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1 ## Copyright (C) 1999-2001 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 ## usage: [Pxx, w] = pwelch(x,n,Fs,window,overlap,ci,range,units,trend) |
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18 ## [Pxx, Pci, w] = pwelch(x,n,Fs,window,overlap,ci,range,units,trend) |
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19 ## |
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20 ## Estimate power spectrum of a stationary signal. This chops the signal |
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21 ## into overlapping slices, windows each slice and applies a Fourier |
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22 ## transform to determine the frequency components at that slice. The |
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23 ## magnitudes of these slices are then averaged to produce the estimate Pxx. |
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24 ## The confidence interval around the estimate is returned in Pci. |
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25 ## |
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26 ## x: vector of samples |
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27 ## n: size of fourier transform window, or [] for default=256 |
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28 ## Fs: sample rate, or [] for default=2 Hz |
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29 ## window: shape of the fourier transform window, or [] for default=hanning(n) |
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30 ## Note: window length can be specified instead, in which case |
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31 ## window=hanning(length) |
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32 ## overlap: overlap with previous window, or [] for default=length(window)/2 |
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33 ## ci: confidence interval, or [] for default=0.95 |
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34 ## ci must be between 0 and 1; if ci is not passed, or if it is |
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35 ## passed as 0, then no confidence intervals will be computed. |
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36 ## range: 'whole', or [] for default='half' |
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37 ## show all frequencies, or just half of the frequencies |
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38 ## units: 'squared', or [] for default='db' |
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39 ## show results as magnitude squared or as log magnitude squared |
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40 ## trend: 'mean', 'linear', or [] for default='none' |
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41 ## remove trends from the data slices before computing spectral estimates |
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42 ## |
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43 ## Example |
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44 ## [b,a] = cheby1(4,3,[0.2, 0.4]); ## define noise colour |
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45 ## pwelch(filter(b,a,randn(2^12,1))); ## estimate noise colour |
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46 |
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47 ## 2001-04-02 Paul Kienzle |
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48 ## * return nfft/2+1 elements rather than nfft/2 for even nfft. |
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49 ## * use more accurate (and faster) computation of confidence intervals |
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50 |
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51 ## TODO: Should be extended to accept a vector of frequencies at which to |
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52 ## TODO: evaluate the fourier transform (via filterbank or chirp |
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53 ## TODO: z-transform). |
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54 ## TODO: What should happen with the final window when it isn't full? |
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55 ## TODO: currently I dump it, but I should probably zero pad and add |
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56 ## TODO: it in. |
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57 ## TODO: Consider returning the whole gamit of Pxx, Pyy, Pxy, Cxy, Txy |
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58 ## TODO: as well as confidence intervals for each; if users tend |
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59 ## TODO: only to use one of these don't bother, but if they ever need |
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60 ## TODO: more than one, then it's free. Alternatively, break out the |
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61 ## TODO: compute engine into a function that the user can call directly. |
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62 ## TODO: Check if Cxy, Txy are computed frame-by-frame or on the average |
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63 ## TODO: of the frames. SpcTools and I do it on the average, |
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64 ## TODO: wdkirby@ix.netcom.com (1998-04-29 octave-sources) computes |
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65 ## TODO: them frame-by-frame. |
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66 function [...] = pwelch(x, ...) |
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67 ## sort out parameters |
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68 if nargin < 1, |
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69 usage("[Pxx, w] = pwelch(x,nfft,Fs,window,overlap,pc,range,units,trend)"); |
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70 endif |
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71 va_start(); |
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72 |
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73 ## Determine if we are called as pwelch, csd, cohere or tfe |
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74 if isstr(x) |
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75 calledby = x; |
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76 else |
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77 calledby = "pwelch"; |
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78 endif |
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79 if !isstr(x) |
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80 ftype = 1; |
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81 elseif strcmp(x, 'csd') |
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82 ftype = 2; |
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83 elseif strcmp(x, 'cohere') |
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84 ftype = 3; |
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85 elseif strcmp(x, 'tfe') |
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86 ftype = 4; |
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87 endif |
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88 |
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89 ## Sort out x and y vectors |
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90 if ftype!=1 |
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91 x=va_arg(); y=va_arg(); |
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92 first = 4; |
