A New Sequential Block Partial Update Normalized Least Mean M-Estimate Algorithm and its Convergence Performance Analysis

Abstract

This paper proposes a new sequential block partial update normalized least mean square (SBP-NLMS) algorithm and its nonlinear extension, the SBP-normalized least mean M-estimate (SBP–NLMM) algorithm, for adaptive filtering. These algorithms both utilize the sequential partial update strategy as in the sequential least mean square (S–LMS) algorithm to reduce the computational complexity. Particularly, the SBP–NLMM algorithm minimizes the M-estimate function for improved robustness to impulsive outliers over the SBP–NLMS algorithm. The convergence behaviors of these two algorithms under Gaussian inputs and Gaussian and contaminated Gaussian (CG) noises are analyzed and new analytical expressions describing the mean and mean square convergence behaviors are derived. The robustness of the proposed SBP–NLMM algorithm to impulsive noise and the accuracy of the performance analysis are verified by computer simulations.

Appendices

Appendix A: Evaluation of H

In this appendix, we evaluate the expectation ${\mathbf{H}} = E_{{{\left\{ {{\mathbf{X}},\eta _{g} } \right\}}}} {\left[ {\psi {\left( e \right)}{S_{{\text{x}}} {\mathbf{X}}} \mathord{\left/ {\vphantom {{S_{{\text{x}}} {\mathbf{X}}} {{\left( {\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}_{{\text{x}}} {\mathbf{X}}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}_{{\text{x}}} {\mathbf{X}}} \right)}}|{\mathbf{v}}} \right]}$. η _g (n) and x(n) are assumed to be statistically independent, and X are jointly Gaussian with autocorrelation matrix R _xx. Assuming there are C different combinations for S _x, each of which is denoted as ${\mathbf{S}}_{\mathbf{X}}^{\left( i \right)}$, i = 1,2,…,C with equal probability p _i  = 1/C, one gets

$$ {\mathbf{H}} = {\sum\limits_{i = 1}^C {\frac{1}{C}C_{R} \iint\limits_{L + 1\;{\text{fold}}} {\frac{{\psi {\left( e \right)}S^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}}}{{\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}}}\exp {\left( { - \frac{1}{2}{\mathbf{X}}^{{\mathbf{T}}} {\mathbf{R}}^{{ - {\text{1}}}}_{{{\text{xx}}}} {\mathbf{X}}} \right)}f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d\eta _{g} d{\mathbf{X}}}} } $$

(A-1)

where $C_R = \left( {2\pi } \right)^{{{ - L} \mathord{\left/ {\vphantom {{ - L} 2}} \right. \kern-\nulldelimiterspace} 2}} |{\mathbf{R}}_{{\text{xx}}} |^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}$ and $f_{{\eta _{g} }}$ (η _g ) is the PDF of the Gaussian noise η _g . |·| denotes the determinant of a matrix. Let’s focus on the integral inside the summation as follows:

$${\mathbf{H}}_{i} = C_{R} \iint\limits_{L + 1\;{\text{fold}}} {\frac{{\psi {\left( {\text{e}} \right)}{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}}} {{\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}^{{{\left( {\text{i}} \right)}}}_{{\mathbf{X}}} {\mathbf{X}}}}\exp {\left( { - \frac{1} {2}{\mathbf{X}}^{T} {\mathbf{R}}^{{{\text{ - 1}}}}_{{{\text{xx}}}} {\mathbf{X}}} \right)}}f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d_{{\eta _{g} }} d{\mathbf{X}}.$$

(A-2)

Similar to [29], let us consider the integral

$${\mathbf{F}}_{i} {\left( \beta \right)} = C_{R} \iint\limits_{L + 1{\text{ fold}}} {\frac{{\psi {\left( e \right)}{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}\exp {\left( { - \beta {\left( {\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}} \right)}} \right)}}} {{\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}}}}\; \cdot \exp {\left( { - \frac{1} {2}{\mathbf{X}}^{T} {\mathbf{R}}^{{ - 1}}_{{{\text{xx}}}} {\mathbf{X}}} \right)}f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d\eta _{g} d{\mathbf{X}}. $$

(A-3)

It can be seen that

$${\mathbf{H}}_i = {\mathbf{F}}_i \left( 0 \right).$$

(A-4)

Differentiating (A-3) with respect to β, one gets

$$\begin{aligned} & \frac{{d{\mathbf{F}}_{i} {\left( \beta \right)}}}{{d\beta }} = - \exp {\left( { - \beta \varepsilon } \right)}C_{R} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\iint\limits_{L + 1{\text{ fold}}} {{\text{ }}\psi {\left( e \right)}{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}}\exp {\left( { - \frac{1}{2}{\mathbf{X}}^{T} {\mathbf{B}}^{{ - 1}}_{i} {\mathbf{X}}} \right)} \cdot f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d\eta _{g} d{\mathbf{X}}, \\ \end{aligned} $$

(A-5)

where

$$\begin{array}{*{20}c} {{\mathbf{B}}_{i} = {\left( {2\beta {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} + {\mathbf{R}}^{{ - 1}}_{{{\text{xx}}}} } \right)}^{{ - 1}} = {\mathbf{R}}_{{{\text{xx}}}} {\left( {2\beta {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{R}}_{{{\text{xx}}}} + {\mathbf{I}}} \right)}^{{ - 1}} } \\ { = {\mathbf{R}}_{{{\text{xx}}}} {\mathbf{P}}^{T}_{i} {\left[ {\begin{array}{*{20}c} {{{\mathbf{I}}_{P} + 2\beta {\mathbf{R}}_{{Xi\_00}} }} & {{2\beta {\mathbf{R}}_{{Xi\_01}} }} \\ {0} & {{{\mathbf{I}}_{{L - P}} }} \\ \end{array} } \right]}^{{ - 1}} {\mathbf{P}}_{i} ,} \\ \end{array} $$

