MAXIMUM LIKELIHOOD AND MAXIMUM A POSTERIORI DIRECTION-OF-ARRIVAL ESTIMATION IN THE PRESENCE OF SIRP NOISE Xin Zhang∗, Mohammed Nabil El Korso∗∗ and Marius Pesavento∗ ∗Communication Systems Group, Technische Universit¨at Darmstadt, Darmstadt, Germany ∗∗LEME EA416, Universit´eParis Ouest Nanterre La D´efense, Ville d’Avray, France ABSTRACT local scattering. A SIRP is fully characterized by its texture param- eter(s) and speckle covariance matrix. The maximum likelihood (ML) and maximum a posteriori (MAP) The existing works addressing the estimation problems in a estimation techniques are widely used to address the direction-of- SIRP context almost exclusively assume the presence of secondary arrival (DOA) estimation problems, an important topic in sensor ar- data (known noise-only realizations) in order to estimate the speckle ray processing. Conventionally the ML estimators in the DOA es- and texture’s parameters [15, 17–22], instead of unknown noise timation context assume the sensor noise to follow a Gaussian dis- realizations embedded in and contaminating the received signal. tribution. In real-life application, however, this assumption is some- In [18], the authors provided a parameter-expanded expectation- times not valid, and it is often more accurate to model the noise as maximization (PX-EM) algorithm to estimate the unknown signal a non-Gaussian process. In this paper we derive an iterative ML as parameters under the SIRP noise. The problem they consider, how- well as an iterative MAP estimation algorithm for the DOA estima- ever, is a linear one. Furthermore, the application of their algorithm tion problem under the spherically invariant random process noise is restricted to a special model, namely, the so-called generalized assumption, one of the most popular non-Gaussian models, espe- multivariate analysis of variance model [23]. To the best of our cially in the radar context. Numerical simulation results are pro- knowledge, there are no algorithms available in the current literature vided to assess our proposed algorithms and to show their advantage for DOA estimation (a highly non-linear problem), nor for signal in terms of performance over the conventional ML algorithm. parameter estimation in general in a comprehensive manner, under Index Terms— Direction-of-arrival estimation, spherically in- the SIRP noise. To fill this gap, and employing a similar methodol- variant random process, maximum likelihood estimation, maximum ogy as in [4] and [9], we devise in this paper an iterative maximum a posteriori estimation, sensor array processing likelihood estimation (IMLE) algorithm, together with an iterative maximum a posteriori estimation (IMAPE) algorithm in this con- text. The latter exploits information of the noise distribution and 1. INTRODUCTION can be seen as a generalization of the former. Finally, we carry out simulation to illustrate the performances of our algorithms. The direction-of-arrival (DOA) estimation problem is an important topic in sensor array processing which has found wide application 2. MODEL SETUP in, among others, radar, sonar, radio astronomy and wireless com- munications [1–4]. Among the numerous techniques developed for Consider an arbitrary sensor array comprising N sensors that re- the DOA estimation, those based on the maximum likelihood (ML) ceive M (M < N) narrowband far-field source signals with un- criterion are known to have the advantage of offering an outstanding known DOAs θ1,...,θM . The array output at the tth snapshot can tradeoff between the asymptotic and threshold performances [3, 5]. be formulated as [3, 5]: Conventionally, a crucial assumption for the ML estimators is that the noise is uniformly white [3, 5]. Nevertheless, this oversimplify- x(t) = A (θ) s(t) + n(t), t = 1,...,T, (1) arXiv:1603.08982v1 [stat.AP] 29 Mar 2016 ing assumption is unrealistic in certain applications [6–8]. Thus, the T authors of [4] and [9, 10] have devised, resorting to the concept of in which θ = θ1,...,θM is the M × 1 vector of unknown signal stepwise numerical concentration, an iterative ML estimator for the DOAs, A θ [= a θ1 ,...,] a θM denotes the N × M steering case of nonuniform white and colored noise, respectively. matrix, s (t )is the[ M( ×) 1 vector( of)] the source waveforms, n t is The problem, however, is that the Gaussian noise assumption it- the N × 1(sensor) noise vector, T denotes the snapshot number,( and) self, colored or not, is based on the central limit theorem, and loses ⋅ T denotes transpose. immediately its validity in certain scenarios when the conditions for ( ) In this paper, we