Reduced-Rank Adaptive Detection of Distributed Sources Using ...
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Reduced-Rank Adaptive Detection of Distributed Sources Using Subarrays Yuanwei Jin, Member, IEEE, and Benjamin Friedlander, Fellow, IEEE
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. Index Terms—Adaptive processing, detection, distributed source, interference cancellation, reduced rank, subarrays.
INTRODUCTION, BACKGROUND, AND MOTIVATION
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HE 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, which is 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] and [10]) 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
Manuscript received August 23, 2003; revised January 13, 2004. This work was supported by the Office of Naval Research under Grant N00014-01-1-0075. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Jean Pierre Delmas. Y. Jin was with the University of California, Santa Cruz, CA 95060 USA. He is now with Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail: [email protected]). B. Friedlander is with the University of California, Santa Cruz, CA 95060 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2004.838941
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 references 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. Second, 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. 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 subarray 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 and Swingler [19], and Dhanatawari [9]. Owsley suggested that a narrow band uniform linear array (ULA) containing elements could be decomposed into, say, nonoverlapped 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 ULA
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comprising elements. Dhanatawar chose a different subarray geometry, where the subarrays were heavily overlapped. The work in [8] and [14] addresses the issue of signal coherence degradation, but 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 nonstationarity of the underwater acoustic environment. The motion of the sources causes a nonstationary 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 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; denotes the statistical expectation; denotes the matrix determinant; denotes the vector (matrices) Frobenius norm; Tr denotes the trace of a matrix; diag denotes diagonal matrix whose diagonal is the vector ; denotes Kronecker product.
I. PROBLEM FORMULATION A. Array Signal Model We consider a general linear array composed of be the coordinates of the Let
sensors. th sensor
measured in half wave-length units and vector of the array in the direction :
be the steering (1)
Consider narrowband radiating sources impinge on the array from distinct direction . 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 , 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 modeled as
(2) where
is array output at sample time (3)
and is the th complex waveforms constituting the signal . We assume that the first with total signal power is the desired signal with and signal that others are considered as the interfering signals. is a We assume that the instantaneous array response complex Gaussian vector with zero mean and covariance matrix . This covariance matrix is related to the array manifold by the following expression (see, e.g., [12]) (4) where is the array manifold at angle . is normalized . is the spatial energy distribution to be Tr . of the source at azimuth , such that is the nominal direction of arrival of the signal. is the signal angular spread. The energy distribution function 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., (5) 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 be the singular value decomposition of . If indeed is a is exactly the subspace low rank matrix with rank , then spanned by the first columns of . More generally, we will can be approximated by a rank matrix (i.e., assume that the singular values from onward are small compared is with the singular values from 1 to ), in which case, approximately the subspace spanned by the first columns of , i.e., (6)
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where . is the complex white Gaussian noise and is uncorrelated with the with zero mean and covariance signal sources. The covariance matrix of interference plus noise is denoted by . B. Optimal Full Array Detector
Fig. 1.
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 purposes of simplicity) is given as follows:
(7) The optimal full array detector for the above detection problem (7) is given as follows (see Appendix A for details) (8) is the subspace matrix, while the matrix 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 a generalized MVDR (GMVDR) beamformer in the sense that it extends the standard rank one MVDR beamformer where weighted
signal
(9) to a multirank case. In fact, the GMVDR beamformer (see Fig. 1) is the optimal detector for a distributed signal source from a detection theory standpoint. The implementation of the full array detector requires a priori , which is often estimated from finite knowledge of is replaced by its maxtraining samples. In this case, imum likelihood estimate , where is a set of independent and identically distributed training data. It is pointed out (see Reed stationary data vectors are required to [17]) that obtain a moderately statistically stable estimate of . 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 Section III. II. 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
Configuration of the GMVDR beamformer.
Fig. 2. Reduced-rank processing architecture. matrix, 2 is a detector.
C
T2
is a transformation
process the received data of each subarray individually. For a general linear array of sensors, a common scheme is dividing the total array into a number of nonoverlapping subarrays with sensors. Without loss of genequal size. Each subarray has where is the number of erality, let us assume that forming the first subarray, subarrays with sensors sensors forming the second subarray, etc. The full -element input vector is given by (3). The -element input data vector for the th subarray, which will be denoted by , is expressed as follows: (10) A. Optimal Subarray Beamformer It is straightforward to show that the optimal subarray beamformer for the th subarray is the GMVDR beamformer based matrix on the subarray data vector . Hence, the is given as follows:
(11) where . is the th diagonal block of , and is the th subarray is then given manifold. The transformed data vector . By grouping , we have as the following linear transformation: (12) . The linear transformation mawhere is a block diagonal matrix with . trix is given as follows: diag
(13)
B. Coherent Detector A natural question arises as how to combine the processed data from each subarray in an optimal fashion. Noticing (12), this question can be easily answered from the reduced rank detection theory standpoint. The block diagram of the processing , where scheme is depicted in Fig. 2. Let matrix
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is the signal subspace rank, be a detector based upon the transformed data . The reduced detection problem is then described by the following binary hypothesis test
(14) The optimal detector for the above detection problem (14) takes the same form as (8), except that the full dimensional matrices , and are replaced by the transformed matrices , and respectively. It is given as follows: Fig. 3. Configuration of the coherent subarray detector.
(15) Hence, the coherent subarray detector for this reduced rank processing architecture appears to be (16) where which leads to the following test statistics:
, (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 in the reduced rank detection linear transformation matrix compresses theory serves two purposes. First, this matrix -dimensional data into a -dimensional subspace prior to constructing a test statistic. This transformation reduces the into . This reduction in the nuisance parameters of number of nuisance parameters tends to improve the accuracy of . Second, removes the dimensions that the estimate contain least “signal-to-interference-plus-noise” components. A desirable should suppress strong interference components while match to the signal. Not surprisingly, the subarray processing scheme is a special form of reduced rank processing has a block diagonal structure. It is composed in that the of two stages of GMVDR beamforming. The interference is and is further cancelled by cancelled at each subarray by the beamformer . The configuration of the coherent subarray detector is shown in Fig. 3, and Fig. 4 depicts the configuration of the noncoherent subarray detector. C. Noncoherent Detector The concept of noncoherent processing has been employed to advantage in application to coherence-degraded signals (see, e.g., [14]). In this case, the outputs of each subarray after the pre-processing matrix are squared and summed, regardless of the coherence of the signal along each subarray. It is an approximation to the coherent subarray processor where . In practical situations, it is a robust detector [7], [14]. is In fact, if indeed each subarray data realization of independent random process, i.e.,
Fig. 4. Configuration of the noncoherent subarray detector.
