Co-prime array geometries have received a great deal of attention due to their ability to discriminate O(MN) sources with only O(M + N) sensors. This has been demonstrated both theoretically and in simulation. However, there are many practical limitations that make it difficult to realize the enhanced degrees of freedom when applying co-prime geometries to real acoustic data taken on a horizontal line array. For instance, co-prime sampling leads to grating lobes that can obscure lower signal-to-noise-ratio acoustic signals making them difficult to detect. In this work, a synthetic aperture (SA) method is presented for filling in holes and increasing redundancy in the difference co-array by exploiting array motion. The SA method is applied to acoustic data collected off the Southeastern shore of Florida on a fixed large aperture horizontal array. Array motion is simulated by taking a co-prime sampled subarray and virtually moving it along the horizontal aperture of the fixed array. It is demonstrated that SA processing on real acoustic data results in reduced side-lobe and grating lobe levels compared to that of the physical co-prime aperture.

Large aperture acoustic arrays in oceanic environments offer the ability to resolve closely separated sources and control side-lobe levels with their many degrees of freedom. Often, individual array elements or segments of elements may fail over time yielding a sparsely sampled sensor array. Alternatively, sparse sensor arrays may be intentionally designed in order to dramatically decrease the cost of deploying large aperture acoustic arrays. However, the immediate trade-offs made when designing sparse arrays are the increased side-lobe levels and grating lobes that occur resulting from the loss of inter-element sensor spacings. The presence of such artifacts can severely limit the ability to passively detect low signal-to-noise-ratio (SNR) sources in the presence of loud interferers such as shipping vessels.1,2

Some of the drawbacks of sparse arrays can be overcome by judiciously designing the array geometry. For example, one approach to realize an advantageous sparse array geometry is to systematically remove elements from a uniform linear array (ULA) until a desired array pattern is achieved.3–6 Alternatively, sparse array geometries can be obtained by defining element positions according to characteristics of either the difference co-array (passive sensing) or the sum co-array (active sensing). In the passive sensing case, the difference co-array characterizes the redundancy and support of the spatial covariates that can be estimated from array channel data. The difference co-array can be used to constrain the removal of sensor elements such that it remains hole-free over the full aperture of the array.7–10 For large aperture arrays, optimizing the difference co-array can be computationally prohibitive.

Here, we focus on a class of sparse arrays, known as co-prime arrays, to perform passive acoustic sensing. Linear co-prime arrays can be realized by nesting two ULA's; where one array has N ( λ / 2 ) element spacing and the other has M ( λ / 2 ) element spacing with the constraint that the spacing parameters M and N are co-prime integers. The co-prime geometry has been shown to provide more degrees of freedom than number of sensors used to construct the array.11,12 In other words, when data are coherently processed, the hole-free section of the difference co-array has size O(MN) derived from only O(M + N) sensors. In principle, one can design a (M,N) co-prime array, where M < N, that guarantees O ( M N ) degrees of freedom in two manners. First by nesting two ULA subarrays where one subarray has 2N sensors with an inter-element spacing of M ( λ / 2 ) and the other subarray having M sensors with an inter-element spacing of N ( λ / 2 ) . This nesting procedure would produce a sparse array having a total of M + 2N − 1 elements that spans an aperture of ( 2 N 1 ) M ( λ / 2 ) . The other nesting strategy is to nest two subarrays where one subarray has 2M sensors with an inter-element spacing of N ( λ / 2 ) and the other subarray having N sensors with an inter-element spacing of M ( λ / 2 ) . This nesting procedure would produce a sparse array having a total of 2M + N − 1 elements that spans an aperture of ( 2 M 1 ) N ( λ / 2 ) . In this work, we use the former approach for nesting ULA components, which provides the maximum aperture for a given pair of co-prime parameters. Figure 1 illustrates a diagram of a prototypical co-prime sensor array for M = 3 and N = 5, where a 2N-element subarray is nested with an M-element subarray.

FIG. 1.

(Color online) Co-prime array geometry: (3,5) co-prime array geometry with ULA subarrays.

FIG. 1.

(Color online) Co-prime array geometry: (3,5) co-prime array geometry with ULA subarrays.

Close modal

A major shortcoming of co-prime array geometries is that the physical aperture required to form the segment of hole-free difference co-array is approximately twice as long as the span of the hole-free portion of the co-array. It is often desirable to use the full extended co-prime aperture for the enhanced resolution; however, holes in the difference co-array cause unwanted side-lobes and grating lobes. Adhikari, Buck, and Wage proposed a technique to reduce side-lobe levels of the co-prime array by taking the product of two extended co-prime subarrays.13,14 The product processor was later demonstrated on acoustic data collected on a large vertical array in a shallow water environment.15 Liu and Buck proposed the minimum processor to further reduce side-lobe levels over the product processor.16 

An alternative approach to incoherently combining subarrays is to exploit array motion in scenarios involving a moving platform. This concept was successfully demonstrated as a method for disambiguating sources coming from the left and right on linear arrays17 and as a method for reducing side-lobes and coherent multipath arrivals.18 Recently, Ramirez and Krolik proposed to apply synthetic aperture (SA) processing to overcome high side-lobes and grating lobes due to holes in the difference co-array by exploiting platform motion.19 There, the focus was to develop a criterion that minimized the integration time required to extend the hole-free difference co-array over the full aperture. It was shown, in an ideal sensing environment, that by virtualizing missing sensor spacings within the aperture of the co-prime array, it is possible to achieve the spatial covariate sensing capability of a ULA despite having an undersampled acoustic aperture. Wu and Liang have also considered SA processing for augmenting the difference co-array to perform angle of arrival estimation with a co-prime geometry when the source frequency is unknown.20 

