless Communications 70

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Multi-Path In a multi-path environment, the receiver sees the combination of multiple scaled and delayed versions of the transmitted signal. TX RX ©2009, B.-P. Paris Wireless Communications 71

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Distortion from Multi-Path 5 Received signal “looks” very different from Amplitude transmitted signal. 0 Inter-symbol interference (ISI). Multi-path is a very serious problem −5 for wireless 0 2 4 6 8 10 systems. Time/T ©2009, B.-P. Paris Wireless Communications 72

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation The Receiver The receiver is designed to reconstruct the original information sequence b. Towards this objective, the receiver uses the received signal R (t ), knowledge about how the transmitter works, Speciﬁcally, the receiver knows how symbols are mapped to signals. the a-priori probability and rate of the source. The transmitted signal typically contains information that allows the receiver to gain information about the channel, including training sequences to estimate the impulse response of the channel, synchronization preambles to determine symbol locations and adjust ampliﬁer gains. ©2009, B.-P. Paris Wireless Communications 73

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation The Receiver The receiver input is an analog signal and it’s output is a sequence of discrete information symbols. Consequently, the receiver must perform analog-to-digital conversion (sampling). Correspondingly, the receiver can be divided into an analog front-end followed by digital processing. Modern receivers have simple front-ends and sophisticated digital processing stages. Digital processing is performed on standard digital hardware (from ASICs to general purpose processors). Moore’s law can be relied on to boost the performance of digital communications systems. ©2009, B.-P. Paris Wireless Communications 74

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Measures of Performance The receiver is expected to perform its function optimally. Question: optimal in what sense? Measure of performance must be statistical in nature. observed signal is random, and transmitted symbol sequence is random. Metric must reﬂect the reliability with which information is reconstructed at the receiver. Objective: Design the receiver that minimizes the probability of a symbol error. Also referred to as symbol error rate. Closely related to bit error rate (BER). ©2009, B.-P. Paris Wireless Communications 75

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Summary We have taken a brief look at the elements of a communication system. Source, Transmitter, Channel, and Receiver. We will revisit each of these elements for a more rigorous analysis. Intention: Provide enough detail to allow simulation of a communication system. ©2009, B.-P. Paris Wireless Communications 76

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Digital Modulation Digital modulation is performed by the transmitter. It refers to the process of converting a sequence of information symbols into a transmitted (analog) signal. The possibilities for performing this process are virtually without limits, including varying, the amplitude, frequency, and/or phase of a sinusoidal signal depending on the information sequence, making the currently transmitted signal on some or all of the previously transmitted symbols (modulation with memory). Initially, we focus on a simple, yet rich, class of modulation formats referred to as linear modulation. ©2009, B.-P. Paris Wireless Communications 78

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Linear Modulation Linear modulation may be thought of as the digital equivalent of amplitude modulation. The instantaneous amplitude of the transmitted signal is proportional to the current information symbol. Speciﬁcally, a linearly modulated signal may be written as N −1 s (t ) = ∑ bn · p (t − nT ) n =0 where, bn denotes the n-th information symbol, and p (t ) denotes a pulse of ﬁnite duration. Recall that T is the duration of a symbol. ©2009, B.-P. Paris Wireless Communications 79

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Linear Modulation Note, that the expression N −1 bn s (t ) s (t ) = ∑ bn · p (t − nT ) × p (t ) n =0 is linear in the symbols bn . Different modulation formats are ∑ δ(t − nT ) constructed by choosing appropriate symbol alphabets, e.g., BPSK: bn ∈ {1, −1} OOK: bn ∈ {0, 1} PAM: bn ∈ {±1, . . . , ±(M − 1)}. ©2009, B.-P. Paris Wireless Communications 80

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Linear Modulation in MATLAB To simulate a linear modulator in MATLAB, we will need a function with a function header like this: function Signal = LinearModulation( Symbols, Pulse, fsT ) % LinearModulation - linear modulation of symbols with given 3 % pulse shape % % A sequence of information symbols is linearly modulated. Pulse % shaping is performed using the pulse shape passed as input % parameter Pulse. The integer fsT indicates how many samples 8 % per symbol period are taken. The length of the Pulse vector may % be longer than fsT; this corresponds to partial-response signali % % Inputs: % Symbols - vector of information symbols 13 % Pulse - vector containing the pulse used for shaping % fsT - (integer) number of samples per symbol period ©2009, B.-P. Paris Wireless Communications 81

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Linear Modulation in MATLAB In the body of the function, the sum of the pulses is computed. There are two issues that require some care: Each pulse must be inserted in the correct position in the output signal. Recall that the expression for the output signal s (t ) contains the terms p (t − nT ). The term p (t − nT ) reﬂects pulses delayed by nT . Pulses may overlap. If the duration of a pulse is longer than T , then pulses overlap. Such overlapping pulses are added. This situation is called partial response signaling. ©2009, B.-P. Paris Wireless Communications 82

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Body of Function LinearModulation 19 % initialize storage for Signal LenSignal = length(Symbols)*fsT + (length(Pulse))-fsT; Signal = zeros( 1, LenSignal ); % loop over symbols and insert corresponding segment into Signal 24 for kk = 1:length(Symbols) ind_start = (kk-1)*fsT + 1; ind_end = (kk-1)*fsT + length(Pulse); Signal(ind_start:ind_end) = Signal(ind_start:ind_end) + ... 29 Symbols(kk) * Pulse; end ©2009, B.-P. Paris Wireless Communications 83

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Testing Function LinearModulation Listing : plot_LinearModRect.m %% Parameters: fsT = 20; Alphabet = [1,-1]; 6 Priors = 0.5*[1 1]; Pulse = ones(1,fsT); % rectangular pulse %% symbols and Signal using our functions Symbols = RandomSymbols(10, Alphabet, Priors); 11 Signal = LinearModulation(Symbols,Pulse,fsT); %% plot tt = (0 : length(Signal)-1 )/fsT; plot(tt, Signal) axis([0 length(Signal)/fsT -1.5 1.5]) 16 grid xlabel(’Time/T’) ylabel(’Amplitude’) ©2009, B.-P. Paris Wireless Communications 84

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Linear Modulation with Rectangular Pulses 1.5 1 0.5 Amplitude 0 −0.5 −1 −1.5 0 2 4 6 8 10 Time/T ©2009, B.-P. Paris Wireless Communications 85

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Linear Modulation with sinc-Pulses More interesting and practical waveforms arise when smoother pulses are used. A good example are truncated sinc functions. The sinc function is defined as: sin(x ) sinc(x ) = , with sinc(0) = 1. x Specifically, we will use pulses defined by sin(πt /T ) p (t ) = sinc(πt /T ) = ; πt /T pulses are truncated to span L symbol periods, and delayed to be causal. Toolbox contains function Sinc( L, fsT ). ©2009, B.-P. Paris Wireless Communications 86

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation A Truncated Sinc Pulse 1.5 1 Pulse is very smooth, 0.5 spans ten symbol periods, is zero at location of 0 other symbols. Nyquist pulse. −0.5 0 2 4 6 8 10 Time/T ©2009, B.-P. Paris Wireless Communications 87

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Linear Modulation with Sinc Pulses 2 Resulting waveform is 1 also very smooth; expect good spectral properties. Amplitude 0 Symbols are harder to discern; partial response signaling induces −1 “controlled” ISI. But, there is no ISI at symbol locations. −2 0 5 10 15 20 Transients at beginning Time/T and end. ©2009, B.-P. Paris Wireless Communications 88

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Passband Signals So far, all modulated signals we considered are baseband signals. Baseband signals have frequency spectra concentrated near zero frequency. However, for wireless communications passband signals must be used. Passband signals have frequency spectra concentrated around a carrier frequency fc . Baseband signals can be converted to passband signals through up-conversion. Passband signals can be converted to baseband signals through down-conversion. ©2009, B.-P. Paris Wireless Communications 89

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Up-Conversion A cos(2πfc t ) sI (t ) The passband signal sP (t ) is × constructed from two (digitally modulated) baseband signals, sI (t ) sP ( t ) and sQ (t ). + Note that two signals can be carried simultaneously! This is a consequence of sQ (t ) cos(2πfc t ) and sin(2πfc t ) being × orthogonal. A sin(2πfc t ) ©2009, B.-P. Paris Wireless Communications 90

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Baseband Equivalent Signals The passband signal sP (t ) can be written as √ √ sP (t ) = 2 · AsI (t ) · cos(2πfc t ) + 2 · AsQ (t ) · sin(2πfc t ). If we deﬁne s (t ) = sI (t ) − j · sQ (t ), then sP (t ) can also be expressed as √ sP (t ) = 2 · A · {s (t ) · exp(j2πfc t )}. The signal s (t ): is called the baseband equivalent or the complex envelope of the passband signal sP (t ). It contains the same information as sP (t ). Note that s (t ) is complex-valued. ©2009, B.-P. Paris Wireless Communications 91

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Illustration: QPSK with fc = 2/T Passband signal (top): 2 segments of sinusoids 1 with different phases. Amplitude 0 Phase changes occur −1 at multiples of T . −2 Baseband signal 0 1 2 3 4 5 6 7 8 9 10 Time/T (bottom) is complex 2 1 valued; magnitude and 1.5 phase are plotted. 0.5 Magnitude is constant Magnitude Phase/π 1 0 (rectangular pulses). 0.5 0 −0.5 0 5 10 0 5 10 Time/T Time/T ©2009, B.-P. Paris Wireless Communications 92

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Code for QPSK Illustration Listing : plot_LinearModQPSK.m %% Parameters: fsT = 20; L = 10; fc = 2; % carrier frequency 7 Alphabet = [1, j, -j, -1];% QPSK Priors = 0.25*[1 1 1 1]; Pulse = ones(1,fsT); % rectangular pulse %% symbols and Signal using our functions 12 Symbols = RandomSymbols(10, Alphabet, Priors); Signal = LinearModulation(Symbols,Pulse,fsT); %% passband signal tt = (0 : length(Signal)-1 )/fsT; Signal_PB = sqrt(2)*real( Signal .* exp(-j*2*pi*fc*tt) ); ©2009, B.-P. Paris Wireless Communications 93

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Code for QPSK Illustration Listing : plot_LinearModQPSK.m subplot(2,1,1) plot( tt, Signal_PB ) grid 22 xlabel(’Time/T’) ylabel(’Amplitude’) subplot(2,2,3) plot( tt, abs( Signal ) ) 27 grid xlabel(’Time/T’) ylabel(’Magnitude’) subplot(2,2,4) 32 plot( tt, angle( Signal )/pi ) grid xlabel(’Time/T’) ylabel(’Phase/pi’) ©2009, B.-P. Paris Wireless Communications 94

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Frequency Domain Perspective In the frequency domain: √ 2 · SP (f + fc ) for f + fc > 0 S (f ) = 0 else. √ Factor 2 ensures both signals have the same power. SP (f ) S (f ) √ 2·A A f f − fc fc − fc fc ©2009, B.-P. Paris Wireless Communications 95

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Baseband Equivalent System The baseband description of the transmitted signal is very convenient: it is more compact than the passband signal as it does not include the carrier component, while retaining all relevant information. However, we are also concerned what happens to the signal as it propagates to the receiver. Question: Do baseband techniques extend to other parts of a passband communications system? ©2009, B.-P. Paris Wireless Communications 96

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Passband System √ √ 2A cos(2πfc t ) 2 cos(2πfc t ) sI (t ) RI (t ) × NP ( t ) × LPF sP ( t ) RP ( t ) + hP (t ) + sQ (t ) RQ (t ) × × LPF √ √ 2A sin(2πfc t ) 2 sin(2πfc t ) ©2009, B.-P. Paris Wireless Communications 97

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Baseband Equivalent System N (t ) s (t ) R (t ) h (t ) + The passband system can be interpreted as follows to yield an equivalent system that employs only baseband signals: baseband equivalent transmitted signal: s (t ) = sI (t ) − j · sQ (t ). baseband equivalent channel with complex valued impulse response: h(t ). baseband equivalent received signal: R ( t ) = RI ( t ) − j · RQ ( t ) . complex valued, additive Gaussian noise: N (t ) ©2009, B.-P. Paris Wireless Communications 98

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Baseband Equivalent Channel The baseband equivalent channel is deﬁned by the entire shaded box in the block diagram for the passband system (excluding additive noise). The relationship between the passband and baseband equivalent channel is hP (t ) = {h(t ) · exp(j2πfc t )} in the time domain. Example: hP ( t ) = ∑ ak · δ(t − τk ) =⇒ h(t ) = ∑ ak · e−j2πf τ c k · δ(t − τk ). k k ©2009, B.-P. Paris Wireless Communications 99

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Baseband Equivalent Channel In the frequency domain HP (f + fc ) for f + fc > 0 H (f ) = 0 else. HP (f ) H (f ) A A f f − fc fc − fc fc ©2009, B.-P. Paris Wireless Communications 100

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Summary The baseband equivalent channel is much simpler than the passband model. Up and down conversion are eliminated. Expressions for signals do not contain carrier terms. The baseband equivalent signals are easier to represent for simulation. Since they are low-pass signals, they are easily sampled. No information is lost when using baseband equivalent signals, instead of passband signals. Standard, linear system equations hold: R (t ) = s (t ) ∗ h(t ) + n(t ) and R (f ) = S (f ) · H (f ) + N (f ). Conclusion: Use baseband equivalent signals and systems. ©2009, B.-P. Paris Wireless Communications 101

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Channel Model The channel describes how the transmitted signal is corrupted between transmitter and receiver. The received signal is corrupted by: noise, distortion, due to multi-path propagation, interference. Interference is not considered in detail. Is easily added to simulations, Frequent assumption: interference can be lumped in with noise. ©2009, B.-P. Paris Wireless Communications 103

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Additive White Gaussian Noise A standard assumption in most communication systems is that noise is well modeled as additive, white Gaussian noise (AWGN). additive: channel adds noise to the transmitted signal. white: describes the temporal correlation of the noise. Gaussian: probability distribution is a Normal or Gaussian distribution. Baseband equivalent noise is complex valued. In-phase NI (t ) and quadrature NQ (t ) noise signals are modeled as independent (circular symmetric noise). ©2009, B.-P. Paris Wireless Communications 104

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation White Noise The term white means specifically that the mean of the noise is zero, E[N (t )] = 0, the autocorrelation function of the noise is ρN (τ ) = E[N (t ) · N ∗ (t + τ )] = N0 δ(τ ). This means that any distinct noise samples are independent. The autocorrelation function also indicates that noise samples have infinite variance. Insight: noise must be filtered before it can be sampled. In-phase and quadrature noise each have autocorrelation N0 2 δ ( τ ). ©2009, B.-P. Paris Wireless Communications 105

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation White Noise The term “white” refers to the spectral properties of the noise. Speciﬁcally, the power spectral density of white noise is constant over all frequencies: SN (f ) = N0 for all f . White light consists of equal components of the visible spectrum. ©2009, B.-P. Paris Wireless Communications 106

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Generating Gaussian Noise Samples For simulating additive noise, Gaussian noise samples must be generated. MATLAB idiom for generating a vector of N independent, complex, Gaussian random numbers with variance VarNoise: Noise = sqrt(VarNoise/2) * ( randn(1,N) + j * randn(1,N)); Note, that real and imaginary part each have variance VarNoise/2. This causes the total noise variance to equal VarNoise. ©2009, B.-P. Paris Wireless Communications 107

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Toolbox Function addNoise We will frequently simulate additive, white Gaussian noise. To perform this task, a function with the following header is helpful: function NoisySignal = addNoise( CleanSignal, VarNoise ) % addNoise - simulate additive, white Gaussian noise channel % % Independent Gaussian noise of variance specified by the second 5 % input parameter is added to the (vector or matrix) signal passed % as the first input. The result is returned in a vector or matrix % of the same size as the input signal. % % The function determines if the signal is real or complex valued 10 % and generates noise samples accordingly. In either case the total % variance is equal to the second input. ©2009, B.-P. Paris Wireless Communications 108

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation The Body of Function addNoise %% generate noisy signal % distinguish between real and imaginary input signals if ( isreal( CleanSignal ) ) 25 NoisySignal = CleanSignal + ... sqrt(VarNoise) * randn( size(CleanSignal) ); else NoisySignal = CleanSignal + sqrt(VarNoise/2) * ... ( randn( size(CleanSignal) ) + j*randn( size(CleanSignal) ) ); 30 end ©2009, B.-P. Paris Wireless Communications 109

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation 4 2 Amplitude 0 −2 −4 −6 0 5 10 15 20 Time/T Es Figure: Linearly modulated Signal with Noise; N0 ≈ 10 dB, noise 20 variance ≈ 2, noise bandwidth ≈ T . ©2009, B.-P. Paris Wireless Communications 110

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Multi-path Fading Channels In addition to corruption by additive noise, transmitted signals are distorted during propagation from transmitter to receiver. The distortion is due to multi-path propagation. The transmitted signal travels to the receiver along multiple paths, each with different attenuation and delay. The receiver observes the sum of these signals. Effect: undesired ﬁltering of transmitted signal. ©2009, B.-P. Paris Wireless Communications 111

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Multi-path Fading Channels In mobile communications, the characteristics of the multi-path channel vary quite rapidly; this is referred to as fading. Fading is due primarily to phase changes due to changes in the delays for the different propagation paths. Multi-path fading channels will be studied in detail later in the course. In particular, the impact of multi-path fading on system performance will be investigated. ©2009, B.-P. Paris Wireless Communications 112

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Receiver The receiver is responsible for extracting the sequence of information symbols from the received signal. This task is difﬁcult because of the signal impairments induced by the channel. At this time, we focus on additive, white Gaussian noise as the only source of signal corruption. Remedies for distortion due to multi-path propagation will be studied extensively later. Structure of receivers for digital communication systems. Analog front-end and digital post-processing. Performance analysis: symbol error rate. Closed form computation of symbol error rate is possible for AWGN channel. ©2009, B.-P. Paris Wireless Communications 114

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Matched Filter It is well known, that the optimum receiver for an AWGN channel is the matched filter receiver. The matched filter for a linearly modulated signal using pulse shape p (t ) is shown below. The slicer determines which symbol is “closest” to the matched filter output. Its operation depends on the symbols being used and the a priori probabilities. R (t ) ˆ b T × 0 (·) dt Slicer p (t ) ©2009, B.-P. Paris Wireless Communications 115

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Shortcomings of The Matched Filter While theoretically important, the matched filter has a few practical drawbacks. For the structure shown above, it is assumed that only a single symbol was transmitted. In the presence of channel distortion, the receiver must be matched to p (t ) ∗ h(t ) instead of p (t ). Problem: The channel impulse response h(t ) is generally not known. The matched filter assumes that perfect symbol synchronization has been achieved. The matching operation is performed in continuous time. This is difficult to accomplish with analog components. ©2009, B.-P. Paris Wireless Communications 116

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Analog Front-end and Digital Back-end As an alternative, modern digital receivers employ a different structure consisting of an analog receiver front-end, and a digital signal processing back-end. The analog front-end is little more than a ﬁlter and a sampler. The theoretical underpinning for the analog front-end is Nyquist’s sampling theorem. The front-end may either work on a baseband signal or a passband signal at an intermediate frequency (IF). The digital back-end performs sophisticated processing, including digital matched ﬁltering, equalization, and synchronization. ©2009, B.-P. Paris Wireless Communications 117

