Performance analysis and DOA estimation method over acoustic vector sensor array in the presence of polarity inconsistency

Abstract

This paper investigates the direction of arrival (DOA) estimation performance in the presence of polarity inconsistency in uniform acoustic vector sensor (AVS) linear array. We analyze the influence of polarity bias on beampattern directivity of the AVS array. The analysis results show that the polarity bias leads to asymptotically biased estimation. Then, the analytical expression for the asymptotic bias based on classical beamforming is derived in the presence of polarity error. Moreover, to improve the DOA estimation performance in the presence of polarity inconsistency, a polarity calibration method is proposed. Numerical simulations reveal the effectiveness and superiority of the proposed calibration method when the polarity error satisfies with the uniform distribution and the normal distribution.

Appendix A

Here, we derive the bias expression of (27) in Sect. 4. By taking the first and second order derivatives of $f(\theta )$ with respect to $\theta$, we obtain

$$\begin{aligned} \frac{{\partial f(\theta )}}{{\partial \theta }}= & {} \frac{{\partial {{{\varvec{ A}}}^H}(\theta )}}{{\partial \theta }}{\hat{{\varvec{ R}}}}{{\varvec{ A}}}(\theta ) + {{{\varvec{ A}}}^H}(\theta ){\hat{{\varvec{ R}}}}\frac{{\partial {{\varvec{ A}}}(\theta )}}{{\partial \theta }}, \end{aligned}$$

(68)

$$\begin{aligned} \frac{{{\partial ^2}f(\theta )}}{{\partial {\theta ^2}}}= & {} \frac{{{\partial ^2}{{{\varvec{ A}}}^H}(\theta )}}{{\partial {\theta ^2}}}{\hat{{\varvec{ R}}}}{{\varvec{ A}}}(\theta ) + 2\frac{{\partial {{{\varvec{ A}}}^H}(\theta )}}{{\partial \theta }}{\hat{{\varvec{ R}}}}\frac{{\partial {{\varvec{ A}}}(\theta )}}{{\partial \theta }} + {{{\varvec{ A}}}^H}(\theta ){\hat{{\varvec{ R}}}}\frac{{{\partial ^2}{{\varvec{ A}}}(\theta )}}{{\partial {\theta ^2}}}. \end{aligned}$$

(69)

By substituting (68) and (69) for (26), we can get

$$\begin{aligned} \begin{aligned} {\hat{b}}(\theta )&= - \frac{\dfrac{{\partial {{{\varvec{ A}}}^H}(\theta )}}{{\partial \theta }}{\hat{{\varvec{ R}}}}{{\varvec{ A}}}(\theta ) + {{{\varvec{ A}}}^H}(\theta ){\hat{{\varvec{ R}}}}\dfrac{{\partial {{\varvec{ A}}}(\theta )}}{\partial \theta }}{\dfrac{{\partial ^2}{{{\varvec{ A}}}^H}(\theta )}{\partial {\theta ^2}}{\hat{{\varvec{ R}}}}{{\varvec{ A}}}(\theta ) + 2\dfrac{\partial {{{\varvec{ A}}}^H}(\theta )}{\partial \theta }{\hat{{\varvec{ R}}}}\dfrac{\partial {{\varvec{ A}}}(\theta )}{{\partial \theta }} + {{{\varvec{ A}}}^H}(\theta ){\hat{{\varvec{ R}}}}\dfrac{{\partial ^2}{{\varvec{ A}}}(\theta )}{\partial {\theta ^2}}}, \end{aligned} \end{aligned}$$

(70)

where ${\hat{{\varvec{ R}}}} = \sigma _s^2{{\varvec{ a}}}(\theta ,{{\varvec{\beta }}} ){{{\varvec{ a}}}^H}(\theta ,{{\varvec{\beta }}}) + \sigma _n^2{{\varvec{ I}}}$. By calculating $\dfrac{{\partial {{{\varvec{ A}}}^H}(\theta )}}{{\partial \theta }}$ and $\dfrac{{{\partial ^2}{{{\varvec{ A}}}^H}(\theta )}}{{\partial {\theta ^2}}}$, we can get

$$\begin{aligned} \frac{{\partial {{{\varvec{ A}}}^H}(\theta )}}{{\partial \theta }}= & {} {\left[ \begin{array}{c} {{a'}_{1p}}(\theta )\\ {{a'}_{1p}}(\theta )\cos (\theta ) - {a_{1p}}(\theta )\sin (\theta )\\ {{a'}_{1p}}(\theta )\sin (\theta ) + {a_{1p}}(\theta )\cos (\theta )\\ \vdots \\ {{a'}_{Mp}}(\theta )\\ {{a'}_{Mp}}(\theta )\cos (\theta ) - {a_{Mp}}(\theta )\sin (\theta )\\ {{a'}_{Mp}}(\theta )\sin (\theta ) + {a_{Mp}}(\theta )\cos (\theta ) \end{array} \right] ^H}, \end{aligned}$$

