Robust CFAR Detection in Clutter with Unknown Covariance Matrix Victor Golikov, Olga Lebedeva and Andres Castillejos Victor Golikov, Doctor of Science, Professor-Researcher, Universidad Autonoma del Carmen, Col. Aviacion, av. 56 x av.Concordia 4, C.P.24180, Ciudad del Carmen, Campeche, México, tel/fax: 52-9383826516, e-mail: [email protected] Olga Lebedeva, Master of Science, Professor-Researcher, Universidad Autonoma del Carmen, Col. Aviacion, av. 56 x av.Concordia 4, C.P.24180, Ciudad del Carmen, Campeche, México, tel/fax: 52-9383826516, e-mail: [email protected] Andres Castillejos, Engineer of Universidad Autonoma del Carmen, , Col. Aviacion, av. 56 x av.Concordia 4, C.P.24180, Ciudad del Carmen, Campeche, México, tel/fax: 52-9383826516. Abstract — We conduct a simulation analysis for coherent signals by resolving the inner product assessing the constant false alarm rate (CFAR) behavior of the measurement and signal; of three coherent radar detectors in the presence of 2) the MSD, which is a χ2 statistic that detects correlated Gaussian clutter with unknown covariance. We subspace signals (including noncoherent signals) establish the conditions a detector must fulfill in order to by computing the energy of the measurement in ensure the CFAR property. We discuss several detectors the signal subspace; with quasi CFAR property. The analysis of CFAR 3) the coherent CFAR MSD, which is a t statistic property and the performance analysis, which has been that detects coherent signals in noise of carried out in the presence of correlated Gaussian clutter unknown variance by resolving the cosine of the with unknown covariance, shows that the proposed angle the measurement makes with the signal; detectors exhibit a quite acceptable loss with respect to 4) the CFAR MSD, which is an F statistic that optimum Neyman-Pearson detector. detects subspace signals (including noncoherent signals) in noise of unknown variance by measuring the fraction of energy the measurement has in the signal subspace. 1. INTRODUCTION Each of the resulting four detectors is a generalized A very challenging problem arising in radar signal likelihood ratio tests (GLRT) for a concrete problem, and processing is to achieve a reliable target detection in the each is UMP-invariant, uniformly most powerful over the presence of severe clutter backgrounds. When the clutter entire class of detectors invariant to an appropriate statistics are unknown or highly variable the false alarm transformation group. rate of the known nonadaptive radar receivers cannot be All of these detectors are compelling. They have clearly controlled and target decisions become unreliable. This is stated optimalities and invariances, and they have due to the lack of robustness of the these receivers with evocative geometrical interpretations. The MSDs use respect to possible mismatched between the design and extra knowledge of the noise variance for some operating conditions as well as variations in the clutter performance gain against the CFAR MSDs, which do not statistics. assume this knowledge. On the other hand, the CFAR There are four (nonadaptive) matched subspace MSDs compensate for this lack of knowledge by detectors (MSD) that form the basis for the subspace providing an extra invariance to data scaling, a property detector with CFAR property in the presence of correlated that the MSDs do not have. Gaussian clutter with unknown covariance. They arise The MSDs and CFAR MSDs all assume prior from two types of generalizations of the matched filter knowledge of noise covariance matrices. However, this detector. First, the inner product of the matched filter may information is often not known, meaning that, in practice, be generalized to a projection of the measurement onto a it must be estimated and then used correctly in an higher dimensional signal subspace, thus producing a adaptive detector. These detectors, however, still need a subspace detector [1,2,4]. Second, the detector may be sufficient amount of data to “learn” the clutter normalized by an estimate of the noise power to make it environment web enough for reliable detection of the have a CFAR with respect to the noise power. The four signal. In a severely no stationary and/or no homogeneous detectors are thus environment such as that surrounding an airborne 1) the coherent MSD (i.e., matched filter), which is surveillance radar system, the detection performance of an a normally distributed statistic that detects adaptive system can fall below what can be expected in a 0-7803-9323-6/05/$20.00 ©2005 IEEE. 201 stationary and homogeneous environment, simply due to 2) MSD [1], which is a χ2 statistic that detects subspace the lack a sufficient amount of