Nonparametric Missing Sample Spectral Analysis and Its Applications to Interrupted SAR

IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012 1

Nonparametric Missing Sample Spectral Analysisand Its Applications to Interrupted SAR

Duc Vu, Member, IEEE, Luzhou Xu, Member, IEEE, Ming Xue, Member, IEEE, and Jian Li, Fellow, IEEE

Abstract—We consider nonparametric adaptive spectral anal-ysis of complex-valued data sequences with missing samplesoccurring in arbitrary patterns. We first present two high-reso-lution missing-data spectral estimation algorithms: the IterativeAdaptive Approach (IAA) and the Sparse Learning via IterativeMinimization (SLIM) method. Both algorithms can significantlyimprove the spectral estimation performance, including enhancedresolution and reduced sidelobe levels. Moreover, we considerfast implementations of these algorithms using the ConjugateGradient (CG) technique and the Gohberg–Semencul-type (GS)formula. Our proposed implementations fully exploit the struc-ture of the steering matrices and maximize the usage of the fastFourier transform (FFT), resulting in much lower computationalcomplexities as well as much reduced memory requirements. Theeffectiveness of the adaptive spectral estimation algorithms isdemonstrated via several numerical examples including both 1-Dspectral estimation and 2-D interrupted synthetic aperture radar(SAR) imaging examples.

Index Terms—Interrupted synthetic aperture radar (SAR), iter-ative adaptive approach (IAA), missing data, sparse learning viaiterative minimization (SLIM), spectral analysis.

I. INTRODUCTION

S PECTRAL estimation is important in many fields in-cluding astronomy, communications, medical imaging,

radar, and underwater acoustics. Most existing spectral esti-mation algorithms are devised for uniformly sampled com-plete-data sequences. However, in many practical applications,the measured data may be incomplete due, for example,

Manuscript received January 24, 2011; revised August 06, 2011; acceptedAugust 31, 2011. Date of publication September 15, 2011; date of current ver-sion January 18, 2012. This work was supported in part by the Departmentof Defense under Grant HM1582-10-1-0017, the National Science Foundationunder Grant ECCS-0729727, and the U.S. Army Research Laboratory and theU.S. Army Research Office under Grant W911NF-07-1-0450. The views andconclusions contained herein are those of the authors and should not be inter-preted as necessarily representing the official policies or endorsements, eitherexpressed or implied, of the U.S. Government. The U.S. Government is autho-rized to reproduce and distribute reprints for Governmental purposes notwith-standing any copyright notation thereon. The associate editor coordinating thereview of this manuscript and approving it for publication was Prof. PhilippeCiblat.

D. Vu and M. Xue are with the Department of Electrical and Com-puter Engineering, University of Florida, Gainesville, FL 32611-6130 USA(e-mail: [email protected]; [email protected]; [email protected];[email protected]).

L. Xu and J. Li are with the Department of Electrical and Computer En-gineering, University of Florida, Gainesville, FL 32611-6130 USA, and alsowith IAA, Inc., Gainesville, FL 32605 USA (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTSP.2011.2168192

to sensor failures, outliers, the data compression needs,etc. Missing data problems are also encountered in modernradar systems. These systems have multiple duties includingsearching, tracking and the automatic classification of targets.Switching in and out of these modes leads to an incompletephase history data for synthetic aperture radar (SAR) imaging[1]–[3]. This loss of the phase history data is also prevalentduring SAR missions encountering high radio frequency (RF)interferences [4], [1].

Many missing-data spectral estimation techniques have beendeveloped previously. A conceptually and computationallysimple method is the matched filter (MF), implemented viausing the fast Fourier transform (FFT), with the missing sam-ples set to zero. However, as a data-independent approach,the FFT approach, as well as its windowed FFT (WFFT)variations, suffers from poor resolution and high sidelobe levelproblems. This high sidelobe level problem is especially severefor the missing data case. The CLEAN algorithm [5] is usedto estimate the spectrum by deconvolving the missing-dataFFT spectrum into the true signal spectrum and the Fouriertransform of the missing pattern windowing function via aniterative approach. Although the CLEAN algorithm works forboth missing and irregularly sampled data sequences, it cannotresolve closely spaced spectral lines, and hence it may notbe a suitable tool for high-resolution spectral estimation. Themulti-taper methods [6], [7] compute spectral estimates byassuming certain quadratic functions of the available data sam-ples. The coefficients in the corresponding quadratic functionsare optimized according to certain criteria, but it appears thatthis type of approaches cannot overcome the resolution limit ofFFT. To achieve high resolution, several parametric algorithms,e.g., those based on autoregressive (AR) or autoregressivemoving-average (ARMA) models, were used to handle themissing-data cite [8]–[11]. Although these parametric methodscan provide improved spectral estimates, they are sensitive tomodel errors.

Several nonparametric adaptive filtering based techniqueshave been developed for the missing-data spectral estimationproblem [3]. For example, in [12] and [3], two nonparametricmissing-data amplitude and phase estimation (MAPES) al-gorithms are derived by using a “maximum-likelihood (ML)fitting” criterion. The well-known expectation–maximization(EM) technique [13] is used to solve the estimation problemiteratively. MAPES works for the general spectral analysisproblem with missing samples occurring in arbitrary patterns.However, MAPES is conceptually complicated and it is com-putationally intensive. Furthermore, it is well-known that EMmay converge rather slowly, and, it may even fail to converge

U.S. Government work not protected by U.S. copyright.

2 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012

globally. As we will show via numerical examples, the spectralestimation performance of MAPES degrades severely in highdata missing ratio cases.

Recently, sparse signal recovery (SSR) techniques (see, e.g.,[14] and the references therein) have attracted much attentionin the signal processing community. Different from the conven-tional data-adaptive approaches, such as Capon [15] and APES[16], SSR can work for very few or even one snapshot, and hencecan be used for spectral analysis of irregularly sampled data.However, most of the existing SSR algorithms, including thewell-known CoSaMP (Compressive Sampling Matching Pur-suit) [14], require the bases of the signal components to obey a“restricted isometry property” [17], which hardly holds in high-resolution spectral estimation problems. As we will show vianumerical examples, CoSaMP fails to work properly in both 1-Dspectral estimation and 2-D interrupted SAR imaging problems.

