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BEC604 -COMMUNICATION ENG II.pdf

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Sunil Manjani
Sunil Manjani

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UNIT UNIT 1 1: Sampling and : Sampling and Quantization Quantization : Sampling and : Sampling and Quantization Quantization
Introduction Introduction • Digital representation of analog signals Analog signal source Waveform Coding (Codec) Analog-to-Digital Encoding Introduction Introduction Digital representation of analog signals Waveform Digital signal destination Digital Encoding
Advantages of Digital Transmissions Advantages of Digital Transmissions Noise immunity Error detection and correction Ease of multiplexing Integration of analog and digital data Use of signal regenerators Data integrity and security Ease of evaluation and measurements More suitable for processing …….. Advantages of Digital Transmissions Advantages of Digital Transmissions Noise immunity Error detection and correction Ease of multiplexing Integration of analog and digital data Use of signal regenerators Data integrity and security Ease of evaluation and measurements More suitable for processing ……..
Disadvantages Disadvantages of Digital Transmissions of Digital Transmissions More bandwidth requirement Need of precise time synchronization Additional hardware for encoding/decoding Integration of analog and digital data Sudden degradation in Incompatible with existing analog facilities of Digital Transmissions of Digital Transmissions More bandwidth requirement Need of precise time synchronization Additional hardware for encoding/decoding Integration of analog and digital data Sudden degradation in QoS Incompatible with existing analog facilities
A Typical Digital Communication Link A Typical Digital Communication Link Fig. 2 Block Diagram A Typical Digital Communication Link A Typical Digital Communication Link Fig. 2 Block Diagram
Basic Digital Communication Transformations – Formatting/Source Coding – Transforms source info into digital symbols (digitization) – Selects compatible waveforms (matching function) – Introduces redundancy which facilitates accurate decoding despite errors It is essential for reliable communication – Modulation/Demodulation – Modulation is the process of modifying the info signal to facilitate transmission – Demodulation reverses the process of modulation. It involves the detection and retrie of the info signal • Types • Coherent: Requires a reference info for detection • Noncoherent: Does not require reference phase information Basic Digital Communication Transformations Transforms source info into digital symbols (digitization) Selects compatible waveforms (matching function) Introduces redundancy which facilitates accurate decoding despite errors It is essential for reliable communication Modulation is the process of modifying the info signal to facilitate transmission Demodulation reverses the process of modulation. It involves the detection and retrie Coherent: Requires a reference info for detection Noncoherent: Does not require reference phase information
Basic Digital Communication Transformations – Coding/Decoding Translating info bits to transmitter data symbols Techniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference) • Two Categories – Waveform Coding • Produces new waveforms with better performance – Structured Sequences Involves the use of redundant bits to determine the occurrence of error (and sometimes correct it) – Multiplexing/Multiple Access Is synonymous with resource sharing with other users – Frequency Division Multiplexing/Multiple Access (FDM/FDMA Basic Digital Communication Transformations Translating info bits to transmitter data symbols Techniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference) Produces new waveforms with better performance Involves the use of redundant bits to determine the occurrence of error (and Multiplexing/Multiple Access Is synonymous with resource sharing with other users Frequency Division Multiplexing/Multiple Access (FDM/FDMA
Practical Aspects of Sampling Practical Aspects of Sampling 1. Sampling Theorem 2 .Methods of Sampling 3. Significance of Sampling Rate 4. Anti-aliasing Filter 5 . Applications of Sampling Theorem Practical Aspects of Sampling Practical Aspects of Sampling . Applications of Sampling Theorem – PAM/TDM
Sampling Sampling is the processes of converting continuous by taking the “samples” at discrete-time intervals – Sampling analog signals makes them discrete in time but still continuous valued – If done properly (Nyquist theorem is satisfied), sampling does not introduce distortion Sampled values: – The value of the function at the sampling points Sampling interval: – The time that separates sampling points (interval b/w samples), – If the signal is slowly varying, then fewer samples per second will be required than if the wavef is rapidly varying – So, the optimum sampling rate depends on the signal Sampling is the processes of converting continuous-time analog signal, xa(t), into a discrete-time si time intervals Sampling analog signals makes them discrete in time but still continuous valued is satisfied), sampling does not introduce distortion The value of the function at the sampling points The time that separates sampling points (interval b/w samples), Ts If the signal is slowly varying, then fewer samples per second will be required than if the wavef So, the optimum sampling rate depends on the maximum frequency component present in the
Analog-to-digital conversion is (basically) a 2 step process: – Sampling • Convert from continuous-time analog signal – Is obtained by taking the “samples” of xa(t) at discrete Quantization – Convert from discrete-time continuous valued signal to discrete time discrete valued signal step process: time analog signal xa(t) to discrete-time continuous value signal x at discrete-time intervals, Ts time continuous valued signal to discrete time discrete valued signal
Sampling s f f  ling Rate (or sampling frequency fs): he rate at which the signal is sampled, expressed as the number of samples per second eciprocal of the sampling interval), 1/Ts = fs st Sampling Theorem (or Nyquist Criterion): the sampling is performed at a proper rate, no info is lost about the original signal and it can be operly reconstructed later on atement: “If a signal is sampled at a rate at least, but not exactly equal to onent of the waveform, then the waveform can be exactly reconstructed from the samples without any distortion” Sampling max 2 f f  he rate at which the signal is sampled, expressed as the number of samples per second the sampling is performed at a proper rate, no info is lost about the original signal and it can be “If a signal is sampled at a rate at least, but not exactly equal to twice the max frequency waveform can be exactly reconstructed from the samples
….. Sampling Theorem ….. Sampling Theorem Sampling Theorem for Bandpass Signal signal containing no frequency outside the specified bandwidth W Hz, it may be reconstructed from its samples at a sequence of points spaced 1/(2W) seconds apart with zero The minimum sampling rate of (2W) samples per second, for an analog signal bandwidth of W Hz, is called the Nyquist rate. The reciprocal of called the 1/( The phenomenon of the presence of high spectrum of the original analog signal is called aliasing or simply ….. Sampling Theorem ….. Sampling Theorem Signal - If an analog information signal containing no frequency outside the specified bandwidth W Hz, it may be reconstructed from its samples at a sequence of points W) seconds apart with zero-mean squared error. The reciprocal of Nyquist rate, 1/(2W), is called the Nyquist interval, that is, Ts = /(2W). The phenomenon of the presence of high-frequency component in the spectrum of the original analog signal is called aliasing or simply foldover.
Sampling Theorem Sampling Theorem Sampling Theorem for Baseband Signal no frequency components higher than recovered from the knowledge of its samples taken at a rate of at least 2 fm samples per second, that is, sampling frequency The minimum sampling rate fs = 2 fm samples per second is called the Nyquist sampling rate. A baseband signal having no frequency components higher than completely described by its sample values at uniform intervals less than or equal to the sampling interval seconds. Sampling Theorem Sampling Theorem Sampling Theorem for Baseband Signal - A baseband signal having no frequency components higher than fm Hz may be completely recovered from the knowledge of its samples taken at a rate of at samples per second, that is, sampling frequency fs ≥ 2 fm. A baseband signal having no frequency components higher than fm Hz is completely described by its sample values at uniform intervals less than or equal to 1/(2fm) seconds apart, that is, the sampling interval Ts ≤ 1/(2fm) seconds.
Methods of Sampling Methods of Sampling Ideal sampling - an impulse at each sampling instant Ideal Sampling Methods of Sampling Methods of Sampling Ideal Sampling
Ideal Sampling ( or Impulse Sampling) Is accomplished by the multiplication of the signal function) Consider the instantaneous sampling of the analog signal Train of impulse functions select sample values at regular intervals Fourier Series representation: ( ) ( ) ( ) s s n x t x t t nT        1 2 ( ) , jn t s s n n s s t n T e T T                ( or Impulse Sampling) accomplished by the multiplication of the signal x(t) by the uniform train of impulses (comb Consider the instantaneous sampling of the analog signal x(t) Train of impulse functions select sample values at regular intervals ( ) ( ) ( ) s s x t x t t nT   1 2 ( ) , s jn t s s s s t n T e T T         
Ideal Sampling ( or Impulse Sampling) his shows that the Fourier Transform of the sampled signal is the Fourier Transform of the original nal at rate of 1/Ts ( or Impulse Sampling) his shows that the Fourier Transform of the sampled signal is the Fourier Transform of the original
Ideal Sampling ( or Impulse Sampling) As long as fs> 2fm,no overlap of repeated replicas X(f Minimum Sampling Condition: Sampling Theorem: A finite energy function ampled value x(nTs) with provided that => 2 ( ) sin 2 ( ) ( ) ( ) s s s n s f t n T T t T x n T t n T                               ( ) sin (2 ( )) s s s s n T x n T c f t n T        ( or Impulse Sampling) X(f - n/Ts) will occur in Xs(f) A finite energy function x(t) can be completely reconstructed from provided that => 2 s m m s m f f f f f     2 ( ) 2 ( ) s s s f t n T T t n T                       ( ) sin (2 ( )) s s s s T x n T c f t n T   1 1 2 s s m T f f  
Ideal Sampling ( or Impulse Sampling) his means that the output is simply the replication of the original signal at discrete intervals, e.g ( or Impulse Sampling) his means that the output is simply the replication of the original signal at discrete intervals, e.g
Ts is called the Nyquist interval: It is the longest time interval that can be used for sampling a ban signal and still allow reconstruction of the signal at the receiver without distortion It is the longest time interval that can be used for sampling a ban signal and still allow reconstruction of the signal at the receiver without distortion
….. Methods of Sampling ….. Methods of Sampling Natural sampling - a pulse of short width with varying amplitude with natural tops Natural Sampling ….. Methods of Sampling ….. Methods of Sampling Natural Sampling
Natural Sampling Natural Sampling If we multiply x(t) by a train of rectang pulses xp(t), we obtain a gated wavefo that approximates the ideal sampled waveform, known as natural sampling gating (see Figure 2.8) ( ) ( ) ( ) s p x t x t x t  2 ( ) s j n f n n x t c e        ( ) [ ( ) ( ) ] s p X f x t x t   2 [ ( ) j n n c x t e         [ n s n c X f n f       
Each pulse in xp(t) has width Ts and amplitude 1/T The top of each pulse follows the variation of the signal being sampled Xs (f) is the replication of X(f) periodically every fs Xs (f) is weighted by Cn  Fourier Series Coeffiecient The problem with a natural sampled waveform is that the tops of the sample pulses are not flat It is not compatible with a digital system since the amplitude of each sample has infinite number o possible values Another technique known as flat top sampling is used to alleviate this problem /Ts The top of each pulse follows the variation of the signal being sampled Hz Fourier Series Coeffiecient The problem with a natural sampled waveform is that the tops of the sample pulses are not flat It is not compatible with a digital system since the amplitude of each sample has infinite number o is used to alleviate this problem
Flat-top sampling - a pulse of short width with varying amplitude with flat tops Flat-top Sampling ….. Methods of Sampling ….. Methods of Sampling top Sampling ….. Methods of Sampling ….. Methods of Sampling
Flat-Top Sampling ere, the pulse is held to a constant height for the whole sample period at top sampling is obtained by the convolution of the signal obtained after ideal mpling with a unity amplitude rectangular pulse, is technique is used to realize Sample-and S/H, input signal is continuously sampled and then the value is held for as long as i kes to for the A/D to acquire its value Top Sampling ere, the pulse is held to a constant height for the whole sample period at top sampling is obtained by the convolution of the signal obtained after ideal mpling with a unity amplitude rectangular pulse, p(t) and-Hold (S/H) operation S/H, input signal is continuously sampled and then the value is held for as long as i
Flat top sampling (Time Domain) '( ) ( ) ( ) x t x t t   ( ) '( ) * ( ) s x t x t p t  ( ) * ( ) ( ) ( ) * ( ) ( ) p t x t t p t x t t n T      Flat top sampling (Time Domain) ( ) * ( ) ( ) ( ) * ( ) ( ) s n p t x t t p t x t t n T                
Taking the Fourier Transform will result to where P(f) is a sinc function ( ) [ ( ) ] s s X f x t   ( ) ( ) ( ) n P f x t t n T               ( ) ( ) * ( ) P f X f f n f          1 ( ) ( ) n s P f X f n f T        ( ) ( ) ( ) s n P f x t t n T                1 ( ) ( ) * ( ) s n s P f X f f n f T                ( ) ( ) s P f X f n f   
Flat top sampling (Frequency Domain) Flat top sampling becomes identical to ideal sampling as the width of the pulses become shorter Flat top sampling (Frequency Domain) Flat top sampling becomes identical to ideal sampling as the width of the pulses become
Recovering the Analog Signal One way of recovering the original signal from sampled signal ilter (LPF) as shown below f fs > 2B then we recover x(t) exactly Else we run into some problems and signal is not fully recovered Recovering the Analog Signal One way of recovering the original signal from sampled signal Xs(f) is to pass it through a Low Pass Else we run into some problems and signal is not fully recovered
Significance of Sampling Rate When fs < 2fm, spectral components of adjacent samples will overlap, known as aliasing An Illustration of Aliasing Significance of Sampling Rate An Illustration of Aliasing
Undersampling and Aliasing – If the waveform is undersampled (i.e. fs < 2B) then there will be signal he signal at the output of the filter will be different from the original signal spectrum This is the outcome of aliasing! his implies that whenever the sampling condition is not met, an irreversible overlap of the spectral licas is produced ) then there will be spectral overlap in the sample ! his implies that whenever the sampling condition is not met, an irreversible overlap of the spectral
This could be due to: 1. x(t) containing higher frequency than were expected 2. An error in calculating the sampling rate Under normal conditions, undersampling of signals causing aliasing is not recommended containing higher frequency than were expected An error in calculating the sampling rate Under normal conditions, undersampling of signals causing aliasing is not recommended
Solution 1: Anti-Aliasing Analog Filter – All physically realizable signals are not completely bandlimited – If there is a significant amount of energy in frequencies above half the sampling frequency (fs/2), aliasing will occur – Aliasing can be prevented by first passing the analog signal through an called a prefilter) before sampling is performed – The anti-aliasing filter is simply a LPF with cutoff frequency equal to half the sample rate All physically realizable signals are not completely bandlimited If there is a significant amount of energy in frequencies above half the sampling frequency Aliasing can be prevented by first passing the analog signal through an anti-aliasing filter (als ) before sampling is performed aliasing filter is simply a LPF with cutoff frequency equal to half the sample rate
Antialiasing Filter An anti-aliasing filter is a low-pass filter of sufficient higher order which is recommended to be used prior to sampling. Minimizing Aliasing Antialiasing Filter Minimizing Aliasing
• Aliasing is prevented by forcing the bandwidth of the sampled signal to satisfy the requirement of the Sampling Theorem Aliasing is prevented by forcing the bandwidth of the sampled signal to satisfy the requirement
Solution 2: Over Sampling and Filtering in the Digital Domain – The signal is passed through a low performance (less costly) analog low the bandwidth. – Sample the resulting signal at a high sampling frequency. – The digital samples are then processed by a high performance digital filter and down sample the resulting signal. : Over Sampling and Filtering in the Digital Domain The signal is passed through a low performance (less costly) analog low-pass filter to lim Sample the resulting signal at a high sampling frequency. The digital samples are then processed by a high performance digital filter and down
Summary Of Sampling Ideal Sampling (or Impulse Sampling) Natural Sampling (or Gating) Flat-Top Sampling For all sampling techniques – If fs > 2B then we can recover x(t) exactly – If fs < 2B) spectral overlapping known as aliasing ( ) ( ) ( ) ( ) ( ) s s x t x t x t x t t n T    ( ) ( ) ( ) ( ) s p n x t x t x t x t c e   ( ) '( ) * ( ) ( ) ( ) * s s x t x t p t x t t n T p    Summary Of Sampling liasing will occur ( ) ( ) ( ) ( ) ( ) ( ) ( s s n s n x t x t x t x t t n T x n T t n T                   2 ( ) ( ) ( ) ( ) j n s p n n x t x t x t x t c e         ( ) '( ) * ( ) ( ) ( ) * s s n x t x t p t x t t n T p               
Quantization Quantization is a non linear transformation which maps elements from a continuous o a finite set. It is also the second step required by A/D conversion. Sample og Signal ontinuous time ontinuous value - Discrete time - Continuous value Quantization Quantization is a non linear transformation which maps elements from a continuous o a finite set. It is also the second step required by A/D conversion. Quantize Digital Signal - Discrete time - Discrete value Discrete time Continuous value
Uniform Quantization on of operation output -V V -V Uniform Quantization For M=2n levels, step size :  = 2V /2n = V(2-n+1) input w1(t) output w2(t) V
Figure 3.10 Two types of quantization: ( Two types of quantization: (a) midtread and (b) midrise.
output -V V -V Error, /2 -/2 Quantization Error, e input w1(t) output w2(t) V input w1(t) Error, e
2 . 3 is power signal average Then the r wave triangula a is signal input the Suppose 3 2 2 2 ) ( 2 1 : power error Average 2 out 2 3 2 0 2 2 n V V N S V dx x ds s e V                                      0 Error is symmetric around zero.   . and between r wave 2 3 12 2 12 2 2 2 1 2 n n V V V V            
Definition. The dynamic range of an input signal is the ratio of the largest to the smallest power levels which the input signal can take on and be reproduced with the acceptable signal distortion. The dynamic range of the quantizer input in the PCM system is Definition. The dynamic range of an input signal is the ratio of the largest to the smallest power levels which the input signal can take on and be reproduced The dynamic range of the quantizer input in the PCM system is 6n dB.
Nonuniform Quantizer Used to reduce quantization error and increase the dynamic range when input signa not uniformly distributed over its allowed range of values. owed lues for Used to reduce quantization error and increase the dynamic range when input signa not uniformly distributed over its allowed range of values. t input
Nonuniform quantizer Compressor rete ples pressing-and-expanding” is called “companding.” • • • • Decoder ved al signals Nonuniform quantizer Uniform Quantizer digital expanding” is called “companding.” Channel Expander outp
1. Unipolar nonreturn-to-zero (NRZ) Signaling 2. Polar nonreturn-to-zero(NRZ) Signaling 3. Unipor nonreturn-to-zero (RZ) Signaling 4. Bipolar nonreturn-to-zero (BRZ) Signaling 5. Split-phase (Manchester code) Line codes: zero (NRZ) Signaling zero(NRZ) Signaling zero (RZ) Signaling zero (BRZ) Signaling phase (Manchester code) Line codes:
Figure 3.15 Line codes for the electrical representations of binary data. (a) Unipolar NRZ signaling. (b) Polar NRZ signaling. (c) Unipolar RZ signaling. (d) Bipolar RZ signaling. (e) Split-phase or Manchester code. Line codes for the electrical representations of binary data. ) Polar NRZ signaling. ) Bipolar RZ signaling.
Application of Sampling Theorem PAM/TDM Design of PAM/TDM System Application of Sampling Theorem – PAM/TDM
UNIT-II DIGITAL MODULATION II DIGITAL MODULATION
Pulse Code Modulation (PCM) Pulse Code Modulation (PCM) 1. Block Diagram of PCM 2 PCM Sampling 3 Quantization of Sampled Signal 4 Encoding of Quantized Sampled Signal Pulse Code Modulation (PCM) Pulse Code Modulation (PCM) 3 Quantization of Sampled Signal Encoding of Quantized Sampled Signal
The basic elements of a PCM system. Pulse Code Modulation The basic elements of a PCM system. Pulse Code Modulation
PCM Sampling PCM Sampling he Process of Natural Sampling PCM Sampling PCM Sampling
Quantization of Quantization of peration of uantization VL L01 L12 L34 L23 L45 L56 L67 VH s(t) sq(t) sq(t) Quantization of Quantization of Sampled Signal Sampled Signal t Δ Δ/2 Δ0 Δ1 Δ2 Δ3 Δ4 Δ5 Δ6 Δ7 s0 s1 s2 s3 s4 s5 s6 s7 s(t)
Quantization is the conversion of an analog sample of the information signal into discrete form. Thus, an infinite number of possible levels are converted to a finite number of conditions. Quantization error difference between rounding off sample values of an analog signal to the nearest permissible level of the the process of quantization. Classification of Quantization Quantization Error and Classification Quantization Error and Classification Quantization error is defined as the difference between rounding off sample values of an analog signal to the nearest permissible level of the quantizer during the process of quantization. Classification of Quantization Process Uniform quantization Non-uniform quantization Quantization Error and Classification Quantization Error and Classification
Characteristics of Compressor, Uniform Characteristics of Compressor, Uniform and Non and Non- -uniform Quantizer uniform Quantizer Characteristics of Compressor, Uniform Characteristics of Compressor, Uniform uniform Quantizer uniform Quantizer
μ μ- -law law and A and A- -law Compression law Compression Characteristics Characteristics law Compression law Compression Characteristics Characteristics
Encoding of Quantized Sampled Signal Encoding of Quantized Sampled Signal PCM – Functional Blocks Encoding of Quantized Sampled Signal Encoding of Quantized Sampled Signal Functional Blocks
PCM Data Rate (bps) = 2nfm PCM Bandwidth (Hz) = (1/2) PCM Data Rate = Dynamic Range (dB) = 20 log (2n – Coding Efficiency (%) = [(minimum bits)/(actual bits)] x Where n is number of PCM encoding bits and frequency component of information signal PCM System Parameters PCM System Parameters ) PCM Data Rate = nfm 1) Coding Efficiency (%) = [(minimum bits)/(actual bits)] x 100 is number of PCM encoding bits and fm is the highest frequency component of information signal PCM System Parameters PCM System Parameters
DELTA MODULATION DELTA MODULATION
Delta modulation (DM) uses a single digital transmission of analog signals Essence of Delta Modulation (DM) Essence of Delta Modulation (DM) An Ideal Delta Modulation Waveform Delta modulation (DM) uses a single-bit DPCM code to achieve digital transmission of analog signals Essence of Delta Modulation (DM) Essence of Delta Modulation (DM) An Ideal Delta Modulation Waveform
DM system. (a) Transmitter. ( ) Transmitter. (b) Receiver.
lator consists of a comparator, a quantizer, and an accumulator utput of the accumulator is       ) sgn( 1 1        n i q n i q i e i e n m wo types of quantization errors : ope overload distortion and granular noise lator consists of a comparator, a quantizer, and an accumulator (3.55) )
a-Sigma modulation (sigma-delta modulation) modulation which has an integrator eve the draw back of delta modulation (differentiator eficial effects of using integrator: Pre-emphasize the low-frequency content Increase correlation between adjacent samples educe the variance of the error signal at the quantizer input ) Simplify receiver design ause the transmitter has an integrator , the receiver sists simply of a low-pass filter. differentiator in the conventional DM receiver is cancelled by the integrator )    delta modulation) integrator can differentiator) frequency content Increase correlation between adjacent samples educe the variance of the error signal at the quantizer input ) ause the transmitter has an integrator , the receiver differentiator in the conventional DM receiver is cancelled by the integrator )
ivalent versions of delta-sigma modulation system.
