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    • EXPERT SYSTEMS AND SOLUTIONS Email: expertsyssol@gmail.com expertsyssol@yahoo.com Cell: 9952749533 www.researchprojects.info PAIYANOOR, OMR, CHENNAI Call For Research Projects Final year students of B.E in EEE, ECE, EI, M.E (Power Systems), M.E (Applied Electronics), M.E (Power Electronics) Ph.D Electrical and Electronics.Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
    • DIGITAL SIGNAL PROCESSING(DSP)FUNDAMENTALS
    • WHAT IS DSP? Converting a continuously changing waveform (analog) into a series of discrete levels (digital)
    • WHAT IS DSP? The analog waveform is sliced into equal segments and the waveform amplitude is measured in the middle of each segment The collection of measurements make up the digital representation of the waveform
    • 0.5 1.5 -1.5 -0.5-2 -1 0 1 2 1 0 0.22 3 0.44 0.64 5 0.82 0.98 7 1.11 1.2 WHAT IS DSP? 9 1.24 1.27 11 1.24 1.2 13 1.11 0.98 15 0.82 0.64 17 0.44 0.22 19 0 -0.22 -0.44 21 -0.64 -0.82 23 -0.98 -1.11 25 -1.2 -1.26 27 -1.28 -1.26 29 -1.2 -1.11 31 -0.98 -0.82 33 -0.64 -0.44 35 -0.22 37 0
    • DSP IS EVERYWHERE Sound applications  Compression, enhancement, special effects, synthesis, recognition, echo cancellation,…  Cell Phones, MP3 Players, Movies, Dictation, Text-to-speech,… Communication  Modulation, coding, detection, equalization, echo cancellation,…  Cell Phones, dial-up modem, DSL modem, Satellite Receiver,… Automotive  ABS, GPS, Active Noise Cancellation, Cruise Control, Parking,… Medical  Magnetic Resonance, Tomography, Electrocardiogram,… Military  Radar, Sonar, Space photographs, remote sensing,… Image and Video Applications  DVD, JPEG, Movie special effects, video conferencing,… Mechanical  Motor control, process control, oil and mineral prospecting,…
    • SIGNAL PROCESSING Humans are the most advanced signal processors  speech and pattern recognition, speech synthesis,… We encounter many types of signals in various applications  Electrical signals: voltage, current, magnetic and electric fields,…  Mechanical signals: velocity, force, displacement,…  Acoustic signals: sound, vibration,…  Other signals: pressure, temperature,… Most real-world signals are analog  They are continuous in time and amplitude  Convert to voltage or currents using sensors and transducers Analog circuits process these signals using  Resistors, Capacitors, Inductors, Amplifiers,… Analog signal processing examples  Audio processing in FM radios  Video processing in traditional TV sets
    • LIMITATIONS OF ANALOG SIGNALPROCESSING Accuracy limitations due to  Component tolerances  Undesired nonlinearities Limited repeatability due to  Tolerances  Changes in environmental conditions  Temperature  Vibration Sensitivity to electrical noise Limited dynamic range for voltage and currents Inflexibility to changes Difficulty of implementing certain operations  Nonlinear operations  Time-varying operations Difficulty of storing information
    • DIGITAL SIGNAL PROCESSING Represent signals by a sequence of numbers  Sampling or analog-to-digital conversions Perform processing on these numbers with a digital processor  Digital signal processing Reconstruct analog signal from processed numbers  Reconstruction or digital-to-analog conversion digital digital signal signal analog analog signal A/D DSP D/A signal• Analog input – analog output – Digital recording of music• Analog input – digital output – Touch tone phone dialing• Digital input – analog output – Text to speech• Digital input – digital output – Compression of a file on computer 9
    • PROS AND CONS OF DIGITALSIGNAL PROCESSING Pros  Accuracy can be controlled by choosing word length  Repeatable  Sensitivity to electrical noise is minimal  Dynamic range can be controlled using floating point numbers  Flexibility can be achieved with software implementations  Non-linear and time-varying operations are easier to implement  Digital storage is cheap  Digital information can be encrypted for security  Price/performance and reduced time-to-market Cons  Sampling causes loss of information  A/D and D/A requires mixed-signal hardware  Limited speed of processors  Quantization and round-off errors
    • DSP APPLICATIONS Image Processing – Robotic vision, FAX, satellite weather Instrumentation – Spectrum analysis, noise reduction Speech & Audio – Speech recognition, equilization Military – Radar processing, missile guidance Telecommunications – Echo cancellation, video conferencing, VoIP Biomedical – ECG analysis, patient monitoring Consumer Electronics – Cell phones, set top box, video cameras
    • DSP APPLICATIONS
    • ANOTHER LOOK AT DSPAPPLICATIONS High-end  Wireless Base Station - TMS320C6000 Increasing  Cable modem  gateways Cost Mid-end  Cellular phone - TMS320C540  Fax/ voice server Low end  Storage products - TMS320C27 volume Increasing  Digital camera - TMS320C5000  Portable phones  Wireless headsets  Consumer audio  Automobiles, toasters, thermostats, ...
