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

# 4 s ystem r esponse t o e xternal i nput t he z ero

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Unformatted text preview: xercise E 9.7 Show t hat (O.S)k+ 1 u[k] * u[k] = 4[1 - O.S(O.S)k]u[k] Use convolution tahle. Recognize t hat (O.S)k+l = O.S(O.S)k l::. 0 ..:: 0 z 0 0:&gt; or.&gt; «; ~ &quot;&quot;r ;; &quot;&quot; '&quot; ~ ~ &quot;&quot;~ r &quot;&quot;~ .... .,., ; :l ;:l r ~ ;:l &quot;&quot;~ r ..., '&quot; ~ ':l&lt; ;:l ~ ;:l ..., ;:l &quot;&quot;r &quot;&quot;r t- oo C1&gt; '&quot; 0 &quot;&quot;&quot; £. S \1 (1 - ~k)3-k]u[k] = (~)k E xercise E 9.9 Using convolution table, show t hat e -ku[k] * 2 - ku[k] Hint: e - k = (~)k and 2 - k = (O.5)k \1 \1 = 2:. [ e- k - ~2-k]u[k] C omputer E xample C 9.4 F ind a nd s ketch t he z ero-state r esponse for t he s ystem d escribed by ( E2 o::t ; :l * (O.2)ku[k] = .!j[(O.2)k - l::. .+ .., ~ (9.55) • ~£ 7 0: 1l + &quot;&quot; &lt;b 1 ..., + 5.81(O.8)k] u [k] - ;:;-0 ---..::::. ; :l ~ = [ -1.26(0.25)k + 0.444( _ 0.2)k + 5 .81(0.8)k] u [k] &quot; '-, + ~ ~ = ')'(')')k Hint: Use convolution table. Recognize t hat 3 - k ~ ;:l OJ l::. E xercise E 9.S Show t hat k 3- k u[k] &quot;&quot;c-. ; :l ~ .+ .., + ~ ' )'k+l £: :::i' + 7 .27(0.8)k+l] u [k] c-. ~ + .... ~ - 2.22( _ 0.2)k+l R ecognizing t hat :;&quot; ~ - (-0.2)k+l]-7.27 [ (0.25)k+l - (O.8)k+l]) u [k] = [ -5.05(0.25)k+l &lt;:!&gt; Ci 4 ( 0.25)k+l - (O.8)k+l ] k 0.25 _ 0.8 u[ ] = (2.22 [ (0.25)k+l 1 '@ + for t he i nput f [k] + 6 E + 9 )y[k] = ( 2E2 + 6 E)f[k] = 4k u[k]. k =O:l1; b =[2 6 0 ]; a =[l 6 9 ]; f =4.'(k); y =filter(b,a,f); s tem(k,y) x labeJ('k');yJabeJ('y[k)'); 0 592 9 T ime-Domain Analysis of Discrete-Time S ystems 9.4 S ystem r esponse t o E xternal I nput: T he Z ero-State Response 9 .4-1 Graphical Procedure for t he Convolution Sum f [k] 593 g [ k] T he s teps in evaluating t he convolution s um a re parallel t o t hose followed in e valuating t he convolution integraL T he c onvolution s um o f causal signals f [k] a nd g [k] is given by (a) ( b) k e[k] = L f [m]g[k - m] m =O W e first plot f [m] a nd g[k - m] a s functions o f m ( not k), b ecause t he s ummation is o ver m . F unctions f [m] a nd g[m] a re t he s ame a s f [k] a nd g[k], p lotted respectively a s f unctions of m (see Fig. 9.3). T he c onvolution o peration c an b e p erformed as follows: 1. I nvert g[m] a bout t he v ertical axis (m = 0) t o o btain g [-m] (Fig. 9.3d). Figure 9.3e shows b oth f [m] a nd g [-m]. 2. T ime s hift g [-m] by k u nits t o o btain g[k - m]. F or k &gt; 0, t he s hift is t o t he r ight (delay); for k &lt; 0 , t he s hift is t o t he left (advance). Figures 9.3f a nd 9.3g show g[k - m] for k &gt; 0 a nd for k &lt; 0, respectively. 3. N ext we multiply f [m] a nd g[k - m] a nd a dd all t he p roducts t o o btain e[k]. T he p rocedure is r epeated for each value o f k over t he r ange - 00 t o 0 0. We shall d emonstrate b y a n e xample t he g raphical procedure for finding t he c onvolution sum. Although b oth t he f unctions in t his e xample are causal, t he p rocedure is applicable t o g eneral case . • E xample 9 .8 Find 0 I 2 4 3 o k- k- (O.8)m f [m...
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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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