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Performance analysis and improvement of direct position determination based on Doppler frequency shifts in presence of model errors: case of known waveforms

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Abstract

The direct position determination (DPD) method based on Doppler frequency shifts for signals with known waveforms is first proposed by Amar and Weiss. Although this method exhibits excellent asymptotic performance, but does not account for the effects of uncertainties in the receiver positions and velocities. These uncertainties may result in a considerable reduction of localization accuracy. In this paper, the statistical performance of the DPD estimator is investigated under receiver position and velocity errors (also called model errors). We derive an analytical expression for the mean square error in the estimated source location in the case where the estimator assumes that the receiver positions and velocities are accurate, but they in fact contain errors. The main difficulty in the mathematical analysis is that the DPD cost function is not explicit with respect to the emitter position. Consequently, some algebraic manipulation is required to derive closed-form expressions for the first- and second-order partial derivatives of the cost function. The Cramér–Rao bounds (CRBs) for the target position estimation are also deduced in the presence and absence of model errors. These CRBs provide insights into the effects of model errors on the localization accuracy. Two improved DPD methods are developed based on our analysis results. The first has lower complexity than the common grid search, and the second exhibits increased robustness to model errors. Simulation results support and corroborate the theoretical developments in this paper.

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Acknowledgements

The authors would like to thank all the anonymous reviewers for their valuable comments and suggestions. The author also acknowledges support from National Natural Science Foundation of China (Grant Nos. 61201381, 61401513 and 61772548), China Postdoctoral Science Foundation (Grant No. 2016M592989), the Self-Topic Foundation of Information Engineering University (Grant No. 2016600701), and the Outstanding Youth Foundation of Information Engineering University (Grant No. 2016603201).

Author information

Authors and Affiliations

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

    Ding Wang, Jiexin Yin, Ruirui Liu, Hongyi Yu & Yunlong Wang

  2. Zhengzhou Information Science and Technology Institute, Zhengzhou, 450002, Henan, People’s Republic of China

    Ding Wang, Jiexin Yin, Ruirui Liu, Hongyi Yu & Yunlong Wang

Authors
  1. Ding Wang
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  2. Jiexin Yin
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  3. Ruirui Liu
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  4. Hongyi Yu
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Correspondence to Jiexin Yin.

Appendices

Appendix A: Proof of Proposition 1

A second-order Taylor-series expansion of \( \phi ({\hat{\varvec{q}}},{\varvec{\xi}}) \) around \( {\hat{\varvec{q}}} = {\varvec{q}} \) and \( {\varvec{\xi}} = {\varvec{O}}_{M \times 1} \) produces

$$ \begin{array}{l} \phi ({\hat{\varvec{q}}},{\varvec{\xi}}) = \phi ({\varvec{q}},{\varvec{O}}_{M \times 1}) + {\dot{\varvec{\varphi}}}_{1}^{\text{T}} ({\varvec{q}},{\varvec{O}}_{M \times 1}) \cdot {\varvec{\delta q}} + {\dot{\varvec{\varphi}}}_{2}^{\text{T}} ({\varvec{q}},{\varvec{O}}_{M \times 1}){\varvec{\xi}} + \frac{1}{2} \cdot ({\varvec{\delta q}})^{\text{T}} \cdot {\varvec{\ddot{\varPhi}}}_{1} ({\varvec{q}},{\varvec{O}}_{M \times 1}) \cdot {\varvec{\delta q}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + ({\varvec{\delta q}})^{\text{T}} \cdot {\varvec{\ddot{\varPhi}}}_{2} ({\varvec{q}},{\varvec{O}}_{M \times 1}){\varvec{\xi}} + \frac{1}{2} \cdot {\varvec{\xi}}^{\text{T}} {\varvec{\ddot{\varPhi}}}_{3} ({\varvec{q}},{\varvec{O}}_{M \times 1}){\varvec{\xi}} + o(||{\varvec{\xi}}||_{2}^{2}) \hfill \\ \end{array} $$
(A.1)

