Outage Probability Based Robust Distributed Beam-Forming in Multi-User Cooperative Networks with Imperfect CSI

Abstract

In this paper, a new robust problem is proposed for relay beamforming in relay system with stochastic perturbation on channels of multi user and relay network. The robust problem aims to minimize the transmission power of relay nodes while the imperfect channel information (CSI) injects stochastic channel uncertainties to the parameters of optimization problem. In the power minimization framework, the relays amplification weights and phases are optimized assuming the availability of Gaussian channel distribution. The power sum of all relays is minimized while the outage probability of the instantaneous capacity (or SINR) at each link is above the outage capacity (or SINR) for each user. The robust problem is a nonconvex SDP problem with Rank constraint. Due to the nonconvexity of the original problem, three suboptimal problems are proposed. Simulation and numerical results are presented to compare the performance of the three proposed solutions with the existing worst case robust method.

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Notes

  1. 1.

    The function \(\lambda _{max} \left( {\mathbf{X}} \right) \) is non-differentiable when the multiplicity order of maximum eigenvalue is more than one.

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Correspondence to Behrad Mahboobi.

Appendices

Appendix I: Proof of (13) and (14)

Proof of (13) and (14) The mean of \(Z_k\) can be written as

$$\begin{aligned} {\mu _{{Z_k}}}&= {\hbox {r}}\left\{ {{{\mathbf{w}}^\mathrm{H}} \hbox {E} \left( {{\mathbf{R}}_\mathrm{h}^\mathrm{k}{\mathbf{- }}{\gamma _\mathrm{k}}{{\mathbf{Q}}_\mathrm{k}}{\mathbf{- }}{\gamma _\mathrm{k}}{{\mathbf{D}}_\mathrm{k}}} \right) {\mathbf{w}}} \right\} \nonumber \\&= \hbox {Tr}\left\{ {{{\mathbf{w}}^\mathrm{H}}{{\mu }_{{{\mathbf{Y}}_\mathrm{k}}}}{\mathbf{w}}} \right\} \nonumber \\&= \hbox {Tr}\left( {{\mathbf{X}}{{\mu }_{{{\mathbf{Y}}_\mathrm{k}}}}} \right) \end{aligned}$$
(58)

Note that, the expectation is taken over fading channel coefficients i.e. \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\). By the above relation (13) is deduced. The variance of \({Z_k}\) is computed as follows

$$\begin{aligned} E\left[ {{Z_k}{Z_k^H}} \right]&= E\left\{ {\mathrm{Tr}\left[ {\left( {{{\mathbf{w}}^\mathrm{H}}{{\mathbf{Y}}_\mathrm{k}}{\mathbf{w}}} \right) {{\left( {{{\mathbf{w}}^\mathrm{H}}{{\mathbf{Y}}_\mathrm{k}}{\mathbf{w}}} \right) }^*}} \right] } \right\} \nonumber \\&= E\left\{ {\mathrm{Tr}\left[ {{{\mathbf{w}}^\mathrm{H}}{{\mathbf{Y}}_\mathrm{k}}{\mathbf{w}}} \right] \mathrm{Tr}{{\left[ {{{\mathbf{w}}^\mathrm{T}}{{\mathbf{Y}}_\mathrm{k}}{\mathbf{w}}} \right] }^*}} \right\} \nonumber \\&= E\left\{ {\mathrm{Tr}\left[ {{\mathbf{X}}{{\mathbf{Y}}_\mathrm{k}}} \right] \mathrm{Tr}{{\left[ {{\mathbf{X}}{{\mathbf{Y}}_\mathrm{k}}} \right] }^*}} \right\} \end{aligned}$$
(59)

To simplify the above expression, consider the following Lemma.

Lemma 1

If \({\mathbf{X}} \in {R^{m \times n}},{\mathbf{Y}} \in {R^{n \times m}}\), then the following equation holds.

$$\begin{aligned} \mathrm{Tr}\left[ {{\mathbf{XY}}} \right] = \mathrm{vec}{\left( {\mathbf{X}} \right) ^\mathrm{T}}\mathrm{vec}\left( {\mathbf{Y}} \right) \end{aligned}$$
(60)

Proof

The proof is easily concluded by using [60]. \(\square \)

Using the above lemma, (59) can be written as

$$\begin{aligned} E\left[ {{Z_k}{Z_k}^H} \right]&= E\left\{ {vec{{\left( {{{\mathbf{X}}^T}} \right) }^T}vec\left( {{{\mathbf{Y}}_k}} \right) vec{{\left( {{{\mathbf{Y}}_k}} \right) }^H}vec\left( {{{\mathbf{X}}^H}} \right) } \right\} \nonumber \\&= vec{\left( {\mathbf{X}} \right) ^H}E\left\{ {vec\left( {{{\mathbf{Y}}_k}} \right) vec{{\left( {{{\mathbf{Y}}_k}} \right) }^H}} \right\} vec\left( {\mathbf{X}} \right) \end{aligned}$$
(61)

Therefore, the variance of \({Z_k}\) is as follows

$$\begin{aligned} \sigma _{^{{Z_k}}}^2&= E\left[ {{Z_k}{Z_k}^H} \right] - {\mu _{{Z_k}}}{\left( {{\mu _{{Z_k}}}} \right) ^H}\nonumber \\&= vec{\left( {\mathbf{X}} \right) ^H}{E\left\{ {{{\mathbf{B}}_k}{\mathbf{B}}_k^H} \right\} } vec\left( {\mathbf{X}} \right) - vec{\left( {\mathbf{X}} \right) ^H}E\left\{ {{{\mathbf{B}}_k}} \right\} E\left\{ {{\mathbf{B}}_k^H} \right\} vec\left( {\mathbf{X}} \right) \end{aligned}$$
(62)

