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Spectral Analysis Course
by Steve Kay
"Spectral Estimation is at the heart of almost all modern-day signal processing systems. Current systems typically employ the FFT (Fast Fourier Transform), which is inherently limited in resolution by the data record length. With the advent of high speed digital computing it is now possible to implement in practice more modern approaches to spectral estimation."*
"Important techniques of spectral estimation including linear prediction, AutoRegressive (AR), Maximum Entropy, AutoRegressive Moving Average (ARMA), Maximum Likelihood Method (MLM), and eigen-analysis methods."*
Areas of spectral interest include:
"1. The Spectral Estimation Problem. Review of random process theory. Spectral estimation and filtering. Applications to sonar, speech, data processing, etc. Parametric vs. nonparametric approaches. Concept of resolution. Comparison of estimators.
2. Classical (Fourier Methods). Periodogram. Averaged periodogram. Bias-variance tradeoffs. Computation of periodogram. Blackman-Tukey. Correlation estimation. Resolution comparisons.
3. Time Series Modeling. Time series model definitions. AR/MA/ARMA. Yule-Walker equations. Levinson algorithm. Model fitting of empirical data. Effect of observation noise.
4. Statistical Estimation Review. Maximum likelihood estimation (MLE). Cramer-Rao lower bound (CRLB).
5. Autoregressive Spectral Estimation. Autocorrelation, covariance, modified covariance (forward/backward), Burg, and recursive maximum likelihood methods. Properties of estimators. Linear prediction/maximum entropy/autoregressive modeling relationships. Reflection coefficients and lattice filters. Autocorrelation matching. Model order selection - Akaike and MDL. Observation noise effects. Performance for sinusoids in noise.
6. Autoregressive Moving Average Spectral Estimation. Akaike MLE and iterative implementations. Modified Yule-Walker. Least squares modified Yule-Walker. Input/output identification approaches (two stage least squares). Choosing the best method for your application.
7. Moving Average Spectral Estimation. Durbin's method and performance.
8. Capon's Method (MLM). Filtering interpretation. Comparison to periodogram. Resolution vs. conventional and autoregressive estimators.
9. Sinusoidal Frequency Estimation. Signal and noise subspaces. Eigenanalysis of covariance matrix. MLE. Periodogram. Principal components/SVD approaches. MUSIC. Pisarenko methods. Iterative filtering method. Statistical performance vs. CRLB for all methods.
10. Empirical Spectral Estimation. Examples using the Modern Interactive Spectral Analysis software package. Hands-on experience by student analysis of mystery data sets. Evaluation of student-supplied data."*
Personal comments:
At Lockheed Missiles & Space Co. we were able to find signals down some 50+ dB by using Kay's AutoCorr method. This was great for our work. Electronic 'Black-Box' manufactures assumed the standard 30 dB and below was noise. Wrong. And these manufactures became our number one problem. Manufactures need to move their definition of noise to below 100 dB or so.
I highly recommend SM Kay's textbook "Modern Spectral Estimation: Theory and Application" by Steven Kay (Prentice-Hall, 1988), which includes Fortran subroutines for all the basic methods, as well as MATLAB subroutines, suggested problems and solutions.
Windowing FFT does -not- do well for a spectral estimation. Back in the 1980s while at Memorex, Corp. we tried Van der Maas, Hanning, Hamming and many other windows that were thought to improve one's FFT but did not.
Sincerely,
Phil Brubaker
Mathematician
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* From newsgroup messages titled "spectral analysis course" by Steve Kay before April 1998.
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