Reduced Rank Adaptive Detection Of Distributed Sources Using Subarrays Yuanwei Jin and Benjamin Friedlander Abstract We introduce a framework for exploring array detection problems in a reduced dimensional space. This involves calculating a structured subarray transformation matrix for the detection of a distributed signal using large aperture linear arrays. We study the performance of the adaptive subarray detector and evaluate its potential improvement in detection performance compared with the full array detector with finite data samples. One would expect that processing on subarrays may result in performance loss in that smaller number of degrees of freedom is utilized. However it also leads to a better estimation accuracy for the interference and noise covariance matrix with finite data samples, which will yield some gain in performance. By studying the subarray detector for general linear arrays, we identify this gain under various scenarios. We show that when the number of samples is small, the subarray detectors have a significant gain over the full array detector. In addition, the subarray processing can also be successfully applied to the problem of detecting moving sources in an underwater acoustic scenario. We validate our results by computer simulations. Keywords Subarrays, Reduced rank, Adaptive processing, Distributed source, Interference cancellation, Detection, Y. Jin is with the University of California at Davis, CA 95616 USA (e-mail: yjin@ece.ucdavis.edu). B. Friedlander is with the University of California at Santa Cruz, CA 956 USA (e-mail: friedlan@cse.ucsc.edu). This work was supported by the Office of Naval Research, under grant no. N14-1-1-75.
2 I. INTRODUCTION, BACKGROUND AND MOTIVATION The problem of detecting underwater acoustic sources using measurements by an array of sensors has been studied extensively in literature. For a large aperture acoustic array, a narrow beam can be formed so as to distinguish two closely spaced emitters. However, the acoustic energy source may be fairly close to the array and may move through several beams during the sonar system s temporal integration time. The effects of source motion on sonar systems have been studied by several authors (see for example [6] and the references therein). One may model the moving transmitter during an integration time as a source with energy scattering in space, or called a distributed source. The distributed source can be described by a subspace array manifold model [12]. One of the enduring problems associated with the adaptive minimum variance distortionless (MVDR) beamformer (see e.g. [5],[1]) lies in the classic dilemma of wanting long observation times for stable covariance matrix estimates yet needing short observation times to track dynamic field behavior. This is especially true for large aperture arrays. This issue has been addressed by many authors (see e.g. [6]) and is one of the research themes of the Acoustic Observatory (AO) Project [1]. There are several ways of dealing with this issue, for instance, the diagonal loading method, which is essentially a weighted projection method by adding a constant value to each of the terms along the diagonal of the sample covariance matrix (see e.g. [6], [13] and the reference therein). Reduced rank processing is another one of the well known data processing methods (see [4], [11] and the reference therein). In this case, the data are mapped into a lower dimensional subspace via a transformation matrix prior to detection. Rank-reduction directly addresses the sample support issue by reducing the number of statistical unknowns associated with the interference. In this paper, we study the problem of detecting distributed sources using subarrays, i.e., a partial collection of sensors of a full array. In this case, the transformation matrix is a structured block diagonal matrix. The motivation for this study lies in the following two observations. First of all, for a general linear array with elements, the beam width is inverse proportional to the array aperture. A subarray with a smaller aperture gives rise to a wider beam which, consequently, is able to cover the distributed source if the subarray beamwidth is chosen to be close to the signal angular spread. Hence, a simple MVDR beamformer can be implemented on each individual subarray. Secondly, implementation of a MVDR beamformer requires of estimating the sample covariance matrix based on data samples. The estimation accuracy is improved for the subarray processing compared with the full array processing based on the same amount of data. This is because we have a smaller amount of unknown parameters to be estimated.
3 This leads to the following conjecture: With short data records, statistical stability dominates detector performance, and subarray detection requires substantially less SNR than full array processing. With large data records, SNR dominates detection performance, and the sub-array detector requires nearly the same SNR as the full array detector. Hence, substantial performance improvements are possible using the subarray detector relative to the full array detector in limited-data situations. It should be noted that the idea of subarray processing has been proposed before, and has been studied by several authors, for instance, Cox [8], Morgan [14], Owsley, Swingler [19], and Dhanatawari [9]. Owsley suggested that a narrow band uniform linear array containing elements could be decomposed into, say non-overlapped but contiguous subarrays of equal length. Each subarray is operated as a simple delay and sum beamformer and the output from each is treated in exactly the same fashion as the output from a single sensor in a uniform linear array comprising elements. Dhanatawar chose a different sub-array geometry, where the subarrays were heavily overlapped. The work in [14] and [8] addresses the issue of signal coherence degradation, however, it does not specifically address the issue of processing with finite samples. The subarray processing is a particular way of reduced rank processing. Our work aims to derive the subarray detectors (coherent and noncoherent) and their variations for distributed sources from the standpoint of reduced dimension detection theory and to study their performance tradeoffs. From the matched subspace filter standpoint, the optimal reduced dimension processing is to preserve the signal component (or matched to the signal subspace) while to suppress the strong interference components. Thus by reducing the data dimension without loosing signal components significantly, we are able to achieve desirable detection performance when the number of data samples is limited. Futhermore, our work is close in spirit to the well studied partially adaptive beamforming (see e.g. Van Veen [21] and the reference therein), where the number of adaptive degrees of freedom may be considerably fewer than the number of sensors, while still providing useful performance. Reducing the number of adaptive degrees of freedom degrades the interference cancellation performance. Thus minimizing the detection performance degradation is an important consideration in designing the optimal subarray detector for detecting signal sources with energy scattering. In addition, the proposed subarray processing scheme attempts to tackle the problem of non-stationarity of the underwater acoustic environment. The motion of the sources causes a non-stationary background that severely limits the number of data snapshots that can be collected, and consequently limits the performance of passive sonar systems [1],[6]. By tracking the subspace that the moving source travels through
