Beamforming interferometry bathymetry method based on multi-coprime sensor array

Abstract

Recently, coprime array has been a popular research field in the application of direction-of-arrival estimation. Compared with uniform linear array, coprime array can be used to expand array aperture with fewer sensors, and it also has a nice direction-of-arrival estimation performance. According to coprime property, the direction of arrival can be obtained by intersecting the candidate estimation sets from several subarrays. However, when the directions of multiple sources meet a particular relation, the unambiguous phase cannot be unwrapped through the coprime array. In this article, a multi-coprime array is proposed to address the problem, utilizing about half number of array elements and reducing hardware complexity compared with uniform linear array. Then, a low-complexity beamforming interferometry algorithm via multi-coprime array is proposed to reduce computational complexity. Numerical results, including simulation and actual data processing, are provided to indicate that the processing on multi-coprime array can successfully resolve phase ambiguity. Specially, when signal-to-noise ratio exceeds about −4 dB and the element number of subarray in multi-coprime array exceeds 15 for the array geometry in this article, the proposed method achieves close estimation performance with fewer sensors compared with uniform linear array.

Keywords

Coprime array multi-coprime array direction of arrival phase ambiguity bathymetric sonar

Introduction

Sound navigation and ranging (Sonar) is a technique which uses sound to navigate,¹ communicate² with, or detect³ objects on or under the surface of the water. Two types of techniques are used: passive sonar⁴ is utilized to listen for signals and noise made by objects; active sonar⁵ emits an acoustic signal and listens for echoes reflected by an obstacle or object. Bathymetric sonar⁶ is a type of active sonar used to obtain depth information. The principle of bathymetry is shown in Figure 1. The range between two blue arrows denotes the cover section of transmission beam, and red arrows denote echoes of scatters in the seafloor. Generally, in real scenarios, two echoes in distinct directions impinge on the array at $t_{1}$ , and only one snapshot is available for estimating the directions of the echo signals.^7,8 The basic model of bathymetric sonar can be summarized as direction-of-arrival (DOA) estimation from a single observation vector corresponding to the received signals. However, since acoustic echo signals reflected by scatters have the same spectrum characteristic, the coherence is another issue for DOA estimation in the field of bathymetry. Recently, multibeam echosounders have been known to provide very accurate bathymetric information and a uniform linear array (ULA) is always used;^9,10 interferometric sonars also refer to phase differencing bathymetric sonars (PDBS) and provide high-resolution and wide-swath bathymetric performance.^7,11–13 Like the majority of DOA estimation systems, the aperture of array is still one of the main factors restricting the improvement of spatial resolution performance.

Figure 1.

The bathymetric concept.

Recently, Pal and Vaidyanathan^14,15 proposed a sparse array called coprime array. For this array configuration, not only is the redundancy reduced but also the degrees of freedom (DOFs) are increased. Based on the configuration, Zhou et al.¹⁶ proposed DECOM, a DOA estimation method by combining the multiple signal classification (MUSIC), to resolve phase ambiguity, and then two-phase adaptive spectrum search scheme is used to reduce computational complexity. Weng and Djurić¹⁷ proposed a search-free DOA estimation algorithm, and Chinese remainder theorem is utilized to eliminate phase ambiguity. According to the linear relationship between the actual DOA and its ambiguous estimation, Sun et al.¹⁸ proposed a partial spectral search (PSS) method to recover all candidate DOAs, and then the computational complexity is reduced greatly. Wu et al.¹⁹ proposed a coprime planar array to address the two-dimensional (2D) direction estimation issue. Zhou et al.²⁰ proposed an adaptive beamforming algorithm via coprime array to reconstruct the interference-plus-noise covariance matrix, achieving a better output performance. The above-stated ambiguity elimination methods can be summarized as matching a pair of DOAs from candidate sets obtained by decomposed sparse uniform linear subarrays, where the coprime property is utilized. However, for multisource cases such as bathymetry, several “singular points” may destroy the matching property, and then the wrong DOAs may be obtained through conventional coprime array. For this study, one of the motivations is to remove these “singular points” for unambiguous DOA estimations.

In this article, a multi-coprime array (MCA) is proposed to address the above-stated problem and then the existence and uniqueness for accurate DOA estimation is proved. Considering the primary motivation of the article, MCA is used to reduce hardware complexity and eliminate phase ambiguity with the coprime property for bathymetric sonar. According to Zhou et al.,¹⁶ the ambiguity elimination algorithm can be summarized as two steps: first, estimate the DOAs through sparse uniform linear subarrays, and the candidate DOA sets are established; second, the candidate DOA sets from three subarrays are combined and overlaps are searched.

Recently, multibeam echosounder sonar (MBES) and PDBS have been the main sonars used for bathymetric survey. The beamforming method is used for MBES to obtain robust DOA estimations, but a dense angle grid leads to an extremely heavy computational complexity. The phase difference between different receiving elements is utilized for PDBS to acquire high-resolution DOA estimation, but a $2 π$ bias ambiguity and a limitation for multisource estimation always exist.²¹ Another motivation for this article is to develop a method incorporating the advantages of both the methods.

