EigenAnalysis Spectrum

The EigenAnalysis Spectrum option in the Spectral menu or the Spectral toolbar offers four high-performance frequency estimation algorithms. These algorithms are use eigendecomposition methods to generate noise subspace frequency estimators. Theses procedures are perhaps the most accurate and robust of all the spectral procedures within AutoSignal 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 FB (Multiple Signal Classification) and EigVec FB (Eigenvector) algorithms are widely used and robust frequency estimators. They are primarily for extracting sinusoidal harmonic frequencies. For estimating the frequencies of damped sinusoids, the MUSIC Fwd and EigVec Fwd algorithms are offered.

The procedures use a robust SVD eigendecomposition of either a forward-backward prediction (FB) or forward prediction (Fwd) data matrix. The only difference between the MUSIC and EigVec algorithms is a weighting function for the noise subspace eigenvectors.

Since frequency refinement to full machine precision is automatic, there is no need for AutoSignal to include those variants of the MUSIC or EigVec algorithms that offer full precision frequency estimation.

Model Order Selection

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 will be twice the number of oscillatory components comprising the spectrum.

In practice, there is usually some level of noise present in the data and a higher order model is needed. The additional coefficients go primarily into modeling the oscillations within the random noise. To achieve a reasonable signal-noise separation, it is necessary to fit a high enough order so that the primary singular vectors (eigenvectors) span only signal space.

An order should be chosen large enough to provide sufficient signal-noise separation. Unlike the other procedures, however, the MUSIC and EigVec algorithms use the noise portion of the eigendecomposition rather than the signal modes. It is possible, therefore, to see enhanced performance and perhaps a reduced variance in using higher orders. Although very sharp peaks will dramatically slow the Adaptive spectrum procedure, it is recommended for capturing the sharp spectral features.

Signal Subspace Selection

The Graphically Select Signal and Noise Sub-Spaces signal selection is an essential part of these eigenanalysis procedures. You can enter the signal space value numerically if you know with certainty the number of spectral components present in the data. 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. Even when the spectral component count is known, you should use this Graphically Select Signal and Noise Sub-Spaces option to insure that a high enough order is being used to achieve the desired signal-noise separation.

When there is sufficient signal-noise separation in the eigenmodes, the singular value plot reveals one or more sharp transitions between the signal subspace and the noise subspace floor. The last eigenmode before the long sloping noise floor represents the last element of signal space. Assuming a high-enough model order is used, this signal-noise space separation does not become difficult until the noise level approaches that of the signal. At this point, the sharp characteristic transition disappears. An earlier diminishing of this transition occurs when the noise is red.

Unlike the signal-noise separation in SVD-based parameter fitting where only the signal eigenmodes are used, the MUSIC and EigVec algorithms use only the noise portion of the eigendecomposition.

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 option is disabled. When the Full Range option is on, only the total spectral count (n) 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 will be fastest if the values in the drop down box for n are used, since these produce power of 2 FFTs. The eigenanalysis procedures use the Best Exact n FFT procedure.

If the Full Range option is off, you can select the desired start and end frequencies as well as the count of spectral frequencies (n) 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.

MUSIC and EigVec 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 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 frequencies reported in the Numeric Summary are refined to full machine precision regardless of the spectrum type generated. The Adaptive spectrum 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.

Plot

There are three plot formats. The Spec option directly plots the estimator. The dB(Spec) plotting option uses a decibel scale. The dB0(Spec) option plots the estimator on a normalized decibel scale (the largest peak is normalized to 0.0 dB). Note that the dB0(Spec) option is of little value unless the adaptive spectrum is used, and even then the peaks will yield only an approximate ordering of power. To get the power of harmonic components, the linear least-squares sinusoidal fit in the Numeric Summary or the Non-Linear Optimization should be used.

The peak labels consist of frequencies only. They are toggled on and off with the Display Maxima button.

Initial frequency estimates are based upon the local maxima in an 8193 count full-range spectrum. 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.

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.

In the example that follows, there are 12 spectral peaks. Six are alternating reference 0dB peaks, and the others are at -10dB, -20dB, -30dB, -40dB, -50dB, and -60dB in relative power. An adaptive eigenanlysis plot does give an approximate power ordering:

Contrast this with the very accurate power rendering of a good windowed FFT:

Note that while the FFT more accurately renders power, the eigenanalysis spectrum more accurately estimates the component frequencies. The linear least-squares sinusoidal fits within AutoSignal's spectral routines use only the frequencies from the spectrum to estimate the amplitudes and phases, and thus the power of components. For this reason, the eigenanalysis procedures will offer amongst the most accurate of the suboptimal sinusoidal linear fits, and it will furnish some of the best initial frequency estimates for the Non-Linear Optimization.

Add Noise

It may be instructive to see where a given procedure starts to break down as a consequence of temporarily adding white observation noise to the input data. The zero noise level is S/N=300dB (fractional noise=1E-15, the IEEE double precision threshold for addition). At this value, no noise is added to the data. A value of 280 would add noise in the 14th significant figure, 260 in the 13th, 240 in the 12th and so on. This option assumes that the current data set is entirely signal, and adds noise accordingly. Typical test values are 40dB(1% noise), 20(10%), 10(31.6%), 6(50.1%), 3(70.8%), and 0(100%).

This noise option is helpful in ascertaining at what level the eigendecomposition can no longer evidence the eigenmode signal to noise transition.

List

The List Data option lists the index, frequency, and the spectral quantity currently plotted. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the frequency and the spectral quantity currently plotted to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors). You can generally find a Paste As option in most applications if you want specific control over the format imported.

Save

The Save Data to Disk option writes the frequency and and the spectral quantity currently plotted to a supported file format. These formats include ASCII, Excel 97, Excel 95, Lotus WK3, Lotus WK1, SPSS, or Systat.

Production Facility

The AutoSignal Automation facility allows unattended processing of large numbers of data sets. The data sets can be consolidated in an Excel file or acquired using a DLL. The numeric summaries and graphs can be exported to an MS Word RTF file, while the extended data summaries or the current spectra can be exported to an Excel 95 or Excel 97 file.

Numeric Summary

The Numeric Summary offers a full eigenanalysis report. The report optionally includes a frequency analysis, a linear sinusoidal least-squares fit summary, a full eigenmode summary, and a singular values table.

Non-Linear Optimization

The Non-Linear Optimization offers the means to refine the parameter estimates given in the linear sinusoidal fit that is reported in the Numeric Summary. Constrained least-squares and robust (maximum likelihood) non-linear fitting is available with either sinusoid or damped sinusoid models.

Rich-Text Format Export

The Export Numeric Summary and Graph to RTF File option writes the numeric summary and spectral plot to an RTF file. The numeric portion of the file is based upon the current settings in the Numeric Summary option. The text data will be written to portrait orientation pages. The graph uses the current settings and size of the spectral plot, and is inserted as a Windows Metafile. The graph always uses a landscape orientation. Beyond a certain size, the graph utilizes a full landscape page.

Local Options

A local option changes the data set for the duration of the current procedure only. The main data table is not altered. AutoSignal offers four local options in most of the spectral procedures.

Section the data to isolate specific regions for processing.

Detrend for removing mean or subtracting a least-squares trend model.

Fourier Filtration for isolating spectral components by frequency.

Eigendecomposition Filtration for isolating spectral components by signal strength. Note that this option is largely redundant in this procedure.

The Reset button restores the data to its state when first entering the procedure. Note that if you implement sequential local procedures, all of the revisions are discarded upon reset. If an Automation Session is in progress, the Reset button can be used to terminate the automated processing.