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93 else |
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94 y=[]; |
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95 first = 2; |
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96 endif |
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97 if (columns(x) != 1 && rows(x) != 1) || ... |
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98 (!isempty(y) && columns(y) != 1 && rows(y) != 1) |
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99 error ([calledby, " data must be a vector"]); |
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100 end |
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101 if columns(x) != 1, x = x'; end |
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102 if columns(y) != 1, y = y'; end |
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103 if !isempty(y) && rows(x)!=rows(y) |
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104 error ([calledby, " x and y vectors must be the same length"]); |
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105 endif |
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106 |
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107 ## interpret remaining arguments |
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108 trend=nfft=Fs=window=overlap=whole=use_dB=[]; |
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109 ci=-1; ## need to do stupid things with ci |
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110 pos=0; ## no positional parameters yet interpreted. |
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111 for i=first:nargin |
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112 arg = va_arg(); |
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113 if isstr(arg), |
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114 arg=tolower(arg); |
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115 if strcmp(arg, 'squared') |
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116 use_dB = 0; |
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117 elseif strcmp(arg, 'db') |
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118 use_dB = 1; |
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119 elseif strcmp(arg, 'whole') |
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120 whole = 1; |
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121 elseif strcmp(arg, 'half') |
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122 whole = 0; |
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123 elseif strcmp(arg, 'none') |
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124 trend = -1; |
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125 elseif strcmp(arg, 'mean') |
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126 trend = 0; |
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127 elseif strcmp(arg, 'linear') |
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128 trend = 1; |
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129 else |
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130 error([calledby, " doesn't understand '", arg, "'"]); |
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131 endif |
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132 elseif pos == 0 |
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133 nfft = arg; |
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134 pos++; |
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135 elseif pos == 1 |
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136 Fs = arg; |
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137 pos++; |
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138 elseif pos == 2 |
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139 window = arg; |
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140 pos++; |
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141 elseif pos == 3 |
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142 overlap = arg; |
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143 pos++; |
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144 elseif pos == 4 |
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145 ci = arg; |
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146 pos++; |
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147 else |
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148 usage(usagestr); |
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149 endif |
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150 endfor |
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151 |
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152 ## Fill in defaults for arguments that aren't specified |
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153 if isempty(nfft), nfft = min(256, length(x)); endif |
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154 if isempty(Fs), Fs = 2; endif |
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155 if isempty(window), window = hanning(nfft); endif |
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156 if isempty(overlap), overlap = length(window)/2; endif |
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157 if isempty(whole), whole = !isreal(x)||(!isempty(y)&&!isreal(y)); endif |
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158 if isempty(trend), trend=-1; endif |
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159 if isempty(use_dB), |
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160 ## don't default to db for cohere, or for returned values |
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161 use_dB = (ftype!=3 && nargout == 0); |
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162 endif |
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163 if isempty(ci), ci=0.95; endif # if ci was not passed in, it would be 0 |
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164 |
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165 ## sort out default confidence intervals |
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166 if (ci < 0) # ci was not passed in |
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167 if nargout > 2 |
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168 ci = 0.95; |
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169 else |
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170 ci = 0.0; |
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171 endif |
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172 endif |
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173 if (ftype > 2 && ci > 0) |
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174 error([ calledby, " can't compute confidence intervals" ]); |
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175 elseif (ci < 0 || ci > 1) |
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176 error([ calledby, " confidence interval must be between 0 and 1" ]); |
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177 endif |
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178 |
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179 ## if only the window length is given, generate hanning window |
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180 if length(window) == 1, window = hanning(window); endif |
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181 if rows(window)==1, window = window.'; endif |
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182 |
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183 ## Normalize the window |
