multidimensional sinusoidal frequency estimation...

5536 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 10, OCTOBER 2012

[3] C. R. Rao, “Information and the accuracy attainable in the estimationof statistical parameters,” Bull. Calcutta Math. Soc., vol. 37, pp. 81–91,1945.

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[6] C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed.New York: Wiley, 1973.

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[9] P. Stoica and B. C. Ng, “On the Cramér-Rao bound under parametricconstraints,” IEEE Signal Process. Lett., vol. 5, pp. 177–179, July 1998.

[10] P. Stoica and T. L. Marzetta, “Parameter estimation problems with sin-gular information matrices,” IEEE Trans. Signal Process., vol. 49, pp.87–90, Jan. 2001.

[11] J. O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nded. New York: Springer-Verlag, 1985.

[12] H. L. Van Trees and K. L. Bell, Bayesian Bounds for Parameter Estima-tion and Nonlinear Filtering/Tracking. Piscataway, NJ: IEEE Press,2007.

[13] E. Song, Y. Zhu, J. Zhou, and Z. You, “Minimum variance in biased es-timation with singular Fisher information matrix,” IEEE Trans. SignalProcess., vol. 57, pp. 376–381, Jan. 2009.

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[15] B. Su and P. P. Vaidyanathan, “Subspace-based blind channel iden-tification for cyclic prefix systems using few received blocks,” IEEETrans. Signal Process., vol. 55, pp. 4979–4993, Oct. 2007.

[16] J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimationwith constraints,” IEEE Trans. Inf. Theory, vol. 26, pp. 1285–1301,Nov. 1990.

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[19] S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Performance anal-ysis of a deterministic channel estimator for block transmission sys-tems with null guard intervals,” IEEE Trans. Signal Process., vol. 50,pp. 684–695, Mar. 2002.

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Multidimensional Sinusoidal Frequency Estimation UsingSubspace and Projection Separation Approaches

Longting Huang, Yuntao Wu, H. C. So, Yanduo Zhang, andLei Huang

Abstract—In this correspondence, a computationally efficient methodthat combines the subspace and projection separation approaches isdeveloped for �-dimensional (�-D) frequency estimation of multiplesinusoids, where � � �, in the presence of white Gaussian noise. Throughextracting a 2-D slice matrix set from the multidimensional data, we devisea covariance matrix associated with one dimension, from which the corre-sponding frequencies are estimated using the root-MUSIC method. Withthe use of the frequency estimates in this dimension, a set of projectionseparation matrices is then constructed to separate all frequencies in theremaining dimensions. Root-MUSIC method is again applied to estimatethese single-tone frequencies while multidimensional frequency pairing isautomatically attained. Moreover, the mean square error of the frequencyestimator is derived and confirmed by computer simulations. It is shownthat the proposed approach is superior to two state-of-the-art frequencyestimators in terms of accuracy and computational complexity.

Index Terms—Multidimensional frequency estimation, subspacemethod, projection separation.

I. INTRODUCTION

The topic of�-dimensional (�-D) frequency estimation, where� ��, has received extensive attention for its widespread applications innumerous fields such as MIMO wireless channel sounding [1], mobilecommunications [2], MIMO radar [3], sonar, seismology and nuclearmagnetic resonance spectroscopy [4]. Many high-resolution subspace-based parameter estimation techniques have been proposed to solvethis problem such as multidimensional folding (MDF) [1], unitary ES-PRIT [5], improved MDF (IMDF) [6], MUSIC [7], rank reduction es-timator (RARE) [8], decoupled root-MUSIC [9] and higher-order sin-gular value decomposition (HOSVD) [10] methods. Unitary ESPRIT,MDF and IMDF techniques are based on the conventional ESPRIT ap-proach where it is difficult to directly utilize the original multidimen-sional data. Typically, they require to enlarge the received data to con-struct a 2-D matrix with larger size, and then employ the ESPRIT-basedmethod to obtain the desired frequency pairs. Consequently, their com-putational burden is very heavy particularly when the data size is large.The RARE algorithm vectorizes the observed data to exploit the Van-dermonde structure and matrix polynomials are constructed to find thefrequencies of each dimension. However, computing the polynomialcoefficients is a highly demanding job. In the decoupled root-MUSIC

Manuscript received April 09, 2012; accepted June 12, 2012. Date of publi-cation July 10, 2012; date of current version September 11, 2012. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Z. Jane Wang. This work was jointly supported by a Grant fromthe National Natural Science Foundation of China (Project No. 61172150) andby the excellent youth science and technology innovation team project of theEducational Department of Hubei Province of China (No. T201206).

