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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 discretetime Fourier transform (DT~T) a nd t he mverse
discretetime Fourier transform ( IDTFT) c an be computed usmg t he D FT. For .an
Nopoint 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 Nopoint 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
10.11 Find the discretetime 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.
10.12 Repeat Prob. 10.11 if j [k] = cos 2.27rk cos 3.31rk.
10.13 Repeat Prob. 10.11 if J[k] = 2 cos 3.21r(k  3).
flkl tI 1 I 1111, r 11 .I!III. I1ll
L
10.14 Find the discretetime Fourier series and the corresponding amplitude and phase
spectra for the j [k] shown in Fig. P1O.14. 664 1 0.15 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. 10.14 for t he f[kJ d epicted in Fig. P1O.1S.
1 0.21 1 0.16 R epeat P rob. 10.14 for t he f[kJ i llustrated i n Fig. P1O.16. (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 0.22 F ind t he D TFT for t he s ignals shown in Fig. P8.29 ( Chapter 8). 3 (a)
 2" F ig. P I0.15 2" " 0 Fig. P I0.23
1 0.23 [r Using t he t imeshifting 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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