In this paper, SA processing is applied to data collected on a large aperture horizontal acoustic array in a shallow water environment off the coast of South Florida. Although, the horizontal array is fixed, motion is synthesized by sampling co-prime subarrays and translating along the aperture of the array. The SA method presented by Ramirez and Krolik19 was demonstrated on simulated data on a ULA with an idealized signal model and precise knowledge of sensor locations. Here, the SA method will be evaluated on acoustic data collected in a shallow water environment with very complex propagation conditions on a hydrophone array that is only approximately linear. The SA method is shown to significantly reduce side-lobes and grating lobes (due to holes in the difference co-array) by coherently combining data over the synthesized motion of the array. The method is compared to coherent processing over the fully populated aperture and performance is quantified as a function of coherent integration time. The purpose of this paper is to explore the trade-offs in performance of moving from an idealized signal model that is known to real data where there may be a mismatch between the assumed signal and actual signal received on the array.

This paper is organized as follows. In Sec. II, we present the experimental data set and physical sensor array configuration and in Sec. III we discuss the co-prime array design process and characteristics of the difference co-array. In Sec. IV, we describe the simulated motion algorithm for approximating the SA process on a fixed location array and in Sec. V we show the bearing time records (BTRs) associated with our processing methods. Finally, in Sec. VI we discuss and conclude our findings.

The data analyzed here were chosen to come from a fixed, large horizontal aperture acoustic array moored to the bottom of a complex shallow water environment. Specifically, the data were collected as part of the Acoustic Observatory Calibration Operations (CalOps) experiment.21 This data set was chosen to analyze the impact of propagation uncertainty, while isolating other variables that may cause mismatch such as array motion, which is typically required for SA processing. Here, we rely on the large fixed aperture to synthesize array motion by translating a smaller subarray along the larger horizontal array aperture. This will allow for more precise knowledge of sensor element location and remove uncertainties and the challenges associated with array shape estimation.

The Acoustic Observatory sits roughly 20 km off the coast of South Florida and is situated between Miami and the heavily trafficked Port Everglades. The acoustic array is designed for an operational frequency of 450 Hz (1.75 m spacing) and has 500 acoustic sensors, amounting to over 800 m of total acoustic aperture. The array is divided into four smaller subarrays having nearly uniform linear spacing. Figure 2(a) shows the location of the array off the Florida coast and the inset image depicts the physical aperture. The physical aperture consists of four subarrays, where each subarray contains 125 sensors and has roughly 210 m of acoustic aperture. For the purposes of this paper, subarray 2 was chosen since it nearly approximates a ULA. An array beampattern for subarray 2 is given in Fig. 2(b) using a uniform shading window. It can be observed that although the subarray approximates a ULA, the unique shape of the array beampattern is due to imperfect uniform spatial sampling.

FIG. 2.

(Color online) Physical array characteristics: (a) The array location off the coast of southern Florida and local bathymetry near the physical array. (b) The array pattern for subarray 2 showing the non-uniformity of the side-lobe structure due to nearly linear sensor spacings.

FIG. 2.

(Color online) Physical array characteristics: (a) The array location off the coast of southern Florida and local bathymetry near the physical array. (b) The array pattern for subarray 2 showing the non-uniformity of the side-lobe structure due to nearly linear sensor spacings.

Close modal

A large amount of work associated with the Acoustic Observatory data set has focused on the environmental characterization, three-dimensional propagation effects, and noise conditions in the region.21–24 The array is moored to the ocean bottom at a depth of 250 m. The segment of data analyzed in this paper comes from the CalOps experiment during a southerly transit with the R/V Seward Johnson towing an echosounder. The echosounder broadcasts a comb of tones and its track follows along the 250 m isobaths as shown in Fig. 2(a). The CW-comb consists of tone triplets centered at 24, 52.5, 106, 206, and 415 Hz that are transmitted in 60 s cycles. The source power at each peak was determined to be 155.8, 166.5, 167.6, 171.4, 170.9 dB re 1 μPa at 1 m estimated from a monitor hydrophone.

The environment has been characterized as having a karst topography defined by sand pools and limestone outcroppings.25,26 The combination of the complex bottom type, shallow-water waveguide, and upslope conditions due to the array's location on the shelf make for very challenging propagation conditions. However, despite these environmental complexities that may cause distortions to the assumed wavefront impinging on the array, a plane-wave model is often used for determining the beamforming steering vectors in these scenarios. The analysis presented here will quantify the performance degradation due to mismatch for the CalOps data that were analyzed using the method presented by Ramirez and Krolik.19 

The primary objective in designing a sparse array is to determine how to distribute a limited number sensor element over a given aperture. The co-prime array geometry provides a simple sparse array design technique based on nesting ULAs that guarantees the support of the hole-free section of the difference co-array exceeds the number of sensors used to form the array. These qualities make the co-prime array geometry an efficient sparse array configuration. Given the co-prime parameters (M,N), with M < N and ULA nesting procedure described in Sec. I, the (M,N) co-prime array will consist of Q = M + 2 N – 1 sensors spanning an aperture of L Q = ( 2 N 1 ) M ( λ / 2 ) (m).27,28 The design of a co-prime array for a given target aperture LT thus reduces to the selection of co-prime parameters (M,N) such that LQ ≤ LT.