Since \(X\) is a positive semi-definite matrix, (62) can be rewritten as

$$\begin{aligned} \sigma _{^{{Z_k}}}^2 = vec{\left( {{{\mathbf{X}}^H}} \right) ^H}{\Omega _k}\,\,vec\left( {{{\mathbf{X}}^H}} \right) \end{aligned}$$
(63)

where

$$\begin{aligned} {{\varvec{\Phi }}_k} \mathop {=}\limits ^{\varDelta }&\, E\left( {{{\mathbf{B}}_k}{\mathbf{B}}_k^H} \right) \end{aligned}$$
(64)
$$\begin{aligned} {{\varvec{\Psi }}_k} \mathop {=}\limits ^{\varDelta }&\, E\left( {{{\mathbf{B}}_k}} \right) E{\left( {{{\mathbf{B}}_k}} \right) ^H}\end{aligned}$$
(65)
$$\begin{aligned} {{\varvec{\Omega }}_k} \mathop {=}\limits ^{\varDelta }&\, {{\varvec{\Phi }}_k} - {{\varvec{\Psi }}_k} = {{\mathbf{L}}_k}{\mathbf{L}}_k^H \end{aligned}$$
(66)

Using \(S = {\mathbf{L}}_k^Hvec\left( {\mathbf{X}} \right) \), the variance of \({Z_k}\) is expressed as follows

$$\begin{aligned} \sigma _{^{{Z_k}}}^2 = vec{\left( {\mathbf{X}} \right) ^H}{{\mathbf{L}}_k}{\mathbf{L}}_k^Hvec\left( {\mathbf{X}} \right) = S_k^H{S_k} \end{aligned}$$
(67)

Some Lemmas

Lemma 2

Let \({\mathbf{A}} = \left( {\begin{array}{cc} {\mathbf{B}}&{}{{{\mathbf{C}}^H}}\\ {\mathbf{C}}&{}{\mathbf{D}} \end{array}} \right) \) be a symmetric matrix with \(k \times k\) block \({\mathbf{B}}\) and \(l \times l\) block \({\mathbf{D}}\). Assume that \({\mathbf{B}}\) is a positive definite matrix. Then \({\mathbf{A}}\) is positive (semi) definite if and only if the matrix \({\mathbf{D}} - {\mathbf{C}}{{\mathbf{B}}^{ - 1}}{{\mathbf{C}}^H}\) is positive (semi) definite (this matrix is called the Schur complement of \({\mathbf{B}}\) in \({\mathbf{A}}\)).

Proof

See [51]. \(\square \)

Lemma 3

Let \(\mathbf {x}\in \mathbb {C}^{m-1}\), \(t \in \mathbb {\mathfrak {R}} \) and \({{\mathbf{I}}_{m - 1}}\) be a \(\left( {m - 1} \right) \times \left( {m - 1} \right) \) identity matrix. Then the cone \({{\mathcal {L}}^m}\), \(m > 1\), is Semi-definite representable, i.e.

$$\begin{aligned} \left( {\begin{array}{c} \mathbf {x}\\ t\end{array}} \right) \in {{\mathcal {L}}^m} \Leftrightarrow \mathbf {A}\left( {\mathbf {x},t} \right) = \left( {\begin{array}{cc} {t{{\mathbf{I}}_{m - 1}}}&{}\mathbf {x}\\ {{\mathbf {x}^T}}&{}t \end{array}} \right) \succeq 0. \end{aligned}$$
(68)

Proof

To prove the this lemma, we should note the definition of Lorentz cone which is \(\left( {\begin{array}{c} \mathbf {x}\\ t \end{array}} \right) \in {{\mathcal {L}}^m}\) if and only if \(\left\| \mathbf {x}\right\| _2 \le t\). By choosing \(\mathbf {C}=\mathbf {x}^T\), \(\mathbf {D}=t\) and \(\mathbf {B}=t.\mathbf {I}_m-1\) and using Lemma 2, the proof is completed. \(\square \)

Appendix II

The aim of this appendix is to show that the values of \({{\mu }_{{{\mathbf{Y}}_k}}}\) and \({{\varvec{\Phi }}_k}\) that are used in (13) and (14) can be computed based on first, second and forth order cumulants of \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\). To proceed with the proof of the above claim, the following lemma is needed.

Lemma 4

For \({\mathbf{x}},{\mathbf{y}} \in {R^{m \times 1}}\),

$$\begin{aligned} \left( {{\mathbf{x}} \odot {\mathbf{y}}} \right) {\left( {{\mathbf{x}} \odot {\mathbf{y}}} \right) ^H} = {\mathbf{x}}{{\mathbf{x}}^H} \odot {\mathbf{y}}{{\mathbf{y}}^H} \end{aligned}$$
(69)

Proof

By defining \({\mathbf{u}} \,{=}\, \left( {{\mathbf{x}} \odot {\mathbf{y}}} \right) {\left( {{\mathbf{x}} \odot {\mathbf{y}}} \right) ^H}\), the \(\left( {i,j} \right) \)th component of \({\mathbf{u}}\) is \({u_{i,j}} \,{=}\, \left( {{x_i}{y_i}} \right) {\left( {{x_j}{y_j}} \right) ^*}\). By rewriting, the \(\left( {i,j} \right) \)th component of \({\mathbf{v}} = {\mathbf{x}}{{\mathbf{x}}^H} \odot {\mathbf{y}}{{\mathbf{y}}^H}\) as \({v_{i,j}} = {x_i}x_j^*.{y_i}y_j^*\), we conclude.