For this study, a beamforming interferometry method via MCA is proposed. First, beamforming is utilized to preprocess the received signals, and rough DOAs can be obtained. It should be noted that the spatial angle grid is sparse in the first step since only rough DOAs are needed, and computational complexity is fairly low. Second, the phase difference between the actual DOA and the rough grid is utilized for off-grid estimation. Unlike the DECOM method by Zhou et al.,¹⁶ the secondary spectrum search scheme is not required for the proposed method, and thus the computational complexity is greatly reduced. Besides, according to Proposition 5, within the rough grid, there is no $2 π$ bias. The proposed method comprehensively utilizes both magnitude and phase information, and both the multisource estimation issue and the computationally intensive problem are addressed. Finally, results of simulation and field data processing will be provided to verify the effectiveness of the proposed method, in terms of both estimation precision and computational complexity.

The motivations of the article can be categorized as two aspects: first, to reduce hardware complexity, coprime array is utilized; however, for multisource case, and especially at “singular points,” it is important to investigate a method to completely resolve the points. Second, since MUSIC-inherited methods usually increase the estimation accuracy using very small, fixed grid searching step at the expense of much more computational complexity, it is meaningful to develop a method to reduce computational complexity.

The main contributions of this article mainly contain the following three aspects:

First, this article proposes a new coprime array configuration to completely resolve phase ambiguity especially at “singular points,” and in the multisource case, the existence and uniqueness proofs for DOA estimation are provided. In addition, the proposed array geometry use about half number of array elements compared to ULA, reducing hardware complexity.

Second, a beamforming interferometry method is proposed for coherent DOA estimation via MCA, which inherits the capacities of multisource DOA estimation and off-grid estimation, and both magnitude and phase information are incorporated to estimate the DOA, reducing computational complexity.

Third, numerical experiments including simulation and sonar data processing verify the validity and superiority of the proposed method in terms of complexity and DOA estimation accuracy. Specially, for the array geometry in this article, when signal-to-noise ratio (SNR) exceeds about −4 dB and the element number of subarray in MCA exceeds 15, the proposed method achieves close performance compared with ULA.

The remainder of this article is organized as follows. In section “Complete ambiguity elimination using MCA,” we introduce the configuration of MCA and the principle of phase ambiguity elimination. In section “Low-complexity DOA estimation algorithm,” a novel beamforming interferometry method is proposed. Section “Simulation and data process” provides simulation and data processing to verify the effectiveness and superiority of the method. Section “Conclusion” provides the conclusion. Appendix provides some proofs for the propositions.

Problem formulation

Signal model

MCA is a special non-ULA configuration, which consists of three sparse uniform linear subarrays. The first subarray has M elements with an inter-element spacing of Pd, and the locations of sensors are given by the set ${pPd | 0 \leq p \leq M - 1}$ , where d is the unit inter-element spacing, which is set as half a wavelength $λ / 2$ . The second subarray has M elements with an inter-element spacing of Qd, and the locations of sensors are given by the set ${qQd | 0 \leq q \leq M - 1}$ . The third subarray has M elements with an inter-element spacing of Rd, and the locations of sensors are given by the set ${rRd | 0 \leq r \leq M - 1}$ , where P, Q, and R are coprime integers. A prototype of MCA is illustrated in Figure 2. Without loss of generality, assume that $P > Q > R$ . Three subarrays share the first sensor element at the 0th location, which is set as the phase reference point. Compared with MCA, any two subarrays can be constituted as a traditional coprime array.

Figure 2.

Multi-coprime array configuration: (a) decomposed—a triplet of sparse uniform linear subarrays; (b) aligned—a non-uniform linear array.

Considering bathymetric model in Figure 1, suppose that two coherent narrowband far-field echoes impinge on the array from distinct directions at a single snapshot. The signal vectors received by three subarrays in Figure 1 are given by

y_{p} (l) = A_{p} (l) x (l) + n_{p} (l), l = 1, \dots, L

(1a)

y_{q} (l) = A_{q} (l) x (l) + n_{q} (l), l = 1, \dots, L

(1b)

y_{r} (l) = A_{r} (l) x (l) + n_{r} (l), l = 1, \dots, L

(1c)