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Analog Front-end Several, roughly equivalent, alternatives exist for the analog front-end. Two common approaches for the analog front-end will be considered brieﬂy. Primarily, the analog front-end is responsible for converting the continuous-time received signal R (t ) into a discrete-time signal R [n]. Care must be taken with the conversion: (ideal) sampling would admit too much noise. Modeling the front-end faithfully is important for accurate simulation. ©2009, B.-P. Paris Wireless Communications 118

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Analog Front-end: Low-pass and Whitening Filter The first structure contains a low-pass filter (LPF) with bandwidth equal to the signal bandwidth, a sampler followed by a whitening filter (WF). The low-pass filter creates correlated noise, the whitening filter removes this correlation. Sampler, rate fs R (t ) R [n] to LPF WF DSP ©2009, B.-P. Paris Wireless Communications 119

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Analog Front-end: Integrate-and-Dump An alternative front-end has the structure shown below. Here, ΠTs (t ) indicates a ﬁlter with an impulse response that is a rectangular pulse of length Ts = 1/fs and amplitude 1/Ts . The entire system is often called an integrate-and-dump sampler. Most analog-to-digital converters (ADC) operate like this. A whitening ﬁlter is not required since noise samples are uncorrelated. Sampler, rate fs R (t ) R [n ] to ΠTs (t ) DSP ©2009, B.-P. Paris Wireless Communications 120

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Output from Analog Front-end The second of the analog front-ends is simpler conceptually and widely used in practice; it will be assumed for the remainder of the course. For simulation purposes, we need to characterize the output from the front-end. To begin, assume that the received signal R (t ) consists of a deterministic signal s (t ) and (AWGN) noise N (t ): R (t ) = s (t ) + N (t ). The signal R [n] is a discrete-time signal. The front-end generates one sample every Ts seconds. The discrete-time signal R [n] also consists of signal and noise R [n ] = s [n ] + N [n ]. ©2009, B.-P. Paris Wireless Communications 121

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Output from Analog Front-end Consider the signal and noise component of the front-end output separately. This can be done because the front-end is linear. The n-th sample of the signal component is given by: 1 (n+1)Ts s [n ] = · s (t ) dt ≈ s ((n + 1/2)Ts ). Ts nTs The approximation is valid if fs = 1/Ts is much greater than the signal band-width. Sampler, rate fs R (t ) R [n ] to ΠTs (t ) DSP ©2009, B.-P. Paris Wireless Communications 122

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Output from Analog Front-end The noise samples N [n] at the output of the front-end: are independent, complex Gaussian random variables, with zero mean, and variance equal to N0 /Ts . The variance of the noise samples is proportional to 1/Ts . Interpretations: Noise is averaged over Ts seconds: variance decreases with length of averager. Bandwidth of front-end ﬁlter is approximately 1/Ts and power of ﬁltered noise is proportional to bandwidth (noise bandwidth). It will be convenient to express the noise variance as N0 /T · T /Ts . The factor T /Ts = fs T is the number of samples per symbol period. ©2009, B.-P. Paris Wireless Communications 123

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Receiver Performance Our primary measure of performance is the symbol error probability. For AWGN channels, it is possible to express the symbol error probability in closed form for many digital modulation formats. To ﬁx ideas, we employ the following concrete signaling format: BPSK modulation; symbols are drawn equally likely from the alphabet {1, −1}, Pulse shaping: 2 2πt p (t ) = · (1 − cos( )) for 0 ≤ t ≤ T . 3 T This is a smooth, full-response pulse; it is referred to as a raised cosine pulse. ©2009, B.-P. Paris Wireless Communications 124

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation BPSK Modulation with Raised Cosine Pulses 2 1 Amplitude 0 −1 −2 0 2 4 6 8 10 Time/T ©2009, B.-P. Paris Wireless Communications 125

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Signal Model In each symbol period, the received signal is of the form: R (t ) = b · A · p (t ) + N (t ) for 0 ≤ t < T . where b ∈ {1, −1} is a BPSK symbol (with equal a priori probabilities), A is the amplitude of the received signal, p (t ) is the raised cosine pulse, and N (t ) is (complex) white Gaussian noise with spectral height N0 . ©2009, B.-P. Paris Wireless Communications 126

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Performance of Matched Filter Receiver With these assumptions, the symbol error rate is 2A2 T 2Es Pe,min = Q( ) = Q( ). N0 N0 Es = A2 T is the symbol energy. We used the fact that p2 (t ) dt = T . ∞ 1 Q(x ) = x √ exp(−y 2 /2) dy is the Gaussian error 2π function. No receiver can achieve a symbol error rate smaller than Pe,min . ©2009, B.-P. Paris Wireless Communications 127

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Suboptimum Receivers We discussed earlier that it is often not possible to use an exact matched ﬁlter in practice. Practical receivers can often be modeled as matching with a pulse shape g (t ) that is (slightly) different than the transmitted pulse shape p (t ). Such receivers are called linear receivers; note that the matched ﬁlter is also a linear receiver. Example: Assume that we are using the integrate-and-dump front-end and sample once per bit period. This corresponds to matching with a pulse 1 T for 0 ≤ t ≤ T g (t ) = 0 else. ©2009, B.-P. Paris Wireless Communications 128

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Performance with Suboptimum Receivers The symbol error rate of linear receivers can be expressed in closed form: 2ρ2 Es g Pe = Q( ) N0 where ρg captures the mismatch between the transmitted pulse p (t ) and the pulse g (t ) used by the receiver. Speciﬁcally p (t )g (t ) dt ρ2 = g . p2 (t ) dt · g 2 (t ) dt ©2009, B.-P. Paris Wireless Communications 129

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Performance with Suboptimum Receivers For our example with raised cosine pulses p (t ) and rectangular pulses g (t ), we ﬁnd ρ2 = 2/3 g and 4Es Pe = Q ( ). 3N0 Comparing with the error probability for the matched ﬁlter: The term 3Ns replaces 2Es . 4E N0 0 Interpretation: to make both terms equal, the suboptimum receiver must spend 1.5 times more energy (1.7 dB loss in performance). ©2009, B.-P. Paris Wireless Communications 130

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Performance Comparison Symbol Error Probability 100 10−2 10−4 10−6 0 2 4 6 8 10 Es /N0 (dB) ©2009, B.-P. Paris Wireless Communications 131

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Code for Performance Comparison Listing : plot_Q.m %% Set parameters EsN0dB = 0:0.1:10; % range of Es over N0 values on dB scale EsN0lin = dB2lin( EsN0dB ); % convert to linear scale 7 %% compute Pe PeMf = Q( sqrt( 2 * EsN0lin ) ); PeSo = Q( sqrt( 4/3 * EsN0lin ) ); %% plot 12 semilogy( EsN0dB, PeMf, EsN0dB, PeSo) xlabel( ’E_s/N_0 (dB)’) ylabel( ’Symbol Error Probability’ ); grid ©2009, B.-P. Paris Wireless Communications 132

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Summary We introduced the matched ﬁlter as the receiver that minimizes the probability of a symbol error. Practical considerations lead to implementing receivers with simple analog front-ends followed by digital post-processing. Integrate-and-dump front-ends found in typical A-to-D converters were analyzed in some detail. Computed the error probability for the matched ﬁlter receiver and provided expressions for the error rate of linear receivers in AWGN channels. ©2009, B.-P. Paris Wireless Communications 133

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Simulation Objective: Simulate a simple communication system and estimate bit error rate. System Characteristics: BPSK modulation, b ∈ {1, −1} with equal a priori probabilities, Raised cosine pulses, AWGN channel, oversampled integrate-and-dump receiver front-end, digital matched ﬁlter. Measure: Bit-error rate as a function of Es /N0 and oversampling rate. ©2009, B.-P. Paris Wireless Communications 135

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation System to be Simulated Sampler, N (t ) rate fs bn s (t ) R (t ) R [n] to × p (t ) × h (t ) + ΠTs (t ) DSP ∑ δ(t − nT ) A Figure: Baseband Equivalent System to be Simulated. ©2009, B.-P. Paris Wireless Communications 136

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation From Continuous to Discrete Time The system in the preceding diagram cannot be simulated immediately. Main problem: Most of the signals are continuous-time signals and cannot be represented in MATLAB. Possible Remedies: 1. Rely on Sampling Theorem and work with sampled versions of signals. 2. Consider discrete-time equivalent system. The second alternative is preferred and will be pursued below. ©2009, B.-P. Paris Wireless Communications 137

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Towards the Discrete-Time Equivalent System The shaded portion of the system has a discrete-time input and a discrete-time output. Can be considered as a discrete-time system. Minor problem: input and output operate at different rates. Sampler, N (t ) rate fs bn s (t ) R (t ) R [n] to × p (t ) × h (t ) + ΠTs (t ) DSP ∑ δ(t − nT ) A ©2009, B.-P. Paris Wireless Communications 138

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Discrete-Time Equivalent System The discrete-time equivalent system is equivalent to the original system, and contains only discrete-time signals and components. Input signal is up-sampled by factor fs T to make input and output rates equal. Insert fs T − 1 zeros between input samples. N [n ] bn R [n ] × ↑ fs T h [n ] + to DSP A ©2009, B.-P. Paris Wireless Communications 139

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Components of Discrete-Time Equivalent System Question: What is the relationship between the components of the original and discrete-time equivalent system? Sampler, N (t ) rate fs bn s (t ) R (t ) R [n] to × p (t ) × h (t ) + ΠTs (t ) DSP ∑ δ(t − nT ) A ©2009, B.-P. Paris Wireless Communications 140

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Discrete-time Equivalent Impulse Response To determine the impulse response h[n] of the discrete-time equivalent system: Set noise signal Nt to zero, set input signal bn to unit impulse signal δ[n], output signal is impulse response h[n]. Procedure yields: 1 ( n + 1 ) Ts h [n ] = p (t ) ∗ h(t ) dt Ts nTs For high sampling rates (fs T 1), the impulse response is closely approximated by sampling p (t ) ∗ h(t ): h[n] ≈ p (t ) ∗ h(t )|(n+ 1 )Ts 2 ©2009, B.-P. Paris Wireless Communications 141

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Discrete-time Equivalent Impulse Response 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Time/T Figure: Discrete-time Equivalent Impulse Response (fs T = 8) ©2009, B.-P. Paris Wireless Communications 142

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Discrete-Time Equivalent Noise To determine the properties of the additive noise N [n] in the discrete-time equivalent system, Set input signal to zero, let continuous-time noise be complex, white, Gaussian with power spectral density N0 , output signal is discrete-time equivalent noise. Procedure yields: The noise samples N [n] are independent, complex Gaussian random variables, with zero mean, and variance equal to N0 /Ts . ©2009, B.-P. Paris Wireless Communications 143

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Received Symbol Energy The last entity we will need from the continuous-time system is the received energy per symbol Es . Note that Es is controlled by adjusting the gain A at the transmitter. To determine Es , Set noise N (t ) to zero, Transmit a single symbol bn , Compute the energy of the received signal R (t ). Procedure yields: 2 Es = σs · A2 |p (t ) ∗ h(t )|2 dt 2 Here, σs denotes the variance of the source. For BPSK, 2 = 1. σs For the system under consideration, Es = A2 T . ©2009, B.-P. Paris Wireless Communications 144

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Simulating Transmission of Symbols We are now in position to simulate the transmission of a sequence of symbols. The MATLAB functions previously introduced will be used for that purpose. We proceed in three steps: 1. Establish parameters describing the system, By parameterizing the simulation, other scenarios are easily accommodated. 2. Simulate discrete-time equivalent system, 3. Collect statistics from repeated simulation. ©2009, B.-P. Paris Wireless Communications 145

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Listing : SimpleSetParameters.m % This script sets a structure named Parameters to be used by % the system simulator. %% Parameters 7 % construct structure of parameters to be passed to system simulator % communications parameters Parameters.T = 1/10000; % symbol period Parameters.fsT = 8; % samples per symbol Parameters.Es = 1; % normalize received symbol energy to 1 12 Parameters.EsOverN0 = 6; % Signal-to-noise ratio (Es/N0) Parameters.Alphabet = [1 -1]; % BPSK Parameters.NSymbols = 1000; % number of Symbols % discrete-time equivalent impulse response (raised cosine pulse) 17 fsT = Parameters.fsT; tts = ( (0:fsT-1) + 1/2 )/fsT; Parameters.hh = sqrt(2/3) * ( 1 - cos(2*pi*tts)*sin(pi/fsT)/(pi/fsT)); ©2009, B.-P. Paris Wireless Communications 146

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Simulating the Discrete-Time Equivalent System The actual system simulation is carried out in MATLAB function MCSimple which has the function signature below. The parameters set in the controlling script are passed as inputs. The body of the function simulates the transmission of the signal and subsequent demodulation. The number of incorrect decisions is determined and returned. function [NumErrors, ResultsStruct] = MCSimple( ParametersStruct ) ©2009, B.-P. Paris Wireless Communications 147

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Simulating the Discrete-Time Equivalent System The simulation of the discrete-time equivalent system uses toolbox functions RandomSymbols, LinearModulation, and addNoise. A = sqrt(Es/T); % transmitter gain N0 = Es/EsOverN0; % noise PSD (complex noise) NoiseVar = N0/T*fsT; % corresponding noise variance N0/Ts Scale = A*hh*hh’; % gain through signal chain 34 %% simulate discrete-time equivalent system % transmitter and channel via toolbox functions Symbols = RandomSymbols( NSymbols, Alphabet, Priors ); Signal = A * LinearModulation( Symbols, hh, fsT ); 39 if ( isreal(Signal) ) Signal = complex(Signal);% ensure Signal is complex-valued end Received = addNoise( Signal, NoiseVar ); ©2009, B.-P. Paris Wireless Communications 148

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Digital Matched Filter The vector Received contains the noisy output samples from the analog front-end. In a real system, these samples would be processed by digital hardware to recover the transmitted bits. Such digital hardware may be an ASIC, FPGA, or DSP chip. The first function performed there is digital matched filtering. This is a discrete-time implementation of the matched filter discussed before. The matched filter is the best possible processor for enhancing the signal-to-noise ratio of the received signal. ©2009, B.-P. Paris Wireless Communications 149

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Digital Matched Filter In our simulator, the vector Received is passed through a discrete-time matched ﬁlter and down-sampled to the symbol rate. The impulse response of the matched ﬁlter is the conjugate complex of the time-reversed, discrete-time channel response h[n]. R [n ] ˆ bn h∗ [−n] ↓ fs T Slicer ©2009, B.-P. Paris Wireless Communications 150

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Code for Digital Matched Filter The signature line for the MATLAB function implementing the matched ﬁlter is: function MFOut = DMF( Received, Pulse, fsT ) The body of the function is a direct implementation of the structure in the block diagram above. % convolve received signal with conjugate complex of % time-reversed pulse (matched filter) Temp = conv( Received, conj( fliplr(Pulse) ) ); 21 % down sample, at the end of each pulse period MFOut = Temp( length(Pulse) : fsT : end ); ©2009, B.-P. Paris Wireless Communications 151

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation DMF Input and Output Signal DMF Input 400 200 0 −200 −400 0 1 2 3 4 5 6 7 8 9 10 Time (1/T) DMF Output 1500 1000 500 0 −500 −1000 0 1 2 3 4 5 6 7 8 9 10 Time (1/T) ©2009, B.-P. Paris Wireless Communications 152

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation IQ-Scatter Plot of DMF Input and Output DMF Input 300 200 Imag. Part 100 0 −100 −200 −800 −600 −400 −200 0 200 400 600 800 Real Part DMF Output 500 Imag. Part 0 −500 −2000 −1500 −1000 −500 0 500 1000 1500 2000 Real Part ©2009, B.-P. Paris Wireless Communications 153

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Slicer The final operation to be performed by the receiver is deciding which symbol was transmitted. This function is performed by the slicer. The operation of the slicer is best understood in terms of the IQ-scatter plot on the previous slide. The red circles in the plot indicate the noise-free signal locations for each of the possibly transmitted signals. For each output from the matched filter, the slicer determines the nearest noise-free signal location. The decision is made in favor of the symbol that corresponds to the noise-free signal nearest the matched filter output. Some adjustments to the above procedure are needed when symbols are not equally likely. ©2009, B.-P. Paris Wireless Communications 154

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation MATLAB Function SimpleSlicer The procedure above is implemented in a function with signature function [Decisions, MSE] = SimpleSlicer( MFOut, Alphabet, Scale ) %% Loop over symbols to find symbol closest to MF output for kk = 1:length( Alphabet ) % noise-free signal location 28 NoisefreeSig = Scale*Alphabet(kk); % Euclidean distance between each observation and constellation po Dist = abs( MFOut - NoisefreeSig ); % find locations for which distance is smaller than previous best ChangedDec = ( Dist < MinDist ); 33 % store new min distances and update decisions MinDist( ChangedDec) = Dist( ChangedDec ); Decisions( ChangedDec ) = Alphabet(kk); end ©2009, B.-P. Paris Wireless Communications 155

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Entire System The addition of functions for the digital matched ﬁlter completes the simulator for the communication system. The functionality of the simulator is encapsulated in a function with signature function [NumErrors, ResultsStruct] = MCSimple( ParametersStruct ) The function simulates the transmission of a sequence of symbols and determines how many symbol errors occurred. The operation of the simulator is controlled via the parameters passed in the input structure. The body of the function is shown on the next slide; it consists mainly of calls to functions in our toolbox. ©2009, B.-P. Paris Wireless Communications 156

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Listing : MCSimple.m %% simulate discrete-time equivalent system % transmitter and channel via toolbox functions Symbols = RandomSymbols( NSymbols, Alphabet, Priors ); 38 Signal = A * LinearModulation( Symbols, hh, fsT ); if ( isreal(Signal) ) Signal = complex(Signal);% ensure Signal is complex-valued end Received = addNoise( Signal, NoiseVar ); 43 % digital matched filter and slicer MFOut = DMF( Received, hh, fsT ); Decisions = SimpleSlicer( MFOut(1:NSymbols), Alphabet, Scale ); 48 %% Count errors NumErrors = sum( Decisions ~= Symbols ); ©2009, B.-P. Paris Wireless Communications 157

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Monte Carlo Simulation The system simulator will be the work horse of the Monte Carlo simulation. The objective of the Monte Carlo simulation is to estimate the symbol error rate our system can achieve. The idea behind a Monte Carlo simulation is simple: Simulate the system repeatedly, for each simulation count the number of transmitted symbols and symbol errors, estimate the symbol error rate as the ratio of the total number of observed errors and the total number of transmitted bits. ©2009, B.-P. Paris Wireless Communications 158

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Monte Carlo Simulation The above suggests a relatively simple structure for a Monte Carlo simulator. Inside a programming loop: perform a system simulation, and accumulate counts for the quantities of interest 43 while ( ~Done ) NumErrors(kk) = NumErrors(kk) + MCSimple( Parameters ); NumSymbols(kk) = NumSymbols(kk) + Parameters.NSymbols; % compute Stop condition 48 Done = NumErrors(kk) > MinErrors || NumSymbols(kk) > MaxSy end ©2009, B.-P. Paris Wireless Communications 159

