B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Subspace Fitting (SSF) algorithm that approximates the maximum likelihood estimator of . The estimator is derived similarly to the SSF estimator presented in [1] and is thus presented in a somewhat condensed form. The SSF estimates of the delays and Doppler shifts can be found by minimizing [10, 11, 12] n[oporq f 3 st 60750 9 tx t p 3 (19) u wNdR denote the trace operator and the conjugate transpose, z is the orthogonal projection matrix z : 2a a a . 3 (20) is the matrix whose columns are the left singular vectors corresponding to the largest singular values ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

.... bin, maximum likelihood (ML) estimates of ; A; and S can be obtained (asymptotically) by minimizing the least squares criterion JML;1 ( S; A) kX Gamma SV( Ak Note that we have modeled A as a deterministic matrix, and so we use the deterministic rather than stochastic ML approach [25]. As the number of rows of X grows (in our application, as N 1) we expect that both deterministic and stochastic ML will yield asymptotically equivalent results, as in [26] The ML criterion of (35) is separable in either S or A, but not both simultaneously. If we set JML = A = 0 and solve for ....

....singular vectors associated with the d largest singular values of X (assuming d m) and is a certain diagonal weighting matrix formed from the singular values of X. The weighting can be chosen so that the MODE estimates approximate either the deterministic ML or stochastic ML solution [25]. Note that if d m, MODE and IQML coincide since the signal is no longer confined to a low rank subspace of spanfXg. There are two other important differences in the way that MODE and IQML are typically implemented, but they are of less consequence for the problem studied here. First, an initial ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

....to infinity. With the deterministic model, consistent estimation of all model parameters is impossible since the time varying path gains are regarded as unknown parameters. It follows that the number of parameters to estimate grows without bound with increasing N. Moreover it has been shown in [13] that the SML algorithm provides more accurate estimates than the deterministic ML technique. The difference is significant for very small number of sensors, low SNR and highly correlated signals. However the difference in performance is negligible in most scenarios of practical interest as we ....

....Derivation of the CRLB In this section, we derive the Cramr Rao lower bound (CRLB) on the covariance matrix of any unbiased estimator of and . The CRLB is derived under SML assumptions, i.e. the path gains are stochastic Gaussian processes. In theory the SML method outperforms the DML method [13]. This justifies the stochastic model being appropriate for the CRLB. The expression of the CRLB has been derived in [18] with the point source model presented in (2.1) in the SIMO case. As we have seen, our reshaped MIMO data model (3.7) is based on the same assumptions and the covariance matrix ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai. Exact and large sample ML techniques for parameter estimation and detection in array processing. In Haykin, Litva and Shepherd editors, Radar Array Processing, pages 99-151, SpringerVerlag, Berlin, 1993.

....covariance is shown in [99, 114] to be minimized by the weighting W,opt = A UA) where U = Es fi.2A E , and the resulting algorithm using this weighting is referred to as MODE. It can easily be shown that both WSF and MODE yield results with identical asymptotic second order error statistics [115]. Note also that the MUSIC algorithm is equivalent to (45) when W = I, and that deterministic ML is asymptotically equivalent to (45) when w 2 = or W = P [114] 8.7 . Identification via Subspace Fitting While the description of the above algorithms has been couched in the problem of DOA ....

....iteration is rel atively small. The number of iterations required for convergence depends of course on the quality of the initial estimates. When ESPRIT is used to obtain the starting point, adequate convergence can be expected in two to three iter ations. A number of empirical studies [53, 115] have indicated that WSF has better convergence properties than both deterministic and stochastic ML. In comparison with Subspace Fitting and Orthogonal Vector Methods (OVM) Single Shift Invariant methods (such as ESPRIT) are computationally more attractive. The number of operations required for ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and large sample ML techniques for parameter estimation and detection in array processing," in Radar Array Processing (S. Haykin, ed.), ch. 4, Springer- Verlag, 1991.

.... listed below are good starting points because of their tutorial nature and their extensive bibliographies: ffl general books [161, 311, 185] ffl connections with spectral analysis [197, 382] ffl adaptive beamforming [77, 440] ffl applications to radar systems [104, 163] ffl subspace methods [299, 34, 214] ffl applications in communications [298, 396, 306] A WWW link to the author of the above section: http: www.ee.byu.edu swindle 13 SSAP with Computational Acoustic and Electromagnetic Propagation Models Jeffrey Krolik Duke University Durham, NC 27708 Statistical signal and array ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, ""Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing"," in Radar Array Processing, Haykin, Litva, and Shepherd, editors, pp. 99--151, Springer-Verlag, Berlin, 1993.

