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peig

Pseudospectrum using eigenvector method

Syntax

Description

[S,w] = peig(x,p) implements the eigenvector spectral estimation method and returns S, the pseudospectrum estimate of the input signal x, and w, a vector 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:

You can specify the second input argument p as either:

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 Data
Length of S and w
Range of the Corresponding Normalized Frequencies
Real-valued
129
[0, pi]
Complex-valued
256
[0, 2pi)

[S,w] = peig(...,nfft) specifies the length of the FFT used to estimate the pseudospectrum with the integer nfft. 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 for this syntax.

S and Frequency Vector Characteristics  
Real/Complex Input Data
nfft Even/Odd
Length of S and w
Range of w
Real-valued
Even
(nfft/2 + 1)
[0, pi]
Real-valued
Odd
(nfft + 1)/2
[0, pi)
Complex-valued
Even or odd
nfft
[0, 2pi)

[S,f] = peig(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 Table , 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] = peig(...,'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] = peig(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 Table , Eigenvector Length Depending on Input Data and Syntax below for related information on this syntax.

[...] = peig(...,'range') specifies the range of frequency values to include in f or w. This syntax is useful when x is real. 'range' can be either:

[...,v,e] = peig(...) 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.

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

Remarks

In the process of estimating the pseudospectrum, peig computes the noise and signal subspaces from the estimated eigenvectors vj and eigenvalues lambdaj 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 peig 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
Comments on the Syntax
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
The string '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.

Examples

Implement the eigenvector method to find the pseudospectrum of the sum of three sinusoids in noise, using the default FFT length of 256. Use the modified covariance method for the correlation matrix estimate:

Algorithm

The eigenvector method estimates the pseudospectrum from a signal or a correlation matrix using a weighted version of the MUSIC algorithm derived from Schmidt's eigenspace analysis method [1][2]. The algorithm performs eigenspace analysis of the signal's correlation matrix in order to estimate the signal's frequency content. The eigenvalues and eigenvectors of the signal's correlation matrix are estimated using svd if you don't supply the correlation matrix. This algorithm is particularly suitable for signals that are the sum of sinusoids with additive white Gaussian noise.

The eigenvector method produces a pseudospectrum estimate given by

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

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

See Also

corrmtx, pburg, periodogram, pmtm, pmusic, prony, pwelch, rooteig, rootmusic

References

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

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

[3] Stoica, P., and R.L. Moses, Introduction to Spectral Analysis, Prentice-Hall, 1997.


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