Chapter 2 Answers
2.1. (a) We know that
0 0
yi[n] = x[n] * h[n] = ̂ h[k]x[nk = - o o
- k ] (S2.1-1)
The signals x[n] and /i[n] are as shown in Figxire S2.1.
, - t WA
2 .4 h W
3
0 \ t ^ - 1 0i )
1
Figure S2.1
Prom this figure, we can easily see that the above convolution sum reduces to
yi[n] = /i[-l]a:[n + l] + /i[l]a:[n= 2x[n + 1] + 2x[n — 1
- 1 ]
This gives
yi [n] = 25[n + 1] + A5[n] + 2<5[n — 1] + 25[n --2] - 1<M1
(b) We know thatO O
y2[n] = x[n + 2] * /i[n] = ̂ h[k]x[nk — - o o
Comparing with eq. (S2.1-1), we see that
+ 2 -- k ]
y2[n] = yi[n + 2;
(c) We may rewrite eq. (S2.1-1) as0 0
yi[n] = x[n] * /i[n] = ̂ x[k]h[nk = - o o
- k ]
Similarly, we may write
0 0
yz[n] = a;[n] * h[n + 2] = ̂ i[/i:]/i[nf c = - o o
+ 2 -- k ]
Comparing this with eq. (S2.1), we see that•
!/3W = yi[n + 2]
30
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