Recently, two high-resolution spectral estimation algorithms,i.e., the iterative adaptive approach (IAA) [18] and the sparselearning via iterative minimization (SLIM) method [19], [20],have been developed. Both algorithms can work for completeor incomplete data samples, and are able to achieve excellentspectral estimation performance under various conditions.However, both IAA and SLIM are computationally prohibitive,especially for high-dimensional spectral estimation problemssuch as interrupted SAR imaging. Several efficient implemen-tations of IAA are proposed for 1-D and 2-D spectral estimationof uniformly sampled data sequences [21]–[25]. These algo-rithms exploit the Vandermonde/block-Vandermonde structuresof the steering matrices, and are able to reduce the compu-tational cost by at least two orders of magnitudes even formoderate data sizes. Unfortunately, for the missing data spec-tral estimation problems, the said steering matrices are notVandermonde/block-Vandermonde any more, and, hence, theproposed fast implementations of IAA in [21]–[25] are nolonger applicable directly.

In this paper, we will consider the fast implementation ofIAA for the spectral estimation with missing samples. More-over, we will propose two fast implementations of SLIM withinthe missing-data spectral estimation framework. We will applythe proposed algorithms to both 1-D spectral estimation and 2-Dinterrupted SAR imaging problems. Via several numerical ex-amples, we show that IAA and SLIM outperform the conven-tional FFT and MAPES approaches, as well as CoSaMP, sig-nificantly. Compared to the original IAA and SLIM algorithms,the fast implementation approaches can dramatically reduce thecomputational complexities and memory requirements.

The remainder of this paper is organized as follows. Section IIintroduces the IAA and SLIM algorithms. In Section III, we pro-pose three fast implementation methods for missing data spec-tral estimation via IAA and SLIM. Several 1-D spectral estima-tion and 2-D interrupted SAR imaging examples are presentedin Section IV. Finally, Section V contains our conclusions.

II. NONPARAMETRIC SPECTRAL ANALYSIS

In this section, we first present two recently developed adap-tive spectral estimation algorithms, i.e., IAA and SLIM, withinthe missing data framework. Both algorithms can work for 1-D

and higher dimensional spectral estimation problems with com-plete or incomplete data samples.

A. Data Model

We consider a spectral estimation problem with givencomplex-valued data samples, and stack the data samplesinto an column vector . Letbe the complex amplitude at the th frequency grid point of thespectrum, and be the normalized con-tribution of the th frequency grid point to the available datasamples. Then, the given data samples can be modeled as fol-lows:

(1)

where denotes the noise vector

(2)

and

(3)

with denoting the transpose. In (1), is referred to asthe steering matrix, and the column vector represents thespectrum to be estimated. Note that the data model in (1) can beused for both 1-D and higher dimensional spectral estimationproblems. In particular, in the SAR imaging application,contains the complex-valued reflection coefficients of scattererswithin an imaging area (or volume) of interest. Note also thatin high-resolution spectral estimation problems, the grid pointnumber is usually much larger than the number of datasamples. Hence, the conventional approaches based on (1),such as the maximum-likelihood (ML) and least-square (LS)approaches, fail to work completely.

B. Iterative Adaptive Approach (IAA)

IAA is an iterative adaptive algorithm based on a nonpara-metric weighted least squares (WLS) approach. Let

with being an intermediate estimate of. The noise and interference covariance matrix for the th

frequency grid point can be estimated as

(4)

where denotes the conjugate transpose

(5)

and

(6)

with being a diagonal matrix formed by the elementsof a given vector.

Given the noise and interference covariance matrix , theestimate of can be refined using the WLS criterion as follows:

(7)

VU et al.: NONPARAMETRIC MISSING SAMPLE SPECTRAL ANALYSIS AND ITS APPLICATIONS TO INTERRUPTED SAR 3

TABLE IIAA ALGORITHM

where . Solving the optimization problem

of (7) yields

(8)

where we have used (4) and the matrix inversion lemma [26].The final spectrum estimate is obtained via iterating (5) and

(8). The initialization of can be done by a standard MF:

for (9)

with denoting the Euclidean norm of a vector.The IAA algorithm is summarized in Table I. We remark that

in the above derivation we have implicitly taken into accountthe effect of noise. A regularized version of IAA, referred to asIAA-R [27], can be used to take into account the noise effect ex-plicitly. The fast implementation of IAA proposed in Section IIIcan be extended to IAA-R straightforwardly. So we will focuson the former herein.

C. Sparse Learning Via Iterative Minimization (SLIM)

The SLIM algorithm [19], [20] considers the following opti-mization problem:

(10)

where

(11)

with denoting the Euclidean norm of a vector. In (11), isa user parameter that ranges between 0 and 1. In our numericalexamples in Section IV, we use for both 1-D spectral esti-mation and 2-D interrupted SAR imaging problems. The SLIMalgorithm can be viewed as a maximum a posteriori (MAP) ap-proach with being the noise variance and the prior distributionof being .

The optimization problem can be solved iteratively by usingthe cyclic minimization (CM) and majorization–minimization

(MM) techniques [28]. Let and be intermediate esti-mates of and , respectively, for the th iteration. For

is a concave function with respectto . Hence, the function is majorizedby its tangent linear function at , i.e.,

(12)

where

(13)

and the equation holds when . Therefore, we have

(14)

where, again, the equation holds when for.

Given , minimizing the majorization function of

, i.e., , with respect to yields:

(15)

where

(16)

and is defined similarly to in (6), i.e.,

(17)

Furthermore, given , minimizing withrespect to yields

(18)

The SLIM algorithm estimates via iterating (15), (17), and(18).

By using the CM and MM properties, we readily have

(19)

i.e., the cost function decreases monotonically.


TABLE IISLIM ALGORITHM

Again, we initialize using MF in (9). The variableis initialized as

(20)

where is a column vector formed by similarlyto (35), and is chosen to be 10 in our numerical examples inSection IV. The SLIM algorithm is summarized in Table II.

