Add a useful package (from Source forge) for octave

[CreaPhase.git] / octave_packages / signal-1.1.3 / bilinear.m

diff --git a/octave_packages/signal-1.1.3/bilinear.m b/octave_packages/signal-1.1.3/bilinear.m
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+## Copyright (C) 1999 Paul Kienzle <pkienzle@users.sf.net>
+##
+## This program is free software; you can redistribute it and/or modify it under
+## the terms of the GNU General Public License as published by the Free Software
+## Foundation; either version 3 of the License, or (at your option) any later
+## version.
+##
+## This program is distributed in the hope that it will be useful, but WITHOUT
+## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
+## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
+## details.
+##
+## You should have received a copy of the GNU General Public License along with
+## this program; if not, see <http://www.gnu.org/licenses/>.
+
+## usage: [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.
+
+function [Zz, Zp, Zg] = bilinear(Sz, Sp, Sg, T)
+
+  if nargin==3
+    T = Sg;
+    [Sz, Sp, Sg] = tf2zp(Sz, Sp);
+  elseif nargin!=4
+    print_usage;
+  end
+
+  p = length(Sp);
+  z = length(Sz);
+  if z > p || p==0
+    error("bilinear: must have at least as many poles as zeros in s-plane");
+  end
+
+## ----------------  -------------------------  ------------------------
+## Bilinear          zero: (2+xT)/(2-xT)        pole: (2+xT)/(2-xT)
+##      2 z-1        pole: -1                   zero: -1
+## S -> - ---        gain: (2-xT)/T             gain: (2-xT)/T
+##      T z+1
+## ----------------  -------------------------  ------------------------
+  Zg = real(Sg * prod((2-Sz*T)/T) / prod((2-Sp*T)/T));
+  Zp = (2+Sp*T)./(2-Sp*T);
+  if isempty(Sz)
+    Zz = -ones(size(Zp));
+  else
+    Zz = [(2+Sz*T)./(2-Sz*T)];
+    Zz = postpad(Zz, p, -1);
+  end
+
+  if nargout==2, [Zz, Zp] = zp2tf(Zz, Zp, Zg); endif
+endfunction