Differential Pulse-Code Modulation (DPCM) PCM has the sampling rate higher than the Nyquist rate ciently remove this redundancy. Figure 3.28 DPCM system. (a) Transmitter. (b) Receiver. Code Modulation (DPCM) rate .The encode signal contains redundant information. D ) Receiver.
Adaptive Differential Pulse- eed for coding speech at low bit rates , we have two aims in mind: Remove redundancies from the speech signal as far as possible. Assign the available bits in a perceptually efficient manner. Figure 3.29 Adaptive quantization with backward estimation (AQB). Adaptive prediction with backward estimation (APB). -Code Modulation (ADPCM) eed for coding speech at low bit rates , we have two aims in mind: Remove redundancies from the speech signal as far as possible. Assign the available bits in a perceptually efficient manner. Adaptive quantization with backward estimation (AQB). prediction with backward estimation (APB).
S. No. Parameter PCM 1. Number of bits per sample 4/8/16 bits More than one bit but less than PCM 2. Number of levels Depends on number of bits Fixed number of levels 3. Step size Fixed or variable Fixed or variable 4. Transmission bandwidth More bandwidth needed Lesser than PCM 5. Feedback Does not exist Exists 6. Quantization noise/distortion Quantization noise depends on number of bits Quantization noise & slope overload 7. Complexity of implementation Complex Simple Comparison of PCM and DM Techniques Comparison of PCM and DM Techniques DPCM DM ADM More than one bit but less than PCM One bit One bit Fixed number of levels Two levels Two levels Fixed or variable Fixed Variable Lesser than PCM Lowest Lowest Exists Exists Exists Quantization noise & slope overload slope overload & granular noise Quantization noise only Simple Simple Simple Comparison of PCM and DM Techniques Comparison of PCM and DM Techniques
UNIT Basband Pulse Transmission UNIT-III Pulse Transmission
Transmit and Receive Formatting Transmit and Receive Formatting
Sources of Error in received Signal Major sources of errors: – Thermal noise (AWGN) • disturbs the signal in an additive fashion (Additive) • has flat spectral density for all frequencies of interest (White) • is modeled by Gaussian random process (Gaussian Noise) – Inter-Symbol Interference (ISI) • Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared”. Sources of Error in received Signal disturbs the signal in an additive fashion (Additive) has flat spectral density for all frequencies of interest (White) is modeled by Gaussian random process (Gaussian Noise) Due to the filtering effect of transmitter, channel and receiver, symbols are
F R E Q U E N C Y D O W N C O N V E R S IO N A N S M I T T E D A V E F O R M A W G N R E C E I V E D W A V E F O R M R E C E IV IN G F IL T E R F O R B A N D P A S S S I G N A L S D E M O D U L A T E & S A M P L E Demodulation/Detection of digital signals Receiver Structure R E C E IV IN G E Q U A L IZ IN G F IL T E R T H R E S H O L D C O M P A R IS O N C O M P E N S A T I O N F O R C H A N N E L I N D U C E D I S I D E M O D U L A T E & S A M P L E S A M P L E a t t = T D E T E C T M E S S A S Y M B O O R C H A N N S Y M B O E S S E N T IA L O P T IO N A L Demodulation/Detection of digital signals Receiver Structure
Receiver Structure contd The digital receiver performs two basic functions – Demodulation – Detection Why demodulate a baseband signal??? – Channel and the transmitter’s filter causes ISI which “smears” the transmitted pulses – Required to recover a waveform to be sampled at t = nT. Detection – decision-making process of selecting possible digital symbol Receiver Structure contd two basic functions: Why demodulate a baseband signal??? Channel and the transmitter’s filter causes ISI which “smears” the Required to recover a waveform to be sampled at t = nT. making process of selecting possible digital symbol
Steps in designing the receiver Find optimum solution for receiver design with the following goals: 1. Maximize SNR 2. Minimize ISI Steps in design: – Model the received signal – Find separate solutions for each of the goals. Steps in designing the receiver Find optimum solution for receiver design with the following goals:
Detection of Binary Signal in Gaussian Noise covery of signal at the receiver consist of two parts ilter • Reduces the received signal to a single variable z(T) • z(T) is called the test statistics Detector (or decision circuit) • Compares the z(T) to some threshold level 0 , i.e., where H1 and H0 are the two possible binary hypothesis 0 ) ( 0 1  H H T z   Detection of Binary Signal in Gaussian Noise are the two
Finding optimized filter for AWGN channel Assuming Channel with response equal to impulse function Finding optimized filter for AWGN channel Assuming Channel with response equal to impulse function
Detection of Binary Signal in Gaussian Noise • For any binary channel, the transmitted signal over a symbol interval ( • The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse resp the channel hc(t), is Where n(t) is assumed to be zero mean AWGN process • For ideal distortionless channel where hc(t) is an impulse function and convolution with produces no degradation, r(t) can be represented as:         0 ) ( 0 ) ( ) ( 1 0 T t t s T t t s t si ) ( * ) ( ) (   t h t s t r c i t n t s t r i   ) ( ) ( ) ( Detection of Binary Signal in Gaussian Noise For any binary channel, the transmitted signal over a symbol interval (0,T) is: The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse resp Where n(t) is assumed to be zero mean AWGN process (t) is an impulse function and convolution with produces no degradation, r(t) can be represented as: 1 0 binary a for binary a for 2 , 1 ) (  i t n T t i    0 2 , 1 )
Design the receiver filter to maximize the SNR Model the received signal Simplify the model: – Received signal in AWGN ) (t hc ) t ) (t n ) (t r ) (t n ) (t si Ideal channels ) ( ) ( t t hc   AWGN AWGN Design the receiver filter to maximize the SNR ) (t r ) ( ) ( ) ( ) ( t n t h t s t r c i    ) ( ) ( ) ( t n t s t r i  
Find Filter Transfer Function H ctive: To maximizes (S/N)T and find h(t) sing signal ai(t) at filter output in terms of filter transfer function H(f) the filter transfer funtion and S(f) is the Fourier transform of input signal s(t) ded PSD of i/p noise is N0/2 t noise power can be expressed as: (S/N)T : f H t a i      ( ) (      H N 0 2 0 | 2              N f H N S T 0 2 ( Find Filter Transfer Function H0(f) (t) at filter output in terms of filter transfer function H(f) the filter transfer funtion and S(f) is the Fourier transform of input signal s(t) df e f S ft j  2 ) ( ) df f H 2 | ) (     df f H df e f S f fT j 2 2 2 | ) ( | ) ( ) 
• For H(f) = Hopt (f) to maximize (S/N)T use Schwarz’s Inequality • Equality holds if f1(x) = k f*2(x) where k is arbitrary constant and * indicates complex conjugate • Associate H(f) with f1(x) and S(f) ej2 fT with f2 • Substitute yields to: df e f S f H fT j 2 2 ) ( ) (      dx x f x f 2 2 1 ) ( ) (          N N S T 0 2        Schwarz’s Inequality: (x) where k is arbitrary constant and * indicates complex conjugate 2(x) to get: df f S df f H 2 2 ) ( ) (          dx x f dx x f 2 2 2 1 ) ( ) (     df f S 2 ) (    
and energy E of the input signal s(t): s (S/N)T depends on input signal energy power spectral density of noise and T on the particular shape of the waveform lity for holds for optimum filter transfer function H that: eal valued s(t): 0 2 max N E N S T        2 max N N S T        f H f H 0 ( ) (  kS t h 1 ) (        t h 0 ) ( and energy E of the input signal s(t): depends on input signal energy on the particular shape of the waveform holds for optimum filter transfer function H0(f) (3.55) df f S E 2 ) (      0 N E fT j e f kS  2 ) ( * )     fT j e f kS  2 ) ( *     where else T t t T kS 0 0 ) (
mpulse response of a filter producing maximum output signal or image of message signal s(t), delayed by symbol time duration T. filter designed is called a MATCHED FILTER ned as: a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform      T kS t h 0 ( ) ( mpulse response of a filter producing maximum output signal-to-noise ratio is the or image of message signal s(t), delayed by symbol time duration T. MATCHED FILTER a linear filter designed to provide the maximum noise power ratio at its output for a given    where else T t t 0 )
Matched Filter Output of a rectangular Pulse Matched Filter Output of a rectangular Pulse
Replacing Matched filter with Integrator Replacing Matched filter with Integrator
Implementation of matched filter receiver ) (t r ( 1 T z ) ( * 1 t T s  ) ( * t T s M  (T z M Bank of M matched filters ) ( ) ( t T s t r z i i     M i ,..., 1  ( )) ( ),..., ( ), ( ( 2 1 M T z T z T z   z Implementation of matched filter receiver           M z z  1 z  ) T ) T z Matched filter output: Observation vector M ) ,..., , ( 2 1 M z z z
Detection Max. Likelihood Detector Probability of Error Detection Max. Likelihood Detector Probability of Error
Detection ed filter reduces the received signal to a single variable z(T), after which the detection of symbol is carried out ncept of maximum likelihood detector is based on Statistical Decision Theory ws us to mulate the decision rule that operates on the data imize the detection criterion 0 ) ( 0 1  H H T z   z(T), after which the detection of symbol is carried out is based on Statistical Decision Theory
P[s1]  a priori probabilities hese probabilities are known before transmission obability of the received sample ), p(z|s1) nditional pdf of received signal z, conditioned on the class s ], P[s1|z]  a posteriori probabilities fter examining the sample, we make a refinement of our previous knowledge 0], P[s0|s1] rong decision (error) 1], P[s0|s0] rrect decision Probabilities Review nditional pdf of received signal z, conditioned on the class si fter examining the sample, we make a refinement of our previous knowledge Probabilities Review
mum Likelihood Ratio test and Maximum a posteriori ( em is that a posteriori probabilities are not known. on: Use Bay’s theorem: How to Choose the threshold? 1 0 ) | ( ) | ( z s p z s p    0 0 1 1 0 1 ) ( ) ( ) | ( ) ( ) ( ) | ( H H z P s P s z p z P s P s z p    0 1 ) | ( ) | ( z s p z s p    | ( ) | ( p s z p z i s p  s means that if received signal is positive, s1 (t) was sent s0 (t) was sent mum Likelihood Ratio test and Maximum a posteriori (MAP) criterion: How to Choose the threshold? 0 H  ) ( ) | ( ) ( ) | ( 0 0 1 1 0 1 ) s P s z p s P s z p H H    1 H  ) ( ) ( ) z p i s p i s (t) was sent, else
Likelihood of So and S 1 and S1
MAP criterion: ) ( ) ( ) | ( ) | ( ) ( 1 0 0 1 0 1 s P s P s z p s z p z L H H     When the two signals, s0(t) and s1(t), are equally likely, i.e., becomes s is known as maximum likelihood ratio test because we are selecting hypothesis that corresponds to the signal with the maximum likelihood. erms of the Bayes criterion, it implies that the cost of both types of error is the same H H s z p s z p z L max 1 ) | ( ) | ( ) ( 0 1 0 1     ) ( LRT test ratio likelihood , are equally likely, i.e., P(s0) = P(s1) = 0.5, then the decision rule because we are selecting hypothesis that corresponds to the signal with the maximum likelihood. erms of the Bayes criterion, it implies that the cost of both types of error is the same test ratio likelihood max
tuting the pdfs  0 0 0 2 1 ) | ( :   s z p H  0 1 1 2 1 ) | ( :   s z p H 2 1 2 1 1 ) | ( ) | ( ) ( 0 0 0 1 0 1 H H s z p s z p z L                           2 0 0 2 1 exp   a z                   2 0 1 2 1 exp  a z     1 2 1 exp 2 1 exp 0 1 2 0 2 0 2 1 2 H H a z a z o                      
) ( )} ( ln{ 2 0 0 1 a a z z L    2 1 0 1 2 0 0 1 2 ( ) (   a H H a a z      ) ( exp 2 0 0 1       a a z Hence: Taking the log, both sides will give 0 2 ) ( 0 1 2 0 2 0 2 1 H H a a      2 0 0 1 0 1 2 0 2 0 2 ) )( ( )   a a a a a     1 2 ) ( 2 0 2 0 2 1         a a
• Hence where z is the minimum error criterion and  0 is optimum threshold • For antipodal signal, s1(t) = - s0 (t)  a1 = - a0 ) ( 2 ) )( ( 0 1 2 0 0 1 0 1 2 0 0 1 a a a a a a H H z        0 0 1 H H z   ptimum threshold 0 0 1 0 1 2 ) (      a a H H z 0
Probability of Error will occur if s sent  s0 is received s sent  s1 is received The total probability of error is sum of the errors dz s z p s e P s e P s H P      0 ) | ( ) | ( ) | ( ) | ( 1 1 1 1 0  dz s z p s e P s e P s H P     0 ) | ( ) | ( ) | ( ) | ( 0 0 0 0 1  ( ) ( ) | ( ) | ( ) , ( 1 1 0 1 2 1 H P s P s H P P s e P s e P P i i B       Probability of Error ) ( ) | ) ( ) | ( ) ( 0 0 1 0 0 1 s P s s P s e P s 
nals are equally probable e, the probability of bit error PB, is the probability that an incorrect hypothesis is made rically, PB is the area under the tail of either of the conditional distributions  ) | ( 2 1 ) ( ) | ( 1 0 1 1 0 P s H P s P s H P PB     s H P P B       0 0 1 exp 2 1 ) | ( 0 0     is the probability that an incorrect hypothesis is made is the area under the tail of either of the conditional distributions p(z|s1) or p(z|s0)  ) | ( ) ( ) | ( 0 1 0 0 1 s H s P s H P  dz a z dz s z p dz                      2 0 0 0 2 1 exp ) | ( 0  
Inter-Symbol Interference (ISI) SI in the detection process due to the filtering effects of the ystem Overall equivalent system transfer function – creates echoes and hence time dispersion – causes ISI at sampling time ) ( ) ( H f H f H t  k k k n s z   Symbol Interference (ISI) SI in the detection process due to the filtering effects of the Overall equivalent system transfer function creates echoes and hence time dispersion ) ( ) ( f H f H r c i k i i k s    
Inter-symbol interference Baseband system model Equivalent model Tx filter Channel 2 x 3 x T T ) ( ) ( f H t h t t H h c c Equivalent system 2 x 3 x T ) ( ) ( f H t h ( ) ( ) ( ) ( f H f H f H f H r c t  symbol interference Channel ) (t n ) (t r Rx. filter Detector k z kT t  ) ( ) ( f H t h r r ) ( ) ( f t c c ) ( ˆ t n ) (t z Detector k z kT t  filtered noise ) f
Nyquist bandwidth constraint Nyquist bandwidth constraint: • The theoretical minimum required system bandwidth to detect without ISI is Rs/2 [Hz]. • Equivalently, a system with bandwidth transmission rate of 2W=1/T=Rs [symbols/s] without ISI. Bandwidth efficiency, R/W [bits/s/Hz] : • An important measure in DCs representing data throughput per hertz of bandw • Showing how efficiently the bandwidth resources are used by signaling techniq 2 2 1     W R W R T s s Nyquist bandwidth constraint The theoretical minimum required system bandwidth to detect Rs [symbols/s] Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz] can support a maxim [symbols/s] without ISI. [bits/s/Hz] : An important measure in DCs representing data throughput per hertz of bandw Showing how efficiently the bandwidth resources are used by signaling techniq Hz] [symbol/s/ 2
Ideal Nyquist pulse (filter) T 2 1 T 2 1  T ) ( f H f 0 T W 2 1  Ideal Nyquist filter Ideal Nyquist pulse (filter) ) / sinc( ) ( T t t h  1 0 T T 2 T  T 2  Ideal Nyquist pulse
Nyquist pulses (filters) Nyquist pulses (filters): – Pulses (filters) which results in no ISI at the Nyquist filter: – Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even Nyquist pulse: – Its shape can be represented by a time function. Example of Nyquist filters: Raised Nyquist pulses (filters) Pulses (filters) which results in no ISI at the sampling time. Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even-symmetric frequency function Its shape can be represented by a sinc(t/T) function multiply by another filters: Raised-Cosine filter
Pulse shaping to reduce ISI Goals and trade-off in pulse-shaping – Reduce ISI – Efficient bandwidth utilization – Robustness to timing error (small side lobes) Pulse shaping to reduce ISI shaping Robustness to timing error (small side lobes)
The raised cosine filter aised-Cosine Filter – A Nyquist pulse (No ISI at the sampling time)               W W W f f H 0 2 | | 4 cos 1 ) ( 0 2  Excess bandwidth: 0 W W  0 0 2 (sinc( 2 ) ( t W W t h  The raised cosine filter A Nyquist pulse (No ISI at the sampling time)          W f W f W W W W W f | | for | | 2 for 2 2 | | for 0 0 0 Roll-off factor 0 0 W W W r   1 0   r 2 0 0 ] ) ( 4 [ 1 ] ) ( 2 cos[ )) t W W t W W t    
The Raised cosine filter 2 ) 1 ( Baseband sSB s R r W   | ) ( | | ) ( | f H f H RC   r 0  r  r T 2 1 T 4 3 T 1 T 4 3  T 2 1  T 1  1 0.5 0 The Raised cosine filter – cont’d r 5 . 0  r 1  r 1 5 . 0 0 ) ( ) ( t h t h RC  1 0.5 0 T T 2 T  T 2  T 3  s R r W ) 1 ( Passband DSB  
Pulse shaping and equalization to remove ISI quare-Root Raised Cosine (SRRC) filter and Equalizer ) ( ) ( RC H f H f H t  No ISI at the sampling time ) ( ) ( ) ( ) ( ) ( ) ( RC RC f H f H f H f H f H f H t r r t     ) ( 1 ) ( f H f H c e  Pulse shaping and equalization to remove ISI Root Raised Cosine (SRRC) filter and Equalizer ) ( ) ( ) ( f H f H f H e r c No ISI at the sampling time ) ( SRRC f H  Taking care of ISI caused by tr. filter Taking care of ISI caused by channel
Example of pulse shaping Square-root Raised-Cosine (SRRC) pulse shaping mp. [V] Data symbol First pulse Second pulse Example of pulse shaping Cosine (SRRC) pulse shaping t/T Baseband tr. Waveform Second pulse Third pulse
Example of pulse shaping … Raised Cosine pulse at the output of matched filter Amp. [V] Example of pulse shaping … Raised Cosine pulse at the output of matched filter t/T Baseband received waveform at the matched filter output (zero ISI)
Eye pattern Eye pattern:Display on an oscilloscope which sweeps the system response to a baseband signal at the rate 1/T (T symbol duration) amplitude scale istortion due to ISI Timing jitter Eye pattern Display on an oscilloscope which sweeps the system response to symbol duration) time scale Noise margin Sensitivity t timing erro
Example of eye pattern: Binary-PAM, SRRQ pulse Perfect channel (no noise and no ISI) Example of eye pattern: PAM, SRRQ pulse Perfect channel (no noise and no ISI)
Correlative Coding Transmit 2W symbols/s with zero ISI, using the theoretical minimum bandwidth of W Hz, without infinitely sharp filters. Correlative coding (or duobinary signaling or partial response signaling) introduces some controlled amount of ISI into the data stream rather than trying to eliminate ISI completely Doubinary signaling Correlative Coding W symbols/s with zero ISI, using the theoretical minimum bandwidth of W Hz, without Correlative coding (or duobinary signaling or partial response signaling) introduces some controlled amount of ISI into the data stream rather than trying to eliminate ISI completely
Duobinary signaling Duobinary signaling (class I partial response) Duobinary signaling Duobinary signaling (class I partial response)
Duobinary signal and Nyguist Criteria Nyguist second criteria: but twice the bandwidth Duobinary signal and Nyguist Criteria Nyguist second criteria: but twice the bandwidth
Differential Coding The response of a pulse is spread over more than one signalin interval. The response is partial in any signaling interval. Detection : – Major drawback : error propagation. To avoid error propagation, need deferential coding (precoding). Differential Coding The response of a pulse is spread over more than one signalin The response is partial in any signaling interval. Major drawback : error propagation. To avoid error propagation, need deferential coding
Modified duobinary signaling Modified duobinary signaling – In duobinary signaling, H(f) is nonzero at the origin. – We can correct this deficiency by using the class IV partial response Modified duobinary signaling In duobinary signaling, H(f) is nonzero at the origin. We can correct this deficiency by using the class IV partial response
Modified duobinary signaling Spectrum Modified duobinary signaling
Modified duobinary signaling Time Sequency: interpretation of receiving Modified duobinary signaling Time Sequency: interpretation of receiving 2, 0, and -2?
