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Improved Active Calibration Algorithms in the Presence of Channel Gain/Phase Uncertainties and Sensor Mutual Coupling Effects

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Abstract

This paper deals with the problem of active calibration when channel gain/phase uncertainties and sensor mutual coupling effects are simultaneously present. The numerical algorithms used to compensate for array error matrix, which is formed by the product of mutual coupling matrix and channel gain/phase error matrix, are presented especially tailored to uniform linear array (ULA) and uniform circular array (UCA). First, the array spatial responses corresponding to different azimuths are numerically evaluated using a set of time-disjoint auxiliary sources at known locations. Subsequently, a least-squares (LS) minimization model with respect to array error matrix is established. To solve this LS problem, two novel algorithms, namely algorithm I and algorithm II, are developed. In algorithm I, the array error matrix is considered as a whole matrix parameter to be optimized and an explicit closed-form solution to the error matrix is obtained. Compared with some existing algorithms with similar computation framework, algorithm I is able to utilize all potentially linear characteristics of ULA’s and UCA’s error matrix, and the calibration accuracy can be increased. Unlike algorithm I, algorithm II decomposes the array error matrix into two matrix parameters (i.e., mutual coupling matrix and channel gain/phase error matrix) to be optimized and all (nonlinear) numerical properties of the error matrix can be exploited. Therefore, algorithm II is able to achieve better calibration precision than algorithm I. However, algorithm II is more computationally demanding as a closed-form solution is no longer available and iteration computation is involved. In addition, the compact Cramér–Rao bound (CRB) expressions for all array error parameters are deduced in the case where auxiliary sources are assumed to be complex circular Gaussian distributed. Finally, the two novel algorithms are appropriately extended to the scenario where non-circular auxiliary sources are used, and the estimation variances of the array error parameters can be further decreased if the non-circularity is properly employed. Simulation experiments show the superiority of the presented algorithms.

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Acknowledgments

The author would like to thank all the anonymous reviewers for their valuable comments and suggestions which vastly improved the content and presentation of this paper. The author also acknowledges support from National Science Foundation of China under Grants 61201381 and the Future Development Foundation of Zhengzhou Information Science and Technology College under Grants YP12JJ202057.

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Authors and Affiliations

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

    Ding Wang

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Correspondence to Ding Wang.

Appendices

Appendix 1: Proof of (14)–(16)

Consider the \(n\)th sub-block in \({\varvec{m}}_{\hbox {ULA}} \) and \(\bar{\varvec{m}}_{\hbox {ULA}} \) as follows:

$$\begin{aligned} \left\{ {\begin{array}{l} {\varvec{m}}_{\hbox {ULA}, n} =\left[ {{\begin{array}{lllllll} {c_n \tau _n } &{} {c_{n-1} \tau _n } &{} \cdots &{} {c_1 \tau _n } &{} {c_2 \tau _n } &{} \cdots &{} {c_{M-n+1} \tau _n } \\ \end{array} }} \right] ^\mathrm{T} \\ \bar{\varvec{m}}_{\hbox {ULA}, n} =\left[ {{\begin{array}{llll} {c_{1} \tau _n } &{} {c_{2} \tau _n } &{} \cdots &{} {c_{\max \left\{ {n\,,\,M-n+1} \right\} } \tau _n } \\ \end{array} }} \right] ^\mathrm{T} \\ \end{array}} \right. \end{aligned}$$
(86)

When \(n\le \left\lfloor {M/2} \right\rfloor \), it follows that \(\hbox {max}\left\{ {n\,,\,M-n+1} \right\} =M-n+1\) and \(\bar{\varvec{m}}_{\hbox {ULA,} n} =\left[ {{\begin{array}{llll} {c_{1} \tau _n } &{} {c_{2} \tau _n } &{} \cdots &{} {c_{M-n+1} \tau _n } \\ \end{array} }} \right] ^\mathrm{T}\). Then, it can be verified that

(87)

Likewise, when \(n>\left\lfloor {M / 2} \right\rfloor \), it can be seen that \(\hbox {max}\left\{ {n\,,\,M-n+1} \right\} =n\) and \(\bar{\varvec{m}}_{\hbox {ULA}, n} =\left[ {{\begin{array}{llll} {c_{1} \tau _n } &{} {c_{2} \tau _n } &{} \cdots &{} {c_n \tau _n } \\ \end{array} }} \right] ^\mathrm{T}\), which produces

(88)

