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Direct Position Determination of Multiple Constant Modulus Sources Based on Direction of Arrival and Doppler Frequency Shift

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Abstract

Direct position determination (DPD) methods are known to outperform classical two-step localization methods under the condition of low signal-to-noise ratios and/or small number of snapshots. Additionally, they can directly exploit the prior knowledge of signal waveforms to achieve higher estimation accuracy. In this paper, we concentrate on the DPD method for locating multiple constant modulus (CM) sources based on a single moving receiving station. Both direction of arrival and Doppler frequency shift are used for source localization. First, a received array signal model with a small time delay that can incorporate the Doppler frequency information is formed. Subsequently, the corresponding maximum likelihood estimation criterion is established. An effective alternating minimization algorithm is developed to solve the optimization problem. Subsequently, we also derive the Cramér–Rao bound expressions for parameter estimation. It is proved that the CM property of the signals is useful in reducing the lower bound for location estimation. Simulation results demonstrate that the proposed method is asymptotically efficient. Moreover, it is also shown that the estimation accuracy of direct localization can be greatly increased if the CM characteristics of the radiated signals are adequately exploited.

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Acknowledgements

The author acknowledges support from National Natural Science Foundation of China (Grant Nos. 61201381, 61401513 and 61772548); China Postdoctoral Science Foundation (Grant No. 2016M592989); Key Scientific and Technological Research Project in Henan Province (Grant No. 192102210092); the Self-Topic Foundation of Information Engineering University (Grant No. 2016600701); and the Outstanding Youth Foundation of Information Engineering University (Grant No. 2016603201).

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Correspondence to Jie-xin Yin.

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Appendices

Appendix 1: Detailed Derivation of (18)–(22)

The first-order derivative of the orthogonal projection matrix \( {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )] \) with respect to \( < {\mathbf{p}} >_{j} \) is given by [26, 40]:

$$ \begin{aligned}\frac{{\partial {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]}}{{\partial < {\mathbf{p}} >_{j} }} &= - {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial {\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{j} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag }\\ & - \left( {{\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial {\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{j} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } } \right)^{\text{T}}\end{aligned} $$
(63)

Using (63), the \( j \)th element of the gradient can be written as:

$$ \begin{aligned} \left\langle {{\mathbf{g}}_{ 1} ({\mathbf{p}})} \right\rangle_{j} & = \frac{{\partial J_{ 4} ({\mathbf{p}})}}{{\partial < {\mathbf{p}} >_{j} }} = - 2\sum\limits_{k = 1}^{K} {{\boldsymbol{\tilde{\bar{x}}}}_{k}^{\text{T}} {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial {\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{j} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } {\boldsymbol{\tilde{\bar{x}}}}_{k} } \\ & = 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {{\mathbf{y}}_{k}^{\text{T}} ({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{H}} \frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{j} }}{\mathbf{y}}_{k} } } \right\} - 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {{\bar{\mathbf{x}}}_{k}^{\text{H}} \frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{j} }}{\mathbf{y}}_{k} } } \right\} \\ \end{aligned} $$
(64)

Inserting the third equality in (12) into (64) yields

$$ \begin{aligned} \left\langle {{\mathbf{g}}_{ 1} ({\mathbf{p}})} \right\rangle_{j} = & 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} - {\bar{\mathbf{x}}}_{k} (t_{l} ))^{\text{H}} \frac{{\partial {\mathbf{B}}_{k} ({\mathbf{p}})}}{{\partial < {\mathbf{p}} >_{j} }}{\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} } } } \right\} \\ & \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} - {\bar{\mathbf{x}}}_{k} (t_{l} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{j} }}} } } \right\} \\ \end{aligned} $$
(65)

which combined with the matrix identity (III) in Table 2 leads to

$$ \begin{aligned} \left\langle {{\mathbf{g}}_{ 1} ({\mathbf{p}})} \right\rangle_{j} = & 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\left( {\sum\limits_{l = 1}^{L} {({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{ * } \otimes ({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} - {\bar{\mathbf{x}}}_{k} (t_{l} ))} } \right)^{\text{H}} \frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial < {\mathbf{p}} >_{j} }}} } \right\} \\ & \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} - {\bar{\mathbf{x}}}_{k} (t_{l} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{j} }}} } } \right\} \\ \end{aligned} $$
(66)

By performing direct algebraic manipulation, the compact expression of gradient \( {\mathbf{g}}_{ 1} ({\mathbf{p}}) \) follows.