where $y_{p} (l), y_{q} (l), and y_{r} (l) \in C^{M \times 1}$ are the observation signal vectors for the three subarrays at the $l th$ snapshot; L is the number of snapshots; $x (l) \in C^{K \times 1}$ denotes the echo signal vector reflected by the scatterers on the seafloor; $n_{p} (l), n_{q} (l), and n_{r} (l) \in C^{M \times 1}$ are assumed to be complex white Gaussian noise; $A_{p} (l) = [a_{p, 1} (l), \dots, a_{p, K} (l)] \in C^{M \times K}$ is the direction matrix with $a_{p, k} (l) = [1, e^{j φ_{p, k} (l)}, \dots, e^{j (M - 1) φ_{p, k} (l)}]^{T} \in C^{M \times 1}$ for subarray 1; $A_{q} (l) = [a_{q, 1} (l), \dots, a_{q, K} (l)] \in C^{M \times K}$ is the direction matrix with $a_{q, k} (l) = [1, e^{j φ_{q, k} (l)}, \dots, e^{j (M - 1) φ_{q, k} (l)}]^{T} \in C^{M \times 1}$ for subarray 2; $A_{r} (l) = [a_{r, 1} (l), \dots, a_{r, K} (l)] \in C^{M \times K}$ is the direction matrix with $a_{r, k} (l) = [1, e^{j φ_{r, k} (l)}, \dots, e^{j (M - 1) φ_{r, k} (l)}]^{T} \in C^{M \times 1}$ for subarray 3, $j = \sqrt{- 1}$ ; and $φ_{p, k} (l) = π P \sin (θ_{k} (l))$ , $φ_{q, k} (l) = π Q \sin (θ_{k} (l))$ , and $φ_{r, k} (l) = π R \sin (θ_{k} (l))$ are the echo signal phase differences between two sensors for the $k th$ scatterer at the $l th$ snapshot.

DOA estimation for sparse ULA

The beamforming method is a fast, robust algorithm to retrieve amplitude and direction information for each snapshot, and it is widely used for bathymetric sonar. According to the beamforming method by Pantzartzis et al.,⁹ beamforming is implemented by complex multiplication and addition from spanned spatial grids, as follows

Z_{p} (l) = B_{p}^{H} y_{p} (l), l = 1, \dots, L

(2a)

Z_{q} (l) = B_{q}^{H} y_{q} (l), l = 1, \dots, L

(2b)

Z_{r} (l) = B_{r}^{H} y_{r} (l), l = 1, \dots, L

(2c)

and the steering matrices are given by $B_{p} = [b ({\hat{φ}}_{p, 0}), \dots, b ({\hat{φ}}_{p, n}), \dots, b ({\hat{φ}}_{p, N - 1})] \in C^{M \times N}$ , $B_{q} = [b ({\hat{φ}}_{q, 0}), \dots, b ({\hat{φ}}_{q, n}), \dots, b ({\hat{φ}}_{q, N - 1})] \in C^{M \times N}$ , and $B_{r} = [b ({\hat{φ}}_{r, 0}), \dots, b ({\hat{φ}}_{r, n}), \dots, b ({\hat{φ}}_{r, N - 1})] \in C^{M \times N}$ , where $b ({\hat{φ}}_{p, n}) = [1 / M, e^{j {\hat{φ}}_{p, n}} / M, \dots, e^{j (M - 1) {\hat{φ}}_{p, n}} / M]^{T} \in C^{M \times 1}$ , $b ({\hat{φ}}_{q, n}) = [1 / M, e^{j {\hat{φ}}_{q, n}} / M, \dots, e^{j (M - 1) {\hat{φ}}_{q, n}} / M]^{T} \in C^{M \times 1}$ , $b ({\hat{φ}}_{r, n}) = [1 / M, e^{j {\hat{φ}}_{r, n}} / M, \dots, e^{j (M - 1) {\hat{φ}}_{r, n}} / M]^{T} \in C^{M \times 1}$ , and M denotes the element number of subarray and scale factor. The spatial DOA grids can be written as ${\hat{φ}}_{n} = 2 π n / N$ , where $n = 0, 1, \dots, N - 1$ . The spatial frequency index n denotes the wavenumber which can be projected to the steer angle

{\hat{φ}}_{p, n} = π P \sin {\hat{θ}}_{p, n} = 2 π n / N

(3a)

{\hat{φ}}_{q, n} = π Q \sin {\hat{θ}}_{q, n} = 2 π n / N

(3b)

{\hat{φ}}_{r, n} = π R \sin {\hat{θ}}_{r, n} = 2 π n / N

(3c)

where ${\hat{θ}}_{p, n}$ , ${\hat{θ}}_{q, n}$ , and ${\hat{θ}}_{r, n}$ are rough angle grids. According to the formula of summation for a geometric sequence, substituting equations (1a)–(1c) and (3a)–(3c) into equations (2a)–(2c), respectively, the approximate beamforming outputs are given by

Z_{p, n} (l) = | Z_{p, n} (l) | e^{- j (M - 1) (φ_{p} - {\hat{φ}}_{p, n}) / 2}

(4a)

Z_{q, n} (l) = | Z_{q, n} (l) | e^{- j (M - 1) (φ_{q} - {\hat{φ}}_{q, n}) / 2}

(4b)

Z_{r, n} (l) = | Z_{r, n} (l) | e^{- j (M - 1) (φ_{r} - {\hat{φ}}_{r, n}) / 2}

(4c)

where $Z_{p} (l) = [Z_{p, 1} (l), \dots, Z_{p, n} (l), \dots, Z_{p, N} (l)]^{T} \in C^{N \times 1}$ , $Z_{q} (l) = [Z_{q, 1} (l), \dots, Z_{q, n} (l), \dots, Z_{q, N} (l)]^{T} \in C^{N \times 1}$ , and $Z_{r} (l) = [Z_{r, 1} (l), \dots, Z_{r, n} (l), \dots, Z_{r, N} (l)]^{T} \in C^{N \times 1}$ . Then, the rough candidate DOA sets ${\hat{θ}}_{p}, {\hat{θ}}_{q}, and {\hat{θ}}_{r}$ can be obtained by finding the peaks from $| Z_{p} (l) |^{2}$ , $| Z_{q} (l) |^{2}$ , and $| Z_{r} (l) |^{2}$ , respectively.

