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annotate main/signal/inst/dct.m @ 2795:7a614782378d octave-forge

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author pkienzle
date Tue, 05 Dec 2006 20:45:13 +0000
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1 ## Copyright (C) 2001 Paul Kienzle
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2 ##
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3 ## This program is free software; you can redistribute it and/or modify
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4 ## it under the terms of the GNU General Public License as published by
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5 ## the Free Software Foundation; either version 2 of the License, or
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6 ## (at your option) any later version.
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7 ##
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8 ## This program is distributed in the hope that it will be useful,
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9 ## but WITHOUT ANY WARRANTY; without even the implied warranty of
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10 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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11 ## GNU General Public License for more details.
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12 ##
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13 ## You should have received a copy of the GNU General Public License
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14 ## along with this program; if not, write to the Free Software
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15 ## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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16
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17 ## y = dct (x, n)
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18 ## Computes the discrete cosine transform of x. If n is given, then
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19 ## x is padded or trimmed to length n before computing the transform.
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20 ## If x is a matrix, compute the transform along the columns of the
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21 ## the matrix. The transform is faster if x is real-valued and even
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22 ## length.
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23 ##
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24 ## The discrete cosine transform X of x can be defined as follows:
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25 ##
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26 ## N-1
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27 ## X[k] = w(k) sum x[n] cos (pi (2n-1) k / 2N ), k = 0, ..., N-1
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28 ## n=0
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29 ##
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30 ## with w(0) = sqrt(1/N) and w(k) = sqrt(2/N), k = 1, ..., N-1. There
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31 ## are other definitions with different scaling of X[k], but this form
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32 ## is common in image processing.
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33 ##
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34 ## See also: idct, dct2, idct2, dctmtx
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35
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36 ## From Discrete Cosine Transform notes by Brian Evans at UT Austin,
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37 ## http://www.ece.utexas.edu/~bevans/courses/ee381k/lectures/09_DCT/lecture9/
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38 ## the discrete cosine transform of x at k is as follows:
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39 ##
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40 ## N-1
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41 ## X[k] = sum 2 x[n] cos (pi (2n-1) k / 2N )
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42 ## n=0
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43 ##
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44 ## which can be computed using:
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45 ##
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46 ## y = [ x ; flipud (x) ]
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47 ## Y = fft(y)
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48 ## X = exp( -j pi [0:N-1] / 2N ) .* Y
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49 ##
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50 ## or for real, even length x
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51 ##
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52 ## y = [ even(x) ; flipud(odd(x)) ]
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53 ## Y = fft(y)
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54 ## X = 2 real { exp( -j pi [0:N-1] / 2N ) .* Y }
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55 ##
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56 ## Scaling the result by w(k)/2 will give us the desired output.
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57
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58 ## Author: Paul Kienzle
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59 ## 2001-02-08
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60 ## * initial release
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61 function y = dct (x, n)
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62
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63 if (nargin < 1 || nargin > 2)
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64 usage ("y = dct(x [, n])");
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65 endif
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66
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67 realx = isreal(x);
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68 transpose = (rows (x) == 1);
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69
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70 if transpose, x = x (:); endif
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71 [nr, nc] = size (x);
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72 if nargin == 1
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73 n = nr;
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74 elseif n > nr
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75 x = [ x ; zeros(n-nr,nc) ];
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76 elseif n < nr
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77 x (nr-n+1 : n, :) = [];
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78 endif
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79
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80 if n == 1
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81 w = 1/2;
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82 else
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83 w = [ sqrt(1/4/n); sqrt(1/2/n)*exp((-1i*pi/2/n)*[1:n-1]') ] * ones (1, nc);
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84 endif
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85 if ( realx && rem (n, 2) == 0 )
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86 y = fft ([ x(1:2:n,:) ; x(n:-2:1,:) ]);
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87 y = 2 * real( w .* y );
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88 else
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89 y = fft ([ x ; flipud(x) ]);
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90 y = w .* y (1:n, :);
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91 if (realx) y = real (y); endif
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92 endif
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93 if transpose, y = y.'; endif
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94
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95 endfunction