In this work, a ULA is spatially subsampled by removing hydrophone channels to form a linear co-prime array. We must first determine the best co-prime configuration to match the physical aperture of the ULA. While a co-prime array geometry cannot be made to fit every array aperture, an approximating aperture can be designed to support the co-prime geometry. The physical ULA is made of 121 hydrophones and spans an aperture of 120 ( λ / 2 ) (m). We therefore seek to find co-prime integers (M,N) such that

(1)

with no constraint on the number of sensors used to form the co-prime array, the range of acceptable co-prime parameters M,N can be found via a simple trial-and-error check of the inequality stated above. Upon determining a range of acceptable parameters for the co-prime geometry, comparisons among sensor count, aperture, and difference co-array can be made to obtain the optimal choice of co-prime parameters for the target aperture.

The co-prime configurations shown in Table I only approximate the target aperture, causing an aperture loss of a few wavelengths or less. In large aperture configurations this loss can be an acceptable trade-off when considering the sensor count of the co-prime array geometry against that of the uniformly sampled array. The fundamental limitation of the co-prime array geometry is that it cannot be realized for any arbitrary aperture. However, its simplistic design along with straightforward parameter search makes generating co-prime geometries for large apertures computationally feasible.

TABLE I.

Partial list of permissible co-prime parameters for the target design aperture, LT = 120(λ/2) (m).

(M, N) Q = M + 2N – 1 LQ=(2N–1)M λ 2 MN + 1
(2,29)  59  114 λ 2   59 
(3,20)  42  117 λ 2   61 
(5,12)  28  115 λ 2   61 
(4,15)  33  116 λ 2   61 
(M, N) Q = M + 2N – 1 LQ=(2N–1)M λ 2 MN + 1
(2,29)  59  114 λ 2   59 
(3,20)  42  117 λ 2   61 
(5,12)  28  115 λ 2   61 
(4,15)  33  116 λ 2   61 

For the remainder of this paper, we focus on the (5,12) co-prime geometry. This array consists of 28 sensors spanning an aperture of 115 ( λ / 2 ) (m). The difference co-array for this co-prime realization is guaranteed to have a MN + 1 = 61 contiguous hole-free section. Of the co-prime parameters listed in Table I, this configuration represents the sparsest solution while imposing a loss in aperture of 3λ (m) (∼10 m) from the physical ULA aperture. The (5,12) co-prime geometry spans an aperture equivalent to a 116 sensor ULA giving it an overall compression ratio of 28/116 = 0.241.

Table I illustrates the trade-offs of the co-prime array geometry. First, co-prime arrays can be designed to span large apertures with a relatively small number of sensors. Second, the support of the contiguous set of spatial covariates that can be estimated from an array snapshot exceeds the number of sensors used to form the array. However, the cost incurred for the increased spatial covariance coverage is physical aperture. Only roughly half the physical aperture worth of spatial covariates can be estimated from the co-prime array with a reduced redundancy when compared to a ULA. These drawbacks can be overcome by augmenting the co-prime array such that the difference co-array becomes hole-free over the full extent of the aperture. An approach to mediate this shortcoming is to apply SA processing to synthesize missing inter-element spacings within the aperture of the co-prime array.19 This typically applies to situations in which the array is mounted on a moving platform.

In this work, we propose to analyze the SA process in a real data signal environment for augmenting the co-prime array to realize the increased degrees of freedom that have been demonstrated in theoretical signal environments. In addition, we consider the impact of the mismatch on the array steering vector arising from environmental uncertainties that are not accounted for in the signal modeling. Using a fixed location array, we first compare different array realizations derived from appending sensors obtained from the SA process and, second, we develop a simulated motion algorithm that allows for the study of temporal coherence over the virtualized SA.

The SA process applied to inter-element sensor synthesis exploits array motion to create virtual inter-element sensor spacings for an array mounted on a moving platform. Recent works have shown that the operational frequency of an array can be increased and completely virtual arrays can be derived from the SA process.29,30 In these works, and references therein, it has been shown that the SA process can be applied in situations where multiple far-field sources of common frequency can be localized. In addition, Ramirez and Hickman showed that multiple far-field sources at different frequencies can be localized via SA techniques.30 Applying the SA process for sensor synthesis involves fusing sensor measurements from different time instants taken over the trajectory of the array.

Suppose the array is towed at a velocity v(m/s) and the signal component model representing the physical array snapshot is given by X ( t ) = exp ( j Ω t ) exp ( j Ω ( v t sin θ / c ) ) v ( θ ) , where v(θ) is the array manifold vector encoding the physical array geometry and exp ( j Ω ( v t sin θ / c ) ) is derived from the Doppler shift between the source and array.19 Then virtual channels at half-wavelength spacings within the array aperture can be realized in the following manner:

  • Time-delay sampling: Let X ( t + 1 τ ) , X ( t + 2 τ ) , , X ( t + η τ ) represent array snapshots taken along the trajectory of the array at different time instants and spatial locations. The sampling instants t + 1 τ , , t + η τ are carefully chosen to measure the wave-field at half-wavelength spatial increments within the array aperture. The time-delay factor, τ, is governed by the signal wavelength of interest and the tow speed velocity. The time-delay factor is given by τ = λ / 2 v (s). When the array is assumed to travel a straight line path, the array snapshots listed above provide measurements on a denser array grid than the physical array itself.

