This file is indexed.

/usr/share/octave/packages/signal-1.3.2/invfreq.m is in octave-signal 1.3.2-5.

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
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
## Copyright (C) 1986, 2000, 2003 Julius O. Smith III <jos@ccrma.stanford.edu>
## Copyright (C) 2007 Rolf Schirmacher <Rolf.Schirmacher@MuellerBBM.de>
## Copyright (C) 2003 Andrew Fitting
## Copyright (C) 2010 Pascal Dupuis <Pascal.Dupuis@uclouvain.be>
##
## 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: [B,A] = invfreq(H,F,nB,nA)
##        [B,A] = invfreq(H,F,nB,nA,W)
##        [B,A] = invfreq(H,F,nB,nA,W,[],[],plane)
##        [B,A] = invfreq(H,F,nB,nA,W,iter,tol,plane)
##
## Fit filter B(z)/A(z) or B(s)/A(s) to complex frequency response at
## frequency points F. A and B are real polynomial coefficients of order
## nA and nB respectively.  Optionally, the fit-errors can be weighted vs
## frequency according to the weights W. Also, the transform plane can be
## specified as either 's' for continuous time or 'z' for discrete time. 'z'
## is chosen by default.  Eventually, Steiglitz-McBride iterations will be
## specified by iter and tol.
##
## H: desired complex frequency response
##     It is assumed that A and B are real polynomials, hence H is one-sided.
## F: vector of frequency samples in radians
## nA: order of denominator polynomial A
## nB: order of numerator polynomial B
## plane='z': F on unit circle (discrete-time spectra, z-plane design)
## plane='s': F on jw axis     (continuous-time spectra, s-plane design)
## H(k) = spectral samples of filter frequency response at points zk,
##  where zk=exp(sqrt(-1)*F(k)) when plane='z' (F(k) in [0,.5])
##     and zk=(sqrt(-1)*F(k)) when plane='s' (F(k) nonnegative)
## Example:
##     [B,A] = butter(12,1/4);
##     [H,w] = freqz(B,A,128);
##     [Bh,Ah] = invfreq(H,F,4,4);
##     Hh = freqz(Bh,Ah);
##     disp(sprintf('||frequency response error|| = %f',norm(H-Hh)));
##
## References: J. O. Smith, "Techniques for Digital Filter Design and System
##      Identification with Application to the Violin, Ph.D. Dissertation,
##      Elec. Eng. Dept., Stanford University, June 1983, page 50; or,
##
## http://ccrma.stanford.edu/~jos/filters/FFT_Based_Equation_Error_Method.html

## FIXME: implement Steiglitz-McBride iterations
## FIXME: improve numerical stability for high order filters (matlab is a bit better)
## FIXME: modify to accept more argument configurations

function [B, A, SigN] = invfreq(H, F, nB, nA, W, iter, tol, tr, plane, varargin)

  if length(nB) > 1, zB = nB(2); nB = nB(1); else zB = 0; endif
  n = max(nA, nB);
  m = n+1; mA = nA+1; mB = nB+1;
  nF = length(F);
  if nF ~= length(H), disp('invfreqz: length of H and F must be the same'); endif
  if nargin < 5 || isempty(W), W = ones(1, nF); endif
  if nargin < 6, iter = []; endif
  if nargin < 7  tol = []; endif
  if nargin < 8 || isempty(tr), tr = ''; endif
  if nargin < 9, plane = 'z'; endif
  if nargin < 10, varargin = {}; endif
  if iter~=[], disp('no implementation for iter yet'),endif
  if tol ~=[], disp('no implementation for tol yet'),endif
  if (plane ~= 'z' && plane ~= 's'), disp('invfreqz: Error in plane argument'), endif

  [reg, prop ] = parseparams(varargin);
  ## should we normalize freqs to avoid matrices with rank deficiency ?
  norm = false;
  ## by default, use Ordinary Least Square to solve normal equations
  method = 'LS';
  if length(prop) > 0
    indi = 1; while indi <= length(prop)
      switch prop{indi}
        case 'norm'
          if indi < length(prop) && ~ischar(prop{indi+1}),
            norm = logical(prop{indi+1});
            prop(indi:indi+1) = [];
            continue
          else
            norm = true; prop(indi) = [];
            continue
          endif
        case 'method'
          if indi < length(prop) && ischar(prop{indi+1}),
            method = prop{indi+1};
            prop(indi:indi+1) = [];
            continue
          else
            error('invfreq.m: incorrect/missing method argument');
          endif
        otherwise # FIXME: just skip it for now
          disp(sprintf("Ignoring unknown argument %s", varargin{indi}));
          indi = indi + 1;
      endswitch
    endwhile
  endif

