## 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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