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184 window = window / norm(window); |
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185 |
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186 ## compute window offsets |
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187 win_size = length(window); |
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188 if (win_size > nfft) |
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189 nfft = win_size; |
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190 warning (sprintf("%s fft size adjusted to %d", calledby, n)); |
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191 end |
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192 step = win_size - overlap; |
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193 |
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194 ## Determine which correlations to compute |
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195 Pxx = Pyy = Pxy = []; |
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196 if ftype!=2, Pxx = zeros(nfft,1); endif # Not needed for csd |
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197 if ftype==3, Pyy = zeros(nfft,1); endif # Only needed for cohere |
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198 if ftype!=1, Pxy = zeros(nfft,1); endif # Not needed for psd |
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199 |
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200 ## Average the slices |
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201 offset = 1:step:length(x)-win_size+1; |
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202 N = length(offset); |
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203 for i=1:N |
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204 a = x(offset(i):offset(i)+win_size-1); |
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205 if trend>=0, a=detrend(a,trend); endif |
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206 a = fft(postpad(a.*window, nfft)); |
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207 if !isempty(Pxx), Pxx = Pxx + a.*conj(a); endif |
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208 if !isempty(Pxy) |
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209 b = y(offset(i):offset(i)+win_size-1); |
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210 if trend>=0, b=detrend(b,trend); endif |
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211 b = fft(postpad(b.*window, nfft)); |
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212 Pxy = Pxy + a .*conj(b); |
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213 if !isempty(Pyy), Pyy = Pyy + b.*conj(b); endif |
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214 endif |
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215 endfor |
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216 if (ftype <= 2) |
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217 ## the factors of N cancel when computing cohere and tfe |
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218 if !isempty(Pxx), Pxx = Pxx / N; endif |
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219 if !isempty(Pxy), Pxy = Pxy / N; endif |
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220 if !isempty(Pyy), Pyy = Pyy / N; endif |
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221 endif |
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222 |
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223 ## Compute confidence intervals |
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224 if ci > 0, Pci = zeros(nfft,1); endif |
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225 if (ci > 0 && N > 1) |
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226 if ftype>2 |
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227 error([calledby, ": internal error -- shouldn't compute Pci"]); |
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228 end |
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229 |
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230 ## c.i. = mean +/- dev |
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231 ## dev = z_ci*std/sqrt(n) |
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232 ## std = sqrt(sumsq(P-mean(P))/(N-1)) |
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233 ## z_ci = normal_inv( 1-(1-ci)/2 ) = normal_inv( (1+ci)/2 ); |
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234 ## normal_inv(x) = sqrt(2) * erfinv(2*x-1) |
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235 ## => z_ci = sqrt(2)*erfinv(2*(1+ci)/2-1) = sqrt(2)*erfinv(ci) |
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236 for i=1:N |
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237 a=x(offset(i):offset(i)+win_size-1); |
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238 if trend>=0, a=detrend(a,trend); endif |
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239 a=fft(postpad(a.*window, nfft)); |
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240 if ftype == 1 # psd |
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241 P = a.*conj(a) - Pxx; |
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242 Pci = Pci + P.*conj(P); |
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243 else # csd |
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244 b=y(offset(i):offset(i)+win_size-1); |
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245 if trend>=0, b=detrend(b,trend); endif |
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246 b=fft(postpad(b.*window, nfft)); |
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247 P = a.*conj(b) - Pxy; |
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248 Pci = Pci + P.*conj(P); |
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249 endif |
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250 endfor |
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251 |
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252 Pci = ( erfinv(ci) * sqrt( 2/N/(N-1) ) ) * sqrt ( Pci ); |
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253 endif |
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254 |
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255 switch (ftype) |
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256 case 1, # psd |
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257 P = Pxx / Fs; |
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258 if ci > 0, Pci = Pci / Fs; endif |
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259 case 2, # csd |
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260 P = Pxy; |
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261 case 3, # cohere |
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262 P = Pxy.*conj(Pxy)./Pxx./Pyy; |
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263 case 4, # tfe |
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264 P = Pxy./Pxx; |
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265 endswitch |
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266 |
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267 ## compute confidence intervals |
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268 if ci > 0, Pci = [ P - Pci, P + Pci ]; endif |
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269 |
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270 if use_dB |
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271 P = 10.0*log10(P); |
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272 if ci > 0, Pci = 10.0*log10(Pci); endif |
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273 endif |