where

$$ {\dot{\varvec{\varphi}}}_{1} ({\varvec{q}},{\varvec{O}}_{M \times 1}) = \frac{{\partial \phi ({\varvec{q}},{\varvec{O}}_{M \times 1})}}{{\partial {\varvec{q}}}}\,\,,\,\,{\dot{\varvec{\varphi}}}_{2} ({\varvec{q}},{\varvec{O}}_{M \times 1}) = \left. {\frac{{\partial \phi ({\varvec{q}},{\varvec{\xi}})}}{{\partial {\varvec{\xi}}}}} \right|_{{{\varvec{\xi}} = {\varvec{O}}_{M \times 1}}} $$
(A.2)
$$ \begin{aligned}{\varvec{\ddot{\varPhi}}}_{1} ({\varvec{q}},{\varvec{O}}_{M \times 1}) &= \frac{{\partial^{2} \phi ({\varvec{q}},{\varvec{O}}_{M \times 1})}}{{\partial {\varvec{q}}\partial {\varvec{q}}^{\text{T}}}}\,\,,\,\,{\varvec{\ddot{\varPhi}}}_{2} ({\varvec{q}},{\varvec{O}}_{M \times 1}) = \left. {\frac{{\partial^{2} \phi ({\varvec{q}},{\varvec{\xi}})}}{{\partial {\varvec{q}}\partial {\varvec{\xi}}^{\text{T}}}}} \right|_{{{\varvec{\xi}} = {\varvec{O}}_{M \times 1}}}, \\ {\varvec{\ddot{\varPhi}}}_{3} ({\varvec{q}},{\varvec{O}}_{M \times 1}) &= \left. {\frac{{\partial^{2} \phi ({\varvec{q}},{\varvec{\xi}})}}{{\partial {\varvec{\xi}}\partial {\varvec{\xi}}^{\text{T}}}}} \right|_{{{\varvec{\xi}} = {\varvec{O}}_{M \times 1}}} \end{aligned} $$
(A.3)

According to the maximum principle, we have

$$ {\dot{\varvec{\varphi}}}_{1} ({\varvec{q}},{\varvec{O}}_{M \times 1}) = \frac{{\partial \phi ({\varvec{q}},{\varvec{O}}_{M \times 1})}}{{\partial {\varvec{q}}}} = {\varvec{O}}_{D \times 1} $$
(A.4)

Then, combining (25), (A.1), and (A.4) yields

$$ {\varvec{\delta q}} \approx \arg \mathop {\max}\limits_{{{\varvec{x}} \in {\varvec{R}}^{D \times 1}}} \left\{{\frac{1}{2} \cdot {\varvec{x}}^{\text{T}} {\varvec{\ddot{\varPhi}}}_{1} ({\varvec{q}},{\varvec{O}}_{M \times 1}){\varvec{x}} + {\varvec{x}}^{\text{T}} {\varvec{\ddot{\varPhi}}}_{2} ({\varvec{q}},{\varvec{O}}_{M \times 1}){\varvec{\xi}}} \right\} $$
(A.5)

where the second- and higher-order error terms (i.e., \( o(||{\varvec{\xi}}||_{2}) \)) are neglected. It is straightforward to deduce from the second equality in (A.5) that

$$ {\varvec{\delta q}} = - {\varvec{\ddot{\varPhi}}}_{1}^{- 1} ({\varvec{q}},{\varvec{O}}_{M \times 1}){\varvec{\ddot{\varPhi}}}_{2} ({\varvec{q}},{\varvec{O}}_{M \times 1}){\varvec{\xi}} + o(||{\varvec{\xi}}||_{2}) $$
(A.6)

which proves Proposition 1.

Appendix B: Proof of Proposition 2

Define \( \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}) \) by

$$ \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}) = \arg \mathop {\max}\limits_{{\varDelta f_{k}}} \{\phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})\} $$
(B.1)

Then, it follows from the first equality in (26) that

$$ \phi ({\varvec{q}},{\varvec{\xi}}) = \sum\limits_{k = 1}^{K} {\mathop {\max}\limits_{{\varDelta f_{k}}} \{\phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})\}} = \sum\limits_{k = 1}^{K} {\phi_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})} $$
(B.2)

The partial derivative of \( \phi ({\varvec{q}},{\varvec{\xi}}) \) with respect to \( {\varvec{q}} \) is given by

$$ \begin{aligned} {\dot{\varvec{\varphi}}}_{1} ({\varvec{q}},{\varvec{\xi}}) &= \frac{{\partial \phi ({\varvec{q}},{\varvec{\xi}})}}{{\partial {\varvec{q}}}} \\ & = \sum\limits_{k = 1}^{K} {\left({\dot{\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}} + {\dot{\varvec{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})} \right)} \end{aligned} $$
(B.3)

where

$$ \dot{\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = \left. {\frac{{\partial \phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial \varDelta f_{k}}}} \right|_{{\varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}} $$
(B.4)
$$ {\dot{\varvec{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = \left. {\frac{{\partial \phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}}} \right|_{{\varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}} $$
(B.5)