Incomplete ambiguity elimination using coprime array

In this section, the phase ambiguity problem for a sparse ULA is explained, and then a general principle of ambiguity elimination is illustrated. In addition, the reason why multisource cases may be invalid is demonstrated.

A coprime array can be formed by selecting any two subarrays from MCA, for example, subarray 1 and subarray 2. In Figure 1, assuming that there are two scatterers located at $θ_{a}$ and $θ_{b}$ , then according to definition of direction vector $a_{p, k}$ and $a_{q, k}$ , the ambiguity angles should follow the relationship as

\sin {\hat{θ}}_{p, a} = \sin θ_{a} + 2 p_{a} / P

(5a)

\sin {\hat{θ}}_{q, a} = \sin θ_{a} + 2 q_{a} / Q

(5b)

\sin {\hat{θ}}_{p, b} = \sin θ_{b} + 2 p_{b} / P

(5c)

\sin {\hat{θ}}_{q, b} = \sin θ_{b} + 2 q_{b} / Q

(5d)

where $p_{a}, p_{b} = {p | p \in [- P + 1, P - 1], p \in Z}$ and $q_{a}, q_{b} = {q | q \in [- Q + 1, Q - 1], q \in Z}$ .

By combining equations (5a)–(5d), the candidate DOA sets of the subarrays are given by

\sin {\hat{θ}}_{p} = {\sin {\hat{θ}}_{p, a}, \sin {\hat{θ}}_{p, b}}

(6a)

\sin {\hat{θ}}_{q} = {\sin {\hat{θ}}_{q, a}, \sin {\hat{θ}}_{q, b}}

(6b)

where ${\hat{θ}}_{p}$ is the candidate DOA set of subarray 1, and ${\hat{θ}}_{q}$ is the candidate DOA set of subarray 2. Therefore, there exist several DOA candidates which are more than the number of actual echoes because array elements are sparsely deployed, which is called the phase ambiguity problem. According to Zhou et al.,¹⁶ the ambiguity elimination method through coprime array can be summarized in two steps: first, estimate DOAs for each uniform linear subarray separately, which has been illustrated in the previous part; second, find the most approximate estimations from the DOA candidate sets.

However, because the method cannot guarantee the accurate solution, it is incomplete to resolve phase ambiguity in “singular points” (see Appendix 1).

Complete ambiguity elimination using MCA

In Figure 1, assuming that there are two narrowband sources located at $θ_{a}$ and $θ_{b}$ , then according to definition of direction vector $a_{r, k}$ , the ambiguity angles of subarray 3 should follow the relationship as

\sin {\hat{θ}}_{r, a} = \sin θ_{a} + 2 r_{a} / R

(7a)

\sin {\hat{θ}}_{r, b} = \sin θ_{b} + 2 r_{b} / R

(7b)

where $r_{a}, r_{b} = {r | r \in [- R + 1, R - 1], r \in Z}$ . By combining equations (7a) and (7b), the candidate DOA sets of subarray 3 are given by

\sin {\hat{θ}}_{r} = {\sin {\hat{θ}}_{r, a}, \sin {\hat{θ}}_{r, b}}

(8)

For MCA, we have the following proposition to completely resolve phase ambiguity.

Proposition 4

Assuming that there exist two echoes reflected by scatterers located at distinct directions, then there exists accurate DOAs obtained by the closest matching operation from candidate sets ${{\hat{θ}}_{p}}$ , ${{\hat{θ}}_{q}}$ , and ${{\hat{θ}}_{r}}$ of the three subarrays.

Proof

See Appendix 3.

According to Proposition 4, by searching for the most approximate angles from the candidate sets, we can obtain the DOA estimations of echoes. To clearly explain the principle of the method, an example can be illustrated as follows.

There exist two echoes from distinct directions $\sin^{- 1} (1 / 2) = 30 °$ and $\sin^{- 1} (- 17 / 30) \approx - 34.518 °$ fulfilling condition (13a). Figure 3 shows the estimated results through coprime array. Red arrows are the candidate DOA estimations of source a, among which in the purple box denotes the actual DOA of source a. Blue arrows are candidate DOA estimations of source b, among which in the purple box denotes the actual DOA of source b. According to Propositions 1, 2, 3, and 4, we have the following conclusions:

For subarray 1 consisting of M sensors with inter-element spacing $5 d$ , there exist 10 candidate DOA estimations ${- 7 / 10, - 3 / 10, 1 / 10, 1 / 2, 9 / 10, - 29 / 30, - 17 / 30, - 1 / 6, 7 / 30, 19 / 30}$ , among which 8 solutions are ambiguous.