Y 4 within an array processing interval, we are able to collect more data and to achieve the desirable detection performance. The rest of the paper is organized as follows. In section II, we present the array signal model, and give the optimal full array detector. In section III, we propose the subarray detectors and their variations under different conditions. We carry out performance analysis for the proposed subarray detector in section IV. In section V, we present computer simulations which serve to illustrate the behavior of the subarray processors. Notation: Vectors (matrices) are denoted by boldface lower (upper) case letters; all vectors are column vectors; superscripts denote the complex conjugate transpose; denotes the identity matrix; E denotes the statistical expectation; denotes the matrix determinant; denotes the vector (matrices) Frobenius norm; denotes the trace of a matrix;!#"%$'&(*)+ denotes diagonal matrix whose diagonal is the vector ) ;, denotes Kronecker product. A. Array Signal Model II. PROBLEM FORMULATION We consider a general linear array composed of sensors. Let *-/.1325476'8 8 8 be the coordinates of the 2 -th sensor measured in half wave-length units and 9 ;:< be the steering vector of the array in the direction : : 9 ;:< =4>@?BABCDFE;GIH@JLK#8 8 8 B?BABCMDBN/GOH@JFKPRQ (1) Consider S narrow-band radiating sources impinge on the array from distinct direction M:UT*V:XW'8 8 8 *V:ZY[. The signals from the sensor outputs are passed through a receiver where they are amplified, shifted to baseband, lowpass filtered, sampled and digitized. We denote the samples of the receiver outputs by \+]R^, where _ is the sensor number and ` is the sample index (different samples at different ` are assumed to be independent). Hence the signal received by the array is modelled as )a I` =4>b 9cT* I` edt I`3 gf where )a I` is array output at sample time `. h i'j W 9b i I`3 ed i I` cflkm I`3 Vn`=476'8 8 8 o (2) )a I` =4p \ ] T*8 8 8 3\ ]%qrq (3) d i I` is the r -th complex waveforms constituting the signal with total signal power E[ d i I` W s. We assume that the first signal dzt I`3 is the desired signal with tuu4 E[ dt I`3 8 W, others are considered as the interfering signals.
{ T C z C { We assume that the instantaneous array response 9<T I` b is a complex Gaussian vector with zero mean and covariance matrix vwu. This covariance matrix is related to the array manifold by the following expression (see e.g. [12]) v u 4yx ~}(V:3 9 ~}Z 9 ~}Z } (4) C { where 9 ~}Z is the array manifold at angle }. vƒu is normalized to be vwü 4. ~}(V:3 is the spatial energy distribution of the source at azimuth :, such that C { z ~}(V:3 } 476. : is the nominal direction of arrival of the signal. is the signal angular spread. The energy distribution function { ~}(V:3 can have different forms. Without loss of generality, throughout the paper we assume that the signal has a uniform distribution which represents the widest spread of signal energy, i.e., ˆ@ ~}(V:3 =4 T Š } ŒŽ ;: ' [V: f ' otherwise The method presented in this paper can also be extended to other distribution functions, for instance, the Gaussian distribution. The subspace manifold can be obtained in this case as follows (see also [12]). We start with the signal source. The interference source follows the same rule. Let vnu 4 ƒš decomposition of vwu. If indeed vwu is a low rank matrix with rank, then 5 (5) be the singular value is exactly the subspace spanned by the first columns of. More generally, we will assume that v u can be approximated by a rank matrix (i.e., the singular values from Ufœ6 onward are small compared to the singular values from 1 to ), in which case is approximately the subspace spanned by the first columns of, i.e., where 4y w*š T ew v umžÿ *šmf 4y (6). km I` is the complex white Gaussian noise with zero mean and covariance W, and is uncorrelated with the signal sources. The covariance matrix of interference plus noise is denoted by v. B. The optimal full array detector A commonly used detection scheme is the binary hypothesis testing, that is, letting the null hypothesis be that the data is signal free and the alternative hypothesis be that the data contains a signal. Hence the detection problem on the basis of full array data vector ) (we drop ` for simplicity purpose) is given as follows, ) «ª( 7 ) «ª( 7 v t uvv uf v (7)
T ¾ 6 The optimal full array detector ± for the above detection problem (7) is given as follows (see appendix I for details) where ~t u v ² ³4Ÿv (8) 4 µ t uv n ~t ub v fÿ F* ew is the weighted signal subspace matrix, while the matrix f«ew is a diagonal dominant matrix and represents how the columns (or beams) of signal subspace are weighted and combined. The weighting matrix consists of columns, where is the rank of the signal subspace. We call this matrix beamformer ¹ a Generalized MVDR (GMVDR) beamformer in the sense that it extends the standard rank one MVDR beamformer º ³4¼»v 9 ;:< (9) to a multi-rank case. In fact, the GMVDR beamformer is the optimal detector for a distributed signal source from a detection theory standpoint. The implementation of the full array detector requires a Fig. 1. Configuration of the GMVDR beamformer priori knowledge of v, which is often estimated from finite training samples. In this case, v by its maximum likelihood estimate v ½¼4 ¾l À j T ) À ) À, where *) À ² 7 vw VBÁÂ4Ã6'8 8 8 ož is replaced is a set of independent and identically distributed training data. It is pointed out (see Reed [17]) that oåäpæ stationary data vectors are required to obtain a moderately statistically stable estimate of v ½. This requirement can be difficult or even impossible to meet in rapidly changing environments, especially for large aperture arrays. Further, real time computation requirements can also be prohibitive for large. Reducing the data dimension through a linear mapping prior to performing detection helps alleviate these problems. Thus, rather than utilize the entire -dimensional data space to obtain a full array detector given in (8), we formulate a subarray detection problem, which in essence is a constrained reduced rank detection problem described in the next section. III. THE SUBARRAY DETECTOR FOR A GENERAL LINEAR ARRAY When we say subarray processing we mean that we divide the full array into many smaller arrays, or called subarrays, and process the received data of each subarray individually. For a general linear array
