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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 [w_i,powers,w_mse,p_mse] = rootmusic(x,p,varargin)
2 %ROOTMUSIC Computes the frequencies and powers of sinusoids via the
3 % Root MUSIC algorithm.
4 % W = ROOTMUSIC(X,P) returns the vector of frequencies W of the complex
5 % sinusoids contained in signal vector X. W is in units of rad/sample.
6 % P is the number of complex sinusoids in X. If X is a data matrix,
7 % each row is interpreted as a separate sensor measurement or trial.
8 % In this case, X must have a number of columns larger than P. You can
9 % use the function CORRMTX to generate data matrices to be used here.
10 %
11 % W = ROOTMUSIC(R,P,'corr') returns the vector of frequencies W, for a
12 % signal whose correlation matrix estimate is given by the positive
13 % definite matrix R. Exact conjugate-symmetry of R is ensured by forming
14 % (R+R')/2 inside the function. The number of rows or columns of R must
15 % be greater than P.
16 %
17 % If P is a two element vector, P(2) is used as a cutoff for signal and
18 % noise subspace separation. All eigenvalues greater than P(2) times
19 % the smallest eigenvalue are designated as signal eigenvalues. In
20 % this case, the signal subspace dimension is at most P(1).
21 %
22 % F = ROOTMUSIC(...,Fs) uses the sampling frequency Fs in the computation
23 % and returns the vector of frequencies, F, in Hz.
24 %
25 % [W,POW] = ROOTMUSIC(...) returns in addition a vector POW containing the
26 % estimates of the powers of the sinusoids in X.
27 %
28 % EXAMPLES:
29 % s1 = RandStream.create('mrg32k3a');
30 % n=0:99;
31 % s=exp(i*pi/2*n)+2*exp(i*pi/4*n)+exp(i*pi/3*n)+randn(s1,1,100);
32 % X=corrmtx(s,12,'mod'); % Estimate the correlation matrix using
33 % % the modified covariance method.
34 % [W,P] = rootmusic(X,3);
35 %
36 % See also ROOTEIG, PMUSIC, PEIG, PMTM, PBURG, PWELCH, CORRMTX, SPECTRUM.
37
38 % Reference: Stoica, P. and R. Moses, INTRODUCTION TO SPECTRAL ANALYSIS,
39 % Prentice-Hall, 1997.
40
41 % Author(s): R. Losada
42 % Copyright 1988-2008 The MathWorks, Inc.
43 % $Revision: 1.1 $ $Date: 2010/02/18 11:16:00 $
44
45 %%%%%%%%%%%%%%%%%%%%%%%%
46 %
47 % Added function to compute approx. MSE for the case of a unique sinusoid
48 %
49 % REFERENCES: Rao, B. Performance Analysis of Root-Music
50 % IEEE Trans. Acoust. Speech and Sig. Proc. 37, 1989
51 %
52 % VERSION: $Id: rootmusic.m,v 1.1 2010/02/18 11:16:00 miquel Exp $
53 %
54 % M Nofrarias 12/02/2010
55 %
56
57 error(nargchk(2,5,nargin,'struct'));
58
59 xIsReal = isreal(x);
60
61 % Check for an even number of complex sinusoids if data is real
62 if xIsReal && rem(p,2),
63 error(generatemsgid('InvalidDimensions'),'Real signals require an even number p of complex sinusoids.');
64 end
65
66 nfft = []; % Root Music doesn't use nfft, but the parser needs it
67 varargin = {nfft,varargin{:}};
68
69 [md,msg] = utils.math.music(x,p,varargin{:});
70 if ~isempty(msg), error(generatemsgid('SigErr'),msg); end
71
72 % Find the Complex Sinusoid Frequencies
73 w_i = compute_freqs(md.noise_eigenvects,md.p_eff,md.EVFlag,md.eigenvals);
74
75 % Estimate the noise variance as the average of the noise subspace eigenvalues
76 sigma_w = sum(md.eigenvals(md.p_eff+1:end))./size(md.noise_eigenvects,2);
77
78 % Estimate the power of the sinusoids
79 [powers] = compute_power(md.signal_eigenvects,md.eigenvals,w_i,md.p_eff,sigma_w,xIsReal);
80
81 % Compute MSE
82 [w_mse,p_mse] = compute_mse(sigma_w,powers,length(x));
83
84 % Convert the estimated frequencies to Hz if Fs was specified
85 if ~isempty(md.Fs),
86 w_i = w_i*md.Fs./(2*pi);
87 w_mse = w_mse*(md.Fs./(2*pi))^2;
88 end
89
90 %---------------------------------------------------------------------------------------------
91 function w_i = compute_freqs(noise_eigenvects,p_eff,EVFlag,eigenvals)
92 %Compute the frequencies via the roots of the polynomial formed with the noise eigenvectors
93 %
94 % Inputs:
95 %
96 % noise_eigenvects - a matrix whose columns are the noise subspace eigenvectors
97 % p_eff - signal subspace dimension
98 % EVFlag - a flag indicating of the eigenvector methos should be used
99 % eigenvals - a vector with all the correlation matrix eigenvalues.
