/usr/share/octave/packages/signal-1.2.2/sftrans.m is in octave-signal 1.2.2-1build1.
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | ## Copyright (C) 1999-2001 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: [Sz, Sp, Sg] = sftrans(Sz, Sp, Sg, W, stop)
##
## Transform band edges of a generic lowpass filter (cutoff at W=1)
## represented in splane zero-pole-gain form. W is the edge of the
## target filter (or edges if band pass or band stop). Stop is true for
## high pass and band stop filters or false for low pass and band pass
## filters. Filter edges are specified in radians, from 0 to pi (the
## nyquist frequency).
##
## Theory: Given a low pass filter represented by poles and zeros in the
## splane, you can convert it to a low pass, high pass, band pass or
## band stop by transforming each of the poles and zeros individually.
## The following table summarizes the transformation:
##
## Transform Zero at x Pole at x
## ---------------- ------------------------- ------------------------
## Low Pass zero: Fc x/C pole: Fc x/C
## S -> C S/Fc gain: C/Fc gain: Fc/C
## ---------------- ------------------------- ------------------------
## High Pass zero: Fc C/x pole: Fc C/x
## S -> C Fc/S pole: 0 zero: 0
## gain: -x gain: -1/x
## ---------------- ------------------------- ------------------------
## Band Pass zero: b ± sqrt(b^2-FhFl) pole: b ± sqrt(b^2-FhFl)
## S^2+FhFl pole: 0 zero: 0
## S -> C -------- gain: C/(Fh-Fl) gain: (Fh-Fl)/C
## S(Fh-Fl) b=x/C (Fh-Fl)/2 b=x/C (Fh-Fl)/2
## ---------------- ------------------------- ------------------------
## Band Stop zero: b ± sqrt(b^2-FhFl) pole: b ± sqrt(b^2-FhFl)
## S(Fh-Fl) pole: ±sqrt(-FhFl) zero: ±sqrt(-FhFl)
## S -> C -------- gain: -x gain: -1/x
## S^2+FhFl b=C/x (Fh-Fl)/2 b=C/x (Fh-Fl)/2
## ---------------- ------------------------- ------------------------
## 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
## ---------------- ------------------------- ------------------------
##
## where C is the cutoff frequency of the initial lowpass filter, Fc is
## the edge of the target low/high pass filter and [Fl,Fh] are the edges
## of the target band pass/stop filter. With abundant 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(S)=g prod(S-Xi)/prod(S-Xj)
##
## The transforms are from the references. The actual pole-zero-gain
## changes I derived myself.
##
## Please note that a pole and a zero at the same place exactly cancel.
## This is significant for High Pass, Band Pass and Band Stop filters
## which create 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 and the
## filter is high pass or band pass. 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 [Sz, Sp, Sg] = sftrans(Sz, Sp, Sg, W, stop)
if (nargin != 5)
print_usage;
end
C = 1;
p = length(Sp);
z = length(Sz);
if z > p || p == 0
error("sftrans: must have at least as many poles as zeros in s-plane");
end
if length(W)==2
Fl = W(1);
Fh = W(2);
if stop
## ---------------- ------------------------- ------------------------
## Band Stop zero: b ± sqrt(b^2-FhFl) pole: b ± sqrt(b^2-FhFl)
## S(Fh-Fl) pole: ±sqrt(-FhFl) zero: ±sqrt(-FhFl)
## S -> C -------- gain: -x gain: -1/x
## S^2+FhFl b=C/x (Fh-Fl)/2 b=C/x (Fh-Fl)/2
## ---------------- ------------------------- ------------------------
if (isempty(Sz))
Sg = Sg * real (1./ prod(-Sp));
elseif (isempty(Sp))
Sg = Sg * real(prod(-Sz));
else
Sg = Sg * real(prod(-Sz)/prod(-Sp));
endif
b = (C*(Fh-Fl)/2)./Sp;
Sp = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
extend = [sqrt(-Fh*Fl), -sqrt(-Fh*Fl)];
if isempty(Sz)
Sz = [extend(1+rem([1:2*p],2))];
else
b = (C*(Fh-Fl)/2)./Sz;
Sz = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
if (p > z)
Sz = [Sz, extend(1+rem([1:2*(p-z)],2))];
end
end
else
## ---------------- ------------------------- ------------------------
## Band Pass zero: b ± sqrt(b^2-FhFl) pole: b ± sqrt(b^2-FhFl)
## S^2+FhFl pole: 0 zero: 0
## S -> C -------- gain: C/(Fh-Fl) gain: (Fh-Fl)/C
## S(Fh-Fl) b=x/C (Fh-Fl)/2 b=x/C (Fh-Fl)/2
## ---------------- ------------------------- ------------------------
Sg = Sg * (C/(Fh-Fl))^(z-p);
b = Sp*((Fh-Fl)/(2*C));
Sp = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
if isempty(Sz)
Sz = zeros(1,p);
else
b = Sz*((Fh-Fl)/(2*C));
Sz = [b+sqrt(b.^2-Fh*Fl), b-sqrt(b.^2-Fh*Fl)];
if (p>z)
Sz = [Sz, zeros(1, (p-z))];
end
end
end
else
Fc = W;
if stop
## ---------------- ------------------------- ------------------------
## High Pass zero: Fc C/x pole: Fc C/x
## S -> C Fc/S pole: 0 zero: 0
## gain: -x gain: -1/x
## ---------------- ------------------------- ------------------------
if (isempty(Sz))
Sg = Sg * real (1./ prod(-Sp));
elseif (isempty(Sp))
Sg = Sg * real(prod(-Sz));
else
Sg = Sg * real(prod(-Sz)/prod(-Sp));
endif
Sp = C * Fc ./ Sp;
if isempty(Sz)
Sz = zeros(1,p);
else
Sz = [C * Fc ./ Sz];
if (p > z)
Sz = [Sz, zeros(1,p-z)];
end
end
else
## ---------------- ------------------------- ------------------------
## Low Pass zero: Fc x/C pole: Fc x/C
## S -> C S/Fc gain: C/Fc gain: Fc/C
## ---------------- ------------------------- ------------------------
Sg = Sg * (C/Fc)^(z-p);
Sp = Fc * Sp / C;
Sz = Fc * Sz / C;
end
end
endfunction
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