Ö 7 of sensors, a common scheme is dividing the total array into a number of non-overlapping subarrays with equal size. Each subarray has Ç sensors. Without loss of generality, let us assume that 4 nç where is the number of subarrays with sensors È6'8 8 8 BÇp forming the first subarray, sensors Dzfn6'8 8 8 *B PÇ> forming the second subarray, etc. The full -element input vector is given by equation (3). The Ç -element input data vector for the r -th subarray, which shall be denoted by ) i, is expressed as follows, A. The optimal subarray beamformer Ï i ) i 4p \É i ÊÌË ÍT 3\É i ÊÌË Í<W 8 8 8 3\ i ËÎ Q r476'8 8 8 (1) It is straightforward to show that, the optimal subarray beamformer for the r -th subarray is the GMVDR beamformer based on the subarray data vector ) i. Hence, the Ç Ï i 4>Ð t u3v i i ~t ub i v i i f ew w i matrix Ï i is given as follows 4¼v i i (11) where i 4 µ t uv i ~t uv i v i i fl ew. v i is the r -th diagonal block of v, and i Œ Ñ Ë ÒZ Ó is the r -th subarray manifold. The transformed data vector Ô i Œ Ñ Ó is then given as Ô i 4¼Ï i ) i 3r4 6'8 8 8 Õ. By grouping Ô i, we have the following linear transformation Ô 4yÏ ) (12) where Ô 4> ÔZ T ÔZ W 8 8 ÔÈ Ö O. The linear transformation matrix Ï ŒƒÑ q Ò[Ø is a block diagonal matrix with Ù 4 i'j T i. Ï is given as follows, ÏÂ4Ÿ!"%$'&X@ÏT*8 8 8 *BÏ Ö (13) B. The coherent detector A natural question arises as how to combine the processed data from each subarray in an optimal fashion. Noticing equation (12), this question can be easily answered from the reduced rank detection theory standpoint. The block diagram of the processing scheme is depicted in figure 2. Let matrix Fig. 2. The reduced rank processing architecture. ÚŸÛ ÜgÝÞPß is a transformation matrix, à¼û ÜcßXÞPá is a detector
T â â 8â ŒlÑ Ø<ÒZ, where is the signal subspace rank, be a detector based upon the transformed data Ô. The reduced detection problem is then described by the following binary hypothesis test Ô «ª/ 7 Ô «ª/ 7 BÏ vï t ulï v uvïãflï vï The optimal detector for the above detection problem (14) takes the same form as equation (8) except that the full dimensional matrices v, vƒu and and Ï respectively. â It is given as follows are replaced by the transformed matrices Ï vï, Ï v ubï 47Ð t u äï vï Ï n ~ af t uv Ïå äï vï Ï w ew Hence, the coherent subarray detector for this reduced rank processing architecture appears to be where uu4 çæéè êë4yï 4yÏå äï vï Ï (14) (15) u (16) µ t ub n ~ af t ub Ï äï vï Ï w ew, which leads to the following test statistics Ù I) =4ã) çæ;èeê' æ;èeê ) 4ì ) çæ;èeêx Wí E îmï (17) The formulation of the subarray detection problem within the framework of the general reduced rank detection theory facilitates the understanding of the subarray processing. The linear transformation matrix Ï in the reduced rank detection theory serves two purposes. Firstly, this matrix Ï compresses - dimensional data into a Ù -dimensional subspace prior to constructing a test statistic. This transformation removes the dimensions that contain least signal-to-interference-plus-noise components. A desirable Ï should suppress strong reduces the nuisance parameters of v ½ into Ï v ½ Ï. This reduction in the number of nuisance parameters tends to improve the accuracy of the estimate Ï v ½ Ï. Secondly, Ï interference components while match to the signal. Not surprisingly, the subarray processing scheme is a special form of reduced rank processing in that the Ï has a block diagonal structure. It is composed of two stages of GMVDR beamforming. The interference is cancelled at each subarray by Ï i, and is further cancelled by the beamformer. The configuration of the coherent subarray detector is shown in Fig. 3. C. The noncoherent detector The concept of noncoherent processing has been employed to advantage in application to coherencedegraded signals (see e.g. [14]). In this case, the outputs of each subarray after the pre-processing
ó â 9 matrix Ï Fig. 3. The configuration of the coherent subarray detector are squared and summed regardless of the coherence of the signal along each subarray. It is an approximation to the coherent subarray processor where detector [14],[7]. 4>. In practical situations, it is a robust In fact, if indeed each subarray data *) i 3r476'8 8 8 Õ is realization of independent random process, i.e., E*). ) i ƒ45ðgbñ 4 ò k, it is straightforward to show that the designed Ï in equation (13) is the optimal subarray detector (see appendix II for details). Hence, the noncoherent subarray detector is given as i è i 4¼Ï (18) Fig. 4. The configuration of the non-coherent subarray detector D. Practical considerations In practical application, we usually do not have a priori knowledge of some of the parameters. Let 4p ~t u8 vw (19)
i i 1 denote the unknown parameter set. Without attempting to estimate those parameters in real time application, we will use an approximation to the subarray processors described before. Notice that the eigenvalues of matrix i v i represent the ratio of signal strength to residue interference plus noise power projected onto the signal subspace i. If we assume that i v i can be approximated as a diagonal matrix and that this diagonal matrix is an identity matrix with a scalar ô i, we can write down the r -th subarray beamformer in a simpler form as follows, Ï i ž¼» i v i i (2) where» i 4 Ð ô i fÿ6 Mt u is a scaling factor. Certainly this scaling factor is unknown because we have no priori knowledge of tu. Further approximation may be made such that» T ž 8 8 žp» Ö implies that the residue SNR on each i is approximately equal. Furthermore, the true covariance matrix v 4õ», which is seldom known and has to be estimated from real data. Let vå½ denote the sample covariance matrix of v. Thus we obtain a practical solution as follows, ö Ï i ö ž»v z ½Z Ti i (21) where ö» is equal to» with v being replaced by v ½. Such an approximation allows us to write down the practical solution of the noncoherent subarray detector as follows ø i è i 4>ù ÏÂ4 ö»!"%$'&x v ½È T 8 8 8 v An approximation to the coherent subarray detector is then given by where ú 4 ø ²æ;èeêë4¼ú ù Ïå ù Ï v ½ Ï ù Ð û fÿ6 Mt u, and Ïå äï vå½(ï Ï üž ½Z Ö!"%$'&X@ T 8 8 8 *B Ö (22) Ï ù (23) û. This approximation is based on the assumption that the ratio of signal strength to residue interference plus noise filtered by Ï is identical along each dimension of the signal subspace. As a comparison, the full array detector under the practical condition takes the form where ýþ4 Ð ÿ fã6 Mt u and v ø ë4ãýv ½ (24) ½ ž ÿ. Again, the approximation is based on the assumption that the residue signal-to-noise is identical in. We notice that the weighting matrix in equation (22)-(24) may hold up to a scaling factor. To be comparable, we can normalize the weighting matrix such that ç <476. Therefore, the scaling factor will not affect our performance measure defined in section IV, and then may be disregarded.
