A neural net algorithm for multidimensional maximum entropy spectrum estimation

Neural Networks, Vol. 4, pp. 619-626, 1991 Printed in the USA, All rights reserved.

0893-6080/91 $3.1X) + .00 Copyright ~' 1991 Pergamon Press plc

ORIGINAL CONTRIBUTION

A Neural Net Algorithm for Multidimensional Maximum Entropy Spectrum Estimation X I N H U A Z H U A N G , I YUNXIN Z H A O , 2 AND THOMAS S. H U A N G 3 ~Universityof Missouri-Columbia,:PanasonicTechnologiesInc., and 3Universityof Illinois (Received 23 February 1990; revised and accepted 6 March 19911

Abstract--As well known, the maximum entropy (ME) method of spectrum estimation has been shown to provide excellent spectral resolution for both one-dimensional (1D ) and multidimensional (roD) signals of short data record length. The maximum entropy spectrum is defined as a solution to a constrained optimization problem, where the constraint being the "'correlation-matching" property. The computation of the 1D ME spectrum is efficient because it can be obtained from the linear equations of autoregressive (AR ) signal modeling. In the mD case, however, the computation of the ME spectrum appears to require the use of nonlinear optimization techniques. The optimization problem has been attacked by many researchers from the point of views of both the primal problem and the dual problem, where the dual problem has the advantages of being finite dimensional and thus attracts most of the efforts. In this paper, we present a neural net algorithm to solve the general mD ME spectral estimation. The problem is formulated as a primal constrained optimization problem and is reduced to solve a well-defined initial value problem of differential equations of Lyapunov type. This initial value problem comprises the basis of our neural net algorithm. Experiments on simulated data showed convincingly that the algorithm did provide mD spectral estimates with high resolution and correlation matching property as well as computation efficiency. The authors believe that the proposed mD spectral estimator will find a broad scope of applications where the high-resolution spectrum is of importance.

Keywords--Maximum entropy, mD spectrum estimation, High resolution, Correlation matching, Primal constrained optimization. I. I N T R O D U C T I O N

spectral density p ( f ) is defined as the Fourier transform of the autocorrelation function r(~),

As well known, the power spectrum is the most widely used concept in the analysis of one-dimensional (1D) or multidimensional (mD) random signals. It is essential in system theory, optics, spectroscopy, oil exploration, earthquake analysis, radar, speech, radio astronomy, and microwave antenna arrays, to name a few. In such applications, a common problem is the estimation of the spectrum from given data. The spectrum analysis of a random process is, in concept, not obtained directly from the process u(x) itself, but is based on knowledge of the autocorrelation function of a zero mean process, r(~) = E{u(x + ~)u*(x)},

P(f)

f r(~)exp(-j2~(f, ~))dd

(2)

where f, ~ E Era, (f, ~) denotes the inner product of the two mD vectors, i.e., (f, ~) = E fi~,. The estimation of the power spectral density of a uniformly sampled process is usually based on procedures employing the fast Fourier transform (FFT). This method is computationally efficient and produces reasonable results for a large class of signal processes. The FFT method does, however, have some limitations in performance. The most serious limitation is frequency resolution, and another is the sidelobe effect, These two limitations of the FFT method become prominent while analyzing short data records. Short data records are commonly seen in reality since many processes are observed brief in duration or have slowly time-varying spectra which may be considered constant only for short record lengths. In an attempt to reduce the inherent limitations of the FFT method, a number of alternate spectral

(1)

where x, ~ each is an mD vector, u:Em ~ El, and the * represents the conjugate operation. The power

Requests for reprints should be sent to Xinhua Zhuang, Department of Electrical and Computer Engineering, Universityof Missouri-Columbia, Columbia, MO 65211. 619

