Abstract
In these lectures we state the spectrum estimation problem (Section 5) and develop periodogram (Section 6) and maximum entropy (Section 7) methods used for its solution. The two methods are really general approaches for dealing with the problem, and most spectrum estimation algorithms fall into one or the other of these categories. Our presentation is decidedly mathematical, but our bibliography contains references for many of the specific engineering and statistical algorithms derived from the techniques outlined herein.
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References
R. Blackman and J.W. Tukey, “The measurement of power spectra,” Dover, New York (1959).
D. Brillinger, “Time series,” expanded edition, Holden- Day, San Francisco, (1981).
D. Brillinger and J.W. Tukey “Spectrum estimation and system identification relying on a Fourier transform”.
D. Childers, ed., “Modern Spectrum Analysis,” IEEE (1978) (many of the “classical” papers).
R. Dudley, Sample functions of the Gaussian process,Ann. of Prob. 1: 66–103 (1973).
H. Dym and I. Gohberg, Extensions of matrix valued functions with rational polynomial inverses,Int.Eq.and Operator Theory2: 503–528 (1979).
H. Dym and I. Gohberg, On an extension problem, generalized Fourier analysis, and an entropy formula,Int.Eq.and Operator Theory3; 143–215 (1980).
H. Dym and H. McKean, “Gaussian processes, function theory, and the inverse spectral problem,” Academic Press, New York, (1976).
P. Fougere, Spontaneous line splitting in multi-channel maximum entropy power spectra, in: “ASSP Workshop on Spectral Estimation,” S.S. Haykin, ed., McMaster University, Hamilton (August, 1981 ).
I. Gelfand and N. Vilenkin, “Generalized functions, Volume 4,” Academic Press, New York (1964).
U. Gretiander, ed., “The Cramer Volume,” John Wiley & Sons, New York, (1959) (esp. the article by Tukey).
B. Harris, ed., “Spectral analysis of time series,” John Wiley & Sons, New York, (1966–1967) (esp. the introduction and the article by Tukey).
F. Harris, On the use of windows for harmonic analysis, Proc IEEE 66: 51–83 (1978).
IEEE Proceedings Volume 70 (Sept. 1982).
E.A. Robinson, A historical perspective of spectrum estimation,Proc.IEEE70: 885–907 (1982).
E.T. Jaynes, On the rationale of maximum entropy methods,Proc. IEEE 70: 939–953 (1982).
J. McClellan, Multidimensional spectral estimation,Proc.IEEE70: 1029–1039 (1982).
D.J. Thomson, Spectrum estimation and harmonic analysis,Proc. IEEE 70: 1055–1096 (1982).
E.T. Jaynes, On the rationale of maximum entropy methods, in: “First ASSP Workshop on Spectral Estimation,” S.S. Haykin, ed., McMaster University, Hamilton (August, 1981 ).
J. Lamperti, “Probability” Benjamin Publishing, New York, (1966).
J. Lamperti, “Stochastic processes” Springer-Verlag, New York, (1977).
H. Landau, H. Pollak, D. Slepian, Prolate spherical wave functions…,Bell System Tech. J. 40: (1961) 43–64 (1961), 40: 65–84 (1961), 41: 1295–1336 (1962), 43: 3009–3058 (1964), 57 1371–1430 (1978).
A. Oppenheim, ed., “Applications of Digital Signal Processing,” Prentice-Hall, Inc., Englewood Cliffs, New Jersey, (1978).
A. Oppenheim and R. Schafer, Digital Signal Processing, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, (1975).
A. Papoulis, “Signal Analysis,” McGraw-Hill, New York, (1977).
A. Papoulis, Maximum entropy and spectral estimation, a review,IEEE Trans ASSP29 (6): 1176–1186 (1981).
A. Papoulis, “Probability, random variables, and stochastic processes,” (there may be a new edition soon).
E. Parzen, Autoregressive spectral estimation,…, in: “First ASSP Workshop on Spectral Estimation,” SS. Haykin, ed., McMaster University, Hamilton (August, 1981 ).
M. Priestley, “Spectral analysis and time series,” Volumes 1 and 2, Academic Press, New York, (1981).
D. Slepian, On bandwidth,Proc.IEFE64: 292–300 (1976).
L. Schwartz, “Théorie des distributions,” Hermann, Paris, (1966).
D.E. Vakman, “Sophisticated signals and the uncertainty principle in radar,” Springer Verlag, Berlin (1968).
N. Wiener, “Extrapolation, interpolation, and smoothing of stationary time series, with engineering application,” MIT Press, Cambridge, Mass. (1949).
N. Wiener, “Cybernetics” 2nd ed., MIT Press (1961).
N. Wiener, “Generalized harmonic analysis and Tauberian Theorems,” MIT Press, Cambridge, Mass. (1966). Also in Wiener’s “Oeuvres, Vol. II,” MIT Press.
N. WienerOeuvres, Volume III, MIT Press.
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© 1985 Plenum Press, New York
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Benedetto, J.J. (1985). Some Mathematical Methods for Spectrum Estimation. In: Price, J.F. (eds) Fourier Techniques and Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-2525-3_5
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DOI: https://doi.org/10.1007/978-1-4613-2525-3_5
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