Duobinary Transfer Function Duobinary Transfer Function
Comparison of Binary with Duobinary Signaling Binary signaling assumes the transmitted pulse amplitude are independent of one another Duobinary signaling introduces correlation between pulse amplitudes Duobinary technique achieve zero ISI signal transmission using a smaller system bandwidth Duobinary coding requires three levels, compared with the usual two levels for binary coding Duobinary signaling requires more power than binary signaling (~ signaling) Comparison of Binary with Duobinary Signaling Binary signaling assumes the transmitted pulse amplitude are independent of one another Duobinary signaling introduces correlation between pulse amplitudes Duobinary technique achieve zero ISI signal transmission using a smaller system bandwidth Duobinary coding requires three levels, compared with the usual two levels for binary coding Duobinary signaling requires more power than binary signaling (~2.5 dB greater SNR than binary
Pass-band Data Transmission band Data Transmission
Block Diagram Functional model of pass-band data transmission system. Block Diagram band data transmission system.
Signaling Illustrative waveforms for the three basic forms of signaling binary information. (a) Amplitude-shift keying. (b) Phase-shift keying. ( continuous phase. Signaling Illustrative waveforms for the three basic forms of signaling binary information. shift keying. (c) Frequency-shift keying with
What do we want to study? We are going to study and compare different modulation techniques in terms of – Probability of errors – Power Spectrum – Bandwidth efficiency B R b   Bits/s/Hz What do we want to study? We are going to study and compare different modulation Bits/s/Hz
Coherent PSK Binary Phase Shift Keying (BPSK) – Consider the system with 2 basis functions – and   T t b  2 cos 2 1    T t b  2 sin 2 2  Coherent PSK basis functions t f c  2 t f c  2
BPSK If we want to fix that for both symbols ( equal, we have 2 We place s0 to minimize probability of error s0 s0 BPSK If we want to fix that for both symbols (0 and 1) the transmitted energies are 1 s1
BPSK We found that phase of s1 and We can rotate s1 and s0  s0 Rotate BPSK and s0 are 180 degree difference. 1 2 s1
BPSK We observe that 2 has nothing to do with signals. Hence, only one basis function is sufficient to represent the signals 2 s0 BPSK has nothing to do with signals. Hence, only one basis function is sufficient to represent the signals 2 s1 1
BPSK Finally, we have   t E t s b  ) ( 1 1    t E t s b  ( 1 0   BPSK t f T E c b b  2 cos 2  t f T E t c b b  2 cos 2 )  
BPSK Signal-space diagram for coherent binary PSK system. The waveforms depicting the transmitted signals s1(t) and s2(t), displayed in the inserts, assume BPSK space diagram for coherent binary PSK system. The waveforms depicting ), displayed in the inserts, assume nc  2.
BPSK • Probability of error calculation. In the case of equally likely (Pr(m0)=Pr(m1)), we have           erfc 2 1 2 erfc 2 1 Pe BPSK Probability of error calculation. In the case of equally likely         0 0 2 N E N d b ik
BPSK Block diagrams for (a) binary PSK transmitter and ( receiver. BPSK ) binary PSK transmitter and (b) coherent binary PSK
Quadriphase-Shift Keying (QPSK)    i t f T E t s c i      2 2 cos 2  T is symbol duration E is signal energy per symbol There are 4 symbols for i = 1, 2, 3, and Shift Keying (QPSK)  T t       0 ; 4 1  , and 4
QPSK          E t i E T i E t si                  4 1 2 cos 2 cos 2 4 1 2 cos 1                1 2 sin 1 2 cos i E i E i s Which we can write in vector format as QPSK          T t t i t f T i E t f c c                  0 ; 4 1 2 sin 2 sin 2 4 1 2 sin 2             4 1 4   Which we can write in vector format as
QPSK i Input Dibit Phase of QPSK signaling 1 10 2 00 3 01 4 11 4 /  4 / 3 4 / 5 / 7 QPSK Phase of QPSK signaling Coordinate of Message point si1 si2 4 4 2 / E 2 / E 2 / E 2 / E 2 / E  2 / E  2 / E  2 / E 
QPSK 2 s3 s2 (10 (00) (01) QPSK 1 s4 s1 10) (11)
QPSK signals QPSK signals
QPSK Block diagrams of (a) QPSK transmitter and (b) coherent QPSK receiver. QPSK
QPSK: Error Probability QPSK Consider signal constellation given in the figure (10 Z3 Z2 E  QPSK: Error Probability QPSK 2 1 s4 s1 s3 s2 (10) (00) 10) (11) Z4 Z1 2 / E 2 / E 2 / E 2 / E 
QPSK       12 2 erfc 2 1 N d P can treat QPSK as the combination of K over the interval T=2Tb ce the first bit is transmitted by nsmitted by 2. bability of error for each channel is given by QPSK              0 0 12 2 erf 2 1 N E c N can treat QPSK as the combination of 2 independent ce the first bit is transmitted by 1 and the second bit is bability of error for each channel is given by
QPSK Pc mbol is to be received correctly both bits must be received ectly. ce, the average probability of correct decision is given by ch gives the probability of errors equal to                        0 2 0 2 fc 2 erfc 4 1 2 erfc N E N E PC QPSK  2 1 P    mbol is to be received correctly both bits must be received ce, the average probability of correct decision is given by ch gives the probability of errors equal to     0 N E
QPSK    erfc symbol per Pe e one symbol of QPSK consists of two bits, we have above probability is the error probability per symbol. The avg. bability of error per bit ch is exactly the same as BPSK .              0 erfc 2 1 symbol per 2 1 it N E Pe b QPSK         0 erfc N E b e one symbol of QPSK consists of two bits, we have E = 2Eb. above probability is the error probability per symbol. The avg.
BPSK vs QPSK 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 P ower s pec trum dens ity of B P S K vs . Q P S K N orm aliz ed frequenc y , Normalized PSD,Sf/2E b BPSK vs QPSK 1.2 1.4 1.6 1.8 2 P ower s pec trum dens ity of B P S K vs . Q P S K N orm aliz ed frequenc y ,fTb B P S K Q P S K
QPSK Conclusion – QPSK is capable of transmitting data twice as faster as BPSK with the same energy per bit. – We will also learn in the future that QPSK has half of the bandwidth of BPSK. QPSK QPSK is capable of transmitting data twice as faster as BPSK with the We will also learn in the future that QPSK has half of the bandwidth
OFFSET QPSK 2 s s3 s2 (10 (00) (01) 90 degree shift in phase 180 degree shift in phase OFFSET QPSK 1 s4 s1 10) (11) degree shift in phase degree shift in phase
OFFSET QPSK OFFSET QPSK
OFFSET QPSK Whenever both bits are changed simultaneously, phase-shift occurs. At 180 phase-shift, the amplitude of the transmitted signal changes very rapidly costing amplitude fluctuation. This signal may be distorted when is passed through the filter or nonlinear amplifier. OFFSET QPSK Whenever both bits are changed simultaneously, 180 degree shift, the amplitude of the transmitted signal changes very rapidly costing amplitude fluctuation. This signal may be distorted when is passed through the filter
OFFSET QPSK 0 1 2 3 4 -2 -1 0 1 2 0 1 2 3 - 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 Original Signal Filtered signal OFFSET QPSK 5 6 7 8 4 5 6 7 8 Filtered signal
OFFSET QPSK To solve the amplitude fluctuation problem, we propose the offset QPSK. Offset QPSK delay the data in quadrature component by T/ seconds (half of symbol). Now, no way that both bits can change at the same time. OFFSET QPSK To solve the amplitude fluctuation problem, we propose the Offset QPSK delay the data in quadrature component by T/2 Now, no way that both bits can change at the same time.
OFFSET QPSK In the offset QPSK, the phase of the signal can change by or 0 degree only while in the QPSK change by 180 90 or 0 degree. OFFSET QPSK In the offset QPSK, the phase of the signal can change by 90 degree only while in the QPSK the phase of the signal can degree.
OFFSET QPSK 0 1 2 3 4 - 2 - 1 0 1 2 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 2 3 4 - 2 - 1 0 1 2 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 1 0 01 10 hase SK phase PSK PSK 0 1 1 0 01 10 SK SK SK OFFSET QPSK 4 5 6 7 8 4 5 6 7 8 4 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 1 0 0 0 10 00 1 0 0 10 10 00
Offset QPSK Possible paths for switching between the message points in (a) QPSK and (b) offset QPSK. Offset QPSK Possible paths for switching between the ) offset QPSK.
OFFSET QPSK      2 erfc Pe dwidths of the offset QPSK and the regular QPSK is the me. m signal constellation we have that ich is exactly the same as the regular QPSK. OFFSET QPSK     0 2 N E dwidths of the offset QPSK and the regular QPSK is the m signal constellation we have that ich is exactly the same as the regular QPSK.
M-array PSK At a moment, there are M possible symbol values being sent for M different phase values, i i 2      M t f T E t s c i 2 2 cos 2        array PSK At a moment, there are M possible symbol values being sent  M / 1     M i i , , 2 , 1 , 1      
M-array PSK Signal-space diagram for octaphase- shift keying (i.e., M  8). The decision boundaries are shown as dashed lines. Signal-space diagram illustrating the application of the union bound for octaphase-shift keying. array PSK
M-array PSK Probability of errors E d d sin 2 18 12     / sin erfc 0      N E Pe  array PSK   M / sin   4 ; /      M M
M-ary PSK 0 5 1 0 1 0 -5 0 1 0 -4 0 1 0 -3 0 1 0 -2 0 1 0 -1 0 1 0 0 E b /N Probability of Symbol errors Q P S K 8 -a ry P S K 1 6 -a ry P S K ary PSK 1 5 2 0 2 5 3 0 /N 0 d B
M-array PSK Power Spectra (M-array) M=2, we have   M E Tf E f S b PSK 2 2 sinc log 2 sinc 2 ) (   E f S b BPSK sinc 2 ) (  array PSK    M f Tb 2 2 log sinc   f Tb 2 sinc
M-array PSK Power spectra of M-ary PSK signals for M array PSK ary PSK signals for M  2, 4, 8. Tbf
M-array PSK Bandwidth efficiency: – We only consider the bandwidth of the main lobe (or null – Bandwidth efficiency of M-ary PSK is given by M T T B b 2 log 2 2    M R R B R b b b 2 log 2    array PSK We only consider the bandwidth of the main lobe (or null-to-null bandwidth) ary PSK is given by M R b 2 log 2 M 2 log 5 . 0 
M-ary QAM QAM = Quadrature Amplitude Modulation Both Amplitude and phase of carrier change according to the transmitted symbol, mi. where ai and bi are integers.     t f a T E t s c i i   2 cos 2 0  ary QAM QAM = Quadrature Amplitude Modulation Both Amplitude and phase of carrier change according to the   T t t f b T E c i   0 ; 2 sin 2 0 
M-ary QAM Again, we have as the basis functions   t f T t c  0 ; 2 cos 2 1     t f T t c b  0 2 sin 2 2   ary QAM T t   T t   0
M-ary QAM QAM square Constellation – Having even number of bits per symbol, denoted by – M=L x L possible values – Denoting M L  ary QAM Having even number of bits per symbol, denoted by 2n.
16-QAM                    , 1 ( ) 3 , 3 ( , 1 ( ) 1 , 3 ( 1 ( ) 1 , 3 ( 1 ( ) 3 , 3 ( , i i b a QAM             ) 3 , 3 ( ) 3 , 1 ( ) 3 , ) 1 , 3 ( ) 1 , 1 ( ) 1 , ) 1 , 3 ( ) 1 , 1 ( ) 1 , 1 ) 3 , 3 ( ) 3 , 1 ( ) 3 , 1
16-QAM L-ary, 4-PAM
16-QAM Calculation of Probability of errors – Since both basis functions are orthogonal, we can treat the as combination of two 4-ary PAM systems. – For each system, the probability of error is given by              0 2 1 1 N d erfc L Pe QAM Calculation of Probability of errors Since both basis functions are orthogonal, we can treat the 16-QAM ary PAM systems. For each system, the probability of error is given by                     0 0 0 1 1 N E erfc M
16-QAM – A symbol will be received correctly if data transmitted on both PAM systems are received correctly. – Probability of symbol error is given by   c symbol P     c e P symbol P      1 1 1 QAM A symbol will be received correctly if data transmitted on both 4-ary PAM systems are received correctly. Hence, we have Probability of symbol error is given by  2 1 e P         e e e e P P P P symbol          2 2 1 1 2 2
16-QAM – Hence, we have – But because average energy is given by – We have        1 1 2 M symbol Pe 2 2 2 2 / 1 0 L E E L i av              1 1 2 M symbol Pe QAM But because average energy is given by            0 0 1 N E erfc M    3 1 2 1 0 2 E M i                     0 1 2 3 N M E erfc M av
Coherent FSK FSK = frequency shift keying Coherent = receiver have information on where the zero phas of carrier. We can treat it as non-linear modulation since information is put into the frequency. Coherent FSK Coherent = receiver have information on where the zero phas linear modulation since information is
Binary FSK Transmitted signals are where          , 0 2 cos 2 b b i T E t s  1 ;    i T i n f b c i Binary FSK    elsewhere 0 , b i T t t f  2 , 1
Binary FSK S1(t) represented symbol “1”. S2(t) represented symbol “0”. This FSK is also known as Sunde’s FSK. It is continuous phase frequency Binary FSK This FSK is also known as Sunde’s FSK. It is continuous phase frequency-shift keying (CPFSK).
Binary FSK There are two basis functions written as As a result, the signal vectors are          , 0 2 cos 2 b i T t         b E and 0 1 s Binary FSK There are two basis functions written as As a result, the signal vectors are    elsewhere 0 , 2 b i T t t f         b E 0 2 s
BFSK From the figure, we have In case of Pr(0)=Pr(1), the probability of error is given by We observe that at a given value of P requires twice as much power as the BPSK system.      2 1 erfc Pe BFSK ), the probability of error is given by We observe that at a given value of Pe, the BFSK system requires twice as much power as the BPSK system. b E d 2 12      0 2 N E b
TRANSMITTER
RECEIVER
Power Spectral density of BFSK Consider the Sunde’s FSK where f1 and f2 We observe that in-phase component does not depend on m     t f T t T E T t t f T E t s c b b b b c b b i     2 cos cos 2 2 cos 2                                b b b b b T E T t T E  cos 2 cos 2 Power Spectral density of BFSK are different by 1/Tb. We can write phase component does not depend on mi since   t f T t T E c b b b   2 sin sin 2                  b T t  cos
Power Spectral density of BFSK We have For the quadrature component                       b b b BI T E T t T E F f S 2 cos 2 2  Half of the symbol power             b b b T t T E t g  sin 2 Power Spectral density of BFSK For the quadrature component                            b b b b T f T f T 2 1 2 1   Half of the symbol power    2 2 2 2 2 1 4 cos 8   f T f T T E S b b b b BQ  
Power Spectral density of BFSK Finally, we obtain S Power Spectral density of BFSK ) ( ) ( ) ( f S f S f BQ BI B  
Phase Tree of BFSK FSK signal is given by At t = 0, we have The phase of Signal is zero.         c b b T t f T E t s   2 cos 2   0 0 2 cos 2 0 b c b b T f T E s              Phase Tree of BFSK     b T t    0 cos 2 b b T E 
Phase Tree of BFSK At t = Tb, we have We observe that phase changes by for symbol “1” and + for symbol “0 We can draw the phase trellis as           2 cos 2 b b c b b b T T T f T E T s Phase Tree of BFSK We observe that phase changes by  after one symbol (Tb seconds). - 0”          cos 2 b b b b T E
Minimum-Shift keying (MSK) MSK tries to shift the phase after one symbol to just half of Sunde’s FSK system. The transmitted signal is given by                2 cos 2   t t f T E c b b Shift keying (MSK) MSK tries to shift the phase after one symbol to just half of Sunde’s FSK system. The transmitted signal is given by           "0" for 0 2 cos 2 "1" for 0 2 cos 2 2 1     t f T E t f T E b b b b
MSK Where Observe that     T h t b      0 b c T h f f and 2 1    1 2 1 f f c  MSK t h b b c T h f f 2 2    2 f 
MSK h = Tb(f1-f2) is called “deviation ratio.” For Sunde’s FSK, h = 1. For MSK, h = 0.5. h cannot be any smaller because the orthogonality between cos( and cos(2f2t) is still held for h < 0.5 Orthogonality guarantees that both signal will not interfere each other in detection process. MSK ) is called “deviation ratio.” cannot be any smaller because the orthogonality between cos(2f1t) 5. Orthogonality guarantees that both signal will not interfere each other
MSK Phase trellis diagram for MSK signal MSK Phase trellis diagram for MSK signal 1101000
MSK Signal s(t) of MSK can be decomposed into where                  t s t f t s f t T E t t f T E t s Q c I b b c b b      2 cos 2 cos cos 2 2 cos 2          b t T t   2 0    MSK (t) of MSK can be decomposed into          t f t f t T E t f c c b b c    2 sin 2 sin sin 2  b T t   0 ;
MSK Symbol (0) 1 0  0 0  MSK (0) (Tb) 0 /2  -/2 0 -/2  /2
MSK For the interval –Tb < t  0, we have Let’s note here that the  for the interval not be the same. We know that     2 0   t T t b                  b T t cos 0 cos 2 0 cos    MSK , we have for the interval -Tb<t 0 and 0< tTb ma 0 ;    t Tb                     b b T t T t 2 sin 0 sin 2 cos    
MSK Since (0) can be either 0 or  depending on the past history. We have “+” for (0) = 0 and “-” for (0) =  Hence, we have               b T t cos 0 cos 2 0 cos    b b b I T t T E t s           2 cos 2 ) (  MSK depending on the past history. We have                   b b T t T t 2 cos 2 cos   b b T t T    ;
MSK Similarly we can write for 0< tTb and Tb < t2Tb. Note the “+” and “ between these intervals. Furthermore, we have that (Tb) can be past history.     b b T T t   2    MSK Note the “+” and “-” may be different ) can be /2 depending on the   b T t 
MSK Hence, we have we have that (Tb) can be /2 depending on the past history.                               cos sin cos sin 2 sin      b b b b b T T T T t T                   b b b b T t T T t T 2 cos 2 sin    MSK depending on the past history.                                           2 2 sin cos 2 2 2 sin cos 2         b b b b b b b b T t T T t T T t T T T t                b T t 2 sin 2  
MSK Hence, we have “+” for (Tb) = +/2 and “-” for (Tb The basis functions change to b b b Q T t T E t s 2 sin 2 ) (               b b T t T t          2 cos 2 cos 2 1      b b T t T t          2 sin 2 sin 2 2   MSK ) = -/2 b T t 2 0 ;        b c T t t f   0 ; 2  b c T t t f   0 ; 2
MSK We write MSK signal as Where and                    ) ( ) ( sin E ) ( 0 cos E 2 cos 2 cos 0 cos 2 2 2 cos cos 2 2 2 1 1 b 1 b t s t s T t f T t T E T E t f t T E t s b b b b b c b b                              0 cos 1  b E s  MSK               ) ( 2 sin 2 sin sin 2 2 sin sin 2 t T t f T t T T E t f t f t b c b b b b c c                    b b T E s  sin 2  
MSK Symbol (0) s 1 0  0 0  E E E  E  MSK s1 (Tb) s2 /2 -/2 -/2 /2 b E b E b E b E b E b E b E  b E 
         0 2 1 N E erfc P b e
0 Phase: /2   /2  /2 0 -/2
MSK We observe that MSK is in fact the QPSK having the pulse shape Block diagrams for transmitter and receiver are given in the next two slides.     b T t 2 cos  MSK We observe that MSK is in fact the QPSK having the Block diagrams for transmitter and receiver are given in the     b
   b b T T dt t t x x ) ( ) ( 1 1    b T dt t t x x 2 0 2 2 ) ( ) ( 
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 Norm aliz ed F requenc y , 1.2 1.4 1.6 1.8 2 Norm aliz ed F requenc y , fTb M S K B P S K Q P S K
MSK Probability of error of MSK system is equal to BPSK and QPSK This due to the fact that MSK observes the signal for two symbol intervals whereas FSK only observes for single signal interval. Bandwidth of MSK system is 50% larger than QPSK. 2 16 cos 32 ) (      E f S b MSK  MSK Probability of error of MSK system is equal to BPSK and QPSK This due to the fact that MSK observes the signal for two symbol intervals whereas FSK only observes for single signal interval. % larger than QPSK.   2 2 2 1 16 2 cos      f T f T b b 
Noncoherent Orthogonal Modulation Noncoherent implies that phase information is not available to the receiver. As a result, zero phase of the receiver can mean any phase of the transmitter. Any modulation techniques that transmits information through the phase cannot be used in noncoherent receivers. Noncoherent Orthogonal Modulation Noncoherent implies that phase information is not available to the As a result, zero phase of the receiver can mean any phase of the Any modulation techniques that transmits information through the phase cannot be used in noncoherent receivers.
