Prony Spectrum

The Prony Spectrum option in the Spectral menu or the Spectral toolbar offers the fitting of a sum of complex exponentials to uniformly sampled data. Using this method, it is possible to fit exponentially damped sines, undamped sines, and damped exponentials. The primary utility of this procedure rests in fitting damped sines or the damped exponentials that occur in multicomponent exponential decays.

The complex exponential model fit is purely deterministic. Unlike the parametric models in the AR, MA, and ARMA procedures, there is no driving white noise within the model. Although the complex exponential model fit is the primary function of this procedure, an energy spectral density plot offers a graphical frequency domain representation.

Components

The Prony model's components are implemented in AutoSignal as follows:

Sine : Y=Ampl*sin(2*Pi*Freq*X+Phase)

Sine, Exp Damped : Y=Ampl*exp(-k*X)*sin(2*Pi*Freq*X+Phase)

Exponential : Y=Ampl*exp(-k*x)

Algorithm

AutoSignal offers three Damped Algorithms for fitting exponentially damped sinusoids. The SVD procedures are recommended for assessing component count and in removing the influence of noise so as to estimate as accurately as possible the component frequencies and damping factors. The Dmp Svd is the recommended algorithm for fitting damped sinusoids or multicomponent exponential decays. The normal equations analog, the Dmp Svd NE algorithm, may offer close to the same estimation accuracy if the Dmp Svd option is too slow with large model orders and data sets. The Damped procedure is fast, but it is also very sensitive to noise and should be used only when very minimal noise is present.

There are also three Undamped Algorithms for fitting sinusoids via a modification of the least-squares Prony method. Similar recommendations apply. The Undmp Svd algorithm should be the first choice, the normal equations Undmp Svd NE algorithm reserved for large model orders and data sets where performance is a factor. The Undamped procedure's speed comes at the price of considerable noise sensitivity.

In terms of fitting undamped sinusoids, there is a strong similarity between the linear sinusoidal fit based on the Data Svd FB algorithm in the AR (AutoRegressive) Spectrum procedure and the Undmp Svd Prony fit. In fact, for a given order and signal subspace, the estimated frequencies are identical. The Prony Undmp Svd procedure involves mapping the signal roots to the unit circle (to insure zero damping coefficients), and then performing the complex amplitude least-squares fit using these modified complex exponential coefficients. The AR Data Svd FB algorithm's linear sinusoidal fit directly performs a least-squares estimate and generally yields slightly more accurate amplitudes and phases.

Model Order Selection

Since the Prony frequencies derive from an AR fit, the model order selection considerations are identical. 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 oscillatory component. Similarly an order of four is needed to fully model two oscillatorycomponents. For real data, positive and negative frequency roots mirror one another. AutoSignal reports only the positive frequencies, but both sides of the spectrum must be taken into account. This is why the minimum order needed must be twice the number of oscillatory components.

Note that the damped exponentials which occur at or near 0 frequency are non-oscillatory and do not exist as pairs mirrored across positive and negative frequencies. Unlike an oscillatory component, each damped exponential adds only one to the minimum order needed. A signal space of one thus describes a single exponential decay.

In practice, there is usually some level of noise present in the data and a higher order model is needed to characterize the signal components. With the SVD 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 accurately determine the signal space threshold. For most data sets, this is far easier than determining an optimum AR order. Note that SVD removes the influence of noise only for the first step in the Prony procedure, the determination of the frequencies and damping factors.

AutoSignal does not blindly include all frequencies associated with roots. If those frequencies fall at or near the Nyquist bound, they are automatically filtered out. Similarly, unless the Allow Real Exp checkbox is checked, the damped exponentials which occur at or near 0 frequency are removed. This box must be checked for fitting multicomponent exponential decay data.

While the SVD signal-noise determination is probably the best way to determine the damped sinusoid count, an alternative exists within the Prony procedure. The Plot Roots option for the damped algorithms displays two sets of complex roots. The forward prediction roots, those used for estimating the frequencies and damping coefficients, will use the ‘+’ symbol. The backward prediction roots are also shown using the ‘o’ symbol. With exponentially-damped sinusoids, the roots for the signal components should be inside the unit circle for forward prediction, and outside for backward prediction. Roots for the noise components are usually inside the unit circle for both. Thus the count of backward prediction roots outside the unit circle represents one estimate of the signal space. This count will be twice the number of expected components.

In a Prony procedure, the model order is used not only for the AR estimation portion of the algorithm, but also for the signal thresholding. A full order linear least-squares fit is made for an initial estimate of the amplitudes and phases. Once this is done, the model is reduced to the most significant components as specified by the signal space value. This is done by retaining the most significant exponential parameters and refitting a reduced component count model.

Signal Subspace Selection

Since the Prony method fits a series of complex exponentials in a deterministic model, it is possible to truncate the signal at a specified number of components simply by discarding the least significant exponentials in the fit and to redo the complex amplitude least-squares fit with fewer components. The final Prony model thus consists of the most significant components as determined by the value in set in the Signal Subspace field. This thresholding occurs for all algorithms, regardless of whether or not SVD is used. To keep this thresholding consistent, the signal space is defined for the non-SVD algorithms in the same way as it is for SVD signal thresholding. To retain 3 damped or undamped sinusoids, for example, the signal subspace must be set to 6.