Further, using (B.3) we obtain

$$ {\varvec{\ddot{\Phi}}}_{ 1} ({\varvec{q}},{\varvec{\xi}}) = \sum\limits_{k = 1}^{K} {\left(\begin{array}{l} {\varvec{\ddot{\varPhi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) + {\ddot{\phi}}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}} \cdot \left({\frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}}} \right)^{\text{T}} \\ + \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}} \cdot {\varvec{\ddot{\varphi}}}_{k,1}^{\text{T}} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) + {\varvec{\ddot{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \left({\frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}}} \right)^{\text{T}} \\ + \dot{\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \frac{{\partial^{2} \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}\partial {\varvec{q}}^{\text{T}}}} \\ \end{array} \right)} $$
(B.6)
$$ {\varvec{\ddot{\Phi}}}_{ 2} ({\varvec{q}},{\varvec{\xi}}) = \sum\limits_{k = 1}^{K} {\left(\begin{array}{l} {\varvec{\ddot{\varPhi}}}_{k,2} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) +{\ddot{\phi}}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}} \cdot \left({\frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{\xi}}_{k}}}} \right)^{\text{T}} \\ + \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}} \cdot {\varvec{\ddot{\varphi}}}_{k,2}^{\text{T}} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) + {\varvec{\ddot{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \left({\frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{\xi}}_{k}}}} \right)^{\text{T}} \\ + \dot{\phi}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \frac{{\partial^{2} \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}\partial {\varvec{\xi}}_{k}^{\text{T}}}} \\ \end{array} \right)({\varvec{i}}_{K}^{{(k){\text{T}}}} \otimes {\varvec{I}}_{{M_{0}}})} $$
(B.7)

where

$$ \left\{\begin{array}{l} {\varvec{\ddot{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = \left. {\frac{{\partial^{2} \phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial \varDelta f_{k} \partial {\varvec{q}}}}} \right|_{{\varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}} \hfill \\ {\varvec{\ddot{\varphi}}}_{k,2} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = \left. {\frac{{\partial^{2} \phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial \varDelta f_{k} \partial {\varvec{\xi}}_{k}}}} \right|_{{\varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}} \hfill \\ \end{array} \right. $$
(B.8)
$$ \left\{\begin{array}{l} {\varvec{\ddot{\varPhi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = \left. {\frac{{\partial^{2} \phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}\partial {\varvec{q}}^{\text{T}}}}} \right|_{{\varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}} \hfill \\ {\varvec{\ddot{\varPhi}}}_{k,2} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = \left. {\frac{{\partial^{2} \phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}\partial {\varvec{\xi}}_{k}^{\text{T}}}}} \right|_{{\varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}} \hfill \\ \end{array} \right. $$
(B.9)

On the other hand, combining the maximum principle with the definition of \( \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}) \) in (B.1) leads to

$$ {\ddot{\phi}}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = \left. {\frac{{\partial \phi_{k} (\varDelta f_{k},{\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial \varDelta f_{k}}}} \right|_{{\varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}} = 0 $$
(B.10)

Taking the derivative with respect to \( {\varvec{q}} \) on both sides of (B.10) yields

$$ {\ddot{\phi}}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}} + {\varvec{\ddot{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = {\varvec{O}}_{D \times 1} $$
(B.11)

which implies

$$ \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{q}}}} = - \frac{{{\varvec{\ddot{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})}}{{\ddot{\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})}} $$
(B.12)

Similarly, taking the derivative with respect to \( {\varvec{\xi}}_{k} \) on both sides of (B.10), we get

$$ {\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) \cdot \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{\xi}}_{k}}} + {\varvec{\ddot{\varphi}}}_{k,2} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) = {\varvec{O}}_{{M_{0} \times 1}} $$
(B.13)

which implies

$$ \frac{{\partial \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k})}}{{\partial {\varvec{\xi}}_{k}}} = - \frac{{{\varvec{\ddot{\varphi}}}_{k,2} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})}}{{\ddot{\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})}} $$
(B.14)

