## 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; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn  {Function File} {[@var{Zb}, @var{Za}] =} bilinear (@var{Sb}, @var{Sa}, @var{T})
## @deftypefnx {Function File} {[@var{Zb}, @var{Za}] =} bilinear (@var{Sz}, @var{Sp}, @var{Sg}, @var{T})
## @deftypefnx {Function File} {[@var{Zz}, @var{Zp}, @var{Zg}] =} bilinear (@dots{})
## 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:
##
## @example
## +---------------+-----------------------+----------------------+
## | 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       |
## +---------------+-----------------------+----------------------+
## @end example
##
## 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:
##
## @example
##    H(Z)=g prod(Z-Xi)/prod(Z-Xj)
## @end example
##
## 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.
## @end deftypefn

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;
  endif

  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");
  endif

## ----------------  -------------------------  ------------------------
## 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);
  endif

  if nargout==2, [Zz, Zp] = zp2tf(Zz, Zp, Zg); endif

endfunction