  • Phase correction: In order to realize a virtual array, measurements taken at sampling instants t + 1 τ , , t + η τ must be phase corrected to create a spatially coherent array snapshot with common time reference t. For the snapshot taken at t + ητ, the signal model can be written as X ( t + η τ ) = exp ( j Ω η τ ) exp ( j Ω ( v η τ sin θ / c ) ) X ( t ) , where exp ( j Ω η τ ) represents a signal-wavelength dependent phase offset from the signal at the desired time instant t and exp ( j Ω ( v η τ sin θ / c ) ) represents the signal-wavelength dependent phase offset required to synthesize virtual half-wavelength array manifold vector entries. To time align the array snapshot from t +  to t the phase correction factor Ψ k = exp ( j Ω k τ ) is applied to X(t + ) producing a phase corrected array snapshot X k ( t ) = Ψ k X ( t + k τ ) . By phase correcting these measurements, the array snapshot X k ( t ) provides the signal received on a subset of virtual channels.

  • Measurement fusion: To create a synthetic array snapshot virtual array channels realized by phase correction must be interleaved in a manner that preserves the spatial relationship between inter-elements sensor spacings. From the collection of measurements taken on the physical array X(t), and the phase corrected virtual channels contained in X 1 ( t ) , , X η ( t ) the synthetic array snapshot XSA(t) can be formed. In this situation, XSA(t) has the same operational wavelength as X(t), however, it has a denser half-wavelength sensor spacing distribution. The primary assumption in the measurement fusion stage is that the spatial coherence span the length of the synthesized aperture and the temporal coherence are at least as long as the time required to create the synthesized aperture.

In the above process, the choice of how many τ-delayed snapshots to use in the time-delay sampling stage determines the extent to how many virtual sensors are realized and consequently the degree to which the corresponding difference co-array is filled. It was shown that the number of snapshots needed to obtain a hole-free difference co-array for a (M,N) co-prime array using the SA process is a function of the co-prime parameters and is given by

(2)

where M < N.28 The SA process, under idealized sampling conditions, applied to the (5,12) co-prime array will produce a hole-free difference co-array by fusing array snapshots from sampling instants t, t + 1τ, and t + 2τ. That is, by using array motion over a distance of λ (m) (∼3.34 m) the hole-free support of the difference co-array can be extended from 61 lags to 117 lags. In this situation, idealized sampling conditions require that the array travel a straight line path and exact sensor placement is known.

Figure 3 shows the difference co-arrays for the (5,12) co-prime array and the corresponding augmented SA array for different levels of SA integration. As the number of τ-augmentation stages increases the holes in the co-prime difference co-array disappear and the measurement redundancy for each spatial covariate increases. After two augmentation stages the difference co-array is hole-free and after four augmentation stages the redundancy in the difference co-array matches that of the ULA. Typical applications of SA processing are used to extend the array aperture creating a composite aperture that is multiple factors longer than the physical array.31–34 Here, the array aperture will be held constant while the SA process is used only to augment the inter-element sensor spacings. In this situation, the hole-free support of the difference co-array is extended from 61 lags to 115 lags while the array aperture is held constant at 115 ( λ / 2 ) (m). In the subsequent analysis, the number of elements in the ULA aperture is held constant at NULA = 116 elements while the number of sensors created during the SA process will be denoted by NSA.

FIG. 3.

(Color online) Difference co-array: (a) (5,12) co-prime array difference co-array, (b) augmented (5,12) co-prime difference co-array. After 2τ array translations the difference co-array is hole-free over the entire array aperture.

FIG. 3.

(Color online) Difference co-array: (a) (5,12) co-prime array difference co-array, (b) augmented (5,12) co-prime difference co-array. After 2τ array translations the difference co-array is hole-free over the entire array aperture.

Close modal

In Sec. III B, we described how to extend the hole-free section of the difference co-array from the half aperture limit of the co-prime array to the full aperture of the array. While the difference co-array is hole-free the composite array retains its sparse array characteristics, not all missing inter-element spacings are realized. In this situation, a direct estimation of the spatial covariance matrix (SCM) on the composite array using outer-product snapshot averaging will result in an array with high side-lobes even though all the spatial covariates are accounted for in the difference co-array. To leverage the structure of the difference co-array, methods have been developed to derive a positive semi-defined covariance matrix from the correlation lags estimated from the array.35,36

Here we consider the spatial covariance estimation method known as rank enhanced spatial smoothing (RESS) developed by Pal and Vaidyanathan.37,38 This method has been widely used for deriving positive semi-definite SCMs from partial and fully hole-free difference co-arrays for different sparse array configurations.19,39,40 This is a technique for the approximation of ULA array pattern and source localization characteristics from sparse arrays up to the degree to which the difference co-array is hole-free. A complete description outlining the different stages of this algorithm can be found in Ref. 37.

The RESS method is an algorithm applied to the snapshot averaged outer-product SCM given by

(3)

where Ns indexes the number of temporal array snapshots used to form the SCM estimate. The RESS algorithm recovers the signal and noise subspace structure of a SCM derived from a fully populated ULA by exploiting the set of spatial covariates estimated from the synthetic array snapshot. The quality of the RESS approximation also depends on the number of instances a spatial covariate is estimated from the array snapshot. The spatial covariate estimation improves with increased measurement redundancy. In general, redundancy increases as new sensors are added to the aperture of the array as can be seen in Fig. 3(b). As more temporal samples are used to construct the SA array, the virtual sensors fused to the physical array increase the redundancy in the spatial covariate measurements.