The second-order derivative of the orthogonal projection matrix \( {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )] \) can be written as [26, 40]:

$$ \frac{{\partial^{2} {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]}}{{\partial < {\mathbf{p}} >_{{j_{1} }} \partial < {\mathbf{p}} >_{{j_{2} }} }} = {\mathbf{Z}}_{k} + {\mathbf{Z}}_{k}^{\text{T}} $$
(67)

where

$$ \begin{aligned} {\mathbf{Z}}_{k} &= {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } \frac{{\partial ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } \\ & \quad + ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{{{\dag }{\kern 1pt} {\kern 1pt} {\text{T}}}} \frac{{\partial {\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } \\ & \quad - {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial^{2} ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} \partial < {\mathbf{p}} >_{{j_{2} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } \\ & \quad - {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{{{\dag }{\kern 1pt} {\kern 1pt} {\text{T}}}} \frac{{\partial {\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )] \\ & \quad + {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } \frac{{\partial ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } \\ \end{aligned} $$
(68)

Under moderate noise level, we can ignore the terms involving \( {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]{\boldsymbol{\tilde{\bar{x}}}}_{k} \) and \( {\boldsymbol{\tilde{\bar{x}}}}_{k}^{\text{T}} {\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )] \). Then, the \( j_{1} \)\( j_{2} \)th element of the Hessian can be approximated as

$$ \begin{aligned} \left\langle {{\mathbf{G}}_{1} ({\mathbf{p}})} \right\rangle_{{j_{1} j_{2} }} & = \frac{{\partial^{2} J_{4} ({\mathbf{p}})}}{{\partial < {\mathbf{p}} >_{{j_{1} }} \partial < {\mathbf{p}} >_{{j_{2} }} }} \approx 2\sum\limits_{k = 1}^{K} {{\tilde{\mathbf{x}}}_{k}^{\text{T}} ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{{{\dag }{\kern 1pt} {\kern 1pt} {\text{T}}}} \frac{{\partial ({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{T}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}{\varvec{\Pi}}^{ \bot } [{\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )]\frac{{\partial {\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}({\tilde{\mathbf{H}}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\dag } {\tilde{\mathbf{x}}}_{k} } \\ & = 2\sum\limits_{k = 1}^{K} {\text{Re} \left\{ {{\mathbf{y}}_{k}^{\text{T}} \frac{{\partial ({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}\frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\mathbf{y}}_{k} } \right\}} \\ & \quad - 2\sum\limits_{k = 1}^{K} {\sum\limits_{d = 1}^{D} {\text{Re} \left\{ {({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ){\mathbf{h}}_{kd} )^{\text{H}} \frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}{\mathbf{y}}_{k} } \right\}\text{Re} \left\{ {{\mathbf{y}}_{k}^{\text{T}} \frac{{\partial ({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ){\mathbf{h}}_{kd} } \right\}} } \\ \end{aligned} $$
(69)

From the third equality in (12) we have

$$ \begin{aligned} & 2\sum\limits_{k = 1}^{K} {\text{Re} \left\{ {{\mathbf{y}}_{k}^{\text{T}} \frac{{\partial ({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}\frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\mathbf{y}}_{k} } \right\}} \\ & \quad = 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{\text{H}} \frac{{\partial ({\mathbf{B}}_{k} ({\mathbf{p}}))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}\frac{{\partial {\mathbf{B}}_{k} ({\mathbf{p}})}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} } } } \right\} \\ & \quad \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{\text{H}} \frac{{\partial ({\mathbf{B}}_{k} ({\mathbf{p}}))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}{\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}} } } \right\} \\ & \quad \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {\frac{{\partial ({\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} )^{\text{H}} \frac{{\partial {\mathbf{B}}_{k} ({\mathbf{p}})}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} } } } \right\} \\ & \quad \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {\frac{{\partial ({\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} )^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}} } } \right\} \\ \end{aligned} $$
(70)

which combined with the matrix identity (III) in Table 2 yields

$$ \begin{aligned} & 2\sum\limits_{k = 1}^{K} {\text{Re} \left\{ {{\mathbf{y}}_{k}^{\text{T}} \frac{{\partial ({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}\frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\mathbf{y}}_{k} } \right\}} \\ & \quad = 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\left( {\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}} \right)^{\text{H}} \left( {\left( {\sum\limits_{l = 1}^{L} {({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{ * } ({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{\text{T}} } } \right) \otimes {\mathbf{I}}_{(N + 1)M} } \right)\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}} } \right\} \\ & \quad \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\left( {\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}} \right)^{\text{H}} \left( {\sum\limits_{l = 1}^{L} {(({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{ * } \otimes {\mathbf{I}}_{(N + 1)M} ){\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}} } \right)} } \right\} \\ & \quad \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\left( {\sum\limits_{l = 1}^{L} {\frac{{\partial ({\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} )^{\text{H}} (({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{\text{T}} \otimes {\mathbf{I}}_{(N + 1)M} )} } \right)\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}} } \right\} \\ & \quad \quad + 2\text{Re} \left\{ {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {\frac{{\partial ({\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} )^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}} } } \right\} \\ \end{aligned} $$
(71)