620

estimators have been developed. Among those estimators, the maximum entropy (ME) method of spectral estimation has been shown to provide excellent spectral resolution for both 1D and mD signals of short data record length (Burg, 1967; Malik & Lira, 1982; McClellan, 1982). The maximum entropy spectrum is defined as a solution to a constrained optimization problem, where the constraint being the "correlation matching" property (Burg 1967). The computation of the 1D ME spectrum is efficient because it can be obtained from the linear equations of A R signal modeling (Burg, 1967; Makhoul, 1981; Papolis, 1981). In the mD case, however. the computation of the ME spectrum appears to require the use of nonlinear optimization techniques (McClellan, 1982). The optimization problem has been attacked by many researchers from the point of views of both the primal problem and the dual problem (Lang & McClellan, 1981; Lang & McClellan, 1982; Lira & Malik, 1981; McClellan & Lang, 1982; Newman, 1979; Newman, 1981; Wernecke & D'Addario, 1977), where the dual problem has the advantage of being finite dimensional and thus attracts most of the efforts. In the special case of isotropic random fields, the 2D ME spectrum has been shown to have a linear solution (Tewfik, Levy, & Willsky, 1988). Although the true ME spectrum was shown to be the inverse of a positive multivariable polynomial, it may not be factorable as the magnitude squared of a finite order polynomial. Thus. the true ME spectral estimate is, in the mD case (m > 1), more general than the autoregressive (AR) spectral estimate. Furthermore, the true mD ME spectrum satisfies the correlation matching property, whereas the mD AR spectral estimate usually not (Malik, 1981). In this paper, we present a neural net algorithm to solve the general mD ME spectral estimates. The problem is formulated as a primal constrained optimization problem and is reduced to solving an initial value problem of differential equations of Lyapunov type (Hopfield & Tank, 1986). This initial value problem comprises the basis of our neural net algorithm. The behavior of the neural net is governed by an energy function that measures the degree of constraint satisfaction, i.e., the correlation matching property. Along the solution path, the energy function is monotonically decreasing and the true mD ME spectrum is quickly reached. In other words, the neural net evolves towards that specific global state which renders the maximal spectral entropy while satisfying the Correlation matching property. Experiments on simulated data showed that the algorithm did provide mD spectral estimators with high resolution and correlation matching property as well as computation efficiency. This paper is organized as follows: In Section 2,

X. Zhuang,

Y. Zhuo. and 7: 5. Hzutng

the general mD ME spectral estimation problem is formulated in a way slightly different from existing ones; Section 3, the problem is reduced into solving a well-defined initial value problem of differential equations; Section 4, we present tl~e neural net a1: gorithm; Section 5, we show numerical results; the final section is the conclusion. 2. PROBLEM FORMULATION

The mD ME spectral estimation problem may be stated briefly in the following way: Given a finite number of autocorrelation measurements, r, = r(~,), i = O, 1 . . . . . M, which specify a stationary and homogeneous zero mean random field u ( x ) , estimate its power spectrum p ( f ) by maximizing the spectral entropy f log p ( f ) dr, subject to the constraint of following correlation condition: ?'i = ~ p ( f ) e x p ( j 2 ~ ( g , , ,

f))dr,

i=

ik ! . . . . .

~1:

C~)

To be practical, we assume that there exists a bounded domain D C E , , so that the power spectral density p ( f ) vanishes or at least is negligible outside of D and is positive in D. Let E ( p ) denote the constraint satisfaction function defined as the sum of magnitude squared correlation matching errors. namely E ( p ) = ~i ~_~ r, ---

,P(f)exp(Bn(~'

f)) df

(4)

then the mD ME spectral estimation problem can be formulated as follows,

t

(5)

[subjecL tO E ( p ) :--: i! The discretized version of eqn (5) is given by N

[subject to E ( p , . . . . .

p,,'~ = O.

where Pk = P ( f * ) with j'k, k = 1. . . . . N, being uniformly distributed over the domain D . E ( p l , . . . . pN) represents the corresponding discretized approximation to E ( p ) , namely E(p, . . . . .

p~) = .~iir " ap~ ~,

where r -- (r,,. r, . . . . . A,.k = ~

ru)',

A .... (A,.~).,~.~

eXp(j2n(¢~, fk)).

tDI - the m - D volume of D, P = (P, . . . . .

P;O'.

(7)

Maximum Entropy Spectrum Estimation

621

As easily seen, introducing the Lagrange multiplier t, the problem formulated in eqn (6) is equivalent to the following one:

{

max { ~

logp, - rE(p)},

subject to E(p)

Assume ~<, = 0, namely, the total spectral power is given: (9)

Then, the eqn (8) is equivalent to the following one max H(p; t, u) subject to E(p)

~ log I)~ - IL ~ p, - tE(p), O,

dVh(p; t, 1~) _ O, t > 0 dt VH(p; 0,/t) 0,

(13)

(8)

O.