Given the estimate of the RESS SCM one can apply array processing techniques for source localization as if a ULA were used to derive the SCM. These methods include spatial power spectrum estimation, multiple source classification algorithm (MUSIC), and or adaptive processing. In the work presented here, we will focus on estimating BTRs for ocean data by computing the spatial power spectrum given by

(4)

where v ( θ ) = [ 1 exp ( j Ω ( d 1 sin θ / c ) ) exp ( j Ω ( d N ULA sin θ / c ) ) ] and R ̂ is the frequency domain SCM derived from overlapping array time-series data.

The objective of this work is to assess the performance of the co-prime array using SA processing on ocean acoustic data. Specifically, we'll evaluate the degradation in performance due to mismatch in the idealized steering vector, v, when applied to a shallow water data set. To this end, we will analyze data collected on a fixed array located off the Southeastern coast of Florida and impose a virtual towed array structure over the data collected on the channels of the array. Creating a towed array with a large aperture fixed array system removes any uncertainties due to sensor position and allows for the analysis of environmental impacts on the steering vector v(θ). In particular, this approach enables us to consider the signal complexities of the ocean environment without being subject to turbulent ocean conditions encountered when towing a physical co-prime array.41 

Array motion is simulated with a fixed array by utilizing array channel data according to the time-delay profile induced by the co-prime array geometry and a hypothesized tow velocity. To approximate the SA process, we first identify the co-prime sensor spacings within the ULA positions of the fixed array, second for each co-prime sensor we determine the subset of virtual sensors created along its trajectory, and finally using the hypothesized tow velocity and inter-element distance between a given co-prime sensor and its virtual sensors we can derive the time-delay profile. The time-delay represents the time it would take a co-prime sensor to travel approximately λ/2 (m). Once the channel data are sampled according to time-delay profile, the data on each virtual channel needs to be appropriately phase corrected to time-align data from those channels to the reference time instant t. After all channels are time-aligned the approximated SA process in complete and the synthetic array snapshot can be used for further processing. Figure 4 shows a schematic of the simulated motion process.

FIG. 4.

(Color online) Simulated motion schematic.

FIG. 4.

(Color online) Simulated motion schematic.

Close modal

Figure 4 shows how array motion can be built into the temporal sampling process. In this schematic we have a ULA segment where sensors x1 and x4 represent co-prime spacings. Suppose via array motion, we require to synthesize array positions x2, x3, and x5 and the hypothesized array tow speed is given by v (m/s). Then sensors x1 and x4 can be used to realize sensor spacings x2, x3 and x5, respectively. Given the inter-element distances and the tow velocity v, the time required for the co-prime sensors to travel to the desired locations is given by τ1,2, τ1,3, and τ4,5. To simulate array motion on a fixed array the channels of the ULA segment are subsequently sampled as x 1 ( t 0 ) , x 2 ( t 0 + τ 1 , 2 ) , x 3 ( t 0 + τ 1 , 3 ) , x 4 ( t 0 ) , and x5(t0 + τ4,5), where t0 is the reference time instant. Depending on the sampling rate of the data acquisition hardware at which channel data are collected to achieve a sampling at t0 + τi,j, the channel data would need to be interpolated to extract the desired sample from the time series data. In the work presented here, channels are interpolated to account for correct time-delay sampling.

In a similar manner as described in Sec. III B the phase correction factor is derived from the signal model representation of the data. Let the data model be given by

(5)

where exp ( j Ω ( v s t sin θ / c ) ) is the physical Doppler shift induced by the R/V Seward Johnson, which is towing an active source at a known velocity of vs (m/s). The time-delayed sampled signal model is given by

(6)

The difference between x(t) and x(t + τi,k) are the terms exp ( j Ω τ i , k ) exp ( j Ω ( v s τ i , k sin θ / c ) ) , which are frequency and arrival angle dependent. The channel phase correction factor to compensate for these terms is thus given by

(7)

which time aligns the received signal on the channels developed from the ith co-prime sensor at the reference time instant t. The phase correction model discussed in Sec. III B assumes the source is static; however, since the data collected in the CalOps experiment involved a moving source the corresponding Doppler shift needs to be accounted for in phase correction. The second term in Eq. (7) is a consequence of the motion due to the tow ship and by phase correcting for this term we can effectively remove this motion from our model.

In the above example the data used to form an array snapshot at t0 is given by x 1 ( t 0 ) , Ψ ( τ 1 , 2 ) x 2 ( t 0 + τ 1 , 2 ) , Ψ ( τ 1 , 3 ) x 3 ( t 0 + τ 1 , 3 ) , x 4 ( t 0 ) , and Ψ(τ4,5)x5(t0 + τ4,5) together these channels represent an approximation to the SA process on a fixed location array. It is worth noting that the ship velocity vs and hypothesized tow velocity of the array v represent different quantities. Since the R/V Seward Johnson is towing the echosounder the velocity vs induces a Doppler shift between the source and array, and the hypothesized tow velocity of the array v is used to determine the time-delay profile used for sampling the channels of the fixed array.

