## Copyright (C) 2001 Paul Kienzle ## ## 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/>. ## y = dct (x, n) ## Computes the discrete cosine transform of x. If n is given, then ## x is padded or trimmed to length n before computing the transform. ## If x is a matrix, compute the transform along the columns of the ## the matrix. The transform is faster if x is real-valued and even ## length. ## ## The discrete cosine transform X of x can be defined as follows: ## ## N-1 ## X[k] = w(k) sum x[n] cos (pi (2n-1) k / 2N ), k = 0, ..., N-1 ## n=0 ## ## with w(0) = sqrt(1/N) and w(k) = sqrt(2/N), k = 1, ..., N-1. There ## are other definitions with different scaling of X[k], but this form ## is common in image processing. ## ## See also: idct, dct2, idct2, dctmtx ## From Discrete Cosine Transform notes by Brian Evans at UT Austin, ## http://www.ece.utexas.edu/~bevans/courses/ee381k/lectures/09_DCT/lecture9/ ## the discrete cosine transform of x at k is as follows: ## ## N-1 ## X[k] = sum 2 x[n] cos (pi (2n-1) k / 2N ) ## n=0 ## ## which can be computed using: ## ## y = [ x ; flipud (x) ] ## Y = fft(y) ## X = exp( -j pi [0:N-1] / 2N ) .* Y ## ## or for real, even length x ## ## y = [ even(x) ; flipud(odd(x)) ] ## Y = fft(y) ## X = 2 real { exp( -j pi [0:N-1] / 2N ) .* Y } ## ## Scaling the result by w(k)/2 will give us the desired output. ## Author: Paul Kienzle ## 2001-02-08 ## * initial release function y = dct (x, n) if (nargin < 1 || nargin > 2) usage ("y = dct(x [, n])"); endif realx = isreal(x); transpose = (rows (x) == 1); if transpose, x = x (:); endif [nr, nc] = size (x); if nargin == 1 n = nr; elseif n > nr x = [ x ; zeros(n-nr,nc) ]; elseif n < nr x (nr-n+1 : n, :) = []; endif if n == 1 w = 1/2; else w = [ sqrt(1/4/n); sqrt(1/2/n)*exp((-1i*pi/2/n)*[1:n-1]') ] * ones (1, nc); endif if ( realx && rem (n, 2) == 0 ) y = fft ([ x(1:2:n,:) ; x(n:-2:1,:) ]); y = 2 * real( w .* y ); else y = fft ([ x ; flipud(x) ]); y = w .* y (1:n, :); if (realx) y = real (y); endif endif if transpose, y = y.'; endif endfunction

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