, , it is straightforward to show that the designed in (13) is the optimal subarray detector (see Appendix B for details). Hence, the noncoherent subarray detector is given as (18) D. Practical Considerations In practical application, we usually do not have a priori knowledge of some of the parameters. Let (19) 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. represent Notice that the eigenvalues of matrix the ratio of signal strength to residue interference plus noise . If we assume power projected onto the signal subspace can be approximated as a diagonal matrix and that that this diagonal matrix is an identity matrix with a scalar , we can write down the th subarray beamformer in a simpler form as follows: (20) where is a scaling factor. Certainly this scaling factor is unknown because we have no a priori knowledge of . Further approximation may be made such that , which implies that the residue SNR on each is approximately equal.
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Furthermore, the true covariance matrix is seldom known denote the sample and has to be estimated from real data. Let covariance matrix of . Thus, we obtain a practical solution as follows:
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the subspace manifold of th diagwhere . Due to the fact that onal block signal covariance matrix (or the the array is a ULA, the th diagonal block matrix ) are identical to interference plus noise covariance matrix , are identical based on (6). each other. Hence,
(21) where is equal to with being replaced by . Such an approximation allows us to write down the practical solution of the noncoherent subarray detector as follows: diag diag
(22)
An approximation to the coherent subarray detector is then given by (23) , and . where This approximation is based on the assumption that the ratio of signal strength to residue interference plus noise filtered by is identical along each dimension of the signal subspace. As a comparison, the full array detector under the practical condition takes the form (24) where , and . Again, the approximation is based on the assumption that the residue signal-to-noise is identical in . We notice that the weighting matrix in (22)–(24) may hold up to a scaling factor. To be such comparable, we can normalize the weighting matrix . Therefore, the scaling factor will not that Tr affect our performance measure defined in Section IV and then may be disregarded. E. Spatial Smoothing for a ULA Spatial smoothing technique [16], [18] 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 identical sensors spaced half wavelengths apart, into overlapping subforming the first subarrays of size with sensors forming the second array, etc. array, sensors denote the total number of subarrays, then Let . The full -element input vector is given by (3), and the -element input vector for the th subarray, which will be denoted by , is expressed as follows: (25) The spatially averaged is given by the form
sample correlation matrix . Thus, the matrix takes (26)
III. 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 at each subarray followed by a combiner . beamformer 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 ULA as an example although the analysis can be, in principle, extended to a non-ULA. A. Interference Cancellation of Coherent and Noncoherent Subarray Detector The matrix represents the covariance matrix of the interference plus noise at the output of th subarray beamformer
diag
(27)
via an eigen-decomposidiag is the eigen-value matrix. represents the residue interference plus noise ap. Furthermore, the pearing at th beam at the output of general covariance matrix at the output of first stage processing with matrix is given by a block matrix being its th block, which indicates the cross-correlation and of the interference plus noise outputs of beamformer . 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 nonco, herent subarray detectors for the following two cases: a) , where is the subarray size, and is the rank and b) of interference subspace. , i.e., the available number of degrees of freedom at If the subarray level is greater than the interference subspace rank, (27) becomes where tion, and
diag
(28)
The above equation indicates that the interference can be fully are uncorrelated. cancelled, and the beam noise outputs of
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To quantify the interference cancellation through different stages of subarray processing, we calculate the deflection of , i.e., the test statistics of different detectors change in mean divided by standard deviation. The deflection (modified based on [3]) is given as follows: Tr
DEFL
Tr
(29)
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 , which leads to based on (20), or . In this case, may be approximated as diag which indicates that the beam noise of fore, utilizing (28)–(30), we obtain DEFL
(30) is uncorrelated. There-
Tr Tr Tr
Tr Tr DEFL
(31)
Hence DEFL DEFL
(32)
The above result is verified by simulations later (see Figs. 5 and 10). 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: DEFL DEFL
(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 combeam outputs coherently and yields a gain of (see bines 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 to the full array detector DEFL DEFL is loss of . Equations (32) and (33) indicate that lower bounded by 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.
Fig. 5. SINR and deflection of subarray and full array processing versus subarray size. Signal angular spread = 4 BW, interference angular spread = 8 BW. SNR = 4 dB, SIR = 30 dB. The effective rank of the interference subspace is q = 8.
0
0
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When , 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 is a diagonal may be significant. Generally the matrix dominant matrix due to weak cross-correlation between difbeam outputs of residue interference. Hence, (32) and ferent becomes extremely small, (33) still hold. However, when DEFL 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 Tr Tr , as our perforSNR . We then mance measure for a simplified case, where , defined as output SNR versus avobtain the SNR gain of erage element SNR, as follows (see Appendix C for details) SNRG
(34)
is the effective rank of subarray signal subspace or, where equivalently, the number of main beams. This result is consistent with the SNR gain for the full array detector reported in [12]. Equation (34) suggests that the SNR gain is the array gain of the subarray normalized by the number of main beams. It shows with appropriate size so that, SNR-wise, using subarrays is generated to obtain its array that a single wide beam gain has little difference than using a full array with multiple . However, the advantage of using subnarrow beams arrays 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 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 denote the SINR with samples is a point source. Let the optimal SINR. This allows us to further quantify and the relative performance between a subarray processor and a full array processor by means of SINR gain defined as follows:
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. We can carry out the similar derivahas the Wishart distribution tion by noticing that with dimension [2], given the fact that is the Wishart distribution with dimension . Caution should be taken in that depends on training is computed from each data. However, due to the fact that has good estimation accuracy when subarray, each block of sample is relative large compared to the subarray size , it is general true that is a relatively constant matrix independent of a particular set of training data. Thus, the normalized for the coherent SINR loss factor subarray processor has the probability density function (37) with mean value of the SINR gain can be written as
. Hence
(38) where the asymptotic SINR gain (39) 2) SINR Gain Without Training Data: In this case, we may define the SINR as the average of signal power to the average interference-plus-noise ratio. Hence, for the full array processor, by citing the results from [22] that (40) Without too much difficulty and assuming that is a relatively constant matrix independent of a particular set of data, we have the following SINR for the coherent subarray detector (41) Thus, the SINR gain in this case is defined as
(35) Similarly 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. 1) SINR Gain with Training Data: For the full array detector, it is well known that the normalized SINR loss factor (see Reed [17]) has the probability density function
where The mean value of
takes the form of
(42)
(36)
where is defined in (39). is the normalized gain function that indicates the potential gain of coherent subarray processing depending on and and is given as follows:
.
