Spectral moment matching in the maximum entropy spectral analysis method

The formula for the spectral moments is expanded in a series consisting of autocorrelation terms. By using the autocorrelation extrapolation inherent in the maximum entropy method (MEM) for spectral analysis, it is found that good estimates of the moments can be found from a very low order autoregressive model. The motivation for this investigation came from results obtained during spectral analysis of ocean wave data [I]. In the characterization of ocean waves the spectral moments up to the fourth order are used to estimate the significant wave height, the average wave period, and the number of crests. However, the results reported here are not limited to ocean wave characterization but are valid for all applications where spectral moments are needed. The maximum entrcpy spectral estimation method is first briefly reviewed, showing the autocorrelation extrapolation implied by the method. Then the spectral moments are expanded in a series consisting of autocorrelation terms. The series converges very quickly and, in an example, we look at how the formula converges with increasing model order. In this method for spectral analysis one desires a spectral estimate which is consistent with the M + 1 available values of the autocorrelation function. At the same time nothing is assumed about the unknown values of the autocorrelation function. That is, the estimate should correspond to the most random time series consistent with the available values of the autocorrelation function, or the underlying time series shall have maximum entropy. It turns out that the solution is a spectral estimate of the all-pole type [2]: where PM is a power term, a, for rn = 1 , 2 , . ., M are a set of autoregressive coefficients, and A t is the sampling period. To find the unknown coefficients one must solve a matrix equation where M + 1 values of the autocorrelation function R ( m ) are the input: In this correspondence the autocorrelation function is assumed to be known. Estimation of this quantity is, however, a very important problem which has led to the development of several parameter estimation algorithms. Manuscript received January 25, 1982; revised May 26, 1982. The author is with the Electronics Research Laboratory. The University of Trondheim, The Norwegian Institute of Technology, 0. S. Bragstads Plass 6, N-7034 Trondheim-NTH, Norway. 001 8-9448/83/0300-03 1 1$01.00 0 1 983 IEEE 312 IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 1~-29, NO. 2, MARCH 1983 The maximum entropy principle implies that the spectral in terms of the autocorrelation function via estimate is consistent with an infinite series of autocorrelation values [2], with the spectral estimate being the Fourier transform of the autocorrelation series: m S ( f ) = At z R(1) exp (-j2alfAt). (3) The coefficients are I = w The lag6 above order M are automatically extended, and the cZ = 2/::51$lncos(2nkx) dx. extension can be found by augmenting the autocorrelation matrix of (2). By inserting zeros for the autoregressive coefficients a,,, n > M the augmented equation is [3] = ~ A ~ ~ ~ / ~ ~ ' x ~ c o s ( ~ ~ ~ t x ) dx. (11) The last line is the desired recursive extrapolation of the autocorrelation function: M R ( M + k ) = 'z R ( M + k n)a, , k >-0. (5) n = l The nth-order spectral moment of a one-sided continuous spectral density is With increasing n the spectral moment pays more and more attention to the high-frequency part of the spectrum. The moments therefore give information about the distribution of power along the frequency axis. The N-point discrete-time approximation to (6) with integration up to the half sampling frequency is This expression is valid for real time series (with symmetrical spectra) when the step size (l/NAt) is small relative to the variations in the spectrum. The expression can be written as an infinite series by expanding the (I/NAt)" term in a cosine series The cosine expansion is symmetric around I = 0 and periodic with period N. Within the range of summation in (8), however, the e:pansion is equal to (I/NAi)" as desired. The autocorrelation function can also be expressed as a cosine-series by using the discrete-time Fourier transform inverse of (3): Combining (8) and (9) the spectral moments can be expressed This is a standard integral [4, p. 4361, giving as a final result for the coefficients The coefficients for k > 0 for the first five spectral moments are The coefficients decay as k with increasing number of terms. The first terms in the series are therefore the most important. With M + 1 values of the autocorrelation function estimated and the rest maximum-entropy extrapolated ( 5 ) , the expression for IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-29, NO. 2, MARCH 1983 3 13