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# pmusic

Pseudospectrum using MUSIC algorithm

## Syntax

[S,w] = pmusic(x,p)
[S,w] = pmusic(x,p,w)
[S,w] = pmusic(...,nfft)
[S,f] = pmusic(x,p,nfft,fs)
[S,f] = pmusic(x,p,f,fs)
[S,f] = pmusic(...,'corr')
[S,f] = pmusic(x,p,nfft,fs,nwin,noverlap)
[...] = pmusic(...,freqrange)
[...,v,e] = pmusic(...)
pmusic(...)

## Description

[S,w] = pmusic(x,p) implements the MUSIC (Multiple Signal Classification) algorithm and returns S, the pseudospectrum estimate of the input signal x, and a vector w of normalized frequencies (in rad/sample) at which the pseudospectrum is evaluated. The pseudospectrum is calculated using estimates of the eigenvectors of a correlation matrix associated with the input data x, where x is specified as either:

• A row or column vector representing one observation of the signal

• A rectangular array for which each row of x represents a separate observation of the signal (for example, each row is one output of an array of sensors, as in array processing), such that x'*x is an estimate of the correlation matrix

 Note   You can use the output of corrmtx to generate such an array x.

You can specify the second input argument p as either:

• A scalar integer. In this case, the signal subspace dimension is p.

• A two-element vector. In this case, p(2), the second element of p, represents a threshold that is multiplied by λmin, the smallest estimated eigenvalue of the signal's correlation matrix. Eigenvalues below the threshold λmin*p(2) are assigned to the noise subspace. In this case, p(1) specifies the maximum dimension of the signal subspace.

 Note:   If the inputs to pmusic are real sinusoids, set the value of p to double the number of input signals. If the inputs are complex sinusoids, set p equal to the number of inputs.

The extra threshold parameter in the second entry in p provides you more flexibility and control in assigning the noise and signal subspaces.

S and w have the same length. In general, the length of the FFT and the values of the input x determine the length of the computed S and the range of the corresponding normalized frequencies. The following table indicates the length of S (and w) and the range of the corresponding normalized frequencies for this syntax.

S Characteristics for an FFT Length of 256 (Default)

Real/Complex Input DataLength of S and wRange of the Corresponding Normalized Frequencies

Real-valued

129

[0, π]

Complex-valued

256

[0, 2π)

[S,w] = pmusic(x,p,w) returns the pseudospectrum in the vector S computed at the normalized frequencies specified in vector w, which has two or more elements

[S,w] = pmusic(...,nfft) specifies the integer length of the FFT nfft used to estimate the pseudospectrum. The default value for nfft (entered as an empty vector []) is 256.

The following table indicates the length of S and w, and the frequency range for w in this syntax.

S and Frequency Vector Characteristics

Real/Complex Input Datanfft Even/OddLength of S and wRange of w

Real-valued

Even

(nfft/2 + 1)

[0, π]

Real-valued

Odd

(nfft + 1)/2

[0, π)

Complex-valued

Even or odd

nfft

[0, 2π)

[S,f] = pmusic(x,p,nfft,fs) returns the pseudospectrum in the vector S evaluated at the corresponding vector of frequencies f (in Hz). You supply the sampling frequency fs in Hz. If you specify fs with the empty vector [], the sampling frequency defaults to 1 Hz.

The frequency range for f depends on nfft, fs, and the values of the input x. The length of S (and f) is the same as in the S and Frequency Vector Characteristics above. The following table indicates the frequency range for f for this syntax.

S and Frequency Vector Characteristics with fs Specified

Real/Complex Input Data

nfft Even/Odd

Range of f

Real-valued

Even

[0, fs/2]

Real-valued

Odd

[0, fs/2)

Complex-valued

Even or odd

[0, fs)

[S,f] = pmusic(x,p,f,fs) returns the pseudospectrum in the vector S computed at the frequencies specified in vector f, which has two or more elements

[S,f] = pmusic(...,'corr') forces the input argument x to be interpreted as a correlation matrix rather than matrix of signal data. For this syntax x must be a square matrix, and all of its eigenvalues must be nonnegative.