L. Huang and H. C. So are with the Department of Electronic Engineering,City University of Hong Kong, Hong Kong 852, China.

Y. Wu and Y. Zhang are with the School of Computer Science and En-gineering, Wuhan Institute of Technology, Wuhan 430073, China (e-mail:[email protected]).

L. Huang is with the Shenzhen Graduate School, Harbin Institute of Tech-nology, Shenzhen 518055, China.

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

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 10, OCTOBER 2012 5537

algorithm, �-D harmonic retrieval is decomposed into � 1-D prob-lems by tensor decomposition [11], [12], which significantly reducesthe computational load, but pairing of the�-D frequencies is required.On the other hand, the HOSVD method utilizes the structure inherentin the received data at the expense of a high computational complexity.In this work, our main contribution is to devise an accurate and com-putationally efficient estimator for multidimensional frequencies in thepresence of white Gaussian noise with the use of the subspace and pro-jection separation techniques.

The rest of this correspondence is organized as follows. The devel-opment of the proposed estimator is presented in Section II. There aretwo basic estimation steps as follows. We first extract a 2-D slice ma-trix set from the�-D signal to construct a covariance matrix associatedwith the first dimension, from which the corresponding frequencies areestimated using the root-MUSIC method. With the use of the frequencyestimates in the first dimension, a set of projection separation matricesis then devised to separate all frequencies in the remaining dimensions.Root-MUSIC method is again utilized to find these single-tone fre-quencies while multidimensional frequency pairing is automatically at-tained. In Section III, the computational complexity and mean squareerror (MSE) of the devised estimator are analyzed. Since our methodexploits covariance matrices whose size is characterized by the numberof sinusoids, its computational requirement is small when comparedwith the conventional schemes. Section IV includes numerical exam-ples for validating the theoretical findings and evaluating the proposedapproach by comparing with the IMDF [6] and HOSVD [10] methodsas well as Cramér-Rao lower bound (CRLB). Finally, conclusions aredrawn in Section V.

II. PROPOSED METHOD

A. Signal Model

The observed �-D signal model is

��

�� (1)

where

��

�

��

�

�

��

�� (2)

with � being the number of sinusoids, which is assumed to be knowna priori [13]. Here, � is the unknown complex amplitude of � th tone, �� is the unknown frequency of � th component in the�th dimension. The �� is a �-D circular complex whiteGaussian noise with mean zero and unknown variance �� . It is assumed

that �� , and the frequencies are distinct in atleast one of the � dimensions. Given the � � �

�� samples of�� , our task is to find the �� unknown frequency param-eters, namely, � �� .

B. Estimation in First Dimension

For ease of presentation but without loss of generality, we startfrequency estimation in the first dimension with the assumption that �� . In tensor form, (1) is

�� (3)

where �� and�� . That is, �� and are thenoise-free and noise-only components of �� , respectively.

To reduce the dimension of �� , we define �� , which is a set of2-D slice matrices extracting from �� , as follows:

��

�� (4)

where

��

��

��

Similarly, �� is a noise-free 2-D matrix set extracting from �� andeach of the �� has theform of (5), shown at the bottom of the page.

It is easy to verify that �� can bewritten as

��

� �� (6)

where

�� (7)

��

��

�

(8)

��

� ��

� � � �� (9)

��

� ��

�

��

��

�

��

��

�

��

�� (10)

�

��

�

�

��

��

�

��

�

�

��

��

�

��

�

�

��

��

�

��

�

�

��

��

.... . .

...�

��

�

�

��

��

�

��

�

�

��

��

� (5)


with � � �� and�� being the conjugate transpose, transpose,and diagonal matrix with vector � as its main diagonal, respectively.

