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Sensor array calibration method in presence of gain/phase uncertainties and position perturbations using the spatial- and time-domain information of the auxiliary sources

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Abstract

This paper deals with the problem of active calibration under the existence of sensor gain/phase uncertainties and position perturbations. Unlike many existing eigenstructure-based (also called subspace-based) calibration methods which using the spatial-domain (i.e., angle) information of the auxiliary sources only, our proposed approach enables exploitation of both the spatial- and time-domain knowledge of the sources, and therefore yields better performance than the eigenstructure-based calibration technology. For the purpose of incorporating the time-domain knowledge of the sources into the error calibration, the maximum likelihood criterion is selected as the optimization principle, and a concentrated alternating iteration procedure (called algorithm II) is developed, which has rapid convergence rate and robustness. As a byproduct of this paper, we also provide an eigenstructure-based calibration approach (termed algorithm I), which alternatively minimizes the weighted signal subspace fitting cost function and weighted noise subspace fitting criterion to update the estimates for sensor position perturbations and gain/phase errors in each iteration, respectively. Similar to some previous subspace-based calibration algorithms in the literature, algorithm I is also asymptotically efficient but is more computationally convenient, and can be introduced as benchmark to be compared to algorithm II. Additionally, the Cramér–Rao bound (CRB) expressions for the sensor gain/phase errors and position perturbations estimates are presented for two situations: (a) the time-domain waveform information of the sources is unavailable, and (b) the time-domain waveform information of the sources is taken as prior knowledge into account. The CRBs for the two cases are also quantitatively compared, and the resulting conclusion demonstrates that by combining the time-domain waveform information of the sources into the calibration algorithm, a significant performance improvement can be achieved. The simulation experiments are conducted to corroborate the advantages of the proposed algorithms as well as the theoretical analysis in this paper.

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Authors and Affiliations

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

    Ding Wang & Ying Wu

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  1. Ding Wang
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  2. Ying Wu
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Correspondence to Ding Wang.

Appendices

Appendix 1: Proof of (31)–(35)

Applying the first-order derivation operator of the orthogonal projection matrix, it follows that

$$\begin{aligned}&\frac{\partial {\varvec{\Pi }} ^{\bot }\left[ {{\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}}\right) } \right] }{\partial \Delta x_n }\nonumber \\&\quad =-{\varvec{\Pi }} ^{\bot }\left[ {{\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] \cdot \frac{\partial {\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) }{\partial \Delta x_n }{\varvec{\varOmega }}^{\mathrm{\dagger }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) \nonumber \\&\qquad -\,\left( {{\varvec{\Pi }}} ^{\bot }\left[ {{\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] \cdot \frac{\partial {\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) }{\partial \Delta x_n }{\varvec{\varOmega }}^{\mathrm{\dagger }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}}\right) \right) ^{\mathrm{H}} \end{aligned}$$
(89)

The substitution of (89) into (30) yields

$$\begin{aligned}&\frac{\partial h_{\mathrm{c-ml}} \left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) }{\partial \Delta x_n }={{\varvec{x}}}^{\mathrm{H}}\frac{\partial {\varvec{\Pi }} ^{\bot }\left[ {{\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] }{\partial \Delta x_n }{{\varvec{x}}}\nonumber \\&\quad =-2\mathrm{Re}\left\{ {{{\varvec{x}}}}^{\mathrm{H}}\cdot {\varvec{\Pi }} ^{\bot }\left[ {{\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] \cdot \frac{\partial {\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) }{\partial \Delta x_n }{\varvec{\varOmega }}^{\mathrm{\dagger }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) {{\varvec{x}}} \right\} \nonumber \\&\quad =2\mathrm{Re}\left\{ \left( {{\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} x} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) {{\varvec{z}}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) -{{\varvec{x}}}} \right) ^{\mathrm{H}}\frac{\partial {\varvec{\varOmega }}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) }{\partial \Delta x_n }{{\varvec{z}}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) \right\} \nonumber \\&\quad =2\mathrm{Re}\left\{ \sum _{k=1}^K \left( {\hbox {diag}\left[ {{{\varvec{B}}}( {\varvec{\theta }}){{\varvec{S}}}_0 ( {t_k }){\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right] \cdot {{\varvec{z}}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) -{{\varvec{x}}}( {t_k })} \right) ^{\mathrm{H}}\right. \nonumber \\&\qquad \left. \cdot \, \hbox {diag}\left[ {{{\varvec{i}}}_M^{\left( n \right) } {{\varvec{i}}}_M^{\left( n \right) \hbox {T}} {\dot{{{\varvec{B}}}}_{{{\varvec{\Delta }}} {{\varvec{x}}}}} ( {\varvec{\theta }}) {{\varvec{S}}}_0 ( {t_k }){\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right] \cdot {{\varvec{z}}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}}\right) \right\} \nonumber \\&\quad =2\mathrm{Re}\left\{ {{{\varvec{i}}}}_M^{\left( n \right) \hbox {T}} {\dot{{{\varvec{B}}}}_{{{\varvec{\Delta }}} {{\varvec{x}}}}} ( {\varvec{\theta }})\left( {{\hat{{{\varvec{R}}}}}_{{{\varvec{ss}}}} {{\varvec{B}}}^{\mathrm{H}}( {\varvec{\theta }})\cdot \hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] -{\hat{{{\varvec{R}}}}}_{{{\varvec{sx}}}} } \right) \right. \nonumber \\&\qquad \left. \cdot \, \hbox {diag}\left[ {{{\varvec{z}}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] \cdot {{\varvec{i}}}_M^{\left( n \right) }\right\} \end{aligned}$$
(90)