    • ADVANTAGES OF DSP Guaranteed Accuracy – Accuracy only limited by bit length Perfect Reproducibility – No component tolerances, no component drift due to temperature or age Greater Flexibility – Functions and algorithms can be changed through software Superior Performance – Adaptive filtering, linear phase response Some Data Naturally Digital – Images, computer files
    • DISADVANTAGES OF DSP Speed and Cost – ADC/DAC, uProc Design Time – Can be tricky Finite Word Length Issues
    • KEY DSP OPERATIONS Convolution Correlation Filtering Transformations Modulation
    • CONVOLUTION  Many uses but a common use is determining a system’s output if system input and system impulse response is known. For continuous system: x( t ) h( t ) y ( t ) = x ( t ) ∗ h( t ) ∞ ∞ y (t ) = x(t ) ∗ h(t ) = ∫ x(τ ) h( t − τ ) dτ = ∫ x( t − τ ) h(τ ) dτ −∞ −∞
    • DISCRETE CONVOLUTION  We may however have a computer sampling a signal so that we have discrete data.  So instead of continuous integration process we have discrete summation. ∞ y ( n ) = x ( n ) ∗ h( n ) = ∑ h( k ) x ( n − k ) n = 0,±1,±2, k = −∞  Practically speaking though we would have finite sequences x(n) and h(n) of lengths N1 and N2 respectively, so this is then: M −1 y ( n ) = x( n ) ∗ h( n ) = ∑ h( k ) x( n − k ) n = 0,1, , ( M − 1) k =0 with M = N1 + N 2 − 1
    • CORRELATION Correlation is essentially the same as convolution (from a computational standpoint). You just don’t “flip” anything. Instead of describing system output, correlation tells us information about the signals. Cross-correlation function  Tellsyou a measure of similarities between two signals.  Application: Identifying radar return signals
    • Signals• What is a signal? – A signal is a function of independent variables such as time, distance, position, temperature, pressure, etc. – Most signals are generated naturally but a signal can also be generated artificially using a computer – Can be in any number of dimensions (1D, 2D or 3D)
    • CLASSIFICATION OF SIGNALS Signals can be classified into various types by  Nature of the independent variables  Value of the function defining the signals Examples:  Discrete/continuous function  Discrete/continuous independent variable  Real/complex valued function  Scalar (single channel)/Vector (multi-channels)  Single/Multi-trial (repeated recordings)  Dimensionality based on the number of independent variables (1D/2D/3D)  Deterministic/random  Periodic/aperiodic  Even/odd  Many more…. Lecture 1 CE804 Autumn 2007/8 (copyright R. Palaniappan, 2008)
    • CLASSIFICATION - DISCRETE/CONTINUOUSSIGNALS Normally, the independent variable is time Continuous time signal  Time is continuous  Defined at every instant of time Discrete time signal  Time is discrete  Defined at discrete instants of time - it is a sequence of numbers Four classifications based on time/amplitude - continuous/discrete:  Analogue, digital, sampled, quantised boxcar Lecture 1 CE804 Autumn 2007/8 (copyright R. Palaniappan, 2008)
    • CONTINUOUS AND DISCRETESIGNALS Continous signal xa(t) Discrete signal (sequence) x[n] T : sampling period x[n] = xa(nT) fs = 1/T : sampling rate