(43)
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TABLE I COMPARISONS OF ADAPTIVE DETECTORS
IV. 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 sensors. The total aperture of the this linear array with array is fixed as where is the half wavelength. However, the rest of the 38 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 ULA of sensors is also considered. The half power beamwith of this . The beamformers of interests under ideal array is BW 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. 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 20 and , respectively. The signal has an angular spread beamwidth, the beamwidth. This interference has an angular spread also implies that the effective rank of the signal subspace and and , respectively. interference subspace are Choosing the subarray size , the detector shows little , the performance performance loss in deflection. When 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 . The signal and interference beamwidth. It shows that, in this have angular spread case, both the coherent and noncoherent subarray detectors have performance loss within 1 dB relative to the optimal full array detector. We notice that the performance difference between the coherent detector and noncoherent detector is insignificant in this case. This observation indicates that the processing matrix 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 is not chosen properly, for instance, , subarray size
Fig. 6. Comparison of detection performance of full array and subarray detectors. The signal source and interfering source are at 20 and 20 , respectively, with angular spread of 12 each. The subarray has size of = 10.
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Fig. 7. Comparison of detection performance of full array and subarray detectors. All the parameters remain the same as in Fig. 6, except that = 4.
the noncoherent detector suffers significant loss indicated by the receiver operating characteristic curve in Fig. 7. Figs. 8 and 9 depict the beampatterns of the optimal detectors. We observe that in Fig. 8, the full array detector has gain of dB at the signal direction. This result is due to the fact that the signal has angular spread of 12 (or equivalently 4 BW), and is consistent with the result in [12]. Choosing the will lead to a significant resolution loss subarray size for the nocoherent subarray detector as is shown in Fig. 9. Fig. 10 shows that the SINR of the coherent subarray detector . In adis very close to that of the full array detector as dition, the deflection gain of coherent subarray detector relative is bounded by 7 dB and 3.5 dB, to while the deflection gain of the noncoherent subarray detector remains at 3.5 dB. There results are in agreement with (32) and (33).
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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 Tr[ ] = 1.
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WW
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 = 4.
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Simulation Test 2—Effects of Sample Support: In this test, we study the performance with different data samples under various practical scenarios. Fig. 11 depicts the SINR gain, defined in (38), for detecting a point source when training data are available. The number of samples ranges from to . 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 (or ten subarrays in this case), the relative SINR gain has 13 dB. This is because the sample covariance matrix is ill conditioned when . 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.
Fig. 10. Comparison of SINR and deflection of subarray and full array processing as the angular spread changes. The SNR = 4 dB, SIR = 30 dB, = 8.
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Fig. 11. Comparison SINR gain of subarray processors and full array changes.the array is a non-ULA, the processor as training sample size SNR = 0 dB, SIR = 30 dB. The source is a point source.
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Fig. 12. SINR gain of the subarray processing relative to the full array processing as sample size changes. All the parameters remain the same as in Fig. 11, except that the training data include signal components.
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This phenomenon demonstrates the asymptotic 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 (42), with different sample size starting from to . 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
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 SNR = 0 dB, SIR = 30 dB, loading level is noise level, and training sample data include signal components.
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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 10 times of noise level.
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out the effects of diagonal loading on detector performance, we experiment the three detectors under practical condition with different loading levels. The results are shown in Figs. 13 and and 14. For three test cases, the diagonal loading level of are utilized. The subarray detector still outperforms the full array detector. 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.
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It shows that diagonal loading on full array GMVDR improves the 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 . 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 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 four beamwidths. We call this beamformer the modified MVDR (MMVDR). In this test, we study the detection performance of the subarray noncoherent detector, full array detector and the modified MVDR beamformer by means of detection of probability. For the three is used. We assume that detectors, a loading level of both the interfering source and signal source move in one direction through four beams (12 ) with different speeds. The experiment setting is summarized in Table II. 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.
V. 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 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.
Fig. 15. Comparison of detection performance of full array and subarray detectors with spatial smoothing (SP) and diagonal loading (DL) for a ULA. = 0:001, The loading level is equal to noise level at different SNR value. P K = 80, and the training data include signal of interest.
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TABLE II EXPERIMENT PARAMETER SETTINGS FOR DETECTION OF MOVING SOURCES
TABLE III COMPARIONS OF DETECTION PERFORMANCE OF PROCESSORS (DETECTION PROBABILITY)
APPENDIX A OPTIMAL FULL ARRAY DETECTOR According to the Neyman–Pearson criterion, the optimal detector for detecting a distributed signal is the one that maxi, and denote mizes the likelihood ratio. Let and , the probability density function under hypotheses respectively
The likelihood ratio test to the detection problem (47) is given . Taking the as logarithm of the above function, we obtain
(44) where . Employing the matrix inversion lemma, it is then straight-forward to write the log-likelihood function as follows [12]:
(48) In the above equation, line one yields result in line two by a simple algebraic manipulation. Hence, we obtain that . APPENDIX C SNR GAIN FOR ANGULAR SPREAD
(45) If we let the
matrix
defined in (8) with We start with the full array processor being replaced by . In this case, the SNR is given by
be (46)
SNR weighting matrix consists of columns, where The is the rank of the signal subspace.