data. In this case no- signals H . In this case, the signal p is known to adaptive radar detection has been utilized. As shown in [5], under this conditions it is possible to obtain the lie in the linear subspace H spanned by the detector (nonadaptive) with a constant probability of a false alarm rate (CFAR). In [5] there were obtained columns of matrix H. The signal is a linear conditions at which the detector guarantees constant combination of modes. It may be represented as p = probability of a false alarm rate under unknown Hθ (3). Here H is a known N x L matrix with interference covariance matrix with Toeplitz’s or columns hn and θ is a L x 1 vector with elements θn. Hermitian structure. MSD test may be represent as 2 H Our aim in this paper is to adapt these detectors of χ = r PH r (4) [1,2,3,4] to unknown clutter covariance in order to 3) The coherent CFAR MSD [1], which is a t statistic produce a robust detector with CFAR property that may that detects coherent signals p in noise of unknown be applied to signal detection for radar, sonar, and data variance σ2 . This test may be represent as communication. For CFAR analysis and assessment of H 2 H p Ppr / σ p p the performance of the detectors we shall use the method t = (5) of Monte-Carlo simulation r H (I − P )r /(N −1)σ 2 p 2. DETECTION STRATEGIES Following the test (5) vector r is projected onto Let us consider the problem of detection of a pulse train orthogonal signal and clutter subspaces. In the signal composed of N coherent pulses in additive clutter. It can subspace, the projection is correlated with be formulated in terms of the following binary hypotheses pH/(pHp)1/2 to produce a matched filter output. In the test: clutter subspace the projection (I-Pp)r is squared, divided by (N-1), and rooted to produce an estimate of . H0 : r = σn, σ 4) The CFAR MSD, which is an F statistic that detects H1 : r = u + σn, (1) subspace signals (including noncoherent signals) in where r, u, and n denote the N-dimensional complex clutter of unknown variance is follows: H 2 vectors of the samples from the baseband equivalents of r P r /σ L F = H , (6) the received signal, the wanted target echo, and the clutter r H (I − P )r /σ 2 (N − L) with unknown covariance matrix. The useful signal u is H modeled as a coherent pulse train, namely u = αp where p where PH is matrix projector onto subspace H , I is the transmitted waveform and α an unknown complex is the unit matrix. This statistic is ratio of quadratic parameter, accounting for both the channel propagation forms in projection matrices. It measures the ratio of effects and the target radar cross section. The clutter the energy of r, per dimension, that lies in the vector n is scaled by the level constant σ. subspace H to the energy of r, per dimension, In the sequel we present several solutions, proposed during the last fifty years, for solving the problem (1). that lies in the orthogonal subspace I – Pp . 1) Coherent MSD (matched filter). Assuming that n is We want to adapt these detectors of (2,4,5,6) to zero-mean, complex, Gaussian vector with known unknown clutter covariance in order to produce a robust covariance matrix M, in [1] has been derived the approximately CFAR detectors. We will obtain a following one-sided threshold test for testing H0 conditions for matrix projections Pp and orthogonal matrix versus H1: projection P⊥ of CFAR test for unknown Hermitian covariance matrix of clutter M and σ. We use the H1 pPpr > obtained in [5] the conditions which provide these m (2) properties: σ (p H p)1/ 2 < 0 tr(Pp M)/tr( P M)=const , (7) H 0 ⊥ here sign tr designates a track of a matrix. According to where Pp is the projection matrix onto vector p; m0 is this equation the sum of all elements in diagonal or in the detection threshold; (.)H denotes conjugate everyone sub-diagonal k of projection matrix are transpose. identical to all projection matrixes Ph of subspace: 202 N −1 i, j+k ∑ Ph ≠ f (h) (8) 3. CFAR ANALYSIS AND DETECTION i=k PERFORMANCE ASSESSMENT The example of such projection matrix for wanted signal The aim of this section is to study the CFAR behavior p is resulted in [5]: of coherent MSD, MSD, coherent CFAR MSD and (a ) K i = a (aT a ) −1 aT , (9) proposed CFAR MMSD in presence of the mismatch i i i i between design and actual value of one-lag clutter here elements of vector ai are: correlation coefficient ρ. As to the clutter correlation N −1 properties, we assume that the returns from a given range- a (τ ) = exp[ j(t ⊕ i) / T ]exp[− j(t ⊕ τ ) / T ] , (10) i ∑ p p cell are exponentially correlated: t=0 2 the operation t ⊕ i is called generalized (logical) shift in M = ||mi,j|| = 2σ || exp|i-j| || (14) an argument t on a value i.