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 \({{\mathbf{F}}_k}\).