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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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1 % WELCH Welch spectral estimation method.
2 %
3 % [pxx, f, info] = welch(x,type,pl)
4 % [pxx, f, info] = welch(x,y,type,pl)
5 %
6 % INPUTS: x - input analysis objects
7 % y - input analysis objects
8 % type - type of spectrum:
9 % 'psd' - compute Power Spectral Denstiy (PSD)
10 % 'ms' - compute Mean-square (Power) Spectrum (MS)
11 % 'cpsd' - compute cross-spectral density
12 % 'tfe' - estimate transfer function between inputs
13 % 'mscohere' - estimate magnitude-squared cross-coherence
14 % 'cohere' - estimate complex cross-coherence
15 % pl - input parameter list
16 %
17 % PARAMETERS: 'Win' - a specwin window object [default: Kaiser -200dB psll]
18 % 'Olap' - segment percent overlap [default: taken from window function]
19 % 'Nfft' - number of samples in each fft [default: length of input data]
20 % 'Scale' - one of
21 % 'ASD' - amplitude spectral density
22 % 'PSD' - power spectral density [default]
23 % 'AS' - amplitude spectrum
24 % 'PS' - power spectrum
25 % * applies only to spectrum 'Type' 'psd'
26 % 'Order' - order of segment detrending
27 % -1 - no detrending
28 % 0 - subtract mean [default]
29 % 1 - subtract linear fit
30 % N - subtract fit of polynomial, order N
31 %
32 % Copied directly from MATLAB and extended to do segment-wise detrending,
33 % compute transfer function variance and to take a plist input
34 %
35 % M Hewitson 08-05-08
36 %
37 % $Id: welch.m,v 1.34 2011/03/14 16:06:05 mauro Exp $
38 %
39
40 % Author(s): P. Pacheco
41 % Copyright 1988-2006 The MathWorks, Inc.
42 % $Revision: 1.34 $ $Date: 2011/03/14 16:06:05 $
43
44 % References:
45 % [1] Petre Stoica and Randolph Moses, Introduction To Spectral
46 % Analysis, Prentice-Hall, 1997, pg. 15
47 % [2] Monson Hayes, Statistical Digital Signal Processing and
48 % Modeling, John Wiley & Sons, 1996.
49 % [3] JS Bendat and AG Piersol, Engineering applications of correlation
50 % and spectral analysis, John Wiley & Sons, 1993.
51
52
53 % Compute the periodogram power spectrum of each segment and average always
54 % compute the whole power spectrum, we force Fs = 1 to get a PS not a PSD.
55
56 function varargout = welch(varargin)
57
58 if nargin == 3
59 a = varargin{1};
60 esttype = varargin{2};
61 pl = varargin{3};
62 x = a.data.y;
63 inunits = a.data.yunits;
64 else
65 a = varargin{1};
66 b = varargin{2};
67 esttype = varargin{3};
68 pl = varargin{4};
69 if a.data.fs ~= b.data.fs
70 error('### Two time-series have different sample rates.');
71 end
72 inunits = b.data.yunits / a.data.yunits;
73 x = {a.data.y, b.data.y};
74 end
75
76 % Parse inputs
77 win = find(pl, 'Win');
78 nfft = find(pl, 'Nfft');
79 olap = find(pl, 'Olap')/100;
80 scale = find(pl, 'scale');
81 Xolap = round(olap*nfft);
82 fs = a.data.fs;
83 order = find(pl, 'order');
84
85 [x,M,isreal_x,y,Ly,win,winName,winParam,noverlap,k,L,options] = ...
86 ao.welchparse(x,esttype,win.win, Xolap, nfft, fs);
87
88 % Initialize
89 Sxx = zeros(options.nfft,1);
90
91 % Frequency vector was specified, return and plot two-sided PSD
92 freqVectorSpecified = false; nrow = 1;
93 if length(options.nfft) > 1,
94 freqVectorSpecified = true;
95 [ncol,nrow] = size(options.nfft);
96 end
97
98 % Compute the periodogram power spectrum of each segment and average always
99 % compute the whole power spectrum, we force Fs = 1 to get a PS not a PSD.