which further implies that

$$\begin{aligned}&\frac{\partial h_{\mathrm{c-ml}} \left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) }{\partial {{\varvec{\Delta }}} {{\varvec{x}}}}\nonumber \\&\quad =2\mathrm{Re}\left\{ \hbox {vecd}\left[ {\bar{{{\varvec{I}}}}}_M \left( {\dot{{{\varvec{B}}}}_{{{\varvec{\Delta }}} {{\varvec{x}}}}} ( {\varvec{\theta }})\left( {\hat{{{\varvec{R}}}}}_{{{\varvec{ss}}}} {{\varvec{B}}}^{\mathrm{H}}( {\varvec{\theta }})\right. \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. \left. \cdot \, \hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] -{\hat{{{\varvec{R}}}}}_{{{\varvec{sx}}}} \right) \cdot \hbox {diag}\left[ {{{\varvec{z}}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] \right) {\bar{{{\varvec{I}}}}}_M^\mathrm{T} \right] \right\} \nonumber \\&\quad =2\mathrm{Re}\left\{ {{\varvec{g}}}_\mathrm{1} \left( {{{\varvec{\Delta }}} x} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) \right\} \end{aligned}$$
(91)

Similarly, it follows that

$$\begin{aligned}&\frac{\partial h_{\mathrm{c-ml}} \left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) }{\partial {{\varvec{\Delta }}} {{\varvec{y}}}}\nonumber \\&\quad =2\mathrm{Re}\left\{ \hbox {vecd}\left[ {\bar{{{\varvec{I}}}}}_M \left( {\dot{{{\varvec{B}}}}_{{{\varvec{\Delta }}} {{\varvec{y}}}}} ( {\varvec{\theta }})\left( {\hat{{{\varvec{R}}}}}_{{{\varvec{ss}}}} {{\varvec{B}}}^{\mathrm{H}}( {\varvec{\theta }})\right. \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. \left. \cdot \, \hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] -{\hat{{{\varvec{R}}}}}_{{{\varvec{sx}}}} \right) \cdot \hbox {diag}\left[ {{{\varvec{z}}}\left( {{{\varvec{\Delta }}} {{\varvec{x}}}} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right] \right) {\bar{{{\varvec{I}}}}}_M^\mathrm{T} \right] \right\} \nonumber \\&\quad =2\mathrm{Re}\left\{ {{{\varvec{g}}}_\mathrm{2} \left( {{{\varvec{\Delta }}} x} , {{\varvec{\Delta }}} {{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} \right) } \right\} \end{aligned}$$
(92)

Combining (91) and (92), equation (32) is proved to be true.

On the other hand, through some algebraic manipulations we can approximately obtain

(93)

which produces

(94)