Since \({{\mu }_{{{\mathbf{Y}}_k}}}\) can be written as

$$\begin{aligned} {{\mu }_{{{\mathbf{Y}}_k}}} =&\, E\left\{ {{\mathbf{R}}_h^k} \right\} - {\gamma _k}E\left\{ {{{\mathbf{Q}}_k}} \right\} - {\gamma _k}E\left\{ {{{\mathbf{D}}_k}} \right\} \nonumber \\=&\, {{\mu }_{{\mathbf{R}}_h^k}} - {\gamma _k}{{\mu }_{{{\mathbf{Q}}_k}}} - {\gamma _k}{{\mu }_{{{\mathbf{D}}_k}}} \end{aligned}$$
(70)

It is sufficient to interpret the terms \({{\mu }_{{\mathbf{R}}_h^k}}\), \(\,{{\mu }_{{{\mathbf{Q}}_k}}}\), \({{\mu }_{{{\mathbf{D}}_k}}}\) versus the statistics \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\). Using the fact that \({\left( {X \odot Y} \right) ^H} = X^H \odot {Y^H}\) and using Lemma 4, the first term in the right hand side of (70) can be written as

$$\begin{aligned} {{\mu }_{{\mathbf{R}}_h^k}} = {P_k}E\left[ {\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right] \end{aligned}$$
(71)

Since \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\) are independent random vectors

$$\begin{aligned} {{\mu }_{{\mathbf{R}}_h^k}} =&\, {P_k}E\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot E\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) \nonumber \\=&\, {P_k}{\mathbf {R}_{{g_k}}} \odot {\mathbf {R}_{{f_k}}} \end{aligned}$$
(72)

where \({\mathbf {R}_{{f_k}}}\) and \({\mathbf {R}_{{g_k}}}\) can be interpreted versus first and second order cumulants of \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\). obviously we can write

$$\begin{aligned} {{\mu }_{{{\mathbf{Q}}_k}}} = \sum \limits _{p \ne k} {{P_p}E\left( {{{\mathbf{g}}_p}{\mathbf{g}}_p^H} \right) \odot E\left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) } = \sum \limits _{p \ne k}^{} {{P_p}{{\mathbf{R}}_{{g_p}}} \odot {{\mathbf{R}}_{{f_p}}}} \end{aligned}$$
(73)

\({{\mu }_{{{\mathbf{D}}_k}}}\) and \({{\mu }_{{{\mathbf{D}}}}}\) can be easily written as

$$\begin{aligned} {{\mu }_{{{\mathbf{D}}_k}}} =&E \, {{\mathbf{D}}_k} = {\sigma }_v^2 {\mathbf{R}}_{\mathbf{g}_\mathbf{k}} \odot \mathbf {I}\end{aligned}$$
(74)
$$\begin{aligned} {{\mu }_{{{\mathbf{D}}}}} =&E \, {{\mathbf{D}}} = {\mathbf{R}}_x \odot \mathbf {I} \end{aligned}$$
(75)

To express \({{\varvec{\Phi }}_k}\) versus cumulants of \({\mathbf {R}_{{f_k}}}\) and \({\mathbf {R}_{{\mathbf{g}_\mathbf{k}}}}\), consider the following definition

$$\begin{aligned} {{\varvec{\Phi }}_k} = E\left\{ {{{\mathbf{B}}_k}{\mathbf{B}}_k^H} \right\} \end{aligned}$$
(76)

where \({{\mathbf{B}}_k} = vec\left( {{\mathbf{R}}_h^k - {\gamma _k}{{\mathbf{Q}}_k} - {\gamma _k}{{\mathbf{D}}_k}} \right) \). By substituting \(\mathbf{R}_\mathbf{h}^\mathbf{k}\), \({\mathbf{Q}_\mathbf{k}}\), and \({{\mathbf{D}}_k}\) in (76), \({{\mathbf{B}}_k}\) can be calculated versus channel coefficients as

$$\begin{aligned} {{\mathbf{B}}_k} =&\, vec\left[ {{P_k}\left( {{{\mathbf{g}}_k} \odot {{\mathbf{f}}_k}} \right) \left( {{\mathbf{g}}_k^H \odot {\mathbf{f}}_k^H} \right) } \right. \nonumber \\&-\, {\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k} \odot {{\mathbf{f}}_p}} \right) \left( {{\mathbf{g}}_k^H \odot {\mathbf{f}}_p^H} \right) - } \left. {{\gamma _k}\sigma _v^2diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right] \end{aligned}$$
(77)

\(\square \)

Using the Lemma 4, (77) can be simplified as

$$\begin{aligned} {{\mathbf{B}}_k}\, =\,&vec\left[ {{P_k}\left( {{{\mathbf{g}}_k}{{\mathbf{f}}_k}} \right) \odot \left( {{\mathbf{g}}_k^H{\mathbf{f}}_k^H} \right) } \right. \nonumber \\ -&{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k}{{\mathbf{f}}_p}} \right) \odot \left( {{\mathbf{g}}_k^H{\mathbf{f}}_p^H} \right) - } \left. {{\gamma _k}\sigma _v^2diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right] \end{aligned}$$
(78)

By substituting (78) in (76) and expanding it, we get to

$$\begin{aligned} {{\varvec{\Phi }}_k} = \,{\varvec{\Phi }}_k^1 + {\varvec{\Phi }}_k^2 + {\varvec{\Phi }}_k^3 - {\varvec{\Phi }}_k^4 - {\varvec{\Phi }}_k^5 - {\varvec{\Phi }}_k^6 - {\varvec{\Phi }}_k^7 + {\varvec{\Phi }}_k^8 + {\varvec{\Phi }}_k^9 \end{aligned}$$
(79)