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Confidence Intervals Question: How many times should the loop be executed? Answer: It depends on the desired level of accuracy (confidence), and (most importantly) on the symbol error rate. Confidence Intervals: Assume we form an estimate of the symbol error rate Pe as described above. ˆ Then, the true error rate Pe is (hopefully) close to our estimate. Put differently, we would like to be reasonably sure that the ˆ absolute difference |Pe − Pe | is small. ©2009, B.-P. Paris Wireless Communications 160

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Confidence Intervals More specifically, we want a high probability pc (e.g., ˆ pc =95%) that |Pe − Pe | < sc . The parameter sc is called the confidence interval; it depends on the confidence level pc , the error probability Pe , and the number of transmitted symbols N. It can be shown, that Pe (1 − Pe ) sc = zc · , N where zc depends on the confidence level pc . Specifically: Q (zc ) = (1 − pc )/2. Example: for pc =95%, zc = 1.96. Question: How is the number of simulations determined from the above considerations? ©2009, B.-P. Paris Wireless Communications 161

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Choosing the Number of Simulations For a Monte Carlo simulation, a stop criterion can be formulated from a desired confidence level pc (and, thus, zc ) an acceptable confidence interval sc , the error rate Pe . Solving the equation for the confidence interval for N, we obtain N = Pe · (1 − Pe ) · (zc /sc )2 . A Monte Carlo simulation can be stopped after simulating N transmissions. Example: For pc =95%, Pe = 10−3 , and sc = 10−4 , we find N ≈ 400, 000. ©2009, B.-P. Paris Wireless Communications 162

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation A Better Stop-Criterion When simulating communications systems, the error rate is often very small. Then, it is desirable to specify the conﬁdence interval as a fraction of the error rate. The conﬁdence interval has the form sc = αc · Pe (e.g., αc = 0.1 for a 10% acceptable estimation error). Inserting into the expression for N and rearranging terms, Pe · N = (1 − Pe ) · (zc /αc )2 ≈ (zc /αc )2 . Recognize that Pe · N is the expected number of errors! Interpretation: Stop when the number of errors reaches (zc /αc )2 . Rule of thumb: Simulate until 400 errors are found (pc =95%, α =10%). ©2009, B.-P. Paris Wireless Communications 163

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Listing : MCSimpleDriver.m 9 % comms parameters delegated to script SimpleSetParameters SimpleSetParameters; % simulation parameters EsOverN0dB = 0:0.5:9; % vary SNR between 0 and 9dB 14 MaxSymbols = 1e6; % simulate at most 1000000 symbols % desired confidence level an size of confidence interval ConfLevel = 0.95; ZValue = Qinv( ( 1-ConfLevel )/2 ); 19 ConfIntSize = 0.1; % confidence interval size is 10% of estimate % For the desired accuracy, we need to find this many errors. MinErrors = ( ZValue/ConfIntSize )^2; Verbose = true; % control progress output 24 %% simulation loops % initialize loop variables NumErrors = zeros( size( EsOverN0dB ) ); NumSymbols = zeros( size( EsOverN0dB ) ); ©2009, B.-P. Paris Wireless Communications 164

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Listing : MCSimpleDriver.m for kk = 1:length( EsOverN0dB ) 32 % set Es/N0 for this iteration Parameters.EsOverN0 = dB2lin( EsOverN0dB(kk) ); % reset stop condition for inner loop Done = false; 37 % progress output if (Verbose) disp( sprintf( ’Es/N0: %0.3g dB’, EsOverN0dB(kk) ) ); end 42 % inner loop iterates until enough errors have been found while ( ~Done ) NumErrors(kk) = NumErrors(kk) + MCSimple( Parameters ); NumSymbols(kk) = NumSymbols(kk) + Parameters.NSymbols; 47 % compute Stop condition Done = NumErrors(kk) > MinErrors || NumSymbols(kk) > MaxSymbol end ©2009, B.-P. Paris Wireless Communications 165

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Simulation Results −1 10 −2 10 Symbol Error Rate −3 10 −4 10 −5 10 −2 0 2 4 6 8 10 Es/N0 (dB) ©2009, B.-P. Paris Wireless Communications 166

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Summary Introduced discrete-time equivalent systems suitable for simulation in MATLAB. Relationship between original, continuous-time system and discrete-time equivalent was established. Digital post-processing: digital matched ﬁlter and slicer. Monte Carlo simulation of a simple communication system was performed. Close attention was paid to the accuracy of simulation results via conﬁdence levels and intervals. Derived simple rule of thumb for stop-criterion. ©2009, B.-P. Paris Wireless Communications 167

Elements of a Digital Communications System Digital Modulation Channel Model Receiver MATLAB Simulation Where we are ... Laid out a structure for describing and analyzing communication systems in general and wireless systems in particular. Saw a lot of MATLAB examples for modeling diverse aspects of such systems. Conducted a simulation to estimate the error rate of a communication system and compared to theoretical results. To do: consider selected aspects of wireless communication systems in more detail, including: modulation and bandwidth, wireless channels, advanced techniques for wireless communications. ©2009, B.-P. Paris Wireless Communications 168

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Part III Digital Modulation and Spectrum ©2009, B.-P. Paris Wireless Communications 169

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Digital Modulation and Spectrum Digital modulation formats and their spectra. Linear, narrow-band modulation (including B/QPSK, PSK, QAM, and variants OQPSK, π/4 DQPSK) Non-linear modulation (including CPM, CPFSK, MSK, GMSK) Wide-band modulation (CDMA and OFDM) The use of pulse-shaping to control the spectrum of modulated signals. Spectrum estimation of digitally modulated signals in MATLAB. ©2009, B.-P. Paris Wireless Communications 170

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Outline Part II: Learning Objectives Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation ©2009, B.-P. Paris Wireless Communications 171

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Learning Objectives Understand choices and trade-offs for digital modulation. Linear modulation: principles and parameters. Non-linear modulation: beneﬁts and construction. The importance of constant-envelope characteristics. Wide-band modulation: DS/SS and OFDM. Visualization of digitally modulated signals. Spectra of digitally modulated signals. Closed-form expressions for the spectrum of linearly modulated signals. Numerical estimation of the spectrum of digitally modulated signals. ©2009, B.-P. Paris Wireless Communications 172

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Linear Modulation We have already introduced Linear Modulation as the digital equivalent of amplitude modulation. Recall that a linearly modulated signal may be written as N −1 s (t ) = ∑ bn · p (t − nT ) n =0 where, bn denotes the n-th information symbol, and p (t ) denotes a pulse of ﬁnite duration. T is the duration of a symbol. We will work with baseband equivalent signals throughout. Symbols bn will generally be complex valued. ©2009, B.-P. Paris Wireless Communications 174

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Linear Modulation Objective: Investigate the impact of the parameters alphabet from which symbols bn are chosen, pulse shape p (t ), bn s (t ) symbol period T × p (t ) on signals in the time and frequency domain. Note: We are interested in the ∑ δ(t − nT ) properties of the analog signals produced by the transmitter. Signals will be signiﬁcantly oversampled to approximate signals closely. ©2009, B.-P. Paris Wireless Communications 175

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Signal Constellations The inﬂuence of the alphabet from which symbols bn are 1000 chosen is well captured by the 500 signal constellation. The signal constellation is Imag. Part 0 simply a plot indicating the location of all possible −500 symbols in the complex plane. −1000 The signal constellation is the −1500 −1000 −500 0 500 1000 1500 noise-free output of the Real Part (digital) matched ﬁlter. ©2009, B.-P. Paris Wireless Communications 176

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Characteristics of Signals Constellations Three key characteristics of a signal constellation: 1. Number of symbols M in constellation determines number of bits transmitted per symbol; Nb = log2 (M ). 2. Average symbol energy is computed as M 1 Es = M ∑ |bk |2 · A2 |p (t )|2 dt; k =1 we will assume that |p (t )|2 dt = 1. 3. Shortest distance dmin between points in constellation has major impact on probability of symbol error. Often expressed in terms of Es . ©2009, B.-P. Paris Wireless Communications 177

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: BPSK A BPSK: Binary Phase Shift Keying Nb = 1 bit per symbol, Imaginary 0 Es = A2 , √ dmin = 2A = 2 Es . Recall that symbol error rate is √ Pe = Q (√ 2Es /N0 ) = −A Q (dmin / 2N0 ). −A 0 A Real ©2009, B.-P. Paris Wireless Communications 178

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: QPSK A QPSK: Quaternary Phase Shift Keying Imaginary 0 Nb = 2 bits per symbol, Es = A2 , √ √ dmin = 2A = 2Es . −A −A 0 A Real ©2009, B.-P. Paris Wireless Communications 179

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: 8-PSK A 8-PSK: Eight Phase Shift Keying Nb = 3 bit per Imaginary 0 symbol, Es = A2 , √ dmin = (2 − 2)A = √ √ (2 − 2) Es . √ 2− 2 ≈ 0.6 −A −A 0 A Real ©2009, B.-P. Paris Wireless Communications 180

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: 16-QAM 3A 16-QAM: 16-Quadrature 2A Amplitude Modulation Nb = 4 bit per symbol, A Es = 10TA2 , √ dmin = 2A = 2 Es . Imaginary 0 5 −A Note that symbols don’t all have the same energy; this is −2A potentially problematic. −3A −3A −2A −A 0 A 2A 3A 16-QAM is not commonly used Real for wireless communications. ©2009, B.-P. Paris Wireless Communications 181

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Pulse Shaping The discrete symbols that constitute the constellation are converted to analog signals s (t ) with the help of pulses. bn s (t ) × p (t ) The symbols bn determine the “instantaneous amplitude” of the; while the pulses determine the ∑ δ(t − nT ) shape of the signals. We will see that the pulse shape has a major impact on the spectrum of the signal. ©2009, B.-P. Paris Wireless Communications 182

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Full-Response Pulses Full-response pulses span exactly one symbol period. 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Time/T ©2009, B.-P. Paris Wireless Communications 183

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Partial-Response Pulses Partial-response pulses are of duration longer than one symbol period. The main benefit that partial-response pulse can provide are better spectral properties. The extreme case, is an infinitely long sinc-pulse which produces a strictly band-limited spectrum. On the negative side, partial-response pulses (can) introduce intersymbol-interference that affects negatively the demodulation of received signals. The special class of Nyquist pulses avoids, in principle, intersymbol interference. In practice, multi-path propagation foils the benefits of Nyquist pulses. ©2009, B.-P. Paris Wireless Communications 184

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Partial-Response Pulses A (truncated) sinc-pulse is a partial-response pulse. 1.5 1 0.5 0 −0.5 0 2 4 6 8 10 Time/T ©2009, B.-P. Paris Wireless Communications 185

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Raised Cosine Nyquist Pulses The sinc-pulse is a special case in a class of important pulse shapes. We will see that raised cosine Nyquist pulses have very good spectral properties and meet the Nyquist condition for avoiding intersymbol interference. Raised cosine Nyquist pulses are given by sin(πt /T ) cos( βπt /T ) p (t ) = · πt /T 1 − (2βt /T )2 The parameter β is called the roll-off factor and determines how quickly the pulses dampens out. ©2009, B.-P. Paris Wireless Communications 186

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Raised Cosine Nyquist Pulses 1.5 β=0 β=0.3 β=0.5 β=1 1 0.5 0 −0.5 0 1 2 3 4 5 6 7 8 9 10 Time/T ©2009, B.-P. Paris Wireless Communications 187

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Visualizing Linearly Modulated Signals To understand the time-domain properties of a linearly modulated signal one would like to plot the signal. However, since baseband-equivalent signals are generally complex valued, this is not straightforward. Plotting the real and imaginary parts of the signal separately does not provide much insight. Useful alternatives for visualizing the modulated signal are Plot the magnitude and phase of the signal; useful because information is generally encoded in magnitude and phase. Plot signal trajectory in the complex plane, i.e., plot real versus imaginary part. Shows how modulated signal moves between constellation points. Plot the up-converted signal (at a low IF frequency). ©2009, B.-P. Paris Wireless Communications 188

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation MATLAB function LinearModulation All examples, rely on the toolbox function LinearModulation to generate the baseband-equivalent transmitted signal. 19 % initialize storage for Signal LenSignal = length(Symbols)*fsT + (length(Pulse))-fsT; Signal = zeros( 1, LenSignal ); % loop over symbols and insert corresponding segment into Signal 24 for kk = 1:length(Symbols) ind_start = (kk-1)*fsT + 1; ind_end = (kk-1)*fsT + length(Pulse); Signal(ind_start:ind_end) = Signal(ind_start:ind_end) + ... 29 Symbols(kk) * Pulse; end ©2009, B.-P. Paris Wireless Communications 189

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: QPSK with Raised Cosine Nyquist Pulses 1.5 1 Magnitude 0.5 0 5 6 7 8 9 10 11 12 13 14 15 Time/T 1 0.5 Phase/π 0 −0.5 −1 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Magnitude and Phase; QPSK, Raised Cosine Nyquist Pulse (β = 0.5) ©2009, B.-P. Paris Wireless Communications 190

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: QPSK with Raised Cosine Nyquist Pulses 1.5 1 0.5 Amplitude 0 −0.5 −1 −1.5 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Up-converted signal, fc = 2/T ; QPSK, Raised Cosine Nyquist Pulse (β = 0.5) ©2009, B.-P. Paris Wireless Communications 191

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: QPSK with Raised Cosine Nyquist Pulses 1 0.5 Imaginary 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 1.5 Real Figure: Complex Plane Signal Trajectory; QPSK, Raised Cosine Nyquist Pulse (β = 0.5) ©2009, B.-P. Paris Wireless Communications 192

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Constant Envelope Signals It is desirable for digitally modulated signals to have constant magnitude. This permits the use of non-linear power amplifiers: Non-linear power amplifiers are more energy efficient; more of the energy they consume is used to amplify the signal. Non-linear: the gain of the amplifier varies with the magnitude of the input signal. This leads to non-linear distortions of the signal — the pulse shape is altered. Constant envelope signals do not experience distortion from non-linear power amplifiers. ©2009, B.-P. Paris Wireless Communications 193

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Towards Constant Envelope Signals The preceding examples show that it is not sufﬁcient for the symbols to have constant magnitude to create constant envelope signals. In particular, abrupt phase changes of 180o lead to signal trajectories through the origin of the complex plane. To reduce the variation of the signal magnitude, one can encode symbols such that 180o phase changes are eliminated. Generally, it is necessary to encode multiple symbols at the same time. We refer to such encoding strategies as linear modulation with memory. ©2009, B.-P. Paris Wireless Communications 194

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: OQPSK OQPSK stands for Offset QPSK. One can think of OQPSK as a modulation format that alternates between using two constellations: each of the two alphabets contains two symbols — one bit is transmitted per symbol period T , in odd-numbered symbol periods, the alphabet A = {1, −1} is used, and in even-numbered symbol periods, the alphabet A = {j, −j } is used. Note that only phase changes of ±90o are possible and, thus, transitions through the origin are avoided. Despite its name, OQPSK has largely the same √ characteristics as BPSK. (Nb = 1, Es = A2 , dmin = 2 Es ) ©2009, B.-P. Paris Wireless Communications 195

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Toolbox function OQPSK The toolbox function OQPSK accepts a vector of BPSK symbols and converts them to OQPSK symbols: function OQPSKSymbols = OQPSK( BPSKSymbols ) %% BPSK -> OQPSK % keep odd-numbered samples, phase-shift even numbered samples OQPSKSymbols = BPSKSymbols; 21 % phase shift even samples OQPSKSymbols(2:2:end) = j*BPSKSymbols(2:2:end); ©2009, B.-P. Paris Wireless Communications 196

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Using the Toolbox Function OQPSK Calls to the toolbox function OQPSK are inserted between the function RandomSymbols for generating BPSK symbols, and the function LinearModulation for creating baseband-equivalent modulated signals. %% symbols and Signal using our functions Symbols = RandomSymbols(Ns, Alphabet, Priors); 13 Symbols = OQPSK(Symbols); Signal = LinearModulation(Symbols,Pulse,fsT); ©2009, B.-P. Paris Wireless Communications 197

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: OQPSK with Raised Cosine Nyquist Pulses 1.5 1 Magnitude 0.5 0 5 6 7 8 9 10 11 12 13 14 15 Time/T 1 0.5 Phase/π 0 −0.5 −1 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Magnitude and Phase; OQPSK, Raised Cosine Nyquist Pulse (β = 0.5) ©2009, B.-P. Paris Wireless Communications 198

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: OQPSK with Raised Cosine Nyquist Pulses 1.5 1 0.5 Amplitude 0 −0.5 −1 −1.5 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Up-converted signal, fc = 2/T ; OQPSK, Raised Cosine Nyquist Pulse (β = 0.5) ©2009, B.-P. Paris Wireless Communications 199

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: OQPSK with Raised Cosine Nyquist Pulses 1 0.8 0.6 0.4 0.2 Imaginary 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Real Figure: Complex Plane Signal Trajectory; OQPSK, Raised Cosine Nyquist Pulse (β = 0.5) ©2009, B.-P. Paris Wireless Communications 200

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation QPSK vs OQPSK 1 0.8 1 0.6 0.4 0.5 0.2 Imaginary Imaginary 0 0 −0.2 −0.5 −0.4 −0.6 −1 −0.8 −1 −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 Real Real ©2009, B.-P. Paris Wireless Communications 201

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation OQPSK with Half-Sine Pulses An important special case 1 arises when OQPSK is used 0.8 in conjunction with pulses of the form 0.6 πt p (t ) = sin( ) for 0 ≤ t ≤ 2T . 0.4 2T 0.2 Note, that these pulse span 0 two symbol periods. 0 0.2 0.4 0.6 0.8 1 Time/T 1.2 1.4 1.6 1.8 2 ©2009, B.-P. Paris Wireless Communications 202

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation OQPSK with Half-Sine Pulses 1 0.8 0.6 0.4 0.2 Imaginary 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Real Figure: Complex Plane Signal Trajectory; OQPSK with Half-Sine Pulses ©2009, B.-P. Paris Wireless Communications 203

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation OQPSK with Half-Sine Pulses The resulting modulated signal is a perfectly constant envelope signal. Very well suited for energy efﬁcient, non-linear ampliﬁers. It can be shown that the resulting signal is equivalent to Minimum Shift Keying (MSK). MSK will be considered in detail later. Relationship is important in practice to generate and demodulate MSK signals. ©2009, B.-P. Paris Wireless Communications 204

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Differential Phase Encoding So far, we have considered only modulation formats where information symbols are mapped directly to a set of phases. Alternatively, it is possible to encode information in the phase difference between consecutive symbols. Example: In Differential-BPSK (D-BPSK), the phase difference between the n-th and (n − 1)-th symbol period is either 0 or π. Thus, one bit of information can be conveyed. ©2009, B.-P. Paris Wireless Communications 205

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Differential Phase Encoding Formally, one can can represent differential encoding with reference to the phase difference ∆θn = θn − θn−1 : ∆θn = φ0 · xn . where φ0 indicates the phase increment (e.g., φ0 = π for D-BPSK) xn is drawn from an alphabet of real-valued integers (e.g., xn ∈ {0, 1} for D-BPSK) The symbols generated by differential phase encoders are of the form bn = exp(jθn ), where θn = θn−1 + ∆θn = θ0 + ∑n =1 ∆θk k = θ0 + φ0 ∑n =1 xn k ©2009, B.-P. Paris Wireless Communications 206