For the undamped case, this ordering will be by decreasing amplitude. For damped exponentials, the ordering is by energy spectral density at the peak frequency. Once the signal components have been pared, the complex amplitude least-squares fit is repeated using only the most significant components. This reduced component fit will be more stable and the energy spectrum smoother with lesser likelihood of spectral peaks arising from noise.

Since a damped sinusoid requires two eigenmodes of signal space to be represented, and an exponential decay requires only a single eigenmode, AutoSignal seeks to automatically set the component count based upon the type of signal elements present. If a component's frequency is less than 1e-8*Nyquist, it is treated as a real exponential. If the Allow Real Exp option is checked, the component is assigned one eigenmode of signal space. Otherwise, the component is not included in the Prony fit. If a component's frequency is non-zero, it is assumed to be a damped or undamped sinusoid, and two eigenmodes of signal space are assigned.

When modeling continuously decreasing multicomponent radioactive decay data, use one unit of signal space for each component present. For three different components that have appreciably different half-lives, the signal space should be set to 3 rather than 6.

SVD Signal-Noise Separation

The Graphically Select Signal and Noise Sub-Spaces signal selection is enabled only when an SVD procedure is being used. It also uses the Signal Subspace value. You can enter this value numerically if you know with certainty the number of spectral components present in the data. To accommodate both positive and negative frequencies, you must enter a value that is twice the number of oscillatory components. If three oscillatory spectral components are known to exist, the signal subspace must be set to 6. Note that damped exponentials require only one unit of signal space. 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 AR 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.

A full signal space SVD fit, one where the signal space equals the model order, produces the same results as the non-SVD algorithms.

Damped Exponentials (First Order Decays)

Spectrum

A Prony energy spectral density (ESD) plot can be generated directly from the model coefficients. The Full Range option locks the 0-0.5 Nyquist range. 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, the Prony option specifies the total frequency count in the output spectrum. The spectrum computation is linear with n.

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.

The Adaptive option uses 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.

Plot

The energy spectrum is always given in dB (decibels). The dB 2-sided option plots a two-sided energy spectrum that will produce a sharper spectral response, although this is more fitting persistent signals. Transient signals are probably better suited to the one-sided spectrum and the dB 1-sided format. A discrete bar chart is used for the undamped case.

For a Prony analysis, the spectra are mainly informational. There is no need to extract parametric components from a Prony spectrum. The parametric damped sinusoids are computed first, and a spectral energy representation follows.

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%).

The SVD procedures will have the greatest noise resistance. This noise option is also helpful in ascertaining at what level the SVD procedures 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 Prony spectral analysis report. The report optionally includes a listing of the complex coefficients and a Prony fit summary.

Non-Linear Optimization

The Non-Linear Optimization offers the means to refine the parameter estimates given in the Prony 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.

Although this optimization is present in all of the spectral options, only this Prony procedure offers starting estimates for the damping coefficients. Although damped exponentials can be fit from any of the spectral procedures, the starting estimates for the exponential rate terms are set to zero (pure sinusoids). Although these fits usually converge successfully, this Prony procedure does offer superior starting estimates.

Multiple real exponentials, such as occur with a sum of first order decays of different rates, can be highly correlated. This can create difficulties in non-linear fitting as rates and amplitudes can shift drastically in response to very small levels of noise. The separate decay rate and amplitude determinations via the Prony procedure may yield more accurate parameters than the simultaneous refinement of both amplitudes and rates using non-linear optimization. In other words, when fitting highly correlated sums of exponential decays, a global least-squares minimum may not be desirable. If you choose to non-linearly fit multiple real exponentials, a robust (maximum likelihood) minimization may offer better results.

Mixed Models with Non-Linear Optimization

Note that real exponentials and undamped sinusoids are subsets of the complex exponentials (damped sinusoids) modeled by the Prony procedure. A complex exponential reduces to a real exponential only when a real root results in a zero frequency. A complex exponential reduces to an undamped sinusoid only when a complex root falls on the unit circle. These reductions occur as part of the fitting procedure and are data dependent. They cannot be specified.

When the non-linear optimization is invoked from the Prony procedure, mixed models are automatically fitted. If a component in the Prony fit produces a frequency less than 1e-8*Nyquist, it is treated as an exponential decay component in the non-linear fitting. If a component's damping factor is such that the sinusoid decays less than 1% in amplitude across the sampling range, it is treated as an undamped sinusoid in the non-linear fitting. When a mixed model results from a Prony fit, the non-linear optimization must be used for valid confidence limits on the fitted parameters.

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.

View Residuals

The residuals from the Prony fit can be inspected to see if they are normally distributed. The SNP plot is particularly useful.

Plot Roots

The Plot Roots option is only available for the damped procedures since the roots for the undamped procedures will all lie exactly on the unit circle. The roots of the AR model used in the first portion of the Prony method can be inspected graphically. The forward prediction roots used with the damped Prony algorithms are shown with the '+', the standard AR pole symbol. Although reverse prediction roots are not used within the Prony computations, they are useful for determining the number of exponentially-damped sinousoids present. These reverse prediction roots are shown with a 'o' symbol. The number of reverse prediction roots that lie outside the unit circle is one indicator of signal subspace.

Toggle Popup Information Window

Because the Prony algorithms are multistep fitting procedures, a host of statistics are available to describe the Prony model fit. The Toggle Popup Information Window is used to show or hide this information. The r² goodness of fit index may be particularly useful, since spectra that visually appear to be well fitted may be the result of a poor deterministic fit. A smooth Prony spectrum is not an indicator of an accurate model fit.

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.

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.