where

$$\begin{aligned}&\displaystyle {\varvec{\Phi }}_k^1 = E\left\{ {vec\left( {{P_k}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right) vec{{\left( {{P_k}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right) }^H}} \right\}&\end{aligned}$$
(80)
$$\begin{aligned}&\displaystyle {\varvec{\Phi }}_k^2 = E\left\{ \!{vec\left( {{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) } } \right) \!\!vec{{\left( {{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) } } \right) }^H}} \right\}&\nonumber \\ \end{aligned}$$
(81)
$$\begin{aligned}&\displaystyle {\varvec{\Phi }}_k^3 = E\left\{ {vec\left( {{\gamma _k}\sigma _v^2diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right) vec{{\left( {{\gamma _k}\sigma _v^2diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right) }^H}} \right\}&\end{aligned}$$
(82)
$$\begin{aligned}&\displaystyle {\varvec{\Phi }}_k^4 = {\left( {{\varvec{\Phi }}_k^5} \right) ^H} \!\!= E\!\!\left\{ \!{vec\left( {{P_k}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right) vec{{\left( {{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) } } \right) }^H}} \right\}&\nonumber \\ \end{aligned}$$
(83)
$$\begin{aligned}&\displaystyle {\varvec{\Phi }}_k^6 = {\left( {{\varvec{\Phi }}_k^7} \right) ^H} = E\left\{ {vec\left( {{P_k}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right) vec{{\left( {{\gamma _k}\sigma _v^2diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right) }^H}} \right\}&\end{aligned}$$
(84)
$$\begin{aligned}&\displaystyle {\varvec{\Phi }}_k^8 = {\left( {{\varvec{\Phi }}_k^9} \right) ^H} = E\left\{ {vec\left( {{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) } } \right) vec{{\left( {{\gamma _k}\sigma _v^2diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right) }^H}} \right\}&\nonumber \\ \end{aligned}$$
(85)

To compute \({{\varvec{\Phi }}_k}\), each of the nine terms will be computed sequentially versus the statistics of complex symmetric Gaussian vectors \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\).

Using Lemma 4, \({\varvec{\Phi }}_k^1\) can be simplified as

$$\begin{aligned} {\varvec{\Phi }}_k^1 =&\, P_k^2E\left[ {vec\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) vec{{\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) }^H}} \right] \odot E\left[ {vec\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) vec{{\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) }^H}} \right] \nonumber \\=&\, P_k^2{{\mathbf{G}}_k} \odot {{\mathbf{F}}_k} \end{aligned}$$
(86)

where \({{\mathbf{G}}_k}\) and \({{\mathbf{F}}_k}\) are

$$\begin{aligned} {{\mathbf{G}}_k}\! =&\,{E\left[ {vec\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) vec{{\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) }^H}} \right] = E\!\left[ {\left( {{\mathbf{g}}_k^* \!\otimes \!{{\mathbf{g}}_k}} \right) \!\!\left( {{{\mathbf{g}}_k}^T \otimes {\mathbf{g}}_k^H} \right) } \right] = E\!\left[ {\left( {{\mathbf{g}}_k^*{{\mathbf{g}}_k}^T} \right) \otimes \left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) } \right] }\nonumber \\ {{\mathbf{F}}_k} =&\, {E\left[ {vec\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) vec{{\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) }^H}} \right] =E\left[ {\left( {{\mathbf{f}}_k^* \otimes {{\mathbf{f}}_k}} \right) \left( {{{\mathbf{f}}_k}^T \otimes {\mathbf{f}}_k^H} \right) } \right] = E\left[ {\left( {{\mathbf{f}}_k^*{{\mathbf{f}}_k}^T} \right) \otimes \left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right] } \end{aligned}$$
(87)

The entries of \({{\mathbf{F}}_k}\) and \({{\mathbf{G}}_k}\) are calculated using up to fourth order moments of random components of \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\), respectively. We will show later that all of the terms of \({\varvec{\Phi }}_k^{j}, j=1,2,\ldots ,9\) can be written as functions of \({{\mathbf{F}}_k}\) and \({{\mathbf{G}}_k}\). For the numerical results section, we need to compute \({{\mathbf{F}}_k}\) and \({{\mathbf{G}}_k}\) for normal distribution, hence we calculate these terms in the sequel. Since \({{\mathbf{G}}_k}\) is similar to \({{\mathbf{F}}_k}\), we only compute the closed form value of \({{\mathbf{F}}_k}\). by denoting \({{\hat{\mathbf{f}}}_k}\) and \({\hat{\mathbf{g}}_k}\) respectively as the perturbation vector of \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\) which are i.i.d complex symmetric gaussian vector with zero mean variance \(\sigma _f^2\), we can write

$$\begin{aligned} {\hat{\mathbf{F}}_k} =&\, E\left[ {\left( {\hat{\mathbf{f}}_k^*{\hat{\mathbf{f}}_k}^T} \right) \otimes \left( {{\hat{\mathbf{f}}_k}\hat{\mathbf{f}}_k^H} \right) } \right] \nonumber \\ =&\, E\left( {\hat{\mathbf{f}}_k^*{\hat{\mathbf{f}}_k}^T} \right) \otimes E\left( {{\hat{\mathbf{f}}_k}\hat{\mathbf{f}}_k^H} \right) + E\left( {\hat{\mathbf{f}}_k^* \otimes {\hat{\mathbf{f}}_k}} \right) E\left( {{\hat{\mathbf{f}}_k}^T \otimes \hat{\mathbf{f}}_k^H} \right) \nonumber \\&+\,\sum \limits _{j = 1}^R {\left( {m_4^{{f_{k,j}}} - 2m_2^{{f_{k,j}}}} \right) \left( {{\mathbf{e}}_j^*{{\mathbf{e}}_j}^T} \right) \otimes \left( {{{\mathbf{e}}_j}{\mathbf{e}}_j^H} \right) } \end{aligned}$$
(88)

where \({\left( {{{\mathbf{e}}_j}} \right) _{R \times 1}} = {[\underbrace{0,0,\ldots ,0}_{j - 1},1,0,\ldots 0]^T}\) and the second and forth order moment of \(f_{k,j}\), \(j{th}\) element of \(\hat{\mathbf{f}}_k\), are respectively \(m_2^{{f_{k,j}}} = \sigma _{{f_{k,j}}}^2\) and \( m_4^{{f_{k,j}}} = 2{\left( {m_2^{{f_{k,j}}}} \right) ^2} = 2\sigma _{{f_{k,j}}}^4\) . The first two terms of the right hand side of (88) can be computed using the following relations