Moreover, using the third equality in (12) leads to

$$ \begin{aligned} & 2\sum\limits_{k = 1}^{K} {\sum\limits_{d = 1}^{D} {\text{Re} \left\{ {({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ){\mathbf{h}}_{kd} )^{\text{H}} \frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}{\mathbf{y}}_{k} } \right\}\text{Re} \left\{ {{\mathbf{y}}_{k}^{\text{T}} \frac{{\partial ({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ){\mathbf{h}}_{kd} } \right\}} } \\ & \quad = 2\sum\limits_{k = 1}^{K} {\sum\limits_{d = 1}^{D} {\left( \begin{aligned} \text{Re} \left\{ {\sum\limits_{l = 1}^{L} {\left( \begin{aligned} ({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} )^{\text{H}} \frac{{\partial {\mathbf{B}}_{k} ({\mathbf{p}})}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}{\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} \\ + ({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} )^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }} \\ \end{aligned} \right)} } \right\} \\ \times \text{Re} \left\{ {\sum\limits_{l = 1}^{L} {\left( \begin{aligned} ({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{\text{H}} \frac{{\partial ({\mathbf{B}}_{k} ({\mathbf{p}}))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} \\ + \frac{{\partial ({\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} )^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} \\ \end{aligned} \right)} } \right\} \\ \end{aligned} \right)} } \\ \end{aligned} $$
(72)

which combined with the matrix identity (III) in Table 2 produces

$$ \begin{aligned} & 2\sum\limits_{k = 1}^{K} {\sum\limits_{d = 1}^{D} {\text{Re} \left\{ {({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ){\mathbf{h}}_{kd} )^{\text{H}} \frac{{\partial {\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} )}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }}{\mathbf{y}}_{k} } \right\}\text{Re} \left\{ {{\mathbf{y}}_{k}^{\text{T}} \frac{{\partial ({\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ))^{\text{H}} }}{{\partial < {\mathbf{p}} >_{{j_{2} }} }}{\mathbf{H}}_{k} ({\mathbf{p}},{\hat{\boldsymbol{\varphi }}}_{k} ){\mathbf{h}}_{kd} } \right\}} } \\ & \quad = 2\sum\limits_{k = 1}^{K} {\sum\limits_{d = 1}^{D} {\left( \begin{aligned} \text{Re} \left\{ {\sum\limits_{l = 1}^{L} {\left( \begin{aligned} (({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{ * } \otimes ({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} ))^{\text{H}} \frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }} \\ +\, ({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} )^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{{j_{1} }} }} \\ \end{aligned} \right)} } \right\} \hfill \\ \times \text{Re} \left\{ {\sum\limits_{l = 1}^{L} {\left( \begin{aligned} (({\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{y}}_{k} )^{ * } \otimes ({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} ))^{\text{H}} \frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }} \\ +\, ({\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ){\mathbf{h}}_{kd} )^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\hat{\mathbf{S}}}_{k} (t_{l} ){\mathbf{Y}}_{k} \frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial < {\mathbf{p}} >_{{j_{2} }} }} \\ \end{aligned} \right)} } \right\} \hfill \\ \end{aligned} \right)} } \\ \end{aligned} $$
(73)

Combining (69), (71), and (73), we can obtain the compact expression of Hessian \( {\mathbf{G}}_{ 1} ({\mathbf{p}}) \).

Appendix 2: Detailed Expressions for \( \frac{{\partial {\mathbf{vec}}({\mathbf{B}}_{{\mathbf{k}}} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{{\mathbf{T}}} }} \) and \( \frac{{\partial {\mathbf{r}}_{{\mathbf{k}}} ({\mathbf{p}},{\mathbf{t}})}}{{\partial {\mathbf{p}}^{{\mathbf{T}}} }} \)