....estimation, maximum likelihood. Introduction Most existing high resolution angles of arrival estimation algorithms, including MUSIC, ESPRIT, and MODE, do not assume any knowledge of the incident signals except for some general statistical properties such as the second order ergodicity (see [1] and the references therein) Recently, there has been a growing interest in developing angle estimators that exploit some a priori knowledge, e.g. the known waveforms, of the incident signals. Such estimators can be used in various applications including wireless communications where known ....

....and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA (Email: hli stevens tech.edu) z Guoqing Liu and Jian Li are with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA. 1 be known to within a constant [1], DEML and CDEML were specifically designed to deal with spatially colored observation noise. However, neither DEML nor CDEML are optimal when the observation noise can be reasonably modeled as spatially white. In this letter, we present a new angle estimator for signals with known waveforms (but ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing," in Radar Array Processing, S. Haykin et. al. (eds.), Chapter 4. New York, NY: Springer-Verlag Inc., 1993.

....noise subspace fitting (NSF) Both algorithms rely on a multidimensional search of roughly the same order of complexity as ML to estimate the DOAs. Since in addition both require an EVD, one may legitimately wonder what is gained by implementing them in lieu of ML. A number of empirical studies [11, 14] have demonstrated one advantage, indicating that WSF has better convergence properties than both conditional and unconditional ML. The primary drawback of using algorithms based on the EVD occurs when operating in a tracking or updating mode where, given a small number of additional snapshots ....

.... the eigenvalues of the array covariance [8, 20, 21, 22, 23, 24] or the asymptotic distribution of a cost function requiring an EVD [11] Two exceptions to this rule are the recently proposed Lanczos based algorithms of Xu and Kailath [25, 26] and the generalized likelihood ratio test proposed in [14]. In this paper, a new optimal algorithm is presented for the simultaneous detection and parameter estimation of narrowband signals. Like WSF, it yields a strongly consistent estimate of the number of signals as well as asymptotically minimum variance DOA estimates. However, it does so without ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

.... Gaussian noise, maximum likelihood (ML) estimates of ; A; and S can be obtained by minimizing the least squares criterion JML ( S; A) kX Gamma SV( Ak 2 F : 32) Note that we have modeled A as a deterministic matrix, and so we use the deterministic rather than stochastic ML approach [22]. As the number of rows of X grows (in our application, as N 1) we expect that both deterministic and stochastic ML will yield asymptotically equivalent results, as in [23] The ML criterion of (32) is separable in either S or A, but not both simultaneously. If we set JML = s( k ) 0, we ....

....singular vectors associated with the d largest singular values of X (assuming d m) and is a certain diagonal weighting matrix formed from the singular values of X. The weighting can be chosen so that the MODE estimates approximate either the deterministic ML or stochastic ML solution [22]. Note that if d m, MODE and IQML coincide since the signal is no longer confined to a low rank subspace of X. There are two other important differences in the way that MODE and IQML are typically implemented, but they are of less consequence for the problem studied here. First, an initial value ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

....Both the signal subspace fitting (SSF) 6] and noise subspace fitting (NSF) 7] methods will be considered. These algorithms have both been shown to have asymptotically the same performance as the standard maximum likelihood (ML) approach, as well as certain computational advantages over ML [8, 9, 10]. For the case of SSF, the optimal subspace weighting remains unchanged, but the form of the criterion function is modified. For NSF, the criterion remains the same, but the weighting is formed by taking a certain sub block of the weighting in the fully calibrated case. In either case, a ....

....d 0 (rank u 0 ) matrices 1 . Thus, for problems involving both CSS and USS signals, the array is simply parameterized by the vector j instead of directly. As long as the above model is identifiable, then both the SSF and NSF estimators of j are guaranteed to be statistically efficient [9, 10]. The conditions necessary for identifiability are examined below. 2.2. Identifiability As in the fully calibrated case, an unambiguous array response is required for the identifiability of all parameters when USS signals are present. As defined elsewhere, an unambiguous array is one for which ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

No context found.

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

....subspaces, a Noise Subspace Fitting (NSF) approach similar to [1, 3, 17, 21] may be taken. In this technique, the estimates are obtained as the minimizing arguments of the cost function V(w, 3 ) trace A (w)I (3 )g.g[I (3 )A(w)U where U = U 0 is a d x d Hermitian weighting matrix. From [10], U should be chosen as U = I (3 )A(co) E,A4(A, a2I)2E(I (3 )A(co) to obtain asymptotically minimum variance estimates. Replacing the weighting with a consistent estimate will not affect the asymptotic properties. Since 1 (3 0 is diagonal, this cost function may be rewritten as = ....

