Source Localization in a Range-Dependent Waveguide with Incomplete Information on Spatial Variability of the Propagation Medium

Abstract

The problem of acoustic source localization in a range-dependent waveguide with interaction of normal waves, which leads to their amplitude coupling, is considered. For the given scenario, an adaptive reduced-rank algorithm is constructed, taking into account the difference between the expected signal replica and the true one and does not require knowledge of the corresponding coupling elements. By statistical modeling, it is found that the proposed estimation method gives a significant advantage both in the accuracy of coordinate measurement and in the probability of correct localization compared to the traditional MUSIC method, which does not take into account the effects of mode interaction.

APPENDIX

THE CRAMER–RAO BOUND

For the chosen problem statement, sample observation vector ${{{\mathbf{x}}}_{l}}$ is a complex Gaussian vector with zero mean and covariance matrix Γ_x determined by Eq. (7). Channel transfer function ${\mathbf{G}}({\boldsymbol{\mathbf{\theta }}})$ appearing in Eq. (7) depends not only on the desired parameter ${\boldsymbol{\mathbf{\theta }}} = {{({\boldsymbol{\mathbf{\theta }}}_{1}^{T}, \ldots ,{\boldsymbol{\mathbf{\theta }}}_{J}^{T})}^{T}},$ but also on set of vectors $\{ {{{\mathbf{c}}}_{j}}\} _{{j = 1}}^{J}$ consisting of the elements of the corresponding coupling matrices. In its turn, as mentioned above, any vector ${{{\mathbf{c}}}_{j}}$ can only be found accurate to an arbitrary complex factor. Consequently, without loss of generality, one of the elements of each individual vector ${{{\mathbf{c}}}_{j}}$ (e.g., the first one) can be considered to be fixed.

Let ${\mathbf{\alpha }}$ be a vector containing $2PJ$ unknown real parameters of the transfer matrix:

$${\mathbf{\alpha }} = [{{{\boldsymbol{\mathbf{\theta }}}}^{T}},{{{\mathbf{\eta }}}^{T}}]{{{\kern 1pt} }^{T}},$$

where ${\mathbf{\eta }} = {{({{{\mathbf{\xi }}}^{T}},{{{\mathbf{\zeta }}}^{T}})}^{T}},$ and quantities ${\mathbf{\xi }} = {{({\mathbf{\xi }}_{2}^{T} \cdots ,{\mathbf{\xi }}_{P}^{T})}^{T}}$ and ${\mathbf{\zeta }} = {{({\mathbf{\zeta }}_{2}^{T}, \cdots ,{\mathbf{\zeta }}_{P}^{T})}^{T}}$ are $(P - 1)J \times 1$ vectors of the following form:

$$\begin{gathered} {{{\mathbf{\xi }}}_{p}} = [\operatorname{Re} \{ {{c}_{p}}({{{\boldsymbol{\mathbf{\theta} }}}_{1}})\} , \ldots ,\operatorname{Re} \{ {{c}_{p}}({{{\boldsymbol{\mathbf{\theta }}}}_{J}})\} ]{{{\kern 1pt} }^{T}}, \\ {{{\mathbf{\zeta }}}_{p}} = [\operatorname{Im} \{ {{c}_{p}}({{{\boldsymbol{\mathbf{\theta} }}}_{1}})\} , \ldots ,\operatorname{Im} \{ {{c}_{p}}({{{\boldsymbol{\mathbf{\theta }}}}_{J}})\} ]{{{\kern 1pt} }^{T}}, \\ p = 2, \ldots ,P. \\ \end{gathered} $$

The covariance matrix of measurement error for vector ${\mathbf{\alpha }}$ satisfies the inequality

$$ < ({\mathbf{\alpha }} - {\mathbf{\hat {\alpha }}}){{({\mathbf{\alpha }} - {\mathbf{\hat {\alpha }}})}^{T}} \geqslant {\mathbf{CRB}}({\mathbf{\alpha }}),$$

where ${\mathbf{CRB}}\,({\mathbf{\alpha }})$ is the lower Cramer–Rao bound, which determines the potentially attainable accuracy for the estimate of this vector. In the case of random signal reception against white noise, the corresponding bound does not depend on the measurement accuracy of both noise level and elements of signal matrix ${\mathbf{S}}$ and is given by the well-known formula [10, 12]:

$$\begin{gathered} {{[{\mathbf{CR}}{{{\mathbf{B}}}^{{ - 1}}}({\mathbf{\alpha }})]}_{{pq}}} = \frac{{2L}}{{{{\sigma }}_{n}^{2}}}\operatorname{Re} \left\{ {Tr{\kern 1pt} {\kern 1pt} \left( {{\mathbf{W}}\frac{{\partial {\kern 1pt} {{{\mathbf{G}}}^{ + }}}}{{\partial {{\alpha }_{q}}}}{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }\frac{{\partial {\kern 1pt} {\mathbf{G}}}}{{\partial {{\alpha }_{p}}}}} \right)} \right\}, \\ p,q = 1, \ldots ,2PJ, \\ \end{gathered} $$