In the remainder of this paper, we will analyze the performance of the co-prime array geometry on data collected during the 2007 CalOps sea experiment. In particular, the performance of SA processing on co-prime arrays in complex shallow water propagation conditions is studied for two specific operating conditions. Operating condition 1 (OC1) considers the case where the co-prime array is augmented with inter-element sensors according to the augmentation schedule defined by the SA process. In this situation, we analyze the performance of the augmented co-prime array where all channels are simultaneously sampled in time, while the array geometry retains the sparse characteristics induced by the SA process. Here the SA process only dictates where sensors are added after each τ-augmentation stage and processing on this array is performed as if all virtual sensors were present at the outset. This approach allows for the analysis of sparse arrays where the difference co-array is hole-free over the full aperture of the array, while relaxing the computational burden incurred by classical sensor placement methods, such as simulated annealing43 to produce hole-free difference co-array arrays. The second operating condition, operating condition 2 (OC2), considers the proposed method for simulating array motion on a fixed location array. In this case the SA process is used to design temporal sampling delays to simulate array motion. Estimates of the temporal sampling delays are subsequently compensated for during the SA process. By considering OC1 and OC2 we can directly compare the impact of uncertainties induced by array motion in a controlled and systematic manner.

Common to both OC1 and OC2 schemes are the time-series data set used and the spatial spectrum estimation techniques. We consider a 15-min segment of data and restrict our attention to the signal energy at 415 Hz. This tone is a component of the CW-comb waveform transmitted during the CalOps experiment. The data segment was chosen to highlight the method described in Secs. III and IV in a higher SNR regime. However, it is worth noting that there is an inherent trade-off between array element sparsity and SNR. By throwing out Na array elements, the array gain against white noise is reduced to 10 log ( N ULA / N a ) . Furthermore, there is a more subtle relationship between SNR and the SA process described above. The overall integration time required to sample the array geometry as it translates is a function of the velocity term, v. For example, having a lower v corresponds to a longer integration time. This allows the SA process to virtualize missing sensors when the spatial statistics remain stationary over the observation interval. It has been shown that the passive SA process does not allow for the recovery of SNR, but instead provides enhanced degrees of freedom that can be used for aperture synthesis and side-lobe reduction.42 A more thorough analysis of the performance versus SNR will be considered in future work.

In this work, a frequency domain beamformer was implemented to estimate a frequency dependent SCM used in spatial spectrum estimation. In particular, we consider two SCM estimation techniques: one derived from the RESS algorithm, R ̂ RESS C N ULA × N ULA and the other given by the array snapshot averaged outer-product shown in Eq. (3), R ̂ Conv C N SA × N SA . The spatial power spectrum estimate derived from each is given in Eq. (4). The peak-to-side-lobe (or peak-to-grating lobe) ratio will be used as a metric to evaluate the performance of the two methods for different coherent integration schemes.

Figure 5 shows the BTR for the analysis data segment as estimated through the physical and the (5,12) co-prime arrays using the conventional SCM estimation method. In addition, a time averaged slice of the BTR over the first 60 s is shown in Fig. 5(b). Figures 5(a) and 5(b) show the ULA response where the tow ship can be seen at approximately θ = 2° relative to array broadside. The BTR also shows returns from other ships traversing the area. Figures 5(c) and 5(d) show the co-prime array response where the BTR is contaminated with aliased returns from the tow ship leaking into the spatial spectrum through the large side-lobes and grating lobes inherent to the co-prime array pattern. The following analysis demonstrates how using SA processing with co-prime sampling can improve the spatial spectrum estimate by reducing side-lobe and grating lobes in the beampattern.

FIG. 5.

(Color online) Common aperture ULA and co-prime array: (a),(b) ULA BTR showing 60 (s) periodicity of source transmission signal and 60 (s) time average spatial spectrum, (c),(d) co-prime (5,12) BTR and 60 (s) time average spatial spectrum.

FIG. 5.

(Color online) Common aperture ULA and co-prime array: (a),(b) ULA BTR showing 60 (s) periodicity of source transmission signal and 60 (s) time average spatial spectrum, (c),(d) co-prime (5,12) BTR and 60 (s) time average spatial spectrum.

Close modal

Given the (5,12) co-prime array geometry the hole-free section of the difference co-array is guaranteed to span the lag range of 61 spatial covariates and requires only 28 sensors over an aperture of 115 ( λ / 2 ) (m), as shown in Table I. To extend the spatial covariate sensing capability of the array, sensors can be appended such that the missing spatial covariates can be recovered. A convenient approach to filling in the spatial covariates is to consider augmenting the co-prime array according to the SA process. Table II shows a summary of the difference co-array characteristics over the different τ-augmentation stages of the SA process. The difference co-array for the (5,12) co-prime array becomes hole-free with 2–τ augmentation stages, which results in increasing the sensor count from 28 to 78 sensors within the co-prime aperture. As more augmentation stages are used the sensor count increases, as well as the redundancy in the measurement of the spatial covariates over the aperture of the array. By utilizing the SA augmentation schedule we can design large aperture sparse arrays with hole-free difference co-arrays without having to rely on computationally expensive/prohibitive optimization procedures for sensor placement.

TABLE II.

SA sensor augmentation schedule.