It follows from the second equality in (5) that

$$ \frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }} = {\text{blkdiag}}\left[ {\frac{{\partial {\mathbf{b}}_{k} ({\mathbf{p}}_{1} )}}{{\partial {\mathbf{p}}_{1}^{\text{T}} }}\frac{{\partial {\mathbf{b}}_{k} ({\mathbf{p}}_{2} )}}{{\partial {\mathbf{p}}_{2}^{\text{T}} }} \cdots \frac{{\partial {\mathbf{b}}_{k} ({\mathbf{p}}_{D} )}}{{\partial {\mathbf{p}}_{D}^{\text{T}} }}} \right] $$
(74)

where

$$ \frac{{\partial {\mathbf{b}}_{k} ({\mathbf{p}}_{d} )}}{{\partial {\mathbf{p}}_{d}^{\text{T}} }} = {\mathbf{c}}_{k} ({\mathbf{p}}_{d} ) \otimes \frac{{\partial {\mathbf{a}}_{k} ({\mathbf{p}}_{d} )}}{{\partial {\mathbf{p}}_{d}^{\text{T}} }} + ({\mathbf{I}}_{N + 1} \otimes {\mathbf{a}}_{k} ({\mathbf{p}}_{d} ))\frac{{\partial {\mathbf{c}}_{k} ({\mathbf{p}}_{d} )}}{{\partial {\mathbf{p}}_{d}^{\text{T}} }} $$
(75)

Using (6), we have

$$ \begin{aligned} \frac{{\partial {\mathbf{c}}_{k} ({\mathbf{p}}_{d} )}}{{\partial {\mathbf{p}}_{d}^{\text{T}} }} &= ( - {\text{j}}2\pi )[0\;\tau_{1} \exp \{ - {\text{j}}2\pi f_{k} ({\mathbf{p}}_{d} )\tau_{1} \}\;\tau_{2} \exp \{ - {\text{j}}2\pi f_{k} ({\mathbf{p}}_{d} )\tau_{2} \} \\&\quad \cdots \tau_{N} \exp \{ - {\text{j}}2\pi f_{k} ({\mathbf{p}}_{d} )\tau_{N} \} ]^{\text{T}} \frac{{\partial f_{k} ({\mathbf{p}}_{d} )}}{{\partial {\mathbf{p}}_{d}^{\text{T}} }} \end{aligned}$$
(76)

where

$$ \frac{{\partial f_{k} ({\mathbf{p}}_{d} )}}{{\partial {\mathbf{p}}_{d}^{\text{T}} }} = \frac{{f_{\text{c}} }}{c}\left( {\frac{{{\mathbf{q}}_{k2}^{\text{T}} }}{{||{\mathbf{p}}_{d} - {\mathbf{q}}_{k1} ||_{2} }} - \frac{{{\mathbf{q}}_{k2}^{\text{T}} ({\mathbf{p}}_{d} - {\mathbf{q}}_{k1} )({\mathbf{p}}_{d} - {\mathbf{q}}_{k1} )^{\text{T}} }}{{||{\mathbf{p}}_{d} - {\mathbf{q}}_{k1} ||_{2}^{3} }}} \right) = \frac{{f_{\text{c}} }}{c}\left( {\left( {\frac{{||{\mathbf{p}}_{d} - {\mathbf{q}}_{k1} ||_{2}^{2} {\mathbf{I}}_{{\dim \{ {\mathbf{p}}_{d} \} }} - ({\mathbf{p}}_{d} - {\mathbf{q}}_{k1} )({\mathbf{p}}_{d} - {\mathbf{q}}_{k1} )^{\text{T}} }}{{||{\mathbf{p}}_{d} - {\mathbf{q}}_{k1} ||_{2}^{3} }}} \right){\mathbf{q}}_{k2} } \right)^{\text{T}} $$
(77)

Moreover, it can be checked from the expression of \( {\mathbf{r}}_{k} ({\mathbf{p}},t) \) that

$$ \begin{aligned}\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t)}}{{\partial {\mathbf{p}}^{\text{T}} }} & = ({\text{j}}2\pi t){\text{diag}}[\exp \{ {\text{j}}2\pi f_{k} ({\mathbf{p}}_{1} )t\}\;\exp \{ {\text{j}}2\pi f_{k} ({\mathbf{p}}_{2} )t\} \\ &\quad \cdots \exp \{ {\text{j}}2\pi f_{k} ({\mathbf{p}}_{D} )t]{\text{blkdiag}}\left[ {\frac{{\partial f_{k} ({\mathbf{p}}_{1} )}}{{\partial {\mathbf{p}}_{ 1}^{\text{T}} }}\frac{{\partial f_{k} ({\mathbf{p}}_{2} )}}{{\partial {\mathbf{p}}_{ 2}^{\text{T}} }} \cdots \frac{{\partial f_{k} ({\mathbf{p}}_{D} )}}{{\partial {\mathbf{p}}_{D}^{\text{T}} }}} \right] \end{aligned} $$
(78)