Combining (87) and (88) leads to

$$\begin{aligned} {\varvec{m}}_{\hbox {ULA}, n} ={\varvec{H}}_{\hbox {ULA}, n} \bar{\varvec{m}}_{\hbox {ULA}, n} \quad ( {{1}\le n\le M}) \end{aligned}$$
(89)

which further implies \({\varvec{m}}_{\hbox {ULA}} ={\varvec{H}}_{\hbox {ULA}} \bar{\varvec{m}}_{\hbox {ULA}} \), where \({\varvec{H}}_{\hbox {ULA}} =\hbox {blkdiag}\left[ {{\begin{array}{llll} {{\varvec{H}}_{\hbox {ULA},{1}} } &{} {{\varvec{H}}_{\hbox {ULA},{2}} } &{} \cdots &{} {{\varvec{H}}_{\hbox {ULA}, M} } \\ \end{array} }} \right] \). Therefore, Eqs. (14)–(16) are proved.

Appendix 2: Proof of (18)–(20)

First, consider the first sub-block in \({\varvec{m}}_{\hbox {UCA}} \) and \(\bar{\varvec{m}}_{\hbox {UCA}} \). When \(M\) is odd, it can be seen that

$$\begin{aligned} \left\{ {\begin{array}{l} {\varvec{m}}_{\hbox {UCA},{1}} =\left[ {{\begin{array}{llllllll} {c_1 \tau _1 } &{} {c_{2} \tau _1 } &{} \cdots &{} {c_L \tau _1 } &{} {c_L \tau _1 } &{} \cdots &{} {c_{3} \tau _1 } &{} {c_{2} \tau _1 } \\ \end{array} }} \right] ^\mathrm{T} \\ \bar{\varvec{m}}_{\hbox {UCA},{1}} =\left[ {{\begin{array}{llll} {c_{1} \tau _1 } &{} {c_{2} \tau _1 } &{} \cdots &{} {c_L \tau _1 } \\ \end{array} }} \right] ^\mathrm{T} \\ \end{array}} \right. \end{aligned}$$
(90)

which leads to

(91)

Similarly, when \(M\) is even, it can be observed that

$$\begin{aligned} \left\{ {\begin{array}{l} {\varvec{m}}_{\hbox {UCA},{1}} =\left[ {{\begin{array}{lllllllll} {c_1 \tau _1 } &{} {c_{2} \tau _1 } &{} \cdots &{} {c_{L-1} \tau _1 } &{} {c_L \tau _1 } &{} {c_{L-1} \tau _1 } &{} \cdots &{} {c_{3} \tau _1 } &{} {c_{2} \tau _1 } \\ \end{array} }} \right] ^\mathrm{T} \\ \bar{\varvec{m}}_{\hbox {UCA},{1}} =\left[ {{\begin{array}{llll} {c_{1} \tau _1 } &{} {c_{2} \tau _1 } &{} \cdots &{} {c_L \tau _1 } \\ \end{array} }} \right] ^\mathrm{T} \\ \end{array}} \right. \end{aligned}$$
(92)

which yields

(93)

Combining (91) and (93), it follows that \({\varvec{m}}_{\hbox {UCA},{1}} ={\varvec{\Delta }}_{1} \bar{\varvec{m}}_{\hbox {UCA},{1}} \). On the other hand, due to the circular Toeplitz structure of matrix \({{\varvec{C}}}\), it can be further found that the \(n\)th sub-block in \({\varvec{m}}_{\hbox {UCA}} \) and \(\bar{\varvec{m}}_{\hbox {UCA}} \) satisfy the following linear relationship

$$\begin{aligned} {\varvec{m}}_{\hbox {UCA}, n} ={\varvec{\Delta }} _2^{n-1} {\varvec{\Delta }}_1 \bar{\varvec{m}}_{\hbox {UCA}, n} \quad ( {{1}\le n\le M}) \end{aligned}$$
(94)

where

figure a

is the circular shift matrix. Consequently, equality \({\varvec{m}}_{\hbox {UCA}} ={\varvec{H}}_{\hbox {UCA}} \bar{\varvec{m}}_{\hbox {UCA}} \) holds, where \({\varvec{H}}_{\hbox {UCA}} =\hbox {blkdiag}\Big [ {{\varvec{\Delta }}_1 } {{\varvec{\Delta }}_2 {\varvec{\Delta }}_1 } \cdots {{\varvec{\Delta }}_2^{M-1} {\varvec{\Delta }}_1 } \Big ]\). Hence, Eqs. (18)–(20) are proved.

