IIR Filter design
(cf. Shenoi, 2006)

The transfer function of the IIR filter is given by
Its frequency responses are (where w is the normalized
frequency ranging in [, ].

When a and b are real, the magnitude response |H(ejw)|
is an even function, and the phase response (jw) is an
odd function.
 Very often it is convenient to compute and plot the log
magnitude of |H(ejw)| as

10log10 | H (e jw ) |2
measured in dB.
 Linear phase:
Consider the ideal delay system. The impulse response is
hid [n]   [n  nd ]
and the frequency response is
Hid [e jw ]  e jwnd

In this case, the magnitude and phase responses are
and
| Hid [e jw ] | 1
Hid [e jw ]  wnd
| w | 
Hence, when time domain is a constant delay, it causes
the frequency a “linear phase distortion.”
 Sometimes we hope the filter response is linear phase,
i.e., the phase response is linear with w.
Eg. an ideal lowpass filter with linear phase (i.e., ideal
low-pass but the output is delayed by nd samples in the
time domain)
 jwnd
e
,
w  wc
| H lp [e ] | 
wc  w  
 0,
jw
However, linear phase is difficult to achieve by using
IIR filters, but it can be easily designed by using FIR
filters.

Group delay: A convenient measure of the linearity of
the phase is the group delay.

d
 ( w)  grd [ H (e )]   {[ H (e jw )]}
dw
jw

For the IIR filter, the group delay is
where
and
(unit: samples)
Since the state-of-the-arts of analog IIR filter is
more advanced, the design of discrete-time IIR filter is
usually an approximation of the analog one.
 Three approximation criteria commonly used:

The Butterworth approximation
 The Chebyshev (minimax) approximation


Analog filters
 represented by Laplace transform H(s).
 Substituting s=jw, we obtain its frequency response
(in terms of continuous Fourier transform).
where |H(jw)| and (jw) can be found by
Maximally Flat and Butterworth Approximation
 Magnitude response of an ideal low-pass analog filter
showing the tolerances:
passband: [0, wp], transition band:[wp, ws], stopband: [ws, ]
passband tolerance: p, stopband tolerance: s.
Butterworth response (or maximally flat magnitude
response):
and now we have
Magnitude responses of Butterworth lowpass filters
normalized passband
D2n and n are the parameters of the Butterworth filter,
where n is the order of this filter.
If, for example, the magnitude at the passband frequency
p, is 1/ 2 , which means that the log magnitude required is
3dB, then we choose D2n = 1.
If the magnitude at the passband frequency  = p = 1 is
required to be 1  p, then we choose D2n, normally denoted
by 2, such that
If the magnitude at the bandwith  = p = 1 is given as
Ap dB, the values 2 is computed by
We get the formula
Let us consider the common case of a Butterworth filter
with a log magnitude of 3dB at the bandwith of p = 1 to
develop the design procedure of Butterworth lowpass
filter. In this case, we use the function for the prototype
filter, in the form
This satisfies the following properties:
Design theory of Butterworth lowpass filters
Let us consider the design of Butterworth lowpass filter
for which (1) the frequency p at which the magnitude is
3DB below the maximum value at =0, and (2) the
magnitude at another frequency s in the stopband are
specified.
2
2n
|
H
(
j

)
|

1
/(
1


) but want to
Since we only know that
infer H(ej). Let us consider the relationship that
|H(ej)|2 = H(ej)H(ej).
We denote  = p/j, so
There are n poles satisfying the following equations:
This gives us the 2n poles of H(p)H(p), which are
and
In general,
These poles have a magnitude of one, and the angle
between adjacent poles is equal to /n.
There are n poles in the left half plane and n poles in the
right plane.
 Note that for a continuous-time linear system, it is
stable iff all poles lie in the left half plane.
Note that, for every pole of H(p) at p=pa, that lies in the
left half-plane, there is a pole of H(p) at p=pa that lies
in the right half-plane.
Because of this property, we identify n poles that are in
the left half of the p planes as the poles of H(p) so that it
is a stable transfer function.
The poles that are in the left half of the p plane are given
by
When we have found these n poles, we construct the
denominator polynomial D(p) of the prototype filter from
H(p) = 1/D(p) from
Pole locations of Butterworth low-pass filters of orders
n=6
The only unknown parameter at this stage of design is the
order n of the filter function H(p). This is calculated using
the specification that at the stopband frequency s, the
log magnitude is required to be no more than As dB.
So we choose n as an integer satisfying
The pairs of poles from left and right half-planes can be
found by the polynomial in the denominator D(p),
Their coefficients can be computed recursively from
where d0=1. The polynomial is referred to as Butterworth
polynomials.
Chebyshev 1 Approximation:
The Chebyshev 1 approximation for an ideal lowpass filter
has equal-valued ripples in the passband . It is known as
minimax approximation and also known as the equiripple
approximation.
The magnitude squared function of Chebyshev
approximation:
where Cn() is the Chebyshev polynomial of degree n. It is
defined by
For n = 2, 3, 4, 5, these polynomials are
Chebyshev II Approximation: (or inverse Chebyshev
filters)
Maximally flat at w=0; decreases monotonically as the
frequency increases and has an equiripple response in the
stopband.
The magnitude square function of the inverse Chebyshev
low pass filter: |H(j)|2 =
Chebyshev I
Chebyshev II
Discrete-time IIR filter
The procedure used for designing discrete-time IIR
filters employ different transformations of the form s =
f(z) to transform H(s) into H(z).
Two methods:
 Impulse invariance
 Bilinear Transform
Please see the copy of book chapters (Oppenheim 1999,
Section 7.1)
General concept on digital filter design (cf. D. Ellis, Columbia
University)
Discrete-time FIR filters
Discrete-time FIR filters
Linear Phase Discrete-time FIR filters
Now we consider the special types of FIR filters in
which the impulse response h[n] are assumed to be
symmetric or antisymmetric.
 Since the order of the polynomial in each of these two
types can be either odd or even, we have four types of
filters with different properties.

Linear phase FIR filters:
Type 1: The coefficients are symmetric, i.e., h[n] = h[Nn],
and the order N is even.
Linear phase response
Linear phase FIR filters:
Type II: The coefficients are symmetric, i.e., h[n] = h[Nn],
and the order N is odd.
Linear phase FIR filters:
Type III: The coefficients are antisymmetric, i.e.,
h[n] =  h[Nn], and the order N is even.
Linear phase FIR filters:
Type IV: The coefficients are antisymmetric, i.e.,
h[n] =  h[Nn], and the order N is odd.
Fourier Series Method Modified by Windows
Let us consider the magnitude response of the ideal
lowpass filter to be HLP(ejw), in which the cutoff frequency
is given as wc. Its inverse Fourier transform (i.e., the
impulse response of the ideal low pass filter) is:
We choose the following finite series to approximate
HLP(ejw):
Since {ejnw} form an set of orthonormal bases, CLP[n] is the
the least squared solution.
The approximation error is
As M increases, the number of ripples in the passband
(and the stopband) increases while the width between the
frequencies at which the the maximum error occurs in the
passband (0wwc) and in the stopband (wcw<) decreases.

Gibbs phenomenon: as M increases, the maximum
deviation from the ideal value decreases except near the
point of discontinuity, where the error remains the same,
however large the value M we choose. (i.e., as M increases,
the maximum amplitude of the oscillation does not approch
zero)