Noncoherent Orthogonal Modulation cos(2ft) sin(2ft) Receiver Noncoherent Orthogonal Modulation cos(2ft) sin(2ft) Transmitter
Noncoherent Orthogonal Modulation It is impossible to draw the signal constellation since we do not know where the axes are. However, we can still determine the distance of the each signal constellation from the origin. As a result, the modulation techniques that put information in the amplitude can be detected. FSK uses the amplitude of signals in two different frequencies. Hence non-coherent receivers can be employed. Noncoherent Orthogonal Modulation It is impossible to draw the signal constellation since we do not know However, we can still determine the distance of the each signal As a result, the modulation techniques that put information in the FSK uses the amplitude of signals in two different frequencies. Hence coherent receivers can be employed.
Noncoherent Orthogonal Modulation Consider the BFSK system where two frequencies f represented two “1” and “0”. The transmitted signal is given by Problem is that  is unknown to the receiver. For the coherent receiver  is precisely known by receiver.   i t f T E t s   2 cos 2 ) (   Noncoherent Orthogonal Modulation Consider the BFSK system where two frequencies f1 and f2 are used to is unknown to the receiver. For the coherent receiver  b T t i    0 , 2 , 1 ;
Noncoherent Orthogonal Modulation Furthermore, we have To get rid of the phase information (       ) ( ) ( 2 2 cos cos 2 2 cos 2 ) ( 2 2 1 1 t s t s T E t f T E t f T E t s i i i i               E s s t s i i    2 2 2 1 cos Noncoherent Orthogonal Modulation To get rid of the phase information (), we use the amplitude     2 sin sin t f i       E E     2 2 sin
Noncoherent Orthogonal Modulation Where The amplitude of the received signal    T i x dt t t s s 1 0 1 1 ) ( ) (     T i x dt t t s s 2 0 2 2 ) ( ) (    2 0 2 cos ) (                    T i i dt t f t x l  Noncoherent Orthogonal Modulation The amplitude of the received signal   T dt t t x 0 1 ) ( ) (    T dt t t x 0 2 ) ( ) (    2 / 1 2 0 2 sin ) (              T i dt t f t x 
Noncoherent Orthogonal Modulation Probability of Errors       2 exp 2 1 N E Pe Noncoherent Orthogonal Modulation     0 N E
Noncoherent: BFSK For BFSK, we have          0 2 cos 2 b b i T E t s Noncoherent: BFSK    elsewhere ; 0 ; 2 b i T t t f 
Noncoherent: BFSK Noncoherent: BFSK
Noncoherent: BFSK Probability of Errors       2 exp 2 1 N E Pe Noncoherent: BFSK     0 N E b
DPSK Differential PSK – Instead of finding the phase of the signal on the interval receiver determines the phase difference between adjacent time intervals. – If “1” is sent, the phase will remain the same – If “0” is sent, the phase will change DPSK Instead of finding the phase of the signal on the interval 0<tTb. Thi receiver determines the phase difference between adjacent time ” is sent, the phase will remain the same ” is sent, the phase will change 180 degree.
DPSK Or we have and           b b b b T E T E t s 2 cos 2 2 cos 2 ) ( 1             b b b b T E T E t s 2 cos 2 2 cos 2 ) ( 2   DPSK       b b c b c T t T t f T t t f 2 ; 2 0 ;          b b c b c T t T t f T t t f 2 ; 2 0 ;   
DPSK In this case, we have T=2Tb and E=2 Hence, the probability of error is given by       0 exp 2 1 N E P b e DPSK 2Eb Hence, the probability of error is given by     0 b
DPSK: Transmitter 1     k k k k k d b d b d DPSK: Transmitter 1 
DPSK bk} 1 0 dk-1} 1 1 Differential encoded dk} 1 1 0 Transmitted Phase 0 0  DPSK 0 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0  0 0  0   
DPSK: Receiver DPSK: Receiver
DPSK: Receiver From the block diagram, we have that the decision rule as If the phase of signal is unchanged (send “ both xi and xQ should not change. Hence, the If the phase of signal is unchanged (send “ both xi and xQ should change. Hence, the   1 0 1 0   Q Q I I x x x x l x DPSK: Receiver From the block diagram, we have that the decision rule as If the phase of signal is unchanged (send “1”) the sign (“+” or “-”) of change. Hence, the l(x) should be positive. If the phase of signal is unchanged (send “0”) the sign (“+” or “-1”) of should change. Hence, the l(x) should be negative. 0 1 say 0 say  
Signal-space diagram of received DPSK signal. space diagram of received DPSK signal.
Unit-V –Introduction to Spread Spectrum Techniques Introduction to Spread Spectrum Techniques
M-ary signaling scheme: In this signaling scheme 2 or more bits are grouped together to form a symbol. One of the M possible signals s1(t) ,s2(t),s3(t),……sM(t) is transmitted during each symbol period of duration Ts. The number of possible signals = M = 2n, where n is an integer.
n M = 2n 1 2 2 4 3 8 4 16 …. …… The symbol values of M for a given value of n: Symbol 0, 1 00, 01, 10, 11 000, 001, 010,011,... 0000, 0001, 0010,0011,…. ………. The symbol values of M for a given value of n:
Fig: waveforms of (a) ASK (b) PSK (c)FSK • Depending on the variation of amplitude, phase or frequency of the carrier, the modulation scheme is calle M-ary ASK, M-ary PSK and M-ary FSK. Fig: waveforms of (a) ASK (b) PSK (c)FSK Depending on the variation of amplitude, phase or frequency of the carrier, the modulation scheme is calle
M-ary Phase Shift Keying(MPSK) In M-ary PSK, the carrier phase takes on one of the M possible values, namely i = 2 * (i - 1) / M where i = 1, 2, 3, …..M. The modulated waveform can be expressed as where Es is energy per symbol = (log2 M) E Ts is symbol period = (log2 M) Tb. ary Phase Shift Keying(MPSK) ary PSK, the carrier phase takes on one of the M possible values, namely The modulated waveform can be expressed as M) Eb b.
he above equation in the Quadrature form is choosing orthogonal basis signals efined over the interval 0  t  Ts he above equation in the Quadrature form is
M-ary signal set can be expressed as  Since there are only two basis signals, the constellation of M dimensional.  The M-ary message points are equally spaced on a circle of radius at the origin.  The constellation diagram of an 8-ary PSK signal set is shown in fig. Since there are only two basis signals, the constellation of M-ary PSK is two ary message points are equally spaced on a circle of radius Es, centered ary PSK signal set is shown in fig.
Fig: Constellation diagram of an M-ary PSK system(m=8) ary PSK system(m=8)
Derivation of symbol error probability: Decision Rule: Fig: Constellation diagram for M=2 (Binary PSK) (Binary PSK)
If a symbol (0,0,0) is transmitted, it is clear that if an error occurs, the transmitted signal is most likely to be mistaken for (0,0,1) and (1,1,1 signal being mistaken for (1,1,0) is remote. The decision pertaining to (0,0,0) is bounded by /8(below 1(t)- axis) to  = + /8 ( above The probability of correct reception is… ) is transmitted, it is clear that if an error occurs, the transmitted signal is most 1) and the ) is remote. ) is bounded by  = - ( above 2(t)- axis)
Fig: Probability density function of Phase .
The average symbol error probability of an coherent M AWGN channel is given by Similarly, The symbol error Probability of a differential M AWGN channel is given by The average symbol error probability of an coherent M-ary PSK system in Similarly, The symbol error Probability of a differential M-ary PSK system in
Fig: The performance of symbol error probability for -different values of M Fig: The performance of symbol error probability for
M-ary Quadrature Amplitude (QAM) It’s a Hybrid modulation As we allow the amplitude to also vary with the phase, a new modulation scheme called quadrature amplitude modulation (QAM) is obtained. The constellation diagram of 16-ary QAM consists of a square lattice of signal points. ary Quadrature Amplitude Modulation (QAM) As we allow the amplitude to also vary with the phase, a new modulation scheme called quadrature amplitude modulation (QAM) is obtained. ary QAM consists of a square lattice of signal
Fig: signal Constellation of M-ary QAM for M= ary QAM for M=16
M-ary Frequency Shift Keying(MFSK) In M-ary FSK modulation the transmitted signals are defined by: where fc = nc/2Ts, for some fixed integer n. The M transmitted signals are of equal energy and equal duration, and the signal frequencies are separated by orthogonal to one another. ary Frequency Shift Keying(MFSK) ary FSK modulation the transmitted signals are defined by: , for some fixed integer n. The M transmitted signals are of equal energy and equal duration, and the signal frequencies are separated by 1/2Ts Hertz, making the signal
The average probability of error based on the union bound is given by Using only the leading terms of the binomial expansion: The average probability of error based on the union bound is given by Using only the leading terms of the binomial expansion:
Power Efficiency and Bandwidth : Bandwidth: The channel bandwidth of a M-ary FSK signal is : Power Efficiency and Bandwidth : ary FSK signal is :
The channel bandwidth of a noncohorent MFSK is : This implies that the bandwidth efficiency of an M with increasing M. Therefore, unlike M- bandwidth inefficient. The channel bandwidth of a noncohorent MFSK is : This implies that the bandwidth efficiency of an M-ary FSK signal decreases -PSK signals, M-FSK signals are
Introduction to Spread Spectrum • Problems such as capacity limits, propagation effects, synchroniza occur with wireless systems • Spread spectrum modulation spreads out the modulated signal bandwidth so it is much greater than the message bandwidth • Independent code spreads signal at transmitter and despreads sig at receiver Introduction to Spread Spectrum Problems such as capacity limits, propagation effects, synchroniza Spread spectrum modulation spreads out the modulated signal bandwidth so it is much greater than the message bandwidth Independent code spreads signal at transmitter and despreads sig
Multiplexing • Multiplexing in 4 dimensions – space (si) – time (t) – frequency (f) – code (c) • Goal: multiple use of a shared medium • Important: guard spaces needed! Multiplexing s2 s3 s1 f t c k2 k3 k4 k5 k6 k1 t c f t c channels ki
Frequency multiplex • Separation of spectrum into smaller frequency bands • Channel gets band of the spectrum for the whole time • Advantages: – no dynamic coordination needed – works also for analog signals • Disadvantages: – waste of bandwidth if traffic distributed unevenly – inflexible – guard spaces Frequency multiplex Separation of spectrum into smaller frequency bands Channel gets band of the spectrum for the whole time k3 k4 k5 t c
t Time multiplex Channel gets the whole spectrum for a certain amount of time Advantages: – only one carrier in the medium at any time – throughput high even for many users Disadvantages: – precise synchronization necessary c k2 k3 k4 k5 k1 Time multiplex Channel gets the whole spectrum for a certain amount of time
Time and frequency multiplex • A channel gets a certain frequency band for a certain amount of time (e.g. GSM) • Advantages: – better protection against tapping – protection against frequency selective interference – higher data rates compared to code multiplex • Precise coordination required t Time and frequency multiplex A channel gets a certain frequency band for a certain amount of time (e.g. c k2 k3 k4 k5 k1
Code multiplex Each channel has unique code All channels use same spectrum at same time Advantages: – bandwidth efficient – no coordination and synchronization – good protection against interference Disadvantages: – lower user data rates – more complex signal regeneration Implemented using spread spectrum technology Code multiplex Implemented using spread spectrum technology k2 k3 k4 k5 k6 k1 t c
Spread Spectrum Technology • Problem of radio transmission: frequency dependent fading can wipe out narrow band signals for duration of the interference • Solution: spread the narrow band signal into a broad band signal using a special code detection at receiver interference spread signal f ower Spread Spectrum Technology Problem of radio transmission: frequency dependent fading can wipe out narrow band signals for duration of the interference Solution: spread the narrow band signal into a broad band signal using a detection at receiver signal spread interferenc f power
Spread Spectrum Technology • Side effects: – coexistence of several signals without dynamic coordination – tap-proof • Alternatives: Direct Sequence (DS/SS), Frequency Hopping (FH/SS) • Spread spectrum increases BW of message signal by a factor Gain P ro cessin g G ain 1 0 lo g N   Spread Spectrum Technology coexistence of several signals without dynamic coordination Alternatives: Direct Sequence (DS/SS), Frequency Hopping (FH/SS) Spread spectrum increases BW of message signal by a factor N, Processing 1 0 P ro cessin g G ain 1 0 lo g ss ss B B N B B        
Effects of spreading and interference P f i) ii) sender P f iii) iv) receiver Effects of spreading and interference P f P f receiver f v) user signal broadband interference narrowband interference P
Spreading and frequency selective fading channel quality 1 2 3 4 5 Narrowband signal guard space channel quality 1 spread spectrum Spreading and frequency selective fading frequency 6 2 2 2 2 2 frequency narrowband channels spread spectrum channels
DSSS (Direct Sequence Spread Spectrum) I • XOR the signal with pseudonoise (PN) sequence (chipping sequence) • Advantages – reduces frequency selective fading – in cellular networks • base stations can use the same frequency range • several base stations can detect and recover the signal • But, needs precise power control DSSS (Direct Sequence Spread Spectrum) I XOR the signal with pseudonoise (PN) sequence (chipping sequence) user dat chippin sequen resulting signal 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 XOR 0 1 1 0 0 1 0 1 1 0 1 0 0 1 = Tb Tc
user data m(t) chipping sequence, c(t) X DSSS (Direct Sequence Spread Spectrum) II radio carrier Spread spectrum Signal y(t)=m(t)c(t) transmitter demodulator received signal radio carrier X Chipping sequence, c(t) receiver DSSS (Direct Sequence Spread Spectrum) II modulator radio carrier Signal y(t)=m(t)c(t) transmit signal X Chipping sequence, c(t) integrator products decision data sampled sums correlator
DS/SS Comments III Pseudonoise(PN) sequence chosen so that its autocorrelation is very narrow => PSD is very wide – Concentrated around t < Tc – Cross-correlation between two user’s codes is very small DS/SS Comments III Pseudonoise(PN) sequence chosen so that its autocorrelation is very narrow => PSD is very wide correlation between two user’s codes is very small
DS/SS Comments IV Secure and Jamming Resistant – Both receiver and transmitter must know c(t) – Since PSD is low, hard to tell if signal present – Since wide response, tough to jam everything Multiple access – If ci(t) is orthogonal to cj(t), then users do not interfere Near/Far problem – Users must be received with the same power DS/SS Comments IV Both receiver and transmitter must know c(t) Since PSD is low, hard to tell if signal present Since wide response, tough to jam everything (t), then users do not interfere Users must be received with the same power
FH/SS (Frequency Hopping Spread Spectrum) • Discrete changes of carrier frequency – sequence of frequency changes determined via PN sequence • Two versions – Fast Hopping: several frequencies per user bit (FFH) – Slow Hopping: several user bits per frequency (SFH) • Advantages – frequency selective fading and interference limited to short period – uses only small portion of spectrum at any time • Disadvantages – not as robust as DS/SS – simpler to detect FH/SS (Frequency Hopping Spread Spectrum) Discrete changes of carrier frequency sequence of frequency changes determined via PN sequence : several frequencies per user bit (FFH) : several user bits per frequency (SFH) frequency selective fading and interference limited to short period uses only small portion of spectrum at any time
FHSS (Frequency Hopping Spread Spectrum) I 0 1 Tb 0 f f1 f2 f3 Td f f1 f2 f3 Td Tb: bit period FHSS (Frequency Hopping Spread Spectrum) I user data slow hopping (3 bits/hop) fast hopping (3 hops/bit) 1 1 t t t Td: dwell time
FHSS (Frequency Hopping Spread Spectrum) III modulator user data narrowband transmitter received signal receiver hopping sequence demodulator frequency synthesizer FHSS (Frequency Hopping Spread Spectrum) III modulator hopping sequence modulator narrowband signal Spread transmit signal demodulator data frequency synthesizer
Applications of Spread Spectrum Cell phones – IS-95 (DS/SS) – GSM Global Positioning System (GPS) Wireless LANs – 802.11b Applications of Spread Spectrum Global Positioning System (GPS)
Performance of DS/SS Systems Pseudonoise (PN) codes – Spread signal at the transmitter – Despread signal at the receiver Ideal PN sequences should be – Orthogonal (no interference) – Random (security) – Autocorrelation similar to white noise (high at equal 0) Performance of DS/SS Systems Autocorrelation similar to white noise (high at t=0 and low for t not
PN Sequence Generation • Codes are periodic and generated by a shift register and XOR • Maximum-length (ML) shift register sequences, bits R( -1/n -nTc + PN Sequence Generation Codes are periodic and generated by a shift register and XOR length (ML) shift register sequences, m-stage shift register, length: n = 2m – 1 R(t) Tc t  nTc Output
Generating PN Sequences Take m=2 =>L=3 cn=[1,1,0,1,1,0, . . .], usually written as bipolar cn=[1,1,-1,1,1,-1, . . .] + Output                1 1 / 1 0 1 1 1 L m L m c c L m L n m n n Generating PN Sequences m Stages connected to modulo-2 adder 2 1,2 3 1,3 4 1,4 5 1,4 6 1,6 8 1,5,6,7
Problems with Cross-correlations with other m different input sequences can be quite high Easy to guess connection setup in secure In practice, Gold codes or Kasami sequences which combine the output of m-sequences are used. Problems with m-sequences m-sequences generated by different input sequences can be quite high Easy to guess connection setup in 2m samples so not too In practice, Gold codes or Kasami sequences which combine sequences are used.
Detecting DS/SS PSK Signals X Bipolar, NRZ m(t) PN sequence, c(t) sqrt(2)cos Spread spectrum Signal y(t)=m(t)c(t) transmitter X received signal X c(t) receiver z(t) sqrt(2)cos (wct + ) LPF x(t) Detecting DS/SS PSK Signals X )cos (wct + ) Signal y(t)=m(t)c(t) transmit signal integrator decision LPF w(t)
Optimum Detection of DS/SS PSK Recall, bipolar signaling (PSK) and white noise give the optimum error probability Not effected by spreading – Wideband noise not affected by spreading – Narrowband noise reduced by spreading b P Q  Optimum Detection of DS/SS PSK Recall, bipolar signaling (PSK) and white noise give the optimum error Wideband noise not affected by spreading Narrowband noise reduced by spreading 2 b b E P Q          
Signal Spectra • Effective noise power is channel noise power plus jamming (NB) signal power divided by N P ro cessin g G ain 1 0 lo g B B T N B B T    T Signal Spectra Effective noise power is channel noise power plus jamming (NB) 1 0 P ro cessin g G ain 1 0 lo g ss ss b c B B T B B T          Tb Tc
Multiple Access Performance Assume K users in the same frequency band, Interested in user 1, other users interfere 4 3 2 Multiple Access Performance users in the same frequency band, , other users interfere 1 5 6
BEC604 -COMMUNICATION ENG II.pdf
BEC604 -COMMUNICATION ENG II.pdf
BEC604 -COMMUNICATION ENG II.pdf

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BEC604 -COMMUNICATION ENG II.pdf

  • 1. UNIT UNIT 1 1: Sampling and : Sampling and Quantization Quantization : Sampling and : Sampling and Quantization Quantization
  • 2. Introduction Introduction • Digital representation of analog signals Analog signal source Waveform Coding (Codec) Analog-to-Digital Encoding Introduction Introduction Digital representation of analog signals Waveform Digital signal destination Digital Encoding
  • 3. Advantages of Digital Transmissions Advantages of Digital Transmissions Noise immunity Error detection and correction Ease of multiplexing Integration of analog and digital data Use of signal regenerators Data integrity and security Ease of evaluation and measurements More suitable for processing …….. Advantages of Digital Transmissions Advantages of Digital Transmissions Noise immunity Error detection and correction Ease of multiplexing Integration of analog and digital data Use of signal regenerators Data integrity and security Ease of evaluation and measurements More suitable for processing ……..
