# EigenAnalysis Algorithms

23.08.2021

Eigenanalysis frequency estimation algorithms are used in the Eigen (MUSIC, EV) spectral procedure. These algorithms offer true Spectral estimators.

## Eigendecomposition

The procedures begin by generating an Eigendecomposition from a data (trajectory) matrix using SVD (singular value decomposition). The data matrices can be forward prediction based (Fwd) or forward-backward prediction based (FB). The FB procedures are generally more accurate when estimating the frequencies of sinusoids.

## MUSIC and EigenVector Algorithms

Once an  is complete, there are a host of frequency estimators that can be constructed using the eigenvectors and eigenvalues.

The MUSIC (Multiple Signal Classification) and EV (EigenVector) algorithms are widely used and robust frequency estimators. These estimators function on the principle that the noise subspace eigenvectors should be orthogonal to the signal vectors. The MUSIC and EV frequency estimators are continuous reciprocal functions of frequency that have sums of products of the noise eigenvectors in the denominator. The signal eigenvectors are nowhere used. The peaks arise because the denominator will approach zero at sinusoidal frequencies, resulting in exceedingly sharp spectral peaks.

The only difference between the algorithms is a weighting function. The EV algorithm weights each noise subspace eigenvector by the inverse of its eigenvalue whereas the MUSIC procedure uses uniform weighting. The inverse eigenvalue weighting may represent a slightly more robust algorithm, although the differences between the algorithms are generally small. It is more important that an effective signal-noise threshold be determined.

The peak frequencies are identified using a two step procedure. First, an 8193 count full-range spectrum is generated using a 16384 size FFT. A 1E-15 fractional error minimization of the estimator is then made for each of the spectral peaks. The spectral peak count will be half the signal subspace value. The spectral peak count will be half the signal subspace value.

The traditional implementation of these algorithms identifies frequencies strictly from the local maxima in the output spectrum. Variants of the MUSIC and Eigenvector algorithms also exist, their primary feature being some form of frequency refinement. In FlexPro, the  and  algorithms automatically offer this full precision frequency estimation.

FlexPro's implementation of the MUSIC and EV procedures are extensions of the algorithms presented by Marple (p. 377).

## References

A good coverage of eigenanalysis spectral algorithms can be found in the following references:

S. Lawrence Marple, Jr., "Digital Spectral Analysis with Applications", Prentice-Hall, 1987, p.361-378.

Steven M. Kay, "Modern Spectral Estimation", Prentice Hall, 1988, p.429-434.