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

# T he s ystem r esponse is obtained as t he c

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Unformatted text preview: b) ( u[k]- u[k - (k + l )u[k]- (k - m + l )u[k - m] 9 Time-Domain Analysis o f Discrete-Time Systems 614 P roblems f [k] a nd f = H -1y 5 g [ k] * j . .. t ttt K nowing h[k] a nd t he o utput y[k], we c an d etermine t he i nput I [k]. T his o peration is t he r everse o f t he c onvolution a nd is known a s t he d econvolution. Moreover, knowing I [k] a nd y[k]' we c an d etermine h[k]. T his c an b e d one b y e xpressing t he a bove m atrix e quation a s k + I s imultaneous e quations i n t erms o f k + 1 u nknowns h[O], h[l], . .. , h[k]. T hese e quations c an r eadily b e solved iteratively. T hus, we c an s ynthesize a s ystem t hat y ields a c ertain o utput y[k] for a given i nput I [k]. ( a) D esign a s ystem ( that is, d etermine h[k] ) t hat will yield t he o utput s equence (8, 12, 14, 15, 15.5, 15.75, . .. ) for t he i nput s equence (1, 1, I , 1, 1, 1, . .. ). ( b) F or a s ystem w ith t he i mpulse response sequence (1, 2, 4, . .. ), t he o utput s equence was (1, 7 /3,43/9, . .. ). D etermine t he i nput sequence. I k- Fig. P 9.4-12 • 1 -10 615 II L * .I l It, "'I .I . . 5 -5 10 5 -5 ,,"I .1. k_ 9 .4-16 F ig. P 9.4-13 g [ k] f [k] * T he s liding-tape m ethod is conceptually q uite v aluable in u nderstanding t he convolution m echanism. Numerical convolution c an also b e p erformed from t he a rrays u sing t he s ets 1[0], t [1], 1[2], . .. , a nd g[O], g[I], g[2], . .. , a s d epicted i n Fig. P9.4-16. T he i jth e lement (element i n t he i th row a nd j th c olumn) is given by g[i]/I.i]. We a dd t he e lements o f t he a rray a long i ts d iagonals t o p roduce e[k] = I [k] *g[k]. F or example, i f we s um t he e lements c orresponding t o t he first diagonal o f t he a rray, we o btain e[O]. Similarly, if we s um a long t he s econd diagonal, we o btain e[I], a nd s o on. D raw t he a rray for t he s ignals I [k] a nd g[k] i n E xample 9.9, a nd find I [k] * g[k]. f (aJ c [ k] [ 0] c [ 1] I P Jg [OJ I [ 3Jg [OJ g ( 1] f [Q.Jt [ I J l [tH[IJ I [ 2Jg [ IJ f [3]g[IJ I [ 01t'[2J I [ I]g [ 2] I [ 2]g [ 2] I [ 3]g [ 2] g [ 3] I [ O]g 1 3] 1 [I]g[3] I [ 2]g [ 3] I [ 3]g [ 3] g [ k] * .111111f. k- I p Jg [OJ g [ 2] 12 6 I [ QIt [OJ -12 [ 2] j ·n] g [ 0] .Irrrrrr.. n O] c -6 (bJ F ig. P 9.4-14 9 .4-12 U sing t he s liding-tape algorithm, find I [k] * g[k] for t he s ignals shown in Fig. P9.4-12. 9 .4-13 R epeat P rob. 9.4-12 for t he s ignals shown in Fig. P9.4-13. 9 .4-14 R epeat P rob. 9.4-12 for t he s ignals depicted i n Fig. P9.4-14. 9 .4-15 T he c onvolution s um i n Eq. (9.48) c an b e e xpressed in a m atrix form a s y[O] h[O] o o y[l] h [l] h[O] o o o F ig. P 9.4-16 1[0] 1[1] 9 .4-17 9 .5-1 U sing Eq. (9.58), show t hat t he t ransfer f unction o f a u nit d elay is H[z] = Using t he classical m ethod, solve y[k + 1] + 2y[k] = I [k + 1] y[k] h[k] h[k - 1] . .. . .. h[O] t[k] ' -v--" v ' '-v--' y H f w ith t he i nput I [k] = e-ku[k], a nd t he a uxiliary condition y[O] = l . 9 .5-2 Using t he classical m ethod, sol...
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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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