/usr/share/octave/packages/signal-1.3.0/bilinear.m is in octave-signal 1.3.0-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | ## 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/>.
## -*- 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
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