$$\begin{aligned} {{\tilde{R}}_{{\hat{\mathbf{f}}_k}}} \mathop {=}\limits ^{\varDelta }&\, E\left( {\hat{\mathbf{f}}_k^* \otimes {\hat{\mathbf{f}}_k}} \right) = {\left[ {E\left( {{\hat{\mathbf{f}}_k}^T \otimes \hat{\mathbf{f}}_k^H} \right) } \right] ^T} = \sum \limits _{j = 1}^R {\sigma _{{f_{k,j}}}^2{\mathbf{e}}_j^* \otimes {{\mathbf{e}}_j}} \end{aligned}$$
(89)
$$\begin{aligned} {R_{{\hat{\mathbf{f}}_k}}} =\,&E\left( {{\hat{\mathbf{f}}_k}\hat{\mathbf{f}}_k^H} \right) = {\left[ {E\left( {\hat{\mathbf{f}}_k^*{\hat{\mathbf{f}}_k}^T} \right) } \right] ^T}=\sum \limits _{j = 1}^R {\sigma _{{f_{k,j}}}^2{\mathbf{e}}_j^* {{\mathbf{e}}_j}^T} \end{aligned}$$
(90)

By writting \({\mathbf{f}}_k\) (and similarly \({\mathbf{g}}_k\)) as the following sum of zero mean random vector and a constant mean vector

$$\begin{aligned} {\mathbf{f}}_k^{}= {{\hat{\mathbf{f}}}}_k + {{\bar{\mathbf{f}}}}_k^{} \end{aligned}$$
(91)

we can obtain \(\mathbf {F}_k\) by some algebraic manipulation

$$\begin{aligned} {{{\mathbf{F}}}_k} =\,&E\left[ {\left( {{\mathbf{f}}_k^*{{{\mathbf{f}}}_k}^T} \right) \otimes \left( {{{{\mathbf{f}}}_k}{\mathbf{f}}_k^H} \right) } \right] = E\left[ {\left( {\left( {{{\hat{\mathbf{f}}}}_k^* + {{\bar{\mathbf{f}}}}_k^*} \right) \left( {{{{\hat{\mathbf{f}}}}_k}^T + {{{{\bar{\mathbf{f}}}}}_k}^T} \right) } \right) \otimes \left( {\left( {{{\hat{\mathbf{f}}}}_k^{} + {{\bar{\mathbf{f}}}}_k^{}} \right) \left( {{{{\hat{\mathbf{f}}}}_k}^H + {{{{\bar{\mathbf{f}}}}}_k}^H} \right) } \right) } \right] \nonumber \\ =&\left( {{{\bar{\mathbf{f}}}}_k^*{{{{\bar{\mathbf{f}}}}}_k}^T} \right) \otimes \left( {{{\bar{\mathbf{f}}}}_k^{}{{{{\bar{\mathbf{f}}}}}_k}^H} \right) + \left( {{{\bar{\mathbf{f}}}}_k^*{{{{\bar{\mathbf{f}}}}}_k}^T} \right) \otimes E\left[ {{{{\hat{\mathbf{f}}}}_k}{{\hat{\mathbf{f}}}}_k^H} \right] + E\left[ {{{\hat{\mathbf{f}}}}_k^*{{{\hat{\mathbf{f}}}}_k}^T} \right] \otimes \left( {{{\bar{\mathbf{f}}}}_k^{}{{{{\bar{\mathbf{f}}}}}_k}^H} \right) \nonumber \\&+ \left( {{{\bar{\mathbf{f}}}}_k^* \otimes {{\bar{\mathbf{f}}}}} \right) E\left( {{{{\hat{\mathbf{f}}}}_k}^T \otimes {{\hat{\mathbf{f}}}}_k^H} \right) + E\left[ {{{\hat{\mathbf{f}}}}_k^* \otimes {{{\hat{\mathbf{f}}}}_k}} \right] \left( {{{{{\bar{\mathbf{f}}}}}_k}^T \otimes {{\bar{\mathbf{f}}}}_k^H} \right) +{{{\hat{\mathbf{F}}}}_k} \end{aligned}$$
(92)

The final simplified form of \({{{\mathbf{F}}}_k}\) can be written as

$$\begin{aligned} {{{\mathbf{F}}_k} }&= {\left( {{{\bar{\mathbf{f}}}}_k^*{{{{\bar{\mathbf{f}}}}}_k}^T} \right) \otimes \left( {{{\bar{\mathbf{f}}}}_k^{}{{{{\bar{\mathbf{f}}}}}_k}^H} \right) + \left( {{{\bar{\mathbf{f}}}}_k^*{{{{\bar{\mathbf{f}}}}}_k}^T} \right) \otimes {R_{{{{{\hat{\mathbf{f}}}}}_k}}} + {R_{{{{{\hat{\mathbf{f}}}}}_k}}}^T \otimes \left( {{{\bar{\mathbf{f}}}}_k^{}{{{{\bar{\mathbf{f}}}}}_k}^H} \right) }\nonumber \\&+\,{\left( {{{\bar{\mathbf{f}}}}_k^* \otimes {{\bar{\mathbf{f}}}}} \right) {{\tilde{R}}_{{{{{\hat{\mathbf{f}}}}}_k}}}^T + {{\tilde{R}}_{{{{{\hat{\mathbf{f}}}}}_k}}}\left( {{{{{\bar{\mathbf{f}}}}}_k}^T \otimes {{\bar{\mathbf{f}}}}_k^H} \right) + {R_{{{{{\hat{\mathbf{f}}}}}_k}}}^T \otimes {R_{{{{{\hat{\mathbf{f}}}}}_k}}} + {{\tilde{R}}_{{{{{\hat{\mathbf{f}}}}}_k}}}{{\tilde{R}}_{{{{{\hat{\mathbf{f}}}}}_k}}}^T} \end{aligned}$$
(93)

As mentioned earlier, calculation of \({{{\mathbf{G}}}_k}\) resembles that of \({{{\mathbf{F}}}_k}\).

