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

# 1114c 2 2ku k 1 60 708 fo 11 d iscrete

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Unformatted text preview: [k] ( d) 1 1.3-1 1 1.3-12 Solve y[k + 1] + 2y[k] = I [k + 1] w ith y[O] = 1 a nd J[k] = e -(k-l)u[k] 1 1.3-3 F ind t he o utput y[k] o f a n L TID s ystem specified by t he e quation 1 1.3-5 1 1.3-14 F ind h[k], t he u nit i mpulse response o f t he s ystems described b y t he following e quations: y[k] + 3y[k - 1] + 2y[k - 2] = J[k] + 3 /[k - 1] + 3 /[k - 2] y[k + 2] + 2y[k + 1] + y[k] = 2 /[k + 2 ]- I[k + 1] y [k]- y[k - 1] + O.Sy[k - 2] = I [k] + 2 /[k - 1] 1 1.3-15 F ind h[k], t he u nit i mpulse response of t he s ystems in P robs. 11.3-9, 11.3-10, a nd ( a) ( b) ( c) 1 1.4-2 = 0, y[-2] = 1, a nd I [k] = u[k]. y[k + 2 ]- 3y[k + 1] + 2y[k] = I [k + 1] = H[z] ( b) ¥. Solve w ith y [-I] Solve 11.3-12. S how a canonical, a cascade a nd a p arallel realization o f t he following t ransfer functions: ( a) 4y[k + 2] + 4y[k + 1] + y[k] = I [k + 1] 1 1.3-6 R epeat P rob. 11.3-9 if I [k] = u[k] a nd F ind t he t ransfer f unctions corresponding t o e ach o f t he s ystems specified by difference e quations i n P robs. 11.3-2, 11.3-3, 11.3-S, a nd 11.3-8. if t he i nitial conditions a re y [-I] = 0, y [-2] = 1, a nd t he i nput I [k] = (4)-ku [k]. Solve P rob. 11.3-3 i f i nstead o f i nitial conditions y [-I],y[-2] y ou a re given t he auxiliary conditions y[O] = a nd y[l] = ! 6(Sz - 1) - Sz + 1 6z 2 1 1.3-13 1 1.4-1 1 1.3-4 = 2z - 1 H[z] = z2 _ 1.6z + 0.8 Solve P rob. 9.4-9 b y t he z -transform m ethod. 1 1.3-2 2z + 3 ( z _ 2 )(z - 3) a nd t he i nput J[k] is ( a) (4)-kU[k] ( b) (4)-(k-2)U[k - 2] ( c) (4)-(k-2)u[k] ( d) (4)-kU[k - 2]. ke- 2k u[k - m] Using only P air 1 i n Table 11.1 a nd a ppropriate p roperties o f t he z -transform, derive iteratively pairs 2 t hrough 9. I n o ther words, first derive P air 2. T hen u se P air 2 ( and P air 1, if needed) t o derive P air 3 , a nd s o on. However, p air 6 s hould b e derived after pair 7. = R epeat P rob. 11.3-9 if F ind t he z -transform o f t he s ignal i llustrated i n Fig. P l1.2-2. Solve t his p roblem in two ways, as i n E xamples 11.2d a nd 11.4. Verify t hat t he t wo answers a re e quivalent. Using only t he f act t hat "(ku[k] z -transform o f 0.8) " (Z ) 2 H[z] 1 1.2-2 z + 0.2}(z - z (3z - 1.8) Z2 - z + 0.16 H[z] = z 2+ z +0.16 H[z] _ 3.8z - 1.1 - (z - 0.2}(z2 - 0.6z Sz + 2.2 ( c) + 0.2S) Give cascade a nd p arallel realizations o f t he following t ransfer functions: ( a) z (1.6z - 1.8) ( z - 0.2)(z2 + z + O.S) + 1.3z + 0.96) + O.S)(z - 0.4)2 ( b) Z(2z2 (z 1 i 714 11 f (tj T D iscrete- Time S ystems A nalysis U sing t he Z - Transform 715 P roblems y (t) f [k] y [k] f (t) y (t) e (t) F ig. P l1.6-l. f (t) y (t) F ig. P 11.6-3. 1 1.7-1 F ind t he z -transform (if i t e xists) a nd t he c orresponding region o f convergence for each o f t he following signals: ( a) (0.8)ku[k] + 2ku[-(k + 1)] ( b) F ig. P 11.6-2. ( c) 1 1.4-3 ( d) Realize a s ystem whose transfer function is ( e) H[z] = 2 z4 3 + z + O. 8 z 2 + 2z + 8 z4 1 1.4-4 ( f) 1 1.7-2 2 ku[k]- 3ku[-(k + 1)] + (0.9)kU[-(k + 1)] [(0.8)k + 3(OA)k] u [-(k + 1)] [(0.8)k + 3(OA)k] u[k] (0.8)kU[k] + 3 (OA)ku[-(k + 1)] (0.8)ku[k] F ind t h...
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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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