Augmentation stage Number of sensors Difference co-array
Physical array  28  Incomplete 
1-τ  54  Incomplete 
2-τ  78  Hole-free 
3-τ  97  Hole-free 
4-τ  116  Hole-free 
Augmentation stage Number of sensors Difference co-array
Physical array  28  Incomplete 
1-τ  54  Incomplete 
2-τ  78  Hole-free 
3-τ  97  Hole-free 
4-τ  116  Hole-free 

Figure 6 shows the (5,12) co-prime augmented array response using the RESS SCM estimation method. The BTRs show a drastic reduction in energy leaking into the spatial spectrum from the side-lobes and grating lobes of the array. As more sensors are appended to the co-prime array energy leaking in from the side-lobes and grating lobes is further reduced. A peak-to-side-lobe power reduction of approximately 13.1 dB and 17.6 dB are realized over the physical co-prime array by appending sensors to the array according to the 2τ and 3τ SA schedule, respectively. Figures 6(b) and 6(d) show the time average of a 60 (s) segment of the BTR. In Figs. 6(b) and 6(d), we compare the RESS and conventional spatial covariance estimation methods. After sensors have been appended to the co-prime array the composite array retains a sparse structure while having a hole-free difference co-array over its entire aperture. In conventional processing, holes in the sensor domain impose an increase in the side-lobe structure of the power spectrum response. However, when applying the RESS algorithm the degrees of freedom available are used to approximate the SCM structure of a ULA resulting in decreased side-lobe levels. An approximate decrease of 5.5 dB and 4.7 dB in peak-to-side-lobe levels are realized by the RESS SCM over the conventional SCM when sensors are appended to the co-prime array according to the 2τ and 3τ SA schedule, respectively.

FIG. 6.

(Color online) Static augmented co-prime arrays: (a),(b) 2-τ augmented array BTR and time average spatial spectrum, (c),(d) 3-τ augmented BTR and time average spatial spectrum. Spatial spectrum shows comparison between RESS and conventional spatial covariance estimation methods. The power spectrums are lower-bounded by the ULA response.

FIG. 6.

(Color online) Static augmented co-prime arrays: (a),(b) 2-τ augmented array BTR and time average spatial spectrum, (c),(d) 3-τ augmented BTR and time average spatial spectrum. Spatial spectrum shows comparison between RESS and conventional spatial covariance estimation methods. The power spectrums are lower-bounded by the ULA response.

Close modal

In Sec. V A, we considered appending sensors to the co-prime array such that the difference co-array becomes hole-free. The BTRs and spatial power spectrums shown in Fig. 6 represent the maximal suppression of side-lobe and grating lobes achievable in terms of the spatial power spectrum response of the augmented co-prime arrays. With these results as a baseline we can compare the performance of the simulated motion procedure described in Sec. IV. The motion simulation process accounts for time-delays incurred when sensors move from one λ / 2 element spacing to the next. By precomputing the sensor dependent time-delays and accounting for phase correction factors on sensors corresponding to virtual sensor positions in the fixed array aperture, the SA process can be simulated on a fixed array. The simulated virtual array thus uses the array data collected over the physical aperture in a manner that approximates what would be seen with a moving array.

Figure 7 shows the BTRs for the simulated motion process. The BTRs in Figs. 7(a) and 7(c) show the response of the 2−τ and 3−τ simulated SA arrays, respectively. Here we see the dominant return from the tow ship is present, but the side-lobes are not reduced to the levels found by appending sensors to the co-prime array. Figures 7(b) and 7(d) show a direct comparison between the spatial power spectrum obtained by appending sensors and using simulated motion to augment the co-prime array. Here, the peak-to-side-lobe levels of the synthetic array developed from simulated motion do not achieve the same level as that obtained by appending sensors to the co-prime array. The peak levels are approximately 1.8 dB and 7.0 dB higher in the simulated motion spatial power spectrums than when sensors are appended according to the 2τ and 3τ SA schedules, respectively. The distinguishing factor among the two approaches is the application of time-delays and phase corrections applied during the simulated motion process. However, both approaches use the same channel data to form the array snapshot. The discrepancy in the side-lobe levels is a direct consequence of the loss of spatial coherence across the virtual aperture as a result of phase correction mismatch.

FIG. 7.

(Color online) Simulated motion augmented co-prime (5,12) arrays: (a),(b) 2-τ augmented BTR and time average spatial spectrum, (c),(d) 3-τ augmented BTR and time average spatial spectrum. Spatial spectrum shows comparison between static and simulated motion methodologies. The power spectrums are lower-bounded by the ULA response.

FIG. 7.

(Color online) Simulated motion augmented co-prime (5,12) arrays: (a),(b) 2-τ augmented BTR and time average spatial spectrum, (c),(d) 3-τ augmented BTR and time average spatial spectrum. Spatial spectrum shows comparison between static and simulated motion methodologies. The power spectrums are lower-bounded by the ULA response.

Close modal

The primary objective in the SA process is to form a temporally and spatially coherent array snapshot using the physical and virtual channels in the augmented array. The key component of this process is the phase correction applied to form virtual channels in the array. Phase correction on the virtual channels allows for the approximation of the wavefront structure at a given time instant, t, over the aperture of the array. When phase correction accounts for all offsets, the time-series data on the SA array can be used to form spatially coherent array snapshots.

To determine the quality of the spatial coherence across the SA we measure the temporal coherence on each virtualized channel of the augmented co-prime array. In particular, we measure the phase error between the ground truth channel data and the time-delayed phase corrected channel data for a given time instant t. As in the simulated motion example shown in Sec. IV, suppose that x3 corresponds to a channel that is used to simulate array motion, then we propose to measure the phase error between the Fourier transform coefficient at f = 415 Hz of x3(t) and Ψ(τ1,3)x3(t + τ1,3). Let χ be the complex error given by

(8)

where F is the Fourier transform operator. By considering the phase angle of χ we can determine the mismatch between the target Fourier transform coefficient and the phase corrected time-delayed Fourier transform coefficient used to augment the array. In the ideal situation, for array snapshots on each virtual channel the coefficients would have a zero phase difference.