Appendix 3: Detailed Derivation of (25)

Applying a first-order Taylor series expansion of \( {\mathbf{e}}_{kl} ({\boldsymbol{\varphi }}_{k} (t_{l} )) \) around \( {\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ) \) produces

$$ {\mathbf{e}}_{kl} ({\boldsymbol{\varphi }}_{k} (t_{l} )) \approx {\mathbf{e}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} )) + {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))({\boldsymbol{\varphi }}_{k} (t_{l} ) - {\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} )) $$
(79)

Then, the cost function \( J_{5,kl} ({\boldsymbol{\varphi }}_{k} (t_{l} )) \) can be approximately expressed as:

$$ \begin{aligned} J_{5,kl} ({\boldsymbol{\varphi }}_{k} (t_{l} )) & \approx ||{\bar{\mathbf{x}}}_{k} (t_{l} ) - {\mathbf{e}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} )) - {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))({\boldsymbol{\varphi }}_{k} (t_{l} ) - {\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))||_{2}^{2} \\ & = \left\| {\left[ {\begin{array}{*{20}c} {{\text{Re}}\{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} } \\ \hline {\text{Im} \{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} } \\ \end{array} } \right]({\boldsymbol{\varphi }}_{k} (t_{l} ) - {\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} )) - \left[ {\begin{array}{*{20}c} {{\text{Re}}\{ {\bar{\mathbf{x}}}_{k} (t_{l} ) - {\mathbf{e}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} } \\ \hline {\text{Im} \{ {\bar{\mathbf{x}}}_{k} (t_{l} ) - {\mathbf{e}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} } \\ \end{array} } \right]} \right\|_{2}^{2} \\ \end{aligned} $$
(80)

It follows from (80) that in the \( i + 1 \)th iteration, the optimum vector \( {\boldsymbol{\varphi }}_{k} (t_{l} ) \) that minimizes \( J_{5,kl} ({\boldsymbol{\varphi }}_{k} (t_{l} )) \) is given by:

$$ \begin{aligned} {\hat{\boldsymbol{\varphi }}}_{k}^{(i + 1)} (t_{l} ) & = {\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ) + (({\text{Re}}\{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} )^{\text{T}} {\text{Re}}\{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} + (\text{Im} \{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} )^{\text{T}} \text{Im} \{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} )^{ - 1} \\ & \quad \times (({\text{Re}}\{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} )^{\text{T}} {\text{Re}}\{ {\bar{\mathbf{x}}}_{k} (t_{l} ) - {\mathbf{e}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} + (\text{Im} \{ {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} )^{\text{T}} \text{Im} \{ {\bar{\mathbf{x}}}_{k} (t_{l} ) - {\mathbf{e}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} ) \\ & = {\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ) + (\text{Re} \{ ({\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} )))^{\text{H}} {\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} ))\} )^{ - 1} \text{Re} \{ ({\mathbf{E}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} )))^{\text{H}} ({\bar{\mathbf{x}}}_{k} (t_{l} ) - {\mathbf{e}}_{kl} ({\hat{\boldsymbol{\varphi }}}_{k}^{(i)} (t_{l} )))\} \\ \end{aligned} $$
(81)

Additionally, in order to improve the stability of the Gauss–Newton iteration, it is necessary to introduce a step length in (81). As a result, we get (25).

Appendix 4: Complexity of the Proposed Method

Table 6 shows the numerical complexity of the proposed algorithm, expressed in the number of multiplication operations.

Table 6 Complexity of the proposed method

Appendix 5: Detailed Expressions for \( {\mathbf{F}}_{{\varvec{\upbeta}}}^{{\mathbf{H}}} {\mathbf{F}}_{{\varvec{\upbeta}}} \), \( {\mathbf{F}}_{{\varvec{\upbeta}}}^{{\mathbf{H}}} {\mathbf{F}}_{{{\mathbf{Re}}\{ {\mathbf{s}}\} }} \), \( {\mathbf{F}}_{{{\mathbf{Re}}\{ {\mathbf{s}}\} }}^{{\mathbf{H}}} {\mathbf{F}}_{{{\mathbf{Re}}\{ {\mathbf{s}}\} }} \), \( {\mathbf{F}}_{{\varvec{\upbeta}}}^{{\mathbf{H}}} {\mathbf{F}}_{{\mathbf{p}}} \), \( {\mathbf{F}}_{{{\mathbf{Re}}\{ {\mathbf{s}}\} }}^{{\mathbf{H}}} {\mathbf{F}}_{{\mathbf{p}}} \), and \( {\mathbf{F}}_{{\mathbf{p}}}^{{\mathbf{H}}} {\mathbf{F}}_{{\mathbf{p}}} \)

