Broadband Field Directionality Mapping with Spatially-Aliased Arrays Jonathan L. Odom and Jeffrey L. Krolik Dept. of Electrical and Computer Engineering, Duke Univ., PO Box 90291 Durham, NC 27708 [email protected], [email protected] Abstract—This paper addresses the problem of broadband spatial spectrum estimation using multiple spatially-aliased arrays. Unlike previous approaches using sparse arrays, the signals here are assumed to be uncorrelated between multiple arrays which, in fact, may be receiving the same source during different time intervals. This paper presents an approach which jointly exploits spatial-orientation and broadband temporal diversity in order to estimate the spatial spectrum even when the inter-element spacing within each array is greater than a half-wavelength. A dynamical model for the spatial spectrum is employed to formulate a maximum likelihood estimate, which is computed via a recursive version of the expectation-maximization algorithm using data from different arrays. Simulation results are presented to demonstrate the ability of the method to suppress spatial grating lobes and increase low SNR target detection in an interference dominated environment.

I. I NTRODUCTION The problem of spatial spectral estimation or field directionality mapping is central to many array processing applications. Typically, the use of distributed arrays assumes the signal is correlated across distinct sub-arrays or, alternatively, each subarray can unambiguously determine angle-of-arrival. In the former case, broadband cross-correlation methods have been used to estimate source locations and in the latter, triangulation provides an unambiguous location estimate. In this paper, we consider the problem of field directionality mapping using data collected from multiple broadband under-sampled sensor arrays with diverse orientations. Data at each sub-array is assumed to be uncorrelated. This may occur when sparse subarrays are separated by greater than the spatial correlation length of the signal and/or a broadband signal is received on a mobile array at disjoint times during a maneuver of its vehicle platform. In addition, this paper addresses the broadband case where the sensors in each sub-array are spaced more than a half-wavelength apart at all frequency bins, creating ambiguous location estimates. For moving sources, field directionality mapping produces an estimate of the bearing versus time record (BTR), which is central to passive sonar. In recent work, a method to combine data from multiple array orientations for time-varying fields has been presented [1]. The work exploits orientation diversity to increase narrowband performance and considers the broadband case by averaging narrowband estimates. This extends the signal covariance matrix estimation methods for a single array [2] and multiple fixed arrays [3]. Previous work also demonstrates the c 2012 IEEE http://dx.doi.org/10.1109/SAM.2012.6250535

possibility of performance gains by using temporal broadband knowledge [4], [5]. Our towed-array work is a spatial spectral implementation of this performance gain. Data snapshots from different array orientations each offer different degrees of information at each spatial frequency. Similarly, different temporal frequency snapshots also offer various degrees of information at each spatial frequency. Broadband spatial spectral information is summarized in the point source assumption, and broadband temporal spectral information is captured in knowledge of the temporal spectrum shape. This paper suggests combining both knowledge across multiple array orientations and a broad band of frequencies to increase detection and estimation performance. In general, use of broadband knowledge alone does not sufficiently mitigate ambiguities due to under-sampling of the array aperture even though incoherent averaging frequency estimates can reduce spatial grating lobes [6]. However, in this paper, we show that broadband processing of arrays with different orientations can provide much greater suppression of bearing ambiguities resulting from spatial aliasing in each array. In this paper, we first provide a stochastic model of the data. Then a model for the spatial spectra across snapshots is given in order to derive a batch estimate. A simulation is used to show the ability of the field directionality map to work with spatial aliased signals. Finally, detection performance is analyzed to compare the proposed estimate with other techniques. II. DATA M ODEL Consider frequency-domain data received by an array at snapshot n and narrowband frequency ω. The data is coherent, meaning the correlation between sensors for a point source is determined by wave propagation. However, data received at different times or snapshots are incoherent, meaning the correlation between snapshots is zero. The model can represent either data from a single array at different locations/orientations or different arrays. For simplicity, consider the case with a constant number of elements, M , and the received data is written as x(n, ω) = D(n, ω)s(n, ω) + η(n, ω).

(1)

The steering matrix, D of size M × Q, is defined [dq (n, ω)]m = exp (−jkTq (n, ω)rm (n)).

(2)

where kq is wavenumber of the qth source and rm is the mth sensor location. Using a circular symmetric complex

Gaussian model for the source signals s and noise η, their field directionality or spatial spectrum is described by the covariance matrices Σ and ση2 I respectively. The received data vector thus has a zero-mean Gaussian distribution with covariance R(n, ω) = D(n, ω)Σ(n, ω)DH (n, ω) + ση2 I.

