Eigenvector method for maximum-likelihood …...Eigenvector method for maximum-likelihood estimation...

Vol. 10, No. 12/December 1993/J. Opt. Soc. Am. A 2539

Eigenvector method for maximum-likelihood estimation ofphase errors in synthetic-aperture-radar imagery

Charles V. Jakowatz, Jr., and Daniel E. Wahl

Sandia National Laboratories, Albuquerque, New Mexico 87815

Received November 25, 1992; revised manuscript received April 14, 1993; accepted May 11, 1993

We develop a maximum-likelihood (ML) algorithm for estimation and correction (autofocus) of phase errors in-duced in synthetic-aperture-radar (SAR) imagery. Here, M pulse vectors in the range-compressed domain areused as input for simultaneously estimating M - 1 phase values across the aperture. The solution involves aneigenvector of the sample covariance matrix of the range-compressed data. The estimator is then used withinthe basic structure of the phase gradient autofocus (PGA) algorithm, replacing the original phase-estimationkernel. We show that, in practice, the new algorithm provides excellent restorations to defocused SAR imagery,typically in only one or two iterations. The performance of the new phase estimator is demonstrated essen-tially to achieve the Cramer-Rao lower bound on estimation-error variance for all but small values of target-to-clutter ratio. We also show that for the case in which M is equal to 2, the ML estimator is similar to that of theoriginal PGA method but achieves better results in practice, owing to a bias inherent in the original PGA phase-estimation kernel. Finally, we discuss the relationship of these algorithms to the shear-averaging and spatial-correlation methods, two other phase-correction techniques that utilize the same phase-estimation kernel butthat produce substantially poorer performance because they do not employ several fundamentalsignal-processing steps that are critical to the algorithms of the PGA class.

1. INTRODUCTION

In 1988 a technique for phase-error correction (autofocus)of synthetic-aperture-radar (SAR) imagery was introducedthat has since proven to be a robust algorithm.' 4 Thephase gradient autofocus (PGA) method employs adjacentpulses of data in the range-compressed phase-historydomain as input for estimating phase differences at eachposition in the aperture. These differences are thensummed so that an estimate for the phase-error correctionfunction across the aperture can be derived. The phase-difference estimator is derived as a certain form of optimalprocessor of adjacent-pulse data and is implemented in aniterative fashion. Typically, five or six iterations are re-quired for achieving convergence on real SAR imagery.The basic algorithmic steps of the PGA method are re-viewed in Fig. 1 in flowchart form. The reader shouldconsult Refs. 1-4 for details.

In this paper we show that the maximum-likelihood (ML)principle can be used for solving the more general phase-estimation problem wherein M contiguous pulse vectorsof range-compressed data are employed for simultaneousestimation of M - 1 phase values across the aperture.This problem has a solution that involves an eigenvector ofthe sample covariance matrix of the range-compressedpulse vectors. When one uses this phase estimator to re-place the original phase-difference estimator within thealgorithmic structure of the PGA method (see Step 5 ofFig. 1), an autofocus technique results that in many casesrequires only one iteration for achievement of excellentimage restoration. We derive the result here and demon-strate with real SAR data the efficacy of the new algo-rithm. We also compute a performance bound in the formof the Cram6r-Rao lower bound for estimation-error vari-ance and then show that the ML eigenvector estimatoressentially achieves this bound, except for cases of poor

target-to-clutter ratio. A similar eigenvector methodhas previously been applied to the closely related problemof time-delay estimation in towed linear passive sonararrays.5 6 Finally, we show that the general eigenvectorsolution for the special case of M = 2 collapses to an esti-mator that is similar to that of the original PGA technique.The new method is superior to the earlier one, however, inthat the typical number of iterations required for achiev-ing convergence is substantially reduced, at no additionalcomputational cost. The deficiency in the original PGAestimator lies in an inherent bias stemming from anassumption in its derivation that requires high target-to-clutter ratios. In many real SAR scenes, however, suchtarget-to-clutter ratios are simply not available. The con-sequence is that the original PGA algorithm requires moreiterations for convergence.