Define �� as the sample covariance matrix of matrix set�� , whichis computed as

��

�

�

� ��

�

� ��

� � �

�

� ��

��

�� (11)

According to the structure of slice matrix �� , theexpected value of ��, denoted by ��, is

��

�

�

� ��

� � �

�

� ��

��

��

��

��

��

��

�

� ��

� � �

�

� ��

��

��

��

��

� ��

�� (12)

where

��

�

�

� ��

� � �

�

� ��

��

��

�� (13)

and �� is the �� identity matrix.In order to circumvent the problem of degraded estimation perfor-

mance in case of closely spaced frequencies in the same dimension,we further propose the forward-backward (FB) smoothing [14], [15]covariance matrix of ��:

��

�

�� (14)

where �� is the exchange matrix with ones on its antidi-agnoal and zeros elsewhere.

The expected value of �� is

��

�

��

�

��

��

�� (15)

where

��

��

�

�� (16)

��

� ��

� � �� (17)

On the other hand, �� can be decomposed using eigenvalue de-

composition (EVD) as

��

��

�� (18)

where the column vectors of �� and �� are the eigenvectors that span the signal and noise

subspaces of ��, respectively, with the associated eigenvalues beingthe diagonal elements of �� and ��.

Let

�� (19)

Employing root-MUSIC method to estimate the frequencies based on��, a null-spectrum function is constructed as:

��

�� (20)

where �� . The polynomial �� has �� roots and the first dimension frequency estimates �� are ob-tained from the � largest-magnitude roots inside the unit circle ac-cording to � � � .

C. Estimation in Remaining Dimensions

With the use of �� , a set of projection matrices �� , is constructed to estimate all � frequencies one by one ineach of the �th dimension, � � � �� . The �� is defined as

��

��

� � � � �� (21)

where

� ��

(22)

��

�

��

�

� ��

� (23)

Analogous to (4), we construct the 2-D slice matrices set ��

and then use the projection matrix �� on �� to obtain �� which contains only the information of � �� inthe �th dimension associated with � ��.

The �� is defined as

��

� ��

� ��

� �� (24)

where � � � �� .Similar to (11), we define a covariance matrix ��

��

�

�

� ��

�

� ��

�

� ��

� � �

�

� ��

��

� �� (25)

where � � � �� .The expected value of �� is calculated as

� ��

�

�

� ��

� � �

�

� ��

�

� ��

� � �

�

� ��

��

� ��

�

��

��

��

� ��

�� (26)


where

��

�

�

� ��

� � �

�

� ��

�

� ��

� � �

�

� ��

��

��

�

�

�

��

� ��

� � �� (27)

and

��

� �� (28)

with � being the �� zeros vector.Furthermore, the FB smoothing version of �� is

��

�

�� (29)

where �� is the exchange matrix.The expected value of ��

�� is

��

�

��

�

��

��

(30)

where��

�

��

�

�� (31)

� � �� (32)

Taking EVD on �� yields

��

��

��

� � �� (33)

where the column vectors of �� and �� are the eigenvectors that span the signal and noise

subspaces of �� , respectively, with the associated eigenvalues beingthe diagonal elements of �� and �� .

Define �� of the form: ��

��

� � �� (34)

We see that the root-MUSIC method can also be used to obtain the� th component in the �th dimension frequency, ��, which are au-tomatically paired with �� with the use of �� . The correspondingnull-spectrum function is

�� (35)

where �� . The polynomial �� has�� roots and the �th dimension frequency of � th component�� is obtained from the largest-magnitude root inside the unit circleaccording to � � �� .

To summarize, the steps in the overall estimation procedure are:1) Compute ��

� in (14) and perform its EVD2) Compute � using (19) and use (20) to compute ��

�� 3) Construct �� in (21) and compute ��

�� using (29)4) Take the EVD on ��

�� , compute �� according to (34) and use(35) to obtain ��

TABLE ICOMPUTATIONAL COMPLEXITY OF FB-ROOT-MUSIC ALGORITHM

III. ALGORITHM ANALYSIS

A. Computational Complexity

The computational complexity of the proposed root-MUSIC algo-rithm is studied and the results are provided in Table I. In summary,its complexity is ��

�� . On the otherhand, the computational requirement in the IMDF method [6] isabout �� and that of the HOSVD algorithm [10] is at least��

�� where �� is the number of spatialsmoothing subarrays in the �th dimension, � � �� . Clearly,the proposed method is more computationally attractive than [6]and [10].