For subarray 2 consisting of M sensors with inter-element spacing $3 d$ , there exist six candidate DOA estimations ${- 5 / 6, - 1 / 6, 1 / 2, - 17 / 30, 1 / 10, 23 / 30}$ , among which four solutions are ambiguous.

For subarray 3 consisting of M sensors with inter-element spacing $2 d$ , there exist four candidate DOA estimations ${- 1 / 2, 1 / 2, - 17 / 30, 13 / 30}$ , among which two solutions are ambiguous.

For coprime array consisting of subarray 1 and subarray 2, except for actual DOAs in the black boxes, two “singular points” can be obtained by searching for the pair of DOAs from different subarrays in green box. As shown, these points are obtained by matching candidate sets corresponding to different sources. It should be emphatically pointed out that the locations of the scatterers are arbitrary in space. For other coprime array consisting of any two subarrays, there always exist the situations as (13a) or (13b), which may lead to wrong matching. And thus a coprime array consisting of two subarrays, for example, subarray 1 and subarray 2, cannot completely resolve phase ambiguity.

As we seen in Figure 3, two actual DOAs ${1 / 2, - 17 / 30}$ in the purple box are obtained incorporating with all candidate estimations corresponding to three subarrays.

Figure 3.

Example of ambiguity elimination using MCA.

Until now, actual DOA estimations could be picked up from candidate sets. However, when there exist more sources, such as three ( $θ_{a}, θ_{b}, θ_{c}$ ) which satisfy the relationship $\sin θ_{a} + 2 p_{a} / P = \sin θ_{b} + 2 q_{b} / Q = \sin θ_{c} + 2 r_{c} / R$ , then the “singular points” problem may influence estimation performance.

Low-complexity DOA estimation algorithm

Beamforming is a robust and low-complexity algorithm widely used for DOA estimation in MBES. Similar to other DOA estimation methods based on prespecified discrete spatial grid, for example, MUSIC, its estimation performance relies on grid density to a certain degree. A dense and fixed grid is always utilized to improve the accuracy of estimation to a certain extent, which leads to a heavy computational complexity. For phase difference bathymetric sonars, the interferometry method is widely used for DOA estimation and there exists a phase ambiguity problem in the processing of unwrapping phase difference of two subarrays. In addition, the Vernier method^7,11,20 is always used to efficiently resolve phase ambiguity. However, the Vernier method is similar to coprime array, and Chinese reminder theory,¹⁷ which uses coprime property to match the most approximate estimations from the candidate DOA sets. However, according to Proposition 3, the method may be invalid for multisource cases. In this article, a beamforming interferometry estimation method via MCA is utilized, which addresses the above-stated problem. First, according to previous part, we can resolve phase ambiguity with beamforming method via MCA. In Figure 4, blue lines denote spatial grids and blue solid lines represent on-grid DOAs estimated with beamforming method. Second, an interferometry method is utilized for off-grid DOA estimation based on rough estimation. In Figure 4, red lines denote the actual directions that can be estimated by the proposed method. A detailed description of the second step is provided below.

Figure 4.

Beamforming interferometry method.

In Figure 4, assuming that there exist two echoes with distinct directions $θ_{a}$ and $θ_{b}$ corresponding to phases $φ_{a}$ and $φ_{b}$ , then the blue solid lines, that is, the on-grid estimations, can be obtained with the beamforming method.

According to the spanned grid, the uncertain degree $δ$ is utilized to describe the estimation bias between the actual angle and the grid. We have $φ \in [{\hat{φ}}_{n} - δ / 2, {\hat{φ}}_{n} + δ / 2]$ , where $δ$ denotes the size of the spanned interval corresponding to $2 π / N$ in Figure 4, and $\hat{φ}$ represents the rough grid, which is shown as a blue line in Figure 4. For the DECOM algorithm by Zhou et al.,¹⁶ more dense grids covering the uncertain interval $δ$ would be utilized to improve the DOA estimation accuracy. However, low efficiency and on-grid are the two main shortages restricting the enhancement of DOA estimation performance. In this article, an interferometry method is used to efficiently estimate the off-grid DOA.

According to equations (4a)–(4c), the phases of beamforming output can be expressed as follows

ψ_{p, n} = (M - 1) (φ_{p} - {\hat{φ}}_{p, n}) / 2 = (M - 1) π P (\sin θ - \sin {\hat{θ}}_{p, n}) / 2

(9a)

ψ_{q, n} = (M - 1) (φ_{q} - {\hat{φ}}_{q, n}) / 2 = (M - 1) π Q (\sin θ - \sin {\hat{θ}}_{q, n}) / 2

(9b)

ψ_{r, n} = (M - 1) (φ_{r} - {\hat{φ}}_{r, n}) / 2 = (M - 1) π R (\sin θ - \sin {\hat{θ}}_{r, n}) / 2

(9c)

According to equations (9a)–(9c), the phases of beamforming contain the information of actual DOAs, which were utilized to estimate the DOAs in phase difference bathymetric sonars. It is important that phase $ψ$ uniquely determines the unambiguous DOA, that is, the phase belongs to interval $(- π, π)$ . A proposition that illustrates the unambiguity of the phase between the spanned interval and the actual DOA is given.