Utilizing Lemma 4, (81) can be written as

$$\begin{aligned} {\varvec{\Phi }}_k^2 =\,&E\left\{ {vec\left( {{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) } } \right) vec{{\left( {{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot \left( {{{\mathbf{f}}_{p'}}{\mathbf{f}}_{p'}^H} \right) } } \right) }^H}} \right\} \nonumber \\ =\,&{{\mathbf{G}}_{\mathbf{k}}} \odot \sum \limits _{p' \in {{\hat{\mathbf{D}}}_k}}^{} {\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_{p'}}{P_p}E\left( {vec\left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) vec{{\left( {{{\mathbf{f}}_{p'}}{\mathbf{f}}_{p'}^H} \right) }^H}} \right) } } \nonumber \\ =\,&{{\mathbf{G}}_{\mathbf{k}}} \odot \left[ {\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}^2{{\mathbf{F}}_p}} + } \right. \left. {\sum \limits _{p' \in {{\hat{\mathbf{D}}}_k}}^{} {{P_{p'}}E\left[ {vec\left( {{{\mathbf{f}}_{p'}}{\mathbf{f}}_{p'}^H} \right) } \right] } \sum \limits _{p \in {{\hat{\mathbf{D}}}_k},p \ne p'}^{} {{P_p}E\left[ {vec{{\left( {{{\mathbf{f}}_p}{\mathbf{f}}_p^H} \right) }^H}} \right] } } \right] \nonumber \\ =\,&{{\mathbf{G}}_{\mathbf{k}}} \odot \left[ {\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}^2{{\mathbf{F}}_p}} + } \right. \left. {\sum \limits _{p' \in {{\hat{\mathbf{D}}}_k}}^{} {{P_{p'}}{\mathbf{R}}_{\mathbf{f}_{p'}}} \sum \limits _{p \in {{\hat{\mathbf{D}}}_k},p \ne p'}^{} {{P_p}E{{\left[ {{\mathbf{R}}_{\mathbf{f}_{p}}} \right] }^H}} } \right] \end{aligned}$$
(94)

where \({{\mathbf{R}}_{{{\mathbf{f}}_{\mathbf{p}}}}} = vec({{\mathbf{f}}_p}{\mathbf{f}}_p^H) = vec({{\mathbf{R}}_{{{{{\hat{\mathbf{f}}}}}_{\mathbf{p}}}}} + {{{\bar{\mathbf{f}}}}_p}{{\bar{\mathbf{f}}}}_p^H)\) consists of up to second order cumulants of \({{\mathbf{f}}_p}\). After that, (82) can be written as

$$\begin{aligned} {\varvec{\Phi }}_k^3&= \gamma _k^2\sigma _v^4E\left\{ {vec\left( {diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right) {} vec{{\left( {diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right) }^H}} \right\} \nonumber \\&=\gamma _k^2\sigma _v^4E\left[ {\left\{ {{{{{\tilde{\mathbf{g}}}}}_k} \odot {{{{\tilde{\mathbf{g}}}}}_k}^*} \right\} \left\{ {{{{{\tilde{\mathbf{g}}}}}_k}^T \odot {{{{\tilde{\mathbf{g}}}}}_k}^H} \right\} } \right] \nonumber \\&= \gamma _k^2\sigma _v^4E\left[ {\left\{ {{{{{\tilde{\mathbf{g}}}}}_k}{{{{\tilde{\mathbf{g}}}}}_k}^H} \right\} \odot \left\{ {{{{{\tilde{\mathbf{g}}}}}_k}^*{{{{\tilde{\mathbf{g}}}}}_k}^T} \right\} } \right] \end{aligned}$$
(95)

where \({{{{\tilde{\mathbf{g}}}}}_k} = vec\left( {diag\left[ {{{\mathbf{g}}_k}} \right] } \right) = \sum \limits _{j = 1}^R {{g_{k,j}}{{\mathbf{q}}_j}}\) and \({{\mathbf{q}}_j} \mathop {=}\limits ^{\varDelta } vec\left( {diag\left[ {{{\mathbf{e}}_j}} \right] } \right) \). By following the same approach used in the computation of \({\mathbf{F}}_k\) and \({\mathbf{G}}_k\), we can write \({{{\tilde{\mathbf{g}}}}_k}\) as sum of its mean and zero mean random part \({{{\tilde{\mathbf{g}}}}_k} = {{{\hat{\tilde{\mathbf{g}}}}}_k} + {{{\bar{\tilde{\mathbf{g}}}}}_k}\). Therefore we have