Performing some algebraic manipulations and using (32) and (33), we have

$$ {\mathbf{F}}_{{\varvec{\upbeta}}}^{\text{H}} {\mathbf{F}}_{{\varvec{\upbeta}}} = {\text{blkdiag[}}{\mathbf{F}}_{{{\varvec{\upbeta}},1}}^{\text{H}} {\mathbf{F}}_{{{\varvec{\upbeta}},1}}\;{\mathbf{F}}_{{{\varvec{\upbeta}}, 2}}^{\text{H}} {\mathbf{F}}_{{{\varvec{\upbeta}}, 2}} \cdots {\mathbf{F}}_{{{\varvec{\upbeta}},K}}^{\text{H}} {\mathbf{F}}_{{{\varvec{\upbeta}},K}} ] $$
(82)
$$ {\mathbf{F}}_{{\varvec{\upbeta}}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} }} = {\text{blkdiag[}}{\mathbf{F}}_{{{\varvec{\upbeta}},1}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,1}}\;{\mathbf{F}}_{{{\varvec{\upbeta}}, 2}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} , 2}} \cdots {\mathbf{F}}_{{{\varvec{\upbeta}},K}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,K}} ] $$
(83)
$$ {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} }}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} }} = {\text{blkdiag[}}{\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,1}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,1}}\;{\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} , 2}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} , 2}} \cdots {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,K}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,K}} ] $$
(84)
$$ {\mathbf{F}}_{{\varvec{\upbeta}}}^{\text{H}} {\mathbf{F}}_{{\mathbf{p}}} = [({\mathbf{F}}_{{{\varvec{\upbeta}},1}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},1}} )^{\text{H}}\;({\mathbf{F}}_{{{\varvec{\upbeta}},2}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},2}} )^{\text{H}} \cdots ({\mathbf{F}}_{{{\varvec{\upbeta}},K}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},K}} )^{\text{H}} ]^{\text{H}} $$
(85)
$$ {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} }}^{\text{H}} {\mathbf{F}}_{{\mathbf{p}}} = [({\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,1}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},1}} )^{\text{H}}\;({\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,2}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},2}} )^{\text{H}} \cdots ({\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,K}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},K}} )^{\text{H}} ]^{\text{H}} $$
(86)
$$ {\mathbf{F}}_{{\mathbf{p}}}^{\text{H}} {\mathbf{F}}_{{\mathbf{p}}} = \sum\limits_{k = 1}^{K} {{\mathbf{F}}_{{{\mathbf{p}},k}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},k}} } $$
(87)

where

$$ {\mathbf{F}}_{{{\varvec{\upbeta}},k}}^{\text{H}} {\mathbf{F}}_{{{\varvec{\upbeta}},k}} = \sum\limits_{l = 1}^{L} {({\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} )} $$
(88)
$$\begin{aligned} {\mathbf{F}}_{{{\varvec{\upbeta}},k}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,k}} &= [({\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{1} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{1} )\\ &\quad\quad\cdots ({\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{L} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{L} )] \end{aligned}$$
(89)
$$ \begin{aligned}{\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,k}}^{\text{H}} {\mathbf{F}}_{{\text{Re} \{ {\mathbf{s}}\} ,k}} &= {\text{blkdiag}}[({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{1} )\\ &\quad\quad\cdots ({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{L} )]\end{aligned} $$
(90)
$$ \begin{aligned} {\mathbf{F}}_{{{\varvec{\upbeta}},k}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},k}} = & \left( {\sum\limits_{l = 1}^{L} {({\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{l} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} } \right)\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }} \\ & \quad + \sum\limits_{l = 1}^{L} {({\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{l} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{l} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{l} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ \end{aligned} $$
(91)
$$ \begin{aligned} {\mathbf{F}}_{{{\text{Re}}\{ {\mathbf{s}}\} ,k}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},k}} = & \left[ {\begin{array}{*{20}c} {({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{1} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{1} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} \\ {({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{2} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{2} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{2} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} \\ \vdots \\ {({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{L} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{L} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} \\ \end{array} } \right]\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }} \\ & \quad + \left[ {\begin{array}{*{20}c} {({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{1} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{1} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ {({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{2} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{2} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{2} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ \vdots \\ {({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{L} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{L} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ \end{array} } \right] \\ \end{aligned} $$
(92)
$$ \begin{aligned} {\mathbf{F}}_{{{\mathbf{p}},k}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},k}} = & \left( {\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }}} \right)^{\text{H}} \left( {\left( {\sum\limits_{l = 1}^{L} {{\varvec{\Delta}}_{k} ({\mathbf{s}}_{k} (t_{l} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{ * } ({\mathbf{s}}_{k} (t_{l} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{T}} {\varvec{\Delta}}_{k} } } \right) \otimes {\mathbf{I}}_{(N + 1)M} } \right)\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }} \\ & \quad + \left( {\sum\limits_{l = 1}^{L} {\left( {\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \right)^{\text{H}} ({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{l} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{l} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} } \right)\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }} \\ & \quad + \left( {\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }}} \right)^{\text{H}} \left( {\sum\limits_{l = 1}^{L} {(({\varvec{\Delta}}_{k} ({\mathbf{s}}_{k} (t_{l} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} ))^{ * } ) \otimes {\mathbf{I}}_{(N + 1)M} ){\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{l} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} } \right) \\ & \quad + \sum\limits_{l = 1}^{L} {\left( {\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \right)^{\text{H}} ({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{l} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{l} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{l} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ \end{aligned} $$
(93)

