Signal Processing Toolbox
peig

Pseudospectrum using eigenvector method

Syntax

• ```[S,w]` = `peig(x,p)
[S,w]` = `peig(...,nfft)
[S,f]` = `peig(x,p,nfft,fs)
[S,f]` = `peig(...,'`corr`')
[S,f]` = `peig(x,p,nfft,fs,nwin,noverlap)
[...]` = `peig(...,'`range`')
[...,v,e]` = `peig(...)
peig(...)
```

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:

• 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 min, the smallest estimated eigenvalue of the signal's correlation matrix. Eigenvalues below the threshold min`*p(2)` are assigned to the noise subspace. In this case, `p(1)` specifies the maximum dimension of the signal subspace.

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.

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

```[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.

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

```[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.

 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.

 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.

 Note    The arguments `nwin` and `noverlap` are ignored when you include the string `'``corr``'` in the 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:

• `'whole'`: Compute the pseudospectrum over the frequency range `[0,fs)`. This is the default for determining the frequency range for complex-valued `x`.
• If you specify `fs` as the empty vector, `[]`, the frequency range is `[0,1)`.
• If you don't specify `fs`, the frequency range is [0, 2).
• `'half'`: Compute the pseudospectrum over the frequency ranges specified for real `x`. This is the default for determining the frequency range for real-valued `x`.
 Note    You can put the string arguments `'``range``'` or `'``corr``'` anywhere in the input argument list after `p`.

```[...,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 j 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.

 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:

• ```randn('state',1); n=0:99;
s=exp(i*pi/2*n)+2*exp(i*pi/4*n)+exp(i*pi/3*n)+randn(1,100);
X=corrmtx(s,12,'mod');
peig(X,3,'whole')        % Uses the default NFFT of 256.

```

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.

`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.

 pcov periodogram