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Robust Direct position determination against sensor gain and phase errors with the use of calibration sources

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Abstract

The direct position determination (DPD) method can provide high localization performance than conventional two-step localization methods. However, the existing DPD methods only consider the scenario of parameters of the receiving arrays, and the localization performance decreases dramatically when the array model is inaccurate in practice. This paper studies the problem for positioning a stationary emitter in the presence of sensor gain and phase errors (SGPEs) aided by calibration sources. To remove these negative effects caused by SGPEs, calibration sources with known positions are introduced. The extended relationship between parameters of calibration sources and errors is used to establish a structural objective function based on the maximum likelihood estimate. The calibration parameters are jointly optimized with target-related parameters and an alternating iterative algorithm is then developed to decouple the multidimensional search into several low-dimensional optimizations. We also derive the Cramér–Rao bound (CRB) to evaluate the performance of the proposed method. Simulation results demonstrate that the proposed method outperforms the existing DPD methods and two-step methods, which incorporates the error information, and the accuracy attains the associated CRB.

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Authors and Affiliations

  1. PLA Strategic Support Force Information Engineering University, Zhengzhou, 450001, People’s Republic of China

    Zeyu Yang, Ding Wang, Bin Yang & Fushan Wei

  2. National Digital Switching System Engineering and Technology Research Center, Zhengzhou, 450002, People’s Republic of China

    Zeyu Yang, Ding Wang & Bin Yang

  3. State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhengzhou, 450001, People’s Republic of China

    Fushan Wei

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  1. Zeyu Yang
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Appendix

Appendix

The partial derivatives with respect to the parameters in (42) are given by

$$ \frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }} = \left[ {\begin{array}{*{20}c} {\varvec{O}_{M \times D} } & {b_{1}^{{(\text{e})}}\varvec{\varGamma}_{1} \left(\frac{{\partial \varvec{a}_{1}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} )}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }} \cdot e^{{ - \text{j}\omega_{k} \tau_{1,1} (\varvec{p}_{{}}^{{(\text{e})}} )}} - \text{j}\omega_{k} e^{{ - \text{j}\omega_{k} \tau_{1,1} (\varvec{p}_{{}}^{{(\text{e})}} )}} \cdot \varvec{a}_{1}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} ) \cdot \frac{{\partial \tau_{1,1} (\varvec{p}_{{}}^{{(\text{e})}} )}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }}\right)} \\ {\varvec{O}_{M \times D} } & {b_{2}^{{(\text{e})}}\varvec{\varGamma}_{2} \left(\frac{{\partial \varvec{a}_{2}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} )}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }} \cdot e^{{ - \text{j}\omega_{k} \tau_{2,1} (\varvec{p}_{{}}^{{(\text{e})}} )}} - \text{j}\omega_{k} e^{{ - \text{j}\omega_{k} \tau_{2,1} (\varvec{p}_{{}}^{{(\text{e})}} )}} \cdot \varvec{a}_{2}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} ) \cdot \frac{{\partial \tau_{2,1} (\varvec{p}_{{}}^{{(\text{e})}} )}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }}\right)} \\ \vdots & \vdots \\ {\varvec{O}_{M \times D} } & {b_{Q}^{{(\text{e})}}\varvec{\varGamma}_{Q} \left(\frac{{\partial \varvec{a}_{Q}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} )}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }} \cdot e^{{ - \text{j}\omega_{k} \tau_{Q,1} (\varvec{p}_{{}}^{{(\text{e})}} )}} - \text{j}\omega_{k} e^{{ - \text{j}\omega_{k} \tau_{Q,1} (\varvec{p}_{{}}^{{(\text{e})}} )}} \cdot \varvec{a}_{Q}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} ) \cdot \frac{{\partial \tau_{Q,1} (\varvec{p}_{{}}^{{(\text{e})}} )}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }}\right)} \\ \end{array} } \right],\;\;(i = 1,2,3). $$
(A.1)

where

$$ \frac{{\partial \tau_{q,1} (\varvec{p}_{{}}^{{(\text{e})}} )}}{{\partial \left\langle {\varvec{p}_{{}}^{{(\text{e})}} } \right\rangle_{i} }} = \frac{1}{c}\left( {\frac{{\left\langle {\varvec{p}_{{}}^{{(\text{e})}} - \varvec{s}_{q} } \right\rangle_{i} }}{{||\varvec{p}_{{}}^{{(\text{e})}} - \varvec{s}_{q} | |}} - \frac{{\left\langle {\varvec{p}_{{}}^{{(\text{e})}} - \varvec{s}_{ 1} } \right\rangle_{i} }}{{||\varvec{p}_{{}}^{{(\text{e})}} - \varvec{s}_{1} | |}}} \right). $$
(A.2)