â 11 E. Spatial smoothing for a uniform linear array Spatial smoothing technique [18], [16] is a preprocessing scheme developed for linear uniformly spaced arrays. The spatial smoothing provides a better estimation accuracy for a covariance matrix which has a Toeplitz structure, and is briefly described below. Let us divide a linear uniformly spaced array with apart, into overlapping sub-arrays of size Ç identical sensors spaced half wavelengths with sensors È6'8 8 8 BÇ> forming the first sub-array, sensors [8 8 8 MBÇÅfy6M forming the second array, etc. Let denote the total number of sub-arrays, then ¹4 Ç fy6. The full -element input vector is given by equation (3), and the Ç -element input vector for the r -th subarray, which shall be denoted by ) i is expressed as follows, ) i 4p \ i 3\ i ÍT*8 8 8 *3\ i ÍcË e Q ƒr476'8 8 8 (25) The spatially averaged Ç Ç sample correlation matrix v b ½ is given by v b ½ 4 matrix Ïù takes the form where i 3r²4 Ï74p ~ ù Ö, b v T Ö Ö i'j T v ½È i. Thus the ½ F ~ Ö,l ƒtb (26) 6'8 8 8 Õ the subspace manifold of r -th diagonal block signal covariance matrix v u Ir. Due to the fact that the array is a uniform linear array, the r -th diagonal block matrix v u Ir (or the interference plus noise covariance matrix vn Ir ) are identical to each other. Hence i 3r4p6'8 8 8 are identical based on equation (6). IV. PERFORMANCE ANALYSIS In this section, we quantify the performance of the subarray detector analytically. It is clear that the derived coherent subarray detector is a cascade of GMVDR beamformers, i.e., the beamformer Ï i at each subarray followed by a combiner. There are three basic issues that need to be understood. One is the interference cancellation through two stages of subarray processing. Smaller number of degrees of freedom is used at the subarray level to cancel out the interference, which causes performance loss, the second stage will gain back some of the loss by combining the outputs from the first stage. The second issue is the potential gain of the subarray processing compared with the full array processing with finite sample size due to a better statistical stability of the interference estimation. The third issue is the effect of signal source angular spread. We use a uniform linear array (ULA) as an example although the analysis can be, in principle, extended to a non-ula.
i E E i 12 A. Interference cancellation of coherent and noncoherent subarray detector The i n i matrix Ï i v i Ï i represents the covariance matrix of the interference plus noise at the output of r -th subarray beamformer Ï i. Ï i v i Ï i 4 ~t u i v 4 i!#"ì$'&x T Í i i f 8 8 8 * Ó T Í ÓÌ where t u i v i 4¼ i gi i via an eigen-decomposition and i 4Ÿ!#"%$'& T 8 8 8 * Ó is the eigenvalue matrix. T Í represents the residue interference plus noise appearing at _ -th beam at the output of Ï i. Furthermore, the general covariance matrix at the output of first stage processing matrix Ï is given by a block matrix Ï vï with Ï. v. i Ï i being its I2 3r -th block, which indicates the cross-correlation of the interference plus noise outputs of beamformer Ï. and Ï i. It is often said that a beamformer requires one adaptive degree of freedom per point interferer to achieve interference cancellation. We extend the point sources to the distributed sources, and study the interference cancellation of the coherent and noncoherent subarray detectors for the following two cases: (a). Ç and (b). Ç, where Ç is the subarray size and is the rank of interference subspace. If Ç, i.e., the available number of degrees of freedom at the subarray level is greater than the interference subspace rank, the equation (27) becomes (27) Ï i v i Ï i žÿ!#"%$'&x T T 8 8 8 * Ó Ó (28) The above equation indicates that the interference can be fully cancelled and the beam noise outputs of Ï i are uncorrelated. To quantify the interference cancellation through different stages of subarray processing, we calculate the deflection of the test statistics of different detectors Ù I) 45 ¹ ) W, i.e., change in mean divided by standard deviation. The deflection (modified based on [3]) is given as follows: 4Ÿt u v u3 ç Ð MR I v ± W Assuming that the antenna placement of the general linear array is around the position of a uniform linear array with a small random offset, we may have the approximation þtž 8 8 ž Ö which leads to Ï T ž7 8 8 /ž¼ï Ö based on (2), or T žâ 8 8 [žã Ö. In this case, Ï vï may be approximated as (29) Ï vïpžÿ Ö,!"%$'&XÈT T88 8 8 Ó ÓP (3)
Ç â ž Ó Ó Ó Ó Ó % * - * Ó Ó * * ' which indicates that the beam noise of Ï is uncorrelated. Therefore utilizing (28)-(3), we obtain # $&%(')+* 4 t u "! µ # É $ %, /1"! Ê.- t u Ó2 # $ % ' É i 3E "! Ê Ö / 4 2Ó 487 4 # $ %(' E65 ž µ t u É i "! Ê µ # É $ É i "! Ê 4 µ ÓMÊ - 13 (31) Hence, 9 ž µ (32) The above result is verified by simulations later (see Fig. 5 and 1). The gain for the coherent subarray detector is rather complicated. Instead, we calculate the bounds of the gain. The gain function is given as follows, µ ;: 9&=<.>@? 