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Toolbox function DPSK The toolbox function DPSK with the following function signature provides generic differential phase encoding: 1 function DPSKSymbols = DPSK( PAMSymbols, phi0, theta0 ) The body of DPSK computes differentially encoded symbols as follows: %% accumulate phase differences, then convert to complex symbols Phases = cumsum( [theta0 phi0*PAMSymbols] ); DPSKSymbols = exp( j*Phases ); ©2009, B.-P. Paris Wireless Communications 207

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: π/4-DQPSK An important modulation format, referred to as π/4-DQPSK results when we choose: φ0 = π/4, and xn ∈ {±1, ±3}. Note that with this choice, the phase difference between consecutive symbols is ±π/4 or ±3π/4. Phase differences of π (180o do not occur — no transitions through origin. The resulting signal has many of the same characteristics as QPSK but has less magnitude variation. Signal is also easier to synchronize, since phase transitions occur in every symbol period. ©2009, B.-P. Paris Wireless Communications 208

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Generating π/4-DQPSK Signals Using our toolbox function, π/4-DQPSK are easily generated. To begin, we deﬁne the appropriate alphabet of symbols for differential encoding: Alphabet = -3:2:3; % 4-PAM Then, the baseband-equivalent signal is generated via calls to the appropriate toolbox functions. %% symbols and Signal using our functions Symbols = RandomSymbols(Ns, Alphabet, Priors); Symbols = DPSK(Symbols, pi/4); 14 Signal = LinearModulation(Symbols,Pulse,fsT); ©2009, B.-P. Paris Wireless Communications 209

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation π/4-DQPSK with Raised Cosine Nyquist Pulses. 1 0.5 Imaginary 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 1.5 Real Figure: Complex Plane Signal Trajectory; π/4-DQPSK with Raised Cosine Nyquist Pulses (β = 0.5) ©2009, B.-P. Paris Wireless Communications 210

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum of Linearly Modulated Signals Having investigated the time-domain properties of linearly modulated signals, we turn now to their spectral properties. Technically, the modulated signals are random processes and the appropriate spectral measure is the power spectral density. Modulated signals are random because the information symbols are random. Speciﬁcally, we compute the power spectral density of the baseband-equivalent signals. The computation of the power spectral density for a modulated signal is lengthy and quite involved. Focus on highlights of results. ©2009, B.-P. Paris Wireless Communications 211

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum of Linearly Modulated Signals The power spectral density is greatly simpliﬁed with the following (reasonable) assumptions: Symbols have zero mean, and are uncorrelated. Full-response pulse shaping. Then, the power spectral density depends only on the Fourier transform of the pulse-shape and is given by Ps (f ) = Es · |H (f )|2 . Note, that the power spectral density does not depend on the modulation format! We will rely on numerical techniques to ﬁnd the spectrum of signals for which this expression does not apply. ©2009, B.-P. Paris Wireless Communications 212

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: Rectangular Pulse Assume, a rectangular pulse is used for pulse shaping: 1 for 0 ≤ t < T p (t ) = 0 else. Then, with the assumptions above, the power spectral density of the transmitted signals equals sin(πfT ) 2 Ps (f ) = Es · ( ) . πfT ©2009, B.-P. Paris Wireless Communications 213

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: Rectangular Pulse 0 −10 Spectral characteristics: −20 Normalization: Es = 1 −30 Zero-to-zero Bandwidth: PSD (dB) −40 2/T Side-lobe decay ∼ 1/f 2 −50 Smoother pulses provide −60 much better spectrum. −70 −10 −8 −6 −4 −2 0 2 4 6 8 10 Normalized Frequency (fT) ©2009, B.-P. Paris Wireless Communications 214

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Summary Detailed discussion of linearly modulated signals and their simulation in MATLAB. Pulse-shaping: full and partial response, Modulation formats with and without memory. Constant envelope characteristics Rationale for constant envelope signals (non-linear power ampliﬁers) Improving the envelope characteristics through offset constellations. Power Spectral Density of Modulated Signals Closed form expressions for simple cases. Next: numerical estimation of power spectral density. ©2009, B.-P. Paris Wireless Communications 215

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum Estimation in MATLAB Closed form expressions for the power spectral density are not always easy to obtain, particularly for Partial-response pulse-shaping, non-linear modulation formats. Objective: Develop a simple procedure for estimating the power spectral density of a digitally modulated signal. Start with samples of a digitally modulated waveform, Estimate the PSD from these samples. With a little care, the above objective is easily achieved. ©2009, B.-P. Paris Wireless Communications 217

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation First Attempt One may be tempted to estimate the spectrum as follows: Generate a vector of samples of the signal of interest: N random symbols, fs T samples per symbol period, for the modulation format and pulse shape of interest. Compute the Discrete Fourier Transform (DFT) of the samples: This is easily done using the MATLAB function fft. A smoothing window improves the estimate. Estimate the PSD as the squared magnitude of the DFT. This estimate is referred to as the periodogram. ©2009, B.-P. Paris Wireless Communications 218

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Periodogram Question: How should the number of symbols N and the normalized sampling rate fs T be chosen? Normalized sampling rate fs T determines the observable frequency range: Observable frequencies range from −fs/2 to fs /2. In terms of normalized frequencies fT , highest observable frequency is fs T /2 · 1/T . Chose fs T large enough to cover frequencies of interest. Typical: fs T = 20 Number of Symbols N determines the frequency resolution. The PSD will be sampled N times for each frequency interval of length 1/T . Frequency sampling period: 1/(Nfs T ). Typical: N = 100 ©2009, B.-P. Paris Wireless Communications 219

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Toolbox function Periodogram function Ps = Periodogram( Signal ) % Periodogram - Periodogram estimate for PSD of the input signal. % % Input: 5 % Signal - samples of signal of interest % % Output: % Ps - estimated PSD % 10 % Example: % Ps = Periodogram( Signal ) %% compute periodogram % window eliminates effects due to abrupt onset and end of signal 15 Window = blackman( length(Signal) )’; Ps = fft( Signal.*Window, length(Signal) ); % swap left and right half, to get freqs from -fs/2 to fs/2 Ps = fftshift( Ps ); % return squared magnitude, normalized to account for window 20 Ps = abs(Ps).^2 / norm(Window)^2; ©2009, B.-P. Paris Wireless Communications 220

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: Rectangular Pulses 10 0 −10 −20 PSD (dB) −30 −40 −50 −60 −70 −10 −8 −6 −4 −2 0 2 4 6 8 10 Normalized Frequency (fT) Figure: Periodogram estimate of PSD for rectangular pulse-shaping ©2009, B.-P. Paris Wireless Communications 221

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Improving the Periodogram Estimate The periodogram estimate is extremely noisy. ˆ It is known that the periodogram Ps (f ) has a Gaussian distribution and ˆ is an unbiased estimate E[Ps (f )] = Ps (f ), ˆ has variance Var[Ps (f )] = Ps (f ). The variance of the periodogram is too high for the estimate to be useful. Fortunately, the variance is easily reduced through averaging: Generate M realizations of the modulated signal, and average the periodograms for the signals. Reduces variance by factor M. ©2009, B.-P. Paris Wireless Communications 222

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Variance Reduction through Averaging The MATLAB fragment below illustrates how reliable estimates of the power spectral density are formed. for kk=1:M % generate signal Symbols = RandomSymbols(Ns, Alphabet, Priors); Signal = LinearModulation(Symbols,Pulse,fsT); 20 % accumulate periodogram Ps = Ps + Periodogram( Signal ); end 25 %average Ps = Ps/M; ©2009, B.-P. Paris Wireless Communications 223

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Averaged Periodogram 5 0 −5 −10 −15 PSD (dB) −20 −25 −30 −35 −40 −45 −50 −10 −8 −6 −4 −2 0 2 4 6 8 10 Normalized Frequency (fT) Figure: Averaged Periodogram for a QPSK signal with Rectangular Pulses; M = 500. ©2009, B.-P. Paris Wireless Communications 224

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Remaining Issues The resulting estimate is not perfect. Near the band-edges, the power spectral density is consistently over-estimated. This is due to aliasing. To improve the estimate, one can increase the normalized sampling rate fs T . For pulses with better spectral properties, aliasing is signiﬁcantly reduced. Additionally, center of band is of most interest — to assess bandwidth, simplicity of estimator is very attractive. ©2009, B.-P. Paris Wireless Communications 225

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum Estimation for Linearly Modulated Signals We apply our estimation technique to illustrate the spectral characteristics of a few of the modulation formats considered earlier. Comparison of BPSK and π/4-DQPSK with rectangular pulses, Inﬂuence of the roll-off factor β of raised cosine pulses, OQPSK with rectangular pulses and half-sine pulses (MSK). ©2009, B.-P. Paris Wireless Communications 226

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation BPSK and π/4-DQPSK Spectrum 5 BPSK 0 π/4−DQPSK −5 −10 −15 PSD (dB) −20 −25 −30 −35 −40 −45 −50 −10 −8 −6 −4 −2 0 2 4 6 8 10 Normalized Frequency (fT) Figure: Spectrum of BPSK and π/4-DQPSK modulated signals with rectangular pulses: Spectrum depends only on pulse shape. ©2009, B.-P. Paris Wireless Communications 227

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Raised Cosine Pulses - Inﬂuence of β 0 β=0 β=0.3 −10 β=0.5 β=1 −20 −30 −40 PSD (dB) −50 −60 −70 −80 −90 −100 −4 −3 −2 −1 0 1 2 3 4 Normalized Frequency (fT) Figure: QPSK modulated signals with Raised Cosine Pulse Shaping. Width of main-lobe increases with roll-off factor β. Side-lobes are due to truncation of pulses. ©2009, B.-P. Paris Wireless Communications 228

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation OQPSK with Rectangular and Half-Sine Pulses Rectangular 0 Half−sine −10 −20 PSD (dB) −30 −40 −50 −60 −70 −10 −8 −6 −4 −2 0 2 4 6 8 10 Normalized Frequency (fT) Figure: OQPSK modulated signals with rectangular and half-sine pulses (MSK). Pulse shape affects the spectrum dramatically. ©2009, B.-P. Paris Wireless Communications 229

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Summary A numerical method for estimating the spectrum of digitally modulated signals was presented. Relies on the periodogram estimate, variance reduction through averaging. Simple and widely applicable. Applied method to several linear modulation formats. Conﬁrmed that spectrum of linearly modulated signals depends mainly on pulse shape; constellation does not affect spectrum. Signiﬁcant improvements of spectrum possible with pulse-shaping. Partial-response pulses (in particular, raised cosine Nyquist pulses) have excellent spectral properties. ©2009, B.-P. Paris Wireless Communications 230

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Non-linear Modulation Linear modulation formats were investigated and two characteristics were emphasized: 1. spectral properties - width of main-lobe and decay of side-lobes; 2. magnitude variations - constant envelope characteristic is desired but difﬁcult to achieve. A broad class of (non-linear) modulation formats will be introduced with 1. excellent spectral characteristics - achieved by eliminating abrupt phase and amplitude changes; 2. constant envelope characteristic. More difﬁcult to demodulate in general. ©2009, B.-P. Paris Wireless Communications 232

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Reminder: Analog Frequency Modulation We will present a broad class of non-linear modulation formats with desirable properties. These formats are best understood by establishing an analogy to analog frequency modulation (FM). Recall: a message signal m (t ) is frequency modulated by constructing the baseband-equivalent signal t s (t ) = A · exp(j2πfd m (τ ) dτ ). −∞ Signal is constant envelope, signal is a non-linear function of message m (t ), instantaneous frequency: fd · m (t ), where fd is called the frequency deviation constant. ©2009, B.-P. Paris Wireless Communications 233

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Continuous Phase Modulation The class of modulation formats described here is referred to as continuous-phase modulation (CPM). CPM signals are constructed by frequency modulating a pulse-amplitude modulated (PAM) signal: PAM signal: is a linearly modulated signal of the information symbols bn ∈ {±1, . . . , ±(M − 1)}: N m (t ) = ∑ hn · bn · pf (t − nT ). n =0 CPM signal: FM of m (t ) t s (t ) = A · exp(j2π m (τ ) dτ ). 0 ©2009, B.-P. Paris Wireless Communications 234

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Parameters of CPM Signals The following parameters of CPM signals can be used to create a broad class of modulation formats. Alphabet size M: PAM symbols are drawn form the alphabet A = {±1, . . . , ±(M − 1)}; generally M = 2K . Modulation indices hn : play the role of frequency deviation constant fd ; Often hn = h, constant modulation index, periodic hn possible, multi-h CPM. Frequency shaping function pf (t ): pulse shape of PAM signal. Pulse length L symbol periods. L = 1: full-response CPM, L > 1: partial response CPM. Normalization: pf (t ) dt = 1/2. ©2009, B.-P. Paris Wireless Communications 235

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Generating CPM Signals in MATLAB We have enough information to write a toolbox function to generate CPM signal; it has the signature function [CPMSignal, PAMSignal] = CPM(Symbols, A, h, Pulse, fsT) The body of the function calls LinearModulation to generate the PAM signal, performs numerical integration through a suitable IIR ﬁlter %% Generate PAM signal, using function LinearModulation PAMSignal = LinearModulation( Symbols, Pulse, fsT ); %% Integrate PAM signal using a filter with difference equation 30 % y(n) = y(n-1) + x(n)/fsT and multiply with 2*pi*h a = [1 -1]; b = 1/fsT; Phase = 2*pi*h*filter(b, a, PAMSignal); 35 %% Baseband equivalent signal CPMSignal = A*exp(j*Phase); ©2009, B.-P. Paris Wireless Communications 236

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Using Toolbox Function CPM A typical invocation of the function CPM is as follows: %% Parameters: fsT = 20; Alphabet = [1,-1]; % 2-PAM 6 Priors = ones( size( Alphabet) ) / length(Alphabet); Ns = 20; % number of symbols Pulse = RectPulse( fsT); % Rectangular pulse A = 1; % amplitude h = 0.75; % modulation index 11 %% symbols and Signal using our functions Symbols = RandomSymbols(Ns, Alphabet, Priors); Signal = CPM(Symbols, A, h, Pulse,fsT); ©2009, B.-P. Paris Wireless Communications 237

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: CPM with Rectangular Pulses and h = 3/4 1.5 1 Magnitude 0.5 0 5 6 7 8 9 10 11 12 13 14 15 Time/T 1 0 Phase/π −1 −2 −3 −4 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Magnitude and Phase; CPM, M = 2, h = 3/4, full-response rectangular pulses. ©2009, B.-P. Paris Wireless Communications 238

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: CPM with Rectangular Pulses and h = 3/4 1.5 1 0.5 Amplitude 0 −0.5 −1 −1.5 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Up-converted signal, fc = 2/T ; M = 2, h = 3/4, full-response rectangular pulses. ©2009, B.-P. Paris Wireless Communications 239

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: CPM with Rectangular Pulses and h = 3/4 1 0.8 0.6 0.4 0.2 Imaginary 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Real Figure: Complex Plane Signal Trajectory; M = 2, h = 3/4, full-response rectangular pulses. ©2009, B.-P. Paris Wireless Communications 240

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Excess Phase The excess phase or instantaneous phase of a CPM signal is given by t Φ(t, b ) = 2π 0 m (τ ) dτ t = 2π 0 ∑N=0 hn · bn · pf (τ − nT ) dτ n N t = 2π ∑n=0 hn · bn · 0 pf (τ − nT ) dτ = 2π ∑N=0 hn · bn · β(t − nT ), n t where β(t ) = 0 pf (τ ) dτ. pf (t ) β (t ) 1/2T 1/2 t t T 2T T 2T ©2009, B.-P. Paris Wireless Communications 241

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Excess Phase The excess phase N Φ(t, b ) = 2π ∑ hn · bn · β(t − nT ) n =0 explains some of the features of CPM. The excess phase Φ(t, b ) is a continuous function of time: due to integration, Consequence: no abrupt phase changes, expect good spectral properties. Excess phase Φ(t, b ) has memory: phase depends on all preceding symbols, similar to differential encoding but with continuous phase changes. ©2009, B.-P. Paris Wireless Communications 242

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Excess Phase for Full-Response CPM For full-response CPM signals, the excess phase for the n-th symbol period, nT ≤ t < (n + 1)T , can be expressed as n −1 Φ(t, b ) = πh ∑ bk + 2πhbn β(t − nT ). k =0 The two terms in the sums are easily interpreted: 1. The term θn−1 = πh ∑n−1 bk accounts for the accumulated k =0 phase from preceding symbols. We saw a similar term in connection with differential phase encoding. 2. The second term, 2πhbn β(t − nT ), describes the (continuous) phase change due to the current symbol bn . This term goes from 0 to πh over the current symbol period Similar analysis is possible for partial response CPM. ©2009, B.-P. Paris Wireless Communications 243

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation CPFSK Continuous-Phase Frequency Shift Keying (CPFSK) is a special case of CPM. CPFSK is CPM with full-response rectangular pulses. The rectangular pulses cause the excess phase to change linearly in each symbol period. A linearly changing phase is equivalent to a frequency shift relative to the carrier frequency). Speciﬁcally, the instantaneous frequency in the n-th symbol period equals fc + bn · (πh)/(2T ). Consequently, information is encoded in the frequency of the signal (FSK). Additionally, the phase is guaranteed to be continuous. ©2009, B.-P. Paris Wireless Communications 244

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Phase Tree for Binary CPFSK 4h 3h 2h h Excess Phase/π 0 −h −2h −3h −4h 0 0.5 1 1.5 2 2.5 3 3.5 4 Time/T Figure: Phase Tree for a binary CPFSK signal; CPM with bn ∈ {1, −1}, full-response rectangular pulses. Slope of phase trajectories equals frequency offset. ©2009, B.-P. Paris Wireless Communications 245

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Example: MSK Minimum Shift Keying (MSK) is CPFSK with modulation index h = 1/2. Instantaneous frequency: fc ± 1/4T . Transmitted signals in each symbol period are of the form s (t ) = A cos(2π (fc ± 1/4T )t + θn ). The two signals comprising the signal set are orthogonal. Enables simple non-coherent reception. Orthogonality is not possible for smaller frequency shifts (⇒ minimum shift keying). ©2009, B.-P. Paris Wireless Communications 246

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation MSK: Magnitude and Phase 1.5 Magnitude 1 0.5 0 5 6 7 8 9 10 11 12 13 14 15 Time/T 1 0 Phase/π −1 −2 −3 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Magnitude and Phase; MSK. ©2009, B.-P. Paris Wireless Communications 247

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation MSK: Up-converted Signal 1.5 1 0.5 Amplitude 0 −0.5 −1 −1.5 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Up-converted signal, fc = 2/T ; MSK. ©2009, B.-P. Paris Wireless Communications 248

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation MSK: Complex Plane Signal Trajectory 1 0.8 0.6 0.4 0.2 Imaginary 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Real Figure: Complex Plane Signal Trajectory; MSK. ©2009, B.-P. Paris Wireless Communications 249

