ECE 431 Digital Signal Processing Lecture Notes

1 ECE 431 Digital Signal ProcessingLecture NotesProf. Dan CobbContents1 Introduction22 Review of the DT Fourier De nition and Properties .. Periodic Convolution .. Fourier Series .. 93 Time and Frequency Domain Analysis .. Aliasing .. The Nyquist Theorem .. Anti-Aliasing Filters .. Downsampling .. Upsampling .. Change of Sampling Frequency .. 204 CT Signal Hybrid Systems .. Ideal Signal Reconstruction .. The Zero-Order Hold .. A/D and D/A Converters .. Digital Filters .. 335 The Discrete Fourier De nition and Properties .. Circular Operations .. Fast Fourier Transform Algorithms .. Zero-Padding .. 416 Applications of the Spectral Analysis .. Linear Convolution .. Windowing .. 4517 The CT Laplace Transform.

2 The DT Laplace Transform and thez-Transform .. Properties .. 568 DT Systems and the LTI Systems .. Di erence Equations .. Rational Transfer Functions .. Poles and Zeros .. Partial Fraction Expansion .. Causality and Stability of Di erence Equations .. Choice of Initial Conditions .. Zeroth-Order Di erence Equations .. 729 Analog Filter Introduction .. The Butterworth Filter .. The Chebyshev Filter .. Causality .. Frequency Scaling, Highpass, and Bandpass Transformations .. Zero Phase Filters .. Phase Delay, Linear Phase, and Phase Distortion .. 8410 IIR Conversion of CT to DT Filters .. Recursive Structures for Causal IIR Filters .. The Anti-Causal Case .. 9611 FIR Causal FIR Filters .. Zero-Phase FIR Filters.

3 Choice of Window Length .. Linear Phase FIR Filters .. Di erence Equation Implementation .. DFT Implementation .. 1071 IntroductionDigital Signal Processing (DSP) is the application of a Digital computer to modify an analog ordigital Signal . Typically, the Signal being processed is either temporal, spatial, or both. For example,an audio Signal is temporal, while an image is spatial. A movie is both temporal and spatial. Theanalysis of temporal signals makes heavy use of the Fourier transform in one time variable and onefrequency variable. Spatial signals require two independent variables. Analysis of such signals relieson the Fourier transform in two frequency variables ( ECE 533). In ECE 431, we will restrictourselves to temporal Signal main goal is to be able to design Digital LTI lters.

4 Such lters are using widely in applica-tions such as audio entertainment systems, telecommunication and other kinds of communicationsystems, radar, video enhancement, and biomedical engineering. The rst half of the course willbe spent reviewing and developing the fundamentals necessary to understand the design of Digital lters. Then we will examine the basic types of lters and the myriad of design issues the outset, the student should recognize that there are two distinct classes of applicationsfor Digital are those where data streams into the lter andmustbe processed immediately. A signi cant delay in generating the lter output data cannot betolerated. Such applications include communication networks of all sorts, musical performance,public address systems, and patient monitoring. Real-time ltering is sometimes calledon-lineprocessing and is based on the theory of causal are those where a lter is used to process a pre-existing ( stored) le of data.

5 In this case, the engineer is typically allotted a large amount of time over which theprocessing of data may be performed. Such applications include audio recording and mastering,image Processing , and the analysis of seismic data. Non-real-time ltering is sometimes calledo -lineprocessing and is based on the theory of noncausal systems. In these applications, thefact that noncausal lters may be employed opens the door to a much wider range of lters andcommensurately better results. For example, one problem typical of real-time ltering is phasedistortion, which we will study in detail in this course. Phase distortion can be eliminated completelyif noncausal lters are rst part of the course will consist of review material from signals and systems. Throughoutthe course, we will rely heavily on the theory of Fourier transforms, since much of Signal processingand lter theory is most easily addressed in the frequency domain.

6 It will be convenient to refer tocommonly used transform concepts by the following acronyms:CTFT: Continuous-Time Fourier TransformDTFT: Discrete-Time Fourier TransformCFS: Continuous-Time Fourier SeriesDFS: Discrete-Time Fourier SeriesLT: Laplace TransformDFT: Discrete Fourier TransformZT:z-TransformAn I preceding an acronym indicates Inverse as in IDTFT and IDFT. All of these conceptsshould be familiar to the student, except the DFT and ZT, which we will de ne and study in Review of the DT Fourier De nition and PropertiesTheCT Fourier transform (CTFT)of a CT signalx(t)isFfx(t)g=X(j!) =Z1 1x(t)e j!tdt:TheInverse CT Fourier Transform (ICTFT)isF 1fX(j!)g=12 Z1 1X(j!)ej!td!:3 Recall the CT unit impulse (t);the DT unit impulse [n];and their basic properties:Z1 1 (t)dt= 1;1Xn= 1 [n] = 1x(t) (t ) =x( ) (t ); x[n] [n m] =x[m] [n m]x(t) (t ) =x(t ); x[n] [n m] =x[n m](sifting property).