B. Mean Square Error

Analogous to (18) and (33), we define

��

��

�� (36)

��

��

�� (37)

where ��

�� and ��

��

�� . Note

that �� , are the signal subspace eigenvectors of thefirst dimension, �� are the associated eigenvalues with�� while�� isthe noise subspace of the first dimension. Meanwhile, �� and ��are the eigenvector and eigenvalue of the signal subspace of the � thcomponent in the �th dimension frequency, respectively, and �� isthe noise subspace corresponding to ��.

Following [15], the MSEs of the first dimension frequency estimatesare computed as

��

��

��

��

�

��

��

��

��

�

��

(38)

where �� , and is the expectation operator.On the other hand, the MSEs of the frequency estimates in the re-

maining dimensions are

��

��

��

��

��

��

�

��

(39)

where �� .

IV. SIMULATION RESULTS

Computer simulations have been conducted to evaluate the fre-quency estimation performance of the proposed approach for multiple3-D and 4-D sinusoids in the presence of white Gaussian noise.The algorithms without and with FB smoothing, denoted by root


Fig. 1. �� versus SNR with distinct frequencies.


MUSIC and FB root MUSIC, are examined. The average MSEof the �th dimension, denoted by �� , is employed as theperformance measure. All results provided are averaged from 200independent runs. Apart from CRLB [6], [17] and [18], the per-formance of the proposed approach is compared with that of theIMDF [6], HOSVD [10] algorithms. The signal power is defined as��

� ��

� �� and ��

is scaled to produce different signal-to-noise ratio (SNR) which isdefined as ��

�

� .In the first test, we consider 3-D frequency estimation with ��

�� and the number of tones is � � �. The sinu-soidal parameters are �� and �� . The results of �� versus SNR are plottedin Figs. 1–3. It is observed that although the MSE of the proposedapproach in the first dimension is comparable with that of HOSVDscheme, the former is superior to [6] and [10], with performance closeto CRLB, in the remaining dimensions. The theoretical calculationsof (38) and (39) also agree very well with the simulation results forsufficiently high SNR conditions. The above test is repeated with iden-tical frequency scenario of ��


Fig. 4. �� versus SNR with identical frequencies.


� �� and �� , and the re-sults are plotted in Figs. 4–6. In this challenging case, the proposed



Fig. 7. Computational time versus� .

method without FB smoothing performs the worst because there areidentical frequencies in two dimensions. Nevertheless, the one with FBsmoothing outperforms [6] and [10] for all dimensions and its perfor-mance attains CRLB in the second and third dimensions.

In the second experiment, the average computational time of the in-vestigated algorithms versus different �� with �� in3-D frequency estimation are plotted in Fig. 7. It is seen that the pro-posed approach is much more computationally efficient than the IMDFand HOSVD algorithms, which aligns with the complexity analysis inSection III.

In the third experiment, �� for different frequency separationby varying �� is studied. We consider two 3-D tones with ��

�� . The signal parameters are ��

�� and �� while�� is varying from �� to ��. The corresponding results at��

�� dB are shown in Figs. 8–10. It is observed that the estimation per-formance of proposed approach is inferior to the IMDF and HOSVDschemes in the first dimension when the frequency separation is lessthan ��. Nevertheless, its performance is comparable to them in theother two dimensions.

Fig. 8. �� versus frequency separation.



Finally, 4-D frequency estimation �� versus SNR is studiedand the results are plotted in Figs. 11–14. The number of sinusoids is


Fig. 11. �� versus SNR for 4D Data.



� � � while �� . The signal parame-ters are ��


�� and ��

�� .Again, we observe that the performance of the proposed estimation

is comparable to those of the other two methods [6] and [10]. However,the proposed method attains the CRLB and is better than other the twomethods for dimensions 2, 3, 4 at sufficiently high SNRs.

