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Fixed-Point Widely Linear MCCC for Bias-Compensated Adaptive Filtering

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Abstract

Recently, the widely linear complex-valued estimated-input adaptive filter (WLC-EIAF) used in processing noisy inputs and outputs has received increasing attention in machine learning and signal processing. In this paper, a fixed-point widely linear complex-valued estimated-input maximum complex correntropy criterion (FPWLC-EIMCCC) algorithm is proposed to deal with noncircular complex signals with noise in input and output. Compared with the existing algorithms, benefiting from the fixed-point method, the proposed FPWLC-EIMCCC can simultaneously have better steady-state performance and faster convergence speed. Furthermore, the theoretical analysis of the FPWLC-EIMCCC is performed by transforming FPWLC-EIMCCC into a gradient-like version based on the matrix inverse lemma and some approximations. Simulation results on system identification, channel estimation, and wind prediction show that FPWLC-EIMCCC significantly improves the filtering performance.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by Fundamental Research Funds for the Central Universities (19CX05003A-14).

Author information

Authors and Affiliations

  1. School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, China

    Chen Qiu

  2. School of Science, China University of Petroleum (East China), Qingdao, China

    Zongli Ruan

  3. Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronic and Information Engineering, Southwest University, Chongqing, China

    Guobing Qian

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  1. Chen Qiu
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Appendices

Appendix

Appendix A: Evaluation of \(T_{1}\)

Taking \({\overline{\mathbf{R}}} = E\left[ {{\hat{\mathbf{x}}}_{c} {\hat{\mathbf{x}}}_{c}^{H} } \right]\) into consideration, we further obtain

$$ {\overline{\mathbf{R}}} = E\left[ {{\hat{\mathbf{x}}}_{c} {\hat{\mathbf{x}}}_{c}^{H} } \right] = \sigma_{x}^{2} \left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{L} } & {(1 - 2\xi^{2} ){\mathbf{I}}_{L} } \\ {(1 - 2\xi^{2} ){\mathbf{I}}_{L} } & {{\mathbf{I}}_{L} } \\ \end{array} } \right]. $$
(46)

Based on (46) and the block matrix inverse lemma [47], \({\overline{\mathbf{R}}}^{ - 1}\) is derived as follows:

$$ \begin{aligned} {\overline{\mathbf{R}}}^{ - 1} & = \frac{1}{{\sigma_{x}^{2} }}\left[ {\begin{array}{*{20}c} {\frac{1}{{1 - (1 - 2\xi^{2} )^{2} }}{\mathbf{I}}_{L} } & {\frac{{1 - 2\xi^{2} }}{{(1 - 2\xi^{2} )^{2} - 1}}{\mathbf{I}}_{L} } \\ {\frac{{1 - 2\xi^{2} }}{{(1 - 2\xi^{2} )^{2} - 1}}{\mathbf{I}}_{L} } & {\frac{1}{{1 - (1 - 2\xi^{2} )^{2} }}{\mathbf{I}}_{L} } \\ \end{array} } \right] \\ & = \frac{1}{{\sigma_{x}^{2} }}\frac{1}{{1 - (1 - 2\xi^{2} )^{2} }}\left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{L} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{I}}_{L} } \\ \end{array} } \right] + \frac{1}{{\sigma_{x}^{2} }}\frac{{1 - 2\xi^{2} }}{{(1 - 2\xi^{2} )^{2} - 1}}\left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{I}}_{L} } \\ {{\mathbf{I}}_{L} } & {\mathbf{0}} \\ \end{array} } \right], \\ & = {\overline{\mathbf{R}}}_{1}^{ - 1} + {\overline{\mathbf{R}}}_{2}^{ - 1} \\ \end{aligned} $$
(47)

where \({\overline{\mathbf{R}}}_{1}^{ - 1} = \frac{1}{{\sigma_{x}^{2} }}\frac{1}{{1 - (1 - 2\xi^{2} )^{2} }}\left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{L} } & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{I}}_{L} } \\ \end{array} } \right]\), \({\overline{\mathbf{R}}}_{2}^{ - 1} = - \frac{1}{{\sigma_{x}^{2} }}\frac{{1 - 2\xi^{2} }}{{1 - (1 - 2\xi^{2} )^{2} }}\)\(\left[ {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{I}}_{L} } \\ {{\mathbf{I}}_{L} } & {\mathbf{0}} \\ \end{array} } \right]\), and L represents the length of the filter.

