Digital filters with an Infinite-duration Impulse Response
(IIR) have characteristics that make them useful in many
applications. This section develops and discusses the properties and
characteristics of these filters[3].
Because of the feedback necessary in an implementation, the
Infinite Impulse Response (IIR) filter is also called a recursive
filter or, sometimes, an autoregressive moving-average filter
(ARMA). In contrast to the FIR filter with a polynomial transfer
function, the IIR filter has a rational transfer function. The
transfer function being a ratio of polynomials means it has finite
poles as well as zeros, and the frequency-domain design problem
becomes a rational-function approximation problem in contrast to the
polynomial approximation for the FIR filter[4]. This gives considerably
more flexibility and power, but brings with it certain problems in
both design and implementation[3], [2], [1].
The defining relationship between the input and output
variables for the IIR filter is given by
y
(
n
)
=
∑
k
=
1
N
a
(
k
)
y
(
n
-
k
)
+
∑
m
=
0
M
b
(
m
)
x
(
n
-
m
)
.
y
(
n
)
=
∑
k
=
1
N
a
(
k
)
y
(
n
-
k
)
+
∑
m
=
0
M
b
(
m
)
x
(
n
-
m
)
.
(1)The second summation in Equation 1 is exactly the same
moving average of the present plus past MM values of the input that
occurs in the definition of the FIR filter. The difference arises
from the first summation, which is a weighted sum of the previous
NN output values. This is the feedback or recursive part which
causes the response to an impulse input theoretically to endure
forever. The calculation of each output term y(n) from Equation 1
requires N+M+1N+M+1 multiplications and N+MN+M additions. There are
other algorithms or structures for calculating y(n)y(n) that may
require more or less arithmetic.
In addition to the number of calculations required to calculate each
output term being a measure of efficiency, the amount of storage for
coefficients and intermediate calculations is important. DSP chips
are designed to efficiently implement calculations such as Equation 1
by having a single cycle operation that multiplies a variable by a
constant and accumulates it. In parallel with that operation, it is
simultaneously calculating the address of the next variable.
Just as in the case of the FIR filter, the output of an IIR
filter can also be calculated by convolution.
y
(
n
)
=
∑
k
=
0
∞
h
(
k
)
x
(
n
-
k
)
y
(
n
)
=
∑
k
=
0
∞
h
(
k
)
x
(
n
-
k
)
(2)In this case, the duration of the impulse response
h(n)h(n) is infinite and, therefore, the number of terms in Equation 2
is infinite. The N+M+1N+M+1 operations required in Equation 1 are
clearly preferable to the infinite number required by Equation 2.
This gives a hint as to why the IIR filter is very efficient. The
details will become clear as the characteristics of the IIR filter
are developed in this section.
The transfer function of a filter is defined as the ratio
Y(z)/X(z)Y(z)/X(z), where Y(z)Y(z) and X(z)X(z) are the z-transforms of the
output y(n)y(n) and input x(n)x(n), respectively. It is also the
z-transform of the impulse response. Using the definition of the
z-transform in Equation 32 from Discrete-Time Signals, the transfer function of the IIR filter
defined in Equation 1 is
H
(
z
)
=
∑
n
=
0
∞
h
(
n
)
z
-
n
H
(
z
)
=
∑
n
=
0
∞
h
(
n
)
z
-
n
(3)This transfer function is also the ratio of the
z-transforms of the a(n)a(n) and b(n)b(n) terms.
H
(
z
)
=
∑
n
=
0
M
b
(
n
)
z
-
n
∑
n
=
0
N
a
(
n
)
z
-
n
=
B
(
z
)
A
(
z
)
H
(
z
)
=
∑
n
=
0
M
b
(
n
)
z
-
n
∑
n
=
0
N
a
(
n
)
z
-
n
=
B
(
z
)
A
(
z
)
(4)The frequency response of the filter is found by
setting z=ejωz=ejω, which gives Equation 1 the form
H
(
ω
)
=
∑
n
=
0
∞
h
(
n
)
e
-
j
ω
n
H
(
ω
)
=
∑
n
=
0
∞
h
(
n
)
e
-
j
ω
n
(5)It should be recalled that this form assumes a
sampling rate of T=1T=1. To simplify notation, H(ω)H(ω) is used
to denote the frequency response rather than H(ejω)H(ejω).
This frequency-response function is complex-valued and
consists of a magnitude and phase. Even though the impulse response
is a function of the discrete variable n, the frequency response is
a function of the continuous-frequency variable ωω and is
periodic with period 2π2π.
Unlike the FIR filter case, exactly linear phase is impossible
for the IIR filter. It has been shown that linear phase is
equivalent to symmetry of the impulse response. This is clearly
impossible for the IIR filter with an impulse response that is zero
for n<0n<0 and nonzero for n going to infinity.
The FIR linear-phase filter allowed removing the phase from
the design process. The resulting problem was a real-valued
approximation problem requiring the solution of linear equations.
The IIR filter design problem is more complicated. Linear phase is
not possible, and the equations to be solved are generally
nonlinear. The most common technique is to approximate the
magnitude of the transfer function and let the phase take care of
itself. If the phase is important, it becomes part of the
approximation problem, which then is often difficult to solve.