B. Ottersten, M. Viberg, P. Stoica, and Arye Nehorai. Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing. In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99-151. Springer-Verlag, Berlin, 1993.

.... A multitude of estimators have been presented for this problem and their asymptotic properties have been investigated, 1, 2] The stochastic maximum likelihood (ML) estimator, derived under the stochastic emitter signal model has been shown to produce asymptotically efficient parameter estimates [3, 4, 5], i.e. to achieve the Cram er Rao bound (CRB) in large samples. This important property of the stochastic ML estimator depends on the assumption that the emitter signals are non coherent. When coherent emitter signals are present, the maximum likelihood estimator must be reformulated and a new ....

....the somewhat intriguing result that the asymptotic estimation accuracy of these parameters is not effected by the knowledge of the rank of the emitter signal covariance. Thus, the original stochastic ML that ignores the rank information, or large samples realizations thereof such as MODE and WSF [8, 6, 4], achieve the lowest possible estimation error variance for the signal parameters. In closing this section, we should mention the related work by Bresler [9] The approach in [9] aims at deriving an exact realization of the ML estimate in either coherent or non coherent scenarios, and it is ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. SpringerVerlag, Berlin, 1993.

....using a parameterization of the noise subspace has a nice structure. W Gamma1 B = oe 2 N i Gamma2 s j Omega (B B) 22) Ignoring signal parameter independent terms, it is now straight forward to show that the cost function in (18) reduces to the well known WSF criterion [10], namely, V B N (j) Tr n B (B B) Gamma1 B U s Gamma1 s 2 U s o (23) Using the signal subspace parameterization, it can be shown that the weighting matrix is given by W Gamma1 A = oe 2 N U Gammac Omega (U n U n ) 24) where U Gamma1 = A U s ....

....U n U n o = Tr n U n A (j) UA (j) U n o : 26) It can be shown that the weighting matrix above is given by U = A (j) y U s Gamma1 s 2 U s A (j) y , where A y = A A) Gamma1 A . The cost function above corresponds to the well known MODE criterion [10]. Both these estimators provide asymptotically efficient parameter estimates under the assumptions stated above. 5 Conclusions Herein, a systematic subspace based estimation procedure is described which may be applied to low rank signal models. Two general subspace parameterizations are described ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, in Radar Array Processing, Haykin, Litva, and Shepherd, eds., Springer-Verlag, Berlin, 1993, pp. 99--151.

....j denotes the determinant. The exact ML criteria presented above, both suffer from the requirement of a nonlinear multidimensional optimization in order to calculate the estimates. This is particularly prominent when m is large, since the computational cost is at least proportional to m 2 , [9]. In general the ML criteria have several local minima and thus convergence to the global minimum cannot be guaranteed. To overcome this difficulty, several suboptimal techniques have been proposed. Perhaps the most natural of these is the traditional delay and sum beamforming method. This ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In S. Haykin, editor, Radar Array Signal Processing. Springer-Verlag, Wien - New York, 1992, To Appear.

....for example, radar or wireless communications, may give rise to coherent signals at the array. The stochastic maximum likelihood (ML) estimator has been derived under the stochastic emitter signal model for non coherent signals. It is known to achieve the Cram er Rao bound (CRB) in large samples [1, 2, 3]. This important property of the stochastic ML estimator depends on the assumption that the emitter signals are non coherent (i.e. no two signals are fully correlated) When coherent emitter signals are present, the maximum likelihood estimator must be reformulated and a new Cram er Rao lower ....

....estimation accuracy of the signal parameters (such as the angles of arrival) is not affected by the knowledge of the rank of the emitter signal covariance matrix. Thus, the original stochastic ML estimator that ignores the rank information, or its large samples realizations (such as MODE and WSF [7, 4, 2]) achieves the lowest possible estimation error variance for the signal parameters. 2. PROBLEM FORMULATION Consider the following narrowband array model y(t) A( x(t) n(t) 1) The measurement vector, y(t) represents the m sensor outputs. The n emitter signals are collected in the vector ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

.... their asymptotic properties have been investigated, 1, 2] The stochastic maximum likelihood (ML) estimator, derived under the stochastic emitter signal model, has been shown to achieve the Cram erRao bound (CRB) in large samples and hence it yields asymptotically efficient parameter estimates [3, 4, 5]. This important property of the stochastic ML estimator depends on the assumption that the emitter signals are non coherent (i.e. no two signals are fully correlated) When coherent emitter signals are present, the maximum likelihood estimator must be reformulated and a new Cram er Rao lower ....