(3)

The extension to broadband is represented by a bar and formed by stacking signal vectors of B frequencies. The received data is written  T x ¯(n) = xT (n, ω1 ), xT (n, ω2 ), · · · , xT (n, ωB ) and a large time-bandwidth product is assumed such that each frequency is uncorrelated. Thus the broadband received data covariance matrix is block diagonal with form given by (4). The signal and noise covariance matrix have similar structure and are denoted with a bar.   R(n, ω1 ) 0   .. ¯ R(n) = (4) . . 0

R(n, ωB )

III. E STIMATION T ECHNIQUE A sliding window of N snapshots is used to estimate the field directionality where the key difference between mapping techniques across time is the time-varying model of the spatial spectrum. In this paper, the spatial spectrum is assumed to be a Markov process of the form Σ(n + 1) = Σ(n) + ∆,

(5)

where ∆ is zero-mean Gaussian noise. The same model can be used when N arrays each generate an independent snapshot. A maximum likelihood estimate is used, defined ˆ n = arg max f (¯ Σ x(n − N + 1) · · · x ¯(n)|Σ(n)).

(6)

Σ(n)

The spatial spectrum is formed by assuming a grid over possible source angles. The source signal variance is estimated at each grid point, thus a large number of effective sources are used to cover the entire map. The estimate is computed using an expectation-maximization (EM) algorithm, where the complete data is the received data and the incomplete data is the source and noise signals. For simplicity, consider the case where the power is uniform across all frequencies, but the non-uniform case is handled with weighted averaging. In the limit where the covariance of ∆ goes to 0, the ML estimate is given by ˆ new ˆ old − Σ ˆ old 1 Σ n =Σ NB

n X

B X

ˆ old , (7) (G(n, ωb ))Σ

ˆ old DH + ση2 I. K(n, ωb ) = DΣ

B X ˆ n (ωb ) ˆn = 1 Σ Σ B

(10)

b=1

Narrowband averaging finds the estimate for each frequency and then collapses the parameter space in post-processing. This means that narrowband averaging requires B × Q parameters. However, the broadband solution proposed in this paper uses only B parameters. Thus the proposed method will have lower variance in general. IV. S PATIAL A LIASING For simple uniform linear omnidirectional sensor arrays, spatial aliasing occurs when the received signal’s wavelength is smaller than twice the inter-element spacing. However, this is not strictly true for the more general case with multiple non-uniformly spaced arrays. Thus the number of sensors can be reduced for a given aperture length or the usable frequency range of an existing array increased. Further work should provide insight into the spatial sampling requirements of multiple and/or non-uniform arrays. However, the possible performance increase can be derived using a Cram´er-Rao bounds. Consider a spatial spectrum estimate using a single static array. The Fisher information matrix is given by   B X ∂R(ωb ) ∂R(ωb ) −1 R (ω ) , (11) tr R−1 (ωb ) [J]pq = b ∂σp2 ∂σq2 b=1

where tr{} refers to the trace operator. Estimation of power of a single source reduces to a trivial solution. For multiple sources, traditional processing forms the broadband estimate by averaging narrowband estimates and results in a bound of the form B 1 X −1 J−1 J (ωb ). (12) c = B2 b=1

Uniform or weighted averaging assumes the shape of the temporal spectrum is known. For the flat temporal spectral case, the estimate that jointly considers all frequencies results in a bound " B #−1 X J−1 J(ωb ) . (13) B = b=1

k=n−N +1 b=1

where the frequency and snapshot dependent terms are   ˆ −1 )D G(n, ωb ) = DH (K−1 − K−1 RK

A more general solution is contained in [7]. The algorithm can be initialized with a noise floor estimate and all following estimates can use the previous estimate. Consider an alternative solution based on averaging narrowband estimates. The narrowband estimate is a simplification of (7) where ˆ b ); the resulting broadband B = 1 and estimate is denoted Σ(ω estimate is given by (10). Both solutions assume the broadband spatial spectrum is flat across frequencies and have similar orders of complexity.

(8) (9)

The bounds are written in terms of the narrowband Fisher information matrices. When J(ωb ) varies as a function of frequency, ωb , spatial grating lobes are frequency dependent variations of information loss. This provides the motivation for considering spatially aliased arrays.

V. S IMULATION A simulation is used to demonstrate the effects of spatial aliasing when combining multiple snapshots. A two dimensional environment is simulated with 3 strong interferers at [-90◦ ,−14◦ ,135◦ ] and a weak interferer at −135◦ . A target moves from 60◦ to 15◦ with maximum instantaneous bearing rate of -0.26◦ /sec over the course of the 5 minute simulation. The signal-to-noise (SNR) ratio or interference-to-noise ratio for each source is defined as the power received by a sensor element over the noise level, or σ 2 /ση2 . The interferers are at 10 dB and 3 dB INR with a target source at 3 dB. A cartoon representing the simulation as a bearing-time-record (BTR) is given by Figure 1. A single maneuvering array is considered