assume the source waveforms s t , t = this are not fulfilled. This is the case, e.g., in the context of low- 1,...,T , to be unknown deterministic complex sequences( [3].) The grazing-angle and/or high-resolution radar [11–13], where the radar sensor noise is modeled as a SIRP, which comprises two terms, clutter shows non-stationarity. Various non-Gaussian noise models statistically independent of each other [14]: have been developed to deal with such problems, among which the so-called spherically invariant random process (SIRP) model has be- n t = »τ t σ t , t = 1,...,T ; (2) come the most notable and popular one [12, 14–16]. A SIRP is a ( ) ( ) ( ) two-scale, compound Gaussian process, formulated as the product in which σ t represents the speckle, a temporally white, complex of two components: the square root of a positive scalar random pro- Gaussian process( ) with zero mean and an unknown N × N covari- cess, namely, the texture, accounting for the local power changing, ance matrix Q = E σ t σH t , where ⋅ H stands for the conju- and a complex Gaussian process, namely, the speckle, describing the gate transpose; whereas ( ) the texture,( ) denoted( ) by τ t , is composed ( ) of independent, identically distributed (i.i.d.) positive random vari- s t and Q are fixed. We denote this estimate by τˆ t , which has ables at each snapshot. To resolve the ambiguity between the texture the( ) following expression: ( ) and the speckle so as to make the noise parameters uniquely identi- 1 H 1 fiable, we assume that tr Q = N, in which tr ⋅ denotes the trace. τˆ t = x t − A θ s t Q− x t − A θ s t . (8) N In this paper, we mainly{ consider} two kinds of{ texture} distributions ( ) ( ( ) ( ) ( )) ( ( ) ( ) ( )) that are most widely used in the literature, for both of which τ t Meanwhile, by applying Lemma 3.2.2. in [28] to Eq. (7), one can is characterized by two parameters, the shape parameter a and( the) obtain the expression of Qˆ , representing the estimate of Q when θ, scale parameter b. The firstis the gamma distribution, leading to the s t and τ t and are fixed, as: ( ) ( ) K-distributed noise [12, 24], where the pdf of τ t is: T 1 1 H ( ) Qˆ = Q x t − A θ s t x t − A θ s t , (9) 1 1 τ(t) T 1 τ t p τ t ; a,b τ t a− e− b , (3) t= ( ( ) ( ) ( )) ( ( ) ( ) ( )) = a ( ) ( ( ) ) Γ a b ( ) in which replacing τ t by the expression of τˆ t in Eq. (8) leads to ( ) the following iterative( ) expression of Qˆ : ( ) in which Γ ⋅ denotes the gamma function. The second kind of H our considered( ) texture distribution is the inverse gamma distribu- 1 T − − ˆ (i+ ) N x t A θ s t x t A θ s t tion, leading to the t-distributed noise [25, 26], for which Q = Q ( ( ) ( ) ( )) ( ( −)1 ( ) ( )) , T (i) t=1 x t − A θ s t H Qˆ x t − A θ s t a b b −a−1 − ( ( ) ( ) ( )) ( ( ) ( ) ( )) p τ t ; a,b = τ t e τ(t) . (4) (10) Γ a ( ( ) ) ( ) for which we choose the identity matrix of size N, denoted by IN , ( ) (0) to serve as the initialization matrix Qˆ . Under the assumptions above, the unknown parameter vector of 1 T ˆ (i+ ) our problem is ξ = θT , χT , ζT ,a,b , where χ is a 2NT -element We further need to normalize Q in Eq. (10) to fulfill the (i+1) vector containing the real and imaginary parts of the elements of assumption that tr Q = N. Let Qˆ denote the normalized esti- 2 n t , t ,...,T N (i+1) { } s = 1 , and ζ is a -element vector containing the real mate Qˆ , which is: and( ) imaginary parts of the entries of the lower triangular part of Q. T T T ˆ (i+1) ˆ (i+1) ˆ (i+1) Let x = x 1 , ..., x T denote the full observation vec- Qn = N Q tr Q . (11) ( ) ( T) tor, and τ = τ 1 ,...,τ T represent the vector of texture real- izations at all[ snapshots.( ) The( )] full observation likelihood conditioned Now we consider the estimate of s t when θ, τ t and on τ can be written as: Q are fixed, which, denoted by sˆ t , can( ) be found by( solving) ∂LC ∂s t = 0, as: ( ) 1 H T − ~ ( ) 1 exp τ(t) ρ t ρ t H − H p x τ ; θ, χ, ζ = M ( ) ( ) ; (5) sˆ t = A˜ θ A˜ θ A˜ θ x˜ t , (12) ( S ) t=1 πτ t Q ( ) ( ) ( ) ( ) ( ) S ( ) S 1 2 1 2 in which A˜ θ = Q− ~ A θ , x˜ t = Q− ~ x t , representing the 1 2 in which ρ t = Q− ~ x t − A θ s t , represents the noise steering matrix( ) and the observation( ) ( at) snapshot t(, both) pre-whitened realization at( ) snapshot t with( ( its) speckle( ) spatially( )) whitened. by the speckle covariance matrix Q, respectively. Eq. (5), multiplied by p τ ; a,b , leads to the joint likelihood of From Eqs. (8), (10) and (12) one can see that the estimates of x and τ : ( ) τ t , Q and s t are mutually dependent, and further dependent on the( ) parameter vector( ) θ.