Appendix 3: Proof of (44)–(46)

First, applying the first-order derivation operator of the orthogonal projection matrix leads to

$$\begin{aligned} \frac{\partial {\varvec{\Pi }}^\bot ( {\varvec{\Omega }})}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }=-{\varvec{\Pi }}^\bot ( {\varvec{\Omega }})\frac{\partial {\varvec{\Omega }}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }{\varvec{\Omega }}^{\dag }-( {{\varvec{\Pi }}^\bot ( {\varvec{\Omega }})\frac{\partial {\varvec{\Omega }}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }{\varvec{\Omega }}^{\dag }})^\mathrm{H} \end{aligned}$$
(95)

which yields

$$\begin{aligned} \begin{array}{l} \frac{\partial g_{\hbox {cost}} }{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }={\varvec{\alpha }}^\mathrm{H}{\varvec{V}}^\mathrm{H}\frac{\partial {\varvec{\Pi }}^\bot ( {\varvec{\Omega }})}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }{\varvec{V\alpha }}=-2\cdot \mathrm{Re}\left\{ {{\varvec{\alpha }}^\mathrm{H}{\varvec{V}}^\mathrm{H}{\varvec{\Pi }}^\bot ( {\varvec{\Omega }})\frac{\partial {\varvec{\Omega }}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }{\varvec{\Omega }}^{\dag }{\varvec{V\alpha }}} \right\} \\ =-2\cdot \mathrm{Re}\left\{ {{\varvec{p}}^\mathrm{H}\frac{\partial {\varvec{\Omega }}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }{{\varvec{Q}}}} \right\} =-2\cdot \mathrm{Re}\left\{ {{\varvec{p}}^\mathrm{H}{\varvec{\Sigma }}( {{\varvec{I}}_M \otimes {\varvec{i}}_{\dim \left[ {{\varvec{c}}} \right] }^{( n)} }){{\varvec{Q}}}} \right\} \\ \end{array} \end{aligned}$$
(96)

where the third equality follows from (36). According to (96), it can be readily verified that

$$\begin{aligned} \frac{\partial g_{\hbox {cost}} }{\partial \hbox {Re}\left\{ {{\varvec{c}}} \right\} }=-2\cdot \mathrm{Re}\left\{ {( {{\varvec{I}}_{\dim \left[ {{\varvec{c}}} \right] } \otimes {\varvec{p}}})^\mathrm{H}{\varvec{G}}_1 ( {{\varvec{c}}}){{\varvec{Q}}}} \right\} =-2\cdot \mathrm{Re}\left\{ {{\varvec{g}}( {{\varvec{c}}})} \right\} \end{aligned}$$
(97)

Similarly, it follows that

$$\begin{aligned} \frac{\partial g_{\hbox {cost}} }{\partial \hbox {Im}\left\{ {{\varvec{c}}} \right\} }=-2\cdot \hbox {Re}\left\{ {\hbox {i}( {{\varvec{I}}_{\dim \left[ {{\varvec{c}}} \right] } \otimes {\varvec{p}}})^\mathrm{H}{\varvec{G}}_1 ( {{\varvec{c}}}){{\varvec{Q}}}} \right\} =2\cdot \mathrm{Im}\left\{ {{\varvec{g}}( {{\varvec{c}}})} \right\} \end{aligned}$$
(98)

On the other hand, it can be approximately obtained through some algebraic manipulations that

$$\begin{aligned} \begin{aligned}&\frac{\partial ^2g_{\hbox {cost}} }{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n \partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _m }\\&\quad \approx 2\cdot \mathrm{Re}\left\{ {{\varvec{\alpha }}^\mathrm{H}{\varvec{V}}^\mathrm{H}{\varvec{\Omega }}^{{\dag \mathrm{H}}}\frac{\partial {\varvec{\Omega }}^\mathrm{H}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }{\varvec{\Pi }}^\bot ( {\varvec{\Omega }})\frac{\partial {\varvec{\Omega }}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _m }{\varvec{\Omega }}^{\dag }{\varvec{V\alpha }}} \right\} \\&\quad =2\cdot \mathrm{Re}\left\{ {{{\varvec{Q}}}^\mathrm{H}\frac{\partial {\varvec{\Omega }}^\mathrm{H}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _n }{\varvec{\Pi }}^\bot ( {\varvec{\Omega }})\frac{\partial {\varvec{\Omega }}}{\partial \left[ {\hbox {Re}\left\{ {{\varvec{c}}} \right\} } \right] _m }{{\varvec{Q}}}} \right\} \\&\quad =2\cdot \mathrm{Re}\left\{ {{{\varvec{Q}}}^\mathrm{H}( {{\varvec{I}}_M \otimes {\varvec{i}}_{\dim \left[ {{\varvec{c}}} \right] }^{( n)\hbox {T}} }){\varvec{\Sigma }}^\mathrm{H}{\varvec{\Pi }}^\bot ( {\varvec{\Omega }}){\varvec{\Sigma }}( {{\varvec{I}}_M \otimes {\varvec{i}}_{\dim \left[ {{\varvec{c}}} \right] }^{( m)} }){{\varvec{Q}}}} \right\} \\ \end{aligned} \end{aligned}$$
(99)