ID~ E P* = r,,. N

eqn (12) can also be defined by

where the first equation leads to the first equation in (11) and the second one gives the initial value in eqn (11). Introducing - I t E p, in eqn (10) makes it easier to find out the initial value p(0; It) by solving VH(p; 0,/z) = 0. Third, we show that, along the path p(t; jL)(t >_ 0), E ( p ( t ; IL)) approaches zero as t ~ z. In fact, we can prove even more, that is,

(10)

which is the formulation we will work on in the remaining part of the paper. In the next section, we show our method of reducing eqn (10) into solving an initial value problem of differential equations so as to get around the direct attack on the optimization problem.

To simplify the proof, we assume that the set U = { p : E ( p ) = 0 with each p , being positive} is nonempty. This assumption is quite reasonable. Take an arbitrary point p(' ~ U, by the definition of p(t; l~), for each t -> 0, H(p(t; /L); t, i~) >~ H(F'; t, l~)

= ~ logp~'

3. ME SPECTRAL ESTIMATION FORMULATED AS AN INITIAL VALUE PROBLEM

/t ~/'~' since E(F' ) = 0

= const, independent of t. Thus,

To solve eqn (10), we first show that for each fixed t -> 0 and for each fixed It > 0, the function H ( p ; t, /1) has a unique maximal point denoted as p(t; l~) with each component being positive and finite. The proof is simple. In fact, when t -> 0 and/~ > 0, the function H ( p ; t, lL) tends to the minus infinity as max p , --, ~ or min p , ~ 0, as easily verified. It means that the maximal point of the function H ( p ; t, IL) must be internal if it exists. Since the Hessian matrix is negative, i.e., ~72H = -diag[1/p~ . . . . . l/p~.] t A ' A < O, we conclude that the function H ( p ; t, l~) does have a unique maximal point internally, i.e., each of its components being positive and finite. Second, we show that, with/~ > 0 fixed, the path defined by {p(t;/t) : t -> 0} can be uniquely determined by the following initial value problem of differential equations:

~] log pk - I~ ~ Pk ~ tE(p(t: jr)) + const. const.

> -~.

(15)

Since the left side of eqn (15) tends to - ~ as p~ --~ 0 or ~, we conclude that there must exist two finite and positive bounds 0 < b < B < :~ independent of t so that, for each t >- 0, b <-pk(t;lt) <- B , k

= 1. . . . .

N.

(16)

Using the first inequality in eqn (15) and inequality in eqn (16), we obtain

~(p(t, ltl) <- 71 { ~ log p~(t; ~t) - IL × ~ pk(t: IL) -- const. } 1

= rE,

--~ - {N log B - /tNb - const.}.

t > 0

t

(1 1) p(0;l~) = 1 (1 . . . . . [/

1)i x

As a matter of fact, the unique maximal point p(t: jr) being internal implies that the path {p(t; It):t > O} coincides with the solution curve defined by the following stationary equations, 7H(p; t,/~) = 0, t -> 0.

(12)

By the knowledge of calculus, the curve defined by

which straightforwardly leads to eqn (14). Finally, we show that the path p(t;/L)(t -> 0) defined by max H ( p ; t , / l ) ( t -> 0) or by the initial value problem, i.e., eqn (11), approaches the maximum entropy spectral estimate p* defined by eqn (6) as t --~ ac.

To prove it, we need the existence and uniqueness of the mD maximum entropy (ME) spectral estimation formulated as eqn (6). As easily seen, eqn (6), which is characterized as maximizing the spectral

622

X, Zhuang, E Zha,), and 1~ S. Huang

entropy E log p~, over the closure U = { p : E ( p ) = 0 with each Pk >- 0} can also be characterized as maximizingthe spectral entropy over the intersection of U with V = {p: E log Pk >- E log p0}, where p" U as assumed before. Because of eqn (9), i.e., the total spectral power being constant, the closure U is b_ounded, namely, each component of any point in U is upper bounded. Using this fact, we can verify that each c__omponent of any point in the intersection set, i.e., U N V, is also positively lower bounded: Pk ->exp -~ logpl ~ - ~ logp~ byp E

4. THE NEURAL NET ALGORITHM FOR mD ME SPECTRAL ESTIMATION In this section, the mD ME spectral estimator or algorithm for solving the initial value problem, i.e., eqn (11), is developed. There arc a variety of computing schemes available to solve cqn (I1).

>--exp{~', logpl' - an upper bounded term / byp~U -> ~ > 0.

