Spectral Estimator Analysis Object and Template – AR (AutoRegressive) Spectral Estimator (Spectral Analysis Option)
The AR (AutoRegressive) procedure offers accurate frequency estimation with short data records.
Algorithm
The AR algorithm list offers six procedures. Unlike the various FFT algorithms, each of the AR methods is likely to produce at least slightly different results. The Data Matrix FB SVD algorithm is the most robust and accurate of the methods, although it is also the slowest. If performance is an issue with large data set sizes, the Normal Equations FB SVD algorithm may be a viable alternative. Although FlexPro offers the traditional non-SVD AR algorithms, an (singular value decomposition) least-squares method is recommended.
In an AR spectral procedure, peaks occur exactly at frequencies corresponding to roots in the AR polynomial. For non-SVD algorithms, all roots are treated as relevant peaks. To pare the spectrum to only the signal components of interest, an SVD algorithm must be used. Apart from longer processing times, there are no disadvantages to using an SVD procedure, and the advantages are numerous when extracting harmonics is the primary aim of the modeling.
There is generally no need to fit an AR model non-linearly. The Data Matrix algorithms, both with and without SVD, usually produce stable estimates where all roots lie on or within the unit circle. This is not assured, however. There is a way to insure the least-squares minimum where all roots are constrained to lie within the unit circle. You can use the ARMA (AutoRegressive Moving Average) Spectral Estimator option with the Non-Linear Spectral Factorization or Non-Linear Spectral Factorization SVD procedures and set the moving average (MA) order to 0. This type of iterative fit will be considerably slower.
Spectrum Type
For ARMA spectra, there are only four spectral formats. The PSD can reflect the three different power normalizations, Integral=TISA, Time-Integral Squared Amplitude, Integral=MSA, Mean Squared Amplitude, Integral=SSA, Sum Squared Amplitude, or it can be expressed in dB. There is no normalized dB scale where the highest peak is set to 0 dB; sharp peaks are likely to be poorly characterized for height and they will not linearly reflect the power of spectral components. In general, ARMA spectra should be regarded primarily as frequency estimators.
Parameters
Fitting AR models to harmonic signals in the absence of noise is a simple matter. A model order of two is needed to fully describe one sinusoid. Similarly, an order of four is needed to fully model two components. For noise-free data, the minimum order needed will be twice the number of sinusoids comprising the spectrum. The Data Matrix procedures will achieve a perfect fit in this instance, exactly resolving the frequencies; the other algorithms will not.
An AR model can have both real and complex roots. The real roots, usually at -0.5, 0, or 0.5 normalized frequencies, are not processed since they represent singularities at the bounds. The complex roots produce finite spectral power, and for real data, the positive and negative frequency roots mirror one another. Both sides of the spectrum must be taken into account. This is why the minimum order needed must be twice the number of component sinusoids.
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 at least some of the noise. To achieve a reasonable signal-noise separation with SVD, it is necessary to fit a high enough order so that the primary singular vectors (eigenvectors) span only signal space.
With the routines, the order of the fit ceases to be critical. A tolerably high order is needed, one that is sufficient to produce an effective partitioning of the signal and noise. The quality of the fit for the noise components is not a consideration, since these eigenvectors are discarded in the SVD processing. All that is needed is to specify the signal space.
The Signal Subspace selection is enabled only when an SVD procedure is being used. To accommodate both positive and negative frequencies, you must enter a value that is twice the number of expected components. If three spectral components are known to exist, the signal subspace must be set to 6.
A full signal space SVD fit, one where the signal space equals the model order, produces the same results as the non-SVD algorithms.
Spectrum
An AR spectrum can be generated directly from the AR coefficients, 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 will be fastest if the values in the Number of Frequencies dropdown box are used, since these produce fast FFTs. The AR 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 AR spectral estimator can consist of astonishingly sharp peaks, 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. Even with a large , it is possible to miss some fraction of the power of a peak. 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.
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. The peak frequencies are determined directly from the AR roots of the model, and are usually computed to at least 1E-12 precision. Unlike the FFT, there is no need for a local maxima detection procedure and there is no specification of peak count. For the non-SVD procedures, each valid frequency derived from a root is treated as a valid spectral peak. Thus the spectral peak count can be as high as half the model order. For the SVD procedures, the spectral peak count should be half the signal subspace value.
Unlike the FFT, it is not possible to compare power by looking at the magnitude of the AR spectral peaks. The areas under the peaks, however, are indicative of estimated power.
Options - Set/Clear Reference (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.
Considerations
An AR model of sufficient order and especially using one of the Data Matrix algorithms is an excellent frequency estimator, since the frequencies depend only on the roots of the fitted polynomial. It is the power within an AR spectrum that will be far less certain.
An AR model emphasizes the narrowband spectral content and de-emphasizes wideband trends and noise. AR models can achieve far greater separation of nearly adjacent spectral components than is possible with FFT methods, again supporting this strength in isolating sinusoidal frequencies. On the other hand, this weakness in power characterization may make it impossible for the AR procedures to find low amplitude components that FFT methods clearly reveal.
In general, an AR method is not a good candidate for high dynamic range signal detection. The prominent signals may be captured by most of the estimation capability of the algorithm, and signals just above the noise threshold can easily be lost.