$$\begin{aligned} {\varvec{\Phi }}_k^3 =&\sum \limits _{j = 1}^R {\left( {m_4^{{g_{k,j}}} - 2m_2^{{g_{k,j}}}} \right) \left( {{{\mathbf{q}}_j}{\mathbf{q}}_j^H} \right) \odot \left( {{\mathbf{q}}_j^*{{\mathbf{q}}_j}^T} \right) } \nonumber \\&+\, \gamma _k^2\sigma _v^4E\left( {{{{\hat{\tilde{\mathbf{g}}}}}_k} \odot {\hat{\tilde{\mathbf{g}}}}_k^*} \right) E{\left( {{{{\hat{\tilde{\mathbf{g}}}}}_k} \odot {{\hat{\tilde{\mathbf{g}}}}}_k^*} \right) ^H}+ \gamma _k^2\sigma _v^4\left( {{{{{\bar{\tilde{\mathbf{g}}}}}}_k} \odot {{\bar{\tilde{\mathbf{g}}}}}_k^*} \right) E{\left( {{{{\hat{\tilde{\mathbf{g}}}}}_k} \odot {\hat{\tilde{\mathbf{g}}}}_k^*} \right) ^H} \nonumber \\&+\, \gamma _k^2\sigma _v^4E\left( {{{{\hat{\tilde{\mathbf{g}}}}}_k} \odot {\hat{\tilde{\mathbf{g}}}}_k^*} \right) {\left( {{{{{\bar{\tilde{\mathbf{g}}}}}}_k} \odot {{\bar{\tilde{\mathbf{g}}}}}_k^*} \right) ^H} + \gamma _k^2\sigma _v^4E\left( {{{{{\hat{\tilde{\mathbf{g}}}}}}_k}{{{{\hat{\tilde{\mathbf{g}}}}}}_k}^H} \right) \odot E\left( {{{{{\hat{\tilde{\mathbf{g}}}}}}_k}^*{{{{\hat{\tilde{\mathbf{g}}}}}}_k}^T} \right) \nonumber \\&+\, \gamma _k^2\sigma _v^4E\left( {{{{{\bar{\tilde{\mathbf{g}}}}}}_k}{{{{\bar{\tilde{\mathbf{g}}}}}}_k}^H} \right) \odot E\left( {{{{{\hat{\tilde{\mathbf{g}}}}}}_k}^*{{{{\hat{\tilde{\mathbf{g}}}}}}_k}^T} \right) + \gamma _k^2\sigma _v^4E\left( {{{{{\hat{\tilde{\mathbf{g}}}}}}_k}{{{{\hat{\tilde{\mathbf{g}}}}}}_k}^H} \right) \odot E\left( {{{{{\bar{\tilde{\mathbf{g}}}}}}_k}^*{{{{\bar{\tilde{\mathbf{g}}}}}}_k}^T} \right) \end{aligned}$$
(96)

for i.i.d complex gaussian symmetric gaussian \({{\mathbf{g}}_k}\), (96) can be further simplified as

$$\begin{aligned} \varvec{\Phi }_k^3&= 2\gamma _k^2\sigma _v^4\left( {\sum \limits _{j = 1}^R {\sigma _{{g_{k,j}}}^2\mathbf{e}_j^{}} } \right) {\left( {\sum \limits _{j = 1}^R {\sigma _{{g_{k,j}}}^2\mathbf{q}}_j^{}} \right) ^H} \!+\! \gamma _k^2\sigma _v^4{\mathbf{R}_{{\hat{\tilde{\mathbf{g}}}}_k}} \odot {\mathbf{R}_{{\hat{\tilde{\mathbf{g}}}}_k}}^T\nonumber \\&+\,\gamma _k^2\sigma _v^4\left( {{{{{\hat{\tilde{\mathbf{g}}}}}}_k} \odot {\bar{\tilde{\mathbf{g}}}}_k^*} \right) {\left( {{{{\bar{\tilde{\mathbf{g}}}}}_k} \odot {\bar{\tilde{\mathbf{g}}}}_k^*} \right) ^H} + \gamma _k^2\sigma _v^4\left( {{{{\bar{\tilde{\mathbf{g}}}}}_k} \!\odot \! {\bar{\tilde{\mathbf{g}}}}_k^*} \right) \!\sum \limits _{j = 1}^R \!{\sigma _{{g_{k,j}}}^2\mathbf{q}}_j^H \nonumber \\&+\,\gamma _k^2\sigma _v^4\!\sum \limits _{j = 1}^R \!{\sigma _{{g_{k,j}}}^2{\!\mathbf{q}}_j^{}} {\left( {{{{\bar{\tilde{\mathbf{g}}}}}_k} \!\odot \! {\bar{\tilde{\mathbf{g}}}}_k^*} \right) ^H}+2\gamma _k^2\sigma _v^4\mathfrak {R}\left( \left( {{{{\bar{\tilde{\mathbf{g}}}}}_k}{{{\bar{\tilde{\mathbf{g}}}}}_k}^H} \right) \odot \mathbf{R}_{{{{\hat{\tilde{\mathbf{g}}}}}_k}}^* \right) \end{aligned}$$
(97)

where \({\mathbf{R}}_{{{\hat{\tilde{\mathbf{g}}}}}_k}\triangleq E\left[ {{{\hat{\tilde{\mathbf{g}}}}}_k}{{{\hat{\tilde{\mathbf{g}}}}}_k}^H\right] =\sum \limits _{j = 1}^R {\sigma _{g_{k,j}}^2}{\mathbf{q}}_j{\mathbf{q}}_j^T\) .

Using Lemma 4, (83) can be written as

$$\begin{aligned} {\varvec{\Phi }}_k^4&= {\left( {{\varvec{\Phi }}_k^5} \right) ^H}\nonumber \\&= E\left\{ \left( {{P_k}vec\left( {{\mathbf{g}_k}\mathbf{g}_k^H} \right) vec{{\left( {{\mathbf{g}_k}\mathbf{g}_k^H} \right) }^H}} \right) \odot \left\{ {\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}vec\left( {{\mathbf{f}_k}\mathbf{f}_k^H} \right) vec{{\left( {{\mathbf{f}_p}\mathbf{f}_p^H} \right) }^H}} \right\} \right\} \nonumber \\&= {P_k}{\mathbf{G}_k} \odot \left( {{\gamma _k}\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {\left( {{P_p}{\mathbf{F}_k}\mathbf{F}_p^H} \right) } } \right) \end{aligned}$$
(98)

The equation in (84), by using Lemma 4, can be rewritten as

$$\begin{aligned} {\varvec{\Phi }}_k^6 =\,&{\left( {{\varvec{\Phi }}_k^7} \right) ^H} = {\gamma _k}\sigma _v^2{P_k}E\left\{ {\left[ {vec\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot vec\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right] vec{{\left( {diag\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] } \right) }^H}} \right\} \nonumber \\ =\,&{\gamma _k}\sigma _v^2{P_k}E\left\{ {\left[ {vec\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot vec\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right] vec{{\left( {\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] \odot {\mathbf{I}}} \right) }^H}} \right\} \end{aligned}$$
(99)