In summary, we proved the existence and uniqueness of the mD ME spectral estimation in a constructive way, namely, the true mD ME spectrum can be quickly reached by tracing along the path p(t: , ) (t ~ 0). The initial value problen:, Le.. eqn (!1), thus comprises the basis for the ml) ME spectral estimator to be developed in the next section

(17)

Thus, eqn (6) can finally be characterized as maximizing the spectral entropy over the compact set Uo = { p : E ( p ) = 0 with each Pk >- ~}, each component of any point in Uo is both upper bounded and positively lower bounded. Because of continuity and concavity, the spectral entropy function, i.e., E log p~, has a unique maximal point denoted by p* in U~. Thus, we proved the existence and uniqueness of the mD ME spectral estimation. Now, we proceed to prove that p(t; lz) ~ p* as t --~ :¢.

(18)

From the definition ofp(t; p), for each t - 0, it holds that

4.1. Energy Function Before starting, we show that the initial value problem. i.e.. eqn (11), essentially defines a neural net. For this. we need only to show that the constraint satisfaction function E ( p ) plays a role of an energy function which governs the evolution of the neural net (Hopfield & Tank, 1986). Using eqn (11), we can easily calculate the derivative o f E ( p ( t ; p ) ) w i t h respective to the time variable t as follows. dE(p(t; p)) dp(t; p) dt - (VE(p(t: /~))) dt

- (VEyIV2H) WEb

H(p*; t, p) ~ H(p(t; p); t, p) -< ~ log pk(t; /~) - # ~ pk(t; p). Because of E ( p * ) = 0, it follows that ZI°gp~' - p Z p ~ ' -< E log pk(t; p) - p ~', pk(t; p).

(19)

By eqn (16), i.e., b <- pk(t; #) - B, k = l , . . . , N, we are sure that, when t ~ ~, the path p(t; p)(t ->0) possesses a nonempty clustering point set. Let/~ be any clustering point. Because of eqn (14), E(/3) is equal to zero. Now both E ( p * ) and E(/5) being equal to zero imply that p : = ~, p , .

Since W-H is negative definite, the derivative dE(p(t; ~))/dt could never be positive, In fact, there are only two possibilities: either the derivative is atways negative or it is identically equal to zero. The latter case occurs if and only if the whole path shrinks into a single point which surely ~s the initial point l/p(1 . . . . . 1)i×U. The proof is omitted. Now, it becomes apparent that. along the path p(t; p)(t > 0), the constraint satisfaction function E is strictly monotonically decreasing to zero unless the initial point solves the mD ME spectral estimation. In the latter case. we have

(2o)

Combining eqn (20) with eqn (19) and taking the limit operation at the right side of eqn (19), we obtain

|

p(t; E

Z log Pt <- Z log/~k,

t~-)

z ) -~ -~ (2 . . . . . /1

(1 . . . . .

t)',~.

i);..,,, =

0.

(21)

which won't be possible unless/5 = p* since both p* and/5 satisfy the correlation matching condition and p* was supposed to be the unique mD ME spectrum over U.

In summary, we conclude that the constraint satisfaction function E is qualified as an energy function and thus the initial value eqn ( H ) defines a neural net.

623

M a x i m u m Entropy Spectrum Estimation

to obtain pk+l from pk. In that case, we have

4.2. The Neural Net Algorithm

p,+' =

Recall that

1-~- [b, - ~ Wii

H(p(t; /t)) : ~', log Pk - P ~ Pk - t E ( p ) ,

i=1 .....

E ( p ) = ½Nr - ApN:,

,ol]

p,, :

-~ ( t . . . . .

L-- 2 A

- A'r.

Thus, from eqn (24) it follows, k -~ 0:

diag

.....

where tk+~ -> tk. When all (tk+~ - t~) are small enough, p~ will approximate p ( t ~ ; p ) very well. Equation (23) can be rewritten as follows, p/, : _1 (1 . . . . .

l)i~,

-27. . . . . P~

= (2t~ - t k ~ , ) g ~ + + (t~.,-

.....

~

tk)l

tk /

+-\ tk

)l.

- 1'

(24) The initial valuesp °, gO g~ for eqn (28) are as follows,

- (tk+, - t~)A'r.

p,':

For each k, this is a large linear system of equations. Fortunately, the coefficient matrix VZH(pk; tk, p) is negative definite and symmetric. The Gauss-Seidel iterative scheme (Franklin, 1968) is used to solve eqn (24) efficiently. Let W represent the matrix VZH(pk; tk, p) and b the vector (t~+~ - 2 t k ) A ' A p k - ( 1 / p ~ . . . . . 1/p~)' - (tk+l - & ) A ' r , Furthermore, let x be the (k + 1)th estimate pk+~ ofp(?+~; p). For m -- 1, we have the following Gauss-Seidel iterative equations for solving W x = b:

l

pk*

(28)

+

k -> 0: V-'H(p~; t~, p)p~+~ = (t~., - 2t~)A'Ap ~

r'

+ tkL

1);~,,,,

(23)

I y"

(27)

l=Ar, gk = L p q k >- O.

k -> 0: V2H(p'; tk, p)(p~+' - pk) = (t,+, - tk)VE(pk),

-

(26)

N.