Tr Tr
(49)
Citing the results given in line one of (45), we obtain that
APPENDIX B NONCOHERENT SUBARRAY PROCESSOR Assume that is independent with each other. Thus, the detection problem (14) reduces to the following binary hypothesis test:
(47)
(50) where diag
JIN AND FRIEDLANDER: REDUCED-RANK ADAPTIVE DETECTION OF DISTRIBUTED SOURCES USING SUBARRAYS
is the eigenvalue matrix. Hence, we have
SINR
Tr Tr
(51)
where . We notice that for a large aperture array and moderate SNR, the dominant eigenvalues have the ap, which leads to proximation
Tr
SNRG
Similarly, for the subarray beamformer,
(52) of size
(53)
SNRG where
is the effective rank of subspace
leads to
.
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[13] Y. Kim, S. Pillai, and J. Guerci, “Optimal loading factor for minimal sample support space-time adaptive radar,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 1998, pp. 2505–2508. [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,” J. Acoust. Soc. Amer., vol. 8, no. 2, pp. 737–747, Feb. 1990. [15] N. L. Owsley, Array Signal Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1985. [16] B. D. Rao and K. V. Hari, “Weighted subspace methods and spatial smoothing: Analysis and comparison,” IEEE Trans. Signal Process., vol. 41, pp. 788–803, Febr. 1993. [17] I. S. Reed, J. D. Mallett, and L. E. Brennan, “Rapid convergence rate in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-10, pp. 853–863, Nov. 1974. [18] T. Shan and T. Kailath, “Adaptive beamforming for coherence signals and interference,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-33, no. 3, pp. 527–536, June 1985. [19] D. N. Swingler, “A low-complexity MVDR beamformer for use with short observation times,” IEEE Trans. Signal Process., vol. 47, pp. 1154–1160, Apr. 1999. [20] H. L. Van Trees, Optimum Array Processing. New York: Wiley, 2002. [21] B. D. Van Veen, “An analysis of several partially adaptive beamformer designs,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, pp. 192–203, Feb. 1989. [22] M. Wax and Y. Anu, “Performance analysis of the minimum variance beamformer,” IEEE Trans. Signal Process., vol. 44, pp. 928–937, Apr. 1996.
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful suggestions that considerably improved the quality of this paper. REFERENCES [1] Acoustic Observatory Project. Office of Naval Research. [Online]. Available: http://www.onr.navy.mil/sci_tech/ocean/321_sensing/ us01_ acousticob/default.htm [2] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Second ed. New York: Wiley, 1971. [3] C. R. Baker, “Optimal quadratic detection of a random vector in Gaussian noise,” IEEE Trans. Commun., vol. COM-14, pp. 802–805, 1966. [4] K. A. Burgess and B. D. Van Veen, “Subspace based adaptive generalized likelihood ratio detection,” IEEE Trans. Signal Process., vol. 44, pp. 912–927, Apr. 1996. [5] J. Capon, “Multidimensional maximum likelihood processing of a large aperture seismic array,” Proc. IEEE, vol. 55, pp. 192–211, 1967. [6] H. Cox, “Multi-rate adaptive beamforming (MRABF),” in IEEE SAM Conf., Mar. 16, 2000. [7] , “Line array performance when the signal coherence is spatially dependent,” J. Acoust. Soc. Amer., vol. 54, pp. 1743–1746, 1973. [8] , “A subarray approach to matched-field processing,” J. Acoust. Soc. Amer., vol. 87, no. 1, pp. 168–178, Jan. 1990. [9] A. Dhanantwari et al., “Adaptive beamforming with near instantaneous convergence for matched filter processing,” in Proc. Canadian Conf. Elect. Eng., vol. 2, 1996. [10] W. F. Gabriel, “Using spectral estimation techniques in adaptive processing antenna systems,” IEEE Trans. Antennas Propag., vol. AP-34, pp. 291–300, Mar. 1986. [11] J. R. Guerci, J. S. Goldstein, and I. S. Reed, “Optimal and adaptive reduced rank STAP,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, pp. 647–663, Apr. 2000. [12] Y. Jin and B. Friedlander, “Detection of distributed sources using sensor arrays,” IEEE Trans. Signal Process., vol. 52, no. 6, pp. 1537–1548, Jun. 2004.
Yuanwei Jin (S’98–M’03) received the B.Sc. and the M.Sc. degrees from East China Normal University, Shanghai, China, in 1993 and 1996, respectively, and the Ph.D. degree in electrical engineering from the University of California, Davis (UC Davis), in 2003. He was a lecturer with the Department of Mathematics, Nanjing University of Science and Technology, Nanjing, China, from 1996 to 1997. In the summer of 1999, he was with Sensys Instruments Corporation, Santa Clara, CA. He was also with the University of California, Santa Cruz. He is now with Carnegie Mellon University, Pittsburgh, PA. His research interests are in signal processing, array processing, estimation, and detection with applications in digital communications and sonar/radar. Dr. Jin was a recipient of the Earle C. Anthony Fellowship from UC Davis from 1997 to 1999.
Benjamin Friedlander (S’74–M’76–SM’82–F’87) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion—Israel Institute of Technology, Haifa, Israel, in 1968 and 1972, respectively, and the Ph.D. degree in electrical engineering and the M.Sc. degree in statistics from Stanford University, Stanford, CA, in 1976. From 1976 to 1985, he was with Systems Control Technology, Inc., Palo Alto, CA. From November 1985 to July 1988, he was with Saxpy Computer Corporation, Sunnyvale, CA. From 1989 to 1999, he was with the University of California, Davis. Currently, he is a professor of electrical engineering with the University of California, Santa Cruz. Dr. Friedlander received the 1983 ASSP Senior Award, the 1985 Award for the Best Paper of the Year from the European Association for Signal Processing (EURASIP), the 1989 Technical Achievement Award of the Signal Processing Society, and the IEEE Third Millennium Medal.
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Reduced-Rank Adaptive Detection of Distributed Sources Using Subarrays Yuanwei Jin, Member, IEEE, and Benjamin Friedlander, Fellow, IEEE
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. Index Terms—Adaptive processing, detection, distributed source, interference cancellation, reduced rank, subarrays.