Inserting (B.10) and (B.12) into (B.6) produces

$$ {\varvec{\ddot{\varPhi}}}_{ 1} ({\varvec{q}},{\varvec{\xi}}) = \sum\limits_{k = 1}^{K} {\left({{\varvec{\ddot{\varPhi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) - \frac{{{\varvec{\ddot{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}){\varvec{\ddot{\varphi}}}_{k,1}^{\text{T}} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})}}{{\ddot{\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})^{{^{{}}}}}}} \right)} $$
(B.15)

Substituting (B.10), (B.12) and (B.14) into (B.7) yields

$$ {\varvec{\ddot{\varPhi}}}_{ 2} ({\varvec{q}},{\varvec{\xi}}) = \sum\limits_{k = 1}^{K} {\left({{\varvec{\ddot{\varPhi}}}_{k,2} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}) - \frac{{{\varvec{\ddot{\varphi}}}_{k,1} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k}){\varvec{\ddot{\varphi}}}_{k,2}^{\text{T}} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})}}{{\ddot{\phi}_{k} (\varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{\xi}}_{k}),{\varvec{q}},{\varvec{\xi}}_{k})^{{^{{}}}}}}} \right)({\varvec{i}}_{K}^{{(k){\text{T}}}} \otimes {\varvec{I}}_{{M_{0}}})} $$
(B.16)

Furthermore, note that \( \varDelta f_{k} = \varDelta f_{k}^{{({\text{o}})}} ({\varvec{q}},{\varvec{O}}_{{M_{0} \times 1}}) \). As a consequence, if the error vector \( {\varvec{\xi}} \) in (B.15) and (B.16) is replaced with \( {\varvec{O}}_{M \times 1} \), both (29) and (30) hold. This completes the proof of Proposition 2.

Appendix C: Detailed derivation of \( {\dot{\varvec{C}}}_{k,j}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) \) and \( {\varvec{\ddot{C}}}_{{k,j_{1} j_{2}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) \)

It can be easily seen from (16) that

$$ {\dot{\varvec{C}}}_{k,j}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = [({\dot{\varvec{A}}}_{j}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{1,k}))^{\text{H}} {\varvec{z}}_{1,k} \,\,\,({\dot{\varvec{A}}}_{j}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{2,k}))^{\text{H}} {\varvec{z}}_{2,k} \,\,\, \cdots \,\,\,({\dot{\varvec{A}}}_{j}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{L,k}))^{\text{H}} {\varvec{z}}_{L,k}] $$
(C.1)
$$ {\varvec{\ddot{C}}}_{{k,j_{1} j_{2}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = [({\varvec{\ddot{A}}}_{{j_{1} j_{2}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{1,k}))^{\text{H}} {\varvec{z}}_{1,k} \,\,\,({\varvec{\ddot{A}}}_{{j_{1} j_{2}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{2,k}))^{\text{H}} {\varvec{z}}_{2,k} \,\,\, \cdots \,\,\,({\varvec{\ddot{A}}}_{{j_{1} j_{2}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{L,k}))^{\text{H}} {\varvec{z}}_{L,k}] $$
(C.2)

where

$$ {\dot{\varvec{A}}}_{j}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{l,k}) = \frac{{\partial {\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{j}}}\,\,,\,\,{\varvec{\ddot{A}}}_{{j_{1} j_{2}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{l,k}) = \frac{{\partial^{2} {\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{{j_{1}}} \partial < {\varvec{q}} >_{{j_{2}}}}} $$
(C.3)

Using the first equality in (9), we have

$$ \begin{aligned} {\dot{\varvec{A}}}_{j}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{l,k}) & = ({\text{j2}}\uppi \,f_{c} T_{s}) \cdot \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{j}}} \\ & \quad \times {\text{diag[0}}\,\,\,{ \exp }\{{\text{j2}}{\uppi}\,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \,\,\,2 \cdot { \exp }\{{\text{j4}}{\uppi}\,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \,\,\, \cdots \,\,\,\\ & \qquad (N - 1) \cdot { \exp }\{{\text{j2}}{\uppi}\,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})(N - 1)T_{s} \} ]\end{aligned} $$
(C.4)

