Abstract
The canonical polyadic decomposition (CPD) of higher-order tensors, a.k.a. PARAFAC, has shown excellent performance in two-dimensional direction of arrival (DOA) estimation using the acoustic vector sensor array (AVSA). However, most existing studies pay little attention to the manifold matrix structure of the AVSA during the CPD and are designed for uncorrelated sources. This paper presents a constrained CPD-based algorithm for DOA estimation using a uniform linear AVSA, whose manifold matrix is highly structured. Specifically, the manifold matrix equals to the Khatri-Rao product of a Vandermonde matrix and a proportional column-norm matrix. We show that DOA estimation accuracy is further improved by incorporating the prior structured information. Besides, we also extend the Toeplitz decorrelation technique to the AVSA to handle possibly correlated sources. The algorithm does not require iteration or peak searching and thus is computationally effective. Numerical simulations verify the effectiveness and superior performance of the algorithm.
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Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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This article was funded by National Natural Science Foundation of China under the Grant Nos. (12074315, 52231013, 61901383).
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Appendices
Appendix A Proof of (16)
Proof
From the construction, one obtains
where \(\bar{{{\textbf {D}}}}_i = \sum _{j=1}^4 {{\textbf {D}}}_{i,j}/4\). Let \({{\textbf {d}}}_i \in {\mathbb {C}}^{K\times 1}\) be the vector consisting of the diagonal elements of the diagonal matrix \(\bar{{{\textbf {D}}}}_i\), then \({{\textbf {d}}}_i(k) = \xi _k {{\textbf {B}}}(i,k)\) with \(\xi _k = \sigma _k^2 (1+u_k+v_k+w_k)/4\). Stacking the \({\mathcal {R}}(:,:,i)\) into a long vector gives
Then, the mode-3 unfolding of \({\mathcal {R}}\) can be expressed by
where \({\tilde{{\textbf {B}}}} = {{\textbf {B}}}\cdot \text {Diag}\{\xi _1,\dots ,\xi _K\}\). Comparing with (5), \({\mathcal {R}}\) can be denoted as \({\mathcal {R}}=[[{{\textbf {A}}}, {{\textbf {A}}}^*, {\tilde{{\textbf {B}}}}]]\). \(\square \)
Appendix B Proof of (25)
Proof
The (l, m)-th element of \({{\textbf {R}}}_{i,j}\) can be expressed as
where
From the construction of \({{\textbf {T}}}_{i,j,m}\) in (24), its \((l_1,l_2)\)-th element equals to \(r_{l_1-l_2,m}\) for \(l_1,l_2 = 1,\dots ,M+1\). Note that the first element of \({{\textbf {R}}}_{i,j}\) is indexed by \((-M,-M)\) while that of \({{\textbf {T}}}_{i,j,m}\) is indexed by (1, 1) for a formal convenience.
On the other hand, the \((l_1,l_2)\)-th entry of \({{\textbf {V}}} {{\textbf {D}}}_{i,j,m} {{\textbf {V}}}^H\) equals to
where \({{\textbf {v}}}_k = [1,z_k,\dots ,z_k^M]^T\) is the k-th column of \({{\textbf {V}}}\). Thus, (25) is proven. \(\square \)
Appendix C Proof of full column rank of \({\tilde{{\textbf {B}}}}\odot {{\textbf {A}}}\)
Proof
The Vandermonde matrix \({{\textbf {A}}} \in {\mathbb {C}}^{N\times K}\) is full column rank if \(K<N\) holds. To show \({\tilde{{\textbf {B}}}} \odot {{\textbf {A}}}\) is full column rank, a lemma is introduced as follows: \(\square \)
Lemma 1
[Full Rank of Khatri-Rao Product [22]] If \(k_{{{\textbf {W}}}}+k_{{{\textbf {Z}}}} \ge J+1\), then \({{\textbf {Z}}}\odot {{\textbf {W}}}\) is full column rank J, where \({{\textbf {W}}}\) is \(I_1\times J\) and \({{\textbf {Z}}}\) is \(I_2 \times J\).
Using Lemma 1 and the fact that \(k_{{\textbf {A}}} = \min (N,K)\) and \(k_{{\tilde{{\textbf {B}}}}}=\min (3,K)\), one obtains that \(k_{{\textbf {A}}} + k_{{\tilde{{\textbf {B}}}}} = \min (N,K) + \min (3,K)\). Therefore, \({\tilde{{\textbf {B}}}} \odot {{\textbf {A}}}\) is full column rank if \(1<K<N\) holds.
Appendix D Proof of the PCN constrained solution
Solving (38) seems to be challenging because of the non-positive definiteness of the PCN constraint. Fortunately, it can be solved by the following lemma.
Lemma 2
[Optimal Scaling Lemma [2]] For any column-wise constraint set \({\mathbb {S}}\) it holds that
where \({{{\textbf {b}}}}^{LS}= {{\textbf {X}}}^H{{\textbf {a}}}/\Vert {{\textbf {a}}}\Vert _2^2\). In other words, the optimal solution of the CLS problem is simply the projection of the (unconstrained) LS solution onto the constraint set \({\mathbb {S}}\).
With Lemma 2, the problem in (38) becomes
where \({\tilde{{\textbf {b}}}}_k^{LS}= {\tilde{{\textbf {U}}}}_k^T {{\textbf {a}}}_k^* /\Vert {{\textbf {a}}}_k\Vert _2^2\) is the LS solution in (37). The next step is to project \({\tilde{{\textbf {b}}}}_k^{LS}\) onto the PCN space. Since \({{\textbf {b}}}\) is in the PCN space, it can be parameterized as
where \(\zeta \) is used to adjust the length of \({{\textbf {b}}}\) and \({\tilde{{\textbf {b}}}}_k^{LS}(1)\) is used to simplify the subsequent objective function. Defining
and substituting (D.3) and (D.4) into the objective function in (D.2), one obtains
where \( \text {const} = \left[ {\tilde{{\textbf {b}}}}_k^{LS}(1)\right] ^2 (1+c_2^2+c_3^2+c_4^2 ) = \left\| {\tilde{{\textbf {b}}}}_k^{LS}\right\| _2^2\). After dropping the constant values and the scale factor in (D.5), the CLS problem in (D.2) is equivalent to find \((\alpha ,\beta ,\zeta )\) that minimizes
The minimal of \(f(\alpha ,\beta ,\zeta )\) is obtained when \({\partial f}/{\partial \zeta } = {\partial f}/{\partial \alpha } ={\partial f}/{\partial \beta } = 0\). Solving the value of \((\alpha , \beta , \zeta )\) and substituting them into (D.3), the optimal estimates of \({\tilde{{\textbf {b}}}}_k\) simplifies to
where \(L = \sqrt{c_2^2+c_3^2+c_4^2}\).
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Lu, D., Duan, R. & Yang, K. Two-Dimensional Direction-of-Arrival Estimation in Acoustic Vector Sensor Array via Constrained Tensor Decomposition. Circuits Syst Signal Process 42, 4197–4220 (2023). https://doi.org/10.1007/s00034-023-02310-9
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DOI: https://doi.org/10.1007/s00034-023-02310-9