[S,f] = pmusic(x,p,nfft,fs,nwin,noverlap) allows you to specify nwin, a scalar integer indicating a rectangular window length, or a real-valued vector specifying window coefficients. Use the scalar integer noverlap in conjunction with nwin to specify the number of input sample points by which successive windows overlap. noverlap is not used if x is a matrix. The default value for nwin is 2 × p(1) and noverlap is nwin – 1.

With this syntax, the input data x is segmented and windowed before the matrix used to estimate the correlation matrix eigenvalues is formulated. The segmentation of the data depends on nwin, noverlap, and the form of x. Comments on the resulting windowed segments are described in the following table.

Windowed Data Depending on x and nwin

Input data x

Form of nwin

Windowed Data

Data vector

Scalar

Length is nwin

Data vector

Vector of coefficients

Length is length(nwin)

Data matrix

Scalar

Data is not windowed.

Data matrix

Vector of coefficients

length(nwin) must be the same as the column length of x, and noverlap is not used.

See the Eigenvector Length Depending on Input Data and Syntax below for related information on this syntax.

 Note   The arguments nwin and noverlap are ignored when you include 'corr' in the syntax.

[...] = pmusic(...,freqrange) specifies the range of frequency values to include in f or w. freqrange can be either:

• 'half' — returns half the spectrum for a real input signal, x. If nfft is even, then S has length nfft/2 + 1 and is computed over the interval [0,π]. If nfft is odd, the length of S is (nfft + 1)/2 and the frequency interval is [0,π). When your specify fs, the intervals are [0,fs/2) and [0,fs/2] for even and odd nfft, respectively.

• 'whole' — returns the whole spectrum for either real or complex input, x. In this case, S has length nfft and is computed over the interval [0,2π). When you specify fs, the frequency interval is [0,fs).

• 'centered' — returns the centered whole spectrum for either real or complex input, x. In this case, S has length nfft and is computed over the interval (–π,π] for even nfft and (–π,π) for odd nfft. When you specify fs, the frequency intervals are (–fs/2,fs/2] and (–fs/2,fs/2) for even and odd nfft, respectively.

 Note   You can put the arguments freqrange or 'corr' anywhere in the input argument list after p.

[...,v,e] = pmusic(...) returns the matrix v of noise eigenvectors, along with the associated eigenvalues in the vector e. The columns of v span the noise subspace of dimension size(v,2). The dimension of the signal subspace is size(v,1)-size(v,2). For this syntax, e is a vector of estimated eigenvalues of the correlation matrix.

pmusic(...) with no output arguments plots the pseudospectrum in the current figure window.

## Examples

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This example analyzes a signal vector, x, assuming that two real sinusoidal components are present in the signal subspace. In this case, the dimension of the signal subspace is 4, because each real sinusoid is the sum of two complex exponentials.

n = 0:199;
x = cos(0.257*pi*n) + sin(0.2*pi*n) + 0.01*randn(size(n));
pmusic(x,4)      % Set p to 4 because there are two real inputs

This example analyzes the same signal vector, x, with an eigenvalue cutoff of 10% above the minimum. Setting p(1) = Inf forces the signal/noise subspace decision to be based on the threshold parameter, p(2). Specify the eigenvectors of length 7 using the nwin argument, and set the sampling frequency, fs, to 8 kHz:

rng default
n = 0:199;
x = cos(0.257*pi*n) + sin(0.2*pi*n) + 0.01*randn(size(n));
[P,f] = pmusic(x,[Inf,1.1],[],8000,7); % Window length = 7
plot(f,20*log10(abs(P)))
xlabel 'Frequency (Hz)', ylabel 'Power (dB)'
title 'Pseudospectrum Estimate via MUSIC', grid on

Supply a positive definite correlation matrix, R, for estimating the spectral density. Use the default 256 samples.

