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firls.m

## Copyright (C) 2006 Quentin Spencer
##
## 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 2 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/>.

## b = firls(N, F, A);
## b = firls(N, F, A, W);
##
##  FIR filter design using least squares method. Returns a length N+1
##  linear phase filter such that the integral of the weighted mean
##  squared error in the specified bands is minimized.
##
##  F specifies the frequencies of the band edges, normalized so that
##  half the sample frequency is equal to 1.  Each band is specified by
##  two frequencies, to the vector must have an even length.
##
##  A specifies the amplitude of the desired response at each band edge.
##
##  W is an optional weighting function that contains one value for each
##  band that weights the mean squared error in that band. A must be the
##  same length as F, and W must be half the length of F.

## The least squares optimization algorithm for computing FIR filter
## coefficients is derived in detail in:
##
## I. Selesnick, "Linear-Phase FIR Filter Design by Least Squares,"
## http://cnx.org/content/m10577

function coef = firls(N, frequencies, pass, weight, str);


  if nargin<3 | nargin>6
    usage("");
  end
  if nargin==3
    weight = ones(1, length(pass)/2);
    str = ;
  end
  if nargin==4
    if ischar(weight)
      str = weight;
      weight = ones (size (pass));
    else
      str = [];
    end
  end
  if length (frequencies) ~= length (pass)
    error("F and A must have equal lengths.");
  end
  if 2 * length (weight) ~= length (pass)
    error("W must contain one weight per band.");
  end

  if ischar(str)
    error("This feature is implemented yet");
  else

    M = N/2;
    w = kron(weight(:), [-1; 1]);
    omega = frequencies * pi;
    i1 = 1:2:length(omega);
    i2 = 2:2:length(omega);

    ## Generate the matrix Q
    ## As illustrated in the above-cited reference, the matrix can be
    ## expressed as the sum of a Hankel and Toeplitz matrix. A factor of
    ## 1/2 has been dropped and the final filter coefficients multiplied
    ## by 2 to compensate.
    cos_ints = ;
    q = ' .* (cos_ints * w);
    Q = toeplitz (q(1:M+1)) + hankel (q(1:M+1), q(M+1:end));

    ## The vector b is derived from solving the integral:
    ##
    ##           _ w
    ##          /   2
    ##  b  =   /       W(w) D(w) cos(kw) dw
    ##   k    /    w
    ##       -      1
    ##
    ## Since we assume that W(w) is constant over each band (if not, the
    ## computation of Q above would be considerably more complex), but
    ## D(w) is allowed to be a linear function, in general the function
    ## W(w) D(w) is linear. The computations below are derived from the
    ## fact that:
    ##     _
    ##    /                          a              ax + b
    ##   /   (ax + b) cos(nx) dx =  --- cos (nx) +  ------ sin(nx)
    ##  /                             2                n
    ## -                             n
    ##
    cos_ints2 = [omega(i1).^2 - omega(i2).^2; ...
             cos((1:M)' * omega(i2)) - cos((1:M)' * omega(i1))] ./ ...
                 ([2, 1:M]' * (omega(i2) - omega(i1)));
    d =  (:);
    b = ' .* ((kron (cos_ints2, [1, 1]) + cos_ints(1:M+1,:)) * d);

    ## Having computed the components Q and b of the  matrix equation,
    ## solve for the filter coefficients.
    a = Q \ b;
    coef = [ a(end:-1:2); 2 * a(1); a(2:end) ];

  end

endfunction

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