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

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Unformatted text preview: 3] = 6 15 ( g) 15 c [4] = 10 F igure 9.4j shows t hat e[kJ = 15 for k 2: 5. Moreover, t he two tapes are nonoverlapping for k < 0, so t hat e[kJ = 0 for k < O. Figure 9.4k shows t he p lot of e[kJ. • o ( h) C omputer E xample e 9.5 Using Matiab, find t he convolution of l[kJ a nd g[kJ depicted in Fig. 9.5. f =[O 1 2 3 2 1J; g =[1 1 1 1 1 1J; k =0:1:length(f)+length(g)-2; c =conv(f,g); s tem(k,c) c[5] = 15 c[k] 15 15 (i) 10 c [6] = 15 0 An Array Form o f Graphical Procedure T he c onvolution s um c an a lso b e o btained f rom t he a rray f ormed b y s equences I [k] a nd g[k]. T his p rocedure, a lthough c onvenient from a c omputational v iewpoint, fails t o give p roper u nderstanding o f t he c onvolution m echanism. T he p rocedure is e xplained i n P rob. 9.4-16. k- (j) (1<) F ig. 9 .4 Sliding t ape algorithm for discrete-time convolution. 596 9 Time-Domain Analysis o f Discrete-Time Systems f [k) Classical Solution o f Linear Difference Equations 597 Note t hat H[zJ is a c onstant for a given z. T hus, t he i nput a nd t he o utput a re t he s ame (within a multiplicative constant) for t he everlasting exponential i nput zk. H[zJ, which is called t he t ransfer f unction o fthe system, is a function of t he complex variable z. We can define t he t ransfer function H[zJ o f a n LTID system from Eq. (9.57a) as g [ k) (a) 9.5 (b) H [zJ o utput signal i nput signal ~ -"'----=:~ I (9.58) I nput=everlasting e xponential zk 9 6. E xercise E 9.12 S how t hat t he t ransfer f unction o f t he d igital differentiator i n E xample 8 .7 ( big s haded b lock i n F ig. 8.20b) is g iven b y H[z] = "i}. V c [ k) 6 9 .4-3 T otal R esponse (c) T he t otal response o f a n LTID system c an b e expressed as a s um o f t he zeroinput a nd z ero-state components: k- Fig. 9 .5 Signals f or E xercise E 9.1l. + Total response = LJ. E xercise E 9.11 Using t he g raphical procedure o f E xample 9.9 (sliding-tape technique), show t hat f [k] *g[k] = e[k] i n Fig. 9.5. Verify t he w idth p roperty o f convolution. V 9 .4-2 l[kJ * h[kJ --------- Zert?state c omponent A Very Special Function For LTID S ystems: T he Everlasting Exponential Z k I n Sec. 2.4-3, we showed t hat t here exists one signal for which t he response o f a n LTIC system is t he same as the input within a multiplicative constant. T he respo~se o f a n LTIC system t o a n everlasting exponential i nput est is H {s ) e st , where H {s) IS th~ s ystem transfer function. We now show t hat for a n LTID system, t he s ame role I S played by an everlasting exponential zk. T he s ystem response y[kJ in t his case is given by 00 L = + 2J + IJ - O.16y[kJ = 5 /[k + 2J = O,y[-2J = ¥ a nd i nput l[kJ = ( 4)-ku [k], - O.6y[k Total response = O.2{ -O.2)k , 00 zk y[k h[mJzk-m m =-oo = I n this expression, t he z ero-input component should be appropriately modified for t he case of repeated roots. We have developed procedures t o d etermine these two components. From t he s ystem equation we find 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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