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation MSK MSK is in many respects a nearly ideal waveform for wireless communications: constant envelope, easily generated, easy to demodulate, either non-coherently, or coherently via interpretation as OQPSK with half-sine pulses. Except: spectrum could be a little better. Remedy: use smoother pulses Gaussian MSK (GMSK). ©2009, B.-P. Paris Wireless Communications 250

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation GMSK To improve the spectral properties, while maintaining the other benefits of MSK, one can lowpass-filter the PAM signal before FM. A filter with impulse response equal to a Gaussian pdf is used to produce a Gaussian MSK (GMSK) signal. Equivalently, frequency shaping with the following pulse: 1 t /T + 1/2 t /T − 1/2 pf (t ) = (Q ( ) − Q( )), 2T σ σ where, ln(2) σ= 2πBT and 1 ∞ Q (x ) = √ exp(−z 2 /2) dz. 2π x ©2009, B.-P. Paris Wireless Communications 251

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Frequency-Shaping Pulse for GMSK The parameter BT is called the time-bandwidth product of the pulse. It controls the shape of the pulse (Typical value: BT ≈ 0.3). The toolbox function GaussPulse generates this pulse. BT=0.3 BT=0.5 BT=1 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Time/T ©2009, B.-P. Paris Wireless Communications 252

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation GMSK: Magnitude and Phase 1.5 Magnitude 1 0.5 0 5 6 7 8 9 10 11 12 13 14 15 Time/T 1 0 Phase/π −1 −2 −3 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Magnitude and Phase; GMSK (BT = 0.3). ©2009, B.-P. Paris Wireless Communications 253

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation GMSK: Up-converted Signal 1.5 1 0.5 Amplitude 0 −0.5 −1 −1.5 5 6 7 8 9 10 11 12 13 14 15 Time/T Figure: Up-converted signal, fc = 2/T ; GMSK (BT = 0.3). ©2009, B.-P. Paris Wireless Communications 254

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation GMSK: Complex Plane Signal Trajectory 0.8 0.6 0.4 0.2 Imaginary 0 −0.2 −0.4 −0.6 −0.8 −1 −0.5 0 0.5 1 Real Figure: Complex Plane Signal Trajectory; GMSK (BT = 0.3). ©2009, B.-P. Paris Wireless Communications 255

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation GMSK compared to MSK Both MSK and GMSK are constant envelope signals. GMSK has smoother phase than MSK. In particular, direction changes are smooth rather than abrupt. Due to use of smooth pulses. Expect better spectral properties for GMSK. For GMSK, phase does not change by exactly ±π/2 in each symbol period. GMSK is partial response CPM; effect is akin to ISI. Complicates receiver design; sequence estimation. For BT 0.25, effect is moderate and can be safely ignored; use linear receiver. Next, we will compare the spectra of MSK and GMSK. ©2009, B.-P. Paris Wireless Communications 256

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum of CPM Signals The periodogram-based tools for computing the power spectral density of digitally modulated signals can be applied to CPM signals. We will use these tools to compare the spectrum of MSK and GMSK, assess the inﬂuence of BT on the spectrum of a GMSK signal, compare the spectrum of full-response and partial response CPM signals. ©2009, B.-P. Paris Wireless Communications 257

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum of MSK and GMSK 0 MSK GMSK (BT=0.3) −20 −40 PSD (dB) −60 −80 −100 −120 −10 −8 −6 −4 −2 0 2 4 6 8 10 Normalized Frequency (fT) Figure: Spectrum of MSK and GMSK modulated signals. ©2009, B.-P. Paris Wireless Communications 258

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Inﬂuence of BT on GMSK Spectrum 20 BT=0.2 BT=0.3 BT=0.5 0 BT=1 −20 −40 PSD (dB) −60 −80 −100 −120 −5 −4 −3 −2 −1 0 1 2 3 4 5 Normalized Frequency (fT) Figure: Spectrum of GMSK modulated signals as a function of time-bandwidth product BT . ©2009, B.-P. Paris Wireless Communications 259

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum of Partial Response CPM Signals 20 L=1 L=2 10 L=3 L=4 0 −10 PSD (dB) −20 −30 −40 −50 −60 −5 −4 −3 −2 −1 0 1 2 3 4 5 Normalized Frequency (fT) Figure: Spectrum of CPM signals as a function of pulse-width; rectangular pulses spanning L symbol periods, h = 1/2. ©2009, B.-P. Paris Wireless Communications 260

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Summary CPM: a broad class of digitally modulated signals. Obtained by frequency modulating a PAM signal. CPM signals are constant envelope signals. Investigated properties of excess phase. MSK and GMSK: MSK is binary CPM with full-response rectangular pulses and modulation index h = 1/2. Spectral properties of MSK can be improved by using smooth, partial response pulses: GMSK. Experimented with the spectrum of CPM signals: GMSK with small BT has very good spectral properties. Smooth pulses improve spectrum. Partial-response pulses lead to small improvements of spectrum. Demodulating CPM signals can be difﬁcult. ©2009, B.-P. Paris Wireless Communications 261

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Wide-band Signals Up to this point, we considered modulation methods generally referred to as narrow-band modulation. The bandwith of the modulated signals is approximately to the symbol rate. In contrast, spread-spectrum modulation produces signals with bandwidth much larger than the symbol rate. Power spectral density is decreased proportionally. Useful for co-existence scenarios or for low probability of detection. OFDM achieves simultaneously high data rate and long symbol periods. Long symbol periods are beneﬁcial in ISI channels. Achieved by clever multiplexing of many narrow-band signals. ©2009, B.-P. Paris Wireless Communications 263

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Direct-Sequence Spread Spectrum Conceptually, direct-sequence spread spectrum (DS/SS) modulation is simply linear modulation using wide-band pulses. 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 DS/SS Pulse −1 Narrowband Pulse 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time/T ©2009, B.-P. Paris Wireless Communications 264

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Direct-Sequence Spread Spectrum The effect of employing a wide-band pulse is that the spectrum of the transmitted signal is much wider than the symbol rate. Signal bandwidth is determined by bandwidth of pulse. The purpose of spreading is to distribute signal power over a larger bandwidth, to achieve: channel sharing with other narrow-band and wide-band users, robustness to jamming or interference, low probability of detection. Wide-band-pulse is generally generated from a pseudo-random sequence. Spreading sequence has M chips per symbol. Spreading sequence may be periodic or not. Then, bandwidth increase is M-fold. ©2009, B.-P. Paris Wireless Communications 265

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Spectrum of DS/SS Signals 5 BPSK 0 DS/SS −5 −10 −15 PSD (dB) −20 −25 −30 −35 −40 −45 −50 −20 −15 −10 −5 0 5 10 15 20 Normalized Frequency (fT) Figure: Spectrum of a DS/SS signal; pseudo-random spreading sequence of length M = 15. Same data rate narrow-band waveform (BPSK with rectangular pulses) shown for comparison. ©2009, B.-P. Paris Wireless Communications 266

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Demodulating DS/SS Signals Principally, demodulation of DS/SS signals employs a matched ﬁlter for the wide-band pulse. Practically, the receiver front-end consists of an A-to-D converter operating at the chip rate (or higher). The subsequent digital matched ﬁlter for the Spreading sequence is called a de-spreader. In additive, white Gaussian noise a DS/SS waveform performs identical to a narrow-band waveform with the same symbol energy. However, in environments with interference a DS/SS waveform is more robust. This includes interference from other emitters and ISI. ©2009, B.-P. Paris Wireless Communications 267

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation OFDM Orthogonal Frequency Division Multiplexing (OFDM) is a modulation format that combines the beneﬁts of narrow-band and wide-band signals. Narrow-band signals are easy to demodulate. Wide-band signals can support high data rates and are robust to multi-path fading. In essence, OFDM signals are constructed by clever multiplexing of many narrow-band signals. Computationally efﬁcient via FFT. ©2009, B.-P. Paris Wireless Communications 268

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Digital Synthesis of a Carrier Modulated Signal To start, assume we wanted to generate the samples for a linearly modulated signal: symbol for the m-th symbol period bm (k ), N samples per symbol period, rectangular pulses, up-conversion to digital frequency k /N (physical frequency fs · k /N). The resulting samples in the m-th symbol period are sm,k [n] = bm (k ) · exp(j2πkn/N ) for n = 0, 1, . . . , N − 1. If these samples were passed through a D-to-A converter, we would observe a signal spectrum centered at frequency fs · k /N, and bandwidth 2fs /N. ©2009, B.-P. Paris Wireless Communications 269

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Combining Carriers The single carrier occupies only a small fraction of the bandwidth fs the A-to-D converter can process. The remaining bandwidth can be used for additional signals. Speciﬁcally, we can combine N such signals. Carrier frequencies: fs · k /N for k = 0, 1, . . . N − 1. Resulting carriers are all orthogonal. Samples of the combined signal in the m-th symbol period are N −1 N −1 sm [ n ] = ∑ sm,k [n] = ∑ bm (k ) · exp(j2πkn/N ). k =0 k =0 The signal sm [n] represents N orthogonal narrow-band signals multiplexed in the frequency domain (OFDM). ©2009, B.-P. Paris Wireless Communications 270

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Constructing an OFDM Signal The expression N −1 sm [ n ] = ∑ bm (k ) · exp(j2πkn/N ). k =0 represents the inverse DFT of the symbols bm (k ). Direct evaluation of this equation requires N 2 multiplications and additions. However, the structure of the expression permits computationally efﬁcient construction of sm [n] through an FFT algorithm. MATLAB’s function ifft can be used. ©2009, B.-P. Paris Wireless Communications 271

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Constructing an OFDM Signal The above considerations suggest the following procedure for constructing an OFDM signal from a sequence of information symbols b: 1. Serial-to-parallel conversion: break the sequence of information symbols b into blocks of length N; denote the k -th symbol in the m-th block as bm (k ). 2. Inverse DFT: Take the inverse FFT of each block m; The output are length-N blocks of complex signal samples denoted sm [n]. 3. Cyclic preﬁx: Prepend the ﬁnal L samples from each block to the beginning of each block. Cyclic protects against ISI. 4. Parallel-to-serial conversion: Concatenate the blocks to form the OFDM signal. ©2009, B.-P. Paris Wireless Communications 272

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Properties of OFDM Signals The signal resulting from the above construction has the following parameters properties: Bandwidth: fs (e.g., 10 MHz) Number of subcarriers: N, should be a power of 2 (e.g., 256) Length of Preﬁx: L, depends on properties of channel (e.g., 8) Number of blocks: M (e.g., 64) Subcarrier bandwidth: fs/N (e.g., 40 KHz) Frame duration: M · (N + L)/fs (e.g., 1.7 ms) Number of symbols in frame: M · N (e.g., 16,896) Baud rate: N/(N + L) · fs (e.g. 9.7 MHz) OFDM signals are not constant envelope signals. Signals look like complex Gaussian noise. ©2009, B.-P. Paris Wireless Communications 273

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation OFDM Signals in MATLAB The toolbox contains functions for modulating and demodulating OFDM signals. Their respective signatures are: 1. function Signal = OFDMMod( Symbols, NCarriers, LPrefix ) function SymbolEst = OFDMDemod( Signal, NCarriers, LPrefix, N 2. The bodies of these functions reﬂect the algorithm above. ©2009, B.-P. Paris Wireless Communications 274

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Toolbox Function OFDMMod %% Serial-to-parallel conversion NBlocks = ceil( length(Symbols) / NCarriers ); % zero-pad, if needed 33 Blocks = zeros( 1, NBlocks*NCarriers ); Blocks(1:length(Symbols)) = Symbols; % serial-to-parallel Blocks = reshape( Blocks, NCarriers, NBlocks ); 38 %% IFFT % ifft works column-wise by default Blocks = ifft( Blocks ); %% cyclic prefix 43 % copy last LPrefix samples of each column and prepend Blocks = [ Blocks( end-LPrefix+1 : end, : ) ; Blocks ]; %% Parallel-to-serial conversion Signal = reshape( Blocks, 1, NBlocks*(NCarriers+LPrefix) ) * ... 48 sqrt(NCarriers); % makes "gain" of ifft equal to 1. ©2009, B.-P. Paris Wireless Communications 275

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Toolbox Function OFDMDemod %% Serial-to-parallel conversion 32 NBlocks = ceil( length(Signal) / (NCarriers+LPrefix) ); if ( (NCarriers+LPrefix)*NBlocks ~= length(Signal) ) error(’Length of Signal must be a multiple of (NCarriers+LPrefix)’ end % serial-to-parallel 37 Blocks = reshape( Signal/sqrt(NCarriers), NCarriers+LPrefix, NBlocks ) %% remove cyclic prefix % remove first LPrefix samples of each column Blocks( 1:LPrefix, : ) = [ ]; 42 %% FFT % fft works column-wise by default Blocks = fft( Blocks ); 47 %% Parallel-to-serial conversion SymbolEst = reshape( Blocks, 1, NBlocks*NCarriers ); % remove zero-padding SymbolEst = SymbolEst(1:NSymbols); ©2009, B.-P. Paris Wireless Communications 276

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Using the OFDM functions The code fragment below illustrates a typical use for the OFDM functions in the toolbox. Listing : ’MCOFDM.m’ %% simulate OFDM system % transmitter and channel via toolbox functions Symbols = RandomSymbols( NSymbols, Alphabet, Priors ); Signal = A * OFDMMod( Symbols, Nc, Lp ); 39 Received = addNoise( Signal, NoiseVar ); % OFDM Receiver MFOut = OFDMDemod( Received, Nc, Lp, NSymbols ); Decisions = SimpleSlicer( MFOut, Alphabet, Scale ); ©2009, B.-P. Paris Wireless Communications 277

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Monte Carlo Simulation with OFDM −1 10 −2 10 Symbol Error Rate −3 10 −4 10 −5 10 −2 0 2 4 6 8 10 E /N (dB) s 0 Figure: Monte Carlo Simulation with OFDM Signals; BPSK Symbols, 256 subcarriers, preﬁx length 8. ©2009, B.-P. Paris Wireless Communications 278

Linear Modulation Formats and their Spectra Spectrum Estimation in MATLAB Non-linear Modulation Wide-Band Modulation Where we are ... Considered wide variety of digital modulation techniques. Linear modulation with pulse shaping, Non-linear modulation: CPM and CPFSK, including MSK and GMSK, Wide-band modulation: DS/SS and OFDM. Spectral properties of digitally modulated signals. Numerical estimation of the spectrum via the periodogram. Evaluation of spectrum for various representative modulation formats. The importance of constant envelope characteristics. Next: characterizing the mobile channel. ©2009, B.-P. Paris Wireless Communications 279

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels The Wireless Channel Characterization of the wireless channel and its impact on digitally modulated signals. From the physics of propagation to multi-path fading channels. Statistical characterization of wireless channels: Doppler spectrum, Delay spread Coherence time Coherence bandwidth Simulating multi-path, fading channels in MATLAB. Lumped-parameter models: discrete-time equivalent channel. Path loss models, link budgets, shadowing. ©2009, B.-P. Paris Wireless Communications 281

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Outline Part III: Learning Objectives Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels ©2009, B.-P. Paris Wireless Communications 282

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Learning Objectives Understand models describing the nature of typical wireless communication channels. The origin of multi-path and fading. Concise characterization of multi-path and fading in both the time and frequency domain. Doppler spectrum and time-coherence Multi-path delay spread and frequency coherence Appreciate the impact of wireless channels on transmitted signals. Distortion from multi-path: frequency-selective fading and inter-symbol interference. The consequences of time-varying channels. ©2009, B.-P. Paris Wireless Communications 283

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Path Loss Path loss LP relates the received signal power Pr to the transmitted signal power Pt : Gr · Gt Pr = Pt · , LP where Gt and Gr are antenna gains. Path loss is very important for cell and frequency planning or range predictions. Not needed when designing signal sets, receiver, etc. ©2009, B.-P. Paris Wireless Communications 285

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Path Loss Path loss modeling is “more an art than a science.” Standard approach: ﬁt model to empirical data. Parameters of model: d - distance between transmitter and receiver, fc - carrier frequency, hb , hm - antenna heights, Terrain type, building density, . . .. ©2009, B.-P. Paris Wireless Communications 286

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Free Space Propagation In free space, path loss LP is given by Friis’s formula: 2 2 4πd 4πfc d LP = = . λc c Path loss increases proportional to the square of distance d and frequency fc . In dB: c LP (dB ) = −20 log10 ( ) + 20 log10 (fc ) + 20 log10 (d ). 4π Example: fc = 1GHz and d = 1km LP (dB ) = −146 dB + 180 dB + 60 dB = 94 dB. ©2009, B.-P. Paris Wireless Communications 287

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Two-Ray Channel Antenna heights: hb and hm . Two propagation paths: 1. direct path, free space propagation, 2. reﬂected path, free space with perfect reﬂection. Depending on distance d, the signals received along the two paths will add constructively or destructively. Path loss: 2 2 1 4πfc d 1 LP = · · . 4 c sin( 2πchd hm ) fc b For d hb hm , path loss is approximately equal to: 2 d2 LP ≈ hb hm Path loss proportional to d 4 is typical for urban environment.©2009, B.-P. Paris Wireless Communications 288

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Okumura-Hata Model for Urban Area Okumura and Hata derived empirical path loss models from extensive path loss measurements. Models differ between urban, suburban, and open areas, large, medium, and small cities, etc. Illustrative example: Model for Urban area (small or medium city) LP (dB ) = A + B log10 (d ), where A = 69.55 + 26.16 log10 (fc ) − 13.82 log10 (hb ) − a(hm ) B = 44.9 − 6.55 log10 (hb ) a(hm ) = (1.1 log10 (fc ) − 0.7) · hm − (1.56 log10 (fc ) − 0.8) ©2009, B.-P. Paris Wireless Communications 289

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Signal and Noise Power Received Signal Power: Gr · Gt Pr = Pt · , LP · LR where LR is implementation loss, typically 2-3 dB. (Thermal) Noise Power: PN = kT0 · BW · F , where k - Boltzmann’s constant (1.38 · 10−23 Ws/K), T0 - temperature in K (typical room temperature, T0 = 290 K), ⇒ kT0 = 4 · 10−21 W/Hz = 4 · 10−18 mW/Hz = −174 dBm/Hz, BW - signal bandwidth, F - noise ﬁgure, ﬁgure of merit for receiver (typical value: 5dB). ©2009, B.-P. Paris Wireless Communications 290

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Signal-to-Noise Ratio The ratio of received signal power and noise power is denoted by SNR. From the above, SNR equals: Pt Gr · Gt SNR = . kT0 · BW · F · LP · LR SNR increases with transmitted power Pt and antenna gains. SNR decreases with bandwidth BW , noise ﬁgure F , and path loss LP . ©2009, B.-P. Paris Wireless Communications 291

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Es /N0 For the symbol error rate performance of communications system the ratio of signal energy Es and noise power spectral density N0 is more relevant than SNR. Pr Since Es = Pr · Ts = Rs and N0 = kT0 · F = PN /BW , it follows that Es B = SNR · W , N0 Rs where Ts and Rs denote the symbol period and symbol rate, respectively. ©2009, B.-P. Paris Wireless Communications 292

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Es /N0 Thus, Es /N0 is given by: Es Pt Gr · Gt = . N0 kT0 · Rs · F · LP · LR in dB: E ( Ns )(dB ) = Pt (dBm) + Gt (dB ) + Gr (dB ) 0 −(kT0 )(dBm/Hz ) − Rs(dBHz ) − F(dB ) − LR (dB ) . ©2009, B.-P. Paris Wireless Communications 293

