Signal Processing and Linear Systems-B.P.Lathi copy

Each block of length l m 1 4 is now convolved using d

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Unformatted text preview: tal signal processing can be conveniently performed with the discrete Fourier transform ( DFT) i ntroduced in C hapter 663 P roblems 5. T he D FT c omputations can be very efficiently executed by using t he fast F~u:ier t ransform ( FFT) algorithm. T he D FT is indeed t he workhorse o f m odern .dlgltal signal processing. T he discrete-time Fourier transform (DT~T) a nd t he mverse discrete-time Fourier transform ( IDTFT) c an be computed usmg t he D FT. For .an No-point signal j [k], i ts D FT yields exactly No samples of F (n) a t frequency.mtervals of 21r/No. We c an o btain a larger number o f s amples o f F (n) by pad~mg sufficient number of zero valued samples t o j [k]. T he No-point D FT of j [k] gIves exact values of t he D TFT samples if j [k] h as a finite length No· I f t he l ength of j [k] is infinite, we need t o t runcate j [k] using t he a ppropriate window function. Because of t he convolution property, we c an c ompute convolution of two signals j [k] a nd h[k] using D FT. For this purpose, we need t o p ad b oth t he signa~ b y a s uitable number of zeros so as t o m ake t he linear convolution of t he two SIgnals identical t o t he circular (or periodic) convolution of t he p added signals. Large blo~ks of d ata may be processed by sectioning t he d ata i nto smaller blocks a nd processmg such smaller blocks in sequence. Such a procedure requires smaller memory a nd reduces t he processing time. References 1. M itra, S.K., Digital S ignal processing: A C omputer B ased Approach, McGrawHill, New York, 1998. Problems 10.1-1 Find the discrete-time Fourier series (DTFS) and sketch their spectra for 0 :0; r :0; No - 1 for the following periodic signal: j [k] = 4 cos 2.41rk IVrl and L Vr + 2sin 3.21rk Hint: Reduce frequencies to the fundamental range (0 :0; 0 :0; 21r). The fundamen~al frequency 0 0 is the largest number of which the frequencies appearing in the Founer series are integral multiples. 10.1-2 Repeat Prob. 10.1-1 if j [k] = cos 2.27rk cos 3.31rk. 10.1-3 Repeat Prob. 10.1-1 if J[k] = 2 cos 3.21r(k - 3). flkl tI 1 I 1111, r 11 .I!III. I1ll L 10.1-4 Find the discrete-time Fourier series and the corresponding amplitude and phase spectra for the j [k] shown in Fig. P1O.1-4. 664 1 0.1-5 1 0 F ourier A nalysis o f D iscrete- Time S ignals 665 P roblems H int: I f w is complex, t hen R epeat P rob. 10.1-4 for t he f[kJ d epicted in Fig. P1O.1-S. 1 0.2-1 1 0.1-6 R epeat P rob. 10.1-4 for t he f[kJ i llustrated i n Fig. P1O.1-6. (S.43). F or t he following signals, find t he D TFT d irectly, u sing t he d efinition in E q. (10.31). (a) J[kJ = a[kJ ( d) f[kJ f [kl Iwl 2 = w w', a nd u se Eq. ( b) a[k - ko] (c) aku[k - lal < 1 IJ = aku[k + I] lal < l . I n e ach c ase, s ketch t he s ignal a nd i ts a mplitude s pectrum. S ketch p hase s pectra for p arts ( a) a nd ( b) only. 3 1 0.2-2 F ind t he D TFT for t he s ignals shown in Fig. P8.2-9 ( Chapter 8). -3 (a) - 2" F ig. P I0.1-5 2" " 0- Fig. P I0.2-3 1 0.2-3 [r Using t he t ime-shifting p roperty a nd t he r esults i n E xamples 10.3 a nd IO.S, find t he D TFT o f ( a) ak{...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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