Fig. 1. Cartoon representation of bearing time record for the simulation showing dynamic target in presence of interference

with independent snapshots sampled periodically at 1/2 Hz, thus each location provides a single snapshot per time frame. The minimum wavelength of the processed data is λ. A M = 8 element rigid uniform linear array is maneuvering over a circle of diameter ≈ 110λ. The inter-element spacing is 3/2λ so that all frequency bins contain aliasing. The array heading changes at +1◦ per second along the circle. Data is generated in 2 Hz bins from 550-750 Hz assuming 1538 m/s propagation. For the ML techniques, the EM algorithm uses 10 iterations and N = 9 snapshot window size. Conventional beamforming is data independent and processes each snapshot individually by ˆ ˆ estimating Σ(n, ω) = D(n, ω)H R(n, ω)D(n, ω), then averaging frequency estimates. This results in typical spatial aliasing as well as left-right or front-back ambiguities due to the linear geometry of the array. The BTR, given in Figure 2(a), shows strong interferers clearly as well as the diagonal lines from ambiguities. There is also aliasing of the ambiguities, which appears as wavy lines as a increased noise floor in the BTR. Note the target source is not easily discernible. The ML estimate formed by averaging narrowband estimates, from (10), provides a method for combining multiple array orientations. Thus it can reduce spatial aliasing using different physical geometries. The ambiguities and grating lobes are greatly reduced so that the target source is clearly

seen, given in Figure 2(b). However, some grating lobes are still visible and appear as spurious noise when averaged across frequency, with the largest at 100◦ from 100-200 seconds. The method also has a few problems estimating target power after 150 seconds as the target passes through end-fire. The proposed broadband method jointly estimates the spectrum across frequency bins, which provides better suppression of frequency dependent spatial aliasing. This reduces the spurious noise seen in the previous method. The new method is shown in Figure 2(c). Both ML methods require some time to converge due to the online use of the previous estimate, which increases the noise floor during 0-50 seconds. VI. P ERFORMANCE A NALYSIS A simulation is used to quantify the performance gain of the proposed method and effects of spatial aliasing for low SNR target detection with interference. The scenario from Figure 1 is used, but the target SNR is reduced to -25 dB SNR. Probability of detection (Pd ) and probability of false alarm (Pf ) are calculated using a M-of-N detector. A detection is declared when 80 out of 100 snapshot estimates along a hypothesized target track exceed a constant threshold. This type of metric is used because ambiguities and spatial aliasing create false alarms along incorrect paths. The generated receiver operating characteristic (ROC) curves use 100 realizations under each condition: target present (H1 ) and target absent (H0 ). First consider an array without aliasing so that spacing is λ/2. The ROC for this case is given in Figure 3(a) and shows the performance gain from the ML estimation technique. Without aliasing, little gain is made using the proposed method compared to averaging ML estimates. The conventional method is near the chance diagonal. Then consider an array with 3λ/2 spacing, with result given in Figure 3(b). With spatial aliasing, the broadband method performs significantly better than the other two methods. VII. C ONCLUSION Spatial aliasing of a single array can be mitigated using multiple array orientations. Additional performance gains are achieved by combining data across frequency bins. Typical incoherent broadband schemes average narrowband estimates. This paper proposes a broadband method that jointly estimates the field directionality or spatial spectrum across frequency. The joint estimation reduces spatial aliasing effects when compared to averaging. This allows sources to be detected when concealed by spatially aliased interferers. ACKNOWLEDGMENT This work was N000140810947.

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R EFERENCES [1] J. S. Rogers and J. L. Krolik, “Time-varying spatial spectrum estimation with a maneuverable towed array,” J. Acoust. Soc. Am., vol. 128, no. 6, pp. 3543–3553, 2010. [2] M. Miller and D. Fuhrmann, “Maximum-likelihood narrow-band direction finding and the em algorithm,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 9, pp. 1560–1577, 1990.

(a) Conventional BTR

(a) Broadband ROC

(b) ML Narrowband Averaging

(b) Broadband ROC with aliasing Fig. 3. ROC curves for a broadband simulation with SNR at -25 dB with (a) maximum half-wavelength spacing and (b) under-sampled array

(c) Reduced ML Fig. 2. BTRs showing power estimates, in dB, with a short maneuvering spatial-aliased array using (a) conventional beamforming, (b) averaged narrowband ML estimates, and (c) proposed algorithm.

[3] A. Lanterman, “Statistical radar imaging of diffuse and specular targets using an expectation-maximization algorithm,” in SPIE Conference Series, vol. 4053, 2000, pp. 20–31.

[4] P. Schultheiss and H. Messer, “Optimal and suboptimal broad-band source location estimation,” IEEE Trans. Signal Process., vol. 41, pp. 2751–2763, 1993. [5] H. Messer, “The potential performance gain in using spectral information in passive detection/localization of wideband sources,” IEEE Trans. Signal Process., vol. 43, no. 12, pp. 2964–2974, 1995. [6] H.-C. Song and B.-w. Yoon, “Direction finding of wideband sources in sparse arrays,” in SAM Signal Processing Workshop Proceedings, 2002, pp. 518–522. [7] J. Odom, “Spatial spectrum estimation with a maneuvering sensor array in a dynamic environment,” Master’s thesis, ECE Dept., Duke Uni., Durham, NC, 2011.