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Chapter 2 Answers 2.1. (a) We know that 0 0
yi[n] = x[n] * h[n] = ^ h[k]x[n - k ]
(S2.1-1)
k = - o o
The signals x[n] and /i[n] are as shown in Figxire S2.1. 2 .
,
-tW
h W
4
A
3
0
\
t
^
-10
1
i )
Figure S2.1
Prom this figure, we can easily see that the above convolution
sum reduces to
This gives
T a his th nd wo o eir is rk w r sa co pro is ill le u vi pr de o rse de ot st f a s d s ec ro n an o te y y p d le d th a a ly by e rt ss fo U in o e r te f t ss th nite gr hi in e ity s w g us d S of or stu e o tat th k ( de f i es e in nt ns co w cl le tr p or ud a uc y r k an ing rnin tors igh d on g. in t la is w D no the iss tea s t p W em ch er or in ing m ld a itt W tio ed id n . e W eb )
yi[n] = /i[-l]a:[n + l] + /i[l]a:[n - 1 ] = 2x[n + 1] + 2x[n — 1
yi [n] = 25[n + 1] + A5[n] + 2<5[n — 1] + 25[n -- 2 ] (b) We know that
1
M <
1
O O
y2[n] = x[n + 2] * /i[n] = ^ h[k]x[n +
2 -- k ]
k — - o o
Comparing with eq. (S2.1-1), we see that
y2[n] = yi[n + 2;
(c) We may rewrite eq. (S2.1-1) as 0 0
yi[n] = x[n] * /i[n] = ^ x[k]h[n - k ] k = - o o
Similarly, we may write 0 0
yz[n] = a;[n] * h[n + 2] = ^ i[/i:]/i[n +
2 -- k ]
fc=-oo
Comparing this with eq. (S2.1), we see that
•
!/3W = yi[n + 2] 30
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