Digital Signal Processing Reference
In-Depth Information
is not applicable for the design of highpass, bandstop, and allpass filters since
their frequency responses are not bandlimited at all. If the impulse-invariant
transformation is applied to a minimum phase analog filter
H(s)
, the resulting
digital filter may or may not be a minimum phase filter. For these reasons, the
impulse-invariant transformation is not used very often in practical applications.
4.6 BILINEAR TRANSFORMATION
The bilinear transformation is the one that is the most often used for designing
IIR filters. It is defined as
z
2
T
−
1
s
=
(4.89)
z
+
1
To find how frequencies on the unit circle in the
z
plane map to those in the
s
plane, let us substitute
z
e
jωT
in (4.89). Note that
ω
is the angular frequency
in radians per second and
ωT
is the normalized frequency in the
z
plane. Instead
of using
ω
as the notation for the normalized frequency of the digital filter, we
may denote
θ
as the normalized frequency to avoid any confusion in this section:
=
e
jωT
e
j(ωT/
2
)
e
−
j(ωT/
2
)
tan
ωT
2
e
−
j(ωT/
2
)
2
T
−
1
2
T
−
j
2
T
s
=
=
=
e
jωT
+
e
j(ωT/
2
)
+
1
j
2
f
s
tan
ωT
2
=
=
jλ
This transformation maps the poles inside the unit circle in the
z
plane to the
inside of the left half of the
s
plane and vice versa. It also maps the frequencies
on the unit circle in the
z
plane to frequencies on the entire imaginary axis of
the
s
plane, where
s
jλ
. So this transformation satisfies both conditions
that we required for the mapping
s
=
σ
+
=
f(z)
mentioned in the previous section or
its inverse relationship
z
b(s)
. This mapping is shown in Figure 4.17 and may
be compared with the mapping shown in Figure 4.16.
To understand the mapping in some more detail, let us consider the frequency
response of an IIR filter over the interval
(
0
,(ω
s
/
2
))
,where
ω
s
/
2
=
=
π/T
is the
Nyquist frequency. As an example, we choose a frequency response
H(e
ωT
)
=
H(e
jθ
)
of a Butterworth bandpass digital filter as shown in Figure 4.18a.
In Figure 4.18, we have also shown the curve depicting the relationship between
ωT
and
λ
=
2
f
s
tan
(ωT /
2
)
. The value of
λ
corresponding to any value of
ωT
=
θ
can be calculated from
λ
=
2
f
s
tan
(θ/
2
)
as illustrated by mapping a few frequen-
cies such as
ω
1
T,ω
2
T
in Figure 4.18. The magnitude of the frequency response
of the digital filter at any normalized frequency
ω
k
T
is the magnitude of
H(s)
at
the corresponding frequency
s
=
2
f
s
tan
(ω
k
T/
2
)
.
The plot in Figure 4.17 shows that the magnitude response of the digital filter
over the Nyquist interval
(
0
,π)
maps over the entire range
(
0
,
=
jλ
k
,where
λ
k
)
of
λ
.Sothere
is a nonlinear mapping whereby the frequencies in the
ω
domain are warped
∞
Search WWH ::
Custom Search