D. Computational Complexities and Memory Requirements

For the IAA algorithm in Table I, it is computationallyintensive to compute the covariance matrix in Step 2 and

(for ) in Step 3, with the compu-tational complexities being per iteration. Similarly,SLIM in Table II requires flops per iteration tocompute in Step 2. Furthermore, both IAA and SLIM needto precompute and store the steering matrix of size .Both the computational complexity and memory requirement ofthe algorithms, if implemented directly, would be prohibitive,especially for high-dimensional spectral estimation problemssuch as SAR imaging, where and could be up to and

, respectively.For spectral estimation of uniformly sampled data sequences,

the steering matrix in (1) has a Vandermonde/block-Van-dermonde structure [29]. This structure can be exploited tosignificantly reduce the computational intensities and memoryrequirements of both IAA and SLIM. Specifically, we canshow that the IAA covariance matrices in (5) are Toeplitzand Toeplitz-Block-Toeplitz matrices, respectively, for the1-D and 2-D spectral estimation problems. By using theGS-type factorization technique [30], [31], one can leveragethe Toeplitz/Toeplitz-Block-Toeplitz structure and thus the useof FFT/IFFT to compute the IAA spectral estimate [21]–[25],and to significantly reduce the computational complexityand memory requirement. However, in the incomplete datacases, and are not Vandermonde/block-Vandermonde andToeplitz/Toeplitz-Block-Toeplitz matrices any more, and hencethe proposed algorithms in [21]–[25] are no longer applicabledirectly.

We present below several fast implementation schemes ofIAA and SLIM to dramatically reduce the computational com-plexities and memory requirements under the missing data con-dition.

III. MISSING DATA SPECTRAL ANALYSIS AND

FAST IMPLEMENTATIONS

Hereafter, we focus our discussion on a 2-D spectral estima-tion problem, since the interrupted SAR imaging application is

of particular interest to us. The 1-D spectral estimation problemis a special case of its 2-D counterpart. Moreover, our resultscan be readily extended to 3- or higher dimensional spectral es-timation problems.

A. Data Model

We consider the following 2-D spectral estimation problem:

(21)

where

and (22)

In (21), ( , and) are the complex-valued data samples,

( , and ) are the com-plex amplitudes of signal spectrum at frequency grid ,and denote the noise.

For notional convenience, we organize the complete datasamples and the complex-valued amplitudes

into an matrix and an matrix, respectively, as follows:

......

. . ....

(23)

and

......

. . ....

(24)

Then, the complete data model in (21) can be rewritten as

(25)

where is defined similarly to in (23) for the noise, andand are and inverse Fourier transform

matrices, respectively, i.e.,

......

.... . .

...

for (26)

By applying the matrix vectorization operator to both sidesof (25), the complete data model can be rewritten as

with and (27)

where is the 2-D inverse Fourier transform matrix,denotes the vectorization operator (i.e., stacking the columns


of a matrix on top of each other), and denotes the Kro-necker product of matrices. In (27), we have used the fact that

(see, e.g., [26]).Now we assume that only out of samples are avail-

able. Let be an column vector formed by the given datasamples, and be the corresponding selectionmatrix, i.e.,

(28)

The selection matrix has elements 0 and 1. Each row of theselection matrix contains one and only one element equal to 1,whose position indicates the selected row/column index. As anexample, let

(29)

It can be easily verified that is a 2-row matrix formed bythe second and fifth rows of a 6-row matrix , while is a2-column matrix formed by the second and fifth columns of a6-column matrix . Note also that expands a 2-row matrix

to a 6-row matrix whose second and fifth rows are equal tothe first and second rows of , respectively, with the remainingrows being zero. Also, expands a 2-column matrix to a6-column matrix similarly.

From (27) and (28), we get the data model for the 2-Dmissing-data spectral estimation problem as follows:

(30)

Comparing (30) with (1), we can see that the aforementionedIAA and SLIM algorithms in Section II can be applied to (30)directly with the steering matrix being . If implementeddirectly, the computational complexity and memory require-ment of IAA and SLIM are and ,respectively, which are prohibitive for practical applications.We present below several computationally efficient implemen-tations of IAA and SLIM that exploit the structure of .

B. Fast Iterative Adaptive Approach (Fast-IAA)

We present below an efficient implementation of IAA viafully exploiting the structure of the steering matrix . Werefer to this fast implementation of IAA as Fast-IAA.

1) Efficient Computation of the IAA Covariance Matrix: Wefirst consider how to compute the IAA covariance matrix inStep 2 of Table I efficiently.

Let be an intermediate estimate of , and bethe corresponding power estimate, where denotes the ele-ment-wise modulus-square operation. Then, the IAA covariancematrix in (5) can be written as

(31)

where

(32)

represents the IAA covariance matrix of the complete data sam-ples corresponding to .

By using the 2-D inverse Fourier transform property of ,it can be readily verified that is a Toeplitz-Block-Toeplitzmatrix [32], which can be written as follows:

......

.... . .

...

(33)

with

......

.... . .

...

(34)In (34), (for ;and ) is the 2-D inverse Fourier trans-form of , which can be computed efficiently by using the 2-Dinverse FFT (IFFT) technique. Furthermore, from (31), by theselection matrix property, is a submatrix of formed bythe rows and columns specified by the selection matrix . Byapplying 2-D IFFT to to obtain first and then get , wecan reduce the computational complexity from to

.2) Efficient Computation of : Now we consider an efficient

method to compute given . Applying the WLS method [i.e.,(8)] to the 2-D spectral estimation data model in (30), after somestraightforward manipulations, we readily have

for

and (35)

where

(36)

and

(37)

with and denoting the th element and the thcolumn of a given matrix, respectively.

Given in (36) can be computed efficiently as fol-lows: 1) compute ; 2) expand the so-obtained vectorto an vector by using the selection matrix property;3) organize the vector into an matrix; and 4) apply2-D FFT to the so-obtained matrix.

We now consider how to compute in (37) efficiently.Let , which, obviously, can be obtained from


TABLE IIIFAST-IAA ALGORITHM

by inserting some zero rows and columns specified by theselection matrix . For convenience, we rewrite as an

block matrix formed by sub-matrices, i.e.,

......

. . ....