INTRODUCTION, BACKGROUND, AND MOTIVATION
T
HE 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, which is 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] and [10]) 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
Manuscript received August 23, 2003; revised January 13, 2004. This work was supported by the Office of Naval Research under Grant N00014-01-1-0075. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Jean Pierre Delmas. Y. Jin was with the University of California, Santa Cruz, CA 95060 USA. He is now with Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail: [email protected]). B. Friedlander is with the University of California, Santa Cruz, CA 95060 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2004.838941
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 references 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. Second, 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. 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 subarray 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 and Swingler [19], and Dhanatawari [9]. Owsley suggested that a narrow band uniform linear array (ULA) containing elements could be decomposed into, say, nonoverlapped 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 ULA
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comprising elements. Dhanatawar chose a different subarray geometry, where the subarrays were heavily overlapped. The work in [8] and [14] addresses the issue of signal coherence degradation, but 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 nonstationarity of the underwater acoustic environment. The motion of the sources causes a nonstationary 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 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; denotes the statistical expectation; denotes the matrix determinant; denotes the vector (matrices) Frobenius norm; Tr denotes the trace of a matrix; diag denotes diagonal matrix whose diagonal is the vector ; denotes Kronecker product.
I. PROBLEM FORMULATION A. Array Signal Model We consider a general linear array composed of be the coordinates of the Let
sensors. th sensor
measured in half wave-length units and vector of the array in the direction :
be the steering (1)
Consider narrowband radiating sources impinge on the array from distinct direction . 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 , 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 modeled as
(2) where
is array output at sample time (3)
and is the th complex waveforms constituting the signal . We assume that the first with total signal power is the desired signal with and signal that others are considered as the interfering signals. is a We assume that the instantaneous array response complex Gaussian vector with zero mean and covariance matrix . This covariance matrix is related to the array manifold by the following expression (see, e.g., [12]) (4) where is the array manifold at angle . is normalized . is the spatial energy distribution to be Tr . of the source at azimuth , such that is the nominal direction of arrival of the signal. is the signal angular spread. The energy distribution function 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., (5) 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 be the singular value decomposition of . If indeed is a is exactly the subspace low rank matrix with rank , then spanned by the first columns of . More generally, we will can be approximated by a rank matrix (i.e., assume that the singular values from onward are small compared is with the singular values from 1 to ), in which case, approximately the subspace spanned by the first columns of , i.e., (6)
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where . is the complex white Gaussian noise and is uncorrelated with the with zero mean and covariance signal sources. The covariance matrix of interference plus noise is denoted by . B. Optimal Full Array Detector
Fig. 1.
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 purposes of simplicity) is given as follows:
(7) The optimal full array detector for the above detection problem (7) is given as follows (see Appendix A for details) (8) is the subspace matrix, while the matrix 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 a generalized MVDR (GMVDR) beamformer in the sense that it extends the standard rank one MVDR beamformer where weighted
signal
(9) to a multirank case. In fact, the GMVDR beamformer (see Fig. 1) is the optimal detector for a distributed signal source from a detection theory standpoint. The implementation of the full array detector requires a priori , which is often estimated from finite knowledge of is replaced by its maxtraining samples. In this case, imum likelihood estimate , where is a set of independent and identically distributed training data. It is pointed out (see Reed stationary data vectors are required to [17]) that obtain a moderately statistically stable estimate of . 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 Section III. II. 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
Configuration of the GMVDR beamformer.
Fig. 2. Reduced-rank processing architecture. matrix, 2 is a detector.
C
T2
is a transformation
process the received data of each subarray individually. For a general linear array of sensors, a common scheme is dividing the total array into a number of nonoverlapping subarrays with sensors. Without loss of genequal size. Each subarray has where is the number of erality, let us assume that forming the first subarray, subarrays with sensors sensors forming the second subarray, etc. The full -element input vector is given by (3). The -element input data vector for the th subarray, which will be denoted by , is expressed as follows: (10) A. Optimal Subarray Beamformer It is straightforward to show that the optimal subarray beamformer for the th subarray is the GMVDR beamformer based matrix on the subarray data vector . Hence, the is given as follows:
(11) where . is the th diagonal block of , and is the th subarray is then given manifold. The transformed data vector . By grouping , we have as the following linear transformation: (12) . The linear transformation mawhere is a block diagonal matrix with . trix is given as follows: diag
(13)
B. Coherent Detector A natural question arises as how to combine the processed data from each subarray in an optimal fashion. Noticing (12), this question can be easily answered from the reduced rank detection theory standpoint. The block diagram of the processing , where scheme is depicted in Fig. 2. Let matrix
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is the signal subspace rank, be a detector based upon the transformed data . The reduced detection problem is then described by the following binary hypothesis test
(14) The optimal detector for the above detection problem (14) takes the same form as (8), except that the full dimensional matrices , and are replaced by the transformed matrices , and respectively. It is given as follows: Fig. 3. Configuration of the coherent subarray detector.
(15) Hence, the coherent subarray detector for this reduced rank processing architecture appears to be (16) where which leads to the following test statistics:
, (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 in the reduced rank detection linear transformation matrix compresses theory serves two purposes. First, this matrix -dimensional data into a -dimensional subspace prior to constructing a test statistic. This transformation reduces the into . This reduction in the nuisance parameters of number of nuisance parameters tends to improve the accuracy of . Second, removes the dimensions that the estimate contain least “signal-to-interference-plus-noise” components. A desirable should suppress strong interference components while match to the signal. Not surprisingly, the subarray processing scheme is a special form of reduced rank processing has a block diagonal structure. It is composed in that the of two stages of GMVDR beamforming. The interference is and is further cancelled by cancelled at each subarray by the beamformer . The configuration of the coherent subarray detector is shown in Fig. 3, and Fig. 4 depicts the configuration of the noncoherent subarray detector. C. Noncoherent Detector The concept of noncoherent processing has been employed to advantage in application to coherence-degraded signals (see, e.g., [14]). In this case, the outputs of each subarray after the pre-processing matrix are squared and summed, regardless of the coherence of the signal along each subarray. It is an approximation to the coherent subarray processor where . In practical situations, it is a robust detector [7], [14]. is In fact, if indeed each subarray data realization of independent random process, i.e.,
Fig. 4. Configuration of the noncoherent subarray detector.
