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Unformatted text preview: 9.2 b, c, d, e, a nd f.
T he c omponent of J[k] a t k = m is J[m]6[k  m], a nd J[k] is t he s um o f all these
c omponents s ummed from m =  00 t o 0 0. T herefore P roperties o f t he C onvolution S um
T he s tructure o f t he c onvolution s um is similar t o t hat of t he c onvolution
integraL Moreover, t he p roperties o f t he c onvolution s um a re similar t o t hose of
t he c onvolution integraL We shall e numerate t hese properties h ere w ithout proof.
T he proofs are similar t o t hose for t he c onvolution integral a nd m ay b e d erived by
t he r eader.
1. T he C ommutative P roperty + J[I]6[k  1] + J[2]6[k  2] + ...
+ J[1]6[k + 1] + J[2]6[k + 2] + ... Jdk] * h [k] = h [k] * ! I[k] (9.43) 2. T he D istributive P roperty 00 m =oo ,m =oo v y[k] = r"T"
L 00 L j [k] k = = (d) rr"HI J[k] = J[0]6[k] J[m]6[k  m] ,m =oo k Fig. 9 .2 Representation of an arbitrary signal = J[m]h[k  m] J[m]6[k  m] (9.40) h [k] * (h[k] + h [kj) = ! I[k] * h [k] + ! I[k] * h [k] (9.44) 5 88 9 T imeDomain Analysis of DiscreteTime Systems 9.4 System response t o E xternal I nput: T he Z eroState R esponse 589 3 . T he Associative Property
h [k] * ( h[k] * h [kj) = ( h[k] * h [kj) * h [k] k (9.45) e[k] = L J[m]g[k  m] m =O 4 . T he Shifting Property
If k * h [k] = h [k] t hen+ h [k  m] * h [k  =L
e[k] n] = e[k  m  n] (0.8)mu[m] (0.3)kmu [k  m] (9.51) m =O (9.46) 5 . T he Convolution with an Impulse In the above summation, m lies between 0 and k (0::; m ::; k ). Therefore, if k ;:0: 0, then
both m and k  m ;:0: 0, so t hat u[m] = u[k  m] = 1. I f k < 0, m is negative because m
lies between 0 and k, and u[m] = O. Therefore, Eq. (9.51) becomes
k * 8[k] f [k] = f [k] e[k] = L (0.8)m (0.3)km (9.47) k;:O:O m =O 6 . The Width Property k <O
=0
I f h [k] a nd h [k] h ave lengths of m a nd n e lements respectively, t hen t he
l ength of e[k] is m + n  1 e lements.t T he w idth p roperty m ay a ppear t o b e
violated in some special cases as explained o n p. 136. and
k elk] = (0.3)k L 8 (~:3) m u[k] m =O Causality and ZeroState Response
I n d eriving Eq. (9.41), we assumed t he s ystem t o b e linear a nd t imeinvariant.
T here were no o ther r estrictions o n e ither t he i nput signal o r t he s ystem. I n p ractice,
a lmost all o f t he i nput s ignals a re causal, a nd a m ajority o f t he s ystems a re also
causal. T hese r estrictions f urther simplify t he l imits of t he s um i n Eq. (9.41). I f
t he i nput f [k] is causal, f [m] = 0 for m < O. Similarly, if t he s ystem is causal ( that
is, if h[k] is causal), t hen h[x] = 0 for negative x, so t hat h[k  m] = 0 w hen m > k.
T herefore, i f f [k] a nd h[k] a re b oth causal, t he p roduct f [m]h[k  m] = 0 for m < 0
a nd for m > k, a nd i t is nonzero only for t he r ange 0 ~ m ~ k. T herefore, Eq.
(9.41) i n t his case reduces t o
k f [m]h[k  m] y[k] = L (9.48) m =O This is a geometric progression with common ratio (0.8/0.3). From Sec. 8.74 we have
e[k] =(0....
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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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