--- /dev/null
+## Copyright (C) 2006 Quentin Spencer <qspencer@ieee.org>
+##
+## 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/>.
+
+## 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
+ print_usage;
+ elseif nargin==3
+ weight = ones(1, length(pass)/2);
+ str = [];
+ elseif 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.");
+ elseif 2 * length (weight) ~= length (pass)
+ error("W must contain one weight per band.");
+ elseif 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 = [omega; sin((1:N)' * omega)];
+ q = [1, 1./(1:N)]' .* (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 = [-weight .* pass(i1); weight .* pass(i2)] (:);
+ b = [1, 1./(1:M)]' .* ((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
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff