Direction finding for unstructured emitters in the presence of structured interferers

structure. These signals typically occur in environments that contain interferers of exploitable structure. The goal This paper addresses the problem of direction finding for of this paper is to introduce a new algorithm, referred to unstructured emitters in environments that contain interas Copy-Aided Joint Maximum Likelihood (CA-JML), that ference signals with exploitable properties. The CA-JML effectively eliminates the presence of structured interferers algorithm, presented here, offers dramatic angle estimation and allows the DF and copy of any unstructured emitters improvement in environments that contain exploitable interthat remain. The CA-JML algorithm is derived as a genference. The performance improvement is equivalent to the eralization of the Copy-Aided DF paradigm, in that it asremoval of the interferers from the environment. The comsumes that a weight subspace containing all the SOIs has plexity of the algorithm, however, is comparable to MUSIC. been obtained. The traditional Copy-Aided DF situation Even when structured interferers are not present, the CAoccurs as a special case of a rank 1 weight subspace. JML algorithm can reduce the angle error bias exhibited in MUSIC in challenging environments. The CA-JML algorithm also admits a simple relaxation technique as each 2. DIRECTION FINDING FOR UNSTRUCTURED emitter is localized, which reduces the probability of angle SIGNALS estimation error due to the presence of ambiguous peaks in the angle objective function. 2.1. CA-JML Signal Model and Derivation The CA-JML algorithm can be formulated using maximum1. INTRODUCTION likelihood estimation, from a signal model that assumes a known signal subspace, but makes no assumption on the A common signal processing task of sonar, radar or sesmic structure of that signal subspace. The signal model can be sensor arrays is to detect and localize narrow band emitdescribed mathematically as follows: ters impinging on the array. The MUSIC algorithm, [6], achieves this goal by exploiting the angles of arrival of the [ Q D V Q L Q (1) J received wavefronts. The algorithm does not need knowledge of the signal structure, but uses a calibrated array to where D is the received aperture vector due to the sigJ estimate the angles of arrival. However, when signals have nal of interest V Q , at sample number n , and L Q is the known and exploitable properties, such as constant modreceived interference vector, which includes environmental ulus, burstiness, or known constellations, it is possible to and receiver noise, and might also include additional interexploit their structure to obtain linear copy weights and diferers. The interference L Q is modeled as a complex cirrection finding (DF). DF obtained by property exploitation, cular Gaussian random vector of zero mean with unknown or Copy-Aided DF, yields inherently superior angle estimacovariance 5 , and the received aperture is modeled by tion than conventional superresolution techniques. This is LL because the angle estimation adheres to a lower CramerD s D w J (2) J Rao-Bound (CRB), (equivalent to a signal model that assumes the signal of interest is known), and does not sufwhere D w is the steering vector as a function of angle fer from DF error bias due to the presence of interferers of arrival (AOA) w , and J is an unknown complex gain having nearly the same angle of arrival as the signal of constant. For brevity we refer to D w as D with an implicit interest (SOI) [1, 2, 4, 3]. Despite these advantages, there dependency on the AOA. is still a need to DF signals of unknown or un-exploitable This work was supported by the Intelligence and Electronic Warfare A model of the steering vector that mitigates against multipath, poDirectorate, contract #DAAB10-93-C-0018 larization diversity and other impairments is developed in [1] After substitution of (2) into (1), maximum likelihood The spectrum is modified by removing Z from the weight [ estimation yields the objective function [3, 1], subspace : . In this situation it should be noted that the [ CA-JML spectrum will contain multiple peaks, each peak … c dc c + + corresponding to a genuine signal within the signal weight MZ D M D 5 D [ [  s ln (3) 0 / subspace. Mathematically, the cancellation of a signal can + c 5 b 5 5 5 V V [ V [ V [ [ be accomplished by using the optimal Y determined previously from the eigenvalue problem. Therefore, if Y and c + where Z s 5 5 , 5 s K[ Q [ Q L , 5 s [ V [ [ Q [ V [ [ w are the optimal scrambling gains and DOA for signal f f K[ Q V Q L , and 5 s KV Q V Q L . This can be Q VV Q 1, then the complexity of the CA-JML spectrum can be rewritten in a more amenable form as follows. Define the duced by forcing the search into the orthogonal complement 5 c + [ V whitened signal subspace weights by Z s 5 , [ [ of the Y weights. This exploits an additional property of 5 V c + where 5 is the Hermitian inverse of the Cholesky facthe weight subspace that constrains the true “unscrambled” [ S tor of 5 and 5 V s 5 . Also define the whitened copy weights to be an orthonormal mixing of the CA-JML [ [ VV c + aperture vector, D s 5 D . Substituting these quantities weight subspace vector : , [ [ [ in (3) yields c : 5 : 9 (6) FR S \ [ [ w x + Z 3 D Z [ [ [  s ln (4) + 0 / + where 9 is orthonormal, 9 9 , . The CA-JML weights b Z Z [ [ obtained in Section 3 have this orthonormal-combining propwhere 3 % is the projection operator defined by 3 % s erty. + + c % % % % and 3 % s , b 3 % . " Equation (4) is the form of the maximum likelihood function used by the copy aided DF paradigm. It assumes knowledge of the copy weights, encapsulated here in the vector Z . The copy weights can be found using a variety [ of techniques. The CA-JML algorithm, however, assumes that Z , is contained in the range space of a subspace [ matrix : . This assumption is consistent with the envi[ ronment typically assumed for DF using the MUSIC algorithm. In mathematical terms, it implies that we can write Z s : Y where : is an 0 d 1 subspace [ [ [ VHQ V R U V LJ V matrix, and Y is an 1 d vector of unknown scramV LJ V Figure 1: CA-JML and Music Spectra bling gains. A technique for obtaining : for unstructured [ signals is provided in Section 3. After substituting : Y [ A comparison between CA-JML and MUSIC in eninto (4) we obtain the CA-JML spectrum as the following vironments that do not contain structured emitters shows eigenvalue problem: 6 s ccc & some distinct advantages for the CA-JML algorithm. The + + + primary mathematical difference between MUSIC and CAmax X 4 3 D 4 X T 3 : T (5) [ [ Z Z D D [ [ + X X X JML is that CA-JML adds an inverse data covariance matrix whitening to the aperture and the subspace weights that D [ where T s is the unit normalized D and 4 is [ D Z MUSIC does not employ. This whitening causes sharper [ N D N [ an orthonormal matrix spanning the column space of : . spectral peaks, reduces false maxima, or spectral ambigu[ The CA-JML spectrum is a function of the angle of arrival ities, and also reduces DF error bias in difficult environw , and has its maxima at the DOAs of the signals captured ments. This can be seen in Figure 1, where the MUSIC in the signal weight subspace matrix : . spectrum is seen to exhibit large ambiguities not present in [ the CA-JML spectrum. Note also that the spectral peak at R is removed in the secondary spectrum CAJML-2. The 2.2. Advantages of CA-JML computer simulated environment used to generate Figure 1 s s Because the CA-JML spectrum is a simple generalization contains three emitters arriving at angles of , and s of the copy-aided DF spectrum, it offers an architectural . The emitters are received at , and dB signal to white noise power ratio (SWNR) respectively, and the advantage to systems which would employ both copy-aided and copy unaided DF. The DF spectra for both copy-aided noise interference is spatially colored, circularly symmetric, complex Gaussian. A six element, isotropic array is used and unaided DF can be dealt with using the same CA-JML to DF each emitter. algorithm. " An additional advantage of the CA-JML spectrum is If 9 is an orthonormal basis for the vector subspace the ability to reduce the complexity of the spectrum aforthogonal to Y , then a new signal weight subspace can " ter the DOA of an unstructured signal has been estimated. be generated by the formula, : s : 9 . By substi[ [ 0 ° 5 0 ° 1 0 0 ° 1 5 0 ° 2 0 0 ° 2 5 0 ° 3 0 0 ° 3 5 0 ° 0 M u s i c