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% error: [Zz, Zp, Zg] = bilinear(Sz, Sp, Sg, T)


function [Zz, Zp, Zg] = bilinear(Sz, Sp, Sg, T)


% error: [Zz, Zp, Zg] = bilinear(Sz, Sp, Sg, T)
%        [Zb, Za] = bilinear(Sb, Sa, T)
% Transform a s-plane filter specification into a z-plane
% specification. Filters can be specified in either zero-pole-gain or
% transfer function form. The input form does not have to match the
% output form. 1/T is the sampling frequency represented in the z plane.
% Note: this differs from the bilinear function in the signal processing
% toolbox, which uses 1/T rather than T.
% Theory: Given a piecewise flat filter design, you can transform it
% from the s-plane to the z-plane while maintaining the band edges by
% means of the bilinear transform.  This maps the left hand side of the
% s-plane into the interior of the unit circle.  The mapping is highly
% non-linear, so you must design your filter with band edges in the
% s-plane positioned at 2/T tan(w*T/2) so that they will be positioned
% at w after the bilinear transform is complete.
% The following table summarizes the transformation:
% +---------------+-----------------------+----------------------+
% | Transform     | Zero at x             | Pole at x            |
% |    H(S)       |   H(S) = S-x          |    H(S)=1/(S-x)      |
% +---------------+-----------------------+----------------------+
% |       2 z-1   | zero: (2+xT)/(2-xT)   | zero: -1             |
% |  S -> - ---   | pole: -1              | pole: (2+xT)/(2-xT)  |
% |       T z+1   | gain: (2-xT)/T        | gain: (2-xT)/T       |
% +---------------+-----------------------+----------------------+
% With tedious algebra, you can derive the above formulae yourself by
% substituting the transform for S into H(S)=S-x for a zero at x or
% H(S)=1/(S-x) for a pole at x, and converting the result into the
% form:
%    H(Z)=g prod(Z-Xi)/prod(Z-Xj)
% Please note that a pole and a zero at the same place exactly cancel.
% This is significant since the bilinear transform creates numerous
% extra poles and zeros, most of which cancel. Those which do not
% cancel have a 'fill-in' effect, extending the shorter of the sets to
% have the same number of as the longer of the sets of poles and zeros
% (or at least split the difference in the case of the band pass
% filter). There may be other opportunistic cancellations but I will
% not check for them.
% Also note that any pole on the unit circle or beyond will result in
% an unstable filter.  Because of cancellation, this will only happen
% if the number of poles is smaller than the number of zeros.  The
% analytic design methods all yield more poles than zeros, so this will
% not be a problem.
% References:
% Proakis & Manolakis (1992). Digital Signal Processing. New York:
% Macmillan Publishing Company.


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