The Eigen (MUSIC, EV) procedure offers two high-performance frequency estimation algorithms. These algorithms use Eigendecomposition to generate noise subspace frequency estimators. Theses procedures are perhaps the most accurate and robust of all the spectral procedures within FlexPro for estimating harmonic frequencies.
In general, algorithms classed as frequency estimators do not give meaningful quantitative information regarding the power associated with signal components. The only quantitative values that can be safely be inferred are component count and frequencies.
Algorithm
The MUSIC (Multiple Signal Classification) and EV (EigenVector) algorithms are widely used and robust frequency estimators. They are primarily for extracting sinusoidal harmonic frequencies.
The procedures use a robust SVD of a forward-backward prediction (FB) data matrix. The only difference between the MUSIC and the EV (EigenVector) algorithms is a weighting function for the noise subspace eigenvectors.
Since frequency refinement to full machine precision is automatic, there is no need for FlexPro to include those variants of the MUSIC or EigVec algorithms that offer full precision frequency estimation.
Spectrum Type
There are three output formats. The Eigen option directly plots the estimator. The dB plotting option uses a decibel scale. The dB normalized optionnormalizes the highest peak to 0 dB. Note that the dB normalized option is of little value unless the adaptive spectrum is used, and even then the peaks will yield only an approximate ordering of power.
Parameters - Model Order, Signal Subspace
Eigendecomposition of signals in the absence of noise is a simple matter. Two eigenmodes are needed to capture any oscillatory component, harmonic or anharmonic. Four eigenmodes are needed to fully describe two oscillatory components. For noise-free data, the minimum order needed for signal-based procedures, such as AR spectra, will be twice the number of oscillatory components comprising the spectrum. In the case of the MUSIC and EigenVector Eigenanlysis algorithms, only the noise eigenmodes are used to generate the spectrum. As such, these procedures are not recommended for simulated noise-free data. For real world data containing some level of noise, it is necessary to select a model order sufficiently high to model the noise in the data stream.
The Signal Subspace selection is a further essential part of these eigenanalysis procedures. To process oscillatory signals, you must enter a value that is twice the number of components. If three spectral components are known to exist, the signal subspace must be set to 6.
Spectrum
The spectrum can be generated directly, or with some performance benefits using an FFT. The Full range option locks the 0-0.5 Nyquist range. It also causes the spectrum to be generated via an FFT if the Adaptive spacing option is disabled. When the option Full range is on, only the total spectral count (Number of Frequencies) can be specified. Unlike the FFT options, which specify the length of the transform, this option specifies the total frequency count in the output spectrum. An FFT of 16384 points produces 8193 spectral frequencies from 0 to 0.5 normalized frequency. For the Full range option, it is best to select one of the fast numbers from the drop-down list, as these result in a power of 2 as the FFT length. The eigenanalysis procedures use the Best Exact N FFT algorithm.
When the option Full range is off, you can select the desired Start and End Frequency as well as the count of spectral frequencies (Number of Frequencies) in this band. It is thus possible to generate a detailed spectrum only in the region of specific interest. This option uses a direct computation for the spectrum and any size can be used.
The Adaptive spacing option always uses a direct computation for the spectrum. An ARMA spectral estimator can consist of astonishingly sharp peaks and nulls, especially in comparison with traditional FFT spectra. For uniform sampling, a size of 8193 uniformly spaced points is not unreasonable in order to get good representation of the peaks and nulls. However, even with a large number of points, some of the power of a peak can still be lost. As an alternative, FlexPro can use a Runge-Kutta procedure to integrate the spectrum adaptively, saving the points used in the computation of the integral. This results in an adaptive frequency set containing frequencies concentrated near the peaks.
The MUSIC and EV spectra usually contain the sharpest peaks of all the spectral algorithms. When harmonic components are present, it is almost impossible to get a good spectral representation with a uniform sampling of frequencies, no matter how large the spectrum. To get good representation of the peaks, the Adaptive spacing option can be used. A Runge-Kutta procedure is used to integrate the spectrum adaptively, saving the points used in the computation of the integral. This results in an adaptive frequency set containing frequencies concentrated near the peaks.
The estimated frequency peaks are refined to full machine precision regardless of the spectrum type generated. The Adaptive spacing option is needed only if you want the more accurate graphical rendering. When true harmonics result in the spectral peaks being nearly impulse functions, the computations for the Adaptive spectrum can become intensive.
Options - Toggle Labels (Analysis Wizard Only)
You can view the Y and/or X values of the peaks in the spectrum by pressing Toggle Labels. Initial frequency estimates are based upon the local maxima in an 8193 count full-range spectrum. These peaks are then adjusted with high accuracy using a one-dimensional optimization process. This precise determination of the frequencies is possible because the estimators are continuous functions of the frequency. Inferring Power
Because of the sharp nature of the peaks, it is generally not possible to infer power from the peak maxima. However, when the Adaptive spectrum is plotted, approximate powers are indicated by the peaks.
Options - Set/Clear Reference, Toggle Labels (Analysis Wizard Only)
This function lets you compare various spectral procedures and settings. You can view a copy of the currently displayed spectrum in the lower pane by pressing Set Reference. Next, you can adjust additional settings that affect the display in the upper pane. With Clear Reference you can remove the copy and the time signal will appear again.