9AB:œ (33) The maximal gain of the coherent subarray detector is obtained when the signal source is a point source. In this case, the detector is essentially a conventional beamformer which combines beam outputs coherently and yields a gain of (see also [8]). The lower bound of the gain function is certainly due to the fact that the coherent subarray detector has a better gain than the noncoherent subarray detector. Notice that the relative loss of Ï i to the full array detector CED FHGJILK A is lower bounded by µ. Equation CED FMGJN (32) and (33) indicate that processing at the second stage yields a constant gain that compensates the loss occurred at the first stage. Consequently, we will also see by computer simulations that the overall performance loss of the subarray detector is insignificant compared with that of the full array detector. When Ç O, the number of adaptive degrees of freedom is smaller than the rank of the interference subspace, the performance loss due to incomplete cancellation of the interference may be significant. Generally the matrix Ï v Ï is a diagonal dominant matrix due to weak cross-correlation between different Ï i beam outputs of residue interference. Hence, Eqn. (32) and (33) still hold. However, when becomes extremely small, A tends to the average element deflection, which indicates a severe loss of interference cancellation capability. B. Effect of signal angular spread In this subsection, we use the output SNR, which is given as PRQS # $ % ' * 4UT "! # $ % *, as our performance measure for a simplified case where v 4 W. We then obtain the SNR gain of Ï i, defined "! as
j m 6 ï b 14 output SNR vs. average element SNR, as follows (see appendix III for details) PRQVSXW Ó1ž Ç i (34) where i is the effective rank of subarray signal subspace, or equivalently, the number of main beams. This result is consistent with the SNR gain for the full array detector reported in [12]. Eqn. (34) suggests that the SNR gain is the array gain of the subarray normalized by the number of main beams. It shows that, SNR-wise, using subarrays (Ï i ) with appropriate size so that a single wide beam ( i 4 6 ) is generated to obtain its array gain has little difference than using a full array with multiple narrow beams (ƒä 6 ). However, the advantage of using subarrays is evident when finite sample size is used because of the better estimation accuracy. This also explains that using subarray processing for the distributed sources makes more sense than for the point sources. C. Analysis of SINR gain with finite samples o In this subsection we will examine the effect of the reduced rank processing with finite samples by comparing the performance between the coherent subarray detector and the full array detector. We consider, for simplicity, the case where the signal is a point source. Let ï ~oþ denote the SINR with o samples and ï ZYœ the optimal SINR. This allows us to further quantify the relative performance between a subarray processor and a full array processor by means of SINR gain defined as below [ ¾ 4 ï u]\_^ ~oþ \_`` ~oþ Similarly [Ra represents the asymptotic SINR gain (or loss) of the subarray detector relative the full array detector. Depending on whether the training data for interference and noise are available, we study two different cases. C.1 SINR gain with training data 4i4 É ¾ Ê For the full array detector, it is well known that the normalized SINR loss factor b \_`` 4dcfehg 4i4 É a Ê (see c ehg Reed [17]) has the probability density function ¾ kb= \_`l` o U4 z q ÍT œ6' o fl \_`l` e6 =b= \_`` q m É where I2 3r 4. Êon É i Ê@n ¾ É. Í i Ê@n. The mean value of b takes the form of Epbm \_`l` 4 ¾z q ÍT carry out the similar derivation by noticing that Ï v ½ Ï has the Wishart distribution q äï vï o Ùm (35) (36). We can with dimension Ù [2] given the fact that v ½ is the Wishart distribution q äï v ÏÎ o Ùm with dimension
j ž ž ï j ï ' ï ï É É m 4 E Ê w w E Ê E E v 4 6 ï ï w ' z E T } w ' z E T É b ' 4 E Ê w E ¾ ¾ T T ' ' T ' ' c ' c 15. Caution should be taken in that Ï depends on training data. However, due to the fact that Ï is computed from each subarray, each block of Ï i has good estimation accuracy when sample o large compared to the subarray size Ç is relative, it is general true that Ï is a relatively constant matrix independent of a particular set of training data. Thus the normalized SINR loss factor b u]\_^ 4 subarray processor has the probability density function ¾ kb u]\_^ oƒ Ù a4 z Ø[ÍT ~ÙÕ œ6' o fœ þùm u]\_^ e6 sb u]\_^ Ø ¾ with mean value of Epb uz\t^ 4 ¾ z Ø[ÍT. Hence the SINR gain can be written as [ ¾ 4 ï u]\_^ ~oþ \t`l` ~oþ where the asymptotic SINR gain [ua 4 ï u]\_^ ZYœ \t`l` ZYœ C.2 SINR gain without training data Epb u]\_^ Epb= \t`l` uz\_^ ZYœ \_`l` ZY 9 Ï äï vï Ï 9 9 v 9 o² þù fÿ6 o² fÿ6 É ¾ Ê gfr É a Ê for the coherent gfr (37) [Ra (38) y6 (39) In this case, we may define the SINR as the average of signal power to the average interference-plusnoise ratio. Hence, for the full array processor, by citing the results from [22] that \_`l` ~o až t u 9 v 9/ f Without too much difficulty and assuming that Ï is a relatively constant matrix independent of a particular set of data, we have the following SINR for the coherent subarray detector uz\t^ ~oþ ž Thus the SINR gain in this case is defined as where [Ra q ¾ t u 9 Ï äï vï Ï 9/ f [ ¾ É ¾ 4 Ê c g r 4i4 É ¾ Ê cfehg #v $%xw v *yw E Í #v $ $ %, $ v N * w E Í { v $& $%, $ v v $ % w T Í v $%xw v É q Ê ¾ (42) T Í v $ $ %, $ v É Ø Ê ¾ 4 [ua j ~o Ùm j É ¾ is defined in equation (39). ~o Ùm a4~} Ê É a Ê is the normalized gain function that indicates the potential gain of coherent subarray processing depending on Ù and o, and is given as follows, 6af 9 v É 9 T q Ê ~o Ùm =4 6Uf 9 Ïå äï vï É Ï 9 Ø Ê t u Ø ¾ t u (4) (41) (43)
Ç 16 V. MONTE CARLO SIMULATIONS In this section we evaluate the performance of subarray detectors described above, using Monte Carlo simulations. We will experiment the subarray detectors under various scenarios to validate our discussions. The simulation is based upon a general linear array with the this array is fixed as 4 sensors. The total aperture of where is the half wavelength. However, the rest of the Æ" elements are randomly placed in a uniform distribution fashion. The element spacings are in units of half wavelength. To be comparable, this general linear array is generated once for all the test scenarios. As a special case, a uniformly linear array (ULA) of sensors is also considered. The half power beamwith of this array is ƒ 4Â u l JM. The beamformers of interests under ideal conditions (i.e. all the statistics are known) and their approximations under practical conditions are listed in table I. The performance of the beamformers under ideal conditions serves as references. However, in most of the simulation tests (except for the first test), we test the performance of their approximations under practical conditions. TABLE I COMPARISONS OF ADAPTIVE DETECTORS Ideal Condition Practical Condition full coherent noncoherent full coherent noncoherent array subarray subarray array subarray subarray Detectors Eq.(8) Eq.(16) Eq.(18) Eq.(24) Eq.(23) Eq.(22) Simulation test 1: Performance under ideal condition In this test we compare detection performance and beampatterns (gain presented to a distributed source) of the optimal full array and subarray detectors under ideal conditions for a uniform spaced array. Fig. 5 depicts the SINR and deflection along two stages of subarray processing with different subarray size. The signal source and interfering source are at and respectively. The signal has an angular spread u ^ˆ 4 beamwidth, the interference has an angular spread ^ i ] 4O beamwidth. This also implies that the effective rank of the signal subspace and interference subspace are ³4 and ë4o respectively. Choosing the subarray size Ç Ä, the detector Ï shows little performance loss in deflection. When Š, the performance degradation becomes substantial due to the partial cancellation of interference. In this case, the plot of the coherent subarray detector shows that the second stage processing will not gain back the loss. Fig. 6 depicts the detection performance of the three detectors with subarray size Ç 476. The signal
q Ë 17 and interference have angular spread gu ^ˆ 4 beamwidth. It shows that, in this case, both the coherent and non-coherent subarray detectors have performance loss within 6 db relative to the optimal full array detector. We notice that the performance difference between the coherent detector and non-coherent detector is insignificant in this case. This observation indicates that the processing matrix Ï projects the signal from element space onto a small amount of beams. A simple noncoherent combining scheme is very effective regardless of coherence of the signal. However, if the subarray size Ç is not chosen properly, for instance, Ç 4Œ, the noncoherent detector suffers significant loss indicated by the receiver operating characteristic curve in Fig. 7. Fig. 8-9 depict the beampatterns of the optimal detectors. We observe that in Fig. 8 the full array detector has gain of q 4 6 db at the signal direction. This result is due to the fact that the signal has angular spread of 6* M degrees (or equivalently BW), and is consistent with the result in [12]. Choosing the subarray size Ç 4Š will lead to a significant resolution loss for the nocoherent subarray detector as is shown in Fig. 9. Fig. 1 shows that the SINR of the coherent subarray detector is very close to that of the full array detector as Ç db ( ±4 ÄŽ. Also the deflection gain of coherent subarray detector relative to Ï i is bounded by 4Œ ) and Æu l db. While the deflection gain of the noncoherent subarray detector remains at Æu l db. There results are in agreement with equation (32) and (33). 1 (a) 1 (b) 5 5 SINR [db] 5 1 Deflection [db] 5 1 15 15 2 T n coherent sub noncoherent sub full array 2 T n coherent sub noncoherent sub full array 25 1 2 3 4 subarray size 25 1 2 3 4 subarray size Fig. 5. SINR and deflection of subarray and full array processing vs. subarray size. Signal angular spread E @ l = BW, interference angular spread R p BW. š" 9œ Ÿž db, š6 Zœ Ÿž( _ db. The effective rank of the interference subspace is x. Simulation test 2: Effects of sample support
18 1.9.8.7 Detection Probability.6.5.4.3.2.1 full array coherent subarray noncoherent subarray 16 14 12 1 8 6 4 2 2 SNR [db] Fig. 6. Comparison of detection performance of full array and subarray detectors. The signal source and interfering source are at p J and ž9 _ t respectively with angular spread of 6 each. The subarray has size of ;ª. 1.9.8.7 Detection Probability.6.5.4.3.2.1 full array noncoherent subarray coherent subarray 16 14 12 1 8 6 4 2 2 4 SNR [db] Fig. 7. Comparison of detection performance of full array and subarray detectors. All the parameters remain the same as in Fig. 6 except that «. In this test we study the performance with different data samples o under various practical scenarios. Fig. 11 depicts the SINR gain, defined in equation (38), for detecting a point source when training data are available. The number of samples ranges from o 4U#6 to o 4² PP. As we can see, with very small number of samples, there exists a very large SINR improvement for the subarray processing compared with the full array processing. At Ùì4 6 (or 6 subarrays in this case), the relative SINR gain has 6*Æ db. This is because the sample covariance matrix is ill conditioned when oå4 #6. A full dimensional processing certainly leads to very bad performance. As the sample size becomes larger, this gain diminishes, eventually goes to the negative domain. This phenomenon demonstrates the asymptotic
