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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 function [music_data,msg] = music(x,p,varargin)
2 %MUSIC Implements the heart of the MUSIC algorithm of line spectra estimation.
3 % MUSIC is called by both PMUSIC and ROOTMUSIC.
4 %
5 % Inputs:
6 %
7 % x - vector or matrix. If vector it is a signal, if matrix it may be either a data
8 % matrix such that x'*x=R, or a correlation matrix R.
9 % p - scalar or two element vector. If scalar, it indicates the dimension of the
10 % signal subspace. If vector, p(2) is a threshold used to determine the
11 % aforementioned dimension.
12 % nfft - (optional) to be used only with PMUSIC. A scalar indicating the number of
13 % points used in the evaluation of the pseudospectrum.
14 % Fs - (optional) a scalar specifying the sampling frequency. If omitted, we work
15 % in rad/sample; if empty it defaults to 1 Hz.
16 % nw - (optional) a scalar or vector indicating either the order of the correlation
17 % matrix or (when a vector) a window whose length is the order of the matrix
18 % and whose values are used to window each column of the data matrix.
19 % noverlap - (optional) a integer indicating the number of samples to overlap from
20 % column to column.
21 % strings - Optional input strings are: 'corr', 'EV' and range ('half' or 'whole').
22 %
23 % Outputs:
24 %
25 % msg - a possible error message.
26 %
27 % music_data - a structure with the following fields:
28 %
29 % noise_eigenvects - a matrix whose columns are the noise subspace eigenvectors.
30 % signal_eigenvects - a matrix whose columns are the signal subspace eigenvectors.
31 % eigenvals - the eigenvalues of the correlation matrix.
32 % p_eff - the effective dimension of the signal subspace.
33 % nfft - number of points used to evaluate the pseudospectrum (only used in PMUSIC).
34 % Fs - sampling freq.
35 % range - string indicating whether 'half' or the 'whole' pseudospectrum should be
36 % computed. (Only used in PMUSIC.)
37 % EVFlag - flag, 0 = MUSIC method; 1 = EigenVector method.
38
39 % Author(s): R. Losada
40 % Copyright 1988-2006 The MathWorks, Inc.
41 % $Revision: 1.1 $ $Date: 2010/02/18 11:16:20 $
42
43 % References:
44 % [1] Petre Stoica and Randolph Moses, Introduction To Spectral
45 % Analysis, Prentice-Hall, 1997, pg. 15
46 % [2] S. J. Orfanidis, Optimum Signal Processing. An Introduction.
47 % 2nd Ed., Macmillan, 1988.
48
49
50 xIsReal = isreal(x);
51 msg = '';
52 music_data = [];
53
54 if isempty(p),
55 msg = 'The signal subspace dimension cannot be empty.';
56 return
57 end
58
59 [opts,msg] = music_options(xIsReal,p,varargin{:});
60 if ~isempty(msg),
61 return
62 end
63
64 % Compute the eigenvalues and eigenvectors of the correlation matrix
65 [eigenvals,eigenvects] = computeeig(x,opts.CorrFlag,opts.CorrMatrOrd,opts.nw,opts.noverlap,opts.window,opts.EVFlag);
66
67 % Determine the effective dimension of the signal subspace
68 p_eff = determine_signal_space(p,eigenvals);
69
70 % Separate the signal and noise eigenvectors
71 signal_eigenvects = eigenvects(:,1:p_eff);
72 noise_eigenvects = eigenvects(:,p_eff+1:end);
73
74 % Generate the output structure
75 music_data.noise_eigenvects = noise_eigenvects;
76 music_data.signal_eigenvects = signal_eigenvects;
77 music_data.eigenvals = eigenvals;
78 music_data.p_eff = p_eff;
79 music_data.nfft = opts.nfft;
80 music_data.Fs = opts.Fs;
81 music_data.EVFlag = opts.EVFlag;
82 music_data.range = opts.range;
83
84
85 %--------------------------------------------------------------------------------------
86 function [options,msg] = music_options(xIsReal,p,varargin)
87 %MUSIC_OPTIONS Parse the optional inputs to the MUSIC function.
88 % MUSIC_OPTIONS returns a structure, OPTIONS, with the following fields:
89 %
90 % options.nfft - number of freq. points at which the psd is estimated
91 % options.Fs - sampling freq. if any
92 % options.range - 'onesided' or 'twosided' pseudospectrum (they correspond to
93 % 'half' and 'whole' respectively, but are returned as is by
94 % psdoptions.m
95 % options.nw - number of columns in the data matrix
96 % options.noverlap - number of samples to overlap
97 % options.window - a vector with window coefficients
98 % options.CorrFlag - a flag indicating whether the input is a correlation matrix
99 % options.EVFlag - flag, 0 = MUSIC method ; 1 = EigenVector method
100 % options.CorrMatrOrd - order of the correlation matrix to be used in computations
101
102
103
104 % Assign Defaults
105 msg = '';
106 options.nw = [];
107 options.noverlap = [];
108 options.window = [];
109 options.nfft = 256;
110 options.Fs = [];
111 options.CorrFlag = 0;
112 options.EVFlag = 0;
113 % Determine if frequency vector specified
114 freqVecSpec = false;
115 if (length(varargin) > 0 && isnumeric(varargin{1}) && length(varargin{1}) > 1)
116 freqVecSpec = true;
117 end
118
119 if xIsReal && ~freqVecSpec,
120 options.range = 'onesided';
121 else
122 options.range = 'twosided';
123 end
124
125 [options,msg] = psdoptions(xIsReal,options,varargin{:});
126
127 if length(options.nfft) > 1,
128 if strcmpi(options.range,'onesided')
129 warning(generatemsgid('InconsistentRangeOption'),...
