Digital Signal Processing Lecture Notes

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ECE431 Digital Signal Processing LectureNotes Prof. Dan Cobb Contents 1 Introduction 2 2 Review of the DT Fourier Transform 3 2.1 De…nition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Periodic Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Sampling 12 3.1 Time and Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 The Nyquist Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Anti-Aliasing Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5 Downsampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 Upsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.7 Change of Sampling Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 CT Signal Reconstruction 21 4.1 Hybrid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Ideal Signal Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 The Zero-Order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 A/D and D/A Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5 Digital Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5 The Discrete Fourier Transform 34 5.1 De…nition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Circular Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.3 Fast Fourier Transform Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.4 Zero-Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Applications of the DFT 41 6.1 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Linear Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.3 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 17 The z-Transform 52 7.1 The CT Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2 The DT Laplace Transform and the z-Transform. . . . . . . . . . . . . . . . . . . . 53 7.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8 DT Systems and the ZT 57 8.1 LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8.2 Di¤erence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 8.3 Rational Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8.4 Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.5 Partial Fraction Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.6 Causality and Stability of Di¤erence Equations . . . . . . . . . . . . . . . . . . . . 68 8.7 Choice of Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.8 Zeroth-Order Di¤erence Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9 Analog Filter Design 73 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9.2 The Butterworth Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9.3 The Chebyshev Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 9.4 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.5 Frequency Scaling, Highpass, and Bandpass Transformations . . . . . . . . . . . . . 78 9.6 Zero Phase Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 9.7 Phase Delay, Linear Phase, and Phase Distortion . . . . . . . . . . . . . . . . . . . 84 10IIR Filters 87 10.1 Conversion of CT to DT Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 10.2 Recursive Structures for Causal IIR Filters . . . . . . . . . . . . . . . . . . . . . . . 92 10.3 The Anti-Causal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 11FIR Filters 98 11.1 Causal FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 11.2 Zero-Phase FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.3 Choice of Window Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11.4 Linear Phase FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 11.5 Di¤erence Equation Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . 