EigenAnalysis Algorithms

Eigenanalysis frequency estimation algorithms are used in the EigenAnalysis Spectrum option. These algorithms offer true high-resolution frequency estimation. 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.

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. For damped sinusoids, the Fwd procedures will typically be more accurate. These matrices are the same as those used in the Data AR procedures and in the Prony option.

MUSIC and EigenVector Algorithms

Once an eigendecomposition is complete, there are a host of frequency estimators that can be constructed using the eigenvectors and eigenvalues. The MUSIC (Multiple Signal Classification) and EigVec (Eigenvector) algorithms are two of the more widely used frequency estimators. These estimators function on the principle that the noise subspace eigenvectors should be orthogonal to the signal vectors. The MUSIC and EigVec 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 EigVec algorithms weight each noise subspace eigenvector by the inverse of its eigenvalue whereas the MUSIC procedures use 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 frequencies are identified using a two step procedure. First, an 8193 count full-range spectrum is generated using a 16384 size FFT. Then, for each peak detected, a one-dimensional minimization follows that resolves the frequency to 1E-15 fractional precision. This precise estimation of frequencies is possible because the estimators are defined as continuous functions of frequency. 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 AutoSignal, the MUSIC and Eigenvector algorithms automatically offer this full precision frequency estimation. The Numeric Summary will always report these refined frequencies, regardless of how the spectrum is generated.

AutoSignal's implementation of the MUSIC and Eigenvector procedures follows the algorithm presented by Marple (p.377).