(38)

Then, can be expressed as a polynomial function ofand as follows:

(39)

where

for and

for and

for and

for and(40)

From (39), (for and) can be computed efficiently via

applying 2-D FFT to the polynomial coefficientsin (40), which reduces the computational complexity from

to .The Fast-IAA algorithm is summarized in Table III. Note that

different from MAPES [12], as well as the subsequent GS-SLIMapproach in Section III-D, the Fast-IAA algorithm does not esti-mate the missing data samples explicitly. However, the missing-data IAA (MIAA) algorithm in [33] can be used to recover themissing data samples from the Fast-IAA spectra estimates, i.e.,

in (35).Note that the computationally most intensive steps of the

Fast-IAA algorithm are the matrix inverse of in Step 3

TABLE IVCG ALGORITHM

and 2-D FFT/IFFT, with computational complexitiesand , respectively. Hence, the overallcomputational complexity of the IAA algorithm is reduced to

, which is much smaller thanof the original IAA algorithm. We remark that in

the complete data case, in (31) is a Toeplitz-Block-Toeplitzmatrix [32], whose inverse can be computed efficiently byusing the Gohberg–Semencul (GS) type formula [30], [31].We refer to this GS-type formula based Fast-IAA algorithmas Fast-IAA-GS [21]. The computational complexity ofFast-IAA-GS is . Furthermore, we notethat Fast-IAA, as well as Fast-IAA-GS, does not use thesteering matrix explicitly. Hence, it does not need to precom-pute and store the steering matrix, which significantly reducesthe memory requirement.

C. Fast SLIM Using Conjugate Gradient (CG-SLIM)

Next, we consider a fast implementation of the SLIM algo-rithm in Table II by using the conjugate gradient (CG) technique[29]. We refer to this algorithm as CG-SLIM.

CG is an efficient method to solve a linear system iteratively.The number of iterations needed by CG is usually much smallerthan the problem size, which can significantly reduce the com-putational complexity. In this section, we employ the CG tech-nique to compute computationally the most intensive step ofSLIM, i.e., in Table II. Let

(41)

Then, the unknown vector in (41) can be computed efficientlyby using the CG approach in Table IV.

For the CG algorithm in Table IV, Step 1 is computation-ally the most intensive, which involves a matrix-vector product.However, in the 2-D spectral estimation problem in (30), thecomputational complexity of this step can be reduced by usingthe structure of the steering vector. Let be an intermediate es-timate of , and . Replacing and in (16) by

and , respectively, we have

(42)

Then, Step 1 of Table IV can be written as

(43)


TABLE VCG-SLIM ALGORITHM

where denotes the Hadamard product of matrices, and thefunctions and are defined, respectively, as fol-lows:

(44)

and

(45)

By the properties of the 2-D inverse Fourier transform matrixand the selection matrix , (44) and (45) can be computed effi-ciently by using the 2-D FFT/IFFT technique. For thefunction in (44), we can reorganize the vector intoan matrix, apply 2-D IFFT to the so-obtained matrix,and then pick up the elements of the 2-D IFFT output speci-fied by the selection matrix to form a column vector. For the

function in (45), we can first expand the columnvector to an vector via inserting zeros at the posi-tions specified by the selection matrix , reorganize the vectorinto an matrix, and then apply 2-D FFT to the so-ob-tained matrix. By using these 2-D FFT and IFFT techniques,the computational complexity of Step 1 of Table IV is reducedto .

By combining Tables II and IV, the CG-SLIM algorithm issummarized in Table V. We terminate the CG iterations whenthe ratio of the square of the Euclidean norm of the residualvector to the square of the Euclidean norm of the data vector

becomes less than a predefined threshold . In the numer-ical examples in Section IV, we choose . Similarto Fast-IAA, CG-SLIM does not estimate the missing samplesexplicitly. Once is obtained in Table V, the missing data sam-ples can be recovered by simply applying 2D FFT to .

We note that computationally the most intensive step inTable V is now 2-D FFT/IFFT, with a computational com-plexity . Hence, the overall computationalcomplexity of CG-SLIM is per SLIMiteration, with being the average CG iteration number, whichis usually much smaller than . For instance, in the interruptedSAR example in Section IV, the data sample number is upto 1600, but the CG algorithm usually converges in less than

iterations.

D. Fast SLIM Using the Gohberg-Semencul-Type Formula(GS-SLIM)

For a complete-data spectral estimation problem, i.e., when, similarly to in (33), the matrix in (42) will

be a Toeplitz-Block-Toeplitz matrix [32], [21]. In this case,the vector in (41) can be computed efficientlyby using the GS-type formula [30], [31], whose complexityis around , which is much lower than

of the original SLIM algorithm. Inspired bythis, we introduce below a new fast implementation of SLIM,referred to as GS-SLIM, for the missing data spectral estima-tion problem. This algorithm includes two steps: 1) estimatemissing samples from an intermediate spectral estimate, and 2)estimate the signal spectrum from the given data samples, aswell as the estimated missing samples. Note that the second stepof the GS-SLIM algorithm is essentially a complete-data spec-tral estimation problem, where the aforementioned GS-typefactorization technique can be utilized.

For the 2-D spectral estimation problem in (30), the SLIMformulation in (10) and (11) can be written as follows:

(46)

where

(47)

Below, we employ the MM technique to solve the optimizationproblem iteratively.

Let and be the missing samples and the associatedselection matrix, respectively. Let and be the inter-mediate estimates of and at the th iteration, respectively.Naturally, the missing data can be estimated as follows:

(48)

We consider below an efficient method to update the andestimates such that the cost function in (47) decreases monoton-ically.

From (47), we readily have

(49)

(50)


(51)

(52)

(53)

where is the complete data composed by the estimatedmissing sample vector , as well as the given sample vector

. The inequality in (52) is obtained similarly to (14), and

(54)

Obviously, the equality in holdswhen . In other words, the function is ma-jorized by at the point . There-fore, we can obtain the estimates of and via minimizing

with respect to and .Similarly to (15), given , minimizing

with respect to yields

(55)

where

(56)

As discussed above, is a Toeplitze-Block-Toeplitz matrix[32], [21] having the form

......

.... . .

...

(57)

and

......

.... . .

...

(58)

where is a Kronecker delta function, and isthe 2-D Fourier transform of and can be computed efficientlyby using 2-D FFT. Furthermore, via exploiting the Toeplitze-

TABLE VIGS-SLIM ALGORITHM

Block-Toeplitz structure of in (55) can be computedefficiently by using the GS-type factorization algorithm [32],[21].

On the other hand, given , minimizingwith respect to yields

(59)

Equation (59) is a natural estimate of given .However, we find empirically that the GS-SLIM algorithm suf-fers from a slow convergence problem using (59). Instead, wepropose to update as follows:

(60)

where is chosen to be the given-to-complete data ratio, i.e.,, in the numerical examples in Section IV.