(A2)

where ${\mathbf{W}}{\kern 1pt} \,\, = {\mathbf{S}}{{{\mathbf{G}}}^{ + }}{\mathbf{\Gamma }}_{{\mathbf{x}}}^{{ - 1}}{\mathbf{GS}} \in {{C}^{{J \times J}}},$ and ${\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot } = {\mathbf{I}} - {\mathbf{G}}{{({{{\mathbf{G}}}^{ + }}{\mathbf{G}})}^{{ - 1}}}{{{\mathbf{G}}}^{ + }} \in {{C}^{{N \times N}}}.$ Calculating the derivatives involved in (A2) and introducing the notation

$$\begin{gathered} {{{\mathbf{D}}}_{{\mathbf{r}}}} = \left[ {{\mathbf{U}}{{{\mathbf{C}}}_{1}}\frac{{\partial {\kern 1pt} {\mathbf{a}}({{{\boldsymbol{\mathbf{\theta}}}}_{1}})}}{{\partial {{r}_{1}}}}, \ldots ,{\mathbf{U}}{{{\mathbf{C}}}_{J}}\frac{{\partial {\kern 1pt} {\mathbf{a}}({{{\boldsymbol{\mathbf{\theta }}}}_{J}})}}{{\partial {{r}_{J}}}}} \right]{\kern 1pt} {\kern 1pt} , \\ {{{\mathbf{D}}}_{{\mathbf{z}}}} = \left[ {{\mathbf{U}}{{{\mathbf{C}}}_{1}}\frac{{\partial {\kern 1pt} {\mathbf{a}}({{{\boldsymbol{\mathbf{\theta} }}}_{1}})}}{{\partial {{z}_{1}}}}, \ldots ,{\mathbf{U}}{{{\mathbf{C}}}_{J}}\frac{{\partial {\kern 1pt} {\mathbf{a}}({{{\boldsymbol{\mathbf{\theta} }}}_{J}})}}{{\partial {{z}_{J}}}}} \right]{\kern 1pt} {\kern 1pt} , \\ \end{gathered} $$

${{{\mathbf{D}}}_{p}} = \left[ {{\mathbf{UT}}({{{\boldsymbol{\mathbf{\theta }}}}_{1}}){\kern 1pt} {\kern 1pt} {{{\mathbf{e}}}_{p}}, \ldots ,{\mathbf{UT}}({{{\boldsymbol{\mathbf{\theta}}}}_{J}}){\kern 1pt} {\kern 1pt} {{{\mathbf{e}}}_{p}}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} ,$ where $p = 2, \ldots ,P,$ and ${{{\mathbf{e}}}_{p}}$ represents the $p$th column of the $P \times P$ identity matrix, for ${\mathbf{CRB}}\,({\mathbf{\alpha }})$ we obtain

$${\mathbf{CRB}}{\kern 1pt} {\kern 1pt} ({\mathbf{\alpha }}) = \frac{{{{\sigma }}_{n}^{2}}}{{2L}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {{\left( {\begin{array}{*{20}{c}} {{{{\mathbf{J}}}_{{{\mathbf{\theta \theta }}}}}}&{{{{\mathbf{J}}}_{{{\mathbf{\theta \eta }}}}}} \\ {{\mathbf{J}}_{{{\mathbf{\theta \eta }}}}^{T}}&{{{{\mathbf{J}}}_{{{\mathbf{\eta \eta }}}}}} \end{array}} \right)}^{{ - 1}}}.$$

(A3)

Here, $\;{{{\mathbf{J}}}_{{{\mathbf{\theta \theta }}}}} = \operatorname{Re} \{ {\mathbf{F}}\} \,,$${{{\mathbf{J}}}_{{{\mathbf{\theta \eta }}}}} = [\operatorname{Re} \{ {\mathbf{K}}\} - \operatorname{Im} \{ {\mathbf{K}}\} ],$ ${{{\mathbf{J}}}_{{{\mathbf{\eta \eta }}}}} = \left( {\begin{array}{*{20}{c}} {\operatorname{Re} \{ {\mathbf{\Sigma }}\} }&{ - \operatorname{Im} \{ {\mathbf{\Sigma }}\} } \\ {\operatorname{Im} \{ {\mathbf{\Sigma }}\} }&{\operatorname{Re} \{ {\mathbf{\Sigma }}\} } \end{array}} \right),$ where