2. DERIVATION OF THE EIGENVECTORSOLUTION FOR MAXIMUM-LIKELIHOODESTIMATION OF PHASE ERRORS

We begin by defining the phase-error-estimation problemto be treated here. Consider samples of range-compressedphase-history-domain data, for which there are N rangelines and M aperture positions, so that a total of M XN samples are used as input. The basic signal model isdeveloped as follows. Consider a simple hypothetical SARimage wherein on each range line there exists a singlepoint reflector located at the center cross-range column.7

The real and imaginary (I and Q) components of the com-plex reflectivity of this point target on each range line aretreated as zero-mean Gaussian random variables that aremutually independent and identically distributed. Theclutter reflectivity I and Q values, which exist at everyother (noncenter) cell in the image space, are also modeledas independent and identically distributed Gaussian ran-

0740-3232/93/122539-08$06.00 C) 1993 Optical Society of America

C. V Jakowatz, Jr., and D. E. Wahl

2540 J. Opt. Soc. Am. A/Vol. 10, No. 12/December 1993

RMS YesPhase ErrorDoThreshod

Fig. 1. Algorithmic steps of the PGA.

dom variables. All clutter and target components aremutually independent across all values of range andcross range. The clutter is thereby modeled as uniform-intensity Gaussian white noise.

This image-domain description of targets and clutterleads to a simple equivalent model in the domain whereinthe cross-range dimension has undergone a discreteFourier transformation but the range dimension is leftunaltered. We refer to this other domain as either therange-compressed phase-history domain, or simply therange-compressed domain. The Fourier-transformedcross-range variable will be referred to as aperture posi-tion and denoted by 1.

In the range-compressed domain the phase errors aremodeled such that between aperture positions I and 1there exists a phase difference, denoted by a1, that is con-stant across all range lines. That is, the phase errors con-stitute a one-dimensional function of aperture position.We then have the following model for the data in thatdomain:

gkl = ak + nki7

gk2 = akej2 + k2,

gkM = akei0m + kM,

with k = 1, ... N and where gkl represents the sample inaperture position I on the kth range bin. The phase at thefirst aperture position is arbitrarily assigned the valueof zero.

Since we modeled the image-domain target structure asa single point reflector at the center of each range line,the corresponding target values in the range-compresseddomain are simply complex constants across the apertureposition dimension, different for each range line, denotedak. The clutter terms in the range-compressed domain,nlk, remain as uniform-intensity white Gaussian noise, asa consequence of the fact that the discrete Fourier trans-form is a unitary linear transformation. Therefore theI and Q components of nlk are independent and identicallydistributed Gaussian random variables (for all k and 1).Also, the I and Q components of the target values ak aremutually independent across the range (k) dimension andare independent of all clutter values. We let the varianceof the components of nlk be cr 2/2 and the variance of thecomponents of ak be a

2/2. The target-to-clutter ratio fora single range line of this canonical image can thereforebe defined as

0'a2

2 (2)0yn

An important step in the PGA algorithm structure isthe windowing (filtering) of the image-domain data beforetransformation to the range-compressed space. The widthof the window is estimated from the support (width) of theblurring (point-spread) function. The effective value of f,in turn, varies inversely as the chosen window width, since

.n2 changes in direct proportion to this width. The moti-vation for windowing is to make /3 as large as possible,short of rejecting data that are interior to the blur width.Since the window function is flat (constant) and we com-pute a discrete Fourier transform on the samples insidethe window without zero padding, this process does notalter the model of Eqs. (1) for the range-compressed data.That is, all the clutter samples of the range-compresseddata are mutually independent, regardless of the chosenwindow width, as a consequence of the unitary property ofthe discrete Fourier transform mentioned above.

Let X denote the entire two-dimensional set of inputsamples, and let xk denote a vector that contains thesamples of X on the kth range line, i.e.,

Xk = [gi, gk2, *.. gkM]

Since samples on one range line are assumed to be statis-tically independent of samples on any other range line,we calculate the logarithm of the conditional probability-density function for the set of samples X, given I as8

NIn p(X I A) = -N ln[7|MICI] - E XkHC iXk,

kohl(3)

where I is the vector of phase errors:

t = [, 2, ... M]and C is the covariance matrix for each range line of data.

(1) An expression for C is obtained by insertion of the signal

Step 1Input Complex Image-Domain Data.

Step 2Center Shift Largest Targets.I +~~~~

Step 3Determine Window Width and Apply

Window.

Step 4Fourier Transform in Cross-Range dimension

(to range-compressed domain).

Step 5Estimate Phase-Error Function Across

Aperture.

Step 6Apply Phase Correction.

i~~~~~~~~~~~~~~~~~~~~~

Step 7Inverse Fourier Transform back to Image

Domain.



model of Eqs. (1) into the definition for the covariance ma-trix: C = E{xxH}. The result is

C = n2 + ra2 VH. (4)

Here I is the identity matrix and v is the phase-only vector:

~1

v= t . .(5)

ei#m

One obtains the ML estimator for the phase-error vec-tor I by finding the particular value of I that maximizesthe expression of Eq. (3). To this end, consider first theterm CI. We can calculate this determinant as the prod-uct of the eigenvalues of C, which are easily shown to be

Ai = MO'a + Un2 ,

k2 = 0,n2i

A2 2

AM = On

Therefore

CI = cn2M(1 + MfJ),

where /3 = 0a 2 /0n2 . Since CI is not a function of theparameter A, the optimization of Eq. (3) is not affected bythe first term on the right-hand side. Therefore it is suf-ficient to maximize the expression

NQ1 = - > XkC Xk,

k-1(6)

subject to the constraint of Eq. (5). At this point we calcu-late an expression for the inverse of C. The special formof C as given by Eq. (4) allows its inverse to be written inthe form'0

C-l= a1I + a2VVH, (7)

where a, and a2 are constants given by

1al= ;-2'

o~n

a 2= - 2 + Ma 2

The constrained optimization of Q, then becomesequivalent to the maximization of the expression

N N N \

Q2 = Xk VVHXk = V HXXkV = VH XkXkH v.k-1 k-1 k-1

Since the sample covariance matrix of the data set is de-fined as

N

C = - 2 XkXk,N k=

(8)

we have the equivalent problem of finding the phase-only

vector v that maximizes the quadratic form

Q3 = vHv, (9)

where C is Hermitian.The solution to the maximization of Q3 subject to a

slightly different constraint on v turns out to provide anapproximate, but nearly exact, solution to the problem ofoptimizing Q3 subject to v being phase only. Specifically,it can be shown that if Q3 is maximized over v when v isconstrained as

11V112 = vHv = M

then the solution is to choose v to be the eigenvector of Ccorresponding to its largest eigenvalue and scaled so thatits squared modulus is equal to M. (We provide a proof ofthis result in Appendix A.)

The link between these two similar optimization prob-lems is the following. When v is required to be phase only,its square modulus must be M. Of course, not all vectorswith this modulus are phase only, but recall that the truecovariance matrix, C, has the form of Eq. (4). The eigen-vector of C corresponding to its largest eigenvalue isprecisely the phase-only v of Eq. (5). As a result, for suf-ficiently large values of N the sample covariance matrixclosely approximates the form of Eq. (4), so that the eigen-vector corresponding to the largest eigenvalue is verynearly phase only. The algorithm for estimating thephase-error values across the M aperture positions, then,is computation of the phases of the components of theeigenvector of C corresponding to the largest eigenvalue.In Section 3 we test the efficacy of this technique by ap-plying controlled phase errors to real SAR imagery.

3. CRAMER-RAO LOWER BOUND ANDEIGENVECTOR ESTIMATOR

In this section we derive an expression for the Cram6r-Rao lower bound on variance for the estimation problemat hand. We then show by way of application of a con-trolled phase-error function to a real SAR scene that theML eigenvector estimator developed here essentiallyachieves this bound, even for reasonably low target-to-clutter ratios.

