## MUSIC and Eigenvector Analysis Methods

The `pmusic`

and `peig`

functions provide two related
spectral analysis methods:

Both of these methods are frequency estimator techniques based on eigenanalysis of the autocorrelation matrix. This type of spectral analysis categorizes the information in a correlation or data matrix, assigning information to either a signal subspace or a noise subspace.

### Eigenanalysis Overview

Consider a number of complex sinusoids embedded in white noise. You can write the
autocorrelation matrix *R* for this system as the sum of the signal
autocorrelation matrix (*S*) and the noise autocorrelation matrix
(*W*): *R* = *S* + *W*. There is a close relationship between the eigenvectors of the signal
autocorrelation matrix and the signal and noise subspaces. The eigenvectors *v*
of *S* span the same signal subspace as the signal vectors. If the system
contains *M* complex sinusoids and the order of the autocorrelation matrix is
*p*, eigenvectors
*v*_{M+1} through
*v*_{p+1} span the noise subspace of
the autocorrelation matrix.

### Frequency Estimator Functions

To generate their frequency estimates, eigenanalysis methods calculate functions of the vectors in the signal and noise subspaces. Both the MUSIC and EV techniques choose a function that goes to infinity (denominator goes to zero) at one of the sinusoidal frequencies in the input signal. Using digital technology, the resulting estimate has sharp peaks at the frequencies of interest; this means that there might not be infinity values in the vectors.

The MUSIC estimate is given by the formula

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

where the *v _{k}* are the eigenvectors of the noise subspace and

*e*(

*f*) is a vector of complex sinusoids:

$$e(f)={[\begin{array}{lllll}1\hfill & {e}^{j2\pi f}\hfill & {e}^{j4\pi f}\hfill & \dots \hfill & {e}^{j2(M-1)\pi f}\hfill \end{array}]}^{T}.$$

Here *v* represents the eigenvectors of the input signal's correlation
matrix; *v _{k }* is the

*k*th eigenvector.

*H*is the conjugate transpose operator. The eigenvectors used in the sum correspond to the smallest eigenvalues and span the noise subspace (

*p*is the size of the signal subspace).

The expression *v _{k}^{H}*

*e*(

*f*) is equivalent to a Fourier transform (the vector

*e*(

*f*) consists of complex exponentials). This form is useful for numeric computation because the FFT can be computed for each

*v*and then the squared magnitudes can be summed.

_{k}The EV method weights the summation by the eigenvalues of the correlation matrix:

$${\widehat{P}}_{\text{EV}}(f)=\frac{1}{{\displaystyle \sum _{k=p+1}^{M}\frac{1}{{\lambda}_{k}}{\left|{v}_{k}^{H}e(f)\right|}^{2}}}.$$

The `pmusic`

and `peig`

functions interpret their first input either as a signal matrix or as a
correlation matrix (if the `'corr'`

input flag is set). In the former case, the
singular value decomposition of the signal matrix is used to determine the signal and noise
subspaces. In the latter case, the eigenvalue decomposition of the correlation matrix is used to
determine the signal and noise subspaces.