  • 4. Disadvantages Disadvantages of Digital Transmissions of Digital Transmissions More bandwidth requirement Need of precise time synchronization Additional hardware for encoding/decoding Integration of analog and digital data Sudden degradation in Incompatible with existing analog facilities of Digital Transmissions of Digital Transmissions More bandwidth requirement Need of precise time synchronization Additional hardware for encoding/decoding Integration of analog and digital data Sudden degradation in QoS Incompatible with existing analog facilities
  • 5. A Typical Digital Communication Link A Typical Digital Communication Link Fig. 2 Block Diagram A Typical Digital Communication Link A Typical Digital Communication Link Fig. 2 Block Diagram
  • 6.
  • 7. Basic Digital Communication Transformations – Formatting/Source Coding – Transforms source info into digital symbols (digitization) – Selects compatible waveforms (matching function) – Introduces redundancy which facilitates accurate decoding despite errors It is essential for reliable communication – Modulation/Demodulation – Modulation is the process of modifying the info signal to facilitate transmission – Demodulation reverses the process of modulation. It involves the detection and retrie of the info signal • Types • Coherent: Requires a reference info for detection • Noncoherent: Does not require reference phase information Basic Digital Communication Transformations Transforms source info into digital symbols (digitization) Selects compatible waveforms (matching function) Introduces redundancy which facilitates accurate decoding despite errors It is essential for reliable communication Modulation is the process of modifying the info signal to facilitate transmission Demodulation reverses the process of modulation. It involves the detection and retrie Coherent: Requires a reference info for detection Noncoherent: Does not require reference phase information
  • 8. Basic Digital Communication Transformations – Coding/Decoding Translating info bits to transmitter data symbols Techniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference) • Two Categories – Waveform Coding • Produces new waveforms with better performance – Structured Sequences Involves the use of redundant bits to determine the occurrence of error (and sometimes correct it) – Multiplexing/Multiple Access Is synonymous with resource sharing with other users – Frequency Division Multiplexing/Multiple Access (FDM/FDMA Basic Digital Communication Transformations Translating info bits to transmitter data symbols Techniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference) Produces new waveforms with better performance Involves the use of redundant bits to determine the occurrence of error (and Multiplexing/Multiple Access Is synonymous with resource sharing with other users Frequency Division Multiplexing/Multiple Access (FDM/FDMA
  • 9. Practical Aspects of Sampling Practical Aspects of Sampling 1. Sampling Theorem 2 .Methods of Sampling 3. Significance of Sampling Rate 4. Anti-aliasing Filter 5 . Applications of Sampling Theorem Practical Aspects of Sampling Practical Aspects of Sampling . Applications of Sampling Theorem – PAM/TDM
  • 10. Sampling Sampling is the processes of converting continuous by taking the “samples” at discrete-time intervals – Sampling analog signals makes them discrete in time but still continuous valued – If done properly (Nyquist theorem is satisfied), sampling does not introduce distortion Sampled values: – The value of the function at the sampling points Sampling interval: – The time that separates sampling points (interval b/w samples), – If the signal is slowly varying, then fewer samples per second will be required than if the wavef is rapidly varying – So, the optimum sampling rate depends on the signal Sampling is the processes of converting continuous-time analog signal, xa(t), into a discrete-time si time intervals Sampling analog signals makes them discrete in time but still continuous valued is satisfied), sampling does not introduce distortion The value of the function at the sampling points The time that separates sampling points (interval b/w samples), Ts If the signal is slowly varying, then fewer samples per second will be required than if the wavef So, the optimum sampling rate depends on the maximum frequency component present in the
  • 11. Analog-to-digital conversion is (basically) a 2 step process: – Sampling • Convert from continuous-time analog signal – Is obtained by taking the “samples” of xa(t) at discrete Quantization – Convert from discrete-time continuous valued signal to discrete time discrete valued signal step process: time analog signal xa(t) to discrete-time continuous value signal x at discrete-time intervals, Ts time continuous valued signal to discrete time discrete valued signal
  • 12. Sampling s f f  ling Rate (or sampling frequency fs): he rate at which the signal is sampled, expressed as the number of samples per second eciprocal of the sampling interval), 1/Ts = fs st Sampling Theorem (or Nyquist Criterion): the sampling is performed at a proper rate, no info is lost about the original signal and it can be operly reconstructed later on atement: “If a signal is sampled at a rate at least, but not exactly equal to onent of the waveform, then the waveform can be exactly reconstructed from the samples without any distortion” Sampling max 2 f f  he rate at which the signal is sampled, expressed as the number of samples per second the sampling is performed at a proper rate, no info is lost about the original signal and it can be “If a signal is sampled at a rate at least, but not exactly equal to twice the max frequency waveform can be exactly reconstructed from the samples
  • 13. ….. Sampling Theorem ….. Sampling Theorem Sampling Theorem for Bandpass Signal signal containing no frequency outside the specified bandwidth W Hz, it may be reconstructed from its samples at a sequence of points spaced 1/(2W) seconds apart with zero The minimum sampling rate of (2W) samples per second, for an analog signal bandwidth of W Hz, is called the Nyquist rate. The reciprocal of called the 1/( The phenomenon of the presence of high spectrum of the original analog signal is called aliasing or simply ….. Sampling Theorem ….. Sampling Theorem Signal - If an analog information signal containing no frequency outside the specified bandwidth W Hz, it may be reconstructed from its samples at a sequence of points W) seconds apart with zero-mean squared error. The reciprocal of Nyquist rate, 1/(2W), is called the Nyquist interval, that is, Ts = /(2W). The phenomenon of the presence of high-frequency component in the spectrum of the original analog signal is called aliasing or simply foldover.
  • 14. Sampling Theorem Sampling Theorem Sampling Theorem for Baseband Signal no frequency components higher than recovered from the knowledge of its samples taken at a rate of at least 2 fm samples per second, that is, sampling frequency The minimum sampling rate fs = 2 fm samples per second is called the Nyquist sampling rate. A baseband signal having no frequency components higher than completely described by its sample values at uniform intervals less than or equal to the sampling interval seconds. Sampling Theorem Sampling Theorem Sampling Theorem for Baseband Signal - A baseband signal having no frequency components higher than fm Hz may be completely recovered from the knowledge of its samples taken at a rate of at samples per second, that is, sampling frequency fs ≥ 2 fm. A baseband signal having no frequency components higher than fm Hz is completely described by its sample values at uniform intervals less than or equal to 1/(2fm) seconds apart, that is, the sampling interval Ts ≤ 1/(2fm) seconds.
  • 15. Methods of Sampling Methods of Sampling Ideal sampling - an impulse at each sampling instant Ideal Sampling Methods of Sampling Methods of Sampling Ideal Sampling
  • 16. Ideal Sampling ( or Impulse Sampling) Is accomplished by the multiplication of the signal function) Consider the instantaneous sampling of the analog signal Train of impulse functions select sample values at regular intervals Fourier Series representation: ( ) ( ) ( ) s s n x t x t t nT        1 2 ( ) , jn t s s n n s s t n T e T T                ( or Impulse Sampling) accomplished by the multiplication of the signal x(t) by the uniform train of impulses (comb Consider the instantaneous sampling of the analog signal x(t) Train of impulse functions select sample values at regular intervals ( ) ( ) ( ) s s x t x t t nT   1 2 ( ) , s jn t s s s s t n T e T T         
  • 17. Ideal Sampling ( or Impulse Sampling) his shows that the Fourier Transform of the sampled signal is the Fourier Transform of the original nal at rate of 1/Ts ( or Impulse Sampling) his shows that the Fourier Transform of the sampled signal is the Fourier Transform of the original
  • 18. Ideal Sampling ( or Impulse Sampling) As long as fs> 2fm,no overlap of repeated replicas X(f Minimum Sampling Condition: Sampling Theorem: A finite energy function ampled value x(nTs) with provided that => 2 ( ) sin 2 ( ) ( ) ( ) s s s n s f t n T T t T x n T t n T                               ( ) sin (2 ( )) s s s s n T x n T c f t n T        ( or Impulse Sampling) X(f - n/Ts) will occur in Xs(f) A finite energy function x(t) can be completely reconstructed from provided that => 2 s m m s m f f f f f     2 ( ) 2 ( ) s s s f t n T T t n T                       ( ) sin (2 ( )) s s s s T x n T c f t n T   1 1 2 s s m T f f  
  • 19. Ideal Sampling ( or Impulse Sampling) his means that the output is simply the replication of the original signal at discrete intervals, e.g ( or Impulse Sampling) his means that the output is simply the replication of the original signal at discrete intervals, e.g
  • 20. Ts is called the Nyquist interval: It is the longest time interval that can be used for sampling a ban signal and still allow reconstruction of the signal at the receiver without distortion It is the longest time interval that can be used for sampling a ban signal and still allow reconstruction of the signal at the receiver without distortion
  • 21. ….. Methods of Sampling ….. Methods of Sampling Natural sampling - a pulse of short width with varying amplitude with natural tops Natural Sampling ….. Methods of Sampling ….. Methods of Sampling Natural Sampling
  • 22. Natural Sampling Natural Sampling If we multiply x(t) by a train of rectang pulses xp(t), we obtain a gated wavefo that approximates the ideal sampled waveform, known as natural sampling gating (see Figure 2.8) ( ) ( ) ( ) s p x t x t x t  2 ( ) s j n f n n x t c e        ( ) [ ( ) ( ) ] s p X f x t x t   2 [ ( ) j n n c x t e         [ n s n c X f n f       
  • 23. Each pulse in xp(t) has width Ts and amplitude 1/T The top of each pulse follows the variation of the signal being sampled Xs (f) is the replication of X(f) periodically every fs Xs (f) is weighted by Cn  Fourier Series Coeffiecient The problem with a natural sampled waveform is that the tops of the sample pulses are not flat It is not compatible with a digital system since the amplitude of each sample has infinite number o possible values Another technique known as flat top sampling is used to alleviate this problem /Ts The top of each pulse follows the variation of the signal being sampled Hz Fourier Series Coeffiecient The problem with a natural sampled waveform is that the tops of the sample pulses are not flat It is not compatible with a digital system since the amplitude of each sample has infinite number o is used to alleviate this problem
  • 24. Flat-top sampling - a pulse of short width with varying amplitude with flat tops Flat-top Sampling ….. Methods of Sampling ….. Methods of Sampling top Sampling ….. Methods of Sampling ….. Methods of Sampling
  • 25. Flat-Top Sampling ere, the pulse is held to a constant height for the whole sample period at top sampling is obtained by the convolution of the signal obtained after ideal mpling with a unity amplitude rectangular pulse, is technique is used to realize Sample-and S/H, input signal is continuously sampled and then the value is held for as long as i kes to for the A/D to acquire its value Top Sampling ere, the pulse is held to a constant height for the whole sample period at top sampling is obtained by the convolution of the signal obtained after ideal mpling with a unity amplitude rectangular pulse, p(t) and-Hold (S/H) operation S/H, input signal is continuously sampled and then the value is held for as long as i
  • 26. Flat top sampling (Time Domain) '( ) ( ) ( ) x t x t t   ( ) '( ) * ( ) s x t x t p t  ( ) * ( ) ( ) ( ) * ( ) ( ) p t x t t p t x t t n T      Flat top sampling (Time Domain) ( ) * ( ) ( ) ( ) * ( ) ( ) s n p t x t t p t x t t n T                
  • 27. Taking the Fourier Transform will result to where P(f) is a sinc function ( ) [ ( ) ] s s X f x t   ( ) ( ) ( ) n P f x t t n T               ( ) ( ) * ( ) P f X f f n f          1 ( ) ( ) n s P f X f n f T        ( ) ( ) ( ) s n P f x t t n T                1 ( ) ( ) * ( ) s n s P f X f f n f T                ( ) ( ) s P f X f n f   
  • 28. Flat top sampling (Frequency Domain) Flat top sampling becomes identical to ideal sampling as the width of the pulses become shorter Flat top sampling (Frequency Domain) Flat top sampling becomes identical to ideal sampling as the width of the pulses become
  • 29. Recovering the Analog Signal One way of recovering the original signal from sampled signal ilter (LPF) as shown below f fs > 2B then we recover x(t) exactly Else we run into some problems and signal is not fully recovered Recovering the Analog Signal One way of recovering the original signal from sampled signal Xs(f) is to pass it through a Low Pass Else we run into some problems and signal is not fully recovered
  • 30. Significance of Sampling Rate When fs < 2fm, spectral components of adjacent samples will overlap, known as aliasing An Illustration of Aliasing Significance of Sampling Rate An Illustration of Aliasing
  • 31. Undersampling and Aliasing – If the waveform is undersampled (i.e. fs < 2B) then there will be signal he signal at the output of the filter will be different from the original signal spectrum This is the outcome of aliasing! his implies that whenever the sampling condition is not met, an irreversible overlap of the spectral licas is produced ) then there will be spectral overlap in the sample ! his implies that whenever the sampling condition is not met, an irreversible overlap of the spectral
  • 32. This could be due to: 1. x(t) containing higher frequency than were expected 2. An error in calculating the sampling rate Under normal conditions, undersampling of signals causing aliasing is not recommended containing higher frequency than were expected An error in calculating the sampling rate Under normal conditions, undersampling of signals causing aliasing is not recommended
  • 33. Solution 1: Anti-Aliasing Analog Filter – All physically realizable signals are not completely bandlimited – If there is a significant amount of energy in frequencies above half the sampling frequency (fs/2), aliasing will occur – Aliasing can be prevented by first passing the analog signal through an called a prefilter) before sampling is performed – The anti-aliasing filter is simply a LPF with cutoff frequency equal to half the sample rate All physically realizable signals are not completely bandlimited If there is a significant amount of energy in frequencies above half the sampling frequency Aliasing can be prevented by first passing the analog signal through an anti-aliasing filter (als ) before sampling is performed aliasing filter is simply a LPF with cutoff frequency equal to half the sample rate
  • 34. Antialiasing Filter An anti-aliasing filter is a low-pass filter of sufficient higher order which is recommended to be used prior to sampling. Minimizing Aliasing Antialiasing Filter Minimizing Aliasing
  • 35. • Aliasing is prevented by forcing the bandwidth of the sampled signal to satisfy the requirement of the Sampling Theorem Aliasing is prevented by forcing the bandwidth of the sampled signal to satisfy the requirement
  • 36. Solution 2: Over Sampling and Filtering in the Digital Domain – The signal is passed through a low performance (less costly) analog low the bandwidth. – Sample the resulting signal at a high sampling frequency. – The digital samples are then processed by a high performance digital filter and down sample the resulting signal. : Over Sampling and Filtering in the Digital Domain The signal is passed through a low performance (less costly) analog low-pass filter to lim Sample the resulting signal at a high sampling frequency. The digital samples are then processed by a high performance digital filter and down
  • 37. Summary Of Sampling Ideal Sampling (or Impulse Sampling) Natural Sampling (or Gating) Flat-Top Sampling For all sampling techniques – If fs > 2B then we can recover x(t) exactly – If fs < 2B) spectral overlapping known as aliasing ( ) ( ) ( ) ( ) ( ) s s x t x t x t x t t n T    ( ) ( ) ( ) ( ) s p n x t x t x t x t c e   ( ) '( ) * ( ) ( ) ( ) * s s x t x t p t x t t n T p    Summary Of Sampling liasing will occur ( ) ( ) ( ) ( ) ( ) ( ) ( s s n s n x t x t x t x t t n T x n T t n T                   2 ( ) ( ) ( ) ( ) j n s p n n x t x t x t x t c e         ( ) '( ) * ( ) ( ) ( ) * s s n x t x t p t x t t n T p               
  • 38. Quantization Quantization is a non linear transformation which maps elements from a continuous o a finite set. It is also the second step required by A/D conversion. Sample og Signal ontinuous time ontinuous value - Discrete time - Continuous value Quantization Quantization is a non linear transformation which maps elements from a continuous o a finite set. It is also the second step required by A/D conversion. Quantize Digital Signal - Discrete time - Discrete value Discrete time Continuous value
  • 39. Uniform Quantization on of operation output -V V -V Uniform Quantization For M=2n levels, step size :  = 2V /2n = V(2-n+1) input w1(t) output w2(t) V
  • 40. Figure 3.10 Two types of quantization: ( Two types of quantization: (a) midtread and (b) midrise.
  • 41. output -V V -V Error, /2 -/2 Quantization Error, e input w1(t) output w2(t) V input w1(t) Error, e
  • 43. Definition. The dynamic range of an input signal is the ratio of the largest to the smallest power levels which the input signal can take on and be reproduced with the acceptable signal distortion. The dynamic range of the quantizer input in the PCM system is Definition. The dynamic range of an input signal is the ratio of the largest to the smallest power levels which the input signal can take on and be reproduced The dynamic range of the quantizer input in the PCM system is 6n dB.
  • 44. Nonuniform Quantizer Used to reduce quantization error and increase the dynamic range when input signa not uniformly distributed over its allowed range of values. owed lues for Used to reduce quantization error and increase the dynamic range when input signa not uniformly distributed over its allowed range of values. t input
  • 45. Nonuniform quantizer Compressor rete ples pressing-and-expanding” is called “companding.” • • • • Decoder ved al signals Nonuniform quantizer Uniform Quantizer digital expanding” is called “companding.” Channel Expander outp
  • 46. 1. Unipolar nonreturn-to-zero (NRZ) Signaling 2. Polar nonreturn-to-zero(NRZ) Signaling 3. Unipor nonreturn-to-zero (RZ) Signaling 4. Bipolar nonreturn-to-zero (BRZ) Signaling 5. Split-phase (Manchester code) Line codes: zero (NRZ) Signaling zero(NRZ) Signaling zero (RZ) Signaling zero (BRZ) Signaling phase (Manchester code) Line codes:
  • 47. Figure 3.15 Line codes for the electrical representations of binary data. (a) Unipolar NRZ signaling. (b) Polar NRZ signaling. (c) Unipolar RZ signaling. (d) Bipolar RZ signaling. (e) Split-phase or Manchester code. Line codes for the electrical representations of binary data. ) Polar NRZ signaling. ) Bipolar RZ signaling.
  • 48. Application of Sampling Theorem PAM/TDM Design of PAM/TDM System Application of Sampling Theorem – PAM/TDM
  • 49. UNIT-II DIGITAL MODULATION II DIGITAL MODULATION
  • 50. Pulse Code Modulation (PCM) Pulse Code Modulation (PCM) 1. Block Diagram of PCM 2 PCM Sampling 3 Quantization of Sampled Signal 4 Encoding of Quantized Sampled Signal Pulse Code Modulation (PCM) Pulse Code Modulation (PCM) 3 Quantization of Sampled Signal Encoding of Quantized Sampled Signal
  • 51. The basic elements of a PCM system. Pulse Code Modulation The basic elements of a PCM system. Pulse Code Modulation
  • 52. PCM Sampling PCM Sampling he Process of Natural Sampling PCM Sampling PCM Sampling
  • 53. Quantization of Quantization of peration of uantization VL L01 L12 L34 L23 L45 L56 L67 VH s(t) sq(t) sq(t) Quantization of Quantization of Sampled Signal Sampled Signal t Δ Δ/2 Δ0 Δ1 Δ2 Δ3 Δ4 Δ5 Δ6 Δ7 s0 s1 s2 s3 s4 s5 s6 s7 s(t)
  • 54. Quantization is the conversion of an analog sample of the information signal into discrete form. Thus, an infinite number of possible levels are converted to a finite number of conditions. Quantization error difference between rounding off sample values of an analog signal to the nearest permissible level of the the process of quantization. Classification of Quantization Quantization Error and Classification Quantization Error and Classification Quantization error is defined as the difference between rounding off sample values of an analog signal to the nearest permissible level of the quantizer during the process of quantization. Classification of Quantization Process Uniform quantization Non-uniform quantization Quantization Error and Classification Quantization Error and Classification
  • 55. Characteristics of Compressor, Uniform Characteristics of Compressor, Uniform and Non and Non- -uniform Quantizer uniform Quantizer Characteristics of Compressor, Uniform Characteristics of Compressor, Uniform uniform Quantizer uniform Quantizer
  • 56. μ μ- -law law and A and A- -law Compression law Compression Characteristics Characteristics law Compression law Compression Characteristics Characteristics
  • 57. Encoding of Quantized Sampled Signal Encoding of Quantized Sampled Signal PCM – Functional Blocks Encoding of Quantized Sampled Signal Encoding of Quantized Sampled Signal Functional Blocks
  • 58. PCM Data Rate (bps) = 2nfm PCM Bandwidth (Hz) = (1/2) PCM Data Rate = Dynamic Range (dB) = 20 log (2n – Coding Efficiency (%) = [(minimum bits)/(actual bits)] x Where n is number of PCM encoding bits and frequency component of information signal PCM System Parameters PCM System Parameters ) PCM Data Rate = nfm 1) Coding Efficiency (%) = [(minimum bits)/(actual bits)] x 100 is number of PCM encoding bits and fm is the highest frequency component of information signal PCM System Parameters PCM System Parameters
  • 60. Delta modulation (DM) uses a single digital transmission of analog signals Essence of Delta Modulation (DM) Essence of Delta Modulation (DM) An Ideal Delta Modulation Waveform Delta modulation (DM) uses a single-bit DPCM code to achieve digital transmission of analog signals Essence of Delta Modulation (DM) Essence of Delta Modulation (DM) An Ideal Delta Modulation Waveform
  • 61. DM system. (a) Transmitter. ( ) Transmitter. (b) Receiver.