Proposition 5

A sparse ULA has M sensors with an inter-element spacing Pd, where d is the unit inter-element spacing which is set as half a wavelength $λ / 2$ . Assume that $θ$ is a DOA of the source. Equally divide steer phase domain into at least $N > (M - 1) / 2$ grids. In addition, the phase difference $ψ$ of beamforming is unambiguous, that is, $| ψ | < π$ , where $ψ = (M - 1) π P (\sin θ - \sin {\hat{θ}}_{p}) / 2$ , in which ${\hat{θ}}_{p}$ denotes the rough DOA estimation with the beamforming method.

Proof

See Appendix 4.

According to Proposition 5, when there exist $N > (M - 1) / 2$ spatial grids, the phase differences $ψ_{p}$ , $ψ_{q}$ , and $ψ_{r}$ would be unambiguous, and then the actual DOA can be obtained by

\sin θ = \sin {\hat{θ}}_{p} + 2 ψ_{p, m} / ((M - 1) π P)

(10a)

\sin θ = \sin {\hat{θ}}_{q} + 2 ψ_{q, m} / ((M - 1) π Q)

(10b)

\sin θ = \sin {\hat{θ}}_{r} + 2 ψ_{r, m} / ((M - 1) π R)

(10c)

At this point, the beamforming peak position and phase of beamforming output are combined, and all candidate DOA estimations for each subarray can be obtained. Then, the actual DOA can be picked up from the candidate sets through coprime property. The proposed beamforming interferometry method is summarized in Table 1.

Table 1.

Process scheme of the proposed algorithm.

Input: Received acoustic signalsOutput: Accurate off-grid DOA estimations
Step 1	Preprocess in a spanned angle domain with beamforming for each sparse uniform linear array, and both magnitude and phase difference information are obtained;
Step 2	Rough on-grid DOAs can be obtained by finding peaks from magnitude;
Step 3	Incorporating with rough DOAs, off-grid estimations for each subarray are obtained according to equations (10a)–(10c);
Step 4	According to $\min_{i, j, c} (\| \sin θ_{p, i} - \sin θ_{q, j} \| + \| \sin θ_{p, i} - \sin θ_{r, c} \| + \| \sin θ_{q, j} - \sin θ_{r, c} \|) < ε$ ,^a find the most approximate estimation from candidate sets.

$ε$ denotes the threshold to determine the approximate degree from DOA candidate set ${\sin θ_{p, i}, \sin θ_{q, j}, \sin θ_{r, c}}$ , where $i, j, and c$ denote the element number of candidate sets for the subarrays 1, 2, and 3, respectively. Proposition 4 provides a strict support for accurate estimations and usually a relaxed threshold is selected.

Simulation and data process

Simulation

Three experiments were conducted to evaluate the performance of the proposed method. In them, several typical scenarios that the bathymetric sonar works in the “singular point” are considered.

In our simulation, we used MCA consisting of three subarrays, as shown in Figure 2: the first one with an inter-element spacing of $Pd = 5 d$ , the second one with an inter-element spacing of $Qd = 3 d$ , and the third one with an inter-element spacing of $Rd = 2 d$ , where d is the unit inter-element spacing, which was set as half a wavelength. The first two subarrays of MCA form a coprime array. The ULA has PM elements with an inter-element spacing of d. It is noted that the MCA and coprime array were formed by selecting the corresponding array elements from the ULA, that is, the MCA and coprime array are subarray of ULA. According to the application background of bathymetry, two coherent echoes from distinct scatterers impinge on the array simultaneously. Their incident angles were set as $θ_{a}$ and $θ_{b}$ , respectively, where $\sin θ_{a} + 2 p / P = \sin θ_{b} + 2 q / Q$ , that is, the “singular point.” The additive noise was modeled as a zero-mean spatially and temporally complex white Gaussian process. We set the total snapshot for DOA estimation as one in the simulation. The grid number of the steer angle was $N = 2048$ , and the spanned interval was $2 π / 2048$ . For each scenario, 2000 Monte-Carlo trials were conducted.

To obtain the off-grid estimation for fair comparison, several Root-MUSIC²² based algorithms were used for comparison with the proposed method. Since the echoes were coherent, forward spatial smoothing–based Root-MUSIC (FSS-RM),²³ forward and backward spatial smoothing–based Root-MUSIC (FBSS-RM),²⁴ and Toeplitz reconstruction–based Root-MUSIC (TR-RM)^25,26 were utilized for comparison. In addition, based on coprime array and MCA, all the above methods were used for comparison to illustrate validity of resolving phase ambiguity.

