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It contains three different spectral estimation techniques: Power Density Spectra, Maximum Likelihood Method, and Maximum Entropy Method.These are all indirect methods, because they use a sample correlation function, rather than the data itself, to estimate the spectral content.

This is a SAC subprocess. A subprocess is like a small program within the main SAC program. You start a subprocess by typing its name (SPE in this case.) You can terminate it and return to the main program using the QUITSUB command. You can also terminate SAC from within a subprocess using the QUIT command. While within a subprocess, you can execute any command belonging to that subprocess plus a limited number of main SAC commands. A list of the allowed main SAC commands is included in this document.

This manual contains the documentation on each of the SPE commands. They are in alphabetical order and are also summarized below. The format and notation is the same as the SAC Command Reference Manual. A list of the allowed main SAC commands is also shown below. You can also use all of the SAC macro features in this subprocess.

SPE Commands

COR	Computes the correlation function.
MEM	Calculates the spectral estimate using Maximum Entropy Method.
MLM	Calculates the spectral estimate using Maximum Likelihood Method
PDS	Calculates the spectral estimate using Power Density Spectra Method.
PLOTCOR	Plots the correlation function.
PLOTPE	Plots the RMS prediction error function.
PLOTSPE	Plots the spectral estimate.
QUITSUB	Terminates a SAC subprocess.
READ	Reads data from a SAC data file into memory.
WRITECOR	Writes a SAC file containing the correlation function.
WRITESPE	Writes a SAC file containing the spectral estimate.

AXES	BEGINDEVICES	BEGINFRAME
BEGINWINDOW	BORDER	COLOR
COMCOR	COPYHDR	DATAGEN
ECHO	ENDDEVICES	ENDFRAME
ERASE	EVALUATE	FLOOR
GETBB	GRID	GTEXT
HELP	INSTALLMACRO	LINE
LINLIN	LINLOG	LOGLAB
LOGLIN	LOGLOG	MACRO
MESSAGE	PAUSE	PLABEL
PLOTC	QDP	QUIT
READALPHA	READBBF	REPORT
SETBB	SETDATADIR	SETDEVICE
SETMACRO	SGF	SYMBOL
SYNTAX	SYSTEMCOMMAND	TICKS
TITLE	TSIZE	VSPACE
WAIT	WINDOW	WRITEBBF
XDIV	XFUDGE	XFULL
XGRID	XLABEL	XLIM
XLIN	XLOG	XVPORT
YDIV	YFUDGE	YFULL
YGRID	YLABEL	YLIM
YLIN	YLOG	YVPORT

Overview

SPE is a spectrum estimation package intended primarily for use with stationary random processes. It implements three different indirect spectral estimators. They are called indirect, because they do not estimate the spectrum directly from the data, but from a sample correlation function that is computed from the data. The choice of indirect methods is a matter of taste, since direct spectral estimation techniques are also available. The correlation function itself is a useful quantity. You may wish to examine it in the course of performing spectral estimation tasks. The choice of indirect techniques is supported by ``Spectral Analysis and Its Application,'' by Jenkins and Watts, a respected reference on the subject of spectrum estimation. The type of spectrum estimated by SPE is properly described as the power density spectrum, with the spectrum defined in the frequency domain. Thus, the estimated power delivered by the random process in some band of frequencies is the integral of the spectral power density estimate over that band of frequencies.

The Estimators

The user has a choice of three spectral estimators: Power Density Spectra ( PDS), Maximum Likelihood Method ( MLM), and Maximum Entropy Method ( MEM).

PDS

The PDS estimator is quite simple: the sample correlation function is multiplied by a correlation window, then the result is transformed with an FFT to obtain the spectral estimate. The user again has a choice of the window type and the size of the window. The above mentioned book by Jenkins and Watts could be considered as the detailed documentation for the PDS technique.

MLM

The MLM estimator generates a spectral estimate which is the power output of a bank of narrow band pass filters which have been optimized to reject out-of-band power. The result is a smoothed, parametric estimate of the power density spectrum. The user can choose the number of parameters. Documentation for this method can be found in the paper by Richard Lacoss in the IEEE book ``Modern Spectrum Analysis'' by Donald Childers.

MEM

The MEM estimator is another parametric method, which uses a prediction error filter to whiten the data. The resulting spectral estimate is proportional to the inverse of the filter's power frequency response. The user is free to choose the order of the prediction error filter. Documentation for this method can be found in the review paper on linear prediction by John Makhoul in ``Modern Spectrum Analysis.'' The formal name of the actual method implemented is the Yule-Walker method.

Initialization

This progression begins when the SPE command is executed. Default values for the various SPE parameters are defined at this time. The data file may have been read in using the READ command before entering SPE or at any time while within SPE. Space for the above mentioned auxiliary functions is created whenever a new file is read.

Correlation

The correlation function is then computed, using the COR command. It may be saved as a SAC data file using the WRITECOR command and later read back into SPE using the READCOR command. This is more efficient than recomputing the correlation each time, especially if the data file is very long. At this point, you may wish to examine the correlation function using the PLOTCOR command. You may also wish to examine the prediction error function using the PLOTPT command if you are going to use the MEM method.

Estimation

Now you are ready to select one of the three spectral estimation techniques using the PDS, MLM, or MEM commands. Each technique has its own options. You may now examine the resulting spectrum using the PLOTSPE command. There are several different scaling options available. You can also save the spectral estimate as a SAC data file using the WRITESPE command.

User Control

SPE affords the user some control over the details of estimation process. For some, with experience in estimating spectra, this is highly desirable. Defaults are provided for those who do not wish to become involved in the details of the theory. The user has a choice of data window type, size, and the number of windows used when estimating the correlation function. Generally these parameters control the resolution of the estimate, and the amount of reduction of variance desired in the final estimate. In addition, prewhitening of the data may be specified as part of the process of estimating the correlation function. Prewhitening often has the effect of mitigating a severe ``window bias'' that can occur in spectral estimates having a high dynamic range. The warping of the spectrum that occurs with prewhitening is compensated for in the final result. In this implementation, low-order prediction error filters are used for prewhitening.

Diagnostics

In addition to the spectrum, several diagnostic functions can be calculated and plotted. The prediction error can be plotted as a function of order. This plot can be used to select a good size for the prediction error filter used in the MEM method. Since much is known about the performance of the PDS estimator, more diagnostic information is available for this method in SPE. The 90`` confidence limits can be estimated theoretically, as can the frequency resolution of the estimate. Both of these quantities can be indicated on a PDS spectral plot.

Termination

At this point you have several options: you can select a different spectral estimate technique, read in a different correlation function, read in a different data file, terminate the subprocess using the QUITSUB command, or terminate SAC using the QUIT command.

Spectral Estimation Suprocess Manual

Introduction