  • 62. lator consists of a comparator, a quantizer, and an accumulator utput of the accumulator is       ) sgn( 1 1        n i q n i q i e i e n m wo types of quantization errors : ope overload distortion and granular noise lator consists of a comparator, a quantizer, and an accumulator (3.55) )
  • 63. a-Sigma modulation (sigma-delta modulation) modulation which has an integrator eve the draw back of delta modulation (differentiator eficial effects of using integrator: Pre-emphasize the low-frequency content Increase correlation between adjacent samples educe the variance of the error signal at the quantizer input ) Simplify receiver design ause the transmitter has an integrator , the receiver sists simply of a low-pass filter. differentiator in the conventional DM receiver is cancelled by the integrator )    delta modulation) integrator can differentiator) frequency content Increase correlation between adjacent samples educe the variance of the error signal at the quantizer input ) ause the transmitter has an integrator , the receiver differentiator in the conventional DM receiver is cancelled by the integrator )
  • 64. ivalent versions of delta-sigma modulation system.
  • 65. Differential Pulse-Code Modulation (DPCM) PCM has the sampling rate higher than the Nyquist rate ciently remove this redundancy. Figure 3.28 DPCM system. (a) Transmitter. (b) Receiver. Code Modulation (DPCM) rate .The encode signal contains redundant information. D ) Receiver.
  • 66. Adaptive Differential Pulse- eed for coding speech at low bit rates , we have two aims in mind: Remove redundancies from the speech signal as far as possible. Assign the available bits in a perceptually efficient manner. Figure 3.29 Adaptive quantization with backward estimation (AQB). Adaptive prediction with backward estimation (APB). -Code Modulation (ADPCM) eed for coding speech at low bit rates , we have two aims in mind: Remove redundancies from the speech signal as far as possible. Assign the available bits in a perceptually efficient manner. Adaptive quantization with backward estimation (AQB). prediction with backward estimation (APB).
  • 67. S. No. Parameter PCM 1. Number of bits per sample 4/8/16 bits More than one bit but less than PCM 2. Number of levels Depends on number of bits Fixed number of levels 3. Step size Fixed or variable Fixed or variable 4. Transmission bandwidth More bandwidth needed Lesser than PCM 5. Feedback Does not exist Exists 6. Quantization noise/distortion Quantization noise depends on number of bits Quantization noise & slope overload 7. Complexity of implementation Complex Simple Comparison of PCM and DM Techniques Comparison of PCM and DM Techniques DPCM DM ADM More than one bit but less than PCM One bit One bit Fixed number of levels Two levels Two levels Fixed or variable Fixed Variable Lesser than PCM Lowest Lowest Exists Exists Exists Quantization noise & slope overload slope overload & granular noise Quantization noise only Simple Simple Simple Comparison of PCM and DM Techniques Comparison of PCM and DM Techniques
  • 69. Transmit and Receive Formatting Transmit and Receive Formatting
  • 70. Sources of Error in received Signal Major sources of errors: – Thermal noise (AWGN) • disturbs the signal in an additive fashion (Additive) • has flat spectral density for all frequencies of interest (White) • is modeled by Gaussian random process (Gaussian Noise) – Inter-Symbol Interference (ISI) • Due to the filtering effect of transmitter, channel and receiver, symbols are “smeared”. Sources of Error in received Signal disturbs the signal in an additive fashion (Additive) has flat spectral density for all frequencies of interest (White) is modeled by Gaussian random process (Gaussian Noise) Due to the filtering effect of transmitter, channel and receiver, symbols are
  • 71. F R E Q U E N C Y D O W N C O N V E R S IO N A N S M I T T E D A V E F O R M A W G N R E C E I V E D W A V E F O R M R E C E IV IN G F IL T E R F O R B A N D P A S S S I G N A L S D E M O D U L A T E & S A M P L E Demodulation/Detection of digital signals Receiver Structure R E C E IV IN G E Q U A L IZ IN G F IL T E R T H R E S H O L D C O M P A R IS O N C O M P E N S A T I O N F O R C H A N N E L I N D U C E D I S I D E M O D U L A T E & S A M P L E S A M P L E a t t = T D E T E C T M E S S A S Y M B O O R C H A N N S Y M B O E S S E N T IA L O P T IO N A L Demodulation/Detection of digital signals Receiver Structure
  • 72. Receiver Structure contd The digital receiver performs two basic functions – Demodulation – Detection Why demodulate a baseband signal??? – Channel and the transmitter’s filter causes ISI which “smears” the transmitted pulses – Required to recover a waveform to be sampled at t = nT. Detection – decision-making process of selecting possible digital symbol Receiver Structure contd two basic functions: Why demodulate a baseband signal??? Channel and the transmitter’s filter causes ISI which “smears” the Required to recover a waveform to be sampled at t = nT. making process of selecting possible digital symbol
  • 73. Steps in designing the receiver Find optimum solution for receiver design with the following goals: 1. Maximize SNR 2. Minimize ISI Steps in design: – Model the received signal – Find separate solutions for each of the goals. Steps in designing the receiver Find optimum solution for receiver design with the following goals:
  • 74. Detection of Binary Signal in Gaussian Noise covery of signal at the receiver consist of two parts ilter • Reduces the received signal to a single variable z(T) • z(T) is called the test statistics Detector (or decision circuit) • Compares the z(T) to some threshold level 0 , i.e., where H1 and H0 are the two possible binary hypothesis 0 ) ( 0 1  H H T z   Detection of Binary Signal in Gaussian Noise are the two
  • 75. Finding optimized filter for AWGN channel Assuming Channel with response equal to impulse function Finding optimized filter for AWGN channel Assuming Channel with response equal to impulse function
  • 76. Detection of Binary Signal in Gaussian Noise • For any binary channel, the transmitted signal over a symbol interval ( • The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse resp the channel hc(t), is Where n(t) is assumed to be zero mean AWGN process • For ideal distortionless channel where hc(t) is an impulse function and convolution with produces no degradation, r(t) can be represented as:         0 ) ( 0 ) ( ) ( 1 0 T t t s T t t s t si ) ( * ) ( ) (   t h t s t r c i t n t s t r i   ) ( ) ( ) ( Detection of Binary Signal in Gaussian Noise For any binary channel, the transmitted signal over a symbol interval (0,T) is: The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse resp Where n(t) is assumed to be zero mean AWGN process (t) is an impulse function and convolution with produces no degradation, r(t) can be represented as: 1 0 binary a for binary a for 2 , 1 ) (  i t n T t i    0 2 , 1 )
  • 77. Design the receiver filter to maximize the SNR Model the received signal Simplify the model: – Received signal in AWGN ) (t hc ) t ) (t n ) (t r ) (t n ) (t si Ideal channels ) ( ) ( t t hc   AWGN AWGN Design the receiver filter to maximize the SNR ) (t r ) ( ) ( ) ( ) ( t n t h t s t r c i    ) ( ) ( ) ( t n t s t r i  
  • 78. Find Filter Transfer Function H ctive: To maximizes (S/N)T and find h(t) sing signal ai(t) at filter output in terms of filter transfer function H(f) the filter transfer funtion and S(f) is the Fourier transform of input signal s(t) ded PSD of i/p noise is N0/2 t noise power can be expressed as: (S/N)T : f H t a i      ( ) (      H N 0 2 0 | 2              N f H N S T 0 2 ( Find Filter Transfer Function H0(f) (t) at filter output in terms of filter transfer function H(f) the filter transfer funtion and S(f) is the Fourier transform of input signal s(t) df e f S ft j  2 ) ( ) df f H 2 | ) (     df f H df e f S f fT j 2 2 2 | ) ( | ) ( ) 
  • 79. • For H(f) = Hopt (f) to maximize (S/N)T use Schwarz’s Inequality • Equality holds if f1(x) = k f*2(x) where k is arbitrary constant and * indicates complex conjugate • Associate H(f) with f1(x) and S(f) ej2 fT with f2 • Substitute yields to: df e f S f H fT j 2 2 ) ( ) (      dx x f x f 2 2 1 ) ( ) (          N N S T 0 2        Schwarz’s Inequality: (x) where k is arbitrary constant and * indicates complex conjugate 2(x) to get: df f S df f H 2 2 ) ( ) (          dx x f dx x f 2 2 2 1 ) ( ) (     df f S 2 ) (    
  • 80. and energy E of the input signal s(t): s (S/N)T depends on input signal energy power spectral density of noise and T on the particular shape of the waveform lity for holds for optimum filter transfer function H that: eal valued s(t): 0 2 max N E N S T        2 max N N S T        f H f H 0 ( ) (  kS t h 1 ) (        t h 0 ) ( and energy E of the input signal s(t): depends on input signal energy on the particular shape of the waveform holds for optimum filter transfer function H0(f) (3.55) df f S E 2 ) (      0 N E fT j e f kS  2 ) ( * )     fT j e f kS  2 ) ( *     where else T t t T kS 0 0 ) (
  • 81. mpulse response of a filter producing maximum output signal or image of message signal s(t), delayed by symbol time duration T. filter designed is called a MATCHED FILTER ned as: a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform      T kS t h 0 ( ) ( mpulse response of a filter producing maximum output signal-to-noise ratio is the or image of message signal s(t), delayed by symbol time duration T. MATCHED FILTER a linear filter designed to provide the maximum noise power ratio at its output for a given    where else T t t 0 )
  • 82. Matched Filter Output of a rectangular Pulse Matched Filter Output of a rectangular Pulse
  • 83. Replacing Matched filter with Integrator Replacing Matched filter with Integrator
  • 84. Implementation of matched filter receiver ) (t r ( 1 T z ) ( * 1 t T s  ) ( * t T s M  (T z M Bank of M matched filters ) ( ) ( t T s t r z i i     M i ,..., 1  ( )) ( ),..., ( ), ( ( 2 1 M T z T z T z   z Implementation of matched filter receiver           M z z  1 z  ) T ) T z Matched filter output: Observation vector M ) ,..., , ( 2 1 M z z z
  • 85. Detection Max. Likelihood Detector Probability of Error Detection Max. Likelihood Detector Probability of Error
  • 86. Detection ed filter reduces the received signal to a single variable z(T), after which the detection of symbol is carried out ncept of maximum likelihood detector is based on Statistical Decision Theory ws us to mulate the decision rule that operates on the data imize the detection criterion 0 ) ( 0 1  H H T z   z(T), after which the detection of symbol is carried out is based on Statistical Decision Theory
  • 87. P[s1]  a priori probabilities hese probabilities are known before transmission obability of the received sample ), p(z|s1) nditional pdf of received signal z, conditioned on the class s ], P[s1|z]  a posteriori probabilities fter examining the sample, we make a refinement of our previous knowledge 0], P[s0|s1] rong decision (error) 1], P[s0|s0] rrect decision Probabilities Review nditional pdf of received signal z, conditioned on the class si fter examining the sample, we make a refinement of our previous knowledge Probabilities Review
  • 88. mum Likelihood Ratio test and Maximum a posteriori ( em is that a posteriori probabilities are not known. on: Use Bay’s theorem: How to Choose the threshold? 1 0 ) | ( ) | ( z s p z s p    0 0 1 1 0 1 ) ( ) ( ) | ( ) ( ) ( ) | ( H H z P s P s z p z P s P s z p    0 1 ) | ( ) | ( z s p z s p    | ( ) | ( p s z p z i s p  s means that if received signal is positive, s1 (t) was sent s0 (t) was sent mum Likelihood Ratio test and Maximum a posteriori (MAP) criterion: How to Choose the threshold? 0 H  ) ( ) | ( ) ( ) | ( 0 0 1 1 0 1 ) s P s z p s P s z p H H    1 H  ) ( ) ( ) z p i s p i s (t) was sent, else
  • 89. Likelihood of So and S 1 and S1
  • 90. MAP criterion: ) ( ) ( ) | ( ) | ( ) ( 1 0 0 1 0 1 s P s P s z p s z p z L H H     When the two signals, s0(t) and s1(t), are equally likely, i.e., becomes s is known as maximum likelihood ratio test because we are selecting hypothesis that corresponds to the signal with the maximum likelihood. erms of the Bayes criterion, it implies that the cost of both types of error is the same H H s z p s z p z L max 1 ) | ( ) | ( ) ( 0 1 0 1     ) ( LRT test ratio likelihood , are equally likely, i.e., P(s0) = P(s1) = 0.5, then the decision rule because we are selecting hypothesis that corresponds to the signal with the maximum likelihood. erms of the Bayes criterion, it implies that the cost of both types of error is the same test ratio likelihood max
  • 91. tuting the pdfs  0 0 0 2 1 ) | ( :   s z p H  0 1 1 2 1 ) | ( :   s z p H 2 1 2 1 1 ) | ( ) | ( ) ( 0 0 0 1 0 1 H H s z p s z p z L                           2 0 0 2 1 exp   a z                   2 0 1 2 1 exp  a z     1 2 1 exp 2 1 exp 0 1 2 0 2 0 2 1 2 H H a z a z o                      
  • 92. ) ( )} ( ln{ 2 0 0 1 a a z z L    2 1 0 1 2 0 0 1 2 ( ) (   a H H a a z      ) ( exp 2 0 0 1       a a z Hence: Taking the log, both sides will give 0 2 ) ( 0 1 2 0 2 0 2 1 H H a a      2 0 0 1 0 1 2 0 2 0 2 ) )( ( )   a a a a a     1 2 ) ( 2 0 2 0 2 1         a a
  • 93. • Hence where z is the minimum error criterion and  0 is optimum threshold • For antipodal signal, s1(t) = - s0 (t)  a1 = - a0 ) ( 2 ) )( ( 0 1 2 0 0 1 0 1 2 0 0 1 a a a a a a H H z        0 0 1 H H z   ptimum threshold 0 0 1 0 1 2 ) (      a a H H z 0
  • 94. Probability of Error will occur if s sent  s0 is received s sent  s1 is received The total probability of error is sum of the errors dz s z p s e P s e P s H P      0 ) | ( ) | ( ) | ( ) | ( 1 1 1 1 0  dz s z p s e P s e P s H P     0 ) | ( ) | ( ) | ( ) | ( 0 0 0 0 1  ( ) ( ) | ( ) | ( ) , ( 1 1 0 1 2 1 H P s P s H P P s e P s e P P i i B       Probability of Error ) ( ) | ) ( ) | ( ) ( 0 0 1 0 0 1 s P s s P s e P s 
  • 95. nals are equally probable e, the probability of bit error PB, is the probability that an incorrect hypothesis is made rically, PB is the area under the tail of either of the conditional distributions  ) | ( 2 1 ) ( ) | ( 1 0 1 1 0 P s H P s P s H P PB     s H P P B       0 0 1 exp 2 1 ) | ( 0 0     is the probability that an incorrect hypothesis is made is the area under the tail of either of the conditional distributions p(z|s1) or p(z|s0)  ) | ( ) ( ) | ( 0 1 0 0 1 s H s P s H P  dz a z dz s z p dz                      2 0 0 0 2 1 exp ) | ( 0  
  • 96. Inter-Symbol Interference (ISI) SI in the detection process due to the filtering effects of the ystem Overall equivalent system transfer function – creates echoes and hence time dispersion – causes ISI at sampling time ) ( ) ( H f H f H t  k k k n s z   Symbol Interference (ISI) SI in the detection process due to the filtering effects of the Overall equivalent system transfer function creates echoes and hence time dispersion ) ( ) ( f H f H r c i k i i k s    
  • 97. Inter-symbol interference Baseband system model Equivalent model Tx filter Channel 2 x 3 x T T ) ( ) ( f H t h t t H h c c Equivalent system 2 x 3 x T ) ( ) ( f H t h ( ) ( ) ( ) ( f H f H f H f H r c t  symbol interference Channel ) (t n ) (t r Rx. filter Detector k z kT t  ) ( ) ( f H t h r r ) ( ) ( f t c c ) ( ˆ t n ) (t z Detector k z kT t  filtered noise ) f
  • 98. Nyquist bandwidth constraint Nyquist bandwidth constraint: • The theoretical minimum required system bandwidth to detect without ISI is Rs/2 [Hz]. • Equivalently, a system with bandwidth transmission rate of 2W=1/T=Rs [symbols/s] without ISI. Bandwidth efficiency, R/W [bits/s/Hz] : • An important measure in DCs representing data throughput per hertz of bandw • Showing how efficiently the bandwidth resources are used by signaling techniq 2 2 1     W R W R T s s Nyquist bandwidth constraint The theoretical minimum required system bandwidth to detect Rs [symbols/s] Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz] can support a maxim [symbols/s] without ISI. [bits/s/Hz] : An important measure in DCs representing data throughput per hertz of bandw Showing how efficiently the bandwidth resources are used by signaling techniq Hz] [symbol/s/ 2
  • 99. Ideal Nyquist pulse (filter) T 2 1 T 2 1  T ) ( f H f 0 T W 2 1  Ideal Nyquist filter Ideal Nyquist pulse (filter) ) / sinc( ) ( T t t h  1 0 T T 2 T  T 2  Ideal Nyquist pulse
  • 100. Nyquist pulses (filters) Nyquist pulses (filters): – Pulses (filters) which results in no ISI at the Nyquist filter: – Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even Nyquist pulse: – Its shape can be represented by a time function. Example of Nyquist filters: Raised Nyquist pulses (filters) Pulses (filters) which results in no ISI at the sampling time. Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even-symmetric frequency function Its shape can be represented by a sinc(t/T) function multiply by another filters: Raised-Cosine filter
  • 101. Pulse shaping to reduce ISI Goals and trade-off in pulse-shaping – Reduce ISI – Efficient bandwidth utilization – Robustness to timing error (small side lobes) Pulse shaping to reduce ISI shaping Robustness to timing error (small side lobes)
  • 102. The raised cosine filter aised-Cosine Filter – A Nyquist pulse (No ISI at the sampling time)               W W W f f H 0 2 | | 4 cos 1 ) ( 0 2  Excess bandwidth: 0 W W  0 0 2 (sinc( 2 ) ( t W W t h  The raised cosine filter A Nyquist pulse (No ISI at the sampling time)          W f W f W W W W W f | | for | | 2 for 2 2 | | for 0 0 0 Roll-off factor 0 0 W W W r   1 0   r 2 0 0 ] ) ( 4 [ 1 ] ) ( 2 cos[ )) t W W t W W t    
  • 103. The Raised cosine filter 2 ) 1 ( Baseband sSB s R r W   | ) ( | | ) ( | f H f H RC   r 0  r  r T 2 1 T 4 3 T 1 T 4 3  T 2 1  T 1  1 0.5 0 The Raised cosine filter – cont’d r 5 . 0  r 1  r 1 5 . 0 0 ) ( ) ( t h t h RC  1 0.5 0 T T 2 T  T 2  T 3  s R r W ) 1 ( Passband DSB  
  • 104. Pulse shaping and equalization to remove ISI quare-Root Raised Cosine (SRRC) filter and Equalizer ) ( ) ( RC H f H f H t  No ISI at the sampling time ) ( ) ( ) ( ) ( ) ( ) ( RC RC f H f H f H f H f H f H t r r t     ) ( 1 ) ( f H f H c e  Pulse shaping and equalization to remove ISI Root Raised Cosine (SRRC) filter and Equalizer ) ( ) ( ) ( f H f H f H e r c No ISI at the sampling time ) ( SRRC f H  Taking care of ISI caused by tr. filter Taking care of ISI caused by channel
  • 105. Example of pulse shaping Square-root Raised-Cosine (SRRC) pulse shaping mp. [V] Data symbol First pulse Second pulse Example of pulse shaping Cosine (SRRC) pulse shaping t/T Baseband tr. Waveform Second pulse Third pulse
  • 106. Example of pulse shaping … Raised Cosine pulse at the output of matched filter Amp. [V] Example of pulse shaping … Raised Cosine pulse at the output of matched filter t/T Baseband received waveform at the matched filter output (zero ISI)
  • 107. Eye pattern Eye pattern:Display on an oscilloscope which sweeps the system response to a baseband signal at the rate 1/T (T symbol duration) amplitude scale istortion due to ISI Timing jitter Eye pattern Display on an oscilloscope which sweeps the system response to symbol duration) time scale Noise margin Sensitivity t timing erro
  • 108. Example of eye pattern: Binary-PAM, SRRQ pulse Perfect channel (no noise and no ISI) Example of eye pattern: PAM, SRRQ pulse Perfect channel (no noise and no ISI)
  • 109. Correlative Coding Transmit 2W symbols/s with zero ISI, using the theoretical minimum bandwidth of W Hz, without infinitely sharp filters. Correlative coding (or duobinary signaling or partial response signaling) introduces some controlled amount of ISI into the data stream rather than trying to eliminate ISI completely Doubinary signaling Correlative Coding W symbols/s with zero ISI, using the theoretical minimum bandwidth of W Hz, without Correlative coding (or duobinary signaling or partial response signaling) introduces some controlled amount of ISI into the data stream rather than trying to eliminate ISI completely
  • 110. Duobinary signaling Duobinary signaling (class I partial response) Duobinary signaling Duobinary signaling (class I partial response)
  • 111. Duobinary signal and Nyguist Criteria Nyguist second criteria: but twice the bandwidth Duobinary signal and Nyguist Criteria Nyguist second criteria: but twice the bandwidth
  • 112. Differential Coding The response of a pulse is spread over more than one signalin interval. The response is partial in any signaling interval. Detection : – Major drawback : error propagation. To avoid error propagation, need deferential coding (precoding). Differential Coding The response of a pulse is spread over more than one signalin The response is partial in any signaling interval. Major drawback : error propagation. To avoid error propagation, need deferential coding
  • 113. Modified duobinary signaling Modified duobinary signaling – In duobinary signaling, H(f) is nonzero at the origin. – We can correct this deficiency by using the class IV partial response Modified duobinary signaling In duobinary signaling, H(f) is nonzero at the origin. We can correct this deficiency by using the class IV partial response
  • 115. Modified duobinary signaling Time Sequency: interpretation of receiving Modified duobinary signaling Time Sequency: interpretation of receiving 2, 0, and -2?
