DIGITAL SIGNAL PROCESSING LECTURE 1 Fall 2010 2K8-5 th Semester Tahir Muhammad tmuhammad_07@yahoo.com Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice Hall Inc.
Introduction 2
Examples 3
Examples 4
Examples 5
Applications 6
Why DSP? 7
Textbook and References Textbook: Oppenheim, A.V., Schafer, R.W, "Discrete-Time Signal Processing", 2 nd Edition, Prentice-Hall, 1999. Reference Books: (4 th Edition), John G. Proakis, Dimitris K Manolakis Vinay K. Ingle, John G. Proakis, Digital Signal Processing using MATLAB, 2 nd Ed., Thomson, 2007. 8
Course Outline Chapter # 1 [Introduction] Chapter # 2 [Discrete-Time Signals and Systems] Chapter # 3 [The Z-Transform] Chapter # 4 [Sampling of Continuous-Time Signals] Chapter # 6 [Structures for Discrete-Time Systems] Chapter # 7 [Filter Design Techniques] Chapter # 8 [The Discrete Fourier Transform] 9
Definitions of DSP Signal A function of independent variables such as time, distance, position, temperature and pressure Signals are analog in nature(continuous) such as human voice, electrical signal(voltage or current), radio wave, optical, audio, and so on which contains a stream of information or data. Or may be discrete such as temperature, stock, etc. Processing Operating in some fashion on signal to extract some useful information 10
Definitions of DSP Concerned with the representation of signals by sequence of numbers or symbols and the processing of these sequence The purpose of such processing may be to estimate characteristic parameters or transform a signal 11
Characterization and classification of signals Depending on number of independent variables 1-D Signals : speech signal 2-D Signals : Image signal M-D Signals : Video signal Based on independent variables Continuous-time time signal: signal is defined at every instant of time Discrete-time signal: takes certain numerical values at specified discrete instants of time, basically a sequence of numbers 12
Types of Signals 13
Types of Signals 14
Definition of Discrete-time Signal & System Define at equally spaced discrete value of time Represented as a sequence of numbers The sequence is denoted as x[n]; where n is an integer in the range of - to. A discrete time is represented as {x(n)} {x(n)} = {... 0.95, -0.2, 2.17, 1.1, 0.2, -3.67, 2.9...} Arrow indicate time index, n = 0 15
Definition of Discrete-time Signal & System Define at equally spaced discrete value of time Represented as a sequence of numbers The sequence is denoted as x[n]; where n is an integer in the range of - to. A discrete time is represented as {x(n)} {x(n)} = {... 0.95, -0.2, 2.17, 1.1, 0.2, -3.67, 2.9...} Arrow indicate time index, n = 0 16
Definition of Discrete-time Signal & System The discrete-time signal is obtained by periodically sampling a continuous-time i signal at uniform time interval. The sampling interval or period is denoted as T s. Thus the sampling frequency can be defined d as reciprocal of T s, namely, F s = 1 / T s. When the analog is sampled at certain period of time, the discrete-time signal can be written as below :- x[n] [ ] = x a [] [t] = x a [nt s ], n =,-2,-1,0,1,2,... 1012 17
Definition of Discrete-time Signal & System Periodic Sampling of an analog signal is shown below: 18
Operation on Sequence If the input signal to the systems is DTS, the output of the systems will be DTS. INPUT x[n] SYSTEM OUTPUT y[n] 19
Operation on Sequence Product/modulation w 1 [n] = x[n].y[n] Multiplication/scaling w 2 [n] = Ax[n] Addition w 3 [n] = x[n] + y[n] 20
Operation on Sequence Time shifting w 4 [n] = x[n N], N is an integer If N > 0 ; it s a delay operation ; is a unit delay If N < 0 ; its an advance operation w 5 [n] = x[n + 1] ; is a unit advance Time reversal w 6 [n] = x[- n] 21
SEQUENCE REPRESENTATION Unit sample/unit impulse δ[n] = {1, n = 0; 0, n 0 } Unit sample shifted by k samples is δ[n- k ] = {1, n = k; 0, n k} 22
SEQUENCE REPRESENTATION Unit Step µ[n] = {1, n 0; 0, n < 0 } Unit step shifted by k samples is µ[n - k] = {1, n k; 0, n < k } 23
Sequence Representation Unit sample and unit step are related as follows µ δ [ n] = δ[ n m] = m= 0 k= [ n ] = µ [ n] µ [ n 1] n δ [ k] 24
Sequence Representation Sinusoidal x [ n ] = A cos( ω n + φ ), < n o < 25
Sinusoidal 26
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Sequence Representation Real Exponential x[n] = Aα n 28
Sequence Representation Complex Exponential x[n] = Aе jωn ; ω frequency of complex exponential sinusoid, A is a constant 29
Introduction to LTI System Discrete-time Systems Function: to process a given input sequence to generate an output sequence x[n] Input sequence Discrete-time time system y[n] Output sequence Fig: Example of a single-input, single-output system 30
Introduction to LTI System Classification of Discrete-time System Linear DTS x[n] [ ] y[n] [ ] = αx 1 [n] + βx 2 [n] = αy 1 [n] + βy 2 [n] 31
Introduction to LTI System Classification of Discrete-time System 32
Classification of Discrete-time System Time Invariant system 33
Introduction to LTI System Classification of Discrete-time System Causal System Changes in output samples do not precede changes in input samples y[n o ] depends only on x[n] for n n o Example: y[n] = x[n]-x[n-1] 34
Introduction to LTI System Classification of Discrete-time System Stable System For every bounded input, the output is also bounded (BIBO) Is the y[n] is the response to x[n], and if x[n] < B x for all value of n then y[n] < B y for all value of n Where B x and B y are finite positive constant 35
Introduction to LTI System Impulse and Step Response If the input to the DTS system is Unit Impulse (δ[n]), then output of the system will be Impulse Response (h[n]). If the input to the DTS system is Unit Step (µ[n]), then output of the system will be Step Response (s[n]). 36
Impulse Response 37
Input-Output Relationship A Linear time-invariant invariant system satisfied both the linearity and time invariance properties. An LTI discrete-time system is characterized by its impulse response Example: x[n] = 0.5δ[n+2] + 1.5δ[n-1] - δ[n-4] will result in y[n] = 0.5h[n+2] + 1.5h[n-1] - h[n-4] 38
Input-Output Relationship x[n] [ ] can be expressed in the form k = x [ n] = x[ k ] δ [ n k ] where x[k] denotes the kth sample of sequence {x[n]} The response to the LTI system is y[ n] = k = or represented as x[ k] h [ n k] = k = x[ n y[ [ n ] = x [ n ] h [ n ] k] h [ k] 39
Computation of Discrete Convolution 40
Input-output Relationship Properties of convolution Properties of convolution Commutative ] [ ] [ ] [ ] [ n x n x n x n x Associative ] [ ] [ ] [ ] [ 1 2 2 1 n x n x n x n x = Di t ib ti ] [ ] [ ] [ ] [ ]) [ ] [ ( ] [ 3 1 2 1 3 2 1 n x n x n x n x n x n x n x + = + Distributive ]) [ ] [ ( ] [ ] [ ]) [ ] [ ( 3 2 1 3 2 1 n x n x n x n x n x n x = 41
Properties of LTI Systems Stability if and only if, sum of magnitude of Impulse Response, h[n] is finite S = h [ n ] = n = h n < 42
Properties of LTI Systems Causality if and only if Impulse Response,h[n] = 0 for all n < 0 43
Properties of LTI Systems 44