, , it is straightforward to show that the designed in (13) is the optimal subarray detector (see Appendix B for details). Hence, the noncoherent subarray detector is given as (18) D. Practical Considerations In practical application, we usually do not have a priori knowledge of some of the parameters. Let (19) 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. represent Notice that the eigenvalues of matrix the ratio of signal strength to residue interference plus noise . If we assume power projected onto the signal subspace can be approximated as a diagonal matrix and that that this diagonal matrix is an identity matrix with a scalar , we can write down the th subarray beamformer in a simpler form as follows: (20) where is a scaling factor. Certainly this scaling factor is unknown because we have no a priori knowledge of . Further approximation may be made such that , which implies that the residue SNR on each is approximately equal.
JIN AND FRIEDLANDER: REDUCED-RANK ADAPTIVE DETECTION OF DISTRIBUTED SOURCES USING SUBARRAYS
Furthermore, the true covariance matrix is seldom known denote the sample and has to be estimated from real data. Let covariance matrix of . Thus, we obtain a practical solution as follows:
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the subspace manifold of th diagwhere . Due to the fact that onal block signal covariance matrix (or the the array is a ULA, the th diagonal block matrix ) are identical to interference plus noise covariance matrix , are identical based on (6). each other. Hence,
(21) where is equal to with being replaced by . Such an approximation allows us to write down the practical solution of the noncoherent subarray detector as follows: diag diag
(22)
An approximation to the coherent subarray detector is then given by (23) , and . where This approximation is based on the assumption that the ratio of signal strength to residue interference plus noise filtered by is identical along each dimension of the signal subspace. As a comparison, the full array detector under the practical condition takes the form (24) where , and . Again, the approximation is based on the assumption that the residue signal-to-noise is identical in . We notice that the weighting matrix in (22)–(24) may hold up to a scaling factor. To be such comparable, we can normalize the weighting matrix . Therefore, the scaling factor will not that Tr affect our performance measure defined in Section IV and then may be disregarded. E. Spatial Smoothing for a ULA Spatial smoothing technique [16], [18] 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 identical sensors spaced half wavelengths apart, into overlapping subforming the first subarrays of size with sensors forming the second array, etc. array, sensors denote the total number of subarrays, then Let . The full -element input vector is given by (3), and the -element input vector for the th subarray, which will be denoted by , is expressed as follows: (25) The spatially averaged is given by the form
sample correlation matrix . Thus, the matrix takes (26)
III. 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 at each subarray followed by a combiner . beamformer 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 ULA as an example although the analysis can be, in principle, extended to a non-ULA. A. Interference Cancellation of Coherent and Noncoherent Subarray Detector The matrix represents the covariance matrix of the interference plus noise at the output of th subarray beamformer
diag
(27)
via an eigen-decomposidiag is the eigen-value matrix. represents the residue interference plus noise ap. Furthermore, the pearing at th beam at the output of general covariance matrix at the output of first stage processing with matrix is given by a block matrix being its th block, which indicates the cross-correlation and of the interference plus noise outputs of beamformer . 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 nonco, herent subarray detectors for the following two cases: a) , where is the subarray size, and is the rank and b) of interference subspace. , i.e., the available number of degrees of freedom at If the subarray level is greater than the interference subspace rank, (27) becomes where tion, and
diag
(28)
The above equation indicates that the interference can be fully are uncorrelated. cancelled, and the beam noise outputs of
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To quantify the interference cancellation through different stages of subarray processing, we calculate the deflection of , i.e., the test statistics of different detectors change in mean divided by standard deviation. The deflection (modified based on [3]) is given as follows: Tr
DEFL
Tr
(29)
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 , which leads to based on (20), or . In this case, may be approximated as diag which indicates that the beam noise of fore, utilizing (28)–(30), we obtain DEFL
(30) is uncorrelated. There-
Tr Tr Tr
Tr Tr DEFL
(31)
Hence DEFL DEFL
(32)
The above result is verified by simulations later (see Figs. 5 and 10). 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: DEFL DEFL
(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 combeam outputs coherently and yields a gain of (see bines 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 to the full array detector DEFL DEFL is loss of . Equations (32) and (33) indicate that lower bounded by 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.
Fig. 5. SINR and deflection of subarray and full array processing versus subarray size. Signal angular spread = 4 BW, interference angular spread = 8 BW. SNR = 4 dB, SIR = 30 dB. The effective rank of the interference subspace is q = 8.
0
0
JIN AND FRIEDLANDER: REDUCED-RANK ADAPTIVE DETECTION OF DISTRIBUTED SOURCES USING SUBARRAYS
When , 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 is a diagonal may be significant. Generally the matrix dominant matrix due to weak cross-correlation between difbeam outputs of residue interference. Hence, (32) and ferent becomes extremely small, (33) still hold. However, when DEFL 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 Tr Tr , as our perforSNR . We then mance measure for a simplified case, where , defined as output SNR versus avobtain the SNR gain of erage element SNR, as follows (see Appendix C for details) SNRG
(34)
is the effective rank of subarray signal subspace or, where equivalently, the number of main beams. This result is consistent with the SNR gain for the full array detector reported in [12]. Equation (34) suggests that the SNR gain is the array gain of the subarray normalized by the number of main beams. It shows with appropriate size so that, SNR-wise, using subarrays is generated to obtain its array that a single wide beam gain has little difference than using a full array with multiple . However, the advantage of using subnarrow beams arrays 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 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 denote the SINR with samples is a point source. Let the optimal SINR. This allows us to further quantify and the relative performance between a subarray processor and a full array processor by means of SINR gain defined as follows:
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. We can carry out the similar derivahas the Wishart distribution tion by noticing that with dimension [2], given the fact that is the Wishart distribution with dimension . Caution should be taken in that depends on training is computed from each data. However, due to the fact that has good estimation accuracy when subarray, each block of sample is relative large compared to the subarray size , it is general true that is a relatively constant matrix independent of a particular set of training data. Thus, the normalized for the coherent SINR loss factor subarray processor has the probability density function (37) with mean value of the SINR gain can be written as
. Hence
(38) where the asymptotic SINR gain (39) 2) SINR Gain Without Training Data: In this case, we may define the SINR as the average of signal power to the average interference-plus-noise ratio. Hence, for the full array processor, by citing the results from [22] that (40) Without too much difficulty and assuming that is a relatively constant matrix independent of a particular set of data, we have the following SINR for the coherent subarray detector (41) Thus, the SINR gain in this case is defined as
(35) Similarly 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. 1) SINR Gain with Training Data: For the full array detector, it is well known that the normalized SINR loss factor (see Reed [17]) has the probability density function
where The mean value of
takes the form of
(42)
(36)
where is defined in (39). is the normalized gain function that indicates the potential gain of coherent subarray processing depending on and and is given as follows:
.