which implies

$$ \begin{aligned} \varvec{\ddot{A}}_{{j_{1} j_{2}}}^{(1)} (\varvec{q},\bar{\varvec{p}}_{l,k}) & = (\hbox{j}2\uppi \,f_{c} T_{s}) \cdot \frac{{\partial^{2} \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})}}{{\partial < \varvec{q} >_{{j_{1}}} \partial < \varvec{q} >_{{j_{2}}}}} \\ & \quad \times {\text{diag}}[0\,\,\,\exp \{\hbox{j}2\uppi \,f_{c} \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})T_{s} \} \,\,\,2 \cdot \exp \{\hbox{j}4\uppi \,f_{c} \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})T_{s} \} \\ & \qquad \cdots (N - 1) \cdot \exp \{\hbox{j}2\uppi \,f_{c} \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})(N - 1)T_{s} \}] \\ & \quad - 4\uppi^{2} f_{c}^{2} T_{s}^{2} \cdot \frac{{\partial \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})}}{{\partial < \varvec{q} >_{{j_{1}}}}} \cdot \frac{{\partial \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})}}{{\partial < \varvec{q} >_{{j_{2}}}}} \\ & \quad \times {\text{diag}}[0\,\,\,\exp \{\hbox{j}2\uppi \,f_{c} \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})T_{s} \} \,\,\,4 \cdot \exp \{\hbox{j}4\uppi \,f_{c} \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})T_{s} \} \\ & \qquad \cdots (N - 1)^{2} \cdot \exp \{\hbox{j}2\uppi \,f_{c} \gamma (\varvec{q},\bar{\varvec{p}}_{l,k})(N - 1)T_{s} \}] \\ \end{aligned} $$
(C.5)

Next, expressions for the gradient vector and Hessian matrix of \( \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k}) \) with respect to \( {\varvec{q}} \) are derived, which lead to \( \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{j}}} \) and \( \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{{j_{1}}} \partial < {\varvec{q}} >_{{j_{2}}}}} \), respectively.

First, it follows from (3) that

$$ \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\varvec{q}}}} = \frac{1}{v} \cdot \left({{\varvec{I}}_{D} - \frac{{({\varvec{q}} - {\varvec{p}}_{l,k})({\varvec{q}} - {\varvec{p}}_{l,k})^{\text{T}}}}{{||{\varvec{q}} - {\varvec{p}}_{l,k} ||_{2}^{2}}}} \right) \cdot \frac{{{\dot{\varvec{p}}}_{l,k}}}{{||{\varvec{q}} - {\varvec{p}}_{l,k} ||_{2}}} $$
(C.6)

which implies

$$ \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\varvec{q}}\partial {\varvec{q}}^{\text{T}}}} = \frac{1}{v} \cdot \left(\begin{array}{l} 3 \cdot \frac{{{\dot{\varvec{p}}}_{{l,{\kern 1pt} k}}^{\text{T}} ({\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}})}}{{||{\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}} ||_{2}^{5}}} \cdot ({\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}})({\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}})^{\text{T}} - \frac{{{\dot{\varvec{p}}}_{{l,{\kern 1pt} k}}^{\text{T}} ({\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}})}}{{||{\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}} ||_{2}^{3}}} \cdot {\varvec{I}}_{D} \\ - \frac{{({\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}}){\dot{\varvec{p}}}_{{l,{\kern 1pt} k}}^{\text{T}}}}{{||{\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}} ||_{2}^{3}}} - \frac{{{\dot{\varvec{p}}}_{{l,{\kern 1pt} k}} ({\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}})^{\text{T}}}}{{||{\varvec{q}} - {\varvec{p}}_{{l,{\kern 1pt} k}} ||_{2}^{3}}} \\ \end{array} \right) $$
(C.7)

Obviously, \( \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{j}}} \) is the jth component of \( \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\varvec{q}}}} \) and \( \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{{j_{1}}} \partial < {\varvec{q}} >_{{j_{2}}}}} \) is the (j1, j2)th element of \( \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\varvec{q}}\partial {\varvec{q}}^{\text{T}}}} \).