where the third equality follows from (36). Then, it can be easily proved that

$$\begin{aligned} \frac{\partial ^2g_{\hbox {cost}} }{\partial \hbox {Re}\left\{ {{\varvec{c}}} \right\} \partial \hbox {Re}^\mathrm{T}\left\{ {{\varvec{c}}} \right\} }\approx 2\cdot \mathrm{Re}\left\{ {{\varvec{G}}_2 ( {{\varvec{c}}}){\varvec{\Pi }}^\bot ( {\varvec{\Omega }}){\varvec{G}}_2^\mathrm{H} ( {{\varvec{c}}})} \right\} =2\cdot \mathrm{Re}\left\{ {{\varvec{G}}( {{\varvec{c}}})} \right\} \end{aligned}$$
(100)

Through a similar mathematical manipulation, it can also be concluded that

$$\begin{aligned} \left\{ {\begin{array}{l} \frac{\partial ^2g_{\hbox {cost}} }{\partial \hbox {Re}\left\{ {{\varvec{c}}} \right\} \partial \mathrm{Im}^\mathrm{T}\left\{ {{\varvec{c}}} \right\} }\approx 2\cdot \mathrm{Re}\left\{ {\hbox {i}{\varvec{G}}_2 ( {{\varvec{c}}}){\varvec{\Pi }}^\bot ( {\varvec{\Omega }}){\varvec{G}}_2^\mathrm{H} ( {{\varvec{c}}})} \right\} =-2\cdot \mathrm{Im}\left\{ {{\varvec{G}}( {{\varvec{c}}})} \right\} \\ \frac{\partial ^2g_{\hbox {cost}} }{\partial \mathrm{Im}\left\{ {{\varvec{c}}} \right\} \partial \mathrm{Re}^\mathrm{T}\left\{ {{\varvec{c}}} \right\} }\approx 2\cdot \mathrm{Re}\left\{ {-\hbox {i}{\varvec{G}}_2 ( {{\varvec{c}}}){\varvec{\Pi }}^\bot ( {\varvec{\Omega }}){\varvec{G}}_2^\mathrm{H} ( {{\varvec{c}}})} \right\} =2\cdot \mathrm{Im}\left\{ {{\varvec{G}}( {{\varvec{c}}})} \right\} \\ \frac{\partial ^2g_{\hbox {cost}} }{\partial \mathrm{Im}\left\{ {{\varvec{c}}} \right\} \partial \mathrm{Im}^\mathrm{T}\left\{ {{\varvec{c}}} \right\} }\approx 2\cdot \mathrm{Re}\left\{ {{\varvec{G}}_2 ( {{\varvec{c}}}){\varvec{\Pi }}^\bot ( {\varvec{\Omega }}){\varvec{G}}_2^\mathrm{H} ( {{\varvec{c}}})} \right\} =2\cdot \mathrm{Re}\left\{ {{\varvec{G}}( {{\varvec{c}}})} \right\} \\ \end{array}} \right. \end{aligned}$$
(101)

Based on the above analysis, Eqs. (44)–(46) hold true.

Appendix 4: Proof of (59)–(62)