  Ruu = zeros(mB, mB); Ryy = zeros(nA, nA); Ryu = zeros(nA, mB);
  Pu = zeros(mB, 1);   Py = zeros(nA,1);
  if strcmp(tr,'trace')
    disp(' ')
    disp('Computing nonuniformly sampled, equation-error, rational filter.');
    disp(['plane = ',plane]);
    disp(' ')
  endif

  s = sqrt(-1)*F;
  switch plane
    case 'z'
      if max(F) > pi || min(F) < 0
        disp('hey, you frequency is outside the range 0 to pi, making my own')
        F = linspace(0, pi, length(H));
        s = sqrt(-1)*F;
      endif
      s = exp(-s);
    case 's'
      if max(F) > 1e6 && n > 5,
        if ~norm,
          disp('Be careful, there are risks of generating singular matrices');
          disp('Call invfreqs as (..., "norm", true) to avoid it');
        else
          Fmax = max(F); s = sqrt(-1)*F/Fmax;
        endif
      endif
  endswitch

  for k=1:nF,
    Zk = (s(k).^[0:n]).';
    Hk = H(k);
    aHks = Hk*conj(Hk);
    Rk = (W(k)*Zk)*Zk';
    rRk = real(Rk);
    Ruu = Ruu + rRk(1:mB, 1:mB);
    Ryy = Ryy + aHks*rRk(2:mA, 2:mA);
    Ryu = Ryu + real(Hk*Rk(2:mA, 1:mB));
    Pu = Pu + W(k)*real(conj(Hk)*Zk(1:mB));
    Py = Py + (W(k)*aHks)*real(Zk(2:mA));
  endfor
  Rr = ones(length(s), mB+nA); Zk = s;
  for k = 1:min(nA, nB),
    Rr(:, 1+k) = Zk;
    Rr(:, mB+k) = -Zk.*H;
    Zk = Zk.*s;
  endfor
  for k = 1+min(nA, nB):max(nA, nB)-1,
    if k <= nB, Rr(:, 1+k) = Zk; endif
    if k <= nA, Rr(:, mB+k) = -Zk.*H; endif
    Zk = Zk.*s;
  endfor
  k = k+1;
  if k <= nB, Rr(:, 1+k) = Zk; endif
  if k <= nA, Rr(:, mB+k) = -Zk.*H; endif

  ## complex to real equation system -- this ensures real solution
  Rr = Rr(:, 1+zB:end);
  Rr = [real(Rr); imag(Rr)]; Pr = [real(H(:)); imag(H(:))];
  ## normal equations -- keep for ref
  ## Rn= [Ruu(1+zB:mB, 1+zB:mB), -Ryu(:, 1+zB:mB)';  -Ryu(:, 1+zB:mB), Ryy];
  ## Pn= [Pu(1+zB:mB); -Py];

  switch method
    case {'ls' 'LS'}
      ## avoid scaling errors with Theta = R\P;
      ## [Q, R] = qr([Rn Pn]); Theta = R(1:end, 1:end-1)\R(1:end, end);
      [Q, R] = qr([Rr Pr], 0); Theta = R(1:end-1, 1:end-1)\R(1:end-1, end);
      ## SigN = R(end, end-1);
      SigN = R(end, end);
    case {'tls' 'TLS'}
      ## [U, S, V] = svd([Rn Pn]);
      ## SigN = S(end, end-1);
      ## Theta =  -V(1:end-1, end)/V(end, end);
      [U, S, V] = svd([Rr Pr], 0);
      SigN = S(end, end);
      Theta =  -V(1:end-1, end)/V(end, end);
    case {'mls' 'MLS' 'qr' 'QR'}
      ## [Q, R] = qr([Rn Pn], 0);
      ## solve the noised part -- DO NOT USE ECONOMY SIZE !
      ## [U, S, V] = svd(R(nA+1:end, nA+1:end));
      ## SigN = S(end, end-1);
      ## Theta = -V(1:end-1, end)/V(end, end);
      ## unnoised part -- remove B contribution and back-substitute
      ## Theta = [R(1:nA, 1:nA)\(R(1:nA, end) - R(1:nA, nA+1:end-1)*Theta)
      ##         Theta];
      ## solve the noised part -- economy size OK as #rows > #columns
      [Q, R] = qr([Rr Pr], 0);
      eB = mB-zB; sA = eB+1;
      [U, S, V] = svd(R(sA:end, sA:end));
      ## noised (A) coefficients
      Theta = -V(1:end-1, end)/V(end, end);
      ## unnoised (B) part -- remove A contribution and back-substitute
      Theta = [R(1:eB, 1:eB)\(R(1:eB, end) - R(1:eB, sA:end-1)*Theta)
               Theta];
      SigN = S(end, end);
    otherwise
      error("invfreq: unknown method %s", method);
  endswitch

  B = [zeros(zB, 1); Theta(1:mB-zB)].';
  A = [1; Theta(mB-zB+(1:nA))].';

  if strcmp(plane,'s')
    B = B(mB:-1:1);
    A = A(mA:-1:1);
    if norm, # Frequencies were normalized -- unscale coefficients
      Zk = Fmax.^[n:-1:0].';
      for k = nB:-1:1+zB, B(k) = B(k)/Zk(k); endfor
      for k = nA:-1:1, A(k) = A(k)/Zk(k); endfor
    endif
  endif

endfunction

%!demo
%! order = 6;  # order of test filter
%! fc = 1/2;   # sampling rate / 4
%! n = 128;    # frequency grid size
%! [B, A] = butter(order,fc);
%! [H, w] = freqz(B,A,n);
%! [Bh, Ah] = invfreq(H,w,order,order);
%! [Hh, wh] = freqz(Bh,Ah,n);
%! plot(w,[abs(H), abs(Hh)])
%! xlabel("Frequency (rad/sample)");
%! ylabel("Magnitude");
%! legend('Original','Measured');
%! err = norm(H-Hh);
%! disp(sprintf('L2 norm of frequency response error = %f',err));