(A-6)

P = L/C, I _k is the k×k identity matrix, and P _i is a permutation matrix with

$$P_{i} R_{{{\text{xx}}}} P^{T}_{i} = {\left[ {\begin{array}{*{20}c} {{R_{{Xi\_00}} }} & {{R_{{Xi\_01}} }} \\ {{R_{{Xi\_10}} }} & {{R_{{Xi\_11}} }} \\ \end{array} } \right]}{\text{and}}\,P_{i} S^{{{\left( i \right)}}}_{{\mathbf{X}}} P^{T}_{i} = {\left[ {\begin{array}{*{20}l} {{I_{p} } \hfill} & {0 \hfill} \\ {0 \hfill} & {0 \hfill} \\ \end{array} } \right]}.$$

Since R _{Xi _00} is symmetric, its eigenvalue decomposition has the following form

$${\mathbf{R}}_{Xi\_00} = {\mathbf{U}}_{Xi} \Lambda _{Xi} {\mathbf{U}}_{Xi}^T ,$$

(A-7)

where $\Lambda _{Xi} = {\text{diag}}\left( {\lambda _{Xi,1} ,\lambda _{Xi,2} , \cdots ,\lambda _{Xi,P} } \right)$ contains the eigenvalues of R _{Xi _00}. Therefore, we have $|{\mathbf{B}}_{i} | = |{\mathbf{R}}_{{{\text{xx}}}} ||{\mathbf{I}}_{P} + 2\beta {\mathbf{R}}_{{Xi\_00}} |^{{ - 1}} = |{\mathbf{R}}_{{{\mathbf{XX}}}} |{\mathop \Pi \limits_{k = 1}^P }(2\beta \lambda _{{Xi,k}} + 1)^{{ - 1}}$. Using the matrix inverse identity, we can rewrite (A-6) as

$${\mathbf{B}}_i = {\mathbf{R}}_{{\text{xx}}} {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c}{\left( {{\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} } \right)^{ - 1} } & { - 2\beta ({\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} )^{ - 1} {\mathbf{R}}_{Xi\_01} } \\0 & {{\mathbf{I}}_{L - P} } \\\end{array} } \right]{\mathbf{P}}_i = {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c}{{\mathbf{A}}_{11} } & {{\mathbf{A}}_{12} } \\{{\mathbf{A}}_{21} } & {{\mathbf{A}}_{22} } \\\end{array} } \right]{\mathbf{P}}_i ,$$

(A-8)

where

$$\begin{array}{*{20}c}{{\mathbf{A}}_{11} = {\mathbf{R}}_{Xi\_00} \left( {{\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} } \right)^{ - 1} ,{\mathbf{A}}_{21} = {\mathbf{R}}_{Xi\_10} \left( {{\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} } \right)^{ - 1} ,} \\{{\mathbf{A}}_{12} = \left( {{\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} } \right)^{ - 1} {\mathbf{R}}_{Xi\_01} {\kern 1pt} ,\,{\text{and}}} \\{{\mathbf{A}}_{22} = {\mathbf{R}}_{Xi\_11} - 2\beta {\mathbf{R}}_{Xi\_10} \left( {{\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} } \right)^{ - 1} {\mathbf{R}}_{Xi\_01} .} \\\end{array} $$

Furthermore, one can rewrite (A-5) as follows

$$\begin{array}{*{20}c} {\frac{{d{\mathbf{F}}_{i} {\left( \beta \right)}}} {{d\beta }} = - \gamma _{i} {\left( \beta \right)}C_{{B_{i} }} \iint\limits_{L + 1\;{\text{fold}}} {\psi {\left( e \right)}{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}\exp {\left( { - \frac{1} {2}{\mathbf{X}}^{T} {\mathbf{B}}^{{ - 1}}_{1} {\mathbf{X}}} \right)} \cdot f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d\eta _{g} d{\mathbf{X}}}} \\ { = - \gamma _{i} {\left( \beta \right)}E_{{{\left\{ {{\text{X}},\eta _{g} } \right\}}}} {\left[ {\psi {\left( e \right)}{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}\left| v \right.} \right]}\left| {_{{E{\left[ {XX^{T} } \right]} = B_{i} }} } \right.} \\ { = - \gamma _{i} {\left( \beta \right)}{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{L}}_{1} ,} \\ \end{array} $$

(A-9)

where $C_{B_i } = \left( {2\pi } \right)^{{{ - L} \mathord{\left/ {\vphantom {{ - L} 2}} \right. \kern-\nulldelimiterspace} 2}} |{\mathbf{B}}_i |^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}$,$\gamma _i \left( \beta \right) = \exp \left( { - \beta \varepsilon } \right)|{\mathbf{B}}_i |^{\frac{1}{2}} |{\mathbf{R}}_{{\text{xx}}} |^{ - \frac{1}{2}}$, and ${\mathbf{L}}_1 = E_{\left\{ {{\mathbf{X}},\eta _g } \right\}} \left[ {\psi \left( e \right){\mathbf{X}}|{\mathbf{v}}} \right]|_{E\left[ {{\mathbf{XX}}^T } \right] = {\mathbf{B}}_i }$ is the expectation of ψ(e)X conditioned on v when x _i ,x _j ∈X are jointly Gaussian with correlation matrix B _i .