19 1 1 Magnitude [db] 2 3 4 5 full array detector coherent subarray detector noncoherent subarray detector 6 8 6 4 2 2 4 6 8 Azimuth [deg] Fig. 8. Beampatterns of full array detector and subarray detectors. All the parameters remain the same as in Fig. 6. The processors s weighting matrix for three cases has been normalized such that ± ² ³ Ŕ 1 1 Magnitude [db] 2 3 4 5 full array coherent subarray noncoherent subarray 6 8 6 4 2 2 4 6 8 Azimuth [deg] Fig. 9. Beampatterns of the full array and the subarray detectors. All the parameters remain the same as in Fig. 8 except that the subarray has size of ;=. performance loss of the subarray processing relative to the full array processing. However, the loss is insignificant for this experiment. When the training data are contaminated by the signal component, we also observe significant performance gain for the subarray processor. Fig. 12 depicts the SINR gain, defined in equation (42), with different sample size starting from o 4Œ /6 to o 4 PP. We observed a similar behavior of SINR gain in this case. However, it shows that the gain decreases with a much slower rate in this case. One may argue that the diagonal loading method will improve the full array detector performance significantly. In order to find out the effects of diagonal loading on detector performance, we experiment
2 14 (a) 14 (b) 12 T n coherent sub noncoherent sub full array 12 T n coherent sub noncoherent sub full array 1 1 SINR [db] 8 6 Deflection [db] 8 4 6 2 4 2 4 6 8 1 Angular spread [BW] 2 2 4 6 8 1 Angular Spread [BW] Fig. 1. Comparison of SINR and deflection of subarray and full array processing as the angular spread changes. The š" œ µž db, š6 Zœsµž( t db, ;. the three detectors under practical condition with different loading levels. The results are shown in Fig. 13 and Fig. 14. For three test cases, the diagonal loading level of ³4¼ W and ³4p6 subarray detector still outperforms the full array detector. W are utilized. The 14 12 L=1,theoretical L=1,numerical L= 4,theoretical L= 4,numerical 1 8 SINR Gain [db] 6 4 2 2 1 2 1 3 Sample size (K) Fig. 11. Comparison SINR gain of subarray processors and full array processor as training sample size changes.the array is a non-ula, the šh 9œss db, š6 Zœsž( _ db. The source is a point source. Simulation test 3: Effects of spatial smoothing for a ULA In this test, we study the effects of spatial smoothing and diagonal loading on detection performance with finite training sample size. Fig. 15 depicts the detection performance for noncoherent subarray processor and full array processor. It shows that diagonal loading on full array GMVDR improves the
21 1 8 L=1,theoretical L=1,numerical L= 4,theoretical L= 4,numerical 6 Gain [db] 4 2 2 1 2 1 3 Sample size (K) Fig. 12. The SINR gain of the subarray processing relative to the full array processing as sample size All the parameters remain the same as in Fig. 11 except that the training data include signal components. changes. 1 NULA,1DL.9.8.7 Detection Probability.6.5.4.3.2.1 full array noncoherent subarray coherent subarray 1 2 sample size (K) Fig. 13. Comparison of detection performance for subarray detectors and full array detector as the training sample size changes. The array is a non-ula, the š" œ db, š6 Zœ ž( t db, loading level is noise level, training sample data include signal components detection performance significantly. The subarray processor with spatial smoothing and diagonal loading has the best performance. This plot demonstrates that combining spatial smoothing and diagonal loading improves the detection performance for a uniform linear array. It is also evident that this subarray processor outperforms the full array processor with limited number of samples where o 4 Simulation test 4: Moving sources Up to this point we restricted our discussion to the case that the sources are stationary but distributed. Now we extend the approach to the case that the source are moving. Due to the dynamic nature of.
22 1 NULA,1DL.9.8.7 Detection Probability.6.5.4.3.2.1 full array noncoherent subarray coherent subarray 1 2 sample size (K) Fig. 14. Comparison of detection performance for subarray processors and full array processor as training sample size changes. All the parameters remain the same as in Fig. 13 except that the loading level is equal to times of noise level. 1.9 (a). Full array processor DL GMVDR GMVDR 1.9 (b). Subarray processor SP DL SP,DL None.8.8.7.7 Detection Probability.6.5.4 Detection Probability.6.5.4.3.3.2.2.1.1 15 1 5 SNR [db] 15 1 5 SNR [db] Fig. 15. Comparison of detection performance of full array and subarray detectors with spatial smoothing (SP) and diagonal loading (DL) for a uniform linear array. The loading level is equal to noise level at different SNR value. ¹Eº¼»½ H¾ _ M, = t, the training data include signal of interest. the environment, the collected data samples contain both the signal of interest and interfering signal components. For the case of strong moving interference, a well noted work [6] is to process data in blocks. In each data block, the interference components are suppressed through an orthogonal projection operation. The processed data are collected to calculate a MVDR beamformer. We modify this approach by generating a wide main beam that covers beamwidths. We call this beamformer the modified MVDR (MMVDR). In this test, we study the detection performance of the subarray noncoherent detector, full
