Digital Filters
51
conformal transformation
presence filter
transition band
2.4
Frequency warping
Section (1.5.2) has shown how the bilinear transformation distorts the frequency
axis while maintaining the "shape" of the frequency response. Such transforma-
tion is a so-called conformal transformation  of the complex plane onto itself.
In this section we are interested in conformal transformations that map the unit
circumference (instead of the imaginary axis) onto itself, in such a way that, if
applied to a discrete-time filter, they give a new discrete-time filter having the
same stability properties.
Indeed, the simplest non-trivial transformation of this kind is a bilinear
transformation
z
-1
=
a +
-1
1 + a
-1
.
(53)
The transformation (53) is allpass and, therefore, it maps the unit circumference
onto itself. Moreover, if the transformation (53) is applied to a discrete-time filter
described by a transfer function in z, it preserves the filter order in the variable
.
The reason for using conformal maps in digital filter design is that it might
be easier to design a filter using a warped frequency axis. For instance, to design
a presence filter it is convenient to start from a second-order resonant filter pro-
totype having center frequency at /2 and tunable bandwidth and boost. Then,
it is possible to compute the coefficient of the conformal transformation (53)
in such a way that the resonant peak gets moved to the desired position .
Conformal transformations of order higher than the first are often used to de-
sign multiband filters starting from the design of a lowpass filter, or to satisfy
demanding specifications on the slope of the transition band that connects the
pass band from the attenuated band.
When designing digital filters to be used in models of acoustic systems,
the transformation (53) can be useful, especially if it is specialized in order to
optimize some psychoacoustic-based quality measure. Namely, the warping of
the frequency axis can be tuned in such a way that it resembles the frequency
distribution of critical bands in the basilar membrane of the ear . Similarly
to what we saw in section 1.5.2 for the bilinear transformation, it can be shown
that a first-order conformal map is determined by setting the correspondence
in three points, two of them being = 0 and = . The mapping of the third
point is determined by the coefficient a to be used in (53). Surprisingly enough, a
simple first-order transformation is capable to follow the distribution of critical
bands quite accurately. Smith and Abel , using a technique that minimizes
the squared equation error, have estimated the value that has to be assigned
to a for sampling frequencies ranging from 1Hz to 50KHz, in order to have a
ear-based frequency distribution. An approximate expression to calculate such
coefficient is
a(F
s
)
1.0211
2
arctan (76 · 10
-6
F
s
)
1/2
- 0.19877 .
(54)
As an exercise, the reader can set a value of the sampling rate F
s
, and compute
the value of a by means of (54). Then the curve that maps the frequencies in
the plane to the frequencies in the z plane can be drawn and compared to
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