We shall evaluate L ₁ by considering its k-th element as follows:

$${\mathbf{L}}_{1,k} = E_{\left\{ {{\mathbf{X}},\eta _g } \right\}} \left. {\left[ {\psi \left( e \right)x_k |{\mathbf{v}}} \right]} \right|_{E\left[ {{\mathbf{XX}}^T } \right] = {\mathbf{B}}_i } .$$

(A-10)

Using Price’s theorem [12, 13], we have

$$\begin{array}{*{20}c} {\frac{{\partial {\mathbf{L}}_{1,k} }}{{\partial r_{x_k e} }} = E_{\left\{ {{\mathbf{X}},\eta _g } \right\}} \left. {\left[ {\frac{{d\psi \left( e \right)}}{{de}}\left| {\mathbf{v}} \right.} \right]} \right|_{E\left[ {{\mathbf{XX}}^T } \right] = {\mathbf{B}}_i } } \\ { = \frac{1}{{\sqrt {2\pi \sigma _e } }}\int_{ - \infty }^\infty {\psi \prime \left( e \right)\exp \left( { - \frac{{e^2 }}{{2\sigma _e^2 }}} \right)de = } \overline {\psi \prime } \left( {\sigma _e^2 } \right),} \\ \end{array} $$

(A-11)

where $\overline {\psi \prime } \left( {\sigma _e^2 } \right)$ is the average of $\psi \prime \left( e \right)$, and e=X ^T v+η _g , which is Gaussian distributed with zero mean and variance $\sigma _e^2 \left( {\mathbf{v}} \right) = E\left. {\left[ {{\mathbf{v}}^T {\mathbf{XX}}^T {\mathbf{v}}|{\mathbf{v}}} \right]} \right|_{E\left[ {{\mathbf{XX}}^T } \right] = {\mathbf{B}}_i } + \sigma _g^2 = {\mathbf{v}}^T {\mathbf{B}}_i {\mathbf{v}} + \sigma _g^2$. To save notation, we shall write $\overline {\psi \prime } \left( {\sigma _e^2 \left( {\mathbf{v}} \right)} \right)$simply as $\overline {\psi \prime } \left( {\sigma _e^2 } \right)$. Integrating (A-11) with respect to $r_{x_k e}$, one gets

$${\mathbf{L}}_{1,k} = \overline {\psi \prime } \left( {\sigma _e^2 } \right)r_{x_k e} ,$$

(A-12)

where the constant of integration is zero because $r_{x_k e}$ is equal to zero when x _k and e are uncorrelated. Since

$$\begin{array}{*{20}c}{r_{x_k e} = \left. {E\left[ {x_k e|{\mathbf{v}}} \right]} \right|_{E\left[ {{\mathbf{XX}}^T } \right] = {\mathbf{B}}_i } } \\{ = \left. {E\left[ {x_k \left( {{\mathbf{X}}^T {\mathbf{v}} + \eta _g } \right)|{\mathbf{v}}} \right]} \right|_{E\left[ {{\mathbf{XX}}^T } \right] = {\mathbf{B}}_i } = {\mathbf{B}}_{i,k} {\mathbf{v}},} \\\end{array} $$

(A-13)

where B _i,k is the k-th row of B _i , we have

$${\mathbf{L}}_1 = \overline {\psi \prime } \left( {\sigma _e^2 } \right){\mathbf{B}}_i {\mathbf{v}}\left( n \right)\,.$$

(A-14)

Substituting (A-14) in (A-9) and integrating with respect to β yields

$${\mathbf{F}}_{i} {\left( \beta \right)} = {\left( { - \smallint ^{\beta } \gamma _{i} {\left( \beta \right)}\overline{{\psi \prime }} {\left( {\sigma ^{2}_{e} } \right)}{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{B}}_{i} d\beta } \right)}{\mathbf{v}}{\left( n \right)} \approx \overline{{\psi \prime }} {\left( {\sigma ^{2}_{e} } \right)}\alpha _{i} {\left( \beta \right)}{\mathbf{v}}{\left( n \right)},$$

(A-15)

where the constant of integration is equal to zero because of the boundary condition F _i (∞) = 0. Here, we have assumed that $\sigma _e^2 \left( {\mathbf{v}} \right)$ depends weakly on β and is taken outside of the integral using mean value theorem. This is a good approximation if the variation of $\overline {\psi \prime } \left( {\sigma _e^2 } \right)$ is limited, such as in the RLM algorithm [21] with ATS or at the steady state of the algorithm.

We now evaluate

$$\alpha _i \left( \beta \right) = - \smallint ^\beta \gamma _i \left( \beta \right){\mathbf{S}}_{\text{X}}^{\left( i \right)} {\mathbf{B}}_i d\beta $$

(A-16)

From (A-6), ${\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} {\mathbf{B}}_i$ can be denoted as

$$\begin{array}{*{20}c}{{\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} {\mathbf{B}}_i = {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c}{{\mathbf{R}}_{Xi\_00} \left( {{\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} } \right)^{ - 1} } & {\left( {{\mathbf{I}}_P + 2\beta {\mathbf{R}}_{Xi\_00} } \right)^{ - 1} {\mathbf{R}}_{Xi\_01} } \\0 & 0 \\\end{array} } \right]{\mathbf{P}}_i } \\{ = {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c}{{\mathbf{U}}_{Xi} \Lambda _{Xi} ({\mathbf{I}}_P + 2\beta \Lambda _{Xi} )^{ - 1} {\mathbf{U}}_{Xi}^T } & {{\mathbf{U}}_{Xi} ({\mathbf{I}}_P + 2\beta \Lambda _{Xi} )^{ - 1} {\mathbf{U}}_{Xi}^T {\mathbf{R}}_{Xi\_01} } \\0 & 0 \\\end{array} } \right]{\mathbf{P}}_i .} \\\end{array} $$