....intriguing result that the asymptotic estimation accuracy of these parameters is not affected by the knowledge of the rank of the emitter signal covariance matrix. Thus, the original stochastic ML estimator that ignores the rank information, or its large samples realizations (such as MODE and WSF [8, 6, 4]) achieves the lowest possible estimation error variance for the signal parameters. In closing this section, we mention the related work by Bresler [9] The approach in [9] aims at deriving an exact realization of the ML estimate in either coherent or non coherent scenarios, and it is somewhat ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. SpringerVerlag, Berlin, 1993.

....In addition, if the signals are zero mean complex Gaussian distributed, the asymptotic variance of the NSF parameter estimates is equal to the Cram er Rao lower bound, and hence is the minimum achievable variance of any estimator. Proof: The proof requires only a slight modification of results in [12, 13], and thus will not be given here. The expression (30) depends on the particular constraint one chooses for the polarization parameters. However, it can be shown that the upper left block of C, i.e. the part that gives the accuracy of the DOA estimates, does not depend on the constraint. It is ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

....83 Linkoping, Sweden. 1 Introduction During the last decade, much work in the area of sensor array processing has been devoted to the problem of making performance predictions for estimators of physical parameters as, e.g. direction of arrivals and emitted source and measurement noise powers, [1, 2, 3, 4, 5, 6]. Also the possibility of using polarization sensitive sensors for the reception has been analyzed, 7, 8, 9, 10] This should be useful when the emitted waveforms are exhibiting different polarizations, thus improving the reception of signal power at the array. However, an issue that not has been ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In S. Haykin, editor, Radar Array Signal Processing. Springer-Verlag, Wien - New York, 1993.

....with the K(i 1) largest eigenvalues and E s are the corresponding eigenvectors. Estimates of the unknown delays can then be found as the values of k that minimizes k E n Hk 2 U , where U is a weighting matrix. This is referred to as weighted Noise Subspace Fitting (NSF) see e.g. [14]. Unfortunately, this results in a complex multidimensional parameter search. Instead, the users delays are found one at a time and the k th user s delay is given by the minimizing argument of [10] J k;MU ( Tr n U k ( H k ( E n E n H k ( o ; 5) where U k 2 C ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing" in Radar Array Processing (Haykin, Litva, and Shepherd, eds.), pp. 99--151, Berlin: SpringerVerlag, 1993.

....properties and the computational load can, in some extent, be improved by replacing the Hessian by its limiting value. This technique is often termed scoring. The reader is referred to [20] for sophisticated algorithms to solve the optimization problem. This subject is also discussed in [21]. Below, the algebraic expressions for the gradient and asymptotic Hessian are presented. The details is deferred to Appendix B and Appendix C respectively. The gradient vector is given by G = vec(R) j vec(R Gamma1 Gamma R Gamma1 RNR Gamma1 ) 18) where the notation vec(X) ....

B. Ottersten, M. Viberg, P. Stoica and A. Nehorai, Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing. In Radar Array Processing, edited by Haykin, Litva and Shepherd, pp. 99--151. Springer-Verlag, Berlin, 1993.

....asymptotically achieve the Cram er Rao bound (CRB) but only for a very special type of array error model. On the other hand, the MAP approach in [24] is asymptotically statistically efficient for very general error models. However, since it is implemented by means of noise subspace fitting [32], if the sources are highly correlated or closely spaced in angle, its finite sample performance may be poor. In fact, the method of [24] is not a consistent estimator of the DOAs if the signals are perfectly coherent. In this paper, we develop a statistically efficient weighted signal subspace ....

....MAPprox estimates and point out the relation to the GWSF estimator derived in Section 4. 3. 3 MAP NSF Another approach to approximate the MAP estimator is the MAP NSF method of [24] In MAPNSF, one uses the fact that the noise subspace fitting (NSF) method is also asymptotically equivalent to ML [32]. Thus, in lieu of using the WSF criterion in (2) the NSF cost function VNSF = TrfA En E n A Ug (6) is employed. Here, U is a consistent estimate of the matrix U = A y E s e 2 Gamma1 s E s A y : The NSF cost function can be rewritten using the following general ....