Thus, we have

$$ \begin{aligned} T_{1} & = E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}^{ - 1} {\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \quad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 1} {\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \\ & = E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}_{1}^{ - 1} {\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \quad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}_{1}^{ - 1} {\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \\ & \quad + E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \quad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \\ & = 2\frac{1}{{\sigma_{x}^{2} }}\frac{1}{{1 - (1 - 2\xi^{2} )^{2} }}E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right] \\ & \quad + E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \quad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right]. \\ \end{aligned} $$
(48)

Furthermore, we get

$$ \begin{aligned} & E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \qquad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \\ & \quad = \frac{1}{{\sigma_{x}^{2} }}\frac{{1 - 2\xi^{2} }}{{(1 - 2\xi^{2} )^{2} - 1}} \\ & \qquad \times E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right)\left[ {\begin{array}{*{20}c} {\left( {{\mathbf{x}}_{i} \left( n \right){\mathbf{x}}_{i}^{T} \left( n \right)} \right)^{*} } \\ {{\mathbf{x}}_{i} \left( n \right){\mathbf{x}}_{i}^{H} \left( n \right)} \\ \end{array} \;\;\begin{array}{*{20}c} {{\mathbf{x}}_{i}^{*} \left( n \right){\mathbf{x}}_{i}^{T} } \\ {{\mathbf{x}}_{i} \left( n \right){\mathbf{x}}_{i}^{T} \left( n \right)} \\ \end{array} } \right]{\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \qquad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right)\left[ {\begin{array}{*{20}c} {{\mathbf{x}}_{i} \left( n \right){\mathbf{x}}_{i}^{T} \left( n \right)} \\ {{\mathbf{x}}_{i} \left( n \right){\mathbf{x}}_{i}^{H} \left( n \right)} \\ \end{array} \;\;\;\begin{array}{*{20}c} {{\mathbf{x}}_{i}^{*} \left( n \right){\mathbf{x}}_{i}^{T} } \\ {\left( {{\mathbf{x}}_{i} \left( n \right){\mathbf{x}}_{i}^{T} \left( n \right)} \right)^{*} } \\ \end{array} } \right]{\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \\ & \quad \ge 2\frac{1}{{\sigma_{x}^{2} }}\frac{{1 - 2\xi^{2} }}{{(1 - 2\xi^{2} )^{2} - 1}}E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right], \\ \end{aligned} $$
(49)

and

$$ \begin{aligned} & E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \quad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \le 0, \\ \end{aligned} $$
(50)

where \({\mathbf{x}}_{c,i} \left( n \right) = \left[ {{\mathbf{x}}_{i}^{T} \left( n \right) {\mathbf{x}}_{i}^{H} \left( n \right)} \right]^{T}\), and \(\frac{{1 - 2\xi^{2} }}{{(1 - 2\xi^{2} )^{2} - 1}} \le {0}\).

Based on (49), we have

$$ \begin{aligned} T_{1} & = 2\frac{1}{{\sigma_{x}^{2} }}\frac{1}{{1 - (1 - 2\xi^{2} )^{2} }}E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right] \\ & \quad + E\left[ {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right. \\ & \quad + \left. {{\tilde{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\mathbf{x}}_{c,i} \left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}_{2}^{ - 1} {\tilde{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \\ & = T_{{0}} E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right] \\ & \ge 2\frac{1}{{\sigma_{x}^{2} }}\frac{{2\xi^{2} }}{{1 - (1 - 2\xi^{2} )^{2} }}E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right]. \\ \end{aligned} $$
(51)

Therefore, the range for \(T_{{0}}\) is

$$ 2\frac{1}{{\sigma_{x}^{2} }}\frac{{2\xi^{2} }}{{1 - (1 - 2\xi^{2} )^{2} }} \le T_{{0}} \le 2\frac{1}{{\sigma_{x}^{2} }}\frac{1}{{1 - (1 - 2\xi^{2} )^{2} }}. $$
(52)

Appendix B: Evaluation of \(T_{2}\) and \(T_{{4}}\)

