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

Multiply fr by h r to obtain yr this step yields yo

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Unformatted text preview: Output blocks F ig. 1 0.15 Overlap and add method of block filtering. Fr 3. y [k) L N o.1 2 rn~P-m~bl-OC-ks~- ~1~I_o~_. N o.3 Output blocks 657 Signal p rocessing Using D FT a nd F FT = Lh[k]e-ir~k k=O :j:We also p ad h[kJ w ith L - 1 zeros s o t hat t he l ength of t he p added hlkJ is L +M - 1. \6110110181212\212\41". k_ F ig. 1 0.16 An Example of overlap a nd a dd method of block filtering. Also, Y r = F rHr . We compute t he vaI ues 0 f F r , H r , a nd Y.r using these equations for each block: For t he first block, and Also Substituting r F o=8 Ho = 4 Yo = 32 y[O] = 6 = 0, 1, 2, 3, we obtain F I=-2j H I = 2 v'2e-it YI = 4 v'2e- i ¥ y [l] = 10 F3 = 2 j H3 = 2 v'2eit Y3 = 4 v'2ei ¥ F 2=4 H2 = 0 Y2 = 0 y[2] = 10 y[3] = 6 Using t he same procedure for the second block, we o btain y[O] = 2, y[l] =2 y[2] = 2, y[3] =2 y[l] =2 y[2] = 4, y[3] =2 For t he t hird block, we o btain y[O] = 0, Figure 10.16 shows the overlapping input and o utput blocks, and the convolution sequence obtained by adding t he o utput blocks. . T he ocedure using D FT given here is much more laborious t han t he direct convolution : : t he sliding t ape method (Sec. 9.4). Th~ reason is t hat we d id not use F:~ algorithm to compute D FT here. In this example, With a r ather small No, even F FT 658 10 F ourier A nalysis o f Discrete- Time S ignals D ata blocks L L No.1 IAI-i] Input Blocks L L N o.2 '7 2 3 0 I 31 II Identical Output Blocks 1M-II L II 6 Discard N o.3 1 10 II M -l Discard 0 110 1 II Discard O utput blocks 659 G eneralization o f t he D TFT t o t he z -transform 3 I,-M~"-l",,-I _ _-"""---...J17. Identical L t p~ 1 0.7 8 2 II II I II 0 21 2 2 14 I Discard L No.1 Final Output L D iscard N o.2 No.3 F ig. 1 0.17 Overlap a nd Save m ethod of block filtering. have more number of computations t han t he direct convolution. In practice, we generally deal wit~ mu~h larger. values of No, where use of D FT (utilizing F FT algorithm) pays off. I t Will b e mformative for t he r eader t o find linear convolution (using t he sliding t ape method) of one unpadded d ata block with unpadded h[k]. Next, find the circular convolution of t he same d ata block padded with 1 zero a nd h[kJ p added with 2 zerOs. Use t he graphical method of circular convolution illustrated in Fig. 5.17. Verify t hat you get t he same answer in b oth cases. • Overlap a nd S ave M ethod I n t he overlap and s ave m ethod also, t he i nput s equence is s ectioned i nto nonoverla~ping b locks o f a m anageable l ength L . A s b efore, e ach b lock is augmented With M - 1 . data p oints. B ut u nlike t he p revious m ethod, t his m ethod p laces a ugmented p omts a t t he b eginning o f e ach b lock. T he a ugmented M - 1 d ata p oints o f a b lock a re t he s ame a s t he l ast M - 1 p oints o f t he p revious b lock s o t hat t he l ast M - 1 d ata p oints o f e ach block also a ppear a s t he f irst M - 1 d ata p oints o f t~e s ucceeding block. T he e xception is t he f irst block, w here t he f irst M - 1 d ata p omts a re t aken a s z eros, as s...
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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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