(43)
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY 2005
TABLE I COMPARISONS OF ADAPTIVE DETECTORS
IV. 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 sensors. The total aperture of the this linear array with array is fixed as where is the half wavelength. However, the rest of the 38 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 ULA of sensors is also considered. The half power beamwith of this . The beamformers of interests under ideal array is BW 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. 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 20 and , respectively. The signal has an angular spread beamwidth, the beamwidth. This interference has an angular spread also implies that the effective rank of the signal subspace and and , respectively. interference subspace are Choosing the subarray size , the detector shows little , the performance performance loss in deflection. When 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 . The signal and interference beamwidth. It shows that, in this have angular spread case, both the coherent and noncoherent subarray detectors have performance loss within 1 dB relative to the optimal full array detector. We notice that the performance difference between the coherent detector and noncoherent detector is insignificant in this case. This observation indicates that the processing matrix 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 is not chosen properly, for instance, , subarray size
Fig. 6. Comparison of detection performance of full array and subarray detectors. The signal source and interfering source are at 20 and 20 , respectively, with angular spread of 12 each. The subarray has size of = 10.
M
0
M
Fig. 7. Comparison of detection performance of full array and subarray detectors. All the parameters remain the same as in Fig. 6, except that = 4.
the noncoherent detector suffers significant loss indicated by the receiver operating characteristic curve in Fig. 7. Figs. 8 and 9 depict the beampatterns of the optimal detectors. We observe that in Fig. 8, the full array detector has gain of dB at the signal direction. This result is due to the fact that the signal has angular spread of 12 (or equivalently 4 BW), and is consistent with the result in [12]. Choosing the will lead to a significant resolution loss subarray size for the nocoherent subarray detector as is shown in Fig. 9. Fig. 10 shows that the SINR of the coherent subarray detector . In adis very close to that of the full array detector as dition, the deflection gain of coherent subarray detector relative is bounded by 7 dB and 3.5 dB, to while the deflection gain of the noncoherent subarray detector remains at 3.5 dB. There results are in agreement with (32) and (33).
JIN AND FRIEDLANDER: REDUCED-RANK ADAPTIVE DETECTION OF DISTRIBUTED SOURCES USING SUBARRAYS
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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 Tr[ ] = 1.
W
WW
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 = 4.
M
Simulation Test 2—Effects of Sample Support: In this test, we study the performance with different data samples under various practical scenarios. Fig. 11 depicts the SINR gain, defined in (38), for detecting a point source when training data are available. The number of samples ranges from to . 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 (or ten subarrays in this case), the relative SINR gain has 13 dB. This is because the sample covariance matrix is ill conditioned when . 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.
Fig. 10. Comparison of SINR and deflection of subarray and full array processing as the angular spread changes. The SNR = 4 dB, SIR = 30 dB, = 8.
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Fig. 11. Comparison SINR gain of subarray processors and full array changes.the array is a non-ULA, the processor as training sample size SNR = 0 dB, SIR = 30 dB. The source is a point source.
K
0
Fig. 12. SINR gain of the subarray processing relative to the full array processing as sample size changes. All the parameters remain the same as in Fig. 11, except that the training data include signal components.
K
This phenomenon demonstrates the asymptotic 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 (42), with different sample size starting from to . 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
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 SNR = 0 dB, SIR = 30 dB, loading level is noise level, and training sample data include signal components.
0
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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 10 times of noise level.
K
out the effects of diagonal loading on detector performance, we experiment the three detectors under practical condition with different loading levels. The results are shown in Figs. 13 and and 14. For three test cases, the diagonal loading level of are utilized. The subarray detector still outperforms the full array detector. 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.
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It shows that diagonal loading on full array GMVDR improves the 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 . 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 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 four beamwidths. We call this beamformer the modified MVDR (MMVDR). In this test, we study the detection performance of the subarray noncoherent detector, full array detector and the modified MVDR beamformer by means of detection of probability. For the three is used. We assume that detectors, a loading level of both the interfering source and signal source move in one direction through four beams (12 ) with different speeds. The experiment setting is summarized in Table II. 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.
V. 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 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.
Fig. 15. Comparison of detection performance of full array and subarray detectors with spatial smoothing (SP) and diagonal loading (DL) for a ULA. = 0:001, The loading level is equal to noise level at different SNR value. P K = 80, and the training data include signal of interest.
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TABLE II EXPERIMENT PARAMETER SETTINGS FOR DETECTION OF MOVING SOURCES
TABLE III COMPARIONS OF DETECTION PERFORMANCE OF PROCESSORS (DETECTION PROBABILITY)
APPENDIX A OPTIMAL FULL ARRAY DETECTOR According to the Neyman–Pearson criterion, the optimal detector for detecting a distributed signal is the one that maxi, and denote mizes the likelihood ratio. Let and , the probability density function under hypotheses respectively
The likelihood ratio test to the detection problem (47) is given . Taking the as logarithm of the above function, we obtain
(44) where . Employing the matrix inversion lemma, it is then straight-forward to write the log-likelihood function as follows [12]:
(48) In the above equation, line one yields result in line two by a simple algebraic manipulation. Hence, we obtain that . APPENDIX C SNR GAIN FOR ANGULAR SPREAD
(45) If we let the
matrix
defined in (8) with We start with the full array processor being replaced by . In this case, the SNR is given by
be (46)
SNR weighting matrix consists of columns, where The is the rank of the signal subspace.