V. CONCLUSION

A new approach for frequency estimation of multidimensional si-nusoids in additive white circular Gaussian noise has been developed.The main idea in our methodology is to rearrange the -D sinusoidsas a series of 2-D slice matrices and combine the subspace and projec-tion separation techniques. The frequencies in one dimension are firstestimated using the root-MUSIC algorithm, which are then utilized toseparate the frequencies in the remaining dimensions. Using the sep-arated data, the remaining frequencies are then estimated one by oneusing the root-MUSIC method such that the multidimensional parame-ters are automatically paired. It is shown that the proposed algorithm issuperior to the IMDF and HOSVD methods in terms of computationalload and estimation performance.

REFERENCES

[1] K. N. Mokios, N. D. Sidiropoulos, M. Pesavento, and C. F. Mecklen-bräuker, “On 3-D harmonic retrieval for wireless channel sounding,” inProc. IEEE Int. Conf. Acoust., Speech, Signal Process., Montreal, QC,Canada, May 2004, vol. 2, pp. II89–II92.

[2] X. Liu, N. D. Sidiropoulos, and A. Swami, “Blind high-resolution lo-calization and tracking of multiple frequency hopped signals,” IEEETrans. Signal Process., vol. 50, no. 4, pp. 889–901, Apr. 2002.

[3] D. Nion and N. D. Sidiropoulos, “Tensor algebra and multidimensionalharmonic retrieval in signal processing for MIMO radar,” IEEE Trans.Signal Process., vol. 58, no. 11, pp. 5693–5705, Nov. 2010.

[4] Y. Li, J. Razavilar, and K. J. R. Liu, “A high-resolution technique formultidimensional NMR spectroscopy,” IEEE Trans. Biomed. Eng., vol.45, no. 1, pp. 78–86, Jan. 1998.

[5] M. Haardt and J. A. Nossek, “Simultaneous schur decomposition ofseveral nonsymmetric matrices to achieve automatic pairing in multidi-mensional harmonic retrieval problems,” IEEE Trans. Signal Process.,vol. 46, no. 1, pp. 161–169, Jan. 1998.

[6] J. Liu and X. Liu, “An eigenvector-based approach for multidimen-sional frequency estimation with improved identifiability,” IEEE Trans.Signal Process., vol. 54, no. 12, pp. 4543–4556, Dec. 2006.


[7] H. L. Van Trees, Optimum Array Processing: Detection, Estimation,and Modulation Theory. New York: Wiley, 2002, pt. IV.

[8] M. Pesavento, C. F. Mecklenbräker, and J. F. Böme, “Multidimen-sional rank reduction estimator for parametric MIMO channel models,”EURASIP J. Appl. Signal Process., vol. 2004, pp. 1354–1363, Jan.2004.

[9] R. Boyer, “Decoupled root-music algorithm for multidimensional har-monic retrieval,” in Proc. IEEE Workshop Signal Process. Adv. Wire-less Commun., Recife, Brazil, Jul. 2008, pp. 16–20.

[10] M. Haardt, F. Roemer, and G. Del Galdo, “Higher-order SVD-basedsubspace estimation to improve the parameter estimation accuracy inmultidimensional harmonic retrieval problems,” IEEE Trans. SignalProcess., vol. 56, no. 7, pp. 3198–3213, Jul. 2008.

[11] L. de Lathauwer, B. de Moor, and J. Vanderwalle, “A multilinear sin-gular-value decomposition,” SIAM J. Matrix Anal. Appl., vol. 21, no.4, pp. 1253–1278, 2000.

[12] L. de Lathauwer, B. de Moor, and J. Vanderwalle, “On the best rank-1and rank-�� approximation of higher-order tensors,”SIAM J. Matrix Anal. Appl., vol. 21, no. 4, pp. 1324–1342, 2000.

[13] J. P. C. L. Da Costa, F. Roemer, M. Haardt, and R. T. de Sousa, Jr.,“Multidimensional model order selection,” EURASIP J. Adv. SignalProcess., 2011, 2011:26.

[14] D. A. Linebarger, R. D. DeGroat, and E. M. Dowling, “Efficient direc-tion-finding methods employing forward-backward averaging,” IEEETrans. Signal Process., vol. 42, no. 8, pp. 2136–2145, Aug. 1994.