The 2-D spectrum is estimated via iterating (48), (55),(57), and (60). We prove below that the cost functiondecreases monotonically. First, we readily have the first-orderderivative of , with respect to , as fol-lows:

(61)

where is defined in (59). From (61), we can see thatfor

and for .In other words, the functionmonotonically decreases with respect to for ,and monotonically increases for . On the other hand,obviously we have when , and

when . That immediately implies

(62)


Fig. 1. Nonparametric spectral estimation without and with missing samples by (a) WFFT, (b) MAPES, (c) CoSaMP, (d) IAA, and (e) SLIM, where the completedata length is 128. The subplots in the left, middle and right columns are obtained with the complete data, 50% data, and 30% data, respectively.

Using (62) and the fact that majorizes[see (53)], we immediately have

(63)

i.e., the cost function is monotonically decreasing.

The GS-SLIM algorithm is summarized in Table VI.We note that computationally the most intensive stepsin Table VI are the 2-D FFT/IFFT and the GS-typefactorization, whose computational complexities are

and , re-spectively. Hence, the overall computational complexity ofGS-SLIM isper iteration.

IV. NUMERICAL EXAMPLES

We present below several 1-D spectral estimation and 2-Dinterrupted SAR imaging examples to demonstrate the perfor-mance of the missing data spectral estimation algorithms andtheir fast implementations.


Fig. 2. Nonparametric spectral estimates for data sequences with missing sam-ples obtained by (a) WFFT, (b) IAA, (c) Fast-IAA, (d) SLIM, (e) CG-SLIM,and (f) GS-SLIM, when 800 out of 1000 data samples are available for spectralanalysis.

A. 1-D Spectral Estimation

To demonstrate the performance improvements of the newmissing data spectral analysis approaches over some existingones, we first consider a simple example in Fig. 1. The com-plete data has 128 samples, and contains eight complex sinu-soids. The frequencies and amplitudes of the sinusoids are indi-cated by circles in Fig. 1. The data is corrupted by a zero-meancircularly symmetric complex white Gaussian noise with vari-ance . In Fig. 1(a)–(e), we provide the spectra es-timates obtained by windowed FFT (WFFT), APES/MAPES,CoSaMP, IAA, and SLIM, respectively. A Taylor window witha 30 dB sidelobe level is used for WFFT. The iteration num-bers for all iterative approaches, including MAPES, CoSaMP,IAA, and SLIM, are fixed to 20. (In our examples, we did notfind significant performance improvement after 20 iterations forall algorithms.) We choose the user parameter of CoSaMP to be32. For SLIM, we choose . The subplots at the left, middleand right columns in Fig. 1 show the results obtained from thecomplete data, 50% and 30% data, respectively. The availabledata samples are selected randomly. For Fig. 1(b), we use APESfor the complete data case, and MAPES for the two missing datacases.

From Fig. 1(a), it is clear that WFFT suffers from the lowresolution and high sidelobe level problems. The sidelobe levelincreases significantly in the missing data cases. Specifically,the peak-sidelobe level (PSL) increases by around 18.5 dB and21 dB, respectively. Due to the high sidelobe levels, severalspectral lines fail to be detected. In Fig. 1(b), APES provides

Fig. 3. Missing samples estimated by (a) CG-SLIM and (b) GS-SLIM. Thesubplots in the left and right columns represent the real and imaginary parts ofthe data samples, respectively.

Fig. 4. Nonparametric spectral estimates for data sequences with missing sam-ples obtained by (a) CG-SLIM-IAA and (b) GS-SLIM-IAA, when 800 out of1000 data samples are available for spectral analysis.

a good spectral estimation performance for the complete datacase. Its missing data variant, i.e., MAPES, behaves well when50% of the data samples are available. However, its performancedegrades significantly when the available data percentage de-creases to 30%. Specifically, the spectral line at 0.28 Hz fails tobe detected. Moreover, MAPES is computationally very inten-sive. The computation times needed by MAPES, when imple-mented on an ordinary personal computer (Intel Xeon CPU at2.67 GHz; 12 GB RAM), for the two missing-data examples inFig. 1(b) are 969 and 847 seconds, respectively, while the com-putation times needed by all the other algorithms are less than1 second. We can also see in Fig. 1(c) that CoSaMP producesmany false peaks in the missing data cases, and it fails to detectthe spectral line at 0.28 Hz at all scenarios. From Fig. 1(d) and(e), we note that both IAA and SLIM provide good spectral es-timation performance under various conditions. All of the eightsinusoids can be identified clearly even with only 30% data sam-ples available. We remark that the amplitude estimates of SLIMare significantly biased downward (up to 15 dB). However, withthe correctly identified sinusoidal number and frequencies, theamplitude estimates can be redetermined readily using, for ex-ample, the standard least-squares (LS) method or the RELAXalgorithm [34].

We next consider a more challenging problem of analyzinga long data sequence where 800 out of 1000 data samples areavailable for spectral estimation. The data sequence contains100 complex-valued sinusoids, and is corrupted by a zero-mean


TABLE VIICOMPUTATION TIMES NEEDED BY IAA, SLIM, AND

THEIR FAST IMPLEMENTATIONS

circularly symmetric complex white Gaussian noise with vari-ance . Fig. 2(a)–(f) shows the spectral estimates(zoomed-in to show details within the [0.21, 0.24] Hz band)obtained by WFFT, IAA, SLIM, Fast-IAA, CG-SLIM, andGS-SLIM, respectively. Again, the data-adaptive algorithmsoutperform the conventional data-independent WFFT approachsignificantly. The performance of the fast implementationsof IAA and SLIM are very close to those of their originalalgorithms. From Fig. 2, we see again that the amplitudeestimates of SLIM and its variants are biased downward. Themaximum amplitude estimation biases of SLIM, CG-SLIM,and GS-SLIM are dB, 11 dB and 14 dB, respectively.On the other hand, IAA and its variant provide rather accurateamplitude estimates, with the maximum biases less than 2.5 dB.

