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

Let ykj b e t he linear convolution of t hese

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Unformatted text preview: ler processing delay. We shall now discuss two such m ethod o f block processing, o verlap a nd add m ethod a nd o verlap a nd s ave m ethod. E ither o f t he m ethods r equires t he s ame n umber o f c omputations; hence, which m ethod is used is a m atter o f choice. Overlap a nd Add M ethod C onsider a filtering operation, where t he o utput y[k] is t he c onvolution of t he i nput J[k] a nd a F IR filter impulse response h[k] o f l ength M . I n p ractice, usually t It c an b e shown t hat i n convolution o f t wo sequences, each o f l ength No, t he n umber o f computations r equired is o n t he o rder o f N6, w hereas for O FT, u sing F FT a lgorithm, t he n umber o f c omputations r equired is only o n t he o rder o f No log2 No. 6 56 10.6 10 Fourier Analysis o f D iscrete- Time S ignals Nonoveriapping data blocks L Zero-padded L L ( a) I [k) JL L data blocks N o.1 L o 0 12345678 N o.2 I k_ 1M-II L 3 to 1M-II o=r\-I'-I-I• ; =1 \ 61 10 1 10 • ( c) L -M+I N o.2 1M-II L -M+I N o.3 Final output 012345678 M is m uch s maller t han t he l ength o f t he i nput. F igure 10.15 i llustrates a l ong i nput s equence sectioned i nto n onoverlapping blocks o f a m anageable l ength L . T he n umber i nside each block i ndicates i ts l ength. L et u s a ssume t hat L » M . We n ow p rocess each block o f t he i nput d ata in sequence. T o b e a ble t o u se t he c ircular c onvolution for performing t he l inear convolution, we n eed t o p ad M - 1 z eros a t t he e nd o f e ach d ata block.:} F igure 10.15 shows e ach d ata b lock a ugmented b y M - 1 zeros. T he a ugmented p ortion o f t he b lock is s hown s haded. O bserve t hat t he a ugmented ( zero-padded) blocks o f t he i nput, e ach o f l ength L + M - 1, now o verlap. T he o utput s equence c orresponding t o e ach block also has a l ength L + M - 1 ( recall t hat t he l ength o f t he c ircular c onvolution o f t wo sequences, e ach o f l ength L +M - 1, is also L +M - 1). T he o utput sequences, therefore, also overlap, a s s hown i n F ig. 10.15. T he t otal o utput is g iven by t he s um o f a ll t hese o verlapping o utput b locks o f l ength L + M - 1. T he c ontents o f t he t wo successive blocks a re a dded w herever t hey o verlap. T his m ethod is known a s t he overlap a nd a dd m ethod. • E xample 1 0.13 Using overlap and add method of block filtering, find the response y[k] of a n LTID system, whose impulse response h[k] and the input f [k] are shown in Fig. 10.16. T he o utput y[k] is a linear convolution of f [k] a nd h[k]. Let us use L = 3 for t he block convolution. Also M = 2. Hence, we need to break t he i nput sequence in blocks of 3 digits and pad each block with M - 1 = 1 zero, as depicted in Fig. 10.16. We convolve each of these blocks with h[k] using DFT, as demonstrated in Example 10.11. I n this case, No = 4 and 110 = -rr/4. The DFTs Fr a nd Hr of the zero-padded sequences f [k] a nd h[k] are given by 2 = LJ[k]e-ir~k k=O I and Hr IIII2G1II . . . .:--2-1 ---.4...
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