(A-17)

Substituting (A-17) into (A-16), we get

$$\alpha _i \left( 0 \right) = {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c}{{\mathbf{U}}_{Xi} \left( {\Lambda _{Xi} I\left( {\Lambda _{Xi} } \right)} \right){\mathbf{U}}_{Xi}^T } & {{\mathbf{U}}_{Xi} \left( {I\left( {\Lambda _{Xi} } \right)} \right){\mathbf{U}}_{Xi}^T {\mathbf{R}}_{Xi\_01} } \\0 & 0 \\\end{array} } \right]{\mathbf{P}}_i $$

(A-18)

where $I\left( {\Lambda _{Xi} } \right) = \int_0^\infty {\gamma _i \left( \beta \right)\left( {{\mathbf{I}}_P + 2\beta \Lambda _{Xi} } \right)^{ - 1} } d\beta$. Since the integrant is a diagonal matrix, it suffices to evaluate the integral for its diagonal entries as follows:

$$\left[ {I\left( {\Lambda _{Xi} } \right)} \right]_{k,k} = \int_0^\infty {\frac{{\exp \left( { - \beta \varepsilon } \right)}}{{\left( {1 + 2\beta \lambda _{Xi,k} } \right)}}\mathop \Pi \limits_{j = 1}^P \left( {2\beta \lambda _{Xi,j} + 1} \right)^{{{ - 1} \mathord{\left/{\vphantom {{ - 1} 2}} \right.\kern-\nulldelimiterspace} 2}} d\beta } ,$$

(A-19)

k = 1,…,P, which is a special integral function defined to facilitate analysis. Finally, from (A-4) and (A-15)–(A-19), we have

$${\mathbf{H}}_i = {\mathbf{F}}_i \left( 0 \right) = \overline {\psi \prime } \left( {\sigma _e^2 \left( n \right)} \right)\alpha _i {\mathbf{v}}\left( n \right),$$

(A-20)

And

$${\mathbf{H}} = \sum\nolimits_{i = 1}^C {\frac{1}{C}\overline {\psi \prime } } \left( {\sigma _e^2 \left( n \right)} \right){\text{ $ \alpha $ }}_i {\mathbf{v}}\left( n \right).$$

(A-21)

where we write α _i (0) as α _i for simplicity.

Appendix B: Evaluation of s₃

${\mathbf{s}}_3 = E_{\left\{ {{\mathbf{X}},\eta _g } \right\}} \left[ {{{\psi ^2 \left( e \right){\mathbf{S}}_{\text{x}} {\mathbf{XX}}^T {\mathbf{S}}_{\text{x}} } \mathord{\left/ {\vphantom {{\psi ^2 \left( e \right){\mathbf{S}}_{\text{x}} {\mathbf{XX}}^T {\mathbf{S}}_{\text{x}} } {\left( {\varepsilon + {\mathbf{X}}^T {\mathbf{S}}_{\text{x}} {\mathbf{X}}} \right)^2 }}} \right. \kern-\nulldelimiterspace} {\left( {\varepsilon + {\mathbf{X}}^T {\mathbf{S}}_{\text{x}} {\mathbf{X}}} \right)^2 }}|{\mathbf{v}}} \right]$ is evaluated in this appendix. As in Appendix A, we can write:

$${\mathbf{s}}_3 = \sum\limits_{i = 1}^C {\frac{1}{C}C_R \iint\limits_{\mathop {L + 1}\limits_{{\text{fold}}} } {\frac{{\psi ^2 \left( e \right){\mathbf{S}}_{\text{X}}^{\left( i \right)} {\mathbf{XX}}^T {\mathbf{S}}_{\text{X}}^{\left( i \right)} }}{{\left( {\varepsilon + {\mathbf{X}}^T {\mathbf{S}}_{\text{X}}^{\left( i \right)} {\mathbf{X}}} \right)^2 }}\exp \left( { - \frac{1}{2}{\mathbf{X}}^{\mathbf{T}} {\mathbf{R}}_{{\text{XX}}}^{{\text{ - 1}}} {\mathbf{X}}} \right)f_{\eta _g } \left( {\eta _g } \right)d\eta _g d{\mathbf{X}}.}} $$

(B-1)

Let us define

$$\overline{{\mathbf{F}}} _{i} {\left( \beta \right)} = C_{R} \iint\limits_{L + 1{\text{ fold}}} {\frac{{\psi ^{2} {\left( e \right)}{\mathbf{XX}}^{T} exp{\left( { - \beta {\left( {\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}} \right)}} \right)}}} {{{\left( {\varepsilon + {\mathbf{X}}^{T} {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{X}}} \right)}^{2} }}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \cdot \exp {\left( { - \frac{1} {2}{\mathbf{X}}^{T} {\mathbf{R}}^{{ - 1}}_{{{\text{xx}}}} {\mathbf{X}}} \right)}f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d\eta _{g} d{\mathbf{X}}.$$

(B-2)

Comparing (B-2) with (B-1), it can be seen that

$${\mathbf{s}}_3 = \frac{1}{C}\sum\nolimits_{i = 1}^C {{\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} \overline {\mathbf{F}} _i \left( 0 \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} .} $$

(B-3)