105 11.6 DFT Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 1 Introduction Digital Signal Processing (DSP) is the application of a digital computer to modify an analog or digitalsignal. Typically,thesignalbeingprocessediseithertemporal,spatial,orboth. Forexample, an audio signal is temporal, while an image is spatial. A movie is both temporal and spatial. The analysis of temporal signals makes heavy use of the Fourier transformin one time variable and one frequencyvariable. Spatialsignalsrequiretwoindependentvariables. Analysisofsuchsignalsrelies on the Fourier transform in two frequency variables (e.g. ECE 533). In ECE 431, we will restrict ourselves to temporal signal processing. 2OurmaingoalistobeabletodesigndigitalLTI…lters. Such…ltersareusingwidelyinapplica- tions such as audio entertainment systems, telecommunication and other kinds of communication systems, radar, video enhancement, and biomedical engineering. The …rst half of the course will be 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 surrounding them. Fromthe outset, the student should recognize that there are two distinct classes of applications for digital …lters. Real-time applications are those where data streams into the …lter and must be processed immediately. A signi…cant delay in generating the …lter output data cannot be tolerated. Such applications include communication networks of all sorts, musical performance, public address systems, and patient monitoring. Real-time …ltering is sometimes called on-line processing and is based on the theory of causal systems. Non-real-time applications are those where a …lter is used to process a pre-existing (i.e. stored) …le of data. In this case, the engineer is typically allotted a large amount of time over which the processing 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 called o¤-line processing and is based on the theory of noncausal systems. In these applications, the fact that noncausal …lters may be employed opens the door to a much wider range of …lters and commensurately better results. For example, one problem typical of real-time …ltering is phase distortion,whichwewillstudyindetailinthiscourse. Phasedistortioncanbeeliminatedcompletely if noncausal …lters are permitted. The…rstpartofthecoursewillconsistofreviewmaterialfromsignalsandsystems. Throughout thecourse, wewill relyheavilyonthetheoryof Fouriertransforms, sincemuchof signal processing and …lter theory is most easily addressed in the frequency domain. It will be convenient to refer to commonly used transform concepts by the following acronyms: CTFT: Continuous-Time Fourier Transform DTFT: Discrete-Time Fourier Transform CFS: Continuous-Time Fourier Series DFS: Discrete-Time Fourier Series LT: Laplace Transform DFT: Discrete Fourier Transform ZT: z-Transform An “I”preceding an acronym indicates “Inverse”as in IDTFT and IDFT. All of these concepts shouldbefamiliartothestudent,excepttheDFTandZT,whichwewillde…neandstudyindetail. 2 Review of the DT Fourier Transform 2.1 De…nition and Properties The CT Fourier transform (CTFT) of a CT signal x (t) is Z 1 jt Ffx (t)g =X (j) = x (t)e dt: 1 The Inverse CT Fourier Transform (ICTFT) is Z 1 1 1 jt F fX (j)g = X (j)e d: 2 1 3Recall the CT unit impulse  (t); the DT unit impulse  n; and their basic properties: Z 1 1 X  (t)dt = 1;  n = 1 1 n=1 x (t) (t) =x () (t); x n nm =x m nm x (t) (t) =x (t); x n nm =x nm (sifting property). ForanyDTsignalx n; wemayde…neitsDT Fourier transform (DTFT) byassociatingwithx n the CT impulse train 1 X x (t) = x n (tn) n=1 and taking the transform Z 1 1 X jt X (j) = x n (tn)e dt 1 n=1 Z 1 1 X jn = x ne  (tn)dt 1 n=1 1 X jn = x ne : n=1 Thus we may write 1 X  n j X (j) = x n e ; n=1 j