    • CLASSIFICATION - DISCRETE/CONTINUOUSSIGNALS (CONT) Amplitude- continuous Amplitude- discrete Time-continuous Time-discrete Amplitude- continuous Amplitude- discrete Time-discrete Time-continuous Lecture 1 CE804 Autumn 2007/8 (copyright R. Palaniappan, 2008)
    • RANDOM VS DETERMINISTIC SIGNAL Deterministic signal  A signal that can be predicted using some methods like a mathematical expression or look-up table  Easier to analyse Random (stochastic)  A signal that is generated randomly and cannot be predicted ahead of time  Most biological signals fall in this category  More difficult to analyse 25
    • CLASSIFICATION –PERIOD/APERIODIC Periodic  Continuous time-signal is periodic if it exhibits periodicity, i.e. x(t+T)=x(t), -∝<t<∝ where T=period of the signal  The smallest value of T is Periodic signal (continuous-time) called the fundamental period, T0  A periodic signal has a definite pattern that repeats over and over with a repetition period of T0 Periodic signal (discrete-time)  For discrete-time signals, x(n+N0)=x(n),-∝<n<∝  A signal, which does not have a repetitive pattern is Lecture 1 CE804 Autumn 2007/8 (copyright R. Palaniappan, aperiodic 2008)
    • SINGULAR FUNCTIONS Singular functions  Important non-periodic signals  Delta/unit-impulse function is the most basic and all other singular functions can be derived from it 0, t < 0 0, n < 0 ∞ 1, n = 0 r (t ) = { r(n ) = { δ (n ) = { t, t > 0 n, n ≥ 0 δ (t ) = 0, t ≠ 0; ∫ δ (t )dt = 1 0, n ≠ 0 −∞ Unit impulse functions Unit ramp functions 0, t < 0 0, n < 0  1  1 u(t ) = { u(n ) = { Π(t ) = u t +  − u t −  1, t > 0 1, n > 0  2  2 Unit step functions Unit pulse function Lecture 1 CE804 Autumn 2007/8 (copyright R. Palaniappan, 2008)
    • CLASSIFICATION –EVEN/ODD Even signal  Signal exhibit symmetry in the time domain  x(t)=x(-t) or x(n)=x(-n) Odd signal  Signal exhibit anti-symmetry in the time domain  x(t)=-x(-t) or x(n)=-x(-n) A signal can be expressed as a sum of its even and odd components  x(t)=xeven(t)+xodd(t)  where xeven(t)=1/2[x(t)+x(-t)], xodd(t)=1/2[x(t)-x(-t)]
    • CLASSIFICATION OF SIGNALSDIMENSIONALITYSIGNAL DESCRIPTION EXAMPLE1–D Signal is a function of Speech a single independent variable2-D Signal is a function of Image 2 independent variablesM-D Signal has more than 2 Video signal independent variables
    • CLASSIFICATION OF SIGNALS Continuous-time signals The signal is defined for every instant of time in a defined range Discrete-time signal The independent variable (time) is discrete. The signal is defined at discrete instants of time
    • CLASSIFICATION OF SIGNALS Analog signalx(t) A continuous-time t and a continuous amplitude A Quantized Signal discrete inxq(t) t amplitude but continuous in time
    • CLASSIFICATION OF SIGNALS  Sampled data signal has a continuous amplitude. Amplitude can take any value within a specified range. • Digital signal is a discrete-time signal with discrete-valued amplitudes
