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Particle-Velocity Coarray Augmentation for Direction Finding with Acoustic Vector Sensors

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Abstract

In this paper, the problem of passive direction finding is addressed using an acoustic vector sensor array (AVS), which may be deployed either in free space or near a reflecting boundary. Building upon the \(4 \times 1\) vector field measured by an AVS, the particle-velocity coarray augmentation (PVCA) is proposed to admit the underdetermined direction finding using the spatial difference coarray derived from the vectorization of the array covariance matrix. Unlike the widely used spatial coarray Toeplitz recovery technique, the PVCA is applicable to arbitrary array geometries and imposes no reduction of the spatial difference coarray aperture. For the array located at or near a reflecting boundary, the PVCA allows resolving up to 13 sources, while for the array located in free space, the PVCA can identify 9 sources at most. By applying to the systematically designed nonuniform arrays, such as coprime arrays and nested arrays, the PVCA can be coupled with the spatial smoothing technique to get the number of resolvable sources multiplied. Finally, the effectiveness of the PVCA is verified by numerical simulations.

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Data Availability Statement

The data used to support the findings of this study are available from the corresponding author upon request.

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Authors and Affiliations

  1. Shanghai Key Laboratory of Intelligent Sensing and Recognition, School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Ting Shu, Jin He & Trieu-Kien Truong

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  1. Ting Shu
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  2. Jin He
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Correspondence to Jin He.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, and there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Particle–Velocity Coarray Augmentation For Direction Finding with Acoustic Vector Sensors.”

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This work was supported by the National Natural Science Foundation of China under grant 61771302.

Appendices

Appendix A: Derivation of the Matrix \({{\varvec{P}}}\) in (7)

Define a \(PQ \times PQ\) matrix \({{\varvec{U}}}_{P \times Q}\) as

$$\begin{aligned} {{\varvec{U}}}_{P \times Q} \triangleq \sum _{p = 1}^P \sum _{q = 1}^Q {{\varvec{E}}}_{pq} \otimes {{\varvec{F}}}_{qp}, \end{aligned}$$
(22)

where \({{\varvec{E}}}_{pq}\) is a \(P \times Q\) matrix with all zeros except a 1 at the (p, q)th position and \({{\varvec{F}}}_{qp}\) is a \(Q \times P\) matrix with all zeros except a 1 at the (q, p)th position.

Using the following three matrix prosperities given in [17]

$$\begin{aligned} {{\varvec{A}}} \odot ({{\varvec{B}}} \odot {{\varvec{C}}})= & {} ({{\varvec{A}}} \odot {{\varvec{B}}}) \odot {{\varvec{C}}}, \end{aligned}$$
(23)
$$\begin{aligned} {{\varvec{U}}}_{P \times Q} ({{\varvec{A}}} \odot {{\varvec{B}}})= & {} ({{\varvec{B}}} \odot {{\varvec{A}}}), \end{aligned}$$
(24)
$$\begin{aligned} ({{\varvec{A}}} \otimes {{\varvec{B}}})({{\varvec{C}}} \odot {{\varvec{D}}})= & {} {{\varvec{A}}} {{\varvec{C}}}\odot {{\varvec{B}}} {{\varvec{D}}}, \end{aligned}$$
(25)

we have

$$\begin{aligned}{} & {} ({{\varvec{U}}}_{4 \times L} \otimes {{\varvec{I}}}_{4L} ) [({{\varvec{Q}}} \odot {{\varvec{C}}})^*\odot ({{\varvec{Q}}} \odot {{\varvec{C}}})] \nonumber \\ {}{} & {} \quad = {{\varvec{U}}}_{4 \times L} ({{\varvec{Q}}}^*\odot {{\varvec{C}}}^*) \odot {{\varvec{I}}}_{4L} ({{\varvec{Q}}} \odot {{\varvec{C}}}) \nonumber \\ {}{} & {} \quad = ({{\varvec{C}}}^*\odot {{\varvec{Q}}}^*) \odot ({{\varvec{Q}}} \odot {{\varvec{C}}}) \end{aligned}$$
(26)

