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

8kuk has infinite length find the dft of this signal

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Unformatted text preview: k =O:l:L-l; s ubplot(2,1,1);stem(k,y2) s tem(k,y2) 0 655 /). Figure 1O.14e shows y[k] and its periodic extension (dotted). The IDFT yields the periodic signal. We need only the first cycle. • o 10.6 Signal processing Using D FT a nd F FT E xercise E IO.6 Show t hat t he c ircular convolution o f two 3 -point sequences 3, 2, 3, a nd 1, 2, 3 , ( both s tarting a t k = 0) is also a 3 -point s equence 15, 17, 16. F ind t he a nswer b y u sing O FT, a nd verify your answer using t he g raphical m ethod e xplained in Fig. 5.17. Using t he s liding t ape m ethod (Fig. 9.4), show t hat t he l inear convolution o f t hese s equences is a 5 -point s equence 3, 8, 16, 12, 9. I ndicate how y ou will o btain t he l inear convolution o f t hese t wo s equences using circular convolution. 'V /). E xercise E IO.7 T he i nput I [k] o f a n L TIO s ystem is a 4-point sequence 1, 1, 1, 1, a nd h[kJ, t he i mpulse response o f t he s ystem, is a 3 -point s equence 1, 2, 1. B oth s equences s tart a t k = O. Using O FT, show t hat t he o utput y[kJ is a 6 -point s equence 1, 3, 4, 4, 3, 1 s tarting a t k = O. Verify y our a nswer by deriving t he o utput a s a linear convolution o f t he t wo sequences using t he s liding t ape m ethod. 'V Efficacy o f O FT in C onvolution C omputation Using t he sliding t ape a lgorithm discussed in C hapter 9, we c an p erform t he c onvolution in E xample 10.12 w ith a m ere 6 (real) multiplications a nd 2 a dditions. T he D FT m ethod discussed in t his s ection a ppears m uch t oo l aborious, a nd t he use o f D FT for convolution may seem questionable. Recall, however, o ur d iscussion in Sec. 5.3, which showed t hat t he u se o f F FT a lgorithm t o c ompute D FT r educes t he n umber o f c omputations d ramatically, especially for large No. F or s mall l ength sequences, t he d irect c onvolution m ethod, s uch as t he s liding t ape m ethod is faster t han t he D FT m ethod. B ut for long sequences, t he D FT m ethod u sing t he F FT a lgorithm is m uch f aster a nd far more efficient. t T he m ethod o f finding convolution using t he fast Fourier t ransform ( FFT) is known as t he f ast convolution. B lock Filtering o r C onvolution I n p ractice, t he l ength of a n i nput signal m ay b e v ery large, whereas a c omputer processing such a signal m ay h ave a limited memory. T o p rocess such long sequences, we c an s ection t he i nput s ignal i nto blocks of a length small enough t o b e p rocessed by a given c omputer, a nd t hen a dd t he o utputs r esulting from all t he i nput blocks. T his p rocedure c an b e u sed because t he o peration is linear. Such a procedure would b e d esirable even if t he p rocessing c omputer h ad a n u nlimited memory. For longer i nputs, we m ust w ait a long t ime b efore t he i nput is fed t o t he c omputer ( before i t even s tarts p rocessing), a nd t hen a n even longer t ime for processing t he l arge a mount o f d ata. C onsequently, t here is a long delay in t he o utput. S ectioning t he i nput allows t he o utput t o h ave a smal...
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