    • CLASSIFICATION OF SIGNALS A deterministic Signal is one that is uniquely determined by a well defined process such as a mathematical expression or a look- up table • A random signal is one that is generated in a random fashion and cannot be predicted or reproduced
    • CLASSIFICATION OF SIGNALSDimension Type Symbol Independent variable1-D Continuous-time v(t) t Copyright © 2001, S. K. Mitra1-D Discrete - time {v(n)} n2-D Continuous-spatial v(x,y) x,y2-D Discrete - spatial {v(m,n)} m,n3-D Continuous-time v(x,y,t) x,y,t and spatial3-D Continuous-time x,y,t  r ( x, y , t )  and spatial u ( x, y , t ) =  g ( x, y , t )     b ( x, y , t )   
    • DISCRETE-TIME SIGNALS:TIME-DOMAIN REPRESENTATION Graphical representation of a discrete-time signal with real-valued samples is as shown below:
    • DISCRETE-TIME SIGNALS: TIME-DOMAIN REPRESENTATION•In some applications, a discrete-time sequence{x[n]} may be generated by sampling a continuous-time signal xa (t ) at uniform intervals of time
    • DISCRETE-TIME SIGNALS: TIME-DOMAIN REPRESENTATION Here, n-th sample is given by x[n] = xa (t ) t = nT = xa (nT ), n = , − 2, − 1,0,1, The spacing T between two consecutive samples is called the sampling interval or sampling period Reciprocal of sampling interval T, denoted as FT , is called the sampling frequency: 1 FT = T
    • DISCRETE-TIME SIGNALS:TIME-DOMAIN REPRESENTATION Unit of sampling frequency is cycles per second, or hertz (Hz), if T is in seconds Whether or not the sequence {x[n]} has been obtained by sampling, the quantity x[n] is called the n-th sample of the sequence {x[n]} is a real sequence, if the n-th sample x[n] is real for all values of n Otherwise, {x[n]} is a complex sequence
    • DISCRETE-TIME SIGNALS: TIME-DOMAIN REPRESENTATION A complex sequence {x[n]} can be written as {x[n]} = {xre [n]} + j{xim [n]} where xre [n] and xim [n] are the real and imaginary parts of x[n] The complex x * [ n]} = {xre [ n]} of {x[n]} is given by { conjugate sequence − j{xim [n]} Often the braces are ignored to denote a sequence if there is no ambiguity
    • DISCRETE-TIME SIGNALS:TIME-DOMAIN REPRESENTATION Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued - Digital signals in which samples are discrete- valued Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation
    • DISCRETE-TIME SIGNALS:TIME-DOMAIN REPRESENTATION A discrete-time signal may be a finite-length or an infinite-length sequence Finite-length (also called finite-duration or finite-extent) sequence is defined only for a finite time interval: N ≤ n ≤ N 1 2 where − ∞ < N1 and N 2 < ∞ with N1 ≤ N 2 Length or duration of the above finite-length sequence is N = N − N + 1 2 1
    • DISCRETE-TIME SIGNALS:TIME-DOMAIN REPRESENTATION A length-N sequence is often referred to as an N-point sequence The length of a finite-length sequence can be increased by zero-padding, i.e., by appending it with zeros A right-sided sequence x[n] has zero-valued samples for n < N1 If N ≥ 0, a right-sided sequence is called a 1 causal sequence
    • DISCRETE-TIME SIGNALS:TIME-DOMAIN REPRESENTATION A left-sided sequence x[n] has zero-valued samples for n > N 2 If N 2 ≤ 0,a left-sided sequence is called a anti- causal sequence A right-sided sequence n N1 A left-sided sequence N2 n