Note that different from MAPES and GS-SLIM, IAA, SLIM,Fast-IAA, and CG-SLIM do not estimate the missing datasamples explicitly. For SLIM and CG-SLIM, the missing datasamples can be recovered by simply applying FFT [see (48)]to their spectral estimates. The missing-data IAA (MIAA)algorithm in [33] can also be used to recover the missingsamples from the IAA or Fast-IAA spectra estimates, which,however, is more complicated both conceptually and compu-tationally than its SLIM counterpart. Fig. 3(a) and (b) showsthe missing samples estimated by CG-SLIM and GS-SLIM, re-spectively. As we can see, the missing samples can be estimatedquite accurately. To mitigate the biased amplitude estimationproblem of CG-SLIM and GS-SLIM, we can apply the effi-cient complete-data IAA algorithm discussed in Section III-B,i.e., Fast-IAA-GS, to the estimated missing samples obtainedusing the SLIM variants, as well as the given samples. Werefer to these two new spectral estimation procedures (i.e.,CG-SLIM followed by Fast-IAA-GS, and GS-SLIM followedby Fast-IAA-GS) as CG-SLIM-IAA and GS-SLIM-IAA, re-spectively. Note that due to the usage of the GS-type formula,Fast-IAA-GS is computationally quite efficient. Furthermore,the Fast-IAA-GS step can be initialized by using the CG-SLIMor GS-SLIM spectral estimate to reduce the needed numberof iterations. Fig. 4(a) and (b) shows the spectral estimatesobtained by CG-SLIM-IAA and GS-SLIM-IAA, respectively.The Fast-IAA-GS iteration number is set to be 5 for both algo-rithms. From Fig. 4(a) and (b), we can see that CG-SLIM-IAAand GS-SLIM-IAA provide more accurate amplitude esti-mates than CG-SLIM and GS-SLIM. In particular, comparingFigs. 2(c) and 4(a), we see that CG-SLIM-IAA provides asimilar spectral estimate as Fast-IAA. However, as we willshow below, CG-SLIM-IAA is computationally much moreefficient than Fast-IAA.

The computation times needed by the 1-D spectral estimationmethods to obtain Figs. 2 and 4 are shown in Table VII. Clearly,

Fig. 5. Slicy object and benchmark SAR image. (a) Photograph of the object(taken at 45 azimuth angle), and (b) benchmark SAR image formed with a288� 288 (not 40� 40) data matrix.

Fig. 6. Modulus of the SAR images of the Slicy object obtained from a 40� 40complete data matrix. (a) Available data pattern, (b) WFFT, (c) IAA, (d) Fast-IAA-GS, (e) CoSaMP, (f) SLIM, (g) CG-SLIM, and (h) GS-SLIM.

the proposed fast implementations of IAA and SLIM arecomputationally much more efficient than their original coun-terparts. In particular, the proposed CG-SLIM and GS-SLIMalgorithms reduce the computation times needed by SLIM from44.34 seconds to 0.4 and 1.4 seconds, respectively. The compu-tational efficiencies of CG-SLIM-IAA and GS-SLIM-IAA areclose to those of CG-SLIM and GS-SLIM, respectively. In par-ticular, CG-SLIM-IAA needs only 0.78 seconds, which amountto only 9.3% of the computation time needed by Fast-IAA(8.39 seconds, while providing a similar spectral estimationperformance to Fast-IAA. Although for this 1-D spectral esti-mation example, CG-SLIM is computationally more efficientthan GS-SLIM, as we will show in Section IV-B, GS-SLIMcan be computationally more efficient than CG-SLIM for 2-Dinterrupted SAR imaging.


Fig. 7. Modulus of the interrupted SAR images of the Slicy object under arandom interruption condition (68% data available compared to Fig. 6). (a)Available data pattern, (b) WFFT, (c) IAA, (d) Fast-IAA, (e) CoSaMP, (f) SLIM,(g) CG-SLIM, and (h) GS-SLIM.

B. 2-D Interrupted SAR Imaging

In the following examples, we illustrate the applications ofthe missing-data spectral analysis algorithms to interruptedSAR imaging using incomplete phase history data.

We use the two-dimensional phase-history data of an objectcalled Slicy generated at 0 azimuth angle using XPATCH [35],a high-frequency electromagnetic scattering prediction code forcomplex 3-D objects. A photo of the Slicy object taken at 45azimuth angle and a SAR image benchmark obtained from acomplete 288 288 data matrix are shown in Fig. 5(a) and (b),respectively. Below, we use only a 40 40 center block of thephase-history data for SAR imaging.

Figs. 6–8 show the SAR images obtained from the 40 40complete data, and two randomly interrupted data. The avail-able data patterns for the two interruption cases are shown inFigs. 7(a) and 8(a), where the available data ratios are 68% and30%, respectively. We provide the SAR images obtained byIAA, SLIM, and their fast variants, i.e., Fast-IAA/Fast-IAA-GS,CG-SLIM and GS-SLIM, as well as WFFT and CoSaMP. Notethat we use Fast-IAA-GS for the complete data case in Fig. 6,and Fast-IAA for the two missing data cases in Figs. 7 and 8.The iteration numbers for all iterative approaches, includingCoSaMP, IAA, and SLIM, are all fixed to 10. We set theCoSaMP user parameter to be 3000 (which appears to giventhe best image for CoSaMP), the SLIM parameter , and

Fig. 8. Modulus of the interrupted SAR images of the Slicy object under arandom interruption condition (30% data available compared to Fig. 6). (a)Available data pattern, (b) WFFT, (c) IAA, (d) Fast-IAA, (e) CoSaMP, (f) SLIM,(g) CG-SLIM, and (h) GS-SLIM.

TABLE VIIICOMPUTATION TIMES NEEDED BY IAA, SLIM, AND THEIR

FAST IMPLEMENTATIONS FOR INTERRUPTED SAR IMAGING

UNDER VARIOUS INTERRUPTION CONDITIONS

for CG-SLIM. The noise variance estimate ofSLIM is initialized using (20).

Figs. 6(b), 7(b), and 8(b) show the SAR images formed byWFFT, where a 2-D Taylor window with a 30 dB sidelobelevel is applied. Again, WFFT has low-resolution and high-sidelobe problems, especially for the interrupted scenarios. Theobject features are smeared under the interrupted data condi-tions. We can also see that the CoSaMP approach fails to workproperly under all conditions. IAA, SLIM, and their fast vari-ants, i.e., Fast-IAA, CG-SLIM and GS-SLIM, significantly im-prove the interrupted SAR imaging quality, in terms of bothhigher imaging resolution and lower sidelobe levels, as evi-denced by sharper SAR images with less ghost artifacts. In this


example, SLIM and its variants provide the best imaging per-formance, as compared to the benchmark in Fig. 5 under all in-terruption conditions. Note that only 0.56% of the data sampleswas used to form the SAR images in Fig. 8, compared to theentire 288 288 data samples in Fig. 5.