Since \(vec\left( {{\mathbf{x}} \odot {\mathbf{y}}} \right) = vec\left( {\mathbf{x}} \right) \odot vec\left( {\mathbf{y}} \right) \), (99) can be written as

$$\begin{aligned} {\varvec{\Phi }}_k^6 = {\left( {{\varvec{\Phi }}_k^7} \right) ^H} = {\gamma _k}\sigma _v^2{P_k}E\left\{ {\left[ {vec\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot vec\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right] {{\left( {vec\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] \odot vec\left( {\mathbf{I}} \right) } \right) }^H}} \right\} \end{aligned}$$
(100)

Again, by using lemma 4, the above expression can be rewritten as

$$\begin{aligned} {\varvec{\Phi }}_k^6 =\,&{\left( {{\varvec{\Phi }}_k^7} \right) ^H} = {\gamma _k}\sigma _v^2{P_k}E\left\{ {\left[ {vec\left( {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right) \odot vec{{\left[ {{{\mathbf{g}}_k}{\mathbf{g}}_k^H} \right] }^H}} \right] \left( {vec\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) \odot vec{{\left( {\mathbf{I}} \right) }^T}} \right) } \right\} \nonumber \\=\,&{\gamma _k}\sigma _v^2{P_k}{{\mathbf{G}}_k}\left( {E\left\{ {vec\left( {{{\mathbf{f}}_k}{\mathbf{f}}_k^H} \right) } \right\} \odot vec{{\left( {\mathbf{I}} \right) }^T}} \right) \nonumber \\ =\,&{\gamma _k}\sigma _v^2{P_k}{{\mathbf{G}}_k}\left( {vec\left\{ {{\mathbf{R}}_f^k} \right\} \odot vec{{\left( {\mathbf{I}} \right) }^T}} \right) \end{aligned}$$
(101)

To compute \({\varvec{\Phi }}_k^8\), we used the fact that \(vec\left( {{\mathbf{x}} \odot {\mathbf{y}}} \right) = vec\left( {\mathbf{x}} \right) \odot vec\left( {\mathbf{y}} \right) \) as follows

$$\begin{aligned} {\varvec{\Phi }}_k^8&= {\left( {{\varvec{\Phi }}_k^9} \right) ^H} = E\left\{ {vec\left( {{\gamma _k}\left( {{\mathbf{g}_k}\mathbf{g}_k^H} \right) \odot \sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{\mathbf{f}_p}\mathbf{f}_p^H} \right) } } \right) vec{{\left( {{\gamma _k}\sigma _v^2diag\left[ {{\mathbf{g}_k}\mathbf{g}_k^H} \right] } \right) }^H}} \right\} \nonumber \\&= \gamma _k^2\sigma _v^2\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_k}} E\left\{ {vec\left( {{\gamma _k}\left( {{\mathbf{g}_k}\mathbf{g}_k^H} \right) \odot \sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\left( {{\mathbf{f}_p}\mathbf{f}_p^H} \right) } } \right) {{\left( {vec\left( {{\mathbf{g}_k}\mathbf{g}_k^H} \right) \!\odot \! vec\!\left( \mathbf{I} \right) } \right) }^H}} \right\} \nonumber \\ \end{aligned}$$
(102)

Again, by using Lemma 4, the above expression can be rewrite as

$$\begin{aligned} {\varvec{\Phi }}_k^8&= \gamma _k^2\sigma _v^2\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}} {{P_k}} E\left\{ {vec\left( {\left( {{\mathbf{g}_k}\mathbf{g}_k^H} \right) \odot vec{{\left( {{\mathbf{g}_k}\mathbf{g}_k^H} \right) }^H}} \right) } \right. \nonumber \\&\left. \times {\left( {vec\left\{ {\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}} {{P_p}\left( {{\mathbf{f}_p}\mathbf{f}_p^H} \right) } } \right\} \odot vec{{\left( \mathbf{I} \right) }^T}} \right) } \right\} \nonumber \\&= \sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}{{P_k}} {\mathbf{G}_k}\!E\!\left( \!{\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}\!\left( {{\mathbf{f}_p}\mathbf{f}_p^H} \right) } \odot vec{{\left( \mathbf{I} \right) }^T}}\!\right) \nonumber \\&= {\gamma _k}\sigma _v^2{P_k}{\mathbf{G}_k}\left( {\sum \limits _{p \in {{\hat{\mathbf{D}}}_k}}^{} {{P_p}} vec\!\left\{ {\mathbf{R}_f^k} \right\} \!\odot \! vec{{\left( \mathbf{I} \right) }^T}}\!\right) \end{aligned}$$
(103)

From the above, it is obvious that the entries of \({\varvec{\Phi }}_k^6\), \({\varvec{\Phi }}_k^7\), \({\varvec{\Phi }}_k^8\) and \({\varvec{\Phi }}_k^9\) consist of up to second and fourth order cumulants of \({{\mathbf{f}}_k}\) and \({{\mathbf{g}}_k}\), respectively. Finally, the expression of (79) can be computed based on \({\mathbf{R}}_f^k\), \({{\mathbf{G}}_k}\) and \({{\mathbf{F}}_k}\).

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Mahboobi, B., Soleimani-Nasab, E. & Ardebilipour, M. Outage Probability Based Robust Distributed Beam-Forming in Multi-User Cooperative Networks with Imperfect CSI. Wireless Pers Commun 77, 1629–1658 (2014). https://doi.org/10.1007/s11277-013-1520-2

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Keywords

  • Multiple unicast
  • Relay beamforming
  • Imperfect channel state information
  • Penalty function
  • Conic quadratic programming
  • Statistical robust optimization
  • Conic programming