In the steady state, assuming that \(\left| {e_{a,i} \left( n \right)} \right|^{2}\) is statistically independent of \({\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c,i} \left( n \right)\) [41] for long adaptive filter, we get

$$ \begin{aligned} T_{2} & = E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} {\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c,i} \left( n \right)} \right] = E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right]E\left[ {{\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c,i} \left( n \right)} \right] \\ & = E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right]E\left[ {{\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c,i} \left( n \right)} \right] \\ & = E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right]E\left[ {{\mathbf{x}}_{c}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c} \left( n \right) + i^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} i\left( n \right)} \right] \\ & = E\left[ {\left| {e_{a,i} \left( n \right)} \right|^{2} } \right]\left[ {Tr\left( {{\overline{\mathbf{R}}}^{ - 1} } \right) + \sigma_{i}^{2} Tr\left( {{\overline{\mathbf{R}}}^{ - 2} } \right)} \right]. \\ \end{aligned} $$
(53)

Similarly, \(T_{{4}}\) can be rewritten as

$$ T_{{4}} = E\left[ {{\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c,i} \left( n \right)} \right] = Tr\left( {{\overline{\mathbf{R}}}^{ - 1} } \right) + \sigma_{i}^{2} Tr\left( {{\overline{\mathbf{R}}}^{ - 2} } \right). $$
(54)

Appendix C: Evaluation of \(T_{{3}}\)

$$ \begin{aligned} T_{{3}} & = E\left[ {{{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} i\left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c,i} \left( n \right)i^{H} \left( n \right){{\varvec{\upomega}}}_{{{\text{opt}}}} } \right] \\ & = E\left[ {{{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} i\left( n \right){\mathbf{x}}_{c}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c} \left( n \right)i^{H} \left( n \right){{\varvec{\upomega}}}_{{{\text{opt}}}} } \right. \\ & \quad + \left. {{{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} i\left( n \right)i^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} i\left( n \right)i^{H} \left( n \right){{\varvec{\upomega}}}_{{{\text{opt}}}} } \right] \\ & = \sigma_{i}^{2} Tr\left( {{\overline{\mathbf{R}}}^{ - 1} } \right)\left\| {{\mathbf{w}}_{{{\text{opt}}}} } \right\|^{{2}} {\text{ + E}}\left[ {{{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} i\left( n \right)i^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} i\left( n \right)i^{H} \left( n \right){{\varvec{\upomega}}}_{{{\text{opt}}}} } \right]. \\ \end{aligned} $$
(55)

Based on the Gaussian fourth-order moment theorem [32], we get

$$ \begin{aligned} & {\text{E}}\left[ {{{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} i\left( n \right)i^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} i\left( n \right)i^{H} \left( n \right){{\varvec{\upomega}}}_{{{\text{opt}}}} } \right] \\ & \quad = {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} E\left[ {i\left( n \right)i^{H} \left( n \right){\overline{\mathbf{R}}}^{{ - {1}}} } \right]E\left[ {{\overline{\mathbf{R}}}^{{ - {1}}} i\left( n \right)i^{H} \left( n \right)} \right]{{\varvec{\upomega}}}_{{{\text{opt}}}} \\ & \qquad + {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} E\left[ {i\left( n \right)i^{H} \left( n \right){\overline{\mathbf{R}}}^{{ - {1}}} } \right]E\left[ {{\overline{\mathbf{R}}}^{{ - {1}}} i\left( n \right)i^{H} \left( n \right)} \right]{{\varvec{\upomega}}}_{{{\text{opt}}}} \\ & \qquad + {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} E\left[ {i\left( n \right)i^{H} \left( n \right)} \right]E\left[ {i^{H} \left( n \right){\overline{\mathbf{R}}}^{{ - {2}}} i\left( n \right)} \right]{{\varvec{\upomega}}}_{{{\text{opt}}}} \\ & \quad { = 2}\sigma_{i}^{4} {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} {\overline{\mathbf{R}}}^{{ - {2}}} {{\varvec{\upomega}}}_{{{\text{opt}}}} { + }\sigma_{i}^{4} {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} Tr\left( {{\overline{\mathbf{R}}}^{{ - {2}}} } \right){{\varvec{\upomega}}}_{{{\text{opt}}}} . \\ \end{aligned} $$
(56)

Substituting (56) into (55), we have

$$ T_{{3}} = \sigma_{i}^{2} Tr\left( {{\overline{\mathbf{R}}}^{ - 1} } \right)\left\| {{\mathbf{w}}_{{{\text{opt}}}} } \right\|^{{2}} { + 2}\sigma_{i}^{4} {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} {\overline{\mathbf{R}}}^{{ - {2}}} {{\varvec{\upomega}}}_{{{\text{opt}}}} { + }\sigma_{i}^{4} {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} Tr\left( {{\overline{\mathbf{R}}}^{{ - {2}}} } \right){{\varvec{\upomega}}}_{{{\text{opt}}}} . $$
(57)

