DOA estimation of moving sound sources in the context of nonuniform spatial noise using acoustic vector sensor

Abstract

In this paper, DOA estimation of moving sound source using an acoustic vector-sensor in the context of spatially nonuniform noise is discussed. We propose a novel method for DOA tracking. In this method, non-uniform noise covariance is first estimated using acoustic vector sensor measurement and then the weighted parameter of conventional maximum power (MP) method is fixed by noise pre-whitening technique. In this way the weighted parameter selection problem of MP is solved when the noise powers of monopole and dipole are unknown. Moreover, under the assumption of constant velocity model of source dynamics, the DOA estimation by the improved maximum energy method is treated as measuring information and the Kalman filter algorithm in polar coordinate system is introduced to improve the accuracy of DOA estimation of moving sources. Theoretical analysis and simulation results demonstrate that the mean square angle error of the proposed method is lower than the traditional Cramer–Rao lower bound which only employs static measurement information.

Appendices

Appendix 1

From the relationship between noise covariance matrix $\varvec{\varOmega }_k$ and received signal covariance matrix ${{\varvec{C}}}_k$, we can get estimation of the noise covariance matrix $\varvec{\varOmega }_k$,namely ${\hat{\varvec{\varOmega }}}_k$, at time step $k$, and Yun-tao and Hou (2006) shows us an useful noise covariance matrix estimating method.

Firstly, the matrix ${{\varvec{C}}}_k^D$ in formula (15) can be expressed as

$$\begin{aligned} {{\varvec{C}}}_k^D&= {\varvec{DC}}_k {\varvec{D}}^{\mathrm{T}} \nonumber \\&= {{\varvec{D}}}\!\left( {{{\varvec{a}}}(\varvec{\varTheta }_k )\delta _{s,k}^2 {{\varvec{a}}}(\varvec{\varTheta }_k )^{\mathrm{T}}+\varvec{\omega }_k} \right) {\varvec{D}}^{\mathrm{T}} \nonumber \\&= {\varvec{D}}{{\varvec{a}}}(\varvec{\varTheta }_k )\delta _{s,k}^2 {{\varvec{a}}}(\varvec{\varTheta }_k)^{\mathrm{T}}{\varvec{D}}^{\mathrm{T}}+{\varvec{D}}\varvec{\omega }_k D^{\mathrm{T}} \nonumber \\&= {{\varvec{B}}}_k \delta _{s,k}^2 {{\varvec{B}}}_k^{\mathrm{T}}+\varvec{\omega }_k^{\varvec{D}} \end{aligned}$$

(30)

where

$$\begin{aligned} {{\varvec{B}}}_k&= {{\varvec{Da}}}(\varvec{\varTheta }_k )=\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {B_{k,1}}&{} {B_{2,k}}&{} {B_{3,k}} \\ \end{array}}} \right] ^{\mathrm{T}},\end{aligned}$$

(31)

$$\begin{aligned} \varvec{\varOmega }_k^{\varvec{D}}&= {\varvec{D}}\varvec{\varOmega }_k {\varvec{D}}^{\mathrm{T}}=\textit{diag}\left\{ {\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {\varOmega _{k,1}^D}&{} {{\varOmega }_{k,2}^D}&{} {{\varOmega }_{k,3}^D} \\ \end{array}}} \right] } \right\} . \end{aligned}$$

(32)

Combine with the formula (15), we have the following

$$\begin{aligned} \hat{{c}}_{11}&= B_{k,1} \delta _{s,k}^2 B_{k,1}^{\mathrm{T}}+{\varOmega }_{k,1}^D,\end{aligned}$$

(33)

$$\begin{aligned} \hat{{c}}_{22}&= B_{k,2} \delta _{s,k}^2 B_{k,2}^{\mathrm{T}}+{\varOmega }_{k,2}^D,\end{aligned}$$

(34)

$$\begin{aligned} \hat{{x}}_2&= B_{k,1} \delta _{s,k}^2 B_{k,3}^{\mathrm{T}},\end{aligned}$$

(35)

$$\begin{aligned} \hat{{x}}_3&= B_{k,2} \delta _{s,k}^2 {B_{k,1}}^{\mathrm{T}},\end{aligned}$$

(36)

$$\begin{aligned} \hat{{x}}_4&= B_{k,2} \delta _{s,k}^2 {B_{k,3}}^{\mathrm{T}}. \end{aligned}$$

(37)

Therefore, we obtain

$$\begin{aligned} {\varOmega }_{k,1}^D&= \hat{{c}}_{11} -B_{k,1} \delta _{s,k}^2 B_{k,1} ^{\mathrm{T}} \nonumber \\&= \hat{{c}}_{11} -B_{k,1} \delta _{s,k}^2 {B_{k,1}}^{\mathrm{T}}-\hat{{x}}_2 \hat{{x}}_4^{-1}\hat{{x}}_3 +\hat{{x}}_2 \hat{{x}}_4^{-1}\hat{{x}}_3 \nonumber \\&= \hat{{c}}_{11} -\hat{{x}}_2 \hat{{x}}_4^{-1}\hat{{x}}_3 -B_{k,1} \delta _{s,k}^2 B_{k,1}^{\mathrm{T}}+B_{k,1} \delta _{s,k}^2 B_{k,3} ^{\mathrm{T}}\left( {B_{k,2} \delta _{s,k}^2 B_{k,3}^{\mathrm{T}}} \right) ^{-1}B_{k,2} \delta _{s,k}^2 B_{k,1}^\mathrm{T} \nonumber \\&= \hat{{c}}_{11} -\hat{{x}}_2 \hat{{x}}_4^{-1}\hat{{x}}_3 \end{aligned}$$

(38)

Similarly,

$$\begin{aligned} {\varOmega }_{k,2}^D =\hat{{c}}_{22} -\hat{{x}}_3 \hat{{x}}_5 ^{-1}\hat{{x}}_6. \end{aligned}$$

(39)

As and , ${\varOmega }_{\mathrm{k,1}} ={\varOmega }_{k,1}^D$ and ${\varOmega }_\mathrm{k,2} ={\varOmega }_{k,2}^D$.