According to (57), it follows that

$$\begin{aligned} \begin{aligned}&\left[ {\mathbf{FISH}_{\mathrm{Re}\left\{ {\varvec{\tau }} \right\} \mathrm{Re}\left\{ {\varvec{\tau }} \right\} } } \right] _{nm}\\&\quad =\frac{{2}K}{\sigma ^2}\cdot \hbox {Re}\left\{ {\sum \limits _{k=1}^D {\omega _k {{\varvec{a}}}^\mathrm{H}( {\theta _k }){{\varvec{C}}}^\mathrm{H}{\varvec{i}}_M^{( n)} {\varvec{i}}_M^{(n){\mathrm{T}}} {{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}{\varvec{i}}_M^{( m)} {\varvec{i}}_M^{( m)\mathrm{T}} {{\varvec{Ca}}}( {\theta _k })} } \right\} \\&\quad =\frac{{2}K}{\sigma ^2}\cdot \hbox {Re}\left\{ {\sum \limits _{k=1}^D {\left[ {{{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}} \right] _{nm} \cdot \left[ {\omega _k {{\varvec{Ca}}}( {\theta _k }){{\varvec{a}}}^\mathrm{H}( {\theta _k }){{\varvec{C}}}^\mathrm{H}} \right] _{mn} } } \right\} \\ \end{aligned}\nonumber \\ \end{aligned}$$
(102)

which implies

$$\begin{aligned} \mathbf{FISH}_{\hbox {Re}\left\{ {\varvec{\tau }} \right\} \hbox {Re}\left\{ {\varvec{\tau }} \right\} }&=\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {( {{{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}})\bullet ( {\omega _k {{\varvec{Ca}}}( {\theta _k }){{\varvec{a}}}^\mathrm{H}( {\theta _k }){{\varvec{C}}}^\mathrm{H}})^\mathrm{T}} } \right\} \nonumber \\&=\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {{\varvec{X}}}_k }\right\} \end{aligned}$$
(103)

Through a similar mathematical manipulation, it can be proved that

$$\begin{aligned} \left\{ {\begin{array}{l} \mathbf{FISH}_{\mathrm{Re}\left\{ {\varvec{\tau }} \right\} \mathrm{Im}\left\{ {\varvec{\tau }} \right\} } =-\mathbf{FISH}_{\mathrm{Im}\left\{ {\varvec{\tau }} \right\} \mathrm{Re}\left\{ {\varvec{\tau }} \right\} } =-\frac{{2}K}{\sigma ^2}\cdot \mathrm{Im}\left\{ {\sum \limits _{k=1}^D {{\varvec{X}}}_k } \right\} \\ \mathbf{FISH}_{\mathrm{Im}\left\{ {\varvec{\tau }} \right\} \mathrm{Im}\left\{ {\varvec{\tau }} \right\} } =\mathbf{FISH}_{\mathrm{Re}\left\{ {\varvec{\tau }} \right\} \mathrm{Re}\left\{ {\varvec{\tau }} \right\} } =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {{\varvec{X}}}_k } \right\} \\ \end{array}} \right. \end{aligned}$$
(104)

Likewise, applying (57) again leads to

$$\begin{aligned} \begin{aligned}&\left[ {\mathbf{FISH}_{\mathrm{Re}\left\{ {\varvec{\tau }} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} } } \right] _{nm}\\&\quad =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {\omega _k {{\varvec{a}}}^\mathrm{H}( {\theta _k }){{\varvec{C}}}^\mathrm{H}{\varvec{i}}_M^{( n)} {\varvec{i}}_M^{( n)\mathrm{T}} {{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}{{\varvec{\Gamma }}}\cdot {\varvec{T}}\left[ {{{\varvec{a}}}( {\theta _k })} \right] \cdot {\varvec{i}}_N^{( m)} } } \right\} \\&\quad =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {\left[ {{{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}{{\varvec{\Gamma }}}\cdot {\varvec{T}}\left[ {{{\varvec{a}}}( {\theta _k })} \right] } \right] _{nm} \cdot \left[ {\omega _k {{\varvec{C}}}^*{{\varvec{a}}}^*( {\theta _k })} \right] _n } } \right\} \end{aligned} \nonumber \\ \end{aligned}$$
(105)

which yields

$$\begin{aligned} \begin{aligned}&\mathbf{FISH}_{\mathrm{Re}\left\{ {\varvec{\tau }} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} }\\&\quad =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {( {{{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}{{\varvec{\Gamma }}}\cdot {\varvec{T}}\left[ {{{\varvec{a}}}( {\theta _k })} \right] })\bullet ( {\omega _k ( {{{\varvec{Ca}}}( {\theta _k })})^\mathrm{H}\otimes {\varvec{1}}_{N\times 1} })^\mathrm{T}} } \right\} \\&\quad =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ \sum \limits _{k=1}^D {{{\varvec{Y}}_k } } \right\} \\ \end{aligned}\nonumber \\ \end{aligned}$$
(106)