The derivative of the array response \( \varvec{a}_{l}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} ) \) with respect to \( \varvec{p}_{{}}^{{(\text{e})}} \) can be obtained from the array geometries directly.

$$ \begin{aligned} \hfill \\ \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}l} {\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Re} (\tilde{\varvec{b}}^{{(\text{e})}} )} \right\rangle_{q - 1} }} = \left[ {\begin{array}{*{20}c} {\varvec{O}_{(M(q - 1)) \times D} } & {\varvec{O}_{(M(q - 1)) \times 1} } \\ {\varvec{O}_{M \times D} } & {e^{{ - \text{j}\omega_{k} \tau_{q,1} (\varvec{p}_{{}}^{{(\text{e})}} )}} \cdot\varvec{\varGamma}_{q} \varvec{a}_{q}^{{(\text{e})}} (\varvec{p}_{{}}^{{(\text{e})}} )} \\ {\varvec{O}_{(M(Q - q)) \times D} } & {\varvec{O}_{(M(Q - q)) \times 1} } \\ \end{array} } \right],} \hfill \\ {\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Im} (\tilde{\varvec{b}}^{{(\text{e})}} )} \right\rangle_{q - 1} }} = \text{j}\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Re} (\tilde{\varvec{b}}^{{(\text{e})}} )} \right\rangle_{q - 1} }}} \hfill \\ \end{array} } & {\quad (q = 2,3, \ldots ,Q).} \\ \end{array} } \right. \hfill \\ \end{aligned} $$
(A.3)
$$ \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}l} {\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Re} (\tilde{\varvec{b}}_{{}}^{{(\text{c})}} )} \right\rangle_{(d - 1)(Q - 1) + (q - 1)} }}} \\{\qquad= \left[ {\begin{array}{*{20}c} {\varvec{O}_{(M(q - 1)) \times (d - 1)} } & {\varvec{O}_{(M(q - 1)) \times 1} } & {\varvec{O}_{(M(q - 1)) \times (D - d + 1)} } \\ {\varvec{O}_{M \times (d - 1)} } & {b_{q,d}^{{(\text{c})}} e^{{ - \text{j}\omega_{k} \tau_{q,1} (\varvec{p}_{D}^{{(\text{c})}} )}}\varvec{\varGamma}_{q} \varvec{a}_{q,d}^{{(\text{c})}} (\varvec{p}_{d}^{{(\text{c})}} )} & {\varvec{O}_{M \times (D - d + 1)} } \\ {\varvec{O}_{(M(Q - q)) \times (d - 1)} } & {\varvec{O}_{(M(Q - q)) \times 1} } & {\varvec{O}_{(M(Q - q)) \times (D - d + 1)} } \\ \end{array} } \right]} \hfill \\ {\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Im} (\tilde{\varvec{b}}_{{}}^{{(\text{c})}} )} \right\rangle_{(d - 1)(Q - 1) + (q - 1)} }} = \text{j}\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Re} (\tilde{\varvec{b}}^{{(\text{e})}} )} \right\rangle_{(d - 1)(Q - 1) + (q - 1)} }},} \hfill \\ \end{array} } & {\quad \left( \begin{aligned} d = 1,2, \ldots ,D \hfill \\ q = 2,3, \ldots ,Q \hfill \\ \end{aligned} \right)} \\ \end{array} } \right. $$
(A.4)
$$ \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}l} {\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Re} (\tilde{\varvec{\rho }})} \right\rangle_{(q - 1)(M - 1) + (m - 1)} }} = \left[ {\begin{array}{*{20}c} {\varvec{O}_{(M(q - 1)) \times (D + 1)} } \\ {\varvec{i}_{M}^{(m)} \varvec{A}_{q}^{{}} {\varvec{\Upsilon}}_{q} (k)} \\ {\varvec{O}_{(M(Q - q)) \times (D + 1)} } \\ \end{array} } \right]} \hfill \\ {\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Im} (\tilde{\varvec{\rho }})} \right\rangle_{(q - 1)(M - 1) + (m - 1)} }} = \text{j}\frac{{\partial {\varvec{\Psi}}(k)}}{{\partial \left\langle {\text{Re} (\tilde{\varvec{\rho }})} \right\rangle_{(q - 1)(M - 1) + (m - 1)} }},} \hfill \\ \end{array} } & {\quad \left( \begin{aligned} m = 2,3, \ldots ,M \hfill \\ q = 1,2, \ldots ,Q \hfill \\ \end{aligned} \right)} \\ \end{array} } \right. $$
(A.5)

After some algebraic manipulations, we can obtain the sub-blocks of the matrices in (24), (31) and (44) using the above derivations.

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Yang, Z., Wang, D., Yang, B. et al. Robust Direct position determination against sensor gain and phase errors with the use of calibration sources. Multidim Syst Sign Process 31, 1435–1468 (2020). https://doi.org/10.1007/s11045-020-00716-1

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