and

$$\begin{aligned}{} & {} ({{\varvec{I}}}_4 \otimes {{\varvec{U}}}_{4 \times L^2}) ({{\varvec{C}}}^*\odot {{\varvec{Q}}}^*) \odot ({{\varvec{Q}}} \odot {{\varvec{C}}}) \nonumber \\{} & {} \quad = {{\varvec{I}}}_4 {{\varvec{C}}}^*\odot {{\varvec{U}}}_{4 \times L^2}[ ({{\varvec{Q}}}^*\odot {{\varvec{Q}}}) \odot {{\varvec{C}}}] \nonumber \\ {}{} & {} \quad = ({{\varvec{C}}}^*\odot {{\varvec{C}}}) \odot ({{\varvec{Q}}}^*\odot {{\varvec{Q}}}). \end{aligned}$$
(27)

Let \({{\varvec{P}}} = ({{\varvec{I}}}_4 \otimes {{\varvec{U}}}_{4 \times L^2}) ({{\varvec{U}}}_{4 \times L} \otimes {{\varvec{I}}}_{4L})\). The relationship (7) is established.

Appendix B: Proof of Theorem 1

The rank of \(\bar{{\varvec{C}}}\) depends on the linear dependence of the particle-velocity coarray steering vector \(\bar{{\varvec{c}}}(\theta , \phi ) \triangleq {{\varvec{c}}}^*(\theta , \phi ) \otimes {{\varvec{c}}}(\theta , \phi )\). In fact, the ith element of \(\bar{{\varvec{c}}}(\theta , \phi )\), denoted by \({\bar{c}}_i\), is the product of the mth element of \({{\varvec{c}}}^*(\theta , \phi )\) and the nth element of \({{\varvec{c}}}(\theta , \phi )\), where the relationship between m, n, and i is

$$\begin{aligned} m = \left\lceil i/4 \right\rceil , \ n = i - 4 \left\lfloor (i - \textrm{eps})/4 \right\rfloor , \end{aligned}$$
(28)

where “eps” represents the floating-point relative accuracy, which is used to ensure the values of n are within 1 to 4. With the foregoing notations, the 16 elements of \(\bar{{\varvec{c}}}\), listed in Table 4, can be computed, where \(\rho \triangleq \mathcal{R}(\theta ) e^{-j \vartheta }\).

Table 4 Entries in the \(16 \times 1\) Vector \({{\varvec{c}}}^*\otimes {{\varvec{c}}}\)
Full size table

From Table 4, we have the following two cases:

  1. (i)

    For \(\mathcal{R}(\theta ) \ne 0, \pm 1, \forall \theta \), \({\bar{c}}_5 = {\bar{c}}_2\), \({\bar{c}}_9 = {\bar{c}}_3\), and \({\bar{c}}_{10} = {\bar{c}}_7\). Therefore, the 5th, 9th, and 10th rows of \({\bar{{\varvec{C}}}}\) are linearly dependent on the remaining 13 rows. For sources of arbitrary but different directions, the rank of \({\bar{{\varvec{C}}}}\) is up to 13 at most.

  2. (ii)

    For \(\mathcal{R}(\theta ) = 0, \forall \theta \), \({\bar{c}}_5 = {\bar{c}}_2\), \({\bar{c}}_9 = {\bar{c}}_3\), \({\bar{c}}_{10} = {\bar{c}}_7\), \({\bar{c}}_{13} = {\bar{c}}_4\), \({\bar{c}}_{14} = {\bar{c}}_8\), \({\bar{c}}_{15} = {\bar{c}}_{12}\), and \({\bar{c}}_{16} = {\bar{c}}_{1} - {\bar{c}}_{6} - {\bar{c}}_{11}\). Therefore, the 5th, 9th, 10th, and 13-16th rows of \({\bar{{\varvec{C}}}}\) are linearly dependent on the remaining 9 rows. For sources of arbitrary but different directions, the rank of \({\bar{{\varvec{C}}}}\) is up to 9 at most.

Based on the above (i) and (ii), the results in Theorem 1 are established.

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Shu, T., He, J. & Truong, TK. Particle-Velocity Coarray Augmentation for Direction Finding with Acoustic Vector Sensors. Circuits Syst Signal Process 42, 3072–3093 (2023). https://doi.org/10.1007/s00034-022-02269-z

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