$${\mathbf{F}} = \operatorname{Re} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {\begin{array}{*{20}{c}} {{\mathbf{D}}_{{\mathbf{r}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{{\mathbf{r}}}} \circ {{{\mathbf{W}}}^{T}}}&{{\mathbf{D}}_{{\mathbf{r}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{{\mathbf{z}}}} \circ {{{\mathbf{W}}}^{T}}} \\ {{\mathbf{D}}_{{\mathbf{z}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{{\mathbf{r}}}} \circ {{{\mathbf{W}}}^{T}}}&{{\mathbf{D}}_{{\mathbf{z}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{{\mathbf{z}}}} \circ {{{\mathbf{W}}}^{T}}} \end{array}} \right){\kern 1pt} \in {{C}^{{2J \times 2J}}},$$

$${\mathbf{{\rm K}}} = \left( {\begin{array}{*{20}{c}} {{\mathbf{D}}_{{\mathbf{r}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{2}} \circ {{{\mathbf{W}}}^{T}} \ldots {\mathbf{D}}_{{\mathbf{r}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{P}} \circ {{{\mathbf{W}}}^{T}}} \\ {{\mathbf{D}}_{{\mathbf{z}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{2}} \circ {{{\mathbf{W}}}^{T}} \ldots {\mathbf{D}}_{{\mathbf{z}}}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{P}} \circ {{{\mathbf{W}}}^{T}}} \end{array}} \right) \in {{C}^{{2J \times (P - 1)J}}},$$

$$\begin{gathered} {\mathbf{\Sigma }} = \left( {\begin{array}{*{20}{c}} {{\mathbf{D}}_{2}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{2}} \circ {{{\mathbf{W}}}^{T}}}& \cdots &{{\mathbf{D}}_{2}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{P}} \circ {{{\mathbf{W}}}^{T}}} \\ \vdots & \ddots & \vdots \\ {{\mathbf{D}}_{P}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{2}} \circ {{{\mathbf{W}}}^{T}}}& \cdots &{{\mathbf{D}}_{P}^{ + }{\mathbf{\Pi }}_{{\mathbf{G}}}^{ \bot }{{{\mathbf{D}}}_{P}} \circ {{{\mathbf{W}}}^{T}}} \end{array}} \right) \\ \in {{C}^{{(P - 1)J \times (P - 1)J}}}, \\ \end{gathered} $$

and the symbol $\circ$ indicates the Hadamard product (or element-by-element multiplication of matrices ${{[{\mathbf{A}} \circ {\mathbf{B}}]}_{{ij}}} = {{[{\mathbf{A}}]}_{{ij}}}{{[{\mathbf{B}}]}_{{ij}}}$).

Below we will be interested in the accuracy limit of measuring the spatial positions of sources. The corresponding $2J \times 2J$ error matrix is determined by inversion of the upper left block in Eq. (A3):

$$\begin{gathered} {\mathbf{CRB}}({\mathbf{\theta }}) = \frac{{{{\sigma }}_{n}^{2}}}{{2L}}{{({{{\mathbf{J}}}_{{{\mathbf{\theta \theta }}}}} - {{{\mathbf{J}}}_{{{\mathbf{\theta \eta }}}}}{\kern 1pt} {\mathbf{J}}_{{{\mathbf{\eta \eta }}}}^{{ - 1}}{\mathbf{J}}_{{{\mathbf{\theta \eta }}}}^{T})}^{{ - {\kern 1pt} {\kern 1pt} 1}}} \\ = \frac{{{{\sigma }}_{n}^{2}}}{{2L}}\operatorname{Re} {{({\mathbf{F}} - {\mathbf{{\rm K}}}{\kern 1pt} {{{\mathbf{\Sigma }}}^{{ - 1}}}{{{\mathbf{K}}}^{ + }})}^{{ - {\kern 1pt} {\kern 1pt} 1}}}. \\ \end{gathered} $$

(A4)

Expression (A4) is obtained using the result of [11], according to which

$$\begin{gathered} \text{[}\operatorname{Re} \{ {\mathbf{K}}\} - \operatorname{Im} \{ {\mathbf{K}}\} ]\,\,{\kern 1pt} \\ \times \,\,{\kern 1pt} {{\left( {\begin{array}{*{20}{c}} {\operatorname{Re} \{ {\mathbf{\Sigma }}\} }&{ - \operatorname{Im} \{ {\mathbf{\Sigma }}\} } \\ {\operatorname{Im} \{ {\mathbf{\Sigma }}\} }&{\operatorname{Re} \{ {\mathbf{\Sigma }}\} } \end{array}} \right)}^{{ - 1}}}{\kern 1pt} \left[ {\begin{array}{*{20}{c}} {\operatorname{Re} {{{\{ {\mathbf{K}}\} }}^{T}}} \\ {\operatorname{Im} {{{\{ {\mathbf{K}}\} }}^{T}}} \end{array}} \right] = \operatorname{Re} ({\mathbf{{\rm K}}}{{{\mathbf{\Sigma }}}^{{ - 1}}}{{{\mathbf{{\rm K}}}}^{ + }}). \\ \end{gathered} $$