Recall that the Cram6r-Rao lower bound for the varianceof the estimation error for multiple-parameter estimationis given as follows2 :

For any unbiased estimate h

var[i(X) - ii] Ž J", (10)

where J" is the iith element in a K-by-K square matrix J'1and K is the dimension of the parameter vector. The ele-ments of matrix J, which is called the Fisher informationmatrix, are calculated as

J, = E n p(X I I) a In p(X I A)]a 4f i a qf

- -E[2 In P(X I 41[ atiaqj

Here, where we are estimating M - 1 phase values, wecan show that the (M - 1)-by-(M - 1) Fisher information



20 -

-25

~-20

-35-20 -15 -10 -5 0 5 10

SNR (dB)

Fig. 2. Estimation-error variance versus Cramdr-Rao lowerbound (CRLB). SNR, signal-to-noise ratio; MSE, mean-squareerror.

matrix is given by

M -

J = -2a2No a2

-1

1 -1 ..-1 -

M-1 ... -1

-1 ... -1-1 ... M-1

tion derived above. This corresponds to the use of onlyadjacent pulses in the range-compressed domain for theestimation procedure. Starting with Eq. (9), we have forthis special case

N

1 2 IgkilQ3= N [1 e Nk-

2 glgkl+lk-1

N

2 glgk,l+1k-1i

2 gk,+k t1

(11)

Instead of calculating the eigenvalues and eigenvec-tors of the sample covariance matrix, we directly obtainthe ML solution by finding the value of qi that maximizesQ3. The argument proceeds as follows. Expansionof Eq. (11) shows that it is sufficient to maximize theexpression

N

> (gklg*,1,1ej + gkgk,1+1e jtk-1

N r N \gkigk,i~ cos q - i 1kk,1+1 JIk-1 1 \k / (12)

Inspection of Eq. (12) reveals that the ML solution isgiven by

This matrix has the same special form as that of Eq. (4):

J = -2a2 Noa 2[MI T],

where

1T= [1, 1 . .. 1] .

Therefore, by the same inversion rule employed for obtain-ing Eq. (7), we can show that

J` = b[I + 1i"1,with

1

2MNa2 a2

Finally, we have from relation (10) that

var[i(X) - ] 1' Mf3-MNfJ 2

The performance of the eigenvector phase estimatorversus the Cram6r-Rao lower bound is shown in Fig. 2.Here we used a Monte Carlo simulation of the estimatorwith the observation model of Eqs. (1) to compute the vari-ance of the error of the eigenvector estimator. The plotsof Fig. 2 show the performance for the case of N = 512,M = 2, and for the case of N = 512, M = 64, comparedwith the corresponding Cramer-Rao lower bound plots.In both cases it is shown that the eigenvector solutionessentially achieves the Cram6r-Rao lower bound plots forall but small values of target-to-clutter ratio.

4. ALGORITHM FOR M = 2 AND ITSRELATION TO OTHER AUTOFOCUSMETHODS

An interesting result may be derived by consideration ofthe special case for which M = 2 in the more general solu-

(13)

where denotes the principal value of the angle of thecomplex quantity, computed on the interval [-r, ir].

The form of the above estimator for the angle betweenadjacent pulses in the range-compressed domain has aninteresting relationship to the phase estimators used inseveral other previously published SAR autofocus al-gorithms. The form is identical to the estimation kernelused in the shear-averaging algorithm of FienupS and alsoin the spatial-correlation technique of Attia and Stein-berg.4 The derivations of these algorithms, however, aread hoc and do not show the estimation kernel to be ML.In addition, both of these algorithms suffer in perfor-mance in comparison with the ML algorithm developedhere, because they fail to employ several key steps that arecentral to the PGA class of algorithms (see Fig. 1). Thesedeficiencies were documented in a recent publication bythe authors and others.4

The other estimator to which Eq. (13) has relevance isthat of the original PGA algorithm, itself. The new MLphase-estimation kernel differs from the PGA kernel,which is given by

N

I19kWt~k*W1)k 1

4GA(t) = (14)N

> Igk(t)12

k-1

where instead of a discrete measurement of the apertureposition 1, the continuous parameter t is used. Both thenew ML estimator for the case of M = 2 and the originalPGA kernel use only data on adjacent pulses in the range-compressed space to estimate a single value of the phasebetween the two pulses. These phase differences canthen be summed to produce an estimate of the phase-errorfunction across the entire aperture. The mathematical