expressing X as a function of e : For this reason, the DTFT is normally written 1 X  j jn X e = x ne : n=1 Technically, this is an abuse of notation, since the two X’s are actually di¤erent functions, but the meaning will usually be clear from context. In order to help distinguish between CT and DT transforms, we will henceforth denote the frequency variable in DT transforms as : 1 X  j j n X e = x ne : (2.1) n=1 Although your text writes frequency as for both CT and DT transforms, the notation has numerous advantages. For example, it keeps the units of frequency straight: is in rad/sec, while is in radians. By Euler’s formula, j e = cos +j sin ;  j j soe is periodic with fundamental period 2: Hence, X e has period 2: We also write  j Ffx ng =X e 4and  j x n X e : The Inverse DTFT is Z 2    1 1 j j j n F X e =x n = X e e d : 2 0 The integral may be evaluated over any interval of length 2: Properties: (See O&S Table 2.1 on p. 55 and Table 2.2 on p. 58.) Periodicity:   j( +2) j X e =X e Linearity:   j x n X e   j j x n +x n X e +X e 1 2 1 2 Time Shift:  j n j 0 x nn e X e 0 Frequency Shift:  j n j( ) 0 0 e x n X e Time/Frequency Scaling:    n n x ; an integer N N x n = (N) 0; else  j N x n X e (N) Convolution: 1 X x nx n = x nmx m 1 2 1 2 m=1   j j x nx n X e X e 1 2 1 2 Multiplication: Z 2   1 j( ) j x nx n X e X e d 1 2 1 2 2 0 Time Di¤erencing:   j j x nx n 1 1e X e Accumulation: n X  1 j x m X e j 1e m=1 Frequency Di¤erentiation:  j dX e nx n j d Conjugation:    j x n X e 5Re‡ection:  j x n X e Real Time Signal:   j X e even  x n real () j \X e odd Even-Odd:   j x n even () X e real  j x n odd () X e imaginary Parseval’s Theorem: Z 1 2 X   1  j  j x nx n = X e Y e d 1 2 2 0 n=1 Example 2.1 The DT unit impulse  1; n = 0  n = 0; n6= 0 has DTFT 1 X j n Ff ng =  ne = 1: n=1 Example 2.2 The unit impulse train in frequency 1 X  j X e =  ( 2k) k=1 has Inverse DTFT Z 1 2 X 1 j n x n =  ( 2k) e d 2 0 k=1 Z 1 2 X 1 j2kn =  ( 2k)e d 2 0 k=1 Z 1 2 X 1 =  ( 2k)d : 2 0 k=1 But Z  2 1; k = 0  ( 2k)d = ; 0; k6= 0 0 so 1 x n = 2 and 1 X 1 2  ( 2k): k=1 6Example 2.3 De…ne the DT rectangular window  1; nN 1 w n = : N 0; else The DTFT is 1 X  j j n W e = w ne N N n=1 N1 X j n = e n=0 N1 X  n j = e n=0 jN 1e = j 1e   (N1) (N+1) (N1) j j j 2 2 2 j e e e 2 e = j j 1e 2 e N N j j 2 2 (N1) e e j 2 = e j j 2 2 e e N sin (N1) 2 j 2 = e : sin 2  j The real factor in W e is the “periodic sinc”function: N Figure 2.1 (See O&S Table 2.3 on p. 62 for further examples.) 2.2 Periodic Convolution The multiplication property involves the periodic convolution Z 2     j j j( ) j X e X e = X e X e d: 1 2 1 2 0 7   j j Since X e and Y e both have period 2; the linear (i.e. ordinary) convolution blows up (except in trivial cases): Z Z 1 1 2(i+1) X     j( ) j j( ) j X e X e d = X e X e d 1 2 1 2 1 2i i=1 Z 1 2 X   j( ) j = X e X e d 1 2 0 i=1 =1: On the other hand, the periodic convolution is well-de…ned with period 2: Example 2.4 Consider the square wave   1; 0  j X e = 0;  2  j with period 2: We wish to convolve X e with itself. We need to look at two cases: 1) 0  Z Z 2   j( ) j X e X e d = 1d = 0 0 Figure 2.2 2)  2 Z Z 2    j( ) j X e X e d = 1d = 2 0  Figure 2.3 The periodic convolution is the triangle wave    ; 0  j j X e X e = 2 ;  2 with period 2: 8Periodic convolution may also be de…ned for sequences. If x n andx n have periodN; then 1 2 N1 X x nx n = x nmx m 1 2 1 2 m=0 has period N: 2.3 Fourier Series Let a be a sequence of complex numbers with period N and k 2 = : 0 N Suppose we restrict attention to DT signals whose DTFT’s are impulse trains of the form 1 X  j X e = 2 a  ( k): (2.2) k 0 k=1 Then Z 2  1 j j n x n = X