The mathematical problem inherent in the frequency-domain
filter design problem is the approximation of a desired complex
frequency-response function Hd(z)Hd(z) by a rational transfer
function H(z)H(z) with an Mth-degree numerator and an Nth-degree
denominator for values of the complex variable zz along the unit
circle of z=ejωz=ejω. This approximation is achieved by
minimizing an error measure between H(ω)H(ω) and Hd(ω)Hd(ω).
For the digital filter design problem, the mathematics are
complicated by the approximation being defined on the unit circle.
In terms of zz, frequency is a polar coordinate variable. It is
often much easier and clearer to formulate the problem such that
frequency is a rectangular coordinate variable, in the way it
naturally occurs for analog filters using the Laplace complex
variable ss. A particular change of complex variable that converts
the polar coordinate variable to a rectangular coordinate variable
is the bilinear transformation[4], [5], [3], [2].
z
=
-
s
+
1
s
-
1
z
=
-
s
+
1
s
-
1
(1)The details of the bilinear and alternative transformations are
covered elsewhere. For the purposes of this section, it is
sufficient to observe[4], [3] that the frequency response of a
filter in terms of the new variable is found by evaluating H(s)H(s)
along the imaginary axis, i.e., for s=jωs=jω. This is exactly
how the frequency response of analog filters is obtained.
There are two reasons that the approximation process is often
formulated in terms of the square of the magnitude of the transfer
function, rather than the real and/or imaginary parts of the
complex transfer function or the magnitude of the transfer
function. The first reason is that the squared-magnitude frequency-
response function is an analytic, real-valued function of a real
variable, and this considerably simplifies the problem of finding a
“best" solution. The second reason is that effects of the signal or
interference are often stated in terms of the energy or power
that is proportional to the square of the magnitude of the signal
or noise.
In order to move back and forth between the transfer function
F(s)F(s) and the squared-magnitude frequency response
|F(jω)|2|F(jω)|2, an intermediate function is defined. The analytic
complex-valued function of the complex variable s is defined by
F
F
(
s
)
=
F
(
s
)
F
(
-
s
)
F
F
(
s
)
=
F
(
s
)
F
(
-
s
)
(2)which is related to the squared magnitude
by
F
F
(
s
)
|
s
=
j
ω
=
|
F
(
j
ω
)
|
2
F
F
(
s
)
|
s
=
j
ω
=
|
F
(
j
ω
)
|
2
(3)If
F
(
j
ω
)
=
R
(
ω
)
+
j
I
(
ω
)
F
(
j
ω
)
=
R
(
ω
)
+
j
I
(
ω
)
(4)then
|
F
(
j
ω
)
|
2
=
R
(
ω
)
2
+
I
(
ω
)
2
|
F
(
j
ω
)
|
2
=
R
(
ω
)
2
+
I
(
ω
)
2
(5)
=
(
R
(
ω
)
+
j
I
(
ω
)
)
(
R
(
ω
)
-
j
I
(
ω
)
)
=
(
R
(
ω
)
+
j
I
(
ω
)
)
(
R
(
ω
)
-
j
I
(
ω
)
)
(6)
=
F
(
s
)
F
(
-
s
)
|
s
=
j
ω
=
F
(
s
)
F
(
-
s
)
|
s
=
j
ω
(7)In this context, the approximation is arrived at in terms of
F(jω)F(jω), and the result is an analytic function FF(s)FF(s) with a
factor F(s)F(s), which is the desired filter transfer function in
terms of the rectangular variable ss. A comparable function can be
defined in terms of the digital transfer function using the polar
variable zz by defining
H
H
(
z
)
=
H
(
z
)
H
(
1
/
z
)
H
H
(
z
)
=
H
(
z
)
H
(
1
/
z
)
(8)which gives the magnitude-squared frequency response when evaluated
around the unit circle, i.e., z=ejωz=ejω.
The next section develops four useful approximations using the
continuous-time Laplace transform formulation in s. These will be
transformed into digital transfer functions by techniques covered in
another module. They can also be used directly for analog filter
design.