(71)

$$\begin{aligned} \frac{{{\partial ^2}{{{\varvec{ A}}}^H}(\theta )}}{{\partial {\theta ^2}}}= & {} {\left[ \begin{array}{c} {{{a''}}_{1p}}(\theta )\\ {{{a''}}_{1p}}(\theta )\cos (\theta ) - 2{{a'}_{1p}}(\theta )\sin (\theta ) - {a_{1p}}(\theta )\cos (\theta )\\ {{{a''}}_{1p}}(\theta )\sin (\theta ) + 2{{a'}_{1p}}(\theta )\cos (\theta ) - {a_{1p}}(\theta )\sin (\theta )\\ \vdots \\ {{{a''}}_{Mp}}(\theta )\\ {{{a''}}_{Mp}}(\theta )\cos (\theta ) - 2{{a'}_{Mp}}(\theta )\sin (\theta ) - {a_{Mp}}(\theta )\cos (\theta )\\ {{{a''}}_{Mp}}(\theta )\sin (\theta ) + 2{{a'}_{Mp}}(\theta )\cos (\theta ) - {a_{Mp}}(\theta )\sin (\theta ) \end{array} \right] ^H}, \end{aligned}$$

(72)

where ${a'}_{mp}(\theta )$ and ${a''}_{mp}(\theta )$ represent the first derivative and the second derivative of ${a_{mp}}(\theta )$ with respect to $\theta$, respectively.

In order to compute the analytic expression of ${\hat{b}}({\theta })$, we also need to compute the following inner product.

$$\begin{aligned} P= & {} \frac{\partial {{{\varvec{ A}}}^H}(\theta )}{{\partial \theta }}{{\varvec{ A}}}(\theta ) = \sum \limits _{m = 1}^M {{2a'}_{mp}^H(\theta ){a_{mp}}(\theta )} , \end{aligned}$$

(73)

$$\begin{aligned} J= & {} {{{\varvec{ B}}}^H}(\theta ,\beta ){{\varvec{ A}}}(\theta ) = \sum \limits _{m = 1}^M {(1 + \cos {\beta _m})} , \end{aligned}$$

(74)

$$\begin{aligned} W= & {} \frac{{\partial {{{\varvec{ A}}}^H}(\theta )}}{{\partial \theta }}{{\varvec{ B}}}(\theta ,\beta ) = \sum \limits _{m = 1}^M {{a'}_{mp}^H(\theta ){a_{mp}}(\theta )} \nonumber \\&+ \sum \limits _{m = 1}^M {{a'}_{mp}^H(\theta ){a_{mp}}(\theta )\cos {\beta _m}} - \sum \limits _{m = 1}^M {\sin {\beta _m}} , \end{aligned}$$

(75)

$$\begin{aligned} Q= & {} \frac{{{\partial ^2}{{{\varvec{ A}}}^H}(\theta )}}{{\partial {\theta ^2}}}{{\varvec{ B}}}(\theta ,\beta ) = \sum \limits _{m = 1}^M {{a''}_{mp}^H(\theta ){a_{mp}}(\theta )} + \sum \limits _{m = 1}^M {{a''}_{mp}^H(\theta ){a_{mp}}(\theta )\cos {\beta _m}} \nonumber \\&- 2\sum \limits _{m = 1}^M {{a'}_{mp}^H(\theta ){a_{mp}}(\theta )\sin {\beta _m}} - \sum \limits _{m = 1}^M {\cos {\beta _m}} , \end{aligned}$$

(76)

$$\begin{aligned} H= & {} \frac{{{\partial ^2}{{{\varvec{ A}}}^H}(\theta )}}{{\partial {\theta ^2}}}{{\varvec{ A}}}(\theta ) = \sum \limits _{m = 1}^M {(2{a''}_{mp}^H(\theta ){a_{mp}}(\theta ) - 1)} , \end{aligned}$$

(77)

$$\begin{aligned} D= & {} \frac{{\partial {{{\varvec{ A}}}^H}(\theta )}}{{\partial \theta }}\frac{{\partial {{\varvec{ A}}}(\theta )}}{{\partial \theta }} = \sum \limits _{m = 1}^M {(2{a'}_{mp}^H(\theta ){{a'}_{mp}}(\theta ) + 1)} , \end{aligned}$$

(78)

Substituting (73)–(78) into (70), finally leads to (27).