e e d 2 0 Z 1 2 X j n = a  ( k)e d k 0 0 k=1 Z 1 2 X j kn 0 = a e  ( k)d : k 0 0 k=1 But  Z 2 1; 0kN 1  ( k) = ; 0 0; else 0 so N1 X j kn 0 x n = a e : (2.3) k k=0 Note that j k(n+N) j kn j kN 0 0 0 e =e +e j kn j2k 0 =e +e j kn 0 =e ; j kn 0 soe and, therefore, x n have period N: Formula (2.3) is the DT Fourier series (DFS) representation of the periodic signal x n: The (complex) numbers a are the Fourier coe¢ cients of x n: In this case, we write k x n a : k 9Every DT signal x n with period N has DTFT (2.2) and DFS (2.3). The Fourier coe¢ cients also have period N and may be derived from x n via the summation N1 X 1 j kn 0 a = x ne : (2.4) k N n=0 In both the DFS (2.3) and its inverse (2.4), the sum may be taken over any interval of length N: The properties of the DFS are similar to those of the DTFT. (See O&S Table 8.1 on p. 634.) Linearity:  x n a k x n +x n a +b 1 2 k k Time-Shift: j kn 0 0 x nn e a 0 k Frequency Shift j k n 0 0 e x n a kk 0 Time/Frequency Scaling: 1 x n a (period MN) (M) k M Convolution: N1 X x nmx m Na b 1 2 k k m=0 Multiplication: N1 X x nx n a b 1 2 ki i i=0 Time Di¤erencing:  j k 0 x nx n 1 1e a k Accumulation: n X 1 x m a (only for a = 0) k 0 j k 0 1e m=1 Frequency Di¤erencing:  j n 0 1e x n a a k k1 Conjugation:   x n a k Re‡ection: x n a k Real Time Signal:  jaj even k x n real () \a odd k Even-Odd:  x n even () a real k x n odd () a imaginary k 10Parseval’s Theorem: N1 N1 X X 1   x nx n = a b 1 k 2 k N n=0 k=0 ManyofthepropertiesoftheDFSappeartobe“mirrorimages”ofoneanother. Thisprincipleis calledduality andistheresultofthesimilarityofequations(2.3)and(2.4). Thesamephenomenon can be seen with regard to transforms of speci…c signals. Example 2.5 Find the DTFT and DFS of 1 X x n =  nmN: m=1 The coe¢ cients are N1 1 1 N1 X X X X 1 1 j kn j kmN 0 0 a =  nmNe = e  nmN : k N N n=0 m=1 m=1 n=0 But  N1 X 1; m = 0  nmN = ; 0; m =6 0 n=0 1 so a = for every k: The DFS is k N N1 X 1 j kn 0 x n = e N k=0 and the DTFT is 1 1 X X  j X e = 2 a  ( k) =  ( k): k 0 0 0 k=1 k=1 Example 2.6 From Example 2.5, the Fourier coe¢ cients corresponding to an impulse train are constant. Now …nd the Fourier coe¢ cients of x n = 1: By duality, we should get an impulse train. N1 X 1 j kn 0 a = x ne k N n=0 N1 X  1 n j k 0 = e N n=0 ( 1; k =mN N j k = 0 1 e ( ) 1 ; else j k N 1e 0 But  N j k j2k 0 e =e = 1; so  1 X 1; k =mN a = =  kiN: k 0; else i=1 113 Sampling 3.1 Time and Frequency Domain Analysis For any T 0; we may sample a CT signal x (t) to generate the DT signal x n =x (nT ): This amounts to evaluating x (t) at uniformly spaced points on the t-axis. The number T is the sampling period, 1 f = s T is the sampling frequency, and 2 = 2f = s s T is the radian sampling frequency. Normally, the units of f are Hertz or samples/sec. The units of s are rad/sec. The time interval nT; (n + 1)T is called the nth sampling interval. The process s of sampling is sometimes depicted as a switch which closes momentarily every T units of time: Figure 3.1 AusefulexpressionfortheDTFTofx ncanbeobtainedbywritingx (t)intermsofitsinverse transform: Z 1 1 jt x (t) = X (j)e d 2 1 x n =x (nT ) Z 1 1 jnT = X (j)e d 2 1 Z 1 (k+1) X s 1 jnT = X (j)e d 2 k s k=1   Z 1 2 X 1 + 2k j( +2k)n = X j e d ( =T 2k) 2T T 0 k=1 Z   1 2 X 1 1 + 2k j n = X j e d 2 T T 0 k=1 The analysis shows that   1 X  1 + 2k j X e = X j : (3.1) DT CT T T k=1 Expression (3.1) is referred to as the Poisson summation formula. 123.2 Aliasing WesayaCTsignalx (t)isbandlimited ifthereexists 1suchthatX (j) = 0forjj : B CT B SupposeX hastransformdepicted(veryroughly)inFigure3.2. (WeuseasignalwithX (0) = 1 CT CT for reference.) Figure 3.2  j Thenumber isthebandwidth ofthesignal. Ifx (t)isbandlimited,(3.1)indicatesthatX e B DT looks like Figure 3.3 Figure 3.3 or Figure 3.4. 