    • OPERATIONS ON SEQUENCES:BASIC OPERATIONSProduct (modulation) operation:  Modulator x[n] × y[n] w[n] y[n] = x[n] ⋅ w[n] An application is in forming a finite-length sequence from an infinite-length sequence by multiplying the latter with a finite-length sequence called a window sequence Process called windowing
    • BASIC OPERATIONS Addition operation:  Adder x[n] + y[n] w[n] y[n] = x[n] + w[n] Multiplication operation  Multiplier A x[n] y[n] y[n] = A ⋅ x[n]
    • BASIC OPERATIONS Time-shifting operation: y[n] = x[n − N ] where N is an integer(i)If N > 0, it is delaying operation  Unit delay x[n] z −1 y[n] y[n] = x[n − 1](ii)If N < 0, it is an advance operation  Unit advance x[n] z y[n] y[n] = x[n + 1]
    • BASIC OPERATIONS Time-reversal (folding) operation: y[n] = x[−n] Branching operation: Used to provide multiple copies of a sequence x[n] x[n] x[n]
    • ALIASING  Aliasing:  If you sample too slow, the high frequency components will become irregular noise at the sampling frequency  They are noises that are in the same frequency range of your signal!!!• Look at the samples alone• Can you tell which of the two frequencies the sampled series represents?• Either of the two signals could produce the samples, i.e., the signals are “aliases” of each other
    • CLASSIFICATION OF SEQUENCES: ENERGY AND POWER SIGNALSPower Signal An infinite energy signal with finite average power is called a power signalExample - A periodic sequence which has a finite average power but infinite energy
    • Energy Signals A finite energy signal with zero averagepower is called an energy signalExample - a finite-length sequence which hasfinite energy but zero average power 3(−1) n ,  others x[n] =   0,  n < 0, n ≥ 10
    • OTHER TYPES OF CLASSIFICATIONS A sequence x[n] is said to be bounded if x[n] ≤ Bx < ∞ A sequence x[n] is said to be absolutely summable if ∞ ∑ x[n] < ∞ n = −∞ A sequence x[n] is said to be square-summable if ∞ 2 ∑ x[n] < ∞ n = −∞
    • BASIC SEQUENCES Unit impulse Unit step Sinusoidal Exponential Periodic Random
    • BASIC SEQUENCESUnit sample sequence - 1, n = 0 δ [ n] =  1 0, n ≠ 0 n –4 –3 –2 –1 0 1 2 3 4 5 6Unit step sequence - 1, n ≥ 0 µ[ n] =  0, n < 0 1 n –4 –3 –2 –1 0 1 2 3 4 5 6
    • BASIC SEQUENCES Real sinusoidal sequence - x[n] = A cos(ω o n + φ ) where A is the amplitude,ω o is the angular frequency, and φ is the phase of x[n] Example - ωo = 0.1 2 1 Amplitude 0 -1 -2 0 10 20 30 40 Time index n
    • DISCRETE-TIME SYSTEMSINTRO. TO DISCRETE-TIME SYSTEMS The difference equation, the impulse response and the system function are equivalent characterization of the input/output relation of a LTI Discrete-time systems. LTI system can be modeled using : 1. A Difference/Differential equation, y(n) = x[n] + x[n-1] + … 2. Impulse Response, h(n) 3. Transfer Function, H(z) The systems that described by the difference equations can be represented by structures consisting of an interconnection of the basic operations of addition, multiplication by a constant or signal multiplication, delay and advance.
