IIR (Infinite Impulse Response) filter structures must be recursive
(use feedback); an infinite number of coefficients could not otherwise
be realized with a finite number of computations per sample.
Hz=NzDz=
b
0
+
b
1
z-1+
b
2
z-2+…+
b
M
z-M1+
a
1
z-1+
a
2
z-2+…+
a
N
z-N
H
z
N
z
D
z
b
0
b
1
z
b
2
z
-2
…
b
M
z
M
1
a
1
z
a
2
z
-2
…
a
N
z
N
The corresponding time-domain difference equation is
yn=-
a
1
yn-1-
a
2
yn-2+…-
a
N
yn-N+
b
0
x0+
b
1
xn-1+…+
b
M
xn-M
y
n
a
1
y
n
1
a
2
y
n
2
…
a
N
y
n
N
b
0
x
0
b
1
x
n
1
…
b
M
x
n
M
Direct-form I IIR Filter Structure
The difference equation above is implemented directly as written by the
Direct-Form I IIR Filter Structure.
Note that this is a cascade of two systems,
Nz
N
z
and
1Dz
1
D
z
. If we reverse the order of the filters, the overall
system is unchanged: The memory elements appear in the middle
and store identical values, so they can be combined, to form
the Direct-Form II IIR Filter Structure.
Direct-Form II IIR Filter Structure
This structure is canonic: (i.e., it requires
the minimum number of memory elements).
Flowgraph reversal gives the
Transpose-Form IIR Filter Structure
Usually we design IIR filters with
N=M
N
M
, but not always.
Obviously, since all these structures have identical
frequency response, filter structures are not unique. We
consider many different structures because
- Depending on the technology or application, one
might be more convenient than another
- The response in a practical realization, in which the
data and coefficients must be quantized,
may differ substantially, and some
structures behave much better than others with quantization.
The Cascade-Form IIR filter structure is one of the least sensitive
to quantization, which is why it is the most commonly used IIR filter
structure.
IIR Cascade Form
The numerator and denominator polynomials can be factored
Hz=
b
0
+
b
1
z-1+…+
b
M
z-m1+
a
1
z-1+…+
a
N
z-N=
b
0
∏k=1Mz-
z
k
zM-N∏i=1Nz-
p
k
H
z
b
0
b
1
z
…
b
M
z
m
1
a
1
z
…
a
N
z
N
b
0
k
1
M
z
z
k
z
M
N
i
1
N
z
p
k
and implemented as a cascade of short IIR filters.
Since the filter coefficients are usually real yet the roots are
mostly complex, we
actually implement these as second-order sections, where
comple-conjugate pole and zero pairs are combined into
second-order sections with real coefficients.
The second-order sections are usually implemented with either
the Direct-Form II or Transpose-Form structure.
Parallel form
A rational transfer function can also be written as
b
0
+
b
1
z-1+…+
b
M
z-m1+
a
1
z-1+…+
a
N
z-N=
c
0
′+
c
1
′z-1+…+
A
1
z-
p
1
+
A
2
z-
p
2
+…+
A
N
z-
p
N
b
0
b
1
z
…
b
M
z
m
1
a
1
z
…
a
N
z
N
c
0
c
1
z
…
A
1
z
p
1
A
2
z
p
2
…
A
N
z
p
N
which by linearity can be implemented as
As before, we combine complex-conjugate pole pairs into
second-order sections with real coefficients.
The cascade and parallel forms are of interest because they
are much less sensitive to coefficient quantization than
higher-order structures, as analyzed in later modules in this course.
Other forms
There are many other structures for IIR filters,
such as wave digital filter
structures, lattice-ladder, all-pass-based forms, and so forth.
These are the result of extensive research to find structures
which are computationally efficient and
insensitive to quantization error. They all represent various
tradeoffs; the best choice in a given context is not yet fully
understood, and may never be.