130 'Ignoring the ''onesided'' option. When a frequency vector is specified, a ''twosided'' PSD is computed');
131 end
132 options.range = 'twosided';
133 end
134
135 % psdoptions doesn't handle this field, assign it separetely
136 options.CorrMatrOrd = 2*p(1);
137
138 %-----------------------------------------------------------------------------------------
139 function [eigenvals,eigenvects] = computeeig(x,CorrFlag,CorrMatrOrd,nw,noverlap,window,EVFlag)
140 %COMPUTEEIG Compute eigenvalues and eigenvectors of correlation matrix.
141 %
142 % Inputs:
143 %
144 % x - input vector or matrix
145 % CorrFlag - (flag) indicates whether x is a correlation matrix
146 % nw - (integer) length of the rows of the data matrix
147 % (only used if x is vector)
148 % noverlap - (integer) overlap between the rows of the data matrix
149 % (used in conjunction with nw)
150 % window - (vector) window to be applied to each column of data
151 % matrix (not used if x is a correlation matrix)
152 % EVFlag - True if eigenvector method, false if MUSIC.
153 %
154 %
155 % Outputs:
156 %
157 % eigenvals
158 % eigenvects
159 %
160 % If x is a matrix,
161 % If CorrFlag = 1, input x is a correlation matrix, we compute the
162 % eigendecomposition and order the eigenvalues and eigenvectors.
163 %
164 % If x is a vector,
165 % a data matrix is formed by calling corrmtx unless a custom nw
166 % and noverlap are specified. In that case, we use buffer to form
167 % the data matrix.
168 %
169 % If window is not empty, each row of the data matrix will be
170 % multiplied by the window.
171
172 % Determine if the input is a matrix
173 xIsMatrix = ~any(size(x)==1);
174
175 if xIsMatrix && CorrFlag,
176 % Input is Correlation matrix
177
178 % Compute the eigenvectors and eigenvalues
179 [E,D] = eig((x+x')/2); % Ensure hermitian
180 [eigenvals,indx] = sort(diag(D),'descend');
181 eigenvects = E(:,indx);
182
183 else
184 if xIsMatrix
185 % Input is already a data matrix
186 [Mx,Nx] = size(x); % Determine size of data matrix
187 if EVFlag && (Nx > Mx),
188 errmsg = 'The number of columns in the data matrix cannot exceed the number of rows.';
189 error(generatemsgid('invalidDataMatrix'),errmsg);
190 end
191
192 else
193 % x is a vector
194 x = x(:); % Make it a column
195 if isempty(nw),
196 x = corrmtx(x,CorrMatrOrd-1,'cov');
197 else
198 if EVFlag && nw > (ceil((length(x)-nw)/(nw-noverlap))+1),
199 errmsg = sprintf(['The segment length and overlap specified result in\n',...
200 'a data matrix with more columns than rows.']);
201 error(generatemsgid('invalidDataMatrix'),errmsg);
202 end
203 Lx = length(x);
204 x = buffer(x,nw,noverlap,'nodelay');
205 if Lx <= nw,
206 error(generatemsgid('invalidSegmentLength'),'The segment length, NW, must be smaller than the signal length.');
207 end
208 x = x'./sqrt(Lx-nw); % Scale appropriately such that X'*X is a scaled estimate of R
209 end
210 end
211 if ~isempty(window),
212 % Apply window to each row of data matrix
213 if length(window) ~= size(x,2),
214 error(generatemsgid('InvalidDimensions'),'Window length must equal the number of columns in the data matrix.');
215 end
216 window = repmat(window(:).',size(x,1),1);
217 x = x.*window;
218 end
219
220 % Compute the eigenvectors and eigenvalues via the SVD
221 [U,S,eigenvects] = svd(x,0);
222 eigenvals = diag(S).^2; % We need to square the singular values here
223 end
224
225
226 %--------------------------------------------------------------------------------------------
227 function p_eff = determine_signal_space(p,eigenvals)
228 %DETERMINE_SIGNAL_SPACE Determines the effective dimension of the signal subspace.
229 %
230 % Inputs:
231 %
232 % p - (scalar or vector) signal subspace dimension
233 % (but may contain a desired threshold).
234 % eigenvals - (vector) contains the eigenvalues (sorted in decreasing order)
235 % of the correlation matrix
236 %
237 % Outputs:
238 %
239 % p_eff - The effective dimension of the signal subspace. If a threshold
240 % is given as p(2), the signal subspace will be equal to the number
241 % of eigenvalues, NEIG, greater than the threshold times the smallest
242 % eigenvalue. However, the dimension of the signal subspace is at most
243 % p(1), so that if NEIG is greater than p(1), p_eff will be equal to
244 % p(1). If the threshold criteria results in an empty signal subspace,
245 % once again we make p_eff = p(1).
246
247
248 % Use the signal space dimension or the threshold to separate the noise subspace eigenvectors
249 if length(p) == 2,
250 % The threshold will be the input threshold times the smallest eigenvalue
251 thresh = p(2)*eigenvals(end);
252 indx = find(eigenvals > thresh);
253 if ~isempty(indx)
254 p_eff = min( p(1), length(indx) );
255 else
256 p_eff = p(1);
257 end
258 else
259 p_eff = p;
260 end
261
262 % [EOF] - music.m