With a similar derivation, it can be easily checked that

$$\begin{aligned} \left\{ {\begin{array}{l} \mathbf{FISH}_{\mathrm{Re}\left\{ {\varvec{\tau }} \right\} \mathrm{Im}\left\{ {{\varvec{c}}} \right\} }=-\mathbf{FISH}_{\mathrm{Im}\left\{ {\varvec{\tau }} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} } =-\frac{{2}K}{\sigma ^2}\cdot \mathrm{Im}\left\{ \sum \limits _{k=1}^D {{{\varvec{Y}}_k } } \right\} \\ \mathbf{FISH}_{\mathrm{Im}\left\{ {\varvec{\tau }} \right\} \mathrm{Im}\left\{ {{\varvec{c}}} \right\} } =\mathbf{FISH}_{\mathrm{Re}\left\{ {\varvec{\tau }} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} } =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ \sum \limits _{k=1}^D {{{\varvec{Y}}_k } } \right\} \\ \end{array}} \right. \end{aligned}$$
(107)

Analogously, it follows from (57) that

$$\begin{aligned}&\left[ {\mathbf{FISH}_{\mathrm{Re}\left\{ {{\varvec{c}}} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} } } \right] _{nm}\nonumber \\&\quad \!=\!\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {\omega _k {\varvec{i}}_M^{( n)\mathrm{T}} \!\cdot \! {\varvec{T}}^\mathrm{H}\left[ {{{\varvec{a}}}( {\theta _k })} \right] \cdot {{\varvec{\Gamma }}}^\mathrm{H}{{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}{{\varvec{\Gamma }}}\cdot {\varvec{T}}\left[ {{{\varvec{a}}}( {\theta _k })} \right] \cdot {\varvec{i}}_N^{( m)} } } \right\} \nonumber \\&\quad =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {\left[ {\omega _k \cdot {\varvec{T}}^\mathrm{H}\left[ {{{\varvec{a}}}( {\theta _k })} \right] \cdot {{\varvec{\Gamma }}}^\mathrm{H}{{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}{{\varvec{\Gamma }}}\cdot {\varvec{T}}\left[ {{{\varvec{a}}}( {\theta _k })} \right] } \right] _{nm} } } \right\} \nonumber \\ \end{aligned}$$
(108)

which gives

$$\begin{aligned}&\mathbf{FISH}_{\mathrm{Re}\left\{ {{\varvec{c}}} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} }\nonumber \\&\quad =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {\omega _k \cdot {\varvec{T}}^\mathrm{H}\left[ {{{\varvec{a}}}( {\theta _k })} \right] \cdot {{\varvec{\Gamma }}}^\mathrm{H}{{\varvec{W}}}^{-\mathrm{H}}{\varvec{\Pi }}^\bot ( {{\varvec{u}}_k }){{\varvec{W}}}^{-1}{{\varvec{\Gamma }}}\cdot {\varvec{T}}\left[ {{{\varvec{a}}}( {\theta _k })} \right] } } \right\} \nonumber \\&\quad =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {{{\varvec{Z}}}_k } } \right\} \end{aligned}$$
(109)

Through a similar derivation as above, it can be readily verified that

$$\begin{aligned} \left\{ {\begin{array}{l} \mathbf{FISH}_{\mathrm{Re}\left\{ {{\varvec{c}}} \right\} \mathrm{Im}\left\{ {{\varvec{c}}} \right\} } =-\mathbf{FISH}_{\mathrm{Im}\left\{ {{\varvec{c}}} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} } =-\frac{{2}K}{\sigma ^2}\cdot \mathrm{Im}\left\{ {\sum \limits _{k=1}^D {{{\varvec{Z}}}_k } } \right\} \\ \mathbf{FISH}_{\mathrm{Im}\left\{ {{\varvec{c}}} \right\} \mathrm{Im}\left\{ {{\varvec{c}}} \right\} } =\mathbf{FISH}_{\mathrm{Re}\left\{ {{\varvec{c}}} \right\} \mathrm{Re}\left\{ {{\varvec{c}}} \right\} } =\frac{{2}K}{\sigma ^2}\cdot \mathrm{Re}\left\{ {\sum \limits _{k=1}^D {{{\varvec{Z}}}_k } } \right\} \\ \end{array}} \right. \end{aligned}$$
(110)

At this point, the proof of (59)–(62) is completed.

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Wang, D. Improved Active Calibration Algorithms in the Presence of Channel Gain/Phase Uncertainties and Sensor Mutual Coupling Effects. Circuits Syst Signal Process 34, 1825–1868 (2015). https://doi.org/10.1007/s00034-014-9926-y

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