Tr Tr
(49)
Citing the results given in line one of (45), we obtain that
APPENDIX B NONCOHERENT SUBARRAY PROCESSOR Assume that is independent with each other. Thus, the detection problem (14) reduces to the following binary hypothesis test:
(47)
(50) where diag
JIN AND FRIEDLANDER: REDUCED-RANK ADAPTIVE DETECTION OF DISTRIBUTED SOURCES USING SUBARRAYS
is the eigenvalue matrix. Hence, we have
SINR
Tr Tr
(51)
where . We notice that for a large aperture array and moderate SNR, the dominant eigenvalues have the ap, which leads to proximation
Tr
SNRG
Similarly, for the subarray beamformer,
(52) of size
(53)
SNRG where
is the effective rank of subspace
leads to
.
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[13] Y. Kim, S. Pillai, and J. Guerci, “Optimal loading factor for minimal sample support space-time adaptive radar,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 1998, pp. 2505–2508. [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,” J. Acoust. Soc. Amer., vol. 8, no. 2, pp. 737–747, Feb. 1990. [15] N. L. Owsley, Array Signal Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1985. [16] B. D. Rao and K. V. Hari, “Weighted subspace methods and spatial smoothing: Analysis and comparison,” IEEE Trans. Signal Process., vol. 41, pp. 788–803, Febr. 1993. [17] I. S. Reed, J. D. Mallett, and L. E. Brennan, “Rapid convergence rate in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-10, pp. 853–863, Nov. 1974. [18] T. Shan and T. Kailath, “Adaptive beamforming for coherence signals and interference,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-33, no. 3, pp. 527–536, June 1985. [19] D. N. Swingler, “A low-complexity MVDR beamformer for use with short observation times,” IEEE Trans. Signal Process., vol. 47, pp. 1154–1160, Apr. 1999. [20] H. L. Van Trees, Optimum Array Processing. New York: Wiley, 2002. [21] B. D. Van Veen, “An analysis of several partially adaptive beamformer designs,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, pp. 192–203, Feb. 1989. [22] M. Wax and Y. Anu, “Performance analysis of the minimum variance beamformer,” IEEE Trans. Signal Process., vol. 44, pp. 928–937, Apr. 1996.
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful suggestions that considerably improved the quality of this paper. REFERENCES [1] Acoustic Observatory Project. Office of Naval Research. [Online]. Available: http://www.onr.navy.mil/sci_tech/ocean/321_sensing/ us01_ acousticob/default.htm [2] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, Second ed. New York: Wiley, 1971. [3] C. R. Baker, “Optimal quadratic detection of a random vector in Gaussian noise,” IEEE Trans. Commun., vol. COM-14, pp. 802–805, 1966. [4] K. A. Burgess and B. D. Van Veen, “Subspace based adaptive generalized likelihood ratio detection,” IEEE Trans. Signal Process., vol. 44, pp. 912–927, Apr. 1996. [5] J. Capon, “Multidimensional maximum likelihood processing of a large aperture seismic array,” Proc. IEEE, vol. 55, pp. 192–211, 1967. [6] H. Cox, “Multi-rate adaptive beamforming (MRABF),” in IEEE SAM Conf., Mar. 16, 2000. [7] , “Line array performance when the signal coherence is spatially dependent,” J. Acoust. Soc. Amer., vol. 54, pp. 1743–1746, 1973. [8] , “A subarray approach to matched-field processing,” J. Acoust. Soc. Amer., vol. 87, no. 1, pp. 168–178, Jan. 1990. [9] A. Dhanantwari et al., “Adaptive beamforming with near instantaneous convergence for matched filter processing,” in Proc. Canadian Conf. Elect. Eng., vol. 2, 1996. [10] W. F. Gabriel, “Using spectral estimation techniques in adaptive processing antenna systems,” IEEE Trans. Antennas Propag., vol. AP-34, pp. 291–300, Mar. 1986. [11] J. R. Guerci, J. S. Goldstein, and I. S. Reed, “Optimal and adaptive reduced rank STAP,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, pp. 647–663, Apr. 2000. [12] Y. Jin and B. Friedlander, “Detection of distributed sources using sensor arrays,” IEEE Trans. Signal Process., vol. 52, no. 6, pp. 1537–1548, Jun. 2004.
Yuanwei Jin (S’98–M’03) received the B.Sc. and the M.Sc. degrees from East China Normal University, Shanghai, China, in 1993 and 1996, respectively, and the Ph.D. degree in electrical engineering from the University of California, Davis (UC Davis), in 2003. He was a lecturer with the Department of Mathematics, Nanjing University of Science and Technology, Nanjing, China, from 1996 to 1997. In the summer of 1999, he was with Sensys Instruments Corporation, Santa Clara, CA. He was also with the University of California, Santa Cruz. He is now with Carnegie Mellon University, Pittsburgh, PA. His research interests are in signal processing, array processing, estimation, and detection with applications in digital communications and sonar/radar. Dr. Jin was a recipient of the Earle C. Anthony Fellowship from UC Davis from 1997 to 1999.
Benjamin Friedlander (S’74–M’76–SM’82–F’87) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion—Israel Institute of Technology, Haifa, Israel, in 1968 and 1972, respectively, and the Ph.D. degree in electrical engineering and the M.Sc. degree in statistics from Stanford University, Stanford, CA, in 1976. From 1976 to 1985, he was with Systems Control Technology, Inc., Palo Alto, CA. From November 1985 to July 1988, he was with Saxpy Computer Corporation, Sunnyvale, CA. From 1989 to 1999, he was with the University of California, Davis. Currently, he is a professor of electrical engineering with the University of California, Santa Cruz. Dr. Friedlander received the 1983 ASSP Senior Award, the 1985 Award for the Best Paper of the Year from the European Association for Signal Processing (EURASIP), the 1989 Technical Achievement Award of the Signal Processing Society, and the IEEE Third Millennium Medal.