13Figure 3.4 Figure 3.3 is drawn assuming 2 T T B B or, equivalently, 2 : s B In this case, (3.1) indicates that    j X e =X j DT CT T for : This establishes the fundamental relationship between the CT and DT frequency variables and under sampling: =T: (3.2) We will encounter equation (3.2) under a variety of circumstances when sampling is involved.  For 2 ; the picture reverts to Figure 3.4. In this case, the shifts of X j overlap s B CT T –a phenomenon called aliasing. As we will see, aliasing is undesirable in most signal processing applications. Theminimumradiansamplingfrequency = 2 requiredtoavoidaliasingiscalled s B the Nyquist rate. 3.3 The Nyquist Theorem Consider the set  of all CT signals x (t) and the set  of all DT signals x n: For a given CT DT sampling period T; the process of sampling may be viewed as a mapping from  into  : CT DT x (t)7x n =x (nT ): That is, each CT signal generates exactly one DT signal. The following example shows that the mapping changes if the sampling period changes.  Example 3.1 Let x (t) = sint and T = : Then 2  n1   2  (1) ; n odd x n = sin n = : 2 0; n even On the other hand, setting T = yields x n = sin (n) = 0: Thus sampling sint results in di¤erent signals for di¤erent T: 14The next example shows that the sampling map may not be 1 1: Example 3.2 Let x (t) = sint and x (t) = 0: For T =; 1 2 x n =x n = 0; 1 2 so the distinct CT signals x (t) and x (t) map into the same DT signal. 1 2 Now let    be the set of all CT signals with bandwidth at most : In Example 3.2, CT B B both x (t) and x (t) belong to  : Yet, they map into the same DT signal for T = : In other 1 2 1 words, the sampling map may not be 1 1 even on  : Also, note that in Example 3.2, B 2 = = 2 = 2 ; s B T so we are sampling at exactly the Nyquist rate. The situation is clari…ed by the Nyquist Sampling Theorem: Theorem 3.1 The sampling map is 1 1 on  i¤ 2 : s B B The Nyquist theorem states that, if we are given a signal x (t) and we sample at greater than theNyquistrate,thenthereisnolossofinformationinreplacingx (t)byitssamplesx n:Inother words, x (t) can be recovered fromx n: However, if we sample at or below the Nyquist rate, then knowledge of x n is insu¢ cient to determine x (t) uniquely. 3.4 Anti-Aliasing Filters Inordertoavoidaliasing, wemaysetthesamplingrate 2 : However, incertainapplications s B it is desirable to achieve the same end by reducing the bandwidth of x (t) prior to sampling. This can be done by passing x (t) through a CT …lter. De…ne the ideal CT low-pass …lter (LPF) to be the CT LTI system with transfer function  1; jj 1 H (j) = : (3.3) LP 0; jj 1   If we pass x (t) through the frequency-scaled …lter H j ; thenX (j) is “chopped”down to LP B bandwidth : An LPF used in this way is called an anti-aliasing …lter andmust be built from B analog components. The impulse response of H (j) is LP 1 h (t) =F fH (j)g LP LP Z 1 1 jt = H (j)e d LP 2 1 Z 1 1 jt = e d 2 1  1 jt jt = e e j2t sint = : t Let  1; t = 0 sinct = : sin(t) ; t6= 0 t 15Figure 3.5 We may write 1 t h (t) = sinc (3.4) LP   Note that h (t) has a couple of unfortunate features: LP 1) h (t)6= 0 fort 0; so the ideal LPF isnoncausal. LP 2) Z Z 1 1 1 sin (t) jh (t)jdt = dt LP  t 1 1 Z 1 2 sin (t) = dt  t 0 Z 1 n X 2 sin (t) = dt  t n1 n=1  Z  1 n X 2 1  jsin (t)jdt  n n1 n=1   Z 1 1 X 2 1 = sin (t)dt  n 0 n=1 1 X 2 1 = 2  n n=1 =1: Hence, the ideal LPF isnot BIBO stable. Although an ideal LPF cannot be realized in practice, we will eventually study approximations to the ideal LPF that can actually be built. 