�M = N_ E 9kZ9k,111 ,

-1


forms are different simply because the estimators werederived with two different optimization criteria. 5 As itturns out, the new ML processor gives better performancewhen it is used in place of the original PGA kernel withinthe same algorithmic structure of the PGA method. Thisis due primarily to a significant bias term that is inherentin the original PGA phase estimator. The source of thisbias lies in an assumption of high target-to-clutter ratio(f3) in the derivation of that kernel,5 a condition that oftenis not met in real SAR imagery. In Section 5 this differ-ence in performance will be demonstrated by means of ex-

a amples of known phase errors applied to real SAR imagery.

5. RESULTS WITH REAL SYNTHETIC-APERTURE-RADAR IMAGERY

In this section we demonstrate the utility of the eigen-vector phase estimator when it is used in conjunction withthe basic signal-processing steps of the original PGA al-gorithm (see Fig. 1). Real radar imagery collected by aSAR built and operated by Sandia National Laboratories,Albuquerque, New Mexico, is used with controlled phaseerrors applied. We compare the performance of the ML

b estimator for the case of M = 2 (adjacent pulses only)to the case in which a much larger number of pulses areemployed simultaneously (M = 20). Finally, we show theperformance penalty incurred by the use of the originalPGA phase estimator versus the use of the ML estimatorfor the adjacent-pulses-only case.

Figure 3a shows a well-focused SAR image, and Fig. 3bshows the defocused image that results when the phase-error function of Fig. 4 is applied. This is a so-calledhigh-order phase-error function, in that it is rich inhigh-spatial-frequency content. Figure 3c shows the re-sult after only one iteration of correction by means of the

c PGA algorithm, with use of the new eigenvector methodfor the phase estimation. In this case, blocks of 20 pulsesof range-compressed data were used. Note that the sceneappears to be nearly fully restored to the correct focus.The result of using only adjacent pulses (M = 2) after oneand three iterations are shown in Figs. 3d and 3e, respec-tively. Note that three iterations of the 2-pulse versiongives essentially the same image as only one iteration ofthe 20-pulse case, as is borne out by the impulse responseplots of Fig. 5. These curves were derived from a singleisolated point reflector in the scene.

d 20

U, 10

.2

0

C- -10

e -20 0 128 256 384 512

Fig. 3. a, Original image; b, degraded image; c, 20 pulses, Aperture Position

1 iteration; d, 2 pulses, 1 iteration; e, 2 pulses, 3 iterations. Fig. 4. High-order phase-error function.



.4'

- - - - Original2 Pulses, Iteration

-60 ........... 20 Pulses, 1 Iteration-60 ... .. 2 Pulses, 3 Iterations a

-4 -3 -2 -1 0 1 2 3 4Azimuth Position (Pixels)

Fig. 5. Restored impulse response plot from image corruptedwith high-order phase-error function.

Figure 6a shows a defocused version of a SAR scenewhich is quite unlike that of Fig. 3a, in that no culturaltargets are present. That is, this scene is highly clutter-like. Figure 6a was produced by application of the phase-error function of Fig. 7 to a well-focused image. Fig-ure 6b shows the result of an attempt to autofocus this bscene with one iteration of the 2-column algorithm (i.e.,with use of adjacent pulse pairs only). Note that the resultdoes not show any noticeable improvement. Figure 6cshows that after six iterations a reasonably well-focusedimage is achieved. Figure 6d, on the other hand, is theresult of the application of one iteration of the eigenvectortechnique, wherein blocks of 20 pulse vectors were usedfor the phase estimation. Note that only one iteration ofthis method nearly restores the image to its original form.Complete restoration is achieved after two iterations, asdepicted in Fig. 6e.