Table VIII provides the computation times needed by theaforementioned algorithms under various interruption condi-tions. Again, the proposed fast implementation approaches havesignificantly enhanced computational efficiencies. In particular,for the interruption example in Fig. 7, GS-SLIM needs only 2.4seconds, which amount to only 0.4% of the computation timeneeded by SLIM (584.1 seconds).

V. CONCLUSION

We have considered using data-adaptive methods, namelyIAA and SLIM, for spectral estimation of data sequenceswith missing samples. Both methods can provide significantlyimproved spectral estimates, in terms of spectral resolutionand lower sidelobe levels, as compared to the conventionaldata-independent approaches. We have introduced severalfast implementations of these algorithms, namely Fast-IAA,CG-SLIM and GS-SLIM, by exploiting the structures of thesteering matrices and maximizing the utility of FFTs. Viaseveral numerical examples of both 1-D spectral analysisand 2-D interrupted SAR imaging, we have demonstrated theeffectiveness of IAA and SLIM for spectral estimation and thecomputational efficiencies of their fast implementations.

REFERENCES

[1] J. Salzman, D. Akamine, R. Lefevre, and J. C. Kirk, “Interrupted syn-thetic aperture radar (SAR),” IEEE Aerosp. Electron. Syst. Mag., vol.17, no. 5, pp. 33–39, May 2002.

[2] J. A. Bruder and R. Schneible, “Interrupted SAR waveforms for highinterrupt ratios,” in Proc. IET Int. Conf. Radar Systems, 2007.

[3] Y. Wang, J. Li, and P. Stoica, Spectral Analysis of Signals: The MissingData Case. San Rafael, CA: Morgan & Claypool, 2005.

[4] J. Salzman, D. Akamine, R. Lefevre, and J. Kirk, Jr., “Interrupted syn-thetic aperture radar (SAR),” in Proc. IEEE Radar Conf., 2001, pp.117–122.

[5] J. A. Högbom, “Guaranteeing practical convergence in algorithms forsensor and source localization,” IEEE Trans. Signal Process., vol. 56,no. 9, pp. 4458–4469, Sep. 2008.

[6] T. Bronez, “Spectral estimation of irregularly sampled multidimen-sional processes by generalized prolate spheroidal sequences,” IEEETrans. Acoust., Speech, Signal Process., vol. 36, no. 12, pp. 1862–1873,Dec. 1988.

[7] I. Fodor and P. Stark, “Multitaper spectrum estimation for timeseries with gaps,” IEEE Trans. Signal Process., vol. 48, no. 12, pp.3472–3483, Dec. 2000.

[8] R. H. Jones, “Maximum likelihood fitting of ARMA models to timesseries with missing observations,” Technometrics, vol. 22, pp. 389–395,Aug. 1980.

[9] B. Porat and B. Friedlander, “ARMA spectral estimation of time se-ries with missing observations,” IEEE Trans. Inf. Theory, vol. 30, pp.601–602, Jul. 1986.

[10] Y. Rosen and B. Porat, “The second-order moments of the sample co-variances for time series with missing observations,” IEEE Trans. Inf.Theory, vol. 35, no. 3, pp. 334–341, Mar. 1989.

[11] P. Broersen, S. de Waele, and R. Bos, “Estimation of autoregressivespectra with randomly missing data,” in Proc. 20th IEEE Instrum.Meas. Technol. Conf., Vail, CO, May 2003, vol. 2, pp. 1154–1159.

[12] Y. Wang, P. Stoica, J. Li, and T. L. Marzetta, “Nonparametric spec-tral analysis with missing data via the EM algorithm,” Digital SignalProcess., vol. 15, pp. 191–206, 2005.

[13] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihoodfrom incomplete data via the EM algorithm,” J. R. Statist. Soc. B, vol.39, pp. 1–18, 1977.

[14] D. Needell and J. Tropp, “CoSaMP: Iterative signal recovery from in-complete and inaccurate samples,” Appl. Comput. Harmon. Anal., vol.26, pp. 301–321, May 2009.

[15] J. Capon, “High resolution frequency-wavenumber spectrum analysis,”Proc. IEEE, vol. 57, no. 8, pp. 1408–1418, Aug. 1969.

[16] J. Li and P. Stoica, “An adaptive filtering approach to spectral estima-tion and SAR imaging,” IEEE Trans. Signal Process., vol. 44, no. 6,pp. 1469–1484, Jun. 1996.

[17] E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from in-complete and inaccurate measurements,” Commun. Pure Appl. Math.,vol. 59, pp. 1207–1223, 2006.

[18] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer, “Sourcelocalization and sensing: A nonparametric iterative adaptive approachbased on weighted least squares,” IEEE Trans. Aerosp. Electron. Syst.,vol. 46, no. 1, pp. 425–443, Jan. 2010.

[19] X. Tan, W. Roberts, and J. Li, “A new sparse sensing approach forMIMO radar imaging,” in Proc. 8th Eur. Conf. Synth. Aperture Radar,Jun. 2010.

[20] X. Tan, W. Roberts, J. Li, and P. Stoica, “Sparse learning via iterativeminimization with application to MIMO radar imaging,” IEEE Trans.Signal Process., vol. 59, no. 3, pp. 1088–1101, Mar. 2011.

[21] M. Xue, L. Xu, J. Li, and P. Stoica, “IAA spectral estimation: Fast im-plementation using the Gohberg-Semencul factorization,” IEEE Trans.Signal Process., vol. 59, no. 7, pp. 3251–3261, Jul. 2011.

[22] G.-O. Glentis and A. Jakobsson, “Time-recursive IAA spectral esti-mation,” IEEE Signal Process. Lett., vol. 18, no. 2, pp. 111–114, Feb.2011.

[23] G.-O. Glentis and A. Jakobsson, “Efficient implementation of iterativeadaptive approach spectral estimation techniques,” IEEE Trans. SignalProcess., vol. 59, no. 9, pp. 4154–4167, Sep. 2011.

[24] G.-O. Glentis and A. Jakobsson, “Superfast approximative implemen-tation of the IAA spectral estimate,” IEEE Trans. Signal Process., 2011,to be published.