R = toeplitz(cos(0.1*pi*(0:6))) + 0.1*eye(7);
pmusic(R,4,'corr')

Enter a signal data matrix, Xm, generated from data using corrmtx.

n = 0:699;
x = cos(0.257*pi*(n)) + 0.1*randn(size(n));
Xm = corrmtx(x,7,'mod');
pmusic(Xm,2)

Use the same signal, but let pmusic form the 100-by-7 data matrix using its windowing input arguments. In addition, specify an FFT of length 512.

n = 0:699;
x = cos(0.257*pi*(n)) + 0.1*randn(size(n));
[PP,ff] = pmusic(x,2,512,[],7,0);
pmusic(x,2,512,[],7,0)

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### Tips

In the process of estimating the pseudospectrum, pmusic computes the noise and signal subspaces from the estimated eigenvectors vj and eigenvalues λj of the signal's correlation matrix. The smallest of these eigenvalues is used in conjunction with the threshold parameter p(2) to affect the dimension of the noise subspace in some cases.

The length n of the eigenvectors computed by pmusic is the sum of the dimensions of the signal and noise subspaces. This eigenvector length depends on your input (signal data or correlation matrix) and the syntax you use.

The following table summarizes the dependency of the eigenvector length on the input argument.

Eigenvector Length Depending on Input Data and Syntax

Form of Input Data x

Length n of Eigenvectors

Row or column vector

nwin is specified as a scalar integer.

nwin

Row or column vector

nwin is specified as a vector.

length(nwin)

Row or column vector

nwin is not specified.

2 × p(1)

l-by-m matrix

If nwin is specified as a scalar, it is not used. If nwin is specified as a vector, length(nwin) must equal m.

m

m-by-m nonnegative definite matrix

'corr' is specified and nwin is not used.

m

You should specify nwin > p(1) or length(nwin) > p(1) if you want p(2) > 1 to have any effect.

### Algorithms

The name MUSIC is an acronym for MUltiple SIgnal Classification. The MUSIC algorithm estimates the pseudospectrum from a signal or a correlation matrix using Schmidt's eigenspace analysis method [1]. The algorithm performs eigenspace analysis of the signal's correlation matrix in order to estimate the signal's frequency content. This algorithm is particularly suitable for signals that are the sum of sinusoids with additive white Gaussian noise. The eigenvalues and eigenvectors of the signal's correlation matrix are estimated if you don't supply the correlation matrix.

The MUSIC pseudospectrum estimate is given by

${P}_{\text{MUSIC}}\left(f\right)=\frac{1}{{e}^{H}\left(f\right)\left(\sum _{k=p+1}^{N}{\text{v}}_{k}{\text{v}}_{k}^{H}\right)e\left(f\right)}=\frac{1}{\sum _{k=p+1}^{N}|{\text{v}}_{k}^{H}e\left(f\right){|}^{2}}$

where N is the dimension of the eigenvectors and vk is the k-th eigenvector of the correlation matrix. The integer p is the dimension of the signal subspace, so the eigenvectors vk used in the sum correspond to the smallest eigenvalues and also span the noise subspace. The vector e(f) consists of complex exponentials, so the inner product

${\text{v}}_{k}^{H}e\left(f\right)$

amounts to a Fourier transform. This is used for computation of the pseudospectrum estimate. The FFT is computed for each vk and then the squared magnitudes are summed.

## References

[1] Marple, S. Lawrence. Digital Spectral Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1987, pp. 373–378.

[2] Schmidt, R. O. "Multiple Emitter Location and Signal Parameter Estimation." IEEE® Transactions on Antennas and Propagation. Vol. AP-34, March, 1986, pp. 276–280.

[3] Stoica, Petre, and Randolph L. Moses. Spectral Analysis of Signals. Upper Saddle River, NJ: Prentice Hall, 2005.