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

# T he p rocedure is a lmost identical t o t hat used

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Unformatted text preview: u[kJ - u[k - 1Oj} ( b) u [k]- u[k - 9J. 1 0.3-2 ,llll F ind t he inverse D TFT for t he s pectrum d epicted i n Fig. PIO.2-3. 1 0.3-1 U sing a ppropriate p roperties a nd t he r esult i n E xample 10.3, find t he D TFT o f ( a) (k + l )a k u[k] (Ial < 1) ( b) ak cos floku[k]. F or t he s pectrum F (fl) i n F ig. PIO.2-3 ( a) F ind a nd s ketch i ts I DTFT f[k]. ( b) S ketch f[2kJ, f[4k], a nd find t heir D TFTs. ( c) S ketch f[k/2J a nd fill in t he a lternate m issing samples using ideal i nterpolation ( upsampling b y a f actor 2). F ind t he D TFT o f t he r esulting i nterpolated ( upsampled) s ignal fi[kJ. 1 0.4-1 F ig. P IO.I-6 1 0.1-7 A s ignal f[kJ is a pproximated i n t erm k :S; N I ) as s 0 f f[kJ "'" cX[kJ h . a not e r sIgnal x[kJ over a n i nterval ( N I NI :s; k :s; :s; N2 1 0.5-1 ( a) Show t hat for t he b est a pproxim t · th ... s ignal e[kJ = f[kJ _ cx[kJ th a .Ion a t mInImIzes t he e nergy o f t he e rror over e s ame I nterval C 1 N2 = - x ~ f[kJx'[kJ E" Using t he D TFT m ethod, find t he z ero-state response y[k] o f a c ausal s ystem w ith f requency response ·0 H (fl) = e3 + 0.32 j20 + e jO + 0 .16 e a nd t he i nput f[k] = (-O.S)ku[k] k =N, ( b) I f c = 0, t he d iscrete-time signals f[kJ d [J . t he i nterval ( NI < k < N ) U h. a n x. k a re s aId t o b e o rthogonal over d iscrete-time sign,;Js. 2· s e t IS observatIOn t o define t he o rthogonality o f 1 0.5-2 H (fl) = ~o L k =<No> If[kJl2 = L 1 0.5-3 E arlier [Eq. (1O.S1)], we o btained t he P areseval's theorem for D TFT. - O.S + O.S)(ejO - 1) R epeat P rob. IO.S-1 if H (fl) = e jO -.-0-- eJ - O.S a nd l1'rl2 r =<No> J· 0 e (e jO a nd ( c) S how t hat t he s et o f s ignals e jrOo k for r = 0 . ' 1, 2, 3, . :., No - 1 IS o rthogonal o ver a n i nterval (0 < k < N _ 1) H ~ing t he r esult in ~art (a)~ . ence, find t he e xponentIal Fourier series ( DTFS) H Int: Recall t hat i f w is complex, t hen Iwl 2 = w w'. 1 0.1-8 A nN . IC s o -peno d ··Ignal f [kJ is r epresented b · t D TFS . as In E q. (10.8). Prove Parsev al's theorem (for D TFS) h ·ch t t h Y 1 S , w 1 s a es t a t R epeat P rob. IO.S-1 if 1 0.6-1 F ind t he D FT o f a 3 -point s ignal f [k] specified b y f [-I] = frO] = 3 , f [l] = 2 a nd f[kJ = 0 o therwise. Now d etermine F (fl), t he D TFT o f f [k], a nd verify t hat D FT values a re t he s amples o f F (fl). -666 1 0 F ourier A nalysis o f D iscrete- Time S ignals 667 P roblems f"JL_ ( b) Show t hat t he 3-point D FT o f t his signal is i dentical t o t hat of t he signal I [k] i n F ig. 1O.l1a. C an you explain why? Does this mean t he D TFTs o f t he two signals are a lso identical? Determine t he D TFTs of t he two signals a nd see i f t hey a re identical ( for all values o f n ). o ( e) F ind t he 8 -point D FT o f I [k]. 1 0.6-2 ( a) F ind t he D FT o f t he signal I [k] = 8[k]. F ind also F (n), t he D TFT o f 8[k], a nd verify t he D FT values from F (n). Note t hat t his is a I -point signal (No = 1). 1 0.6-7 F ind t he o utput o...
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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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