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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ] g [ -m] (e) (d) 0 g [ -m] I 2 4 3 4 m- 3 2 o m_ .--f[m] :" .......... / (e) e[k] = f [k] * g[k] --_ .. - o where f [k] and g[k] are depicted in Figs. 9.3a and 9.3b, respectively. We are given f [k] = (0.8)k and g[k] = (0.3)k Therefore f[m] = (0.8)m and g[k - m] = ( 0.3)k-m I 2 3 4 g [ k-m] m- k> 0 ~! Figure 9.3f shows the general situation for k 2: O. T he two functions f [m] and g[k - m] overlap over the interval 0 :S m :S k. Therefore (f) k e[k] = L f[m]g[k - m] k m- m =O k = L (O.8)m(0.3)k-m e [ k] g [ k-m] m =Q ..... f [m] k <0 (h) k = (0.3)k L m =O ( 0.8)m 0.3 k 2:0 (see Sec. B.7-4) For k < 0, there is no overlap between f[m] and g[k - mJ, as shown in Fig. 9.3g so t hat e[k] = 0 k <O e[k] = 2 [(0.8)k+l - (0.3)k+l] u[k] o ( g) m- o 2 r 345 k- Fig. 9 .3 Graphical understanding of convolution of f [k] and g[k] for Example 9.7. a nd which agrees with the earlier result in Eq. (9.52). • k (9.56) 594 t::. 9 T ime-Domain A nalysis o f D iscrete-Time S ystems E~ercise 9.4 S ystem r esponse t o E xternal I nput: T he Z ero-State R esponse 595 E 9.10 F ind (O.S)ku[k] * u[k] graphically and sketch the result. Answer: 5(1 - (O.S)k+l )u[k] \ l J [k] An A lternative Form o f Graphical Procedure: T he Sliding Tape Method 5 T his a lgorithm is convenient w hen t he s equences I [k] a nd g[k] a re s hort o r w hen t hey a re a vailable o nly i n g raphical f orm. T he a lgorithm is b asically t he s ame a s t he g rapb.ical p rocedure i n F ig. 9.3. T he o nly difference is t hat i nstead o f p resenting t he d ata a s g raphical p lots, we display i t as a s equence o f n umbers o n t apes. O therwise t he p rocedure is t he s ame, a s will b ecome c lear i n t he following e xample . • ( a) * , [':1 e[OJ [[[[[ k- 5 E~ample 9 .9 Using t he sliding t ape method, convolve t he two sequences l[kJ andg[kJ depicted in Fig. 9.4a and 9.4b, respectively. In this procedure we write t he sequences l[kJandg[kJ in t he s lots of two tapes: I t ape a nd 9 t ape (Fig. 9.4c). Now leave t he I t ape s tationary (to correspond t o l[mJ). T he g[-mJ t ape is o btained by time inverting t he g[mJ t ape a bout the origin (k = 0) so t hat t he slots corresponding t o f(OJ a nd g[OJ remain aligned (Fig. 9.4d). We now shift the inverted t ape by k slots, multiply values on two tapes in adjacent slots, a nd a dd all t he p roducts t o find e[kJ. Figures 9.4d, e, f, g, h, i, a nd j show t he cases for k = 0 , 1 ,2,3,4,5, a nd 6, respectively. For t he case of k = 0, for example (Fig. 9.4d) (b) k- m=O .j, Jlml f tape c[O] = 0 -+ g tape -+ ( d) c [1] = I rotate the g-tape about the verticle axis as showen in (d) =0 x 1=0 ( e) F or k = 1 (Fig. 9.4e) c[lJ c [2] = 3 = (0 x 1) + (1 x 1) = 1 Similarly, + (1 x 1) + (2 x 1) = 3 e[3J = (0 x 1) + (1 x 1) + (2 x 1) + (3 x 1) = 6 e[4J = (0 x 1) + (1 x 1) + (2 x 1) + (3 x 1) + (4 x 1) = 10 e[5J = (0 x l) + (1 x 1) + (2 x l) + (3 x l) + (4 x l) + (5 x l) = e[6J = (0 x 1) + (1 x 1) + (2 x 1) + (3 x 1) + (4 x 1) + (5 x 1) = ( f) e[2J = (0 x 1) c [...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online