100
101 % Initialize
102 Mnxx = 0; Mn2xx = 0;
103 if freqVectorSpecified,
104 Sxx = zeros(length(options.nfft),1);
105 else
106 Sxx = zeros(options.nfft,1);
107 end
108 range = options.range;
109
110 LminusOverlap = L-noverlap;
111 xStart = 1:LminusOverlap:k*LminusOverlap;
112 xEnd = xStart+L-1;
113 switch lower(esttype)
114 case {'ms','psd'}
115 for ii = 1:k,
116 if order < 0
117 seg = x(xStart(ii):xEnd(ii));
118 else
119 [seg,coeffs] = ltpda_polyreg(x(xStart(ii):xEnd(ii)), order);
120 end
121 [Sxxk,w] = ao.computeperiodogram(seg,win,...
122 options.nfft,esttype,options.Fs);
123 % Welford's algorithm for updating mean and variance
124 if ii == 1
125 Mnxx = Sxxk;
126 else
127 Qxx = Sxxk - Mnxx;
128 Mnxx = Mnxx + Qxx/ii;
129 Mn2xx = Mn2xx + Qxx.*(Sxxk - Mnxx);
130 end
131 end
132 Sxx = Mnxx;
133 if k == 1
134 Svxx = [];
135 else
136 Svxx = Mn2xx/(k-1)/k;
137 end
138 case 'cpsd'
139 for ii = 1:k,
140 if order < 0
141 xseg = x(xStart(ii):xEnd(ii));
142 else
143 [xseg,coeffs] = ltpda_polyreg(x(xStart(ii):xEnd(ii)), order);
144 end
145 if order < 0
146 yseg = y(xStart(ii):xEnd(ii));
147 else
148 [yseg,coeffs] = ltpda_polyreg(y(xStart(ii):xEnd(ii)), order);
149 end
150 [Sxxk,w] = ao.computeperiodogram({xseg,...
151 yseg},win,options.nfft,esttype,options.Fs);
152 % Welford's algorithm to update mean and variance
153 Qxx = Sxxk - Mnxx;
154 Mnxx = Mnxx +Qxx/ii;
155 Mn2xx = Mn2xx + abs(Qxx.*conj(Sxxk - Mnxx));
156 end
157 Sxx = Mnxx;
158 if k ==1
159 Svxx = [];
160 else
161 Svxx = Mn2xx/(k-1)/k;
162 end
163 case 'tfe'
164 % compute transfer function
165 Sxy = zeros(options.nfft,1); % Initialize
166 Syy = zeros(options.nfft,1); % Initialize
167 for ii = 1:k,
168 if order < 0
169 xseg = x(xStart(ii):xEnd(ii));
170 else
171 [xseg,coeffs] = ltpda_polyreg(x(xStart(ii):xEnd(ii)), order);
172 end
173 if order < 0
174 yseg = y(xStart(ii):xEnd(ii));
175 else
176 [yseg,coeffs] = ltpda_polyreg(y(xStart(ii):xEnd(ii)), order);
177 end
178 [Sxxk,w] = ao.computeperiodogram(xseg,...
179 win,options.nfft,esttype,options.Fs);
180 [Sxyk,w] = ao.computeperiodogram({yseg,...
181 xseg},win,options.nfft,esttype,options.Fs);
182 [Syyk,w] = ao.computeperiodogram(yseg,...
183 win,options.nfft,esttype,options.Fs);
184 Sxx = Sxx + Sxxk;
185 Sxy = Sxy + Sxyk;
186 Syy = Syy + Syyk;
187 % don't need to be divided by k because only rations are used here
188 end
189 case {'mscohere','cohere'}
190 % Note: (Sxy1+Sxy2)/(Sxx1+Sxx2) ~= (Sxy1/Sxy2) + (Sxx1/Sxx2)
191 % ie, we can't push the computation of Cxy into computeperiodogram.
192 Sxy = zeros(options.nfft,1); % Initialize
193 Syy = zeros(options.nfft,1); % Initialize
194 for ii = 1:k,
195 if order < 0
196 xseg = x(xStart(ii):xEnd(ii));
197 else
198 [xseg,coeffs] = ltpda_polyreg(x(xStart(ii):xEnd(ii)), order);
199 end
200 if order < 0
201 yseg = y(xStart(ii):xEnd(ii));
202 else
203 [yseg,coeffs] = ltpda_polyreg(y(xStart(ii):xEnd(ii)), order);
204 end
205 [Sxxk,w] = ao.computeperiodogram(xseg,...
206 win,options.nfft,esttype,options.Fs);
207 [Syyk,w] = ao.computeperiodogram(yseg,...
208 win,options.nfft,esttype,options.Fs);
209 [Sxyk,w] = ao.computeperiodogram({xseg,...