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274 |
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275 ## extract the positive frequency components |
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276 if whole |
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277 ret_n = nfft; |
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278 elseif rem(nfft,2)==1 |
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279 ret_n = (nfft+1)/2; |
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280 else |
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281 ret_n = nfft/2 + 1; |
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282 end |
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283 P = P(1:ret_n, :); |
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284 if ci > 0, Pci = Pci(1:ret_n, :); endif |
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285 f = [0:ret_n-1]*Fs/nfft; |
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286 |
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287 ## Plot if there is no |
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288 if nargout==0, |
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289 unwind_protect |
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290 if Fs==2 |
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291 xlabel("Frequency (rad/pi)"); |
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292 else |
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293 xlabel("Frequency (Hz)"); |
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294 endif |
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295 if ftype==1 |
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296 title ("Welch's Spectral Estimate Pxx/Fs"); |
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297 ytext="Power Spectral Density"; |
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298 elseif ftype==2 |
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299 title ("Cross Spectral Estimate Pxy"); |
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300 ytext="Cross Spectral Density"; |
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301 elseif ftype==3 |
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302 title ("Coherence Function Estimate |Pxy|^2/(PxxPyy)"); |
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303 ytext="Coherence "; |
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304 else |
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305 title ("Transfer Function Estimate Pxy/Pxx"); |
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306 ytext="Transfer"; |
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307 endif |
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308 if use_dB, |
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309 ylabel(strcat(ytext, " (dB)")); |
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310 else |
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311 ylabel(ytext); |
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312 endif |
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313 grid("on"); |
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314 if ci>0 |
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315 plot(f, [P, Pci], ";;"); |
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316 else |
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317 plot(f, P, ";;"); |
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318 endif |
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319 unwind_protect_cleanup |
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320 grid("off"); |
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321 title(""); |
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322 xlabel(""); |
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323 ylabel(""); |
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324 end_unwind_protect |
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325 endif |
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326 |
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327 if nargout>=1, vr_val(P); endif |
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328 if nargout>=2 && ci>0, vr_val(Pci); endif |
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329 if nargout>=2 && ci==0, vr_val(f); endif |
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330 if nargout>=3 && ci>0, vr_val(f); endif |
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331 |
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332 endfunction |
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333 |
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334 %!demo |
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335 %! Fs=8000; |
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336 %! [b,a] = cheby1(4,3,2*[500, 1000]/Fs); ## define spectral envelope |
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337 %! s=0.05*randn(2^11,1); ## define noise |
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338 %! idx=fix(1:Fs/70:length(s))'; |
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339 %! s(idx)=s(idx)+ones(size(idx)); ## add 70 Hz excitation |
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340 %! x=filter(b,a,s); ## generate signal |
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341 %! |
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342 %! figure(1); subplot(221); |
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343 %! text(0,0.9,'basic estimate','Units','Normalized'); |
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344 %! pwelch(x',[],Fs); text; # slip in a test for row vs. column vector |
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345 %! subplot(222); |
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346 %! text(0,0.9,'nfft=1024 instead of 256','Units','Normalized'); |
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347 %! pwelch(x,1024); text; |
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348 %! subplot(223); |
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349 %! text(0,0.9,'boxcar instead of hanning','Units','Normalized'); |
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350 %! pwelch(x,[],[],boxcar(256)); text; |
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351 %! subplot(224); |
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352 %! text(0,0.9,'no overlap','Units','Normalized'); |
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353 %! pwelch(x,[],[],[],0); text; |
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354 %! |
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355 %! figure(2); subplot(121); |
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356 %! text(0,0.9,'magnitude units, whole range','Units','Normalized'); |
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357 %! pwelch(x,'whole','squared'); text; |
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358 %! subplot(122); |
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359 %! text(0,0.9,'90% confidence intervals','Units','Normalized'); |
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360 %! pwelch(x,[],[],[],[],0.9); text; |
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361 %! oneplot(); |
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362 %! %---------------------------------------------------------- |
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363 %! % plots should show a chebyshev bandpass filter shape |