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Multi-path Propagation The transmitted signal propagates from the transmitter to the receiver along many different paths. These paths have different path attenuation ak , path delay τk , TX RX phase shift φk , angle of arrival θk . For simplicity, we assume a 2-D model, so that the angle of arrival is the azimuth. In 3-D models, the elevation angle of arrival is an additional parameter. ©2009, B.-P. Paris Wireless Communications 295

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Channel Impulse Response From the above parameters, one can easily determine the channel’s (baseband equivalent) impulse response. Impulse Response: K h (t ) = ∑ ak · ejφ k · e−j2πfc τk · δ(t − τk ) k =1 Note that the delays τk contribute to the phase shifts φk . ©2009, B.-P. Paris Wireless Communications 296

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Received Signal Ignoring noise for a moment, the received signal is the convolution of the transmitted signal s (t ) and the impulse response K R (t ) = s (t ) ∗ h (t ) = ∑ ak · ejφ k · e−j2πfc τk · s (t − τk ). k =1 The received signal consists of multiple scaled (by ak · ejφk · e −j2πfc τk ), delayed (by τk ) copies of the transmitted signal. ©2009, B.-P. Paris Wireless Communications 297

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Channel Frequency Response Similarly, one can compute the frequency response of the channel. Direct Fourier transformation of the expression for the impulse response yields K H (f ) = ∑ ak · ejφ k · e−j2πfc τk · e−j2πf τk . k =1 For any given frequency f , the frequency response is a sum of complex numbers. When these terms add destructively, the frequency response is very small or even zero at that frequency. These nulls in the channel’s frequency response are typical for wireless communications and are refered to as frequency-selective fading. ©2009, B.-P. Paris Wireless Communications 298

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Frequency Response in One Line of MATLAB The Frequency response K H (f ) = ∑ ak · ejφ k · e−j2πfc τk · e−j2πf τk . k =1 can be computed in MATLAB via the one-liner HH = PropData.Field.*exp(-j*2*pi*fc*tau) * exp(-j*2*pi*tau’*ff); Note that tau’*ff is an inner product; it produces a matrix (with K rows and as many columns as ff). Similarly, the product preceding the second complex exponential is an inner product; it generates the sum in the expression above. ©2009, B.-P. Paris Wireless Communications 299

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Ray Tracing 1300 1250 1200 1150 Transmitter 1100 1050 y (m) 1000 950 Receiver 900 850 800 750 450 500 550 600 650 700 750 800 850 900 950 x (m) Figure: All propagation paths between the transmitter and receiver in the indicated located were determined through ray tracing. ©2009, B.-P. Paris Wireless Communications 300

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Impulse Response −5 x 10 4 3 Attenuation 2 1 0 0.8 1 1.2 1.4 1.6 1.8 2 Delay (µs) 4 2 Phase Shift/π 0 −2 −4 0.8 1 1.2 1.4 1.6 1.8 2 Delay (µs) Figure: (Baseband equivalent) Impulse response shows attenuation, delay, and phase for each of the paths between receiver and transmitter. ©2009, B.-P. Paris Wireless Communications 301

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Frequency Response −78 −80 −82 −84 |Frequency Response| (dB) −86 −88 −90 −92 −94 −96 −98 −5 −4 −3 −2 −1 0 1 2 3 4 5 Frequency (MHz) Figure: (Baseband equivalent) Frequency response for a multi-path channel is characterized by deep “notches”. ©2009, B.-P. Paris Wireless Communications 302

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Implications of Multi-path Multi-path leads to signal distortion. The received signal “looks different” from the transmitted signal. This is true, in particular, for wide-band signals. Multi-path propagation is equivalent to undesired filtering with a linear filter. The impulse response of this undesired filter is the impulse response h(t ) of the channel. The effects of multi-path can be described in terms of both time-domain and frequency-domain concepts. In either case, it is useful to distinguish between narrow-band and wide-band signals. ©2009, B.-P. Paris Wireless Communications 303

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Transmission of a Linearly Modulated Signal Transmission of a linearly modulated signal through the above channel is simulated. BPSK, (full response) raised-cosine pulse. Symbol period is varied; the following values are considered Ts = 30µs ( bandwidth approximately 60 KHz) Ts = 3µs ( bandwidth approximately 600 KHz) Ts = 0.3µs ( bandwidth approximately 6 MHz) For each case, the transmitted and (suitably scaled) received signal is plotted. Look for distortion. Note that the received signal is complex valued; real and imaginary part are plotted. ©2009, B.-P. Paris Wireless Communications 304

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Transmission of a Linearly Modulated Signal 2 Transmitted Real(Received) 1.5 Imag(Received) 1 0.5 Amplitude 0 −0.5 −1 −1.5 −2 0 50 100 150 200 250 300 Time (µs) Figure: Transmitted and received signal; Ts = 30µs. No distortion is evident. ©2009, B.-P. Paris Wireless Communications 305

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Transmission of a Linearly Modulated Signal 2 Transmitted Real(Received) 1.5 Imag(Received) 1 0.5 Amplitude 0 −0.5 −1 −1.5 −2 0 5 10 15 20 25 30 35 Time (µs) Figure: Transmitted and received signal; Ts = 3µs. Some distortion is visible near the symbol boundaries. ©2009, B.-P. Paris Wireless Communications 306

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Transmission of a Linearly Modulated Signal 2 Transmitted Real(Received) 1.5 Imag(Received) 1 0.5 Amplitude 0 −0.5 −1 −1.5 −2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (µs) Figure: Transmitted and received signal; Ts = 0.3µs. Distortion is clearly visible and spans multiple symbol periods. ©2009, B.-P. Paris Wireless Communications 307

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Eye Diagrams for Visualizing Distortion An eye diagram is a simple but useful tool for quickly gaining an appreciation for the amount of distortion present in a received signal. An eye diagram is obtained by plotting many segments of the received signal on top of each other. The segments span two symbol periods. This can be accomplished in MATLAB via the command plot( tt(1:2*fsT), real(reshape(Received(1:Ns*fsT), 2*fsT, [ ]))) Ns - number of symbols; should be large (e.g., 1000), Received - vector of received samples. The reshape command turns the vector into a matrix with 2*fsT rows, and the plot command plots each column of the resulting matrix individually. ©2009, B.-P. Paris Wireless Communications 308

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Eye Diagram without Distortion 0.5 Amplitude 0 −0.5 0 10 20 30 40 50 60 Time (µs) 1 0.5 Amplitude 0 −0.5 −1 0 10 20 30 40 50 60 Time (µs) Figure: Eye diagram for received signal; Ts = 30µs. No distortion: “the eye is fully open”. ©2009, B.-P. Paris Wireless Communications 309

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Eye Diagram with Distortion 2 1 Amplitude 0 −1 −2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (µs) 2 1 Amplitude 0 −1 −2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (µs) Figure: Eye diagram for received signal; Ts = 0.3µs. Signiﬁcant distortion: “the eye is partially open”. ©2009, B.-P. Paris Wireless Communications 310

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Inter-Symbol Interference The distortion described above is referred to as inter-symbol interference (ISI). As the name implies, the undesired ﬁltering by the channel causes energy to be spread from one transmitted symbol across several adjacent symbols. This interference makes detection mored difﬁcult and must be compensated for at the receiver. Devices that perform this compensation are called equalizers. ©2009, B.-P. Paris Wireless Communications 311

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Inter-Symbol Interference Question: Under what conditions does ISI occur? Answer: depends on the channel and the symbol rate. The difference between the longest and the shortest delay of the channel is called the delay spread Td of the channel. The delay spread indicates the length of the impulse response of the channel. Consequently, a transmitted symbol of length Ts will be spread out by the channel. When received, its length will be the symbol period plus the delay spread, Ts + Td . Rule of thumb: if the delay spread is much smaller than the symbol period (Td Ts ), then ISI is negligible. If delay is similar to or greater than the symbol period, then ISI must be compensated at the receiver. ©2009, B.-P. Paris Wireless Communications 312

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Frequency-Domain Perspective It is interesting to compare the bandwidth of the transmitted signals to the frequency response of the channel. In particular, the bandwidth of the transmitted signal relative to variations in the frequency response is important. The bandwidth over which the channel’s frequency response remains approximately constant is called the coherence bandwidth. When the frequency response of the channel remains approximately constant over the bandwidth of the transmitted signal, the channel is said to be ﬂat fading. Conversely, if the channel’s frequency response varies signiﬁcantly over the bandwidth of the signal, the channel is called a frequency-selective fading channel. ©2009, B.-P. Paris Wireless Communications 313

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Narrow-Band Signal −75 −80 |Frequency Response| (dB) −85 −90 −95 −100 −5 −4 −3 −2 −1 0 1 2 3 4 Frequency (MHz) Figure: Frequency Response of Channel and bandwidth of signal; Ts = 30µs, Bandwidth ≈ 60 KHz; the channel’s frequency response is approximately constant over the bandwidth of the signal. ©2009, B.-P. Paris Wireless Communications 314

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Wide-Band Signal −75 −80 |Frequency Response| (dB) −85 −90 −95 −100 −5 −4 −3 −2 −1 0 1 2 3 4 Frequency (MHz) Figure: Frequency Response of Channel and bandwidth of signal; Ts = 0.3µs, Bandwidth ≈ 6 MHz; the channel’s frequency response varies signiﬁcantly over the bandwidth of the channel. ©2009, B.-P. Paris Wireless Communications 315

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Frequency-Selective Fading and ISI Frequency-selective fading and ISI are dual concepts. ISI is a time-domain characterization for signiﬁcant distortion. Frequency-selective fading captures the same idea in the frequency domain. Wide-band signals experience ISI and frequency-selective fading. Such signals require an equalizer in the receiver. Wide-band signals provide built-in diversity. Not the entire signal will be subject to fading. Narrow-band signals experience ﬂat fading (no ISI). Simple receiver; no equalizer required. Entire signal may be in a deep fade; no diversity. ©2009, B.-P. Paris Wireless Communications 316

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Time-Varying Channel Beyond multi-path propagation, a second characteristic of many wireless communication channels is their time variability. The channel is time-varying primarily because users are mobile. As mobile users change their position, the characteristics of each propagation path changes correspondingly. Consider the impact a change in position has on path gain, path delay. Will see that angle of arrival θk for k -th path is a factor. ©2009, B.-P. Paris Wireless Communications 317

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Path-Changes Induced by Mobility Mobile moves by ∆d from old position to new position. distance: |∆d | angle: ∠∆d = δ Angle between k -th ray and ∆d is denoted ψk = θk − δ. Length of k -th path increases by |∆d | cos(ψk ). k -th ray k -th ray |∆d | sin(ψk ) |∆d | cos(ψk ) ψk ∆d Old Position New Position ©2009, B.-P. Paris Wireless Communications 318

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Impact of Change in Path Length We conclude that the length of each path changes by |∆d | cos(ψk ), where ψk denotes the angle between the direction of the mobile and the k -th incoming ray. Question: how large is a typical distance |∆d | between the old and new position is? The distance depends on the velocity v of the mobile, and the time-scale ∆T of interest. In many modern communication system, the transmission of a frame of symbols takes on the order of 1 to 10 ms. Typical velocities in mobile systems range from pedestrian speeds (≈ 1m/s) to vehicle speeds of 150km/h( ≈ 40m/s). Distances of interest |∆d | range from 1mm to 400mm. ©2009, B.-P. Paris Wireless Communications 319

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Impact of Change in Path Length Question: What is the impact of this change in path length on the parameters of each path? We denote the length of the path to the old position by dk . Clearly, dk = c · τk , where c denotes the speed of light. Typically, dk is much larger than |∆d |. Path gain ak : Assume that path gain ak decays inversely − proportional with the square of the distance, ak ∼ dk 2 . Then, the relative change in path gain is proportional to (|∆d |/dk )2 (e.g., |∆d | = 0.1m and dk = 100m, then path gain changes by approximately 0.0001%). Conclusion: The change in path gain is generally small enough to be negligible. ©2009, B.-P. Paris Wireless Communications 320

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Impact of Change in Path Length Delay τk : By similar arguments, the delay for the k -th path changes by at most |∆d |/c. The relative change in delay is |∆d |/dk (e.g., 0.1% with the values above.) Question: Is this change in delay also negligible? ©2009, B.-P. Paris Wireless Communications 321

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Relating Delay Changes to Phase Changes Recall: the impulse response of the multi-path channel is K h (t ) = ∑ ak · ejφ k · e−j2πfc τk · δ(t − τk ) k =1 Note that the delays, and thus any delay changes, are multiplied by the carrier frequency fc to produce phase shifts. ©2009, B.-P. Paris Wireless Communications 322

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Relating Delay Changes to Phase Changes Consequently, the phase change arising from the movement of the mobile is ∆φk = −2πfc /c |∆d | cos(ψk ) = −2π |∆d |/λc cos(ψk ), where λc = c/fc - denotes the wave-length at the carrier frequency (e.g., at fc = 1GHz, λc ≈ 0.3m), ψk - angle between direction of mobile and k -th arriving path. Conclusion: These phase changes are signiﬁcant and lead to changes in the channel properties over short time-scales (fast fading). ©2009, B.-P. Paris Wireless Communications 323

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Illustration To quantify these effects, compute the phase change over a time interval ∆T = 1ms as a function of velocity. Assume ψk = 0, and, thus, cos(ψk ) = 1. fc = 1GHz. v (m/s) |∆d | (mm) ∆φ (degrees) Comment 1 1 1.2 Pedestrian; negligible phase change. 10 10 12 Residential area vehicle speed. 100 100 120 High-way speed; phase change signiﬁ- cant. 1000 1000 1200 High-speed train or low-ﬂying aircraft; receiver must track phase changes. ©2009, B.-P. Paris Wireless Communications 324

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Doppler Shift and Doppler Spread If a mobile is moving at a constant velocity v , then the distance between an old position and the new position is a function of time, |∆d | = vt. Consequently, the phase change for the k -th path is ∆φk (t ) = −2πv /λc cos(ψk )t = −2πv /c · fc cos(ψk )t. The phase is a linear function of t. Hence, along this path the signal experiences a frequency shift fd ,k = v /c · fc · cos(ψk ) = v /λc · cos(ψk ). This frequency shift is called Doppler shift. Each path experiences a different Doppler shift. Angles of arrival θk are different. Consequently, instead of a single Doppler shift a number of shifts create a Doppler Spectrum. ©2009, B.-P. Paris Wireless Communications 325

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Illustration: Time-Varying Frequency Response −70 |Frequency Response| (dB) −80 −90 −100 −110 −120 −130 200 150 5 100 0 50 Time (ms) 0 −5 Frequency (MHz) Figure: Time-varying Frequency Response for Ray-Tracing Data; velocity v = 10m/s, fc = 1GHz, maximum Doppler frequency ≈ 33Hz. ©2009, B.-P. Paris Wireless Communications 326

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Illustration: Time-varying Response to a Sinusoidal Input −80 Magnitude (dB) −100 −120 −140 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) 10 0 Phase/π −10 −20 −30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) Figure: Response of channel to sinusoidal input signal; base-band equivalent input signal s (t ) = 1, velocity v = 10m/s, fc = 1GHz, maximum Doppler frequency ≈ 33Hz. ©2009, B.-P. Paris Wireless Communications 327

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Doppler Spread and Coherence Time The time over which the channel remains approximately constant is called the coherence time of the channel. Coherence time and Doppler spectrum are dual characterizations of the time-varying channel. Doppler spectrum provides frequency-domain interpretation: It indicates the range of frequency shifts induced by the time-varying channel. Frequency shifts due to Doppler range from −fd to fd , where fd = v /c · fc . The coherence time Tc of the channel provides a time-domain characterization: It indicates how long the channel can be assumed to be approximately constant. Maximum Doppler shift fd and coherence time Tc are related to each through an inverse relationship Tc ≈ 1/fd . ©2009, B.-P. Paris Wireless Communications 328

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels System Considerations The time-varying nature of the channel must be accounted for in the design of the system. Transmissions are shorter than the coherence time: Many systems are designed to use frames that are shorter than the coherence time. Example: GSM TDMA structure employs time-slots of duration 4.6ms. Consequence: During each time-slot, channel may be treated as constant. From one time-slot to the next, channel varies signiﬁcantly; this provides opportunities for diversity. Transmission are longer than the coherence time: Channel variations must be tracked by receiver. Example: use recent symbol decisions to estimate current channel impulse response. ©2009, B.-P. Paris Wireless Communications 329

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Illustration: Time-varying Channel and TDMA −80 −90 −100 Magnitude (dB) −110 −120 −130 −140 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (s) Figure: Time varying channel response and TDMA time-slots; time-slot duration 4.6ms, 8 TDMA users, velocity v = 10m/s, fc = 1GHz, maximum Doppler frequency ≈ 33Hz. ©2009, B.-P. Paris Wireless Communications 330

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Summary Illustrated by means of a concrete example the two main impairments from a mobile, wireless channel. Multi-path propagation, Doppler spread due to time-varying channel. Multi-path propagation induces ISI if the symbol duration exceeds the delay spread of the channel. In frequency-domain terms, frequency-selective fading occurs if the signal bandwidth exceeds the coherence band-width of the channel. Doppler Spreading results from time-variations of the channel due to mobility. The maximum Doppler shift fd = v /c · fc is proportional to the speed of the mobile. In time-domain terms, the channel remains approximately constant over the coherence-time of the channel. ©2009, B.-P. Paris Wireless Communications 331

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Statistical Characterization of Channel We have looked at the characterization of a concrete realization of a mobile, wire-less channel. For different locations, the properties of the channel will likely be very different. Objective: develop statistical models that capture the salient features of the wireless channel for areas of interest. Models must capture multi-path and time-varying nature of channel. Approach: Models reﬂect correlations of the time-varying channel impulse response or frequency response. Time-varying descriptions of channel are functions of two parameters: Time t when channel is measured, Frequency f or delay τ. ©2009, B.-P. Paris Wireless Communications 333

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Power Delay Profile The impulse response of a wireless channel is time-varying, h(t, τ ). The parameter t indicates when the channel is used, The parameter τ reflects time since the input was applied (delay). Time-varying convolution: r (t ) = h(t, τ ) · s (t − τ )dτ. The power-delay profile measures the average power in the impulse response over delay τ. Thought experiment: Send impulse through channel at time t0 and measure response h(t0 , τ ). Repeat K times, measuring h(tk , τ ). Power delay profile: K 1 Ψh (τ ) = ∑ |h(tk , τ )|2 . K + 1 k =0 ©2009, B.-P. Paris Wireless Communications 334

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Power Delay Proﬁle The power delay proﬁle captures the statistics of the multi-path effects of the channel. The underlying, physical model assumes a large number of propagation paths: each path has a an associated delay τ, the gain for a path is modeled as a complex Gaussian random variable with second moment equal to Ψh (τ ). If the mean of the path loss is zero, the path is said to be Rayleigh fading. Otherwise, it is Ricean. The channel gains associated with different delays are assumed to be uncorrelated. ©2009, B.-P. Paris Wireless Communications 335