To evaluate $\overline {\mathbf{F}} _i \left( \beta \right)$, differentiating (B-2) twice with respect to β, one gets

$$\begin{array}{*{20}c} {\frac{{d^{2} \overline{{\mathbf{F}}} _{i} {\left( \beta \right)}}}{{d\beta ^{2} }} = C_{R} exp{\left( { - \beta \varepsilon } \right)}\iint\limits_{L + 1{\text{ fold}}} {\psi ^{2} {\left( e \right)}{\mathbf{XX}}^{T} }exp{\left( { - \tfrac{1}{2}{\mathbf{X}}^{T} {\mathbf{B}}^{{ - 1}}_{i} {\mathbf{X}}} \right)}{\kern 1pt} \cdot f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d\eta _{g} d{\mathbf{X}}} \\ { = \gamma _{i} {\left( \beta \right)}{\left[ {C_{{B_{i} }} \iint\limits_{L + 1{\text{ fold}}} {\psi ^{2} {\left( e \right)}{\mathbf{XX}}^{T} exp{\left( { - \tfrac{1}{2}{\mathbf{X}}^{T} {\mathbf{B}}^{{ - 1}}_{i} {\mathbf{X}}} \right)}f_{{\eta _{g} }} {\left( {\eta _{g} } \right)}d\eta _{g} d{\mathbf{X}}}} \right]}} \\ { = \gamma _{i} {\left( \beta \right)}E_{{{\left\{ {{\mathbf{X}},\eta _{g} } \right\}}}} \left. {{\left[ {\psi ^{2} (e){\mathbf{XX}}^{T} |{\mathbf{v}}} \right]}} \right|_{{E{\left[ {{\mathbf{XX}}^{T} } \right]} = {\mathbf{B}}_{i} }} } \\ { = \gamma _{i} {\left( \beta \right)}{\mathbf{L}}_{{2i}} } \\ \end{array} $$

(B-4)

where γ _i (β), $C_{B_i }$, and the correlation matrix B _i have been defined in Appendix A. ${\mathbf{L}}_{2i} = E_{\left\{ {{\mathbf{X}},\eta _g } \right\}} \left. {\left[ {\psi ^2 \left( e \right){\mathbf{XX}}^T |{\mathbf{v}}} \right]} \right|_{{\mathbf{B}}_i }$ is the expectation of ψ ²(e)XX ^T taken over {X,η _g } and conditioned on v where x _m , x _n ∈X, m,n=1,2…,L are jointly Gaussian distributed variables with correlation matrix B _i . Let’s evaluate the (m, n)-th element of L _2i as follows:

$${\mathbf{L}}_{{2i.m,n}} = E_{{{\left\{ {X,\eta _{g} } \right\}}}} {\left[ {\psi ^{2} {\left( e \right)}x_{m} x_{n} \left| v \right.} \right]}\left| {_{{{\text{B}}_{i} }} } \right..$$

(B-5)

Using Price’s theorem, we have

$$\begin{array}{*{20}c}{\frac{{\partial {\mathbf{L}}_{2i,m,n} }}{{\partial r_{x_m x_n } }} = E_{\left\{ {{\mathbf{X}},\eta _g } \right\}} \left. {\left[ {\psi ^2 \left( e \right)|{\mathbf{v}}} \right]} \right|_{{\mathbf{B}}_i } } \\{ = \frac{1}{{\sqrt {2\pi } \sigma _e }}\int_{ - \infty }^\infty {\psi ^2 \left( e \right)\exp \left( { - \frac{{e^2 }}{{2\sigma _e^2 }}} \right)de} = \overline {\psi ^2 } \left( {\sigma _e^2 } \right).} \\\end{array} $$

(B-6)

For notational simplicity, we shall write $\overline {\psi ^2 } \left( {\sigma _e^2 } \right)$ as $B_\psi \left( {\sigma _e^2 } \right)$. Integrating with respect to $r_{x_m x_n }$ gives

$${\mathbf{L}}_{2i,m,n} = B_\psi \left( {\sigma _e^2 } \right)r_{x_m x_n } + c_{m,n} ,$$

(B-7)

where $c_{m,n} = E\left. {\left[ {\psi ^2 \left( e \right)x_m x_n } \right]} \right|_{r_{x_m x_n = 0} }$ is the integration constant. Using Price’s theorem again, we have

$$\begin{array}{*{20}c}{\frac{{\partial c_{m,n} }}{{\partial r_{x_m e} }} = E\left. {\left[ {\frac{{d\psi ^2 \left( e \right)}}{{de}}x_n } \right]} \right|_{r_{x_m x_n = 0} } = E\left[ {\frac{{d^2 \psi ^2 \left( e \right)}}{{de^2 }}} \right]r_{x_n e} } \\{ = 2C_\psi \left( {\sigma _e^2 } \right)r_{x_n e} .} \\\end{array} $$

(B-8)

(From Price’s theorem, $\frac{1}{2}E\left[ {\frac{{d^2 \psi ^2 \left( e \right)}}{{de}}} \right] = \frac{d}{{d\sigma _e^2 }}E\left[ {\psi ^2 \left( e \right)} \right]$, and we write it as $C_\psi \left( {\sigma _e^2 } \right)$ hereafter.)