3.5 Downsampling Let x n be a DT signal and N 0 an integer, and de…ne x n =x nN: d 16x n is obtained from x n be selecting every N values and discarding the rest. Hence, we may d viewdownsampling as “sampling a DT signal”. If x n was obtained by sampling a CT signalx (t) with period T; then x n =x nN =x (nNT ) d corresponds to sampling x (t) at a lower rate with period NT: Downsampling leads to a version of the Poisson summation formula. This may be derived by mimicking the analysis leading to (3.1), but in DT: Z 2  1 j j n x n = X e e d 2 0 x n =x (nN) d Z 2  1 j j nN = X e e d 2 0 Z 2 N1 (k+1) X N  1 j j nN = X e e d 2 2 k N k=0 Z N1 2   X 1 +2k j j(+2k)n N = X e e d ( = N 2k) 2N 0 k=0 Z N1 2   X 1 1 +2k j jn N = X e e d 2 N 0 k=0 N1   X  1 +2k j j N X e = X e : (3.5) d N k=0 Expression(3.5)isanalogousto(3.1). Theybothstatethatsamplingintimecorrespondstoadding shifts of the original transform. However, unlike sampling a CT signal, downsampling x n results  j in a…nite sum of shifts of X e : Suppose x n has bandwidth  : B Figure 3.6 Thenx n has spectrum d 17Figure 3.7 To avoid aliasing, we need N 2N B B or  N : B If x n comes from sampling a CT signal x (t) with bandwidth ; then = T and we need B B B  s N = : T 2 B B 3.6 Upsampling ForanyDTsignalx n;theTime/FrequencyScalingpropertyoftheDTFTstatesthattheexpanded signal    n n x ; an integer N N x n = (N) 0; else has DTFT   j N F x n =X e : (N) This is only one kind of interpolation scheme obtained by setting missing data values to 0: An alternative is to assume that x n was obtained by sampling a CT signal x (t) with bandwidth B at some sampling frequency 2 : Resampling x (t) at N times the original rate yields the s B upsampled signal   T x n =x n : (3.6) u N We would like to develop a method for computing x n directly from x n; without resorting to u explicit construction of x (t): Suppose x (t) has CTFTX (j) as in Figure 3.2. Then from the Poisson formula (3.1), CT   j N F x n =X e (3.7) (N) DT   1 X 1 N + 2k = X j CT T T k=1 1 k X + 2 1 N = X j : CT T T N k=1 18Figure 3.8 and 1 X  N + 2k j Ffx ng =X e = X j (3.8) u u CT T T N k=1 Figure 3.9 Note that (3.8) is a scaled version of (3.7), but containing only 1 out of every N terms. De…ne the ideal DT low-pass …lter:    1; j j j N H e = (period 2) LP  0; j j N Figure 3.10 19   j j N Passing from (3.7) to (3.8) is the same as applying NH e toX e : LP DT    j j j N X e =NH e X e u LP DT Figure 3.11 Upsampling from x n to x n amounts to Figure 3.11, where the …rst block indicates expansion u by N: As with the ideal CT LPF, the ideal DT LPF is not realizable in practice. However, close approximations are achievable. 3.7 Change of Sampling Frequency More generally, suppose we are given a DT signal x n and we wish to replace everyN consecutive values by M without changing the “character”of the signal. A common application of this idea is that of resampling a CT signal: Suppose x n consists of samples of a CT signal x (t) at frequency 2 : We may wish to resample x (t) at a new rate r ; where r is a rational number. The s B s numberr must be chosen to avoid aliasing –i.e. r 2 : s B Resampling can be achieved through upsampling and downsampling. Write M r = ; N where M andN are coprime integers. Upsampling by M; we obtain   T x n =x n : u M Downsampling by N yields     T T x n =x nN =x nN =x n : r u M r The blockdiagraminFigure 3.12depicts the process, where the last blockindicates downsampling byN: Figure 3.12 Examination of Figures 3.6-3.10 shows that no aliasing occurs at any step. The …nal spectrum is shown in Figure 3.13: 20

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