Appendix D: Evaluation of \(T_{{5}}\)

$$ \begin{aligned} T_{5} & = E\left[ {\varphi^{2} (n){\hat{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}^{ - 2} {\hat{\varvec{\upomega}}}\left( {n - 1} \right)} \right] \\ & \approx E\left[ {\left( {\frac{{\tau g\left( {e_{i} \left( n \right)} \right)\left| {e_{i} \left( n \right)} \right|^{2} }}{{1 + \tau g\left( {e_{i} \left( n \right)} \right)\left\| {{\hat{\varvec{\upomega}}}\left( {n - 1} \right)} \right\|^{2} }}} \right)^{{2}} {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} {\overline{\mathbf{R}}}^{ - 2} {{\varvec{\upomega}}}_{{{\text{opt}}}} } \right] \\ & \approx \sigma_{i}^{4} {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} {\overline{\mathbf{R}}}^{ - 2} {{\varvec{\upomega}}}_{{{\text{opt}}}} . \\ \end{aligned} $$
(58)

Appendix E: Evaluation of \(T_{{6}}\), \(T_{{7}}\), \(T_{{8}}\) and \(T_{{9}}\)

Based on the independence assumption and \({\hat{\varvec{\upomega}}}^{H} \left( {n - 1} \right) \approx {{\varvec{\upomega}}}_{{{\text{opt}}}}\), we can get the evaluation of \(T_{{6}}\), \(T_{{7}}\), \(T_{{8}}\) and \(T_{{9}}\) as follows:

$$ T_{{6}} \approx {0,}T_{{7}} \approx {0,}T_{{8}} \approx {0,}T_{{9}} \approx {0}{\text{.}} $$
(59)

Appendix F: Evaluation of \(T_{{{10}}}\)

$$ \begin{aligned} T_{{{10}}} & = E\left[ {\varphi (n)\Re \left[ {{{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} i\left( n \right){\mathbf{x}}_{c,i}^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {\hat{\varvec{\upomega}}}\left( {n - 1} \right)} \right]} \right] \\ & \approx E\left[ {\frac{{\tau g\left( {e_{i} \left( n \right)} \right)\left| {e_{i} \left( n \right)} \right|^{2} }}{{1 + \tau g\left( {e_{i} \left( n \right)} \right)\left\| {{\hat{\varvec{\upomega}}}\left( {n - 1} \right)} \right\|^{2} }}} \right]E\left[ {{{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} i\left( n \right)i^{H} \left( n \right){\overline{\mathbf{R}}}^{ - 2} {{\varvec{\upomega}}}_{{{\text{opt}}}} } \right] \\ & \approx \sigma_{i}^{4} {{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} {\overline{\mathbf{R}}}^{ - 2} {{\varvec{\upomega}}}_{{{\text{opt}}}} . \\ \end{aligned} $$
(60)

Appendix G: Evaluation of \(T_{{{11}}}\)

$$ \begin{aligned} T_{{{11}}} & = E\left[ {\varphi (n)\Re \left[ {o^{*} \left( n \right){\hat{\varvec{\upomega}}}^{H} \left( {n - 1} \right){\overline{\mathbf{R}}}^{ - 2} {\mathbf{x}}_{c,i}^{H} \left( n \right)} \right]} \right] \\ & \approx E\left[ {\frac{{\tau g\left( {e_{i} \left( n \right)} \right)\left| {e_{i} \left( n \right)} \right|^{2} }}{{1 + \tau g\left( {e_{i} \left( n \right)} \right)\left\| {{\hat{\varvec{\upomega}}}\left( {n - 1} \right)} \right\|^{2} }}} \right]E\left[ {\Re \left[ {o^{*} \left( n \right){{\varvec{\upomega}}}_{{{\text{opt}}}}^{H} {\overline{\mathbf{R}}}^{ - 2} i^{H} \left( n \right)} \right]} \right] \\ & = 0. \\ \end{aligned} $$
(61)

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Qiu, C., Ruan, Z. & Qian, G. Fixed-Point Widely Linear MCCC for Bias-Compensated Adaptive Filtering. Circuits Syst Signal Process 42, 2959–2985 (2023). https://doi.org/10.1007/s00034-022-02247-5

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