To illustrate results of the example in Figure 3, the first experiment can be described as follows: in the simulation, the first incident angle was set as $θ_{a} = 30 °$ . According to “singular point” defined in equation (13a), we thus have $θ_{b} = \sin^{- 1} (- 17 / 30) \approx - 34.518 °$ . The SNR was set as −4 dB, and the element number of subarray was set as 64. In addition, the DOA estimations of 2000 Monte-Carlo trials are plotted in Figure 5. As shown in the figure, all the coprime array–based methods suffered from ambiguity errors because wrong angles are matched, and by contrast, all MCA-based methods could completely resolve phase ambiguity. These results are in agreement with the results of the example in section “Complete ambiguity elimination using MCA.”

Figure 5.

DOA estimation results for several methods.

To further evaluate the estimation performance, we computed the root-mean-square error (RMSE) of the methods using 2000 independent trials. The RMSE of the DOA estimations can be expressed as follows

RMSE (θ_{k}) = 20 \lg (\sqrt{\frac{1}{T} \sum_{t = 1}^{T} {({\hat{θ}}_{k} (t) - θ_{k} (t))}^{2}})

(11)

where ${\hat{θ}}_{k} (t)$ represents the estimated DOA of the $k th$ scatterer in the $t th$ Monte-Carlo round, and T is the total number of Monte-Carlo simulation rounds, set as 2000. To illustrate the performance, all the above methods were used for comparison via coprime array, MCA, and ULA in the following two experiments.

In the second experiment, we considered an approximate flat terrain scenario, where the DOAs of two echoes were approximately symmetric, that is, $| θ_{a} | \approx | θ_{b} |$ . The first incident angle was set as $θ_{a} = 30 °$ , and thus, according to “singular point” case in equation (13a), we have $θ_{b} = \sin^{- 1} (\sin θ_{a} + 2^{*} (- 1) / 5 - 2^{*} 1 / 3) \approx - 34.518 °$ . For the comparison of RMSE with the SNR, the number of subarray elements was fixed at 64 when the SNR was varied from −10 to 30 dB with a 2-dB interval. For the comparison of RMSE with the number of subarray elements, the SNR was fixed at 10 dB when we varied the number of subarray elements from 5 to 65 with interval of 5.

In the third experiment, we considered a slope terrain scenario, where the DOAs of two echoes were asymmetrical. The first incident angle was set as $θ_{a} = 10 °$ , and thus we have $θ_{b} = \sin^{- 1} (\sin θ_{a} + 2^{*} (- 1) / 5 - 2^{*} 1 / 3) \approx - 63.255 °$ . To compare performance, the DOA estimation methods were same as the second experiment.

Figures 6 and 8 present the RMSE performance versus SNR in the flat terrain scenario and the slope terrain scenario, respectively. Figures 7 and 9 represent the RMSE performance versus the number of subarray elements in the flat terrain scenario and the slope terrain scenario, respectively.

Figure 6.

RMSE performance versus SNR: (a) first scatterer is located at $θ_{a} = 30 °$ ; (b) second scatterer is located at $θ_{b} \approx - 34.518 °$ .

Figure 7.

RMSE performance versus the number of subarray elements: (a) first scatterer is located at $θ_{a} = 30 °$ ; (b) second scatterer is located at $θ_{b} \approx - 34.518 °$ .

Figure 8.

RMSE performance versus SNR: (a) first scatterer is located at $θ_{a} = 10 °$ ; (b) second scatterer is located at $θ_{b} \approx - 63.255 °$ .

Figure 9.

RMSE performance versus the number of subarray elements: (a) first scatterer is located at $θ_{a} = 10 °$ ; (b) second scatterer is located at $θ_{b} \approx - 63.255 °$ .

For the second experiment, according to the simulation results depicted in Figure 6, the coprime array–based methods suffered from performance degradation since the “singular points” always exist, which is in agreement with the results of experiment 1 in Figure 5. In contrast, the performance of the methods via MCA kept increasing with the increase in the SNR, as expected. The estimation performance of the proposed method via MCA with less array elements and the Root-MUSIC-based methods via ULA were approximate when the SNR was larger than −4 dB as shown in Figure 6. In comparison with three Root-MUSIC-based methods with same array structure, the accuracy of the proposed method was better. As shown in Figure 7, as there is an increase in the subarray element number, the performance of different methods via MCA became enhanced. While the number of subarray elements was more than 15, the performance of the proposed method was better than the Root-MUSIC-based methods. The reasons for the superiority can be categorized into three aspects. First, benefiting from the coprime property of the MCA, both magnitude and phase information are used to improve the performance. Second, the Root-MUSIC-based methods rely on the estimation of the covariance matrix. In general, the standard sample covariance matrix cannot exactly represent the true covariance of an ensemble mean, but it comes close enough for reasonable snapshots. Therefore, the performance would be degraded in the single snapshot scene. Third, when the number of snapshots is small, or alternatively, when the SNR is small, the performance of the high-resolution algorithms, such as Root-MUSIC, declines.^27,28

For the third experiment, we observed similar performances as in experiment 2. The performance of the small angle direction was better than that of the large angle in Figures 8 and 9. In contrast to the second experiment, as shown in Figures 6 and 7, the performances of the two directions in the second experiment were approximate. The reason for this is that as there is an increase in the incident angle, as expected, the estimation performance is degraded.²⁹ The above comparisons demonstrate that the proposed method can provide a lower RMSE than the other methods for same array when the number of subarray elements is enough.