Also, as ${{\varvec{a}}}(\varvec{\varTheta }_k )=\left[ {{\begin{array}{l} {a_1 (\varvec{\varTheta }_k )} \\ {a_2 (\varvec{\varTheta }_k )} \\ {{{\varvec{a}}}_3 (\varvec{\varTheta }_k )_{2\times 1}} \\ \end{array}}} \right]$, $C_k ={{\varvec{a}}}(\varvec{\varTheta }_k )\delta _{s,k}^2 {{\varvec{a}}}(\varvec{\varTheta }_k )^{\mathrm{T}}+\hat{\varvec{\varOmega }}_k$ can be developed and expressed as formula (16), where

$$\begin{aligned} \hat{{c}}_1&= a_2 ({\varvec{\varTheta }}_k )\delta _{\mathrm{s,k}}^2 a_1 (\varvec{\varTheta }_k )^{\mathrm{T}},\end{aligned}$$

(40)

$$\begin{aligned} \hat{{\varvec{c}}}_2&= {{\varvec{a}}}_3 ({\varvec{\varTheta }}_k )\delta _{\mathrm{s,k}}^2 a_1 (\varvec{\varTheta }_k )^{\mathrm{T}},\end{aligned}$$

(41)

$$\begin{aligned} \hat{{\varvec{c}}}_3&= a_2 (\varvec{\varTheta }_k )\delta _{\mathrm{s,k}}^2 {{\varvec{a}}}_3 (\varvec{\varTheta }_k )^{\mathrm{T}},\end{aligned}$$

(42)

$$\begin{aligned} \hat{{\varvec{c}}}_4&= {{\varvec{a}}}_3 (\varvec{\varTheta }_k )\delta _{\mathrm{s,k}}^2 {{\varvec{a}}}_3 (\varvec{\varTheta }_k )^{\mathrm{T}}+\varvec{\varOmega }_{k,3}. \end{aligned}$$

(43)

Here we just provide the elements of formula (16) above because other elements is irrelevant to estimate the noise covariance matrix $\hat{\varvec{\varOmega }}_k$.

Based on the previous assumptions, the following is easily derived by

$$\begin{aligned} \varvec{\varOmega }_{k,3}&= \hat{{\varvec{c}}}_4 -{{\varvec{a}}}_3 (\varvec{\varTheta }_k )\delta _{\mathrm{s,k}}^2 {{\varvec{a}}}_3 (\varvec{\varTheta }_k )^{\mathrm{T}} \nonumber \\&= \hat{{\varvec{c}}}_4 -\hat{{\varvec{c}}}_2 \hat{{c}}_1^{-1} \hat{{\varvec{c}}}_3 +\hat{{\varvec{c}}}_2 \hat{{c}}_1 ^{-1}\hat{{\varvec{c}}}_3 -{{\varvec{a}}}_3 (\varvec{\varTheta }_k )\delta _{\mathrm{s,k}}^2 {{\varvec{a}}}_3 (\varvec{\varTheta }_k )^{\mathrm{T}} \nonumber \\&= \hat{{\varvec{c}}}_4 -\hat{{\varvec{c}}}_2 \hat{{c}}_1^{-1} \hat{{\varvec{c}}}_3 +{{\varvec{a}}}_3 (\varvec{\varTheta }_k )\delta _{\mathrm{s,k}}^2 a_1 (\varvec{\varTheta }_k )^{\mathrm{T}}\left[ {a_2 (\varvec{\varTheta }_k )\delta _{\mathrm{s,k}}^2 a_1 (\varvec{\varTheta }_k )^{\mathrm{T}}} \right] ^{-1}a_2 ({\varvec{\varTheta }}_k )\delta _{\mathrm{s,k}}^2 {{\varvec{a}}}_3 (\varvec{\varTheta }_k )^{\mathrm{T}}\nonumber \\&-\,{{\varvec{a}}}_3 (\varvec{\varTheta }_k )\delta _{\mathrm{s,k}}^2 {{\varvec{a}}}_3 (\varvec{\varTheta }_k )^{\mathrm{T}} \nonumber \\&= \hat{{\varvec{c}}}_4 -\hat{{\varvec{c}}}_2 \hat{{c}}_1^{-1} \hat{{\varvec{c}}}_3 \end{aligned}$$

(44)

Thus, estimation of noise covariance matrix $\hat{\varvec{\varOmega }}_k$ is worked out.

Appendix 2

In this appendix, we present a brief sketch of the algorithm presented in Levin et al. (2011) for MP method. And in IMP method, we pre-whiten the AVS received data and fix the weighted parameter $\alpha _{k}^{p}$ to 0.5, the MP method is shown as follows: