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

For analysis o ur approach is parallel t o t hat used

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Unformatted text preview: f a s ystem w ith impulse response h[k] a nd t he i nput I [k] shown in Fig. P10.6-7 by t he following methods: (i)using linear convolution o f J[k] a nd g[k] by sliding t ape m ethod (ii) using circular convolution of suitably p added I [k] a nd g[k] using t he g raphical method, depicted i n Fig. 5.17 (iii) using O FT. f:JrL ( c) R epeat p art ( a) for t he No-point O FT (found b y p adding No - 1 zeros t o 8[kj). E xplain t his O FT from F (n) found in p art ( a). 0 123 ( a) F ind t he O FT o f t he No-point signal I [k] = u[k] - u[k - No]. F ind F (n), t he O TFT o f J[k] a nd verify t hat t he O FT values are t he u niform samples of O TFT a t f requency intervals o f n o = 27r / No. '~ k-+ 0 123 k -+ F ig. P I0.6-8 ( b) Is t he O TFT found in p art ( a) a n a dequate frequency-domain description o f I[k]. I f n ot, what needs t o b e done to o btain a r easonably a dequate O FT? 1 0.6-5 k -+ F ig. P I0.6-7 ( b) Show t hat t he O FT o f I [k] = 8[k - m] is t he s ame as t he O FT o f 8[k] for any integral value of m. Explain this behavior. 1 0.6-4 0 123 k -+ ( 8) F ind t he 4-point and S-point D FT o f a 4 -point signal specified by t he sequence 1 , 2, 2, 1 s tarting a t k = O. ( b) F ind F (n), t he D TFT o f I [k], a nd verify t he D FT values from F (n). 1 0.6-3 I2 3 '~ f [k] ( a) F ind t he 5-point a nd 8 -point O FT o f t he signal I [k] i llustrated i n Fig. PS.2-9d. ( b) F ind F (n), t he O TFT o f I [k], a nd verify t he O FT values from F (n). h [k] 2 q . . . o I • •• 123 -I -2 k -+ F ig. P I0.6-9 10.6-8 10.6-9 f [k] ... ... 0 1234567 k-+ 1 0.6-10 F ind t he 16-point I DFT o f F (n) in Fig. 10.7a. 0 0 t he values of I DFT a gree w ith t he values of I [k] found in Eq. 10.45. I f n ot, why not? ''1 . .. ITTFTTlltTI . .. 0 123456 k -+ F ig. P I0.6-6 1 0.6-6 ( a) R epeat P rob. 10.6-7 for t he signals I [k] a nd h[k] i llustrated in Fig. P1O.6-S. Using b oth t he m ethods o f block filtering (overlap a nd a dd, an~ o verlap a nd sav~), find t he o utput o f a filter w ith impulse response h[k] a nd t he m put I [k] shown I II Fig. P10.6-9. T ake L = 3. Verify t hat t he l inear conv~lution o f t he i nput sequence J[k] = { I, - 2, 3, 0, - 1, 2, . .. } w ith h[k] = {2, 2} gIves t he s ame o utput as t hat found by t he block filtering methods. ( a) Using t he g raphical method shown in Fig. 5.17, find t he c ircular convolution of t he sequences I [k] a nd g[k], d epicted in Fig. P10.6-6. ( b) Using t he sliding t ape m ethod (Fig. 9.4), find t he l inear convolution o f t he first cycles (over t he range 0 :'0 k :'0 3) o f t he sequences I [k] a nd g[k]. Is t he r esult same as t hat found in p art ( a)? ( c) T he circular convolution can be made equivalent t o t he linear convolution by s uitably p adding ( the first cycle of) t he sequences I [k] a nd g[k] w ith zeros. How many zeros d o you need t o p ad t o J[k] a nd h[k]? After s uitably p adding these sequences, p erform t he c ircular convolution using t he g raphical m ethod i llustrated in Fig...
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