Integrating again, one gets

$$c_{m,n} = 2C_\psi \left( {\sigma _e^2 } \right)r_{x_n e} r_{x_m e} .$$

(B-9)

Combining (B-7) and (B-9), we have

$${\mathbf{L}}_{2i,m,n} = B_\psi \left( {\sigma _e^2 } \right)r_{x_m x_n } + 2C_\psi \left( {\sigma _e^2 } \right)r_{x_n e} r_{x_m e} ,$$

(B-10)

$${\text{and}}\quad {\mathbf{L}}_{2i} = B_\psi \left( {\sigma _e^2 } \right){\mathbf{B}}_i + 2C_\psi \left( {\sigma _e^2 } \right){\mathbf{B}}_i {\mathbf{vv}}^T {\mathbf{B}}_i .$$

(B-11)

Substituting (B-11) into (B-4) gives

$$\frac{{d^2 \overline {\mathbf{F}} _i \left( \beta \right)}}{{d\beta ^2 }} = \gamma _i \left( \beta \right)B_\psi \left( {\sigma _e^2 } \right){\mathbf{B}}_i + 2\gamma _i \left( \beta \right)C_\psi \left( {\sigma _e^2 } \right){\mathbf{B}}_i {\mathbf{vv}}^T {\mathbf{B}}_i $$

(B-12)

Double integrating (B-12) with respect to β and multiplying ${\mathbf{S}}_{\mathbf{X}}^{\left( i \right)}$ on both sides yield

$${\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} \overline {\mathbf{F}} _i \left( 0 \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} = \int_0^\infty {\int_{\beta _1 }^\infty {\gamma _i \left( {\beta _2 } \right)B_\psi \left( {\sigma _e^2 } \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} {\mathbf{B}}_i {\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} d\beta _2 d\beta _1 } } + \int_0^\infty {\int_{\beta _1 }^\infty {2\gamma _i \left( {\beta _2 } \right)C_\psi \left( {\sigma _e^2 } \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} {\mathbf{B}}_i {\mathbf{vv}}^T {\mathbf{B}}_i {\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} d\beta _2 d\beta _1 } } .$$

(B-13)

For SBP–NLMS algorithms with general nonlinearity, the integrals are rather difficult to evaluate because of the presence of $\sigma _e^2$. To simplify the analysis, we shall assume that $\sigma _e^2$ depends weakly on β and is taken outside the integral. Like $A_\psi \left( {\sigma _e^2 } \right)$, this is a good approximation if the variations of $B_\psi \left( {\sigma _e^2 } \right)$ and $C_\psi \left( {\sigma _e^2 } \right)$ are limited. Therefore, we have

$${\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} \overline {\mathbf{F}} _i \left( 0 \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} \approx B_\psi \left( {\sigma _e^2 } \right){\mathbf{I}}_{1i} + C_\psi \left( {\sigma _e^2 } \right){\mathbf{I}}_{2i} ,$$

(B-14)

with the boundary conditions: $\overline {\mathbf{F}} _i \left( \infty \right) = 0$, $\partial \overline {\mathbf{F}} _i {{\left( \beta \right)} \mathord{\left/ {\vphantom {{\left( \beta \right)} \partial }} \right. \kern-\nulldelimiterspace} \partial }\left. \beta \right|_{\beta = \infty } = 0$, and ${\mathbf{I}}_{1i} = \int_0^\infty {\int_{\beta _1 }^\infty {\gamma _i \left( {\beta _2 } \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} {\mathbf{B}}_i {\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} d\beta _2 d\beta _1 } }$, ${\mathbf{I}}_{2i} = \int_0^\infty {\int_{\beta _1 }^\infty {2\gamma _i \left( {\beta _2 } \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} {\mathbf{B}}_i {\mathbf{vv}}^T {\mathbf{B}}_i {\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} d\beta _2 d\beta _1 } }$.

For the SBP–NLMS algorithm, $\psi \left( e \right) = e$, $C_\psi \left( {\sigma _e^2 } \right) = 1$, and $B_\psi \left( {\sigma _e^2 } \right) = \sigma _e^2 = {\mathbf{v}}^T {\mathbf{B}}_i {\mathbf{v}} + \sigma _g^2$. Thus,

$${\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} \overline {\mathbf{F}} _i \left( 0 \right){\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} = \sigma _e^2 {\mathbf{I}}_{1i} + {\mathbf{I}}_{2i} .$$

(B-15)

2.1 Derivation of I_1i

$$\begin{array}{*{20}c} {{\mathbf{I}}_{1i} = \int_0^\infty {\int_{\beta _1 }^\infty {\gamma _i \left( {\beta _2 } \right){\mathbf{P}}_i^T } } \cdot \left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi} \Lambda _{Xi} \left( {{\mathbf{I}}_P + 2\beta \Lambda _{Xi} } \right)^{ - 1} {\mathbf{U}}_{Xi}^T } & {{\mathbf{U}}_{Xi} \left( {{\mathbf{I}} + 2\beta \Lambda _{Xi} } \right)^{ - 1} {\mathbf{U}}_{Xi}^T {\mathbf{R}}_{Xi\_01} } \\ 0 & 0 \\ \end{array} } \right] \cdot {\mathbf{P}}_i {\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} d\beta _2 d\beta _1 } \\ { = \int_0^\infty {\int_{\beta _1 }^\infty {\gamma _i \left( {\beta _2 } \right){\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi} \Lambda _{Xi} \left( {{\mathbf{I}} + 2\beta \Lambda _{Xi} } \right)^{ - 1} {\mathbf{U}}_{Xi}^T } & 0 \\ 0 & 0 \\ \end{array} } \right]{\mathbf{P}}_i d\beta _2 d\beta _1 } } } \\ { = {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi} \left( {\Lambda _{Xi} {\mathbf{I}}\prime \left( {\Lambda _{Xi} } \right)} \right){\mathbf{U}}_{Xi}^T } & 0 \\ 0 & 0 \\ \end{array} } \right]{\mathbf{P}}_i ,} \\ \end{array} $$

(B-16)

where $\left[ {{\mathbf{I}}\prime \left( {\Lambda _{Xi} } \right)} \right]_{k,k} = \int_0^\infty {\frac{{\beta \exp \left( { - \beta \varepsilon } \right)}}{{\left( {2\beta \lambda _{Xi,k} + 1} \right)}}\mathop \Pi \limits_{j = 1}^p \left( {2\beta \lambda _{Xi,j} + 1} \right)^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} d\beta } ,$

$$k = 1,2, \cdots ,P,$$

(B-17)

which is another defined special integral function.

