Bilinear Transformation

The bilinear transform is defined by the substitution

where is some positive constant [83,329]. That is, given a continuous-time transfer function , we apply the bilinear transform by defining

where the `` '' subscript denotes ``digital,'' and `` '' denotes ``analog.''

It can be seen that analog dc (
) maps to digital dc (
) and
the highest analog frequency (
) maps to the highest digital
frequency (
). It is easy to show that the entire
axis
in the
plane (where
) is mapped exactly
*once* around the unit circle in the
plane (rather than
summing around it infinitely many times, or ``aliasing'' as it does in
ordinary sampling). With
real and positive, the left-half
plane maps to the interior of the unit circle, and the right-half
plane maps outside the unit circle. This means *stability is
preserved* when mapping a continuous-time transfer function to
discrete time.

Another valuable property of the bilinear transform is that
*order is preserved*. That is, an
th-order
-plane transfer
function carries over to an
th-order
-plane transfer function.
(*Order* in both cases equals the maximum of the degrees of the
numerator and denominator polynomials [452]).^{8.6}

The constant
provides one remaining degree of freedom which can be used
to map any particular finite frequency from the
axis in the
plane to a particular desired location on the unit circle
in the
plane. All other frequencies will be *warped.* In
particular, approaching half the sampling rate, the frequency axis
compresses more and more. Note that at most one resonant frequency can be
preserved under the bilinear transformation of a mass-spring-dashpot
system. On the other hand, filters having a single transition frequency,
such as lowpass or highpass filters, map beautifully under the bilinear
transform; one simply uses
to map the cut-off frequency where it
belongs, and the response looks great. In particular, ``equal ripple'' is
preserved for optimal filters of the elliptic and Chebyshev types because
the values taken on by the frequency response are identical in both cases;
only the frequency axis is warped.

The *frequency-warping* of the bilinear transform is readily found by
looking at the frequency-axis mapping in Eq.
(7.7), *i.e.*, by setting
and
in the bilinear-transform
definition:

Thus, we may interpret as a

The bilinear transform is often used to design digital filters from analog prototype filters [346]. An on-line introduction is given in [452].

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