A new DOA estimation approach using Volterra signal model

Abstract

In this paper, we proposed a new direction-of-arrival estimation approach using Volterra signal model in spatial domain. The new technique basically uses additionally the second-order terms of Volterra series to produce augmented Volterra snapshots, the extension for higher-order case is straightforward. The resolution of the proposed method is high comparing with the standard multiple signal classification (MUSIC) algorithm and the 2q-MUSIC algorithm. Simulation results are presented to show the performance of the proposed technique.

Appendix: The dimension of the signal subspace \(N_s\)

The dimension of the signal subspace $N_s$ is equivalent to the Rank of $\mathbf{R}_v$ for the noiseless case. In order to analysis the Rank of $\mathbf{R}_v$, let us look at some of the properties of Rank:

• (1) For a matrix $\mathbf{{B}}\in \mathbb{C }^{m\times n}$, $\mathrm{Rank}(\mathbf{{B}})\le \min \left(m,n\right).$

• (2) For two matrices $\mathbf{{B}}\in \mathbb{C }^{m\times n}$ and $\mathbf{{C}}\in \mathbb{C }^{n\times t}$, $\mathrm{Rank}(\mathbf{{BC}})\le \min \big (\mathrm{Rank}(\mathbf{{B}}),\mathrm{Rank}(\mathbf{{C}})\big ).$

• (3) For two matrics $\mathbf{{B}}\in \mathbb{C }^{m\times n}$ and $\mathbf{{D}}\in \mathbb{C }^{t\times n}$, $\mathrm{Rank}\left(\left[\begin{array}{c} \mathbf{{B}}\\ \mathbf{{D}}\end{array}\right]\right)\le \min \left(m+t,n\right).$

For the noiseless case, the Volterra snapshots $\mathbf{{X}}_{v}$ is given by

$$\begin{aligned} \mathbf{{X}}_{v}=\left[ \begin{array}{c} \mathbf{{A}}(\theta )\mathbf{{S}}\\ \left(\mathbf{{A}}(\theta )\otimes \mathbf{{A}}^*(\theta )\right)\cdot \left(\mathbf{{S}\odot {S}^*}\right) \end{array} \right], \end{aligned}$$

(33)

where $\mathbf{S} = \left[\mathbf{s}(1),\, \mathbf{s}(2), \ldots , \mathbf{s}(k), \ldots , \mathbf{s}(K)\right]$ and $\odot$ denotes Khatri-Rao product, which is a column-wise Kronecker product.

The estimated noiseless covariance matrix is obtained by

$$\begin{aligned} \mathbf{R}_{v}=\frac{1}{K}\Big (\mathbf{{X}}_{v}\mathbf{{X}}_{v}^{H}\Big ). \end{aligned}$$

(34)

Based on the properties (1) and (2) of Rank, $\mathrm{Rank}\big (\mathbf{{A}}(\theta )\big )\le L$ and $\mathrm{Rank}(\mathbf{{S}})\le L$, then $\mathrm{Rank}\big (\mathbf{{A}}(\theta )\mathbf{{S}}\big )\le L$.

The Kronecker product can be rewritten as

$$\begin{aligned} \mathbf{{s}}(k)\otimes \mathbf{{s}}^*(k) = \mathrm{vec}\Big (\mathbf{{s}}(k)\mathbf{{s}}^*(k)^{T}\Big ), \end{aligned}$$

(35)

where $\mathrm{vec}\left(\mathbf{X}\right)$ denotes the vectorization of the matrix $\mathbf{X}$ formed by stacking the columns of $\mathbf{X}$ into a single column vector.

Since

$$\begin{aligned} \mathbf{{s}}(k)\mathbf{{s}}^*(k)^{T}&= \left(\mathbf{{s}}(k)\cdot \mathbf{{s}}^*(k)^{T}\right)^{T}\nonumber \\&= \left[\begin{array}{cccc} s_{1}(k)s_{1}^*(k)\,&s_{1}(k)s_{2}^*(k)&\cdots&s_{1}(k)s_{L}^*(k)\\ s_{2}(k)s_{1}^*(k) \,&s_{2}(k)s_{2}^*(k)&\vdots&\vdots \\ \vdots \,&\vdots \,&\ddots&\vdots \\ s_{L}(k)s_{1}^*(k) \,&\cdots&\cdots&s_{L}(k)s_{L}^*(k)\end{array}\right], \end{aligned}$$

(36)

then the rank of $\mathbf{{S}~\odot }~\mathbf{{S}}^*$ is

$$\begin{aligned} \mathrm{Rank}\big (\mathbf{{S}~\odot }~\mathbf{{S}}^*\big )\le L^2. \end{aligned}$$

(37)

and

$$\begin{aligned} \mathrm{Rank}\big (\mathbf{{A}}(\theta )\otimes \mathbf{{A}}^*(\theta )\big ) \le L^2, \end{aligned}$$

(38)

and

$$\begin{aligned} \mathrm{Rank}\Big (\big (\mathbf{{A}}(\theta )\otimes \mathbf{{A}}^*(\theta )\big )\cdot \big (\mathbf{{S}\odot {S}}^*\big )\Big ) \le L^2. \end{aligned}$$

(39)

Based on the above property $(3)$, we obtain

$$\begin{aligned} \mathrm{Rank}\left[\begin{array}{c} \mathbf{{A}}(\theta )\mathbf{{S}}\\ \big (\mathbf{{A}}(\theta )\otimes \mathbf{{A}}^*(\theta )\big ) \cdot \big (\mathbf{{S}\odot {S}}^*\big )\end{array} \right]\le L+L^2. \end{aligned}$$

(40)

The dimension of the signal subspace

$$\begin{aligned} N_s=L+L^2. \end{aligned}$$

(41)

The number of noise eigenvectors $N_{e}$ used to perform DOA estimation is given by

$$\begin{aligned} N_e=N-N_s=\frac{M(M+3)}{2}-L(L+1). \end{aligned}$$

(42)

Table 1 shows the dimension of the signal subspace $N_s$ and the number of noise eigenvectors $N_{e}$ used to perform DOA estimation, for different number of sensors $M$ and different number of sources $L$.

Table 1 The dimension of the signal subspace $N_s$ and the number of noise eigenvectors $N_{e}$ used to perform DOA estimation, for different number of sensors $M$ and different number of sources $L$