{

tA'A,

From eqn (11) we have the following iterative equations, 1

W,p~]

t >i

In the sequel we let

VZH = _diag [ 1 . . . . . VE - A'Ap

W,p~*' - ~

i
i[glg " =

1

~ (1 . . . . . L p ~,

1);.,,,

(29)

Lpl.

Thus, eqn (29), which involves matrix vector calculations, is used to find gO, g~, and recursion formula eqn (28) involving only vector calculations is used for gk, k --- 2. Finally, we summarize this subsection by providing the following iterative algorithm for mD ME spectral estimation:

p~

±[b,= W,, L -

~

(25)

/', ]

W.x;"

~ ,i = 1.....

N.

It is proved (Franklin, 1968) that the convergence and the limit point both are independent of the choice o f x °. Thus x m ~ pk+~ as m --> zc. We choose x 0 = pk since p~+~ should be close to pk when tk+~ - tk is small. This choice of x ° greatly speeds up the convergence of (25). Experiments indicate that usually only one iteration for each k is enough

Step 1. Set k = 0, tk = 0, and compute p~, gO, g~ by eqn (29). Step 2. If E ( p k) < ~, then stop. Here ~, is a prespecified small positive number. Step 3. If k is large enough, then stop. Step 4. Solve the first part of eqn (28) for pk+~ by using Gauss-Seidel iterative scheme eqn (25). Step 5. If k -> 1, then compute gk+~ by the second part of eqn (28). Step 6. Set k = k + 1, tk = tk ~ + Gt > 0 with At>0. Step 7. Go to Step 2.

.V Zhuang. }: Z,w
624 4 . 3 Initial V a l u e S e l e c t i o n

Although any positive value of it will work for our purpose, we select the value I~ which minimizes E(1/IL(1 . . . . . t) [. 4,) which is positive. 5. E X P E R I M E N T S

To evaluate the performance of the proposed mD ME spectral estimator, a number of experiments were conducted on simulated 1D or 2D signals. In each, the assumed true spectrum is given and autocorrelations at a number of lags, which are equally spaced both horizontally and vertically, are calculated from the Fourier transformation of the true spectrum; then the maximum entropy spectrum is

!

\

i

M=5 i

A i!

calculated

calculated i assumed

M = 10

FIGURE 2. The assumed and calculated s ~ r a correspondi n g t o N = 30; M = 5, 1 0 , , = 01:5, / ~ : - 0 . 0 6 2 5 ;

calculated from those autocorrelations by using the proposed mD ME spectral estimator. There are two parameters to be controlled. They are the sample size N and the number of autocorrelations used in the experiments, i.e., M. A . I D case The assumed spectrum is defined by the function pl f') as in eqn (30): 4

calculated ,

assumed

P(f) =

(1 -

4/,~) '#

do~

)<

', - I;

(a' -

1)(f . 4~)

if I = /~

f ~::',

_(1 - o0( f - }) + ~

if{ _~ f < :~ + t?

P (41 - 2)

(4l~- l)

FIGURE 1. The assumed and calculated spectra corresponding to N = 20, M = 5, 10, ~ = 0.5,/3 = 0.055.

if '~ + [ / ~ .f <" .~.

Maximum Entropy Spectrum Estimation

625

where 0 < a < 1 and 0 < fl < ¼ are used to shape the notch. For each N = 20, 30, estimate the maximum entropy spectrum against M = 5, 10. The step size At = 1. The results after 20 iterations are shown in Figures 1 and 2, respectively. The average energy E~ N is less than 0.01. B. 2D case: The assumed spectrum is defined by P(.f~, f2) as in eqn (3l) and shown in Figure 3. Ai

P(.f,, .L~) = 1+

(.f, - B , F + (f~ Di

c,):

A:

+ 1 +

(.L

-

B2) ~ +

(L,

-

C_,) -~'

(31)

D~

where A l = A2 = 5.0, B1 = B2 = 0.25, Ci = 0.25, (7., = -(t.25, and DI = D, = 0.005. For e a c h N = 7 × 13, 10 × 19, estimate the 2D maximum entropy spectrum against M = 3 × 3, 5 × 5. The step size At = 2. The results after 15 iterations are shown in Figs. 4 and 5, respectively. The average energy E / N is less than 0.01. The overall results indicate a very good performance in estimation of maximum entropy spectra even in cases where there are only a few autocorrelation measurements available. As seen, the proposed mD ME spectral estimator provides the mD spectral estimates with high-resolution and correlation-matching property. The latter is theoretically guaranteed by eqn (14) and experimentally observed.