Appendix 6: Detailed Expressions for \( {\mathbf{F}}_{{\varvec{\upbeta}}}^{{\mathbf{H}}} {\mathbf{F}}_{{\boldsymbol{\varphi }}} \), \( {\mathbf{F}}_{{\boldsymbol{\varphi }}}^{{\mathbf{H}}} {\mathbf{F}}_{{\boldsymbol{\varphi }}} \), and \( {\mathbf{F}}_{{\boldsymbol{\varphi }}}^{{\mathbf{H}}} {\mathbf{F}}_{{\mathbf{p}}} \)

Performing some algebraic manipulations and combining (32), (33), (52) and (53) yield

$$ {\mathbf{F}}_{{\varvec{\upbeta}}}^{\text{H}} {\mathbf{F}}_{{\boldsymbol{\varphi }}} = {\text{blkdiag[}}{\mathbf{F}}_{{{\varvec{\upbeta}},1}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }},1}}\;{\mathbf{F}}_{{{\varvec{\upbeta}}, 2}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }}, 2}} \cdots {\mathbf{F}}_{{{\varvec{\upbeta}},K}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }},K}} ] $$
(94)
$$ {\mathbf{F}}_{{\boldsymbol{\varphi }}}^{\text{H}} {\mathbf{F}}_{{\boldsymbol{\varphi }}} = {\text{blkdiag[}}{\mathbf{F}}_{{{\boldsymbol{\varphi }},1}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }},1}}\;{\mathbf{F}}_{{{\boldsymbol{\varphi }}, 2}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }}, 2}} \cdots {\mathbf{F}}_{{{\boldsymbol{\varphi }},K}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }},K}} ] $$
(95)
$$ {\mathbf{F}}_{{\boldsymbol{\varphi }}}^{\text{H}} {\mathbf{F}}_{{\mathbf{p}}} = [({\mathbf{F}}_{{{\boldsymbol{\varphi }},1}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},1}} )^{\text{H}}\;({\mathbf{F}}_{{{\boldsymbol{\varphi }},2}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},2}} )^{\text{H}} \cdots ({\mathbf{F}}_{{{\boldsymbol{\varphi }},K}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},K}} )^{\text{H}} ]^{\text{H}} $$
(96)

