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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. 9.416. k (j) (1<) F ig. 9 .4 Sliding t ape algorithm for discretetime convolution. 596 9 TimeDomain Analysis o f DiscreteTime 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 .43 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 erostate 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 (slidingtape 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 .42 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. 2.43, 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[mJzkm m =oo = I n this expression, t he z eroinput 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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