Digital Signal Processing Reference
In-Depth Information
1
1
0
0
−1
−1
0
5
10
0
2
4
6
8
10
(a) Sig[0]
δ
[n]
(b) Sig[0](Imp)
1
1
0
0
−1
−1
0
5
10
0
2
4
6
8
10
(c) Sig[1]
δ
[n−1]
(d) Sig[1](Imp)
1
1
0
0
−1
−1
0
5
10
0
2
4
6
8
10
(e) Sig[2]
δ
[n−2]
(f) Sig[2](Imp)
1
1
0
0
−1
−1
0
5
10
0
2
4
6
8
10
(g) Superpos. (a),(c),& (e)
(h) Superposition of (b),(d),& (f)
Figure 2.21:
The convolution depicted in Fig. 2.20, with the roles of signal and impulse response
reversed. (a) First signal sample, multiplied by
δ (n)
; (b) Impulse response, scaled by first sample of signal;
(c) Second sample of signal; (d) Impulse response, scaled by second signal sample, delayed by one sample;
(e) Third sample of signal; (f ) Impulse response scaled by third signal sample, delayed by two samples;
(g) Input signal, the superposition of its components shown in (a), (c), and (e); (h) The convolution, i.e.,
the superposition of responses shown in (b), (d), and (f ).
y
[
2
]=
x
[
0
]
h
[
2
]+
x
[
1
]
h
[
1
]+
x
[
2
]
h
[
0
]=
1
(
0
.
75
)
+
0
.
7
(
−
0
.
5
)
+
0
.
49
(
1
)
=
0
.
89
If we additionally compute the output for
k
=3, we get
y
[
3
]=
x
[
0
]
h
[
3
]+
x
[
1
]
h
[
2
]+
x
[
2
]
h
[
1
]+
x
[
3
]
h
[
0
]
which yields
y
[
3
]=
0
+
0
.
7
(
0
.
75
)
+
0
.
49
(
−
0
.
5
)
+
0
.
343
(
1
)
=
0
.
623
Thus we see that the roles of the two sequences (signal and impulse response) make no difference
to the resultant convolution sequence.
This can also easily be shown using MathScript's
conv
function by computing the convolution
both ways and taking the difference, which proves to be zero for all corresponding output samples.