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

....(NSF) approach similar to [1, 3, 17, 21] may be taken. In this technique, the estimates are obtained as the minimizing arguments of the cost function V ( fl) trace n A ( Gamma (fl) E n E n Gamma(fl )A( U o where U = U 0 is a d Theta d Hermitian weighting matrix. From [10], U should be chosen as U = Gamma(fl)A( y E s Gamma1 s ( s Gamma oe 2 I) 2 E s ( Gamma(fl )A( y ; to obtain asymptotically minimum variance estimates. Replacing the weighting with a consistent estimate will not affect the asymptotic properties. Since Gamma(fl ) is ....

B. Ottersten, M. Viberg, P. Stoica, and Arye Nehorai. Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing. In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

....sense if only a single signal is received (d = 1) However, neither (18) nor (19) is consistent when multiple arrivals are present. In Section 3, we present several methods that overcome the drawbacks of (19) in the multiple echo case. These methods are counterparts to the subspace fitting [5, 15, 16, 17], MUSIC [3, 4] and ESPRIT [14] algorithms developed for DOA estimation. 2.3 Identifiability The parameters of the model in (4) are said to be identifiable if Q t ( A 6= Q t ( 0 ; 0 )A 0 (20) whenever 6= 0 , 6= 0 or A 6= A 0 . In other words, the unknown parameters ....

....a more computationally efficient manner. Some of these methods have accuracy comparable to that of the ML approach (and the CRB) 4 Subspace Based Estimation Methods In this section we describe algorithms for time delay and frequency doppler offset estimation based on Noise Subspace Fitting (NSF) [15, 17], Signal Subspace Fitting (SSF) 15, 16, 17] MUSIC [3] and ESPRIT [14] It will be shown that due to the special structure of the signal manifold in the frequency domain, both NSF and SSF reduce to a d dimensional search for the delay parameters. Of the two, SSF is expected to be more robust when ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

.... the subspace fitting paradigm has been used to evaluate the asymptotic (in the data) performance of the above algorithms [25, 26, 27, 28, 29] A byproduct of this analysis was the development of an optimal weighted subspace fitting algorithm that achieves parameter estimates of minimum variance [29, 30, 31]. 1.3. Robustness Issues All of the methods for direction finding (DF) listed above rely on the availability of information about the array response, and assume that the signal wavefronts impinging on the array have perfect spatial coherence (e.g. perfect plane waves) The array response may ....

....oe 2 I) 2 Gamma1 s (27) U OPT = A y ( 0 ) E s W OPT E s A y ( 0 ) 28) where oe 2 and 0 are consistent estimates of the noise power and DOAs, respectively. An excellent side by side derivation of the optimality of both the SSF and NSF methods can be found in [31]. The optimal weight matrices given in (27) 28) were derived assuming that only the finite sample effects of noise and not those due to array model errors are present (i.e. ae = ae 0 ) A different set of weights results if model errors are accounted for and finite sample effects ignored [28] ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, "Exact and Large Sample ML Techniques for Parameter Estimation and Detection in Array Processing", In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

....The ML criterion function is known to be separable in P and oe 2 . For fixed A = A( ae) the minimizing signal covariance matrix and noise power are [28] P = A y ( R Gamma oe 2 I)A y (21) oe 2 = 1 m Gamma d Trf Pi Rg : 22) Substituting (21) 22) into (20) leads to [29] VML ( ae) N log fi fi fiA PA oe 2 I fi fi fi const : 23) Clearly, VMAP ( ae; P; oe 2 ) is also separable in P and oe 2 , and ignoring constant terms the concentrated MAP criterion function is VMAP ( ae) VML ( ae) 1 2 (ae Gamma ae 0 ) T Omega Gamma1 (ae ....

....with the signal parameters. In such cases the prior distribution has a crucial influence on the asymptotic properties of the estimates of both and ae. 3.2 The MAP NSF Method It has been assumed that the signal covariance matrix has full rank; i.e. the signals are non coherent. Then it is known [29] that in the absence of model errors, the ML criterion is asymptotically equivalent to the following noise subspace fitting (NSF) criterion VNSF = N TrfA E n E n A Ug ; 25) where U denotes a consistent estimate of the matrix U = oe Gamma2 A y E s e 2 Gamma1 s E s ....

[Article contains additional citation context not shown here]

B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, Exact and large sample ML techniques for parameter estimation and detection in array processing, In Haykin, Litva, and Shepherd, editors, Radar Array Processing, pages 99--151. Springer-Verlag, Berlin, 1993.

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