For the purposes of comparing the original PGA phase-estimation kernel to that of the ML estimator with M = 2,consider the images of Fig. 8. Figure 8a shows the resultof degrading the building scene of Fig. 3a with a low-orderphase-error function, as plotted in Fig. 7. By low order wemean that the spatial-frequency content of this function islower than that of the function of Fig. 4. Figures 8b and8c compare one iteration of the two-pulse ML techniqueand the two-pulse (original) PGA algorithm, respectively.Note the marked superiority of the ML estimator. Fi-nally, note that near-total restoration can be achieved inthree iterations of the ML technique (Fig. 8d), while fiveiterations of the original PGA method are required for ap-proximately the same quality (Fig. 8e). This is confirmed dby the impulse response plots corresponding to these twocases, as shown in Fig. 9. Comparison of Eqs. (13) and (14)shows that the computational burden of the ML method isactually less than that of the original PGA algorithm, thusmaking the ML algorithm the clear choice if one choosesto process only adjacent pulses.

6. SUMMARY AND CONCLUSIONSA maximum-likelihood (ML) method for estimation ofphase errors for use in a SAR autofocus algorithm by thesimultaneous processing of multiple-pulse vectors of erange-compressed data has been developed here. The Fig. 6. a, Degraded image; b, 2 pulses, 1 iteration; c, 2 pulses,mathematics is straightforward and results in a solution 6 iterations; d, 20 pulses, 1 iteration; e, 20 pulses, 2 iterations.



-l uC

0

a-10

a

-20 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

0 128 256 384 512Aperture Position

Fig. 7. Low-order phase-error function.

involving an eigenvector of the sample covariance matrix.The new phase estimator, when used within the structureof the phase gradient autofocus algorithm (PGA), resultsin an autofocus technique that appears to perform ex-tremely well. In many cases the number of iterations re-quired for focus to be achieved is only one. The algorithm

that results when the number of pulses is chosen to be two bis also an effective tool for autofocus. While this versionin general requires more iterations than are needed whenlarger values of M are employed, the resulting simplicityof calculation may in fact make it the algorithm of choice.A comparative study of the net computational complexityof obtaining the eigenvector solution for a small numberof iterations versus the simpler adjacent-pulse-only solu-tion for a larger number of iterations remains to be done.Finally, although the ML algorithm for M = 2 appears tobe similar to the original PGA algorithm, superior perfor-mance is attained by the ML technique, mainly as a resultof better bias properties of this estimator. C

APPENDIX A

In this appendix we prove that the maximization of thequadratic form

Q3 = vH v, (Al)

subject to the constraint that

|V112 = VHV = M

and where C is Hermitian, is obtained when v is chosen to dbe the appropriately scaled eigenvector of C correspondingto the largest eigenvalue. The proof proceeds as follows:

Proof Since C is Hermitian, its eigenvalues are real.Also, if P is the matrix of normalized eigenvectors of C(each column of P is one eigenvector of C), then P is aunitary matrix such that

C = pAPH, (A2)

where P' = P' and A is a diagonal matrix of eigenval-ues. Substituting the expression of Eq. (A2) into that ofEq. (Al), we have that e

Fig. 8. a, Degraded image; b, ML, 1 iteration; c, original PGA,= v"PAPfv. (A3) 1 iteration; d, ML, 3 iterations; e, original PGA, 5 iterations.


TV _ _


-40 IL A

M'L (3)........... LUMV (5)---- ORIG-60_

-4 -3 -2 -1 0 1Azimuth Position (Pixels)

Fig. 9. Restored impulse response plot fromwith low-order phase-error function.

Next, with the substitution z = PHv, onside of Eq. (A3), we have

Q = zHAz.

Since P is a unitary matrix, for everV111

2= M, its corresponding image z, under

pH will have the same modulus. That is,

I z112 = zI = VHppHv = VHV =

As a result, by finding the value of z hmodulus equal to M that maximizes exprEwill have found the desired value of vmizes expression (Al).

From expression (A4) we have that

NQ3 = >Ak|Zk,

k-1

where Zk is the kth component of z. Cle,expression is maximized for Zma. such thatIzkI2 = 0 for k = 2,.. M, where Al is thevalue. That is, Zmax

T= Mej[1, 0,... 0], whei

trary rotation angle. Therefore the correEof v must be vma = Pzmax. Finally, we haMej' 9 F, where 01 is the eigenvector of C co]Al. This completes the proof.