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example 1 2 0.9 1.5 |h(τ)|2 0.8 1 Power Delay Profile 0.7 0.5 0.6 0 0 2 4 6 0.5 Delay τ (µs) 1 0.4 0.3 0.5 Phase of h(τ) 0.2 0 0.1 −0.5 0 −1 0 2 4 6 0 2 4 6 Delay τ (µs) Delay τ (µs) Figure: Power Delay Proﬁle and Channel Impulse Response; the power delay proﬁle (left) equals Ψh (τ ) = exp(−τ/Th ) with Th = 1µs; realization of magnitude and phase of impulse response (left). ©2009, B.-P. Paris Wireless Communications 336

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels RMS Delay Spread From a systems perspective, the extent (spread) of the delays is most signiﬁcant. The length of the impulse response of the channel determines how much ISI will be introduced by the channel. The spread of delays is measured concisely by the RMS delay spread Td : ∞ ∞ 2 (n ) (n ) Td = Ψh (τ )τ 2 dτ − ( Ψh (τ )τdτ )2 , 0 0 where ∞ (n ) Ψh = Ψh / Ψh (τ )dτ. 0 Example: For Ψh (τ ) = exp(−τ/Th ), RMS delay spread equals Th . In urban environments, typical delay spreads are a few µs. ©2009, B.-P. Paris Wireless Communications 337

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Frequency Coherence Function The Fourier transform of the Power Delay Spread Ψh (τ ) is called the Frequency Coherence Function ΨH (∆f ) Ψh (τ ) ↔ ΨH (∆f ). The frequency coherence function measures the correlation of the channel’s frequency response. Thought Experiment: Transmit two sinusoidal signal of frequencies f1 and f2 , such that f1 − f2 = ∆f . The gain each of these signals experiences is H (t, f1 ) and H (t, f2 ), respectively. Repeat the experiment many times and average the products H (t, f1 ) · H ∗ (t, f2 ). ΨH (∆f ) indicates how similar the gain is that two sinusoids separated by ∆f experience. ©2009, B.-P. Paris Wireless Communications 338

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Coherence Bandwidth The width of the main lobe of the frequency coherence function is the coherence bandwidth Bc of the channel. Two signals with frequencies separated by less than the coherence bandwidth will experience very similar gains. Because of the Fourier transform relationship between the power delay proﬁle and the frequency coherence function: 1 Bc ≈ . Td Example: Fourier transform of Ψh (τ ) = exp(−τ/Th ) Th ΨH (∆f ) = ; 1 + j2π∆fTh the 3-dB bandwidth of ΨH (∆f ) is Bc = 1/(2π · Th ). For urban channels, coherence bandwidth is a few 100KHz. ©2009, B.-P. Paris Wireless Communications 339

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Time Coherence The time-coherence function ΨH (∆t ) captures the time-varying nature of the channel. Thought experiment: Transmit a sinusoidal signal of frequency f through the channel and measure the output at times t1 and t1 + ∆t. The gains the signal experiences are H (t1 , f ) and H (t1 + ∆t, f ), respectively. Repeat experiment and average the products H (tk , f ) · H ∗ (tk + ∆t, f ). Time coherence function measures, how quickly the gain of the channel varies. The width of the time coherence function is called the coherence-time Tc of the channel. The channel remains approximately constant over the coherence time of the channel. ©2009, B.-P. Paris Wireless Communications 340

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Isotropic Scatterer Old location: H (t1 , f = 0) = ak · exp(−j2πfc τk ). At new location: the gain ak is unchanged; phase changes by fd cos(ψk )∆t: H (t1 + ∆t, f = 0) = ak · exp(−j2π (fc τk + fd cos(ψk )∆t )). k -th ray k -th ray |∆d | sin(ψk ) |∆d | cos(ψk ) ψk ∆d Old Position New Position ©2009, B.-P. Paris Wireless Communications 341

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Example: Isotropic Scatterer The average of H (t1 , 0) · H ∗ (t1 + ∆t, 0) yields the time-coherence function. Assume that the angle of arrival ψk is uniformly distributed. This allows computation of the average (isotropic scatterer assumption: ΨH (∆t ) = |ak |2 · J0 (2πfd ∆t ) ©2009, B.-P. Paris Wireless Communications 342

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Time-Coherence Function for Isotropic Scatterer 1 0.5 Ψ (∆t) H 0 −0.5 0 50 100 150 200 250 300 Time ∆t (ms) Figure: Time-Coherence Function for Isotropic Scatterer; velocity v = 10m/s, fc = 1GHz, maximum Doppler frequency fd ≈ 33Hz. First zero at ∆t ≈ 0.4/fd . ©2009, B.-P. Paris Wireless Communications 343

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Doppler Spread Function The Fourier transform of the time coherence function ΨH (∆t ) is the Doppler Spread Function Ψd (fd ) ΨH (∆t ) ↔ Ψd (fd ). The Doppler spread function indicates the range of frequencies observed at the output of the channel when the input is a sinusoidal signal. Maximum Doppler shift fd ,max = v /c · fc . Thought experiment: Send a sinusoidal signal of The PSD of the received signal is the Doppler spread function. ©2009, B.-P. Paris Wireless Communications 344

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Doppler Spread Function for Isotropic Scatterer Example: The Doppler spread function for the isotropic scatterer is |ak |2 1 Ψd (fd ) = for |f | < fd . 4πfd 1 − (f /fd )2 ©2009, B.-P. Paris Wireless Communications 345

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Doppler Spread Function for Isotropic Scatterer 7 6 5 4 Ψd(fd) 3 2 1 0 −40 −30 −20 −10 0 10 20 30 40 Doppler Frequency (Hz) Figure: Doppler Spread Function for Isotropic Scatterer; velocity v = 10m/s, fc = 1GHz, maximum Doppler frequency fd ≈ 33Hz. First zero at ∆t ≈ 0.4/fd . ©2009, B.-P. Paris Wireless Communications 346

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Simulation of Multi-Path Fading Channels We would like to be able to simulate the effects of time-varying, multi-path channels. Approach: The simulator operates in discrete-time; the sampling rate is given by the sampling rate for the input signal. The multi-path effects can be well modeled by an FIR (tapped delay-line)filter. The number of taps for the filter is given by the product of delay spread and sampling rate. Example: With a delay spread of 2µs and a sampling rate of 2MHz, four taps are required. The taps should be random with a Gaussian distribution. The magnitude of the tap weights should reflect the power-delay profile. ©2009, B.-P. Paris Wireless Communications 347

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Simulation of Multi-Path Fading Channels Approach (cont’d): The time-varying nature of the channel can be captured by allowing the taps to be time-varying. The time-variations should reﬂect the Doppler Spectrum. ©2009, B.-P. Paris Wireless Communications 348

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Simulation of Multi-Path Fading Channels The taps are modeled as Gaussian random processes with variances given by the power delay proﬁle, and power spectral density given by the Doppler spectrum. s [n ] D D a0 (t ) × a1 ( t ) × a2 (t ) × r [n ] + + ©2009, B.-P. Paris Wireless Communications 349

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Channel Model Parameters Concrete parameters for models of the above form have been proposed by various standards bodies. For example, the following table is an excerpt from a document produced by the COST 259 study group. Tap number Relative Time (µs) Relative Power (dB) Doppler Spectrum 1 0 -5.7 Class 2 0.217 -7.6 Class 3 0.512 -10.1 Class . . . . . . . . . . . . 20 2.140 -24.3 Class ©2009, B.-P. Paris Wireless Communications 350

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Channel Model Parameters The table provides a concise, statistical description of a time-varying multi-path environment. Each row corresponds to a path and is characterized by the delay beyond the delay for the shortest path, the average power of this path; this parameter provides the variance of the Gaussian path gain. the Doppler spectrum for this path; The notation Class denotes the classical Doppler spectrum for the isotropic scatterer. The delay and power column specify the power-delay proﬁle. The Doppler spectrum is given directly. The Doppler frequency fd is an additional parameter. ©2009, B.-P. Paris Wireless Communications 351

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Toolbox Function SimulateCOSTChannel The result of our efforts will be a toolbox function for simulating time-varying multi-path channels: function OutSig = SimulateCOSTChannel( InSig, ChannelParams, fs) Its input arguments are % Inputs: % InSig - baseband equivalent input signal % ChannelParams - structure ChannelParams must have fields 11 % Delay - relative delay % Power - relative power in dB % Doppler - type of Dopller spectrum % fd - max. Doppler shift % fs - sampling rate ©2009, B.-P. Paris Wireless Communications 352

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Discrete-Time Considerations The delays in the above table assume a continuous time axis; our time-varying FIR will operate in discrete time. To convert the model to discrete-time: Continuous-time is divided into consecutive “bins” of width equal to the sampling period, 1/fs. For all paths arriving in same “bin,” powers are added. This approach reﬂects that paths arriving closer together than the sampling period cannot be resolved; their effect is combined in the receiver front-end. The result is a reduced description of the multi-path channel: Power for each tap reﬂects the combined power of paths arriving in the corresponding “bin”. This power will be used to set the variance of the random process for the corresponding tap. ©2009, B.-P. Paris Wireless Communications 353

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Converting to a Discrete-Time Model in MATLAB %% convert powers to linear scale Power_lin = dB2lin( ChannelParams.Power); %% Bin the delays according to the sample rate 29 QDelay = floor( ChannelParams.Delay*fs ); % set surrogate delay for each bin, then sum up the power in each bin Delays = ( ( 0:QDelay(end) ) + 0.5 ) / fs; Powers = zeros( size(Delays) ); 34 for kk = 1:length(Delays) Powers( kk ) = sum( Power_lin( QDelay == kk-1 ) ); end ©2009, B.-P. Paris Wireless Communications 354

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Generating Time-Varying Filter Taps The time-varying taps of the FIR filter must be Gaussian random processes with specified variance and power spectral density. To accomplish this, we proceed in two steps: 1. Create a filter to shape the power spectral density of the random processes for the tap weights. 2. Create the random processes for the tap weights by passing complex, white Gaussian noise through the filter. Variance is adjusted in this step. Generating the spectrum shaping filter: % desired frequency response of filter: HH = sqrt( ClassDoppler( ff, ChannelParams.fd ) ); % design filter with desired frequency response 77 hh = Persistent_firpm( NH-1, 0:1/(NH-1):1, HH ); hh = hh/norm(hh); % ensure filter has unit norm ©2009, B.-P. Paris Wireless Communications 355

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Generating Time-Varying Filter Taps The spectrum shaping filter is used to filter a complex white noise process. Care is taken to avoid transients at the beginning of the output signal. Also, filtering is performed at a lower rate with subsequent interpolation to avoid numerical problems. Recall that fd is quite small relative to fs . % generate a white Gaussian random process 93 ww = sqrt( Powers( kk )/2)*... ( randn( 1, NSamples) + j*randn( 1, NSamples) ); % filter so that spectrum equals Doppler spectrum ww = conv( ww, hh ); ww = ww( length( hh )+1:NSamples ).’; 98 % interpolate to a higher sampling rate % ww = interp( ww, Down ); ww = interpft(ww, Down*length(ww)); % store time-varying filter taps for later use ©2009, B.-P. Paris Wireless Communications 356

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Time-Varying Filtering The final step in the simulator is filtering the input signal with the time-varying filter taps. MATLAB’s filtering functions conv or filter cannot be used (directly) for this purpose. The simulator breaks the input signal into short segments for which the channel is nearly constant. Each segment is filtered with a slightly different set of taps. while ( Start < length(InSig) ) EndIn = min( Start+QDeltaH, length(InSig) ); EndOut = EndIn + length(Powers)-1; 118 OutSig(Start:EndOut) = OutSig(Start:EndOut) + ... conv( Taps(kk,:), InSig(Start:EndIn) ); kk = kk+1; Start = EndIn+1; ©2009, B.-P. Paris Wireless Communications 357

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Testing SimulateCOSTChannel A simple test for the channel simulator consists of “transmitting” a baseband equivalent sinusoid. %% Initialization ChannelParameters = tux(); % COST model parameters 6 ChannelParameters.fd = 10; % Doppler frequency fs = 1e5; % sampling rate SigDur = 1; % duration of signal 11 %% generate input signal and simulate channel tt = 0:1/fs:SigDur; % time axis Sig = ones( size(tt) ); % baseband-equivalent carrier Received = SimulateCOSTChannel(Sig, ChannelParameters, fs); ©2009, B.-P. Paris Wireless Communications 358

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Testing SimulateCOSTChannel 1.8 1.6 1.4 1.2 Magnitude 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) Figure: Simulated Response to a Sinusoidal Signal; fd = 10Hz, baseband equivalent frequency f = 0. ©2009, B.-P. Paris Wireless Communications 359

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Summary Highlighted unique aspects of mobile, wireless channels: time-varying, multi-path channels. Statistical characterization of channels via power-delay proﬁle (RMS delay spread), frequency coherence function (coherence bandwidth), time coherence function (coherence time), and Doppler spread function (Doppler spread). Relating channel parameters to system parameters: signal bandwidth and coherence bandwidth, frame duration and coherence time. Channel simulator in MATLAB. ©2009, B.-P. Paris Wireless Communications 360

Pathloss and Link Budget From Physical Propagation to Multi-Path Fading Statistical Characterization of Channels Where we are ... Having characterized the nature of mobile, wireless channels, we can now look for ways to overcome the detrimental effects of the channel. The importance of diversity to overcome fading. Sources of diversity: Time, Frequency, Space. Equalizers for overcoming frequency-selective fading. Equalizers also exploit freqeuncy diversity. ©2009, B.-P. Paris Wireless Communications 361

The Importance of Diversity Frequency Diversity: Wide-Band Signals Mitigating the Impact of the Wireless Channel Description and analysis of techniques to overcome the detrimental inﬂuence of the wireless channel through various forms of diversity. The importance of diversity. Sources of diversity: time, frequency, and space. Equalization: overcoming ISI and exploiting diversity ©2009, B.-P. Paris Wireless Communications 363

The Importance of Diversity Frequency Diversity: Wide-Band Signals Outline Part IV: Learning Objectives The Importance of Diversity Frequency Diversity: Wide-Band Signals ©2009, B.-P. Paris Wireless Communications 364

The Importance of Diversity Frequency Diversity: Wide-Band Signals Learning Objectives The Importance of Diversity. BER performance in a Rayleigh fading channel without diversity. Diversity to the rescue ... What is diversity? Acceptable performance in Rayleigh fading channels requires diversity. Creating and Exploiting Diversity. spatial diversity through multiple antennas, frequency diversity through wide-band signaling. Equalization ©2009, B.-P. Paris Wireless Communications 365

The Importance of Diversity Frequency Diversity: Wide-Band Signals The Importance of Diversity We have taken a detailed look at the detrimental effects of time-varying multi-path channels. Question: How do time-varying multi-path channels affect the performance of mobile, wireless communications channels? In particular, how is the symbol error rate affected by these channels? Example: Analyze the simple communication system analyzed before in a time-varying multi-path environment. BPSK raised-cosine pulses, low data rate, i.e., narrow-band signals. ©2009, B.-P. Paris Wireless Communications 367

The Importance of Diversity Frequency Diversity: Wide-Band Signals System to be Analyzed The simple communication system in the diagram below will be analyzed. Focus on the effects of the multi-path channel h(t ). Assumption: low baud rate, i.e., narrow-band signal. Sampler, N (t ) rate fs bn s (t ) R (t ) R [n] to × p (t ) × h (t ) + ΠTs (t ) DSP ∑ δ(t − nT ) A ©2009, B.-P. Paris Wireless Communications 368

The Importance of Diversity Frequency Diversity: Wide-Band Signals Assumptions and Implications For our analysis, we make the following speciﬁc assumptions: 1. Narrow-band signals: The symbol period T is assumed to be much smaller than the delay spread of the channel. Implication: ISI is negligible; ﬂat-fading channel. The delay spread of our channel is approximately 2µs; choose symbol period T = 40µs (Baud rate 25KHz). 2. Slow fading: The duration of each transmission is much shorter than the coherence-time of the channel. Implication: the channel remains approximately constant for each transmission. Assuming a Doppler frequency of 30Hz, the coherence time is approximately 20ms; transmitting 60 symbols per frame leads to frame durations of 60 · 40µs = 2.4ms. ©2009, B.-P. Paris Wireless Communications 369

The Importance of Diversity Frequency Diversity: Wide-Band Signals Implications on Channel Model With the above assumptions, the multi-path channel reduces effectively to attenuation by a factor a. a is complex Gaussian (multiplicative noise), The magnitude |a| is Rayleigh distributed (Rayleigh fading). Short frame duration implies a is constant during a frame. Sampler, N (t ) rate fs bn s (t ) R (t ) R [n] to × p (t ) × × + ΠTs (t ) DSP ∑ δ(t − nT ) A a ©2009, B.-P. Paris Wireless Communications 370

The Importance of Diversity Frequency Diversity: Wide-Band Signals Modiﬁcations to the Receiver For the narrow-band channel model, only a minor modiﬁcation to the receiver is required. The effective impulse response of the system is a · p (t ). Hence, the receiver should match with a · p (t ) instead of just p (t ). Most importantly, this reverses any phase rotations introduced by the channel. Problem: The channel attenuation a is unknown and must be estimated. This is accomplished with the help of a training sequence embedded in the signal. To be discussed shortly. ©2009, B.-P. Paris Wireless Communications 371

The Importance of Diversity Frequency Diversity: Wide-Band Signals Instantaneous Symbol Error Rate Assuming that we have successfully determined, the channel attenuation a, the symbol error rate for a given a is easily determined. The attenuation a changes the received energy per symbol to |a|2 · Es . Consequently, the instantaneous symbol error rate for our BPSK system is 2|a|2 · Es Pe (a) = Q( ). N0 Note that for each frame, the system experiences a different channel attenuation a and, thus, a different instantaneous symbol error rate. ©2009, B.-P. Paris Wireless Communications 372

The Importance of Diversity Frequency Diversity: Wide-Band Signals Average Symbol Error Rate The instantaneous symbol error rate varies from frame to frame and depends on a. The average symbol error rate provides a measure of performance that is independent of the attenuation a. It is obtained by averaging over a: Pe = E[Pe (a)] = Pe (a) · PA (a) da, where PA (a) is the pdf for the attenuation a. For a complex Gaussian attenuation a with zero mean and 2 variance σa : 1 SNR Pe = (1 − ), 2 1 + SNR 2 where SNR = σa · Es /N0 . ©2009, B.-P. Paris Wireless Communications 373

The Importance of Diversity Frequency Diversity: Wide-Band Signals Symbol Error Rate with Rayleigh Fading 0 10 Rayleigh AWGN −1 10 −2 10 Symbol Error Rate −3 10 −4 10 −5 10 −6 10 0 1 2 3 4 5 6 7 8 9 10 E /N0 (dB) s 2 Figure: Symbol Error Rate with and without Rayleigh Fading; σa = 1 ©2009, B.-P. Paris Wireless Communications 374

The Importance of Diversity Frequency Diversity: Wide-Band Signals Conclusions The symbol error rate over a Rayleigh fading channel is much worse than for an AWGN channel. Example: To achieve a symbol error rate of 10−4 on an AWGN channel, Es /N0 ≈ 8.2dB is required; on a Rayleigh fading channel, Es /N0 ≈ 34dB is required! The poor performance results from the fact that the probability that channel is in a deep fade is signiﬁcant: 1 Pr(|a|2 Es /N0 < 1) ≈ . Es /N0 Question: What can be done to improve performance? ©2009, B.-P. Paris Wireless Communications 375