It can be easily checked from (94) that

$$\begin{aligned}&\frac{\partial ^{2}h_{\mathrm{c-ml}} \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) }{\partial {\varvec{\Delta }}{{\varvec{x}}}\partial {\varvec{\Delta }}{{\varvec{x}}}^{\mathrm{T}}}\nonumber \\&\approx 2\mathrm{Re}\left\{ {{\bar{{\varvec{I}}}}_M \left( {\left( {{\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{x}}}} \left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{x}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right) ^{\mathrm{T}}{\bullet }\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{{\varvec{z}}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] } \right) {\bar{{\varvec{I}}}}_M^\mathrm{T} } \right\} \nonumber \\&\qquad -\,2\mathrm{Re}\left\{ {{\bar{{\varvec{I}}}}_M \left( {\sum _{l=1}^M {\left( {\begin{array}{ll} \hbox {vecd}\left[ {\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{\varvec{\varphi }}_l \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] \cdot {{\varvec{B}}}\left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{x}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right] \times \\ \hbox {vecd}^{\mathrm{H}}\left[ {\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{\varvec{\varphi }}_l \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] \cdot {{\varvec{B}}}\left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{x}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right] \\ \end{array}} \right) } } \right) {\bar{{\varvec{I}}}}_M^\mathrm{T} } \right\} \nonumber \\&=\hbox {2Re}\left\{ {{{\varvec{G}}}_{11} \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right\} \end{aligned}$$
(95)

Likewise, it can be proved that

$$\begin{aligned}&\frac{\partial ^{2}h_{\mathrm{c-ml}} \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) }{\partial {\varvec{\Delta }}{{\varvec{x}}}\partial {\varvec{\Delta }}{{\varvec{y}}}^{\mathrm{T}}}\nonumber \\&\approx 2\mathrm{Re}\left\{ {{\bar{{\varvec{I}}}}_M \left( {\left( {{\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{y}}}} \left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{x}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right) ^{\mathrm{T}}{\bullet }\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{{\varvec{z}}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] } \right) {\bar{{\varvec{I}}}}_M^\mathrm{T} } \right\} \nonumber \\&\quad -\,2\mathrm{Re}\left\{ {{\bar{{\varvec{I}}}}_M \left( {\sum _{l=1}^M {\left( {\begin{array}{c} \hbox {vecd}\left[ {\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{\varvec{\varphi }}_l \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] \cdot {{\varvec{B}}}\left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{x}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right] \times \\ \hbox {vecd}^{\mathrm{H}}\left[ {\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{\varvec{\varphi }}_l \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] \cdot {{\varvec{B}}}\left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{y}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right] \\ \end{array}} \right) } } \right) {\bar{{\varvec{I}}}}_M^\mathrm{T} } \right\} \nonumber \\&=\hbox {2Re}\left\{ {{{\varvec{G}}}_{12} \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right\} \end{aligned}$$
(96)
$$\begin{aligned}&\frac{\partial ^{2}h_{\mathrm{c-ml}} \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) }{\partial {\varvec{\Delta }}{{\varvec{y}}}\partial {\varvec{\Delta }}{{\varvec{y}}}^{\mathrm{T}}}\nonumber \\&\approx 2\mathrm{Re}\left\{ {{\bar{{\varvec{I}}}}_M \left( {\left( {{\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{y}}}} \left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{y}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right) ^{\mathrm{T}}{\bullet }\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{{\varvec{z}}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] } \right) {\bar{{\varvec{I}}}}_M^\mathrm{T} } \right\} \nonumber \\&-\,2\mathrm{Re}\left\{ {{\bar{{\varvec{I}}}}_M \left( {\sum _{l=1}^M {\left( {\begin{array}{c} \hbox {vecd}\left[ {\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{\varvec{\varphi }}_l \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] \cdot {{\varvec{B}}}\left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{y}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right] \times \\ \hbox {vecd}^{\mathrm{H}}\left[ {\hbox {diag}\left[ {{{\varvec{z}}}^{{*}}\left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) {\bullet }{\varvec{\varphi }}_l \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right] \cdot {{\varvec{B}}}\left( {\varvec{\theta }} \right) {\hat{{\varvec{R}}}}_{{{\varvec{ss}}}} {\dot{{\varvec{B}}}}_{{\varvec{\Delta }}{{\varvec{y}}}}^\mathrm{H} \left( {\varvec{\theta }} \right) } \right] \\ \end{array}} \right) } } \right) {\bar{{\varvec{I}}}}_M^\mathrm{T} } \right\} \nonumber \\&=\hbox {2Re}\left\{ {{{\varvec{G}}}_{\mathrm{22}} \left( {{\varvec{\Delta }}{{\varvec{x}}} , {\varvec{\Delta }}{{\varvec{y}}} , {\hat{\varvec{\alpha }}}_{\mathrm{ml}} } \right) } \right\} \end{aligned}$$
(97)

Combining (95)–(97), equality (33) holds true and hence the proof is completed.