where

$$\begin{aligned} {\mathbf{F}}_{{{\varvec{\upbeta}},k}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }},k}} & = [{\text{j}}({\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{1} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{1} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ) \cdots \\ &\qquad {\text{j}}({\mathbf{B}}_{k} ({\mathbf{p}}){\mathbf{S}}_{k} (t_{L} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{L} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{L} )] \end{aligned} $$
(97)
$$ \begin{aligned} {\mathbf{F}}_{{{\boldsymbol{\varphi }},k}}^{\text{H}} {\mathbf{F}}_{{{\boldsymbol{\varphi }},k}} & = {\text{blkdiag[(}}{\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{1} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{1} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ) \cdots\\ &\qquad ({\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{L} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{L} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ) ]\\ \end{aligned} $$
(98)
$$ \begin{aligned} {\mathbf{F}}_{{{\boldsymbol{\varphi }},k}}^{\text{H}} {\mathbf{F}}_{{{\mathbf{p}},k}} = & \left[ {\begin{array}{*{20}c} {({\text{j}}{\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{1} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{1} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{1} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} \\ {({\text{j}}{\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{2} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{2} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{2} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{2} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} \\ \vdots \\ {({\text{j}}{\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{L} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} ((({\mathbf{s}}_{k} (t_{L} ) \odot {\mathbf{r}}_{k} ({\mathbf{p}},t_{L} ))^{\text{T}} {\varvec{\Delta}}_{k} ) \otimes {\mathbf{I}}_{(N + 1)M} )} \\ \end{array} } \right]\frac{{\partial {\text{vec}}({\mathbf{B}}_{k} ({\mathbf{p}}))}}{{\partial {\mathbf{p}}^{\text{T}} }} \\ & \quad + \left[ {\begin{array}{*{20}c} {({\text{j}}{\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{1} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{1} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{1} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{1} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ {({\text{j}}{\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{2} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{2} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{2} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{2} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ \vdots \\ {({\text{j}}{\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{L} ){\mathbf{R}}_{k} ({\mathbf{p}},t_{L} ))^{\text{H}} {\mathbf{B}}_{k} ({\mathbf{p}}){\varvec{\Delta}}_{k} {\mathbf{S}}_{k} (t_{L} )\frac{{\partial {\mathbf{r}}_{k} ({\mathbf{p}},t_{L} )}}{{\partial {\mathbf{p}}^{\text{T}} }}} \\ \end{array} } \right] \\ \end{aligned} $$
(99)

Appendix 7: Proof of Proposition 3

Notice that

$$ \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{1} {\mathbf{A}}_{2}^{ - 1} {\mathbf{A}}_{1}^{\text{H}} }\;& \,\, {{\mathbf{A}}_{1} } \\ {{\mathbf{A}}_{1}^{\text{H}} }\;& \,\, {{\mathbf{A}}_{ 2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{1} {\mathbf{A}}_{2}^{ - 1} } \\ {{\mathbf{I}}_{n} } \\ \end{array} } \right]{\mathbf{A}}_{ 2} [{\mathbf{A}}_{2}^{ - 1} {\mathbf{A}}_{1}^{\text{H}} \,\,{\mathbf{I}}_{n} ] \ge {\mathbf{O}} $$
(100)

which implies

$$ {\text{Re}}\left\{ {\left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{1} {\mathbf{A}}_{2}^{ - 1} {\mathbf{A}}_{1}^{\text{H}} } \;& \,\, {{\mathbf{A}}_{1} } \\ {{\mathbf{A}}_{1}^{\text{H}} }\;& \,\, {{\mathbf{A}}_{ 2} } \\ \end{array} } \right]} \right\} = \left[ {\begin{array}{*{20}c} {\text{Re} \{ {\mathbf{A}}_{1} {\mathbf{A}}_{2}^{ - 1} {\mathbf{A}}_{1}^{\text{H}} \} }\;& {\text{Re} \{ {\mathbf{A}}_{1} \} } \\ \,\, {\text{Re} \{ {\mathbf{A}}_{1}^{\text{H}} \} }\;& \,\, {\text{Re} \{ {\mathbf{A}}_{ 2} \} } \\ \end{array} } \right] \ge 0 $$
(101)

Moreover, it can be checked that

(102)

Then, we have

$$ \text{Re} \{ {\mathbf{A}}_{1} {\mathbf{A}}_{2}^{ - 1} {\mathbf{A}}_{1}^{\text{H}} \} - \text{Re} \{ {\mathbf{A}}_{1} \} (\text{Re} \{ {\mathbf{A}}_{2} \} )^{ - 1} \text{Re} \{ {\mathbf{A}}_{1}^{\text{H}} \} \ge {\mathbf{O}} $$
(103)

which is equivalent to

$$ \text{Re} \{ {\mathbf{A}}_{1} {\mathbf{A}}_{2}^{ - 1} {\mathbf{A}}_{1}^{\text{H}} \} \ge \text{Re} \{ {\mathbf{A}}_{1} \} (\text{Re} \{ {\mathbf{A}}_{2} \} )^{ - 1} \text{Re} \{ {\mathbf{A}}_{1}^{\text{H}} \} $$
(104)

At this point, the proof is completed.

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Wang, D., Yin, J., Chen, X. et al. Direct Position Determination of Multiple Constant Modulus Sources Based on Direction of Arrival and Doppler Frequency Shift. Circuits Syst Signal Process 39, 268–306 (2020) doi:10.1007/s00034-019-01170-6

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Keywords

  • Source localization
  • DPD
  • DOA
  • Doppler frequency
  • Constant modulus
  • Alternating minimization algorithm
  • CRB