Complexity analysis

First, we compared the complexity of the Root-MUSIC method and the proposed method. According to Marple,³⁰ the approximate relative computational complexity of an eigenanalysis-based method, such as MUSIC and Root-MUSIC, is $M^{3}$ , where M is the number of the array elements, whereas the beamforming one is $M^{2}$ . Thus, in comparison with the proposed method, eigenanalysis-based methods are computationally intensive.

Second, to demonstrate the efficiency of the proposed method via MCA, we compared the computational complexity with the beamforming algorithm via ULA. For a fair comparison, according to Figure 2, a ULA with half a wavelength spacing was considered with the proposed method, whose element number was $(M - 1) E + 1$ to ensure the same aperture as MCA, where $E = max {P, Q, R}$ . Therefore, the hardware complexity was reduced by half, and the computational complexity of the traditional beamforming algorithm was $O (((M - 1) E + 1)^{2} J)$ , whereas the computational complexity of the proposed method was reduced to $O ((3 M)^{2} N)$ by incorporating the off-grid phase-based DOA estimation method, where $J >> N$ represents the number of spanned angle domain grids. For all positive integers, ${2, 3, 5}$ was a triplet of the smallest coprime integers, that is, at least $E = 5 > 3$ was satisfied. Therefore, the proposed method via MCA was more efficient than the beamforming algorithm via ULA.

Sonar data processing

To verify the performance of the proposed method, raw data recorded in a lake by an MBES were utilized. The major parameters of the sonar are shown in Table 2. The sonar system is shown in Figure 10(a), and it was deployed in the side of ship (see Figure 10(b)).

Table 2.

Major parameters of the MBES.

Entry	Parameter
Sonar frequency	200kHz
Channel number	100
Inter-element spacing	3.75 mm (half a wavelength)
Beamwidth	$1.0 ° \times 1.0 °$
Maximum swath width	$140 °$
Maximum swath depth	500 m

Figure 10.

Multibeam echosounder sonar system: (a) system compositions of the sonar; (b) deployment of the sonar.

For a fair comparison, a ULA with half a wavelength spacing was also considered with the proposed method, whose element number was $(M - 1) P + 1$ to ensure the same aperture as MCA, where $M = 20$ and $P = 5$ . In addition, the inter-element spacing of three subarrays was set as $P = 5, Q = 3, and R = 2$ , respectively. According to MCA configuration in Figure 2(b), corresponding array elements were selected from the ULA with 100 elements to form coprime array and MCA, and then the performance among ULA, coprime array, and MCA was compared. The DOA estimations in a specific snapshot number via coprime array, ULA, and MCA are shown in Figure 11.

Figure 11.

Comparison with different methods: (a) raw results via coprime array (CA), (b) raw results via ULA and MCA, and (c) filtered results via ULA and MCA.

As shown in Figure 11(a), numerous “singular points” can be observed. The unambiguous DOAs can be obtained, as shown in Figure 11(b). In addition, a smooth filter¹³ was utilized to improve the performance to a certain extent in Figure 11(c). By constructing with a conventional ULA structure, the performance of the proposed method based on MCA was close. In all, an analysis of the performance and results validate that the proposed method is effective and feasible.

Conclusion

In this article, MCA was proposed for completely resolving phase ambiguity. In addition, a low-complexity beamforming interferometry method was proposed via MCA. The proposed method not only inherits multisource estimation capacity from the beamforming method but also significantly reduces the complexity for off-grid DOA estimation with the interferometry method. Moreover, extensive experimental results demonstrated that the proposed method is feasible for multisource estimation and achieves a better performance than the other Root-MUSIC-based methods in terms of computational complexity and estimation performance. Finally, sonar data processing demonstrated that the performance of the proposed method with fewer array elements was approximate to the conventional method.

Footnotes

Appendix 1

Appendix 2

Appendix 3

Appendix 4 Acknowledgements

The authors specially thank the associate editor and the anonymous reviewers for providing constructive feedback to help improve the quality and presentation of this manuscript.

Handling Editor: Jaime Lloret

Author contribution

The idea was proposed by J.S.; J.Z. and W.Z. simulated the algorithm; T.Z. and D.P. analyzed the data; and J.S. designed the experiments and polished the English.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Research in this paper was supported by the National Natural Science Foundation of China (grant nos U1709203 and 41376103), the Fundamental Research Funds for the Central Universities (no. HEUCFD1502) and the Fund of Acoustics Science and Technology Laboratory.

ORCID iD

Jiajun Shen

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