As shown in another module, LL equally spaced samples of
H(ω)H(ω) can be approximately calculated by taking an LL-length
DFT of h(n)h(n) given in Equation 5. However, unlike for the FIR
filter, this requires that the infinitely long impulse response be
truncated to at least length-L. A more satisfactory alternative is
to use the DFT to evaluate the numerator and denominator of
Equation 4 separately rather than to approximately evaluate
Equation 5. This is accomplished by appending L-NL-N zeros to the
a(n)a(n) and L-ML-M zeros to the b(n)b(n) from Equation 1, and taking
length-L DFTs of both to give
H
(
2
π
k
/
L
)
=
DFT
{
b
(
n
)
}
DFT
{
a
(
n
)
}
H
(
2
π
k
/
L
)
=
DFT
{
b
(
n
)
}
DFT
{
a
(
n
)
}
(6)where the division is a term-wise division of each of the LL values
of the DFTs as a function of kk. This direct method of calculation
is a straightforward and flexible technique that does not involve
truncation of h(n)h(n) and the resulting error. Even nonuniform
spacing of the frequency samples can be achieved by altering the DFT
as was suggested for the FIR filter. Because IIR filters are
generally lower in order than FIR filters, direct use of the DFT is
usually efficient enough and use of the FFT is not necessary. Since
the a(n)a(n) and b(n)b(n) do not generally have the symmetries of the
FIR h(n)h(n), the DFTs cannot be made real and, therefore, the
shifting and stretching techniques of other modules are not
applicable.
As an example, the frequency-response plot of a third-order
elliptic-function lowpass filter with a transfer function of
H
(
z
)
=
0
.
1335
z
3
+
0
.
056
z
2
+
0
.
056
z
+
0
.
1335
z
3
-
1
.
507
z
2
+
1
.
2646
z
-
0
.
3786
H
(
z
)
=
0
.
1335
z
3
+
0
.
056
z
2
+
0
.
056
z
+
0
.
1335
z
3
-
1
.
507
z
2
+
1
.
2646
z
-
0
.
3786
(7)is given in Figure 1a. The details for designing this
filter are discussed in elsewhere. A similar performance for the
magnitude response would require a length of 18 for a linear-phase
FIR filter.
The possible locations of the zeros of the transfer function of an
FIR linear-phase filter were analyzed elsewhere. For the IIR filter,
there are poles as well as zeros. For most applications, the
coefficients a(n)a(n) and b(n)b(n) are real and, therefore, the poles
and zeros occur in complex conjugate pairs or are real. A filter is
stable if for any bounded input, the output is bounded. This implies
the poles of the transfer function must be strictly inside the unit
circle of the complex zz plane. Indeed, the possibility of an
unstable filter is a serious problem in IIR filter design, which
does not exist for FIR filters. An important characteristic of any
design procedure is the guarantee of stable designs, and an
important ability in the analysis of a given filter is the
determination of stability. For a linear filter analysis, this
involves the zeros of the denominator polynomial of Equation 4. The
location of the zeros of the numerator, which are the zeros of
H(z)H(z), are important to the performance of the filter, but have no
effect on stability.
If both the poles and zeros of a transfer function are all
inside or on the unit circle of the zz-plane, the filter is called
minimum phase. The effects of a pole or zero at a radius of rr from
the origin of the zz-plane on the magnitude of the transfer
function are exactly the same as one at the same angle but at a
radius of 1/r1/r. However, the effect on the phase characteristics is
different. Since only stable filters are generally used in practice,
all the poles must be inside the unit circle. For a given magnitude
response, there are two possible locations for each zero that is not
on the unit circle. The location that is inside gives the least
phase shift, hence the name “minimum- phase" filter.
The locations of the poles and zeros of the example in Equation 7
are given in Figure 1b.
Since evaluating the frequency response of a transfer function is
the same as evaluating H(z)H(z) around the unit circle in the
zz-plane, a comparison of the frequency-response plot in Figure 1a
and the pole-zero locations in Figure 1b gives insight into the
effects of pole and zero location on the frequency response. In the
case where it is desirable to reject certain bands of frequencies,
zeros of the transfer function will be located on the unit circle at
locations corresponding to those frequencies.
By having both poles and zeros to describe an IIR filter, much
more can be done than in the FIR filter case where only zeros
exist. Indeed, an FIR filter is a special case of an IIR filter
with a zero-order denominator. This generality and flexibility does
not come without a price. The poles are more difficult to realize
than the zeros, and the design is more complicated.
This section has given the basic definition of the IIR or recursive
digital filter and shown it to a generalization of the FIR filter
described in the previous chapters. The feedback terms in the IIR
filter cause the transfer function to be a rational function with
poles as well as zeros. This feedback and the resulting poles of the
transfer function give a more versatile filter requiring fewer
coefficients to be stored and less arithmetic. Unfortunately, it
also destroys the possibility of linear phase and introduces the
possibility of instability and greater sensitivity to the effects of
quantization. The design methods, which are more complicated than
for the FIR filter, are discussed in another section, and the
implementation, which also is more complicated, is discussed in
still another section.
-
Mitra, Sanjit K. (2006). Digital Signal Processing, A Computer-Based Approach. (Third). [First edition in 1998, second in 2001]. New York: McGraw-Hill.
-
Oppenheim, A. V. and Schafer, R. W. (1999). Discrete-Time Signal Processing. (Second). [Earlier editions in 1975 and 1989]. Englewood Cliffs, NJ: Prentice-Hall.
-
Parks, T. W. and Burrus, C. S. (1987). Digital Filter Design. New York: John Wiley & Sons.
-
Rader, Charles M. (2006, November). The Rise and Fall of Recursive Digital Filters. IEEE Signal Processing Magazine, 23(6), 46–49.