  • 117. Comparison of Binary with Duobinary Signaling Binary signaling assumes the transmitted pulse amplitude are independent of one another Duobinary signaling introduces correlation between pulse amplitudes Duobinary technique achieve zero ISI signal transmission using a smaller system bandwidth Duobinary coding requires three levels, compared with the usual two levels for binary coding Duobinary signaling requires more power than binary signaling (~ signaling) Comparison of Binary with Duobinary Signaling Binary signaling assumes the transmitted pulse amplitude are independent of one another Duobinary signaling introduces correlation between pulse amplitudes Duobinary technique achieve zero ISI signal transmission using a smaller system bandwidth Duobinary coding requires three levels, compared with the usual two levels for binary coding Duobinary signaling requires more power than binary signaling (~2.5 dB greater SNR than binary
  • 118. Pass-band Data Transmission band Data Transmission
  • 119. Block Diagram Functional model of pass-band data transmission system. Block Diagram band data transmission system.
  • 120. Signaling Illustrative waveforms for the three basic forms of signaling binary information. (a) Amplitude-shift keying. (b) Phase-shift keying. ( continuous phase. Signaling Illustrative waveforms for the three basic forms of signaling binary information. shift keying. (c) Frequency-shift keying with
  • 121. What do we want to study? We are going to study and compare different modulation techniques in terms of – Probability of errors – Power Spectrum – Bandwidth efficiency B R b   Bits/s/Hz What do we want to study? We are going to study and compare different modulation Bits/s/Hz
  • 122. Coherent PSK Binary Phase Shift Keying (BPSK) – Consider the system with 2 basis functions – and   T t b  2 cos 2 1    T t b  2 sin 2 2  Coherent PSK basis functions t f c  2 t f c  2
  • 123. BPSK If we want to fix that for both symbols ( equal, we have 2 We place s0 to minimize probability of error s0 s0 BPSK If we want to fix that for both symbols (0 and 1) the transmitted energies are 1 s1
  • 124. BPSK We found that phase of s1 and We can rotate s1 and s0  s0 Rotate BPSK and s0 are 180 degree difference. 1 2 s1
  • 125. BPSK We observe that 2 has nothing to do with signals. Hence, only one basis function is sufficient to represent the signals 2 s0 BPSK has nothing to do with signals. Hence, only one basis function is sufficient to represent the signals 2 s1 1
  • 126. BPSK Finally, we have   t E t s b  ) ( 1 1    t E t s b  ( 1 0   BPSK t f T E c b b  2 cos 2  t f T E t c b b  2 cos 2 )  
  • 127. BPSK Signal-space diagram for coherent binary PSK system. The waveforms depicting the transmitted signals s1(t) and s2(t), displayed in the inserts, assume BPSK space diagram for coherent binary PSK system. The waveforms depicting ), displayed in the inserts, assume nc  2.
  • 128. BPSK • Probability of error calculation. In the case of equally likely (Pr(m0)=Pr(m1)), we have           erfc 2 1 2 erfc 2 1 Pe BPSK Probability of error calculation. In the case of equally likely         0 0 2 N E N d b ik
  • 129. BPSK Block diagrams for (a) binary PSK transmitter and ( receiver. BPSK ) binary PSK transmitter and (b) coherent binary PSK
  • 130. Quadriphase-Shift Keying (QPSK)    i t f T E t s c i      2 2 cos 2  T is symbol duration E is signal energy per symbol There are 4 symbols for i = 1, 2, 3, and Shift Keying (QPSK)  T t       0 ; 4 1  , and 4
  • 131. QPSK          E t i E T i E t si                  4 1 2 cos 2 cos 2 4 1 2 cos 1                1 2 sin 1 2 cos i E i E i s Which we can write in vector format as QPSK          T t t i t f T i E t f c c                  0 ; 4 1 2 sin 2 sin 2 4 1 2 sin 2             4 1 4   Which we can write in vector format as
  • 132. QPSK i Input Dibit Phase of QPSK signaling 1 10 2 00 3 01 4 11 4 /  4 / 3 4 / 5 / 7 QPSK Phase of QPSK signaling Coordinate of Message point si1 si2 4 4 2 / E 2 / E 2 / E 2 / E 2 / E  2 / E  2 / E  2 / E 
  • 135. QPSK Block diagrams of (a) QPSK transmitter and (b) coherent QPSK receiver. QPSK
  • 136. QPSK: Error Probability QPSK Consider signal constellation given in the figure (10 Z3 Z2 E  QPSK: Error Probability QPSK 2 1 s4 s1 s3 s2 (10) (00) 10) (11) Z4 Z1 2 / E 2 / E 2 / E 2 / E 
  • 137. QPSK       12 2 erfc 2 1 N d P can treat QPSK as the combination of K over the interval T=2Tb ce the first bit is transmitted by nsmitted by 2. bability of error for each channel is given by QPSK              0 0 12 2 erf 2 1 N E c N can treat QPSK as the combination of 2 independent ce the first bit is transmitted by 1 and the second bit is bability of error for each channel is given by
  • 138. QPSK Pc mbol is to be received correctly both bits must be received ectly. ce, the average probability of correct decision is given by ch gives the probability of errors equal to                        0 2 0 2 fc 2 erfc 4 1 2 erfc N E N E PC QPSK  2 1 P    mbol is to be received correctly both bits must be received ce, the average probability of correct decision is given by ch gives the probability of errors equal to     0 N E
  • 139. QPSK    erfc symbol per Pe e one symbol of QPSK consists of two bits, we have above probability is the error probability per symbol. The avg. bability of error per bit ch is exactly the same as BPSK .              0 erfc 2 1 symbol per 2 1 it N E Pe b QPSK         0 erfc N E b e one symbol of QPSK consists of two bits, we have E = 2Eb. above probability is the error probability per symbol. The avg.
  • 140. BPSK vs QPSK 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 P ower s pec trum dens ity of B P S K vs . Q P S K N orm aliz ed frequenc y , Normalized PSD,Sf/2E b BPSK vs QPSK 1.2 1.4 1.6 1.8 2 P ower s pec trum dens ity of B P S K vs . Q P S K N orm aliz ed frequenc y ,fTb B P S K Q P S K
  • 141. QPSK Conclusion – QPSK is capable of transmitting data twice as faster as BPSK with the same energy per bit. – We will also learn in the future that QPSK has half of the bandwidth of BPSK. QPSK QPSK is capable of transmitting data twice as faster as BPSK with the We will also learn in the future that QPSK has half of the bandwidth
  • 142. OFFSET QPSK 2 s s3 s2 (10 (00) (01) 90 degree shift in phase 180 degree shift in phase OFFSET QPSK 1 s4 s1 10) (11) degree shift in phase degree shift in phase
  • 144. OFFSET QPSK Whenever both bits are changed simultaneously, phase-shift occurs. At 180 phase-shift, the amplitude of the transmitted signal changes very rapidly costing amplitude fluctuation. This signal may be distorted when is passed through the filter or nonlinear amplifier. OFFSET QPSK Whenever both bits are changed simultaneously, 180 degree shift, the amplitude of the transmitted signal changes very rapidly costing amplitude fluctuation. This signal may be distorted when is passed through the filter
  • 145. OFFSET QPSK 0 1 2 3 4 -2 -1 0 1 2 0 1 2 3 - 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 Original Signal Filtered signal OFFSET QPSK 5 6 7 8 4 5 6 7 8 Filtered signal
  • 146. OFFSET QPSK To solve the amplitude fluctuation problem, we propose the offset QPSK. Offset QPSK delay the data in quadrature component by T/ seconds (half of symbol). Now, no way that both bits can change at the same time. OFFSET QPSK To solve the amplitude fluctuation problem, we propose the Offset QPSK delay the data in quadrature component by T/2 Now, no way that both bits can change at the same time.
  • 147. OFFSET QPSK In the offset QPSK, the phase of the signal can change by or 0 degree only while in the QPSK change by 180 90 or 0 degree. OFFSET QPSK In the offset QPSK, the phase of the signal can change by 90 degree only while in the QPSK the phase of the signal can degree.
  • 148. OFFSET QPSK 0 1 2 3 4 - 2 - 1 0 1 2 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 2 3 4 - 2 - 1 0 1 2 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 2 3 4 - 1 - 0 . 5 0 0 . 5 1 0 1 1 0 01 10 hase SK phase PSK PSK 0 1 1 0 01 10 SK SK SK OFFSET QPSK 4 5 6 7 8 4 5 6 7 8 4 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 1 0 0 0 10 00 1 0 0 10 10 00
  • 149. Offset QPSK Possible paths for switching between the message points in (a) QPSK and (b) offset QPSK. Offset QPSK Possible paths for switching between the ) offset QPSK.
  • 150. OFFSET QPSK      2 erfc Pe dwidths of the offset QPSK and the regular QPSK is the me. m signal constellation we have that ich is exactly the same as the regular QPSK. OFFSET QPSK     0 2 N E dwidths of the offset QPSK and the regular QPSK is the m signal constellation we have that ich is exactly the same as the regular QPSK.
  • 151. M-array PSK At a moment, there are M possible symbol values being sent for M different phase values, i i 2      M t f T E t s c i 2 2 cos 2        array PSK At a moment, there are M possible symbol values being sent  M / 1     M i i , , 2 , 1 , 1      
  • 152. M-array PSK Signal-space diagram for octaphase- shift keying (i.e., M  8). The decision boundaries are shown as dashed lines. Signal-space diagram illustrating the application of the union bound for octaphase-shift keying. array PSK
  • 153. M-array PSK Probability of errors E d d sin 2 18 12     / sin erfc 0      N E Pe  array PSK   M / sin   4 ; /      M M
  • 154. M-ary PSK 0 5 1 0 1 0 -5 0 1 0 -4 0 1 0 -3 0 1 0 -2 0 1 0 -1 0 1 0 0 E b /N Probability of Symbol errors Q P S K 8 -a ry P S K 1 6 -a ry P S K ary PSK 1 5 2 0 2 5 3 0 /N 0 d B
  • 155. M-array PSK Power Spectra (M-array) M=2, we have   M E Tf E f S b PSK 2 2 sinc log 2 sinc 2 ) (   E f S b BPSK sinc 2 ) (  array PSK    M f Tb 2 2 log sinc   f Tb 2 sinc
  • 156. M-array PSK Power spectra of M-ary PSK signals for M array PSK ary PSK signals for M  2, 4, 8. Tbf
  • 157. M-array PSK Bandwidth efficiency: – We only consider the bandwidth of the main lobe (or null – Bandwidth efficiency of M-ary PSK is given by M T T B b 2 log 2 2    M R R B R b b b 2 log 2    array PSK We only consider the bandwidth of the main lobe (or null-to-null bandwidth) ary PSK is given by M R b 2 log 2 M 2 log 5 . 0 
  • 158. M-ary QAM QAM = Quadrature Amplitude Modulation Both Amplitude and phase of carrier change according to the transmitted symbol, mi. where ai and bi are integers.     t f a T E t s c i i   2 cos 2 0  ary QAM QAM = Quadrature Amplitude Modulation Both Amplitude and phase of carrier change according to the   T t t f b T E c i   0 ; 2 sin 2 0 
  • 159. M-ary QAM Again, we have as the basis functions   t f T t c  0 ; 2 cos 2 1     t f T t c b  0 2 sin 2 2   ary QAM T t   T t   0
  • 160. M-ary QAM QAM square Constellation – Having even number of bits per symbol, denoted by – M=L x L possible values – Denoting M L  ary QAM Having even number of bits per symbol, denoted by 2n.
  • 161. 16-QAM                    , 1 ( ) 3 , 3 ( , 1 ( ) 1 , 3 ( 1 ( ) 1 , 3 ( 1 ( ) 3 , 3 ( , i i b a QAM             ) 3 , 3 ( ) 3 , 1 ( ) 3 , ) 1 , 3 ( ) 1 , 1 ( ) 1 , ) 1 , 3 ( ) 1 , 1 ( ) 1 , 1 ) 3 , 3 ( ) 3 , 1 ( ) 3 , 1
  • 163. 16-QAM Calculation of Probability of errors – Since both basis functions are orthogonal, we can treat the as combination of two 4-ary PAM systems. – For each system, the probability of error is given by              0 2 1 1 N d erfc L Pe QAM Calculation of Probability of errors Since both basis functions are orthogonal, we can treat the 16-QAM ary PAM systems. For each system, the probability of error is given by                     0 0 0 1 1 N E erfc M
  • 164. 16-QAM – A symbol will be received correctly if data transmitted on both PAM systems are received correctly. – Probability of symbol error is given by   c symbol P     c e P symbol P      1 1 1 QAM A symbol will be received correctly if data transmitted on both 4-ary PAM systems are received correctly. Hence, we have Probability of symbol error is given by  2 1 e P         e e e e P P P P symbol          2 2 1 1 2 2
  • 165. 16-QAM – Hence, we have – But because average energy is given by – We have        1 1 2 M symbol Pe 2 2 2 2 / 1 0 L E E L i av              1 1 2 M symbol Pe QAM But because average energy is given by            0 0 1 N E erfc M    3 1 2 1 0 2 E M i                     0 1 2 3 N M E erfc M av
  • 166. Coherent FSK FSK = frequency shift keying Coherent = receiver have information on where the zero phas of carrier. We can treat it as non-linear modulation since information is put into the frequency. Coherent FSK Coherent = receiver have information on where the zero phas linear modulation since information is
  • 167. Binary FSK Transmitted signals are where          , 0 2 cos 2 b b i T E t s  1 ;    i T i n f b c i Binary FSK    elsewhere 0 , b i T t t f  2 , 1
  • 168. Binary FSK S1(t) represented symbol “1”. S2(t) represented symbol “0”. This FSK is also known as Sunde’s FSK. It is continuous phase frequency Binary FSK This FSK is also known as Sunde’s FSK. It is continuous phase frequency-shift keying (CPFSK).
  • 169. Binary FSK There are two basis functions written as As a result, the signal vectors are          , 0 2 cos 2 b i T t         b E and 0 1 s Binary FSK There are two basis functions written as As a result, the signal vectors are    elsewhere 0 , 2 b i T t t f         b E 0 2 s
  • 170.
  • 171. BFSK From the figure, we have In case of Pr(0)=Pr(1), the probability of error is given by We observe that at a given value of P requires twice as much power as the BPSK system.      2 1 erfc Pe BFSK ), the probability of error is given by We observe that at a given value of Pe, the BFSK system requires twice as much power as the BPSK system. b E d 2 12      0 2 N E b
  • 174. Power Spectral density of BFSK Consider the Sunde’s FSK where f1 and f2 We observe that in-phase component does not depend on m     t f T t T E T t t f T E t s c b b b b c b b i     2 cos cos 2 2 cos 2                                b b b b b T E T t T E  cos 2 cos 2 Power Spectral density of BFSK are different by 1/Tb. We can write phase component does not depend on mi since   t f T t T E c b b b   2 sin sin 2                  b T t  cos
  • 175. Power Spectral density of BFSK We have For the quadrature component                       b b b BI T E T t T E F f S 2 cos 2 2  Half of the symbol power             b b b T t T E t g  sin 2 Power Spectral density of BFSK For the quadrature component                            b b b b T f T f T 2 1 2 1   Half of the symbol power    2 2 2 2 2 1 4 cos 8   f T f T T E S b b b b BQ  
  • 176. Power Spectral density of BFSK Finally, we obtain S Power Spectral density of BFSK ) ( ) ( ) ( f S f S f BQ BI B  
  • 177. Phase Tree of BFSK FSK signal is given by At t = 0, we have The phase of Signal is zero.         c b b T t f T E t s   2 cos 2   0 0 2 cos 2 0 b c b b T f T E s              Phase Tree of BFSK     b T t    0 cos 2 b b T E 
  • 178. Phase Tree of BFSK At t = Tb, we have We observe that phase changes by for symbol “1” and + for symbol “0 We can draw the phase trellis as           2 cos 2 b b c b b b T T T f T E T s Phase Tree of BFSK We observe that phase changes by  after one symbol (Tb seconds). - 0”          cos 2 b b b b T E
  • 179.
  • 180. Minimum-Shift keying (MSK) MSK tries to shift the phase after one symbol to just half of Sunde’s FSK system. The transmitted signal is given by                2 cos 2   t t f T E c b b Shift keying (MSK) MSK tries to shift the phase after one symbol to just half of Sunde’s FSK system. The transmitted signal is given by           "0" for 0 2 cos 2 "1" for 0 2 cos 2 2 1     t f T E t f T E b b b b
  • 181. MSK Where Observe that     T h t b      0 b c T h f f and 2 1    1 2 1 f f c  MSK t h b b c T h f f 2 2    2 f 
  • 182. MSK h = Tb(f1-f2) is called “deviation ratio.” For Sunde’s FSK, h = 1. For MSK, h = 0.5. h cannot be any smaller because the orthogonality between cos( and cos(2f2t) is still held for h < 0.5 Orthogonality guarantees that both signal will not interfere each other in detection process. MSK ) is called “deviation ratio.” cannot be any smaller because the orthogonality between cos(2f1t) 5. Orthogonality guarantees that both signal will not interfere each other
  • 183. MSK Phase trellis diagram for MSK signal MSK Phase trellis diagram for MSK signal 1101000
  • 184. MSK Signal s(t) of MSK can be decomposed into where                  t s t f t s f t T E t t f T E t s Q c I b b c b b      2 cos 2 cos cos 2 2 cos 2          b t T t   2 0    MSK (t) of MSK can be decomposed into          t f t f t T E t f c c b b c    2 sin 2 sin sin 2  b T t   0 ;
  • 185. MSK Symbol (0) 1 0  0 0  MSK (0) (Tb) 0 /2  -/2 0 -/2  /2
  • 186. MSK For the interval –Tb < t  0, we have Let’s note here that the  for the interval not be the same. We know that     2 0   t T t b                  b T t cos 0 cos 2 0 cos    MSK , we have for the interval -Tb<t 0 and 0< tTb ma 0 ;    t Tb                     b b T t T t 2 sin 0 sin 2 cos    
  • 187. MSK Since (0) can be either 0 or  depending on the past history. We have “+” for (0) = 0 and “-” for (0) =  Hence, we have               b T t cos 0 cos 2 0 cos    b b b I T t T E t s           2 cos 2 ) (  MSK depending on the past history. We have                   b b T t T t 2 cos 2 cos   b b T t T    ;
  • 188. MSK Similarly we can write for 0< tTb and Tb < t2Tb. Note the “+” and “ between these intervals. Furthermore, we have that (Tb) can be past history.     b b T T t   2    MSK Note the “+” and “-” may be different ) can be /2 depending on the   b T t 
  • 189. MSK Hence, we have we have that (Tb) can be /2 depending on the past history.                               cos sin cos sin 2 sin      b b b b b T T T T t T                   b b b b T t T T t T 2 cos 2 sin    MSK depending on the past history.                                           2 2 sin cos 2 2 2 sin cos 2         b b b b b b b b T t T T t T T t T T T t                b T t 2 sin 2  
  • 190. MSK Hence, we have “+” for (Tb) = +/2 and “-” for (Tb The basis functions change to b b b Q T t T E t s 2 sin 2 ) (               b b T t T t          2 cos 2 cos 2 1      b b T t T t          2 sin 2 sin 2 2   MSK ) = -/2 b T t 2 0 ;        b c T t t f   0 ; 2  b c T t t f   0 ; 2
  • 191. MSK We write MSK signal as Where and                    ) ( ) ( sin E ) ( 0 cos E 2 cos 2 cos 0 cos 2 2 2 cos cos 2 2 2 1 1 b 1 b t s t s T t f T t T E T E t f t T E t s b b b b b c b b                              0 cos 1  b E s  MSK               ) ( 2 sin 2 sin sin 2 2 sin sin 2 t T t f T t T T E t f t f t b c b b b b c c                    b b T E s  sin 2  
  • 192. MSK Symbol (0) s 1 0  0 0  E E E  E  MSK s1 (Tb) s2 /2 -/2 -/2 /2 b E b E b E b E b E b E b E  b E 
  • 194. 0 Phase: /2   /2  /2 0 -/2
  • 195. MSK We observe that MSK is in fact the QPSK having the pulse shape Block diagrams for transmitter and receiver are given in the next two slides.     b T t 2 cos  MSK We observe that MSK is in fact the QPSK having the Block diagrams for transmitter and receiver are given in the     b
  • 196.