2.2 Derivation of I _2i

Since

$$\begin{array}{*{20}l} {{{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{B}}_{i} } \hfill} \\ {{ = {\mathbf{P}}^{T}_{i} {\left[ {\begin{array}{*{20}c} {{{\mathbf{U}}_{{Xi}} \Lambda _{{Xi}} {\left( {{\mathbf{I}}_{p} + 2\beta \Lambda _{{xi}} } \right)}^{{ - 1}} {\mathbf{U}}^{T}_{{Xi}} }} & {{{\mathbf{U}}_{{Xi}} {\left( {{\mathbf{I}}_{p} + 2\beta \Lambda _{{Xi}} } \right)}^{{ - 1}} {\mathbf{U}}^{T}_{{Xi}} {\mathbf{R}}_{{Xi\_01}} }} \\ {0} & {0} \\ \end{array} } \right]}{\mathbf{P}}_{i} } \hfill} \\ {{ = {\mathbf{P}}^{T}_{i} {\left[ {\begin{array}{*{20}l} {{{\mathbf{U}}_{{Xi}} } \hfill} & {0 \hfill} \\ {0 \hfill} & {0 \hfill} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{{\left( {{\mathbf{I}}_{p} + 2\beta \Lambda _{{Xi}} } \right)}^{{ - 1}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{\Lambda _{{Xi}} {\mathbf{U}}^{T}_{{Xi}} }} & {{{\mathbf{U}}^{T}_{{Xi}} {\mathbf{R}}_{{Xi\_01}} }} \\ {0} & {0} \\ \end{array} } \right]}{\mathbf{P}}_{i} ,} \hfill} \\ \end{array} $$

Similarly,

$${\mathbf{B}}_i {\mathbf{S}}_{\mathbf{X}}^{\left( i \right)} = {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c}{{\mathbf{U}}_{Xi} \Lambda _{Xi} } & 0 \\{{\mathbf{R}}_{Xi\_01}^T {\mathbf{U}}_{Xi} } & 0 \\\end{array} } \right]\left[ {\begin{array}{*{20}c}{\left( {{\mathbf{I}}_P + 2\beta \Lambda _{Xi} } \right)^{ - 1} } & 0 \\0 & 0 \\\end{array} } \right]\left[ {\begin{array}{*{20}c}{{\mathbf{U}}_{Xi}^T } & 0 \\0 & 0 \\\end{array} } \right]{\mathbf{P}}.$$

Thus

$$\begin{array}{*{20}l} {{{\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} {\mathbf{B}}_{i} {\mathbf{vv}}^{T} {\mathbf{B}}_{i} {\mathbf{S}}^{{{\left( i \right)}}}_{{\mathbf{X}}} } \hfill} \\ {\begin{aligned} & = {\mathbf{P}}^{T}_{i} {\left[ {\begin{array}{*{20}c} {{{\mathbf{U}}_{{Xi}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{{\left( {{\mathbf{I}} + 2\beta \Lambda _{{Xi}} } \right)}^{{ - 1}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{\Lambda _{{Xi}} {\mathbf{U}}^{T}_{{Xi}} }} & {{{\mathbf{U}}^{T}_{{Xi}} {\mathbf{R}}_{{Xi\_01}} }} \\ {0} & {0} \\ \end{array} } \right]}{\mathbf{P}}_{i} \\ & \,\,\,\,\, \cdot {\mathbf{vv}}^{T} {\mathbf{P}}^{T}_{i} {\left[ {\begin{array}{*{20}c} {{{\mathbf{U}}_{{Xi}} \Lambda _{{Xi}} }} & {0} \\ {{{\mathbf{R}}^{T}_{{Xi\_01}} {\mathbf{U}}_{{Xi}} }} & {0} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{{\left( {{\mathbf{I}}_{p} + 2\beta \Lambda _{{Xi}} } \right)}^{{ - 1}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{{\mathbf{U}}^{T}_{{Xi}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\mathbf{P}}_{i} \\ \end{aligned} \hfill} \\ {{ = {\mathbf{P}}^{T}_{i} {\left[ {\begin{array}{*{20}c} {{{\mathbf{U}}_{{Xi}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{{\mathbf{D}}_{{Xi}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\mathbf{R}}_{{i,v}} {\left[ {\begin{array}{*{20}c} {{{\mathbf{D}}_{{Xi}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\left[ {\begin{array}{*{20}c} {{{\mathbf{U}}^{T}_{{Xi}} }} & {0} \\ {0} & {0} \\ \end{array} } \right]}{\mathbf{P}},} \hfill} \\ \end{array} $$