Appendix 2: Proof of Lemma 1

To prove Lemma 1, we begin by supposing that there exits a nonzero real vector \({{\varvec{y}}}\) of conformable dimension satisfying

$$\begin{aligned} {{\varvec{y}}}^{\mathrm{T}}\left( {\hbox {Re}\left\{ {{{\varvec{X}}}_1 } \right\} -\hbox {Re}\left\{ {{{\varvec{X}}}_2 } \right\} } \right) {{\varvec{y}}}<0 \end{aligned}$$
(98)

it follows that

$$\begin{aligned}&{{\varvec{y}}}^{\mathrm{T}}\left( {{{\varvec{X}}}_1 -{{\varvec{X}}}_2 } \right) {{\varvec{y}}}={{\varvec{y}}}^{\mathrm{T}}\left( {\left( {\hbox {Re}\left\{ {{{\varvec{X}}}_1 } \right\} -\hbox {Re}\left\{ {{{\varvec{X}}}_2 } \right\} } \right) +\hbox {i}\left( {\hbox {Im}\left\{ {{{\varvec{X}}}_1 } \right\} -\hbox {Im}\left\{ {{{\varvec{X}}}_2 } \right\} } \right) } \right) {{\varvec{y}}}\nonumber \\&={{\varvec{y}}}^{\mathrm{T}}\left( {\hbox {Re}\left\{ {{{\varvec{X}}}_1 } \right\} -\hbox {Re}\left\{ {{{\varvec{X}}}_2 } \right\} } \right) {{\varvec{y}}}<0 \end{aligned}$$
(99)

which contradicts the assumption that \({{\varvec{X}}}_1 \ge {{\varvec{X}}}_2 \). Thus, Lemma 1 holds true.

Appendix 3: Proof of Lemma 2

Performing the singular value decomposition (SVD) on \({{\varvec{X}}}_1 \) leads to

$$\begin{aligned} {{\varvec{X}}}_1 ={{\varvec{U}}}{\varvec{\varSigma }}{{\varvec{V}}}^{\mathrm{H}} \end{aligned}$$
(100)

where \({{\varvec{U}}}^{\mathrm{H}}{{\varvec{U}}}={{\varvec{I}}}\) such that \(\hbox {range}\left\{ {{{\varvec{X}}}_1 } \right\} =\hbox {range}\left\{ {{\varvec{U}}} \right\} \). Since \(\hbox {range}\left\{ {{{\varvec{X}}}_1 } \right\} \subseteq \hbox {range}\left\{ {{{\varvec{X}}}_2 } \right\} \), we can construct a orthogonal matrix \({\bar{{\varvec{U}}}}\) satisfying that \({\bar{{\varvec{U}}}}^{\mathrm{H}}{\bar{{\varvec{U}}}}={{\varvec{I}}}\), \({\bar{{\varvec{U}}}}^{\mathrm{H}}{{\varvec{U}}}={{\varvec{O}}}\) and \(\hbox {range}\left\{ {{{\varvec{X}}}_\mathrm{2} } \right\} =\hbox {range}\left\{ {\left[ {{\begin{array}{ll} {{\varvec{U}}}&{} {{\bar{{\varvec{U}}}}} \\ \end{array} }} \right] } \right\} \), which produces

$$\begin{aligned} \Pi \left[ {{{\varvec{X}}}_2 } \right] =\left[ {{\begin{array}{ll} {{\varvec{U}}}&{} {{\bar{{\varvec{U}}}}} \\ \end{array} }} \right] \cdot \left[ {{\begin{array}{ll} {{\varvec{U}}}&{} {{\bar{{\varvec{U}}}}} \\ \end{array} }} \right] ^{\mathrm{H}}={{\varvec{UU}}}^{\mathrm{H}}+{{\bar{{\varvec{U}}}}}{{\bar{{\varvec{U}}}}}^{\mathrm{H}}\ge \Pi \left[ {{{\varvec{X}}}_\mathrm{1} } \right] ={{\varvec{UU}}}^{\mathrm{H}} \end{aligned}$$
(101)

It follows immediately from (101) that \(\Pi ^{\bot }\left[ {{{\varvec{X}}}_1 } \right] \ge \Pi ^{\bot }\left[ {{{\varvec{X}}}_\mathrm{2} } \right] \) and, therefore, Lemma 2 is proved.

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Wang, D., Wu, Y. Sensor array calibration method in presence of gain/phase uncertainties and position perturbations using the spatial- and time-domain information of the auxiliary sources. Multidim Syst Sign Process 26, 835–868 (2015). https://doi.org/10.1007/s11045-014-0284-5

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