FIGURE 4. The calculated spectra N=7x 13, M = 3 x 3 , 5 x 5.

corresponding

to

6. C O N C L U S I O N

FIGURE 3. The assumed 2-D spectrum.

In this paper, we present a general algorithm for mD maximum entropy spectral estimation with computation efficiency. The proposed estimator is actually equivalent to a neural net that is governed by an energy function measuring the degree of constraint satisfaction, i.e., correlation matching property. It remains interesting to see the use of this algorithm on larger scales of real data and extend the algorithm to handle noisy measurements. We are currently constructing an mD robust spectral estimator within the general maximum entropy framework. Along the direction, an interesting work (Hanson Jr., 1988) has been reported for 2D robust spectral estimation within the model-based framework.

626

X. Z h u a n g , E Z h a o , ~md [~ ,5, Huang

REFERENCES Burg, J. (1967). Maximum entropy spectral a~udysis, Orlando, FI ,: Meeting Soc. Explor. Geophys. Franklin, J. (19681 Matrix theory. Englewuod Cliffs: Prentice,. Hall. Hanson Jr.. R.. & Chellappa. R. (1988). l~-~dimensional robust spectrum estimation. IEEE Tran,~ on ,-~:~ustics Speech, Signal Processing, ASSP-36. No. Hopfield. J.. & Tank. D. (19861. computing,, w~th neural czrcuits. a model. Science. 233, 625-632 Lang, S.. & McClellan. J. (19811. MEM spectral cstimauon for sensor arrays, Proceedings c)f the International Conjerence ~.,pp Digital Signal Processing (pp. 383 -390) Florence, ltaiv Lang, S., & McClellan. J. (19821. Multidime nsional MEM spectral estimation. IEEE Transactions on Acoustics, Speech, :md Signal Processing ASSP-30, 88(1-,q.~7 Lira J.. & Malik. N (1981 ). A new algorithm tot two-dimensional maximum entropy power spectrum estimation. IEEE Transactions on Acoustics, Speech_ and Stgnal Processing ASSP-29. 401-413 Makhoul, J, (198l). Linear prediction: ~ tutorial rcvte~ Pr~ ceedings o f the 1EEE, 63, 561--58(I. Malik. N. (I981), One. and two-dimensional maximum entropy spectral estimation Ph.D dissertation Cambridge, MA: M.I.T. Malik. N.. & Lira. J 119821. Properties oi ~wu-dimenslonal maximum entropy power spectrum estimates IEEE Transaction,s on Acoustics. Speech and Signal Processing ASSP-3O, 798-7t~7

t FIGURE 5. The calculated spectra N = 10 x 19, M = 3 x 3 , 5 x 5.

corresponding

to

McClellan J. {19821. Multidimensional >pectral estimation Proceedings of IEEE 70, 1029-1039. McClellan. J.. & Lang, S, (19821. Multi-dimensional MEM spectral estimation. International Con ference on Spectral Analysts' and Its' Use in Underwater Acoustics. London. England. Newman. W. (19791. Extension to the maximum entropy method II. IEEE Transactions Information Theory, IT-25,705-708. Newman. W. (19811. Extension to the maximum entropy method 111 Proceedings o f the Ist A S S P Workshop on Spectral Estimation (pp. 1,7. l--1.7.61. Hamilton. On~. ('anada. Papoulis, A, (19811 Maximum entropy and spectral estimation. A review. IEEE Transactions on Acou,~tics, Speech and Signal Processing ASSP-29(6), 1176-1186. Tewfik. A., Levy, B.. & Willskv. A (19881. An efficient max~mum entropy technique for 2-D lsotrt~plc random fields. Pro ceedings o f the International Con (erence on Acoustics, Speech and Signal Processing (pp. 741-744"1. New York. Wernecke, S.. & D'Addario, L. (1977L Maximum entropy image reconstruction. IEEE Transactions, ,m (o,nputer, C-26, 35t