(B-18)

where ${\mathbf{R}}_{i,v} = \left[ {\begin{array}{*{20}c} {\Lambda _{Xi} {\mathbf{U}}_{Xi}^T } & {{\mathbf{U}}_{Xi}^T {\mathbf{R}}_{Xi\_01} } \\ 0 & 0 \\ \end{array} } \right]{\mathbf{P}}_i {\mathbf{vv}}^T {\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi} \Lambda _{Xi} } & 0 \\ {{\mathbf{R}}_{Xi\_01}^T {\mathbf{U}}_{Xi} } & 0 \\ \end{array} } \right]$, and ${\mathbf{D}}_{Xi} = \left( {{\mathbf{I}} + 2\beta \Lambda _{Xi} } \right)^{ - 1}$. Thus, the (m, n)-th element of ${\mathbf{D}}_i^\prime = \int_0^\infty {\int_{\beta _1 }^\infty {\gamma _i \left( {\beta _2 } \right)\left[ {\begin{array}{*{20}c} {{\mathbf{D}}_{Xi} } & 0 \\ 0 & 0 \\ \end{array} } \right]{\mathbf{R}}_{i,v} \left[ {\begin{array}{*{20}c} {{\mathbf{D}}_{Xi} } & 0 \\ 0 & 0 \\ \end{array} } \right]d\beta _2 d\beta _1 } }$ is given by

$$\begin{array}{*{20}c} {{\left[ {{\mathbf{D}}^{\prime }_{i} } \right]}_{{m,n}} = {\left[ {{\mathbf{R}}_{{i,v}} } \right]}_{{m,n}} {\left( {{\int_0^\infty {{\int_{\beta _{1} }^\infty {\gamma _{i} {\left( {\beta _{2} } \right)}d_{{mm}} {\left( n \right)}d_{{nn}} {\left( n \right)}d\beta _{2} d\beta _{1} } }} }} \right)}} \\ { = {\left[ {{\mathbf{R}}_{{i,v}} } \right]}_{{m,n}} I^{{\prime \prime }}_{{mn}} {\left( {\Lambda _{{Xi}} } \right)},{\text{ m, n = 1,}}...{\text{,P, and 0 elsewhere}}{\text{.}}} \\ \end{array} $$

(B-19)

We now evaluate $I_{mn}^{\prime \prime } \left( {\Lambda _{Xi} } \right)$:

$$\begin{array}{*{20}c} {I_{mn}^{\prime \prime } \left( {\Lambda _{Xi} } \right) = \int_0^\infty {\int_{\beta _1 }^\infty {\gamma _i \left( {\beta _2 } \right)} d_{mm} \left( n \right)d_{nn} \left( n \right)d\beta _2 d\beta _1 } } \\ { = \int_0^\infty {\beta _2 \gamma _i \left( {\beta _2 } \right)d_{mm} \left( n \right)d_{nn} \left( n \right)} d\beta _2 } \\ { = \int_0^\infty {\frac{{\beta \exp \left( { - \beta \varepsilon } \right)}}{{\left( {2\beta \lambda _{Xi,m} + 1} \right)\left( {2\beta \lambda _{Xi,n} + 1} \right)}}\mathop \Pi \limits_{j = 1}^P \left( {2\beta \lambda _{Xi,j} + 1} \right)^{ - \tfrac{1}{2}} d\beta } } \\ \end{array} $$

(B-20)

is the third defined special integral function. Summing up the above results yields

$${\mathbf{I}}_{2i} = 2{\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi} } & 0 \\ 0 & 0 \\ \end{array} } \right]\left( {{\mathbf{R}}_{i,v} \circ \left[ {\begin{array}{*{20}c} {{\mathbf{I}}\prime \prime \left( {\Lambda _{Xi} } \right)} & 0 \\ 0 & 0 \\ \end{array} } \right]} \right)\left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi}^T } & 0 \\ 0 & 0 \\ \end{array} } \right]{\mathbf{P}}_i ,$$

(B-21)

where $\left[ {{\mathbf{I}}\prime \prime \left( {\Lambda _{Xi} } \right)} \right]$is a P×P matrix whose (m, n)-th element is $I_{mn}^{\prime \prime } \left( {\Lambda _{Xi} } \right)$ and $\circ$ denotes element-wise (Hadamard) product of two matrices.

Substituting (B-14), (B-16) and (B-21) into (B-3) yields

$$\begin{array}{*{20}c} {{\mathbf{s}}_3 = \frac{1}{C}\sum\limits_{i = 1}^C {\left( {B_\psi \left( {\sigma _e^2 } \right){\mathbf{I}}_{1i} + C_\psi \left( {\sigma _e^2 } \right){\mathbf{I}}_{2i} } \right)} } \\ { = \frac{1}{C}\sum\limits_{i = 1}^C {\left\{ {B_\psi \left( {\sigma _e^2 } \right){\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi} \left( {\Lambda _{Xi} {\mathbf{I}}\prime \left( {\Lambda _{Xi} } \right)} \right){\mathbf{U}}_{Xi}^T } & 0 \\ 0 & 0 \\ \end{array} } \right]{\mathbf{P}}_i } \right. + \left. {2C_\psi \left( {\sigma _e^2 } \right){\mathbf{P}}_i^T \left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi} } & 0 \\ 0 & 0 \\ \end{array} } \right]\left( {{\mathbf{R}}_{i,v} \circ \left[ {\begin{array}{*{20}c} {{\mathbf{I}}\prime \prime \left( {\Lambda _{Xi} } \right)} & 0 \\ 0 & 0 \\ \end{array} } \right]} \right)\left[ {\begin{array}{*{20}c} {{\mathbf{U}}_{Xi}^T } & 0 \\ 0 & 0 \\ \end{array} } \right]{\mathbf{P}}_i } \right\}.} } \\ \end{array} $$

(B-22)