Digital Signal Processing Reference
In-Depth Information
This recursive type of equation represents an IIR filter. The output depends on the
inputs as well as past outputs (with feedback). The output
y
(
n
), at time
n
, depends
not only on the current input
x
(
n
), at time
n
, and on past inputs
x
(
n
-
1),
x
(
n
-
2),
...,
x
(
n
M
).
If we assume all initial conditions to be zero in (5.2), the
z
-transform of (5.2)
becomes
-
N
), but also on past outputs
y
(
n
-
1),
y
(
n
-
2),...,
y
(
n
-
()
=
()
+
()
+
()
+◊◊◊+
()
Yz aXzazXzazXz
-
1
-
2
azXz
-
N
0
1
2
N
()
-
()
-◊◊◊-
()
-
bz Yz
-
1
bz Yz
-
2
b z
-
M
Yz
(5.3)
1
2
M
Let
N
=
M
in (5.3); then the transfer function
H
(
z
) is
()
()
=
()
()
Yz
Xz
aaz az
+
-
1
+
-
2
+◊◊◊+
az
-
N
Nz
Dz
()
=
0
1
2
N
Hz
=
(5.4)
1
+
bz
-
1
+
bz
-
2
+◊◊◊+
b z
-
N
1
2
N
where
N
(
z
) and
D
(
z
) represent the numerator and denominator polynomial, respec-
tively. Multiplying and dividing by
z
N
,
H
(
z
) becomes
N
N
-
1
N
-
2
N
az
+
az
+
az
+◊◊◊+
a
zz
zp
-
-
=
'
()
=
0
1
2
N
i
Hz
=
C
(5.5)
zbz
N
+
N
-
1
+
bz
N
-
2
+◊◊◊+
b
1
2
N
i
i
1
which is a transfer function with
N
zeros and
N
poles. If all the coefficients
b
j
in
(5.5) are zero, this transfer function reduces to the transfer function with
N
poles
at the origin in the
z
-plane representing the FIR filter discussed in Chapter 4. For
a system to be stable, all the poles must reside inside the unit circle, as discussed in
Chapter 4. Hence, for an IIR filter to be stable, the magnitude of each of its poles
must be less than 1, or:
1.
If |
P
i
|
<
1, then
h
(
n
)
Æ
0, as
n
Æ•
, yielding a stable system.
2.
If |
P
i
|
>
1, then
h
(
n
)
Æ•
, as
n
Æ•
, yielding an unstable system.
1, the system is marginally stable, yielding an oscillatory response.
Furthermore, multiple-order poles on the unit circle yield an unstable system.
Note again that with all the coefficients
b
j
=
If |
P
i
|
=
0, the system reduces to a nonrecursive
and stable FIR filter.
5.2 IIR FILTER STRUCTURES
There are several structures that can represent an IIR filter, as discussed next.
5.2.1 Direct Form I Structure
With the direct form I structure shown in Figure 5.1, the filter in (5.2) can be real-
ized. There is an implied summer (not shown) in Figure 5.1. For an
N
th-order filter,
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