210 yseg},win,options.nfft,esttype,options.Fs);
211 Sxx = Sxx + Sxxk;
212 Sxy = Sxy + Sxyk;
213 Syy = Syy + Syyk;
214 % don't need to be divided by k because only rations are used here
215 end
216 end
217 % Generate the freq vector directly in Hz to avoid roundoff errors due to
218 % conversions later.
219 if ~freqVectorSpecified,
220 w = psdfreqvec('npts',options.nfft, 'Fs',options.Fs);
221 else
222 if strcmpi(options.range,'onesided')
223 warning(generatemsgid('InconsistentRangeOption'),...
224 'Ignoring ''onesided'' option. When a frequency vector is specified, a ''twosided'' PSD is computed');
225 end
226 options.range = 'twosided';
227 end
228
229 switch lower(esttype)
230 case {'psd','cpsd'}
231 % Compute the 1-sided or 2-sided PSD [Power/freq] or mean-square [Power].
232 % Also, corresponding freq vector and freq units.
233 % Here we use our own 'computepsd' to scale correctly the variance
234 if k == 1
235 [Pxx,w,units] = computepsd(Sxx,w,options.range,options.nfft,options.Fs,esttype);
236 P = Pxx;
237 dP = [];
238 else
239 [Pxx,Pvxx,w,units] = utils.math.computepsd(Sxx,Svxx,w,options.range,options.nfft,options.Fs,esttype);
240 P = Pxx;
241 % the 1/k factor should come after welchscale if we don't want to apply sqrt() to it.
242 % we correct for that here. It is only needed for 'asd','as' in
243 % psd/cpsd, the other cases go always through 'PSD'.
244 if (strcmpi(scale,'PSD') || strcmpi(scale,'PS'))
245 dP = Pvxx;
246 elseif (strcmpi(scale,'ASD') || strcmpi(scale,'AS'))
247 dP = Pvxx/k;
248 else
249 error('### Unknown scale')
250 end
251 end
252 case 'tfe'
253 % Compute the 1-sided or 2-sided PSD [Power/freq] or mean-square [Power].
254 % Also, corresponding freq vector and freq units.
255 % In the Cross PSD, the frequency vector and xunits are not used.
256 [Pxx,w,units] = computepsd(Sxx,w,options.range,options.nfft,options.Fs,esttype);
257 [Pxy,w,units] = computepsd(Sxy,w,options.range,options.nfft,options.Fs,esttype);
258 [Pyy,w,units] = computepsd(Syy,w,options.range,options.nfft,options.Fs,esttype);
259 % mean and std
260 P = Pxy ./ Pxx; % Txy
261 if k == 1
262 dP =[];
263 else
264 dP = (k/(k-1)^2)*(Pyy./Pxx).*(1 - (abs(Pxy).^2)./(Pxx.*Pyy));
265 end
266 case 'mscohere'
267 % Magnitude Square Coherence estimate.
268 % Auto PSD for 2nd input vector. The freq vector & xunits are not
269 % used.
270 [Pxx,w,units] = computepsd(Sxx,w,options.range,options.nfft,options.Fs,esttype);
271 [Pxy,w,units] = computepsd(Sxy,w,options.range,options.nfft,options.Fs,esttype);
272 [Pyy,w,units] = computepsd(Syy,w,options.range,options.nfft,options.Fs,esttype);
273 % mean and std
274 P = (abs(Pxy).^2)./(Pxx.*Pyy); % Magnitude-squared coherence
275 dP = (2*P/k).*(1-P).^2;
276 case 'cohere'
277 % Complex Coherence estimate.
278 % Auto PSD for 2nd input vector. The freq vector & xunits are not
279 % used.
280 [Pxx,w,units] = computepsd(Sxx,w,options.range,options.nfft,options.Fs,esttype);
281 [Pxy,w,units] = computepsd(Sxy,w,options.range,options.nfft,options.Fs,esttype);
282 [Pyy,w,units] = computepsd(Syy,w,options.range,options.nfft,options.Fs,esttype);
283 P = Pxy./sqrt(Pxx.*Pyy); % Complex coherence
284 dP = (2*abs(P)/k).*(1-abs(P)).^2;
285 end
286 % end
287
288 % Scale to required units
289 [P, dP, info] = ao.welchscale(P, dP, win, fs, scale, inunits);
290 info.navs = k;
291
292 if k ==1
293 dev = [];
294 else
295 dev = sqrt(dP);
296 end
297
298 varargout = {P, w, info, dev};
299
300 end
301