Lumped Models

Digitization of Lumped Models

Bilinear Transformation

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The bilinear transform is defined by the substitution

where is some positive constant [83,326]. That is, given a continuous-time transfer function , we apply the bilinear transform by defining

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 [449]).^{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 bilinear transform is often used to design digital filters from analog prototype filters [343]. An on-line introduction is given in [449].

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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