In this section, we compare the phase correction error and corresponding BTR results for three different levels of phase correction. First, we consider the situation where no-phase correction is applied. In this case the phase correction factor given in Eq. (7) becomes, Ψ(τi,k) = 1, whereby channel data from time t + τi,k is fused directly to channel data from time t to form the SA. Second, we consider the situation where only knowledge of the source frequency is known leading to a partial phase correction. In this case, the phase correction factor becomes Ψ ( τ i , k ) = exp(−ȷΩτi,k). Finally, we consider the situation where full source parameters are known and the phase correction factor is given by Eq. (7), namely, Ψ ( τ i , k )  = exp(−ȷΩτi,k) exp ( j Ω ( v s τ i , k sin θ / c ) ) . In this case it is assumed that knowledge of the source frequency, bearing, and source velocity is known to the processor.

Figure 8 shows a comparison of the phase error histograms over three distinct τ-translation stages for the (5,12) co-prime array and the corresponding BTRs derived from using these three translation stages to form the augmented SA. Each τ-translation stage accounts for the subset of channels virtualized during that period and the phase angle errors are measured for the entire 15 min of time-series data. Figures 8(a) and 8(b) show the histogram and BTR for the case where no-phase correction is applied. This histogram shows that when no-phase correction is applied the variance among the phase errors across all the translation stages is large. Consequently, the corresponding BTR shows the spatial power spread across all bearing angles as a result of the poor spatial coherence of the synthetic array. Figures 8(c) and 8(f) show the phase correction errors and BTRs for two different phase correction scenarios. First, applying a partial phase correction, which only accounts for the source frequency, the variance in the phase errors decreases when compared to the no-phase correction case. In addition, the corresponding spatial power spectrum shown in the BTR reveals the bearing of the tow ship. Second, applying the full phase correction factor, where the source frequency, bearing angle, and velocity are assumed known, the variance in the phase error among all translation stages decreases beyond that of the previous two cases. The reduction in the variance of the phase correction error leads to improved spatial coherence across the SA allowing for the localization of the primary return of the tow ship in the time-series data segment at approximately θ = 2°. The knowledge of the source bearing and velocity provides the lowest phase error variance; however, the spatial power spectrums developed from the partial and full phase correction model are nearly identical. This suggests it may be possible to relax the phase correction model given in Eq. (7) to ignore the Doppler shift term induced from the motion of the source in the experimental data. This allows for forming the SA without knowledge of the source velocity or bearing angle.

FIG. 8.

(Color online) Phase correction error comparison.

FIG. 8.

(Color online) Phase correction error comparison.

Close modal

While the phase correction does not perfectly account for all data mismatch, it does serve to allow for the localization of the primary return of the tow ship in the time-series data segment. The residual phase error in Fig. 8(e) can be attributed to undulations in the wavefront caused by the complex propagation channel. If this were properly accounted for, the co-prime array with 3τ SA processing would approach the performance of the ULA.

In general, over large apertures sparse arrays can be computationally prohibitive to design. The fundamental problem encountered in the design of sparse arrays is where to place the sensors within the aperture to obtain desirable side-lobe and grating lobe levels, as well as sufficiently sample the spatial covariance domain. In this work, we considered a class of linear sparse arrays known as linear co-prime arrays. This class of arrays avoids the design pitfalls of general sparse arrays by allowing for a parametric design that can be used to approximate nearly any desired aperture size. The linear co-prime arrays are inherently designed to be arrays that can estimate spatial covariates over a contiguous support that is larger than the number of sensors used to form the array. By using SCM estimation techniques that take advantage of the contagious set of covariates offered by the co-prime array, the high side-lobes and grating lobes of the co-prime array can be mitigated.

The focus here was to examine the spatial power spectrum estimation capability of the linear co-prime array on real-world ocean data and apply SA processing to augment the sensors spacings in the co-prime array geometry. In this application, sensor augmentation was designed such that the extent of the contagious range where the spatial covariates can be measured matches the aperture of the array. To this end, we proposed a method to simulate array motion over a fixed location array allowing for the use of SA processing concepts. This allowed for isolating the effects the propagation environment may have on the method from other factors like uncertainty in array shape from a moving platform with a towed array.

Over the SA process, sensors are virtualized along the aperture of the array resulting in a cascading of subsets of virtual sensors realized at different time instants along the path of the array. Using the sensor synthesis schedule developed from the co-prime array geometry, we have designed two sensor augmentation approaches, one that appends sensors directly to the co-prime array and the other that relies on simulated motion. In this work, we have found the peak side-lobe levels of the co-prime array can be significantly reduced allowing for accurate source localization for high SNR targets. However, the performance is still worse than that of a ULA due to the sensitivity of the SA process to assumed wavefront mismatch. In addition, the knowledge of source parameters determines the degree to which phase errors are minimized during the SA process. However, it may be possible to develop sufficiently accurate phase correction models with knowledge of only the source frequency. In both sensor augmentation schemes, the linear co-prime array offers a simplified approach to sparse array design and can be used as a template geometry where sensors can then be appended to achieve a desired processing goal.

This research was supported by the Office of Naval Research.

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