    • DISCRETE-TIME SYSTEMS The Adder, Multiplier, Delay & Advance is shown below: 1. Adder : 2. Multiplier : Modulator:
    • 3. Delay :4. Advance :
    •  Time In-variant & Time-variant block diagram : => Time-Invariant => Time-variant => Time-variant => Time-variant
    •  Aliasing  Unable to distinguish two continuous signals with different frequencies based on samples  Frequencies higher than Nyquist frequency Anti-aliasing  Low-pass filter the frequencies above Nyquist frequency
    • DISCRETE-TIME SYSTEM Discrete-time system has discrete-time input and output signals
    • DIGITAL SYSTEM A discrete-time system is digital if it operates on discrete-time signals whose amplitudes are quantized Quantization maps each continuous amplitude level into a number The digital system employs digital hardware 1. explicitly in the form of logic circuits 2. implicitly when the operations on the signals are executed by writing a computer program
    • Discrete-time (DT) system is `sampled data’ system: Input signal u[k] is a sequence of samples (=numbers) ..,u[-2],u[-1],u[0],u[1],u[2],… System then produces a sequence of output samples y[k] ..,y[-2],y[-1],y[0],y[1],y[2],…Will consider linear time-invariant (LTI) DT systems: Linear : input u1[k] -> output y1[k] u[k] y[k] input u2[k] -> output y2[k] hence a.u1[k]+b.u2[k]-> a.y1[k]+b.y2[k] Time-invariant (shift-invariant) input u[k] -> output y[k], hence input u[k-T] -> output y[k-T]
    • Causal systems: iff for all input signals with u[k]=0,k<0 -> output y[k]=0,k<0Impulse response: input …,0,0,1,0,0,0,...-> output …,0,0,h[0],h[1],h[2],h[3],...General input u[0],u[1],u[2],u[3]: (cfr. linearity & shift- invariance!)  y[0] h[0] 0 0 0   y[1]   h[1] h[0] 0 0  u[0]      y[2] h[2] h[1] h[0] 0   u[1]   = .   y[3]  0 h[2] h[1] h[0] u[2]  y[4]  0   0 h[2] h[1]  u[3]      y[5]  0    0 0 h[2] `Toeplitz’ matrix
    • Convolution: u[0],u[1],u[2],u[3] y[0],y[1],... y[0]  h[0] 0 0 0  y[1]   h[1] h[0] 0 0  u[0] h[0],h[1],h[2],0,0,...    y[2] h[2] h[1] h[0] 0   u[1]  = .  y[3]  0 h[2] h[1] h[0] u[2] y[4]  0   0 h[2] h[1]  u[3]    y[5]  0   0 0 h[2]y[k ] = ∑h[k − i ].u[i ] = h[k ] * u[k ] = `convolution sum’ i
    • Z-Transform: ∆ ∆ ∆H ( z ) = ∑ h[i ]. z −i U ( z ) = ∑ u[i ]. z −i Y ( z ) = ∑ y[i ]. z −i i i i  y[0] h[0] 0 0 0   y[1]   h[1] h[0] 0 0  u[0]      y[2] h[2] h[1] h[0] 0   u[1] [1 z −1 z −2 z −3 z −4 −5 z . ] =1 z −1 [z −2 z −3 z −4 −5 z . ] . h[2] h[1] h[0] u[2]   y[3]  0  y[4]  0   0 h[2] h[1]  u[3]      y[5]  0 0 0 h[2]            Y ( z) H ( z).1 z−1 z−2 z−3            ⇒ Y ( z ) = H ( z ).U ( z ) H(z) is `transfer function’
    • Z-Transform : input-output relation Y ( z ) = H ( z ).U ( z ) may be viewed as `shorthand’ notation (for convolution operation/Toeplitz-vector product) stability bounded input u[k] -> bounded output y[k] ∑ h[k ]  ∞ --iff k --iff poles of H(z) inside the unit circle (for causal,rational systems)
    • Example-1 : `Delay operator’ u[k] y[k]=u[k-1] Impulse response is …,0,0,0, 1,0,0,0,… Transfer function is ∆ −1 H ( z) = zExample-2 : Delay + feedback u[k] y[k] Impulse response is …,0,0,0, 1,a,a^2,a^3… + ∆ Transfer function is H ( z ) = z −1 + a. z −2 + a 2 . z −3 + a 3. z −4 + ... a x ⇒ H ( z ) − a. z −1H ( z ) = z −1 z −1 ⇒ H ( z) = 1 − a. z −1
    • LINEAR TIME-INVARIANT(LTI) SYSTEM Discrete-time system is LTI if its input-output relationship can be described by the linear constant coefficients difference equationThe output sample y(ν) might depend on all input samplesthat can be represented as y (ν ) = Φ ( x(k ))