[15] M. Pesavento, A. B. Gershman, and M. Haardt, “Unitary root-MUSICwith a real-valued eigendecomposition: A theoretical and experimentalperformance study,” IEEE Trans. Signal Process., vol. 48, no. 5, pp.1306–1314, May 2000.

[16] Y.-Y. Wang, J.-T. Chen, and W.-H. Fang, “TST-MUSIC for jointDOA-delay estimation,” IEEE Trans. Signal Process., vol. 46, pp.721–729, Apr. 2001.

[17] R. Boyer, “Deterministic asymptotic Cramér-Rao lower bound for themultidimensional harmonic model,” Signal Process., vol. 88, no. 12,pp. 2869–2877, Dec. 2008.

[18] X. Liu and N. D. Sidiropoulos, “Cramér-Rao lower bound for low-rank decomposition of multidimensional arrays,” IEEE Trans. SignalProcess., vol. 49, no. 9, pp. 2074–2086, Sep. 2001.

[19] A. Thakre, M. Haardt, F. Roemer, and K. Giridhar, “Tensor-based spa-tial smoothing (TB-SS) using multiple snapshots,” IEEE Trans. SignalProcess., vol. 58, no. 5, pp. 2715–2728, May 2010.

Computing Constrained Cramér-Rao Bounds

Paul Tune

Abstract—We revisit the problem of computing submatrices of theCramér-Rao bound (CRB), which lower bounds the variance of any un-biased estimator of a vector parameter ��. We explore iterative methodsthat avoid direct inversion of the Fisher information matrix, which canbe computationally expensive when the dimension of �� is large. Thecomputation of the bound is related to the quadratic matrix program,where there are highly efficient methods for solving it. We present severalmethods, and show that algorithms in prior work are special instancesof existing optimization algorithms. Some of these methods convergeto the bound monotonically, but in particular, algorithms convergingnonmonotonically are much faster. We then extend the work to encom-pass the computation of the CRB when the Fisher information matrixis singular and when the parameter �� is subject to constraints. As anapplication, we consider the design of a data streaming algorithm fornetwork measurement.

Index Terms—Cramér-Rao bound, Fisher information, matrix functions,optimization, quadratic matrix program.

I. INTRODUCTION

The Cramér-Rao bound (CRB) [11] is important in quantifying thebest achievable covariance bound on unbiased parameter estimation of� parameters ��. Under mild regularity conditions, the CRB is asymp-totically achievable by the maximum likelihood estimator. The compu-tation of the CRB is motivated by its importance in various engineeringdisciplines: medical imaging [6], blind system identification [18], andmany others.

A related quantity is the Fisher information matrix (FIM), whose in-verse is the CRB. Unfortunately, direct inversion techniques are knownfor their high complexity in space (�� bytes of storage) and time(�� floating point operations or flops). Often, one is just interestedin a portion of the covariance matrix. In medical imaging applications,for example, only a small region is of importance, which is related tothe location of a tumor or lesion. In this instance, computing the fullinverse of the FIM becomes especially tedious and intractable whenthe number of parameters is large. In some applications, the FIM it-self is singular, and the resulting Moore-Penrose pseudoinverse com-putation is even more computationally demanding. Avoiding the addi-tional overhead incurred from direct inversion or other forms of matrixdecompositions (Cholesky, QR, LU decompositions, for example) be-comes a strong motivation.

Prior work [7], [8] proves the tremendous savings in memory andcomputation by presenting several recursive algorithms computingonly submatrices of the CRB. Hero and Fessler [7] developed algo-rithms based on matrix splitting techniques, and statistical insightfrom the Expectation-Maximization (EM) algorithm. Only ��

flops are required per iteration, which is advantageous if convergenceis rapid, and the algorithms produce successive approximations thatconverge monotonically to the CRB. Exponential convergence was

Manuscript received January 23, 2012; revised April 23, 2012; accepted June05, 2012. Date of publication June 11, 2012; date of current version September11, 2012. The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Jean Pierre Delmas.

The author is with the School of Mathematical Sciences, The University ofAdelaide, Australia (e-mail: [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/TSP.2012.2204258

1053-587X/$31.00 © 2012 IEEE

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