The Importance of Diversity Frequency Diversity: Wide-Band Signals Diversity Diversity refers to the ability to observe the transmitted signal over multiple, independent Rayleigh fading channels. Example: Multiple receiver antennas Assume the receiver is equipped with L separate antennas and corresponding receiver front-ends. The antennas are spaced sufﬁciently to ensure that the channels from transmitter to each of the receiver antennas is independent. Antenna spacing approximately equal to the wavelength of the carrier is sufﬁcient. Then, the receiver observes L versions of the transmitted signal, each with a different attenuation al and a different additive noise Nl (t ). ©2009, B.-P. Paris Wireless Communications 376

The Importance of Diversity Frequency Diversity: Wide-Band Signals Diversity Receiver Question: How should the L received signals be processed to minimize the probability of symbol errors. Maximum-Ratio Combining: Assume again that the channel attenuation al are known and that the noise PSD is equal on all channels. Then, the optimum receiver performs matched filtering for each received signal: the l-th received signal is filtered with filter al∗ · p ∗ (−t ). The L matched filter outputs are added and fed into the slicer. Note, since the matched filter includes the attenuation al , the sum is weighted by the attenuation of the channel. ©2009, B.-P. Paris Wireless Communications 377

The Importance of Diversity Frequency Diversity: Wide-Band Signals Symbol Error Rate The instantaneous error probability for given channel gains a = {a1 , a2 , . . . , aL } is ¯ ¯ Pe ( a ) = Q ( 2||a||2 Es /N0 ), ¯ where ||a||2 = ∑L =1 |ak |2 . ¯ k The average error probability is obtained by taking the expectation with respect to the random gains a ¯ 1−µ L L L−k 1 + µ k −1 Pe = ( ) ·∑ ( ) , 2 k =1 k −1 2 where SNR 2 µ= and SNR = σa · Es /N0 . 1 + SNR ©2009, B.-P. Paris Wireless Communications 378

The Importance of Diversity Frequency Diversity: Wide-Band Signals Symbol Error Rate with Receiver Diversity 0 10 L=1 −1 L=2 10 L=3 L=4 −2 L=5 10 AWGN −3 10 Symbol Error Rate −4 10 −5 10 −6 10 −7 10 −8 10 −9 10 0 2 4 6 8 10 12 E /N0 (dB) s Figure: Symbol Error Rate with Diversity over a Rayleigh Fading 2 Channel; σa = 1, AWGN channel without diversity. ©2009, B.-P. Paris Wireless Communications 379

The Importance of Diversity Frequency Diversity: Wide-Band Signals Conclusions The symbol error rate with diversity is much better than without. Example: To achieve a symbol error rate of 10−4 on an AWGN channel, Es /N0 ≈ 8.2dB is required; without diversity on a Rayleigh fading channel, Es /N0 ≈ 34dB is required! with 5-fold diversity on a Rayleigh fading channel, Es /N0 ≈ 5dB is required! The improved performance stems primarily from the fact that all L channels are unlikely to be in a deep fade at the same time. Performance better than an AWGN channel (without diversity) is possible because diversity is provided by the receiver. multiple receiver antennas exploit the same transmitted energy - array gain. ©2009, B.-P. Paris Wireless Communications 380

The Importance of Diversity Frequency Diversity: Wide-Band Signals Symbol Error Rate with Transmitter Diversity If diversity is provided by the transmitter, e.g., the signal is transmitted multiple times, then each transmission can only use symbol energy Es /L. 0 10 −1 10 −2 10 −3 10 Symbol Error Rate −4 10 −5 10 −6 10 −7 L=1 10 L=2 L=3 −8 L=4 10 L=5 AWGN −9 10 0 2 4 6 8 10 12 E /N (dB) s 0 ©2009, B.-P. Paris Wireless Communications 381

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation Objective: Perform a Monte Carlo simulation of a narrow-band communication system with diversity and time-varying multi-path. Approach: As before, we break the simulation into three parts 1. System parameters are set with the script ﬁle NarrowBandSetParameters. 2. The simulation is controlled with the driver script MCNarrowBandDriver. 3. The actual system simulation is carried out in the function MCNarrowBand. All three ﬁles are in the toolbox. ©2009, B.-P. Paris Wireless Communications 382

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation The simulation begins with the generation of the transmitted signal. New facet: a known training sequence is inserted in the middle of the frame. %% simulate discrete-time equivalent system % transmitter and channel via toolbox functions InfoSymbols = RandomSymbols( NSymbols, Alphabet, Priors ); % insert training sequence 45 Symbols = [ InfoSymbols(1:TrainLoc) TrainingSeq ... InfoSymbols(TrainLoc+1:end)]; % linear modulation ©2009, B.-P. Paris Wireless Communications 383

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation To simulate a diversity system, L received signals are generated. Each is sent over a different multi-path channel and experiences different noise. Each received signal is matched ﬁltered and the channel gain is estimated via the training sequence. % loop over diversity channels for kk = 1:L 52 % time-varying multi-path channels and additive noise Received(kk,:) = SimulateCOSTChannel( Signal, ChannelParams, fs); Received(kk,:) = addNoise( Received(kk,:), NoiseVar ); % digital matched filter, gain estimation 57 MFOut(kk,:) = DMF( Received(kk,:), hh, fsT ); GainEst(kk) = 1/TrainLength * ... MFOut( kk, TrainLoc+1 : TrainLoc+TrainLength) * TrainingSeq’; ©2009, B.-P. Paris Wireless Communications 384

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation The ﬁnal processing step is maximum-ratio combining. The matched ﬁlter outputs are multiplied with the conjugate complex of the channel gains and added. Multiplying with the conjugate complex of the channel gains reverses phase rotations by the channel, and gives more weight to strong channels. 61 % delete traning, MRC, and slicer MFOut(:, TrainLoc+1 : TrainLoc+TrainLength) = [ ]; MRC = conj(GainEst)*MFOut; Decisions = SimpleSlicer( MRC(1:NSymbols), Alphabet, ... ©2009, B.-P. Paris Wireless Communications 385

The Importance of Diversity Frequency Diversity: Wide-Band Signals Simulated Symbol Error Rate with Transmitter Diversity −1 10 −2 10 −3 10 Symbol Error Rate −4 10 −5 10 −6 10 Simulated L=2 L=1 L=2 −7 L=3 10 L=4 L=5 −8 AWGN 10 0 2 4 6 8 10 12 E /N (dB) s 0 Figure: Symbol Error Rate with Diversity over a Rayleigh Fading 2 Channel; σa = 1, simulated system has diversity of order L = 2. ©2009, B.-P. Paris Wireless Communications 386

The Importance of Diversity Frequency Diversity: Wide-Band Signals Summary The strong, detrimental impact of mobile, wireless channels on the error rate performance of narrow-band systems was demonstrated. Narrow-band system do not have inherent diversity and are subject to flat Rayleigh fading. To mitigate Rayleigh fading, diversity is required. Quantified the benefits of diversity. Illustrated diversity through antenna (spatial) diversity at the receiver. A complete system, including time-varying multi-path channel and diversity was simulated. Good agreement between theory and simulation. ©2009, B.-P. Paris Wireless Communications 387

The Importance of Diversity Frequency Diversity: Wide-Band Signals Frequency Diversity through Wide-Band Signals We have seen above that narrow-band systems do not have built-in diversity. Narrow-band signals are susceptible to have the entire signal affected by a deep fade. In contrast, wide-band signals cover a bandwidth that is wider than the coherence bandwidth. Beneﬁt: Only portions of the transmitted signal will be affected by deep fades (frequency-selective fading). Disadvantage: Short symbol duration induces ISI; receiver is more complex. The beneﬁts, far outweigh the disadvantages and wide-band signaling is used in most modern wireless systems. ©2009, B.-P. Paris Wireless Communications 389

The Importance of Diversity Frequency Diversity: Wide-Band Signals Illustration: Built-in Diversity of Wide-band Signals We illustrate that wide-band signals do provide diversity by means of a simple thought experiments. Thought experiment: Recall that in discrete time a multi-path channel can be modeled by an FIR ﬁlter. Assume ﬁlter operates at symbol rate Ts . The delay spread determines the number of taps L. Our hypothetical system transmits one information symbol in every L-th symbol period and is silent in between. At the receiver, each transmission will produce L non-zero observations. This is due to multi-path. Observation from consecutive symbols don’t overlap (no ISI) Thus, for each symbol we have L independent observations, i.e., we have L-fold diversity. ©2009, B.-P. Paris Wireless Communications 390

The Importance of Diversity Frequency Diversity: Wide-Band Signals Illustration: Built-in Diversity of Wide-band Signals We will demonstrate shortly that it is not necessary to leave gaps in the transmissions. The point was merely to eliminate ISI. Two insights from the thought experiment: Wide-band signals provide built-in diversity. The receiver gets to look at multiple versions of the transmitted signal. The order of diversity depends on the ratio of delay spread and symbol duration. Equivalently, on the ratio of signal bandwidth and coherence bandwidth. We are looking for receivers that both exploit the built-in diversity and remove ISI. Such receiver elements are called equalizers. ©2009, B.-P. Paris Wireless Communications 391

The Importance of Diversity Frequency Diversity: Wide-Band Signals Equalization Equalization is obviously a very important and well researched problem. Equalizers can be broadly classified into three categories: 1. Linear Equalizers: use an inverse filter to compensate for the variations in the frequency response. Simple, but not very effective with deep fades. 2. Decision Feedback Equalizers: attempt to reconstruct ISI from past symbol decisions. Simple, but have potential for error propagation. 3. ML Sequence Estimation: find the most likely sequence of symbols given the received signal. Most powerful and robust, but computationally complex. ©2009, B.-P. Paris Wireless Communications 392

The Importance of Diversity Frequency Diversity: Wide-Band Signals Maximum Likelihood Sequence Estimation Maximum Likelihood Sequence Estimation provides the most powerful equalizers. Unfortunately, the computational complexity grows exponentially with the ratio of delay spread and symbol duration. I.e., with the number of taps in the discrete-time equivalent FIR channel. ©2009, B.-P. Paris Wireless Communications 393

The Importance of Diversity Frequency Diversity: Wide-Band Signals Maximum Likelihood Sequence Estimation The principle behind MLSE is simple. Given a received sequence of samples R [n], e.g., matched ﬁlter outputs, and a model for the output of the multi-path channel: r [n] = s [n] ∗ h[n], where ˆ s [n] denotes the symbol sequence, and h[n] denotes the discrete-time channel impulse response, i.e., the channel taps. Find the sequence of information symbol s [n] that minimizes N D2 = ∑ |r [n] − s[n] ∗ h[n]|2 . n ©2009, B.-P. Paris Wireless Communications 394

The Importance of Diversity Frequency Diversity: Wide-Band Signals Maximum Likelihood Sequence Estimation The criterion N D2 = ∑ |r [n] − s[n] ∗ h[n]|2 . n performs diversity combining (via s [n] ∗ h[n]), and removes ISI. The minimization of the above metric is difﬁcult because it is a discrete optimization problem. The symbols s [n] are from a discrete alphabet. A computationally efﬁcient algorithm exists to solve the minimization problem: The Viterbi Algorithm. The toolbox contains an implementation of the Viterbi Algorithm in function va. ©2009, B.-P. Paris Wireless Communications 395

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation A Monte Carlo simulation of a wide-band signal with an equalizer is conducted to illustrate that diversity gains are possible, and to measure the symbol error rate. As before, the Monte Carlo simulation is broken into set simulation parameter (script VASetParameters), simulation control (script MCVADriver), and system simulation (function MCVA). ©2009, B.-P. Paris Wireless Communications 396

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation: System Parameters Listing : VASetParameters.m Parameters.T = 1/1e6; % symbol period Parameters.fsT = 8; % samples per symbol Parameters.Es = 1; % normalize received symbol energy to 1 Parameters.EsOverN0 = 6; % Signal-to-noise ratio (Es/N0) 13 Parameters.Alphabet = [1 -1]; % BPSK Parameters.NSymbols = 500; % number of Symbols per frame Parameters.TrainLoc = floor(Parameters.NSymbols/2); % location of t Parameters.TrainLength = 40; 18 Parameters.TrainingSeq = RandomSymbols( Parameters.TrainLength, ... Parameters.Alphabet, [0.5 0.5] % channel Parameters.ChannelParams = tux(); % channel model 23 Parameters.fd = 3; % Doppler Parameters.L = 6; % channel order ©2009, B.-P. Paris Wireless Communications 397

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation The ﬁrst step in the system simulation is the simulation of the transmitter functionality. This is identical to the narrow-band case, except that the baud rate is 1 MHz and 500 symbols are transmitted per frame. There are 40 training symbols. Listing : MCVA.m 41 % transmitter and channel via toolbox functions InfoSymbols = RandomSymbols( NSymbols, Alphabet, Priors ); % insert training sequence Symbols = [ InfoSymbols(1:TrainLoc) TrainingSeq ... InfoSymbols(TrainLoc+1:end)]; 46 % linear modulation Signal = A * LinearModulation( Symbols, hh, fsT ); ©2009, B.-P. Paris Wireless Communications 398

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation The channel is simulated without spatial diversity. To focus on the frequency diversity gained by wide-band signaling. The channel simulation invokes the time-varying multi-path simulator and the AWGN function. % time-varying multi-path channels and additive noise Received = SimulateCOSTChannel( Signal, ChannelParams, fs); 51 Received = addNoise( Received, NoiseVar ); ©2009, B.-P. Paris Wireless Communications 399

The Importance of Diversity Frequency Diversity: Wide-Band Signals MATLAB Simulation The receiver proceeds as follows: Digital matched ﬁltering with the pulse shape; followed by down-sampling to 2 samples per symbol. Estimation of the coefﬁcients of the FIR channel model. Equalization with the Viterbi algorithm; followed by removal of the training sequence. % is long enough so that VA below produces the right number of symbols MFOut = zeros( 1, 2*length(Symbols)+L-1 ); Temp = DMF( Received, hh, fsT/2 ); 57 MFOut( 1:length(Temp) ) = Temp; % channel estimation MFOutTraining = MFOut( 2*TrainLoc+1 : 2*(TrainLoc+TrainLength) ); ChannelEst = EstChannel( MFOutTraining, TrainingSeq, L, 2); 62 % VA over MFOut using ChannelEst ©2009, B.-P. Paris Wireless Communications 400

The Importance of Diversity Frequency Diversity: Wide-Band Signals Channel Estimation Channel Estimate: h = (S S)−1 · S r, ˆ where S is a Toeplitz matrix constructed from the training sequence, and r is the corresponding received signal. TrainingSPS = zeros(1, length(Received) ); 14 TrainingSPS(1:SpS:end) = Training; % make into a Toepliz matrix, such that T*h is convolution TrainMatrix = toeplitz( TrainingSPS, [Training(1) zeros(1, Order-1)]); 19 ChannelEst = Received * conj( TrainMatrix) * ... inv(TrainMatrix’ * TrainMatrix); ©2009, B.-P. Paris Wireless Communications 401

The Importance of Diversity Frequency Diversity: Wide-Band Signals Simulated Symbol Error Rate with MLSE Equalizer −1 10 −2 10 −3 Symbol Error Rate 10 −4 10 −5 10 −6 10 Simulated VA L=1 L=2 −7 L=3 10 L=4 L=5 −8 AWGN 10 0 2 4 6 8 10 12 E /N (dB) s 0 Figure: Symbol Error Rate with Viterbi Equalizer over Multi-path Fading Channel; Rayleigh channels with transmitter diversity shown for comparison. Baud rate 1MHz, Delay spread ≈ 2µs. ©2009, B.-P. Paris Wireless Communications 402

The Importance of Diversity Frequency Diversity: Wide-Band Signals Conclusions The simulation indicates that the wide-band system with equalizer achieves a diversity gain similar to a system with transmitter diversity of order 2. The ratio of delay spread to symbol rate is 2. comparison to systems with transmitter diversity is appropriate as the total average power in the channel taps is normalized to 1. Performance at very low SNR suffers, probably, from inaccurate estimates. Higher gains can be achieved by increasing bandwidth. This incurs more complexity in the equalizer, and potential problems due to a larger number of channel coefﬁcients to be estimated. Alternatively, this technique can be combined with additional diversity techniques (e.g., spatial diversity). ©2009, B.-P. Paris Wireless Communications 403

The Importance of Diversity Frequency Diversity: Wide-Band Signals More Ways to Create Diversity A quick look at three additional ways to create and exploit diversity. 1. Time diversity. 2. Frequency Diversity through OFDM. 3. Multi-antenna systems (MIMO) ©2009, B.-P. Paris Wireless Communications 404

The Importance of Diversity Frequency Diversity: Wide-Band Signals Time Diversity Time diversity: is created by sending information multiple times in different frames. This is often done through coding and interleaving. This technique relies on the channel to change sufﬁciently between transmissions. The channel’s coherence time should be much smaller than the time between transmissions. If this condition cannot be met (e.g., for slow-moving mobiles), frequency hopping can be used to ensure that the channel changes sufﬁciently. The diversity gain is (at most) equal to the number of time-slots used for repeating information. Time diversity can be easily combined with frequency diversity as discussed above. The combined diversity gain is the product of the individual diversity gains. ©2009, B.-P. Paris Wireless Communications 405

The Importance of Diversity Frequency Diversity: Wide-Band Signals OFDM OFDM has received a lot of interest recently. OFDM can elegantly combine the beneﬁts of narrow-band signals and wide-band signals. Like for narrow-band signaling, an equalizer is not required; merely the gain for each subcarier is needed. Very low-complexity receivers. OFDM signals are inherently wide-band; frequency diversity is easily achieved by repeating information (really coding and interleaving) on widely separated subcarriers. Bandwidth is not limited by complexity of equalizer; High signal bandwidth to coherence bandwidth is possible; high diversity. ©2009, B.-P. Paris Wireless Communications 406

The Importance of Diversity Frequency Diversity: Wide-Band Signals MIMO We have already seen that multiple antennas at the receiver can provide both diversity and array gain. The diversity gain ensures that the likelihood that there is no good channel from transmitter to receiver is small. The array gain exploits the beneﬁts from observing the transmitted energy multiple times. If the system is equipped with multiple transmitter antennas, then the number of channels equals the product of the number of antennas. Very high diversity. Recently, it has been found that multiple streams can be transmitted in parallel to achieve high data rates. Multiplexing gain The combination of multi-antenna techniques and OFDM appears particularly promising. ©2009, B.-P. Paris Wireless Communications 407

The Importance of Diversity Frequency Diversity: Wide-Band Signals Summary A close look at the detrimental effect of typical wireless channels. Narrow-band signals without diversity suffer poor performance (Rayleigh fading). Simulated narrow-band system. To remedy this problem, diversity is required. Analyzed systems with antenna diversity at the receiver. Veriﬁed analysis through simulation. Frequency diversity and equalization. Introduced MLSE and the Viterbi algorithm for equalizing wide-band signals in multi-path channels. Simulated system and veriﬁed diversity. A brief look at other diversity techniques. ©2009, B.-P. Paris Wireless Communications 408

Simulation of Wireless Communication Systems

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