Digital Signal Processing Reference
In-Depth Information
2
0
−2
0
50
100
150
200
250
(a) Sample Index
2
0
−2
0
50
100
150
200
250
(b) Sample Index
10
0
−10
0
50
100
150
(c) Lag Number
Figure 4.28:
(a) First digitized sound; (b) Second digitized sound, not shifted; (c) Correlation sequence
(initialized with zeros).
4.12.2 RECURSIVE COMPUTATION
In the single pole IIR, the current (or
n
th
) output of the filter is equal to the current (
n
th
) input, weighted
by coefficient
b
, plus the previous (or
(n
1
)
th
) output weighted by coefficient
a
(which, for this simple
single pole case, is equal to the pole). This can be written as
−
y
[
n
]=
bx
[
n
]+
ay
[
n
−
1
]
(4.17)
Example 4.17.
Filter the sequence s =
[
s
0
]
with a single pole IIR having
b
= 1 and
a
=p.
=1,
p
,
p
2
,p
3
, ...etc. and the output sequence is
s
0
,
ps
0
,
p
2
s
0
,
p
3
s
0
... , which is clearly the impulse response weighted by
s
0
, i.e.,
s
0
h
[
]
The filter impulse response is
h
n
.
If, for example,
s
0
= 2, then the output sequence is 2, 2
p
,2
p
2
, ....= 2
h
[
n
]
[
n
]
.
Example 4.18.
with the same IIR as used immediately above and show
that the output is the superposition of weighted, delayed versions of the filter impulse response.
Filter the sequence s =
[
s
0
s
1
]