  • 197.    b b T T dt t t x x ) ( ) ( 1 1    b T dt t t x x 2 0 2 2 ) ( ) ( 
  • 198. 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 Norm aliz ed F requenc y , 1.2 1.4 1.6 1.8 2 Norm aliz ed F requenc y , fTb M S K B P S K Q P S K
  • 199. MSK Probability of error of MSK system is equal to BPSK and QPSK This due to the fact that MSK observes the signal for two symbol intervals whereas FSK only observes for single signal interval. Bandwidth of MSK system is 50% larger than QPSK. 2 16 cos 32 ) (      E f S b MSK  MSK Probability of error of MSK system is equal to BPSK and QPSK This due to the fact that MSK observes the signal for two symbol intervals whereas FSK only observes for single signal interval. % larger than QPSK.   2 2 2 1 16 2 cos      f T f T b b 
  • 200. Noncoherent Orthogonal Modulation Noncoherent implies that phase information is not available to the receiver. As a result, zero phase of the receiver can mean any phase of the transmitter. Any modulation techniques that transmits information through the phase cannot be used in noncoherent receivers. Noncoherent Orthogonal Modulation Noncoherent implies that phase information is not available to the As a result, zero phase of the receiver can mean any phase of the Any modulation techniques that transmits information through the phase cannot be used in noncoherent receivers.
  • 201. Noncoherent Orthogonal Modulation cos(2ft) sin(2ft) Receiver Noncoherent Orthogonal Modulation cos(2ft) sin(2ft) Transmitter
  • 202. Noncoherent Orthogonal Modulation It is impossible to draw the signal constellation since we do not know where the axes are. However, we can still determine the distance of the each signal constellation from the origin. As a result, the modulation techniques that put information in the amplitude can be detected. FSK uses the amplitude of signals in two different frequencies. Hence non-coherent receivers can be employed. Noncoherent Orthogonal Modulation It is impossible to draw the signal constellation since we do not know However, we can still determine the distance of the each signal As a result, the modulation techniques that put information in the FSK uses the amplitude of signals in two different frequencies. Hence coherent receivers can be employed.
  • 203. Noncoherent Orthogonal Modulation Consider the BFSK system where two frequencies f represented two “1” and “0”. The transmitted signal is given by Problem is that  is unknown to the receiver. For the coherent receiver  is precisely known by receiver.   i t f T E t s   2 cos 2 ) (   Noncoherent Orthogonal Modulation Consider the BFSK system where two frequencies f1 and f2 are used to is unknown to the receiver. For the coherent receiver  b T t i    0 , 2 , 1 ;
  • 204. Noncoherent Orthogonal Modulation Furthermore, we have To get rid of the phase information (       ) ( ) ( 2 2 cos cos 2 2 cos 2 ) ( 2 2 1 1 t s t s T E t f T E t f T E t s i i i i               E s s t s i i    2 2 2 1 cos Noncoherent Orthogonal Modulation To get rid of the phase information (), we use the amplitude     2 sin sin t f i       E E     2 2 sin
  • 205. Noncoherent Orthogonal Modulation Where The amplitude of the received signal    T i x dt t t s s 1 0 1 1 ) ( ) (     T i x dt t t s s 2 0 2 2 ) ( ) (    2 0 2 cos ) (                    T i i dt t f t x l  Noncoherent Orthogonal Modulation The amplitude of the received signal   T dt t t x 0 1 ) ( ) (    T dt t t x 0 2 ) ( ) (    2 / 1 2 0 2 sin ) (              T i dt t f t x 
  • 206. Noncoherent Orthogonal Modulation Probability of Errors       2 exp 2 1 N E Pe Noncoherent Orthogonal Modulation     0 N E
  • 207. Noncoherent: BFSK For BFSK, we have          0 2 cos 2 b b i T E t s Noncoherent: BFSK    elsewhere ; 0 ; 2 b i T t t f 
  • 209. Noncoherent: BFSK Probability of Errors       2 exp 2 1 N E Pe Noncoherent: BFSK     0 N E b
  • 210. DPSK Differential PSK – Instead of finding the phase of the signal on the interval receiver determines the phase difference between adjacent time intervals. – If “1” is sent, the phase will remain the same – If “0” is sent, the phase will change DPSK Instead of finding the phase of the signal on the interval 0<tTb. Thi receiver determines the phase difference between adjacent time ” is sent, the phase will remain the same ” is sent, the phase will change 180 degree.
  • 211. DPSK Or we have and           b b b b T E T E t s 2 cos 2 2 cos 2 ) ( 1             b b b b T E T E t s 2 cos 2 2 cos 2 ) ( 2   DPSK       b b c b c T t T t f T t t f 2 ; 2 0 ;          b b c b c T t T t f T t t f 2 ; 2 0 ;   
  • 212. DPSK In this case, we have T=2Tb and E=2 Hence, the probability of error is given by       0 exp 2 1 N E P b e DPSK 2Eb Hence, the probability of error is given by     0 b
  • 213. DPSK: Transmitter 1     k k k k k d b d b d DPSK: Transmitter 1 
  • 214. DPSK bk} 1 0 dk-1} 1 1 Differential encoded dk} 1 1 0 Transmitted Phase 0 0  DPSK 0 0 1 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0  0 0  0   
  • 216. DPSK: Receiver From the block diagram, we have that the decision rule as If the phase of signal is unchanged (send “ both xi and xQ should not change. Hence, the If the phase of signal is unchanged (send “ both xi and xQ should change. Hence, the   1 0 1 0   Q Q I I x x x x l x DPSK: Receiver From the block diagram, we have that the decision rule as If the phase of signal is unchanged (send “1”) the sign (“+” or “-”) of change. Hence, the l(x) should be positive. If the phase of signal is unchanged (send “0”) the sign (“+” or “-1”) of should change. Hence, the l(x) should be negative. 0 1 say 0 say  
  • 217. Signal-space diagram of received DPSK signal. space diagram of received DPSK signal.
  • 218. Unit-V –Introduction to Spread Spectrum Techniques Introduction to Spread Spectrum Techniques
  • 219. M-ary signaling scheme: In this signaling scheme 2 or more bits are grouped together to form a symbol. One of the M possible signals s1(t) ,s2(t),s3(t),……sM(t) is transmitted during each symbol period of duration Ts. The number of possible signals = M = 2n, where n is an integer.
  • 220. n M = 2n 1 2 2 4 3 8 4 16 …. …… The symbol values of M for a given value of n: Symbol 0, 1 00, 01, 10, 11 000, 001, 010,011,... 0000, 0001, 0010,0011,…. ………. The symbol values of M for a given value of n:
  • 221. Fig: waveforms of (a) ASK (b) PSK (c)FSK • Depending on the variation of amplitude, phase or frequency of the carrier, the modulation scheme is calle M-ary ASK, M-ary PSK and M-ary FSK. Fig: waveforms of (a) ASK (b) PSK (c)FSK Depending on the variation of amplitude, phase or frequency of the carrier, the modulation scheme is calle
  • 222. M-ary Phase Shift Keying(MPSK) In M-ary PSK, the carrier phase takes on one of the M possible values, namely i = 2 * (i - 1) / M where i = 1, 2, 3, …..M. The modulated waveform can be expressed as where Es is energy per symbol = (log2 M) E Ts is symbol period = (log2 M) Tb. ary Phase Shift Keying(MPSK) ary PSK, the carrier phase takes on one of the M possible values, namely The modulated waveform can be expressed as M) Eb b.
  • 223. he above equation in the Quadrature form is choosing orthogonal basis signals efined over the interval 0  t  Ts he above equation in the Quadrature form is
  • 224. M-ary signal set can be expressed as  Since there are only two basis signals, the constellation of M dimensional.  The M-ary message points are equally spaced on a circle of radius at the origin.  The constellation diagram of an 8-ary PSK signal set is shown in fig. Since there are only two basis signals, the constellation of M-ary PSK is two ary message points are equally spaced on a circle of radius Es, centered ary PSK signal set is shown in fig.
  • 225. Fig: Constellation diagram of an M-ary PSK system(m=8) ary PSK system(m=8)
  • 226. Derivation of symbol error probability: Decision Rule: Fig: Constellation diagram for M=2 (Binary PSK) (Binary PSK)
  • 227. If a symbol (0,0,0) is transmitted, it is clear that if an error occurs, the transmitted signal is most likely to be mistaken for (0,0,1) and (1,1,1 signal being mistaken for (1,1,0) is remote. The decision pertaining to (0,0,0) is bounded by /8(below 1(t)- axis) to  = + /8 ( above The probability of correct reception is… ) is transmitted, it is clear that if an error occurs, the transmitted signal is most 1) and the ) is remote. ) is bounded by  = - ( above 2(t)- axis)
  • 228. Fig: Probability density function of Phase .
  • 229. The average symbol error probability of an coherent M AWGN channel is given by Similarly, The symbol error Probability of a differential M AWGN channel is given by The average symbol error probability of an coherent M-ary PSK system in Similarly, The symbol error Probability of a differential M-ary PSK system in
  • 230. Fig: The performance of symbol error probability for -different values of M Fig: The performance of symbol error probability for
  • 231. M-ary Quadrature Amplitude (QAM) It’s a Hybrid modulation As we allow the amplitude to also vary with the phase, a new modulation scheme called quadrature amplitude modulation (QAM) is obtained. The constellation diagram of 16-ary QAM consists of a square lattice of signal points. ary Quadrature Amplitude Modulation (QAM) As we allow the amplitude to also vary with the phase, a new modulation scheme called quadrature amplitude modulation (QAM) is obtained. ary QAM consists of a square lattice of signal
  • 232. Fig: signal Constellation of M-ary QAM for M= ary QAM for M=16
  • 233. M-ary Frequency Shift Keying(MFSK) In M-ary FSK modulation the transmitted signals are defined by: where fc = nc/2Ts, for some fixed integer n. The M transmitted signals are of equal energy and equal duration, and the signal frequencies are separated by orthogonal to one another. ary Frequency Shift Keying(MFSK) ary FSK modulation the transmitted signals are defined by: , for some fixed integer n. The M transmitted signals are of equal energy and equal duration, and the signal frequencies are separated by 1/2Ts Hertz, making the signal
  • 234. The average probability of error based on the union bound is given by Using only the leading terms of the binomial expansion: The average probability of error based on the union bound is given by Using only the leading terms of the binomial expansion:
  • 235. Power Efficiency and Bandwidth : Bandwidth: The channel bandwidth of a M-ary FSK signal is : Power Efficiency and Bandwidth : ary FSK signal is :
  • 236. The channel bandwidth of a noncohorent MFSK is : This implies that the bandwidth efficiency of an M with increasing M. Therefore, unlike M- bandwidth inefficient. The channel bandwidth of a noncohorent MFSK is : This implies that the bandwidth efficiency of an M-ary FSK signal decreases -PSK signals, M-FSK signals are
  • 237. Introduction to Spread Spectrum • Problems such as capacity limits, propagation effects, synchroniza occur with wireless systems • Spread spectrum modulation spreads out the modulated signal bandwidth so it is much greater than the message bandwidth • Independent code spreads signal at transmitter and despreads sig at receiver Introduction to Spread Spectrum Problems such as capacity limits, propagation effects, synchroniza Spread spectrum modulation spreads out the modulated signal bandwidth so it is much greater than the message bandwidth Independent code spreads signal at transmitter and despreads sig
  • 238. Multiplexing • Multiplexing in 4 dimensions – space (si) – time (t) – frequency (f) – code (c) • Goal: multiple use of a shared medium • Important: guard spaces needed! Multiplexing s2 s3 s1 f t c k2 k3 k4 k5 k6 k1 t c f t c channels ki
  • 239. Frequency multiplex • Separation of spectrum into smaller frequency bands • Channel gets band of the spectrum for the whole time • Advantages: – no dynamic coordination needed – works also for analog signals • Disadvantages: – waste of bandwidth if traffic distributed unevenly – inflexible – guard spaces Frequency multiplex Separation of spectrum into smaller frequency bands Channel gets band of the spectrum for the whole time k3 k4 k5 t c
  • 240. t Time multiplex Channel gets the whole spectrum for a certain amount of time Advantages: – only one carrier in the medium at any time – throughput high even for many users Disadvantages: – precise synchronization necessary c k2 k3 k4 k5 k1 Time multiplex Channel gets the whole spectrum for a certain amount of time
  • 241. Time and frequency multiplex • A channel gets a certain frequency band for a certain amount of time (e.g. GSM) • Advantages: – better protection against tapping – protection against frequency selective interference – higher data rates compared to code multiplex • Precise coordination required t Time and frequency multiplex A channel gets a certain frequency band for a certain amount of time (e.g. c k2 k3 k4 k5 k1
  • 242. Code multiplex Each channel has unique code All channels use same spectrum at same time Advantages: – bandwidth efficient – no coordination and synchronization – good protection against interference Disadvantages: – lower user data rates – more complex signal regeneration Implemented using spread spectrum technology Code multiplex Implemented using spread spectrum technology k2 k3 k4 k5 k6 k1 t c
  • 243. Spread Spectrum Technology • Problem of radio transmission: frequency dependent fading can wipe out narrow band signals for duration of the interference • Solution: spread the narrow band signal into a broad band signal using a special code detection at receiver interference spread signal f ower Spread Spectrum Technology Problem of radio transmission: frequency dependent fading can wipe out narrow band signals for duration of the interference Solution: spread the narrow band signal into a broad band signal using a detection at receiver signal spread interferenc f power
  • 244. Spread Spectrum Technology • Side effects: – coexistence of several signals without dynamic coordination – tap-proof • Alternatives: Direct Sequence (DS/SS), Frequency Hopping (FH/SS) • Spread spectrum increases BW of message signal by a factor Gain P ro cessin g G ain 1 0 lo g N   Spread Spectrum Technology coexistence of several signals without dynamic coordination Alternatives: Direct Sequence (DS/SS), Frequency Hopping (FH/SS) Spread spectrum increases BW of message signal by a factor N, Processing 1 0 P ro cessin g G ain 1 0 lo g ss ss B B N B B        
  • 245. Effects of spreading and interference P f i) ii) sender P f iii) iv) receiver Effects of spreading and interference P f P f receiver f v) user signal broadband interference narrowband interference P
  • 246. Spreading and frequency selective fading channel quality 1 2 3 4 5 Narrowband signal guard space channel quality 1 spread spectrum Spreading and frequency selective fading frequency 6 2 2 2 2 2 frequency narrowband channels spread spectrum channels
  • 247. DSSS (Direct Sequence Spread Spectrum) I • XOR the signal with pseudonoise (PN) sequence (chipping sequence) • Advantages – reduces frequency selective fading – in cellular networks • base stations can use the same frequency range • several base stations can detect and recover the signal • But, needs precise power control DSSS (Direct Sequence Spread Spectrum) I XOR the signal with pseudonoise (PN) sequence (chipping sequence) user dat chippin sequen resulting signal 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 XOR 0 1 1 0 0 1 0 1 1 0 1 0 0 1 = Tb Tc
  • 248. user data m(t) chipping sequence, c(t) X DSSS (Direct Sequence Spread Spectrum) II radio carrier Spread spectrum Signal y(t)=m(t)c(t) transmitter demodulator received signal radio carrier X Chipping sequence, c(t) receiver DSSS (Direct Sequence Spread Spectrum) II modulator radio carrier Signal y(t)=m(t)c(t) transmit signal X Chipping sequence, c(t) integrator products decision data sampled sums correlator
  • 249. DS/SS Comments III Pseudonoise(PN) sequence chosen so that its autocorrelation is very narrow => PSD is very wide – Concentrated around t < Tc – Cross-correlation between two user’s codes is very small DS/SS Comments III Pseudonoise(PN) sequence chosen so that its autocorrelation is very narrow => PSD is very wide correlation between two user’s codes is very small
  • 250. DS/SS Comments IV Secure and Jamming Resistant – Both receiver and transmitter must know c(t) – Since PSD is low, hard to tell if signal present – Since wide response, tough to jam everything Multiple access – If ci(t) is orthogonal to cj(t), then users do not interfere Near/Far problem – Users must be received with the same power DS/SS Comments IV Both receiver and transmitter must know c(t) Since PSD is low, hard to tell if signal present Since wide response, tough to jam everything (t), then users do not interfere Users must be received with the same power
  • 251. FH/SS (Frequency Hopping Spread Spectrum) • Discrete changes of carrier frequency – sequence of frequency changes determined via PN sequence • Two versions – Fast Hopping: several frequencies per user bit (FFH) – Slow Hopping: several user bits per frequency (SFH) • Advantages – frequency selective fading and interference limited to short period – uses only small portion of spectrum at any time • Disadvantages – not as robust as DS/SS – simpler to detect FH/SS (Frequency Hopping Spread Spectrum) Discrete changes of carrier frequency sequence of frequency changes determined via PN sequence : several frequencies per user bit (FFH) : several user bits per frequency (SFH) frequency selective fading and interference limited to short period uses only small portion of spectrum at any time
  • 252. FHSS (Frequency Hopping Spread Spectrum) I 0 1 Tb 0 f f1 f2 f3 Td f f1 f2 f3 Td Tb: bit period FHSS (Frequency Hopping Spread Spectrum) I user data slow hopping (3 bits/hop) fast hopping (3 hops/bit) 1 1 t t t Td: dwell time
  • 253. FHSS (Frequency Hopping Spread Spectrum) III modulator user data narrowband transmitter received signal receiver hopping sequence demodulator frequency synthesizer FHSS (Frequency Hopping Spread Spectrum) III modulator hopping sequence modulator narrowband signal Spread transmit signal demodulator data frequency synthesizer
  • 254. Applications of Spread Spectrum Cell phones – IS-95 (DS/SS) – GSM Global Positioning System (GPS) Wireless LANs – 802.11b Applications of Spread Spectrum Global Positioning System (GPS)
  • 255. Performance of DS/SS Systems Pseudonoise (PN) codes – Spread signal at the transmitter – Despread signal at the receiver Ideal PN sequences should be – Orthogonal (no interference) – Random (security) – Autocorrelation similar to white noise (high at equal 0) Performance of DS/SS Systems Autocorrelation similar to white noise (high at t=0 and low for t not
  • 256. PN Sequence Generation • Codes are periodic and generated by a shift register and XOR • Maximum-length (ML) shift register sequences, bits R( -1/n -nTc + PN Sequence Generation Codes are periodic and generated by a shift register and XOR length (ML) shift register sequences, m-stage shift register, length: n = 2m – 1 R(t) Tc t  nTc Output
  • 257. Generating PN Sequences Take m=2 =>L=3 cn=[1,1,0,1,1,0, . . .], usually written as bipolar cn=[1,1,-1,1,1,-1, . . .] + Output                1 1 / 1 0 1 1 1 L m L m c c L m L n m n n Generating PN Sequences m Stages connected to modulo-2 adder 2 1,2 3 1,3 4 1,4 5 1,4 6 1,6 8 1,5,6,7
  • 258. Problems with Cross-correlations with other m different input sequences can be quite high Easy to guess connection setup in secure In practice, Gold codes or Kasami sequences which combine the output of m-sequences are used. Problems with m-sequences m-sequences generated by different input sequences can be quite high Easy to guess connection setup in 2m samples so not too In practice, Gold codes or Kasami sequences which combine sequences are used.
  • 259. Detecting DS/SS PSK Signals X Bipolar, NRZ m(t) PN sequence, c(t) sqrt(2)cos Spread spectrum Signal y(t)=m(t)c(t) transmitter X received signal X c(t) receiver z(t) sqrt(2)cos (wct + ) LPF x(t) Detecting DS/SS PSK Signals X )cos (wct + ) Signal y(t)=m(t)c(t) transmit signal integrator decision LPF w(t)
  • 260. Optimum Detection of DS/SS PSK Recall, bipolar signaling (PSK) and white noise give the optimum error probability Not effected by spreading – Wideband noise not affected by spreading – Narrowband noise reduced by spreading b P Q  Optimum Detection of DS/SS PSK Recall, bipolar signaling (PSK) and white noise give the optimum error Wideband noise not affected by spreading Narrowband noise reduced by spreading 2 b b E P Q          
  • 261. Signal Spectra • Effective noise power is channel noise power plus jamming (NB) signal power divided by N P ro cessin g G ain 1 0 lo g B B T N B B T    T Signal Spectra Effective noise power is channel noise power plus jamming (NB) 1 0 P ro cessin g G ain 1 0 lo g ss ss b c B B T B B T          Tb Tc
  • 262. Multiple Access Performance Assume K users in the same frequency band, Interested in user 1, other users interfere 4 3 2 Multiple Access Performance users in the same frequency band, , other users interfere 1 5 6