100 % However, we use only the noise eigenvalues as weights
101 % in the eigenvector method.
102 %
103 % Outputs:
104 %
105 % w_i - frequencies of the complex sinusoids
106
107
108 % compute weights
109 if EVFlag,
110 % Eigenvector method, use eigenvalues as weights
111 weights = eigenvals(end-size(noise_eigenvects,2)+1:end); % Use the noise subspace eigenvalues
112 else
113 weights = ones(1,size(noise_eigenvects,2));
114 end
115
116 % Form a polynomial D, consisting of a sum of polynomials given by the product of
117 % the noise subspace eigenvectors and the reversed and conjugated version.
118 D = 0;
119 for i = 1:length(weights),
120 D = D + conv(noise_eigenvects(:,i),conj(flipud(noise_eigenvects(:,i))))./weights(i);
121 end
122
123 roots_D = roots(D);
124 % Because D is formed from the product of a polynomial and its conjugated and reversed version,
125 % every root of D inside the unit circle, will have a "reflected" version outside the unit circle.
126 % We choose to use the ones inside the unit circle, because the distance from them to the unit
127 % circle will be smaller than the corresponding distance for the "reflected" root.
128 roots_D1 = roots_D(abs(roots_D) < 1);
129
130 % Sort the roots from closest to furthest from the unit circle
131 [not_used,indx] = sort(abs(abs(roots_D1)-1)); %#ok
132 sorted_roots = roots_D1(indx);
133
134 % Use the first p_eff roots to determine the frequencies
135 w_i = angle(sorted_roots(1:p_eff));
136
137 %-----------------------------------------------------------------------------------------------
138 function [powers] = compute_power(signal_eigenvects,eigenvals,w_i,p_eff,sigma_w,xIsReal)
139 %COMPUTE_POWER Solves the system of linear eqs. to calculate the power of the sinusoids.
140 %
141 % Inputs:
142 %
143 % signal_eigenvects - the matrix whose columns are the signal subspace eigenvectors
144 % eigenvals - a vector containing all eigenvalues of the correlation matrix
145 % w_i - a vector of frequency estimates of the sinusoids
146 % p_eff - the dimension of the signal subspace
147 % sigma_w - the estimate of the variance of the white noise
148 % xIsReal - a flag indicating wether we have real or complex sinusoids
149 %
150 % Outputs:
151 %
152 % powers - a vector that contains the power of each sinusoid
153
154 %This is just the solution of a linear system of eqs, Ax=b
155
156 % For real sinusoids, the system of eqs. has half the number of unknowns
157 if xIsReal,
158 w_i = reshape(w_i,2,length(w_i)./2);
159 w_i = w_i(1,:); % Use only the positive freqs.
160 w_i = w_i(:);
161 p_eff = p_eff./2;
162 end
163
164 % Form the A matrix
165 if length(w_i) == 1,
166 % FREQZ does not compute the gain at a single frequency, handle this separately
167 A = polyval(signal_eigenvects(:,1),exp(1i*w_i));
168 else
169 for n = 1:p_eff,
170 A(:,n) = freqz(signal_eigenvects(:,n),1,w_i);
171 end
172 end
173
174 A = abs(A.').^2;
175
176 % Form the b vector
177 b = eigenvals(1:p_eff) - sigma_w;
178
179 % The powers are simply the solution to the set of eqs.
180 powers = A\b;
181
182 %--------------------------------------------------------------------------
183 function [w_mse,p_mse] = compute_mse(sigma_w,powers,N)
184 % implements eq.30 in Reference
185
186 L = 1; % one element array
187
188 p_mse = 12 * (sigma_w/(powers*N*L^2));
189 % first term of eq.30 in paper is to pass from frequency to DOA
190 % this sigma_w^2 could be wrong
191 w_mse = 12/(2*L)* (sigma_w^2/(powers*N*L^2));
192
193 % [EOF] rootmusic.m
194