23 array detector and the modified MVDR beamformer by means of detection of probability. For the three detectors, a loading level of ³4ã W is used. We assume that both the interfering source and signal source move in one direction through beams (6* u ) with different speeds. The experiment setting is summarized in Table II. TABLE II EXPERIMENT PARAMETER SETTINGS FOR DETECTION OF MOVING SOURCES  ŠParameter Value Parameter Value Parameter Value Array size (P) ÀÁ Speed range (meter/sec) ÃÄ Á Sampling rate (Hz) Angular range (deg) ÅÆ (À BW) Processing interval (sec) ÅÆ Loading level noise level SNR (db) Ç SIR (db) È ÅhÁ ÃÉÈÊ Á Distance to array (meter) w6î Å Á Á Á Sample size (K) ÅÆ ÃÉÅÆÂfÁ False alarm rate (˼ÌMÍ ) ÅhÁ The detection probability results are listed in Table III. The results show that, the subarray beamformer outperforms consistently other two beamformers in all the test cases, especially when the number of samples is limited. This experiment clearly demonstrates that, the subarray processing scheme is a relatively effective way of processing data in a dynamic environment. TABLE III COMPARIONS OF DETECTION PERFORMANCE OF PROCESSORS (DETECTION PROBABILITY) Moving Ï_ÐkÑ½Ò È ÅhÁ db ÏpÐkÑ½Ò ÈÇ Á db Ï_ÐkѽÒÉÈÊ Á db Ó Speed full array subarray modified full array subarray modified full array subarray modified 17.4533 12.365.5945.3292.389.2645.645.4258.4277.948 5.8178 36.617.6494.29.2467.824.1625.59.6821.39 3.497 6.5313.7663.267.538.817.3923.663.657.2693 2.944 1.361.7234.31.512.7513.5326.6511.5229.5588 1.7453 12.6791.813.218.679.8226.6814.611.8121.3984 VI. CONCLUSION In this paper we studied the subarray processing as a special form of reduced dimensional processing scheme where the dimension reduction transformation matrix is a block diagonal matrix. We derived the optimal subarray detector from the detection theory standpoint and studied its performance for large aperture arrays. The subarray processing offers a tradeoff: better statistical accuracy at the cost of reduced number of degrees of freedom. With finite number of data samples, the subarray detectors offer
24 a significant gain. We identified this performance gain by analysis and by computer simulations. Both results demonstrate that the subarray processing scheme is an effective way of dealing with the problem of detection under limited number of samples. Furthermore, this method is shown to be promising in dealing with moving sources in the underwater acoustic scenario. REFERENCES [1] Acoustic Observatory Project, Office of Naval Research, available online at http://www.onr.navy.mil/sci tech /ocean/321 sensing/ us1 acousticob/default.htm [2] T.W. Anderson, An Introduction To Multivariate Statistical Analysis, Second Edition, 1971 [3] C. R. Baker, Optimal quadratic detection of a random vector in Gaussian noise, IEEE Transactions on Communications, COM-14, pp.82-85, 1966. [4] K. A. Burgess and B. D. Van Veen, Subspace based adaptive generalized likelihood ratio detection, IEEE Transactions on Signal Processing, vol. 44, no. 4, pp.912-927, April 1996. [5] J. Capon, et al, Multidimensional maximum likelihood processing of a large aperture seismic array, Proceedings of IEEE, vol. 55, pp.192-211. 1967 [6] H. Cox, Multi-rate adaptive beamforming (MRABF), IEEE SAM 2 Conference, March 16, 2. [7] H. Cox, Line array performance when the signal coherence is spatially dependent, Journal of Acoustics Society of America, vol. 54, pp.1743-1746, 1973. [8] H. Cox, A subarray approach to matched-field processing, Journal of Acoustics Society of America, vol. 87 (1), pp.168-178, January 199. [9] A. Dhanantwari, et al., Adaptive beamforming with near instantaneous convergence for matched filter processing, Candian Conference on Electrical Engineering, 1996. vol.2. [1] W. F. Gabriel, Using spectral estimation techniques in adpative processing antenna systems, IEEE Transactions on Antennas Propagation, vol. AP34, pp.291-3, March, 1986. [11] J. R. Guerci, J. S. Goldstein and I. S. Reed, Optimal and adaptive reduced rank STAP, IEEE Transactions on Aerospace and Electronic Systems, vol. 36, no.2, April 2. [12] Y. Jin and B. Friedlander, Detection of distribted sources using sensor arrays, submitted to IEEE Transactions on Signal Processing (to appear) [13] Y. Kim, S. Pilai and J. Guerci, Optimal loading factor for minimal sample support sapce-time adaptive radar, IEEE 1998 [14] D. R. Morgan and T. M. Smith, Coherence effects on the detection performance of quadratic array processors, with applications to large-array matched-field beamforming, Journal of Acoustical Society of America, vol. 8 (2), pp.737-747, February 199. [15] N. L. Owsley, Array Signal Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentivce-Hall, 1985. [16] B. D. Rao and K. V. Hari, Weighted subspace methods and spatial smoothing: analysis and comparison, IEEE Transactions on Signal Processing, vol. 41, no. 2, pp. 788-83, February 1993. [17] I. S. Reed, J. D. Mallett, and L.E. Brennan, Rapid convergence rate in adaptive arrays, IEEE Transactions on Aerospace and Electronic Systems, vol. AES-1, pp.853-863, Nov. 1974.
j j ž ž ž 25 [18] T. Shan and T. Kailath, Adaptive beamforming for coherence signals and interference, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-33, no.3, pp527-536, June 1985. [19] D. N. Swingler, A low-complexity MVDR beamformer for use with short observation times, IEEE Transactions on Signal Processing, vol.47, no.4, April 1999. [2] H. L. Van Trees, Optimum Array Processing, John Wiley & Sons, Inc, New York 22. [21] B. D. Van Veen, An analysis of several partiallyadaptive beamformer designs, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no.2, pp.192-23, Febuary 1989. [22] M. Wax and Y. Anu, Performance analysis of the minimum variance beamformer, IEEE Transactions On Signal Processing, vol.44, no.4, pp.928-937, April 1996 APPENDIX I. THE OPTIMAL FULL ARRAY DETECTOR According to the Neyman-Pearson criterion, the optimal detector for detecting a distributed signal is the one that maximizes the likelihood Î j ratio. Let I) function under hypotheses where vƒt 4 T and respectively, TB j Î and I) I) TV Ã4 R oô q v TM ÆÕÖR È ) v I) Ã4 R oô q vn ÕÖR È ) v T )+ )+ denote the probability density t ubv u fœv. Employing the matrix inversion lemma, it is then straight-forward to write the log-likelihood function as follows [12]: (44) Ù I) 4 ) ~v þv T ) ) Øäv ~t uv f vw Ù ) ) 1 ~v þv ft ü v n ~t ul v pf v ) ) Øät u v ~t u v pf v Ù ) (45) If we let the w matrix be ² ë47ð t u v n ~t u v >f ew (46) The w weighting matrix consists of columns, where is the rank of the signal subspace.