7 For any DT signalx[n];we may de ne itsDT Fourier transform (DTFT)by associating withx[n]the CT impulse trainx(t) =1Xn= 1x[n] (t n)and taking the transformX(j!) =Z1 11Xn= 1x[n] (t n)e j!tdt=1Xn= 1x[n]e j!nZ1 1 (t n)dt=1Xn= 1x[n]e j!n:Thus we may writeX(j!) =1Xn= 1x[n] ej! n;expressingXas a function ofej!:For this reason, the DTFT is normally writtenX ej! =1Xn= 1x[n]e j!n:Technically, this is an abuse of notation, since the twoX s are actually di erent functions, butthe meaning will usually be clear from context. In order to help distinguish between CT and DTtransforms, we will henceforth denote the frequency variable in DT transforms as :X ej =1Xn= 1x[n]e j n:( )Although your text writes frequency as!for both CT and DT transforms, the notation hasnumerous advantages. For example, it keeps the units of frequency straight:!

8 Is in rad/sec, while is in Euler s formula,ej = cos +jsin ;soej is periodic with fundamental period2 :Hence,X ej has period2 :We also writeFfx[n]g=X ej 4andx[n] !X ej :TheInverse DTFTisF 1 X ej =x[n] =12 Z2 0X ej ej nd :The integral may be evaluated over any interval of length2 :Properties:(See O&S Table on p. 55 and Table on p. 58.)Periodicity:X ej( +2 ) =X ej Linearity: x[n] ! X ej x1[n] +x2[n] !X1 ej +X2 ej Time Shift:x[n n0] !e j n0X ej Frequency Shift:ej 0nx[n] !X ej( 0) Time/Frequency Scaling:x(N)[n] = x nN ;nNan integer0;elsex(N)[n] !X ej N Convolution:x1[n] x2[n] =1Xm= 1x1[n m]x2[m]x1[n] x2[n] !X1 ej X2 ej Multiplication:x1[n]x2[n] !12 Z2 0X1 ej( ) X2 ej d Time Di erencing:x[n] x[n 1] ! 1 e j X ej Accumulation:nXm= 1x[m] !11 e j X ej Frequency Di erentiation:nx[n] !jdX ej d Conjugation:x [n] !

9 X e j 5Re ection:x[ n] !X e j Real Time Signal :x[n]real() X ej even\X ej oddEven-Odd: x[n]even()X ej realx[n]odd()X ej imaginaryParseval s Theorem:1Xn= 1x1[n]x 2[n] =12 Z2 0X ej Y ej d Example DT unit impulse [n] = 1; n= 00; n6= 0has DTFTFf [n]g=1Xn= 1 [n]e j n= 1:Example unit impulse train in frequencyX ej =1Xk= 1 ( 2 k)has Inverse DTFTx[n] =12 Z2 0 1Xk= 1 ( 2 k)!ej nd =12 1Xk= 1Z2 0 ( 2 k)ej2 knd =12 1Xk= 1Z2 0 ( 2 k)d :ButZ2 0 ( 2 k)d = 1; k= 00; k6= 0;sox[n] =12 and1 !2 1Xk= 1 ( 2 k):6 Example ne the DT rectangular windowwN[n] = 1; n N 10;else:The DTFT isWN ej =1Xn= 1wN[n]e j n=N 1Xn=0e j n=N 1Xn=0 e j n=1 e jN 1 e j =ej 2ej 2 ej(N 1) 2 e j(N+1) 2 e j(N 1) 21 e j =ejN 2 e jN 2ej 2 e j 2e j(N 1) 2=sinN 2sin 2e j(N 1) 2:The real factor inWN ej is the periodicsinc function:Figure (See O&S Table on p.)

10 62 for further examples.) Periodic ConvolutionThe multiplication property involves theperiodic convolutionX1 ej X2 ej =Z2 0X1 ej( ) X2 ej d :7 SinceX ej andY ej both have period2 ;the linear ( ordinary) convolution blows up(except in trivial cases):Z1 1X1 ej( ) X2 ej d =1Xi= 1Z2 (i+1)2 iX1 ej( ) X2 ej d =1Xi= 1Z2 0X1 ej( ) X2 ej d =1:On the other hand, the periodic convolution is well-de ned with period2 :Example the square waveX ej = 1;0 < 0; <2 with period2 :We wish to convolveX ej with itself. We need to look at two cases:1)0 < Z2 0X ej( ) X ej d =Z 01d = Figure ) <2 Z2 0X ej( ) X ej d =Z 1d = 2 Figure periodic convolution is the triangle waveX ej X ej = ;0 < 2 ; <2 with period2 :8 Periodic convolution may also be de ned for sequences. Ifx1[n]andx2[n]have periodN;thenx1[n] x2[n] =N 1Xm=0x1[n m]x2[m]has Fourier SeriesLetakbe a sequence of complex numbers with periodNand 0=2 N:Suppose we restrict attention to DT signals whose DTFT s are impulse trains of the formX ej = 2 1Xk= 1ak ( 0k):( )Thenx[n] =12 Z2 0X ej ej nd =Z2 0 1Xk= 1ak ( 0k)ej nd =1Xk= 1akej 0knZ2 0 ( 0k)d :ButZ2 0 ( 0k) = 1;0 k N 10;else;sox[n] =N 1Xk=0akej 0kn:( )Note thatej 0k(n+N)=ej 0kn+ej 0kN=ej 0kn+ej2 k=ej 0kn;soej 0knand, therefore,x[n]have periodN:Formula ( ) is theDT Fourier series (DFS)representation of the periodic signalx[n]:The(complex) numbersakare the Fourier coe cients ofx[n]:In this case, we writex[n] !