ACKNOWLEDGMENTS

The authors thank their colleagues PaulThompson, and Terry Calloway for useful di

taining to this work. This work was performed at SandiaNational Laboratories, supported by the U.S. Departmentof Energy under contract DE-AC04-76DP0789.

REFERENCES AND NOTES1. P. H. Eichel, D. C. Ghiglia, and C. V Jakowatz, Jr., 'A new

phase correction method for synthetic aperture radar," inProceedings of the Digital Signal Processing Workshop atStanford Sierra Lodge (Institute of Electrical and Electron-ics Engineers, New York, 1988), pp. 2.12.1-2.12.2.

2. P. H. Eichel, D. C. Ghiglia, and C. V Jakowatz, Jr., "Speckleprocessing method for synthetic-aperture-radar phase cor-rection," Opt. Lett. 14, 1-3 (1989).

2 3 4 3. P. H. Eichel, D. C. Ghiglia, C. V Jakowatz, Jr., and D. E. Wahl,"Phase-gradient autofocus for SAR phase correction: ex-

image corrupted planation and demonstration of algorithmic steps," in Pro-ceedings of the Digital Signal Processing Workshop atStarved Rock State Park (Institute of Electrical and Elec-tronics Engineers, New York, 1992), pp. 6.6.1-6.6.2.

the right-hand 4. D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and C. V Jakowatz, Jr.,"Phase gradient autofocus: a robust tool for high resolutionSAR phase correction," IEEE Trans. Aerosp. Electron. Syst.

(A4) (to be published).5. D. A. Gray, W 0. Wolfe, and J. L. Riley, 'An eigenvector

y v such that method for estimating the positions of the elements of an ar-tranformation ray of receivers," in Proceedings of the Australian Sympo-sium on Signal Processing Applications (Adelaide,

Australia, 1989), pp. 391-393.6. D. A. Gray and J. L. Riley, "Maximum likelihood estimatelvi . and Cramer-Rao bound for a complex signal vector," in Pro-

ceedings of the International Symposium on Signal Process-laving a square ing Applications (Gold Coast, Australia, 1990), pp. 352-355.ession (A4), we 7. Although this assumption is unreasonable for real SARPz that maxi- scenes, the PGA algorithm utilizes shifting of the strongestreflector of each range line to the scene center to approxi-

mate this condition.8. N. R. Goodman (Ref. 9) discusses the general conditions un-

der which Eq. (3) is valid.9. N. R. Goodman, "Statistical analysis based on a certain mul-

tivariate complex Gaussian distribution (an introduction),"Ann. Statist. Anal. 34, 152-177 (1963).

arly, the above 10. The relevant formula here is that of Sherman-Morrison."1zi 2 M and 11. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns

Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, p. 3.largest eigen- 12. H. Van Trees, Detection, Estimation, and Modulation The-re 0 is an arbi- ory, Part I (Wiley, New York, 1968), Chap. 2, pp. 79-81.iponding value 13. J. Fienup, "Phase error correction by shear averaging," inve that vm = Signal Recovery and Synthesis II, Vol. 15 of 1989 OSA Tech-rrespoding t nical Digest Series (Optical Society of America, Washington,rresponding to D.C., 1989), pp. 134-137.

14. E. H. Attia and B. D. Steinberg, "Self-cohering large antennaarrays using the spatial correlation properties of radar clut-ter," IEEE Trans. Antennas Propagat. 37, 30-38 (1989).

15. P. H. Eichel, "The phase gradient autofocus algorithm: anoptimal estimator of the phase derivative," Tech. Rep.

I Eichel, Paul SAND89-0761 UC-706 (Sandia National Laboratories, Albu-iscussions per- querque, N.M., 1989).


Eigenvector method for maximum-likelihood …...Eigenvector method for maximum-likelihood estimation...

Documents

Transcript of Eigenvector method for maximum-likelihood …...Eigenvector method for maximum-likelihood estimation...