Appendix D: Detailed derivation of \( {\dot{\varvec{C}}}_{k,j}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) \) and \( {\varvec{\ddot{C}}}_{{k,j_{1} j_{2}}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) \)

Using (16) and the definition of the error vector \( {\varvec{\xi}}_{k} \) yields

$$ \left\{\begin{array}{l} {\dot{\varvec{C}}}_{k,j}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = {\varvec{i}}_{L}^{{(N_{1} (j)){\text{T}}}} \otimes (({\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{{N_{1} (j),k}}))^{\text{H}} {\varvec{i}}_{N}^{{(N_{2} (j))}})\quad \,\,\,\,\,{\kern 1pt} {\kern 1pt} (1 \le j \le 2NL\,;\,1 \le N_{2} (j) \le N) \\ {\dot{\varvec{C}}}_{k,j}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = {\varvec{i}}_{L}^{{(N_{1} (j)){\text{T}}}} \otimes ({\text{j(}}{\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{{N_{1} (j),k}}))^{\text{H}} {\varvec{i}}_{N}^{{(N_{3} (j))}})\quad (1 \le j \le 2NL\,;\,N + 1 \le N_{2} (j) \le 2N) \\ {\dot{\varvec{C}}}_{k,j}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = {\varvec{i}}_{L}^{{(D_{1} (j)){\text{T}}}} \otimes \left({\left({\frac{{\partial {\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j),k}})}}{{\partial < {\bar{\varvec{p}}}_{{D_{1} (j),k}} >_{{D_{2} (j)}}}}} \right)^{\text{H}} {\varvec{z}}_{{D_{1} (j),k}}} \right) \\ = {\varvec{i}}_{L}^{{(D_{1} (j)){\text{T}}}} \otimes (({\dot{\varvec{A}}}_{{D_{2} (j)}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j),k}}))^{\text{H}} {\varvec{z}}_{{D_{1} (j),k}})\quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (2NL + 1 \le j \le 2(N + D)L) \\ \end{array} \right. $$
(D.1)

where

$$ \left\{\begin{array}{l} N_{1} (j) = \left\lceil {j/2N} \right\rceil \,\,,\,\,N_{2} (j) = j - 2N(\left\lceil {j/2N} \right\rceil - 1)\,\,,\,\,N_{3} (j) = j - 2N(\left\lceil {j/2N} \right\rceil - 1) - N \hfill \\ D_{1} (j) = \left\lceil {(j - 2NL)/2D} \right\rceil \,\,,\,\,D_{2} (j) = j - 2NL - 2D(\left\lceil {(j - 2NL)/2D} \right\rceil - 1) \hfill \\ {\dot{\varvec{A}}}_{{D_{2} (j)}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j),k}}) = \frac{{\partial {\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j),k}})}}{{\partial < {\bar{\varvec{p}}}_{{D_{1} (j),k}} >_{{D_{2} (j)}}}} \hfill \\ \end{array} \right. $$
(D.2)

Furthermore, from (D.1) and the definition of \( {\varvec{\xi}}_{k} \), we have that

$$ \left\{\begin{array}{l} {\varvec{\ddot{C}}}_{{k,j_{1} j_{2}}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = {\varvec{i}}_{L}^{{(N_{1} (j_{2})){\text{T}}}} \otimes (({\dot{\varvec{A}}}_{{j_{1}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{{N_{1} (j_{2}),k}}))^{\text{H}} {\varvec{i}}_{N}^{{(N_{2} (j_{2}))}})\quad \,\,\,\,\,{\kern 1pt} {\kern 1pt} (1 \le j_{2} \le 2NL\,;\,1 \le N_{2} (j_{2}) \le N) \\ {\varvec{\ddot{C}}}_{{k,j_{1} j_{2}}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = {\varvec{i}}_{L}^{{(N_{1} (j_{2})){\text{T}}}} \otimes ({\text{j(}}{\dot{\varvec{A}}}_{{j_{1}}}^{(1)} ({\varvec{q}},{\bar{\varvec{p}}}_{{N_{1} (j_{2}),k}}))^{\text{H}} {\varvec{i}}_{N}^{{(N_{3} (j_{2}))}})\quad (1 \le j_{2} \le 2NL\,;\,N + 1 \le N_{2} (j_{2}) \le 2N) \\ {\varvec{\ddot{C}}}_{{k,j_{1} j_{2}}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{k}) = {\varvec{i}}_{L}^{{(D_{1} (j_{2})){\text{T}}}} \otimes \left({\left({\frac{{\partial^{2} {\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j_{2}),k}})}}{{\partial < {\varvec{q}} >_{{j_{1}}} \partial < {\bar{\varvec{p}}}_{{D_{1} (j_{2}),k}} >_{{D_{2} (j_{2})}}}}} \right)^{\text{H}} {\varvec{z}}_{{D_{1} (j_{2}),k}}} \right) \\ \quad \quad = {\varvec{i}}_{L}^{{(D_{1} (j_{2})){\text{T}}}} \otimes (({\varvec{\ddot{A}}}_{{j_{1},D_{2} (j_{2})}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j_{2}),k}}))^{\text{H}} {\varvec{z}}_{{D_{1} (j_{2}),k}})\quad {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (2NL + 1 \le j_{2} \le 2(N + D)L) \\ \end{array} \right. $$
(D.3)