[25] G.-O. Glentis and A. Jakobsson, “Preconditioned conjugate gradientIAA spectral estimation,” in Proc. 19th Eur. Signal Process. Conf.,Barchelona, Spain, Aug. 29–Sep. 2 2011.

[26] D. A. Harville, Matrix Algebra From a Statistician’s Perspective.New York: Springer, 1997.

[27] W. Roberts, T. Yardibi, J. Li, P. Stoica, and R. A. Sadjadi, “Iterativeadaptive approaches to MIMO radar imagings,” IEEE J. Sel. TopicsSignal Process., vol. 4, no. 1, pp. 5–10, Feb. 2010.

[28] P. Stoica and Y. Selén, “Cyclic minimizers, majorization techniques,and expectation-maximization algorithm: A refresher,” IEEE SignalProcess. Mag., vol. 21, no. 1, pp. 112–114, Jan. 2004.

[29] G. H. Golub and C. F. V. Loan, Matrix Computations. Baltimore,MD: Johns Hopkins Univ. Press, 1989.

[30] J. R. Jain, “An efficient algorithm for a large Toeplitz set of linear equa-tions,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-27, no.6, pp. 612–615, Dec. 1979.

[31] G. Glentis, “A fast algorithm for APES and Capon spectral estimation,”IEEE Trans. Signal Process., vol. 56, no. 9, pp. 4207–4220, Sep. 2008.

[32] M. Wax and T. Kailath, “Efficient inversion of Toeplitz-block Toeplitzmatrix,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-31,no. 10, pp. 1218–1221, Oct. 1983.

[33] P. Stoica, J. Li, and J. Ling, “Missing data recovery via a nonparametriciterative adaptive approach,” IEEE Signal Process. Lett., vol. 16, no. 4,pp. 241–244, Apr. 2009.

[34] J. Li and P. Stoica, “Efficient mixed-spectrum estimation with appli-cations to target feature extraction,” IEEE Trans. Signal Process., vol.44, no. 2, pp. 281–295, Feb. 1996.

[35] D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, andY. Chu, “XPATCH: A high-frequency electromagnetic scattering pre-diction code and environment for complex three-dimensional objects,”IEEE Antennas Propag. Mag., vol. 36, pp. 65–69, Feb. 1994.


Duc Vu (S’04-M’06) received the B.S. degree inelectrical engineering from Iowa State University,Iowa City, in 2005, and the M.S. degree in electricalengineering from University of Southern California,Los Angeles, in 2008. He is currently workingtoward the Ph.D. degree in electrical engineering atthe University of Florida, Gainesville.

His research interests are in the areas of arraysignal processing, MIMO radar, and SAR imagereconstruction.

Luzhou Xu (S’04–M’06) received the B.Eng. andM.S. degrees in electrical engineering from ZhejiangUniversity, Hangzhou, China, in 1996 and 1999, re-spectively, and the Ph.D. degree in electrical engi-neering from the University of Florida, Gainesville,in 2006.

He is currently a Senior Engineer at the IntegratedAdaptive Applications (IAA), Inc., and an AdjunctAssistant Professor at the Department of Electricaland Computer Engineering, University of Florida. Hewas with Zhongxing Research and Development In-

stitute, Shanghai, China, from 1999 to 2001, with Philips Research East Area,Shanghai, China, from 2001 to 2003, and with ArrayComm LLC, San Jose, CA,from 2006 to 2008. His research interests include statistical and array signal pro-cessing and their applications.

Ming Xue (S’06) received his Ph.D. and M.S.degrees in electrical engineering from University ofFlorida, Gainesville, FL, in 2011 and 2008, respec-tively, and the B.S. degree in electrical engineeringfrom the University of Science and Technology ofChina, Hefei, China, in 2006. He is currently a PostDoctoral Fellow in the Division of Medical Physics,Department of Radiation Oncology, at University ofMaryland School of Medicine, Baltimore, MD.

His research interests includes statistical and arraysignal processing, multiple-input multiple-output

(MIMO) radar signal processing, and fast algorithms.

Jian Li (S’87–M’91–SM’97–F’05) received theM.Sc. and Ph.D. degrees in electrical engineeringfrom Ohio State University, Columbus, in 1987 and1991, respectively.

From April 1991 to June 1991, she was an AdjunctAssistant Professor with the Department of ElectricalEngineering, Ohio State University. From July 1991to June 1993, she was an Assistant Professor withthe Department of Electrical Engineering, Universityof Kentucky, Lexington. Since August 1993, she hasbeen with the Department of Electrical and Computer

Engineering, University of Florida, Gainesville, where she is currently a Pro-fessor. In Fall 2007, she was on sabbatical leave at the Massachusetts Instituteof Technology (MIT), Cambridge. Her current research interests include spec-tral estimation, statistical and array signal processing, and their applications.

Dr. Li is a Fellow of IET. She is a member of Sigma Xi and Phi Kappa Phi. Shereceived the 1994 National Science Foundation Young Investigator Award andthe 1996 Office of Naval Research Young Investigator Award. She was an Ex-ecutive Committee Member of the 2002 International Conference on Acoustics,Speech, and Signal Processing, Orlando, FL, in May 2002. She was an AssociateEditor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1999 to 2005,an Associate Editor of the IEEE Signal Processing Magazine from 2003 to 2005,and a member of the Editorial Board of Signal Processing, a publication of theEuropean Association for Signal Processing (EURASIP), from 2005 to 2007.She has been a member of the Editorial Board of Digital Signal Processing—AReview Journal, a publication of Elsevier, since 2006. She is currently a memberof the Sensor Array and Multichannel (SAM) Technical Committee of IEEESignal Processing Society. She is a coauthor of the papers that have receivedthe First and Second Place Best Student Paper Awards, respectively, at the 2005and 2007 Annual Asilomar Conferences on Signals, Systems, and Computers inPacific Grove, CA. She is also a coauthor of the paper that has received the M.Barry Carlton Award for the best paper published in the IEEE TRANSACTIONS

ON AEROSPACE AND ELECTRONIC SYSTEMS in 2005.

Nonparametric Missing Sample Spectral Analysis and Its Applications to Interrupted SAR

Documents

Transcript of Nonparametric Missing Sample Spectral Analysis and Its Applications to Interrupted SAR