Digital Signal Processing Reference
In-Depth Information
1
1
0
0
−1
−1
−10
−5
0
5
10
−10
−5
0
5
10
(a) n (Seq1)
(b) n (Seq2)
1
1
0
0
−1
−1
−10
−5
0
5
10
−10
−5
0
5
10
(c) n
(d) n
1
1
0
0
−1
−1
−10
−5
0
5
10
−10
−5
0
5
10
(e) n
(f) n
Figure 2.23:
(a) First Sequence (second sequence in previous figure); (b) Second Sequence (first sequence
in previous figure); (c) Second sequence time reversed (TR) and oriented to compute the first value of
the convolution sequence (arrow shows direction the second sequence will slide, sample-by-sample, to
perform convolution); (d) TR second sequence oriented to compute the second value of the convolution
sequence; (e) TR second sequence oriented to compute the third value of the convolution sequence; (f ) TR
second sequence oriented to compute the fourth value of the convolution sequence.
We see that the impulse response is a decaying positive-valued geometric sequence that sums to
1/(1 - 0.9) = 10. Thus we see that the impulse response is absolutely summable and hence the IIR is a
stable system.
Causality is based on the idea that something occurring now must depend only on things that have
happened in the past, up to the present time. In other words, future events cannot influence events in the
present or the past. An LTI system output
y
must depend only on previous or current values of input
and output. More particularly, a system is causal if
[
n
]
h
[
n
]=
0 if
n<
0
Example 2.10.
Determine if the following impulse response is causal:
0
.
5
n
+
1
u
h
[
n
]=
[
n
+
1
]