where

$$ {\varvec{\ddot{A}}}_{{j_{1},D_{2} (j_{2})}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j_{2}),k}}) = \frac{{\partial^{2} {\varvec{A}}({\varvec{q}},{\bar{\varvec{p}}}_{{D_{1} (j_{2}),k}})}}{{\partial < {\varvec{q}} >_{{j_{1}}} \partial < {\bar{\varvec{p}}}_{{D_{1} (j_{2}),k}} >_{{D_{2} (j_{2})}}}} $$
(D.4)

Similar to the derivation in (C.4) and (C.5), it can be checked that

$$ \begin{aligned} {\dot{\varvec{A}}}_{j}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{l,k}) & = ({\text{j2}}\uppi \,f_{c} T_{s}) \cdot \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\bar{\varvec{p}}}_{l,k} >_{j}}} \\ & \quad \times {\text{diag[0}}\,\,\,{ \exp }\{{\text{j2}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \,\,\,2 \cdot { \exp }\{{\text{j4}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \\ & \qquad \cdots (N - 1) \cdot { \exp }\{{\text{j2}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})(N - 1)T_{s} \} ]\\ \end{aligned} $$
(D.5)
$$ \begin{aligned} {\varvec{\ddot{A}}}_{{j_{1} j_{2}}}^{(2)} ({\varvec{q}},{\bar{\varvec{p}}}_{l,k}) & = ({\text{j2}}\uppi \,f_{c} T_{s}) \cdot \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{{j_{1}}} \partial < {\bar{\varvec{p}}}_{l,k} >_{{j_{2}}}}} \\ & \quad \times {\text{diag[0}}\,\,\,{ \exp }\{{\text{j2}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \,\,\,2 \cdot { \exp }\{{\text{j4}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \\ & \qquad \cdots \,\,\,(N - 1) \cdot { \exp }\{{\text{j2}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})(N - 1)T_{s} \} ]\\ & \quad - 4{\uppi}^{ 2} f_{c}^{2} T_{s}^{2} \cdot \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{{j_{1}}}}} \cdot \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\bar{\varvec{p}}}_{l,k} >_{{j_{2}}}}} \\ & \quad \times {\text{diag[0}}\,\,\,{ \exp }\{{\text{j2}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \,\,\,4 \cdot { \exp }\{{\text{j4}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})T_{s} \} \\ & \qquad \cdots (N - 1)^{2} \cdot { \exp }\{{\text{j2}}\uppi \,f_{c} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})(N - 1)T_{s} \} ]\\ \end{aligned} $$
(D.6)

In the following, we derive expressions for the first-order partial derivative vector \( \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\varvec{q}}}} \) and second-order partial derivative matrix \( \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\varvec{q}}\partial {\bar{\varvec{p}}}_{l,k}^{\text{T}}}} \).

First, combining (3) and the definition of \( {\bar{\varvec{p}}}_{l,k} \) leads to

(D.7)

which implies

(D.8)

It is evident that \( \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\bar{\varvec{p}}}_{l,k} >_{j}}} \) is the jth component of \( \frac{{\partial \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\bar{\varvec{p}}}_{l,k}}} \) and \( \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial < {\varvec{q}} >_{{j_{1}}} \partial < {\bar{\varvec{p}}}_{l,k} >_{{j_{2}}}}} \) is the (j1, j2)th entry of \( \frac{{\partial^{2} \gamma ({\varvec{q}},{\bar{\varvec{p}}}_{l,k})}}{{\partial {\varvec{q}}\partial {\bar{\varvec{p}}}_{l,k}^{\text{T}}}} \).

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Wang, D., Yin, J., Liu, R. et al. Performance analysis and improvement of direct position determination based on Doppler frequency shifts in presence of model errors: case of known waveforms. Multidim Syst Sign Process 30, 749–790 (2019). https://doi.org/10.1007/s11045-018-0579-z

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