### 4.1 Minimum conditional MSE criterion

Taking CSI mismatch of **H**
_{1} and **H**
_{2} into account, the proposed objective function is to minimize the conditional MSE between the transmitted and received signals subject to relay power constraint, which is given as follows:

$$\begin{array}{*{20}l} &\min_{\mathbf{Q},\mathbf{W}}\ E\left[\parallel\mathbf{\widehat{s}}-\mathbf{s}{\parallel_{2}^{2}}|\widehat{\mathbf{H}_{1}},\widehat{\mathbf{H}_{2}}\right], \end{array} $$

(9a)

$$\begin{array}{*{20}l} &\text{subject to}\ tr\left(\mathbf{Q}\left({\sigma_{s}^{2}}E\left[\mathbf{H}_{1}\mathbf{H}_{1}^{H}|\mathbf{\widehat{H}}_{1}\right] +{\sigma_{1}^{2}}\mathbf{I}_{L}\right)\mathbf{Q}^{H}\right)\leq P_{r}, \end{array} $$

(9b)

where *P*
_{
r
} is the upper bound of relay transmission power.

Assuming that **s** and **n** are independent, based on (1), setting the gradient of (9a) with respect to **W**
^{H} to zero yields the optimal **W**

$$\begin{array}{*{20}l} \mathbf{W}_{\text{opt}}={\sigma_{s}^{2}}E^{H}\left[\mathbf{H}|\mathbf{\widehat{H}}_{1},\mathbf{\widehat{H}}_{2}\right] \left(\mathbf{R}_{n}\right)^{-1}, \end{array} $$

(10)

where **W**
_{opt} denotes the optimal solution of **W**, and \(\mathbf {R}_{n}={\sigma _{s}^{2}}E\left [\mathbf {HH}^{H}|\mathbf {\widehat {H}}_{1},\mathbf {\widehat {H}}_{2}\right ] +{\sigma _{1}^{2}}E\left [\mathbf {H}_{2}\mathbf {QQ}^{H}\mathbf {H}_{2}^{H}|\mathbf {\widehat {H}}_{2}\right ] +{\sigma _{2}^{2}}\mathbf {I}_{N_{s}}\).

Define \(\mathbf {\overline {H}}=E\left [\mathbf {H}|\mathbf {\widehat {H}}_{1},\mathbf {\widehat {H}}_{2}\right ]\), \(\mathbf {\overline {H}}_{1}=E\left [\mathbf {H}_{1}|\mathbf {\widehat {H}}_{1}\right ]\) and \(\mathbf {\overline {H}}_{2}=E\left [\mathbf {H}_{2}|\mathbf {\widehat {H}}_{2}\right ]\). Substituting (1) and (10) into (9a) yields (11) (shown on the top of next page).

$$\begin{array}{*{20}l} &\max_{\mathbf{Q}}\ tr \left(\mathbf{\overline{H}}^{H}\left({\sigma_{s}^{2}} E\left[\mathbf{HH}^{H}|\mathbf{\widehat{H}}_{1},\mathbf{\widehat{H}}_{2}\right] +\mathbf{R}_{n}\right)^{-1} \mathbf{\overline{H}}\right), \end{array} $$

(11a)

$$\begin{array}{*{20}l} &\text{subject to}\ tr\left(\mathbf{Q}\left({\sigma_{s}^{2}}E\left[\mathbf{H}_{1}\mathbf{H}_{1}^{H}|\mathbf{\widehat{H}}_{1}\right] +{\sigma_{1}^{2}}\mathbf{I}_{L}\right)\mathbf{Q}^{H}\right)\leq P_{r}. \end{array} $$

(11b)

It is observed from (11a) that **Q** is contained in the inversion manipulation; therefore, direct optimization of (11) is difficult. To facilitate the solution of (11), the structure of optimal **Q** is analyzed and it is found that optimal **Q** has the form of

$$\begin{array}{@{}rcl@{}} \mathbf{Q}_{\text{opt}}=\mathbf{V}_{2}\boldsymbol{\Phi}_{1}\mathbf{U}_{1}^{H}, \end{array} $$

(12)

where **Q**
_{opt} is the optimal **Q**, *Φ*
_{1} is an *M*×*M* diagonal matrix, and \(\mathbf {V}_{2}^{H}\) and **U**
_{1} are unitary matrices constituted by right- and left-singular vectors of \(\overline {\mathbf {H}}_{2}\) and \(\overline {\mathbf {H}}_{1}\), respectively. The proof of (12) is provided in Appendix A. Using (12), (1) is equivalently expressed as

$$\begin{array}{*{20}l} \max_{\boldsymbol{\Phi}_{1}}J\left(\boldsymbol{\Phi}_{1}\right), \end{array} $$

(13a)

$$\begin{array}{*{20}l} \text{subject to }\sum_{i=1}^{M}\gamma_{i}|\phi_{i}|^{2}\leq P_{r}, \end{array} $$

(13b)

where

$$\begin{array}{*{20}l} J\left(\boldsymbol{\Phi}_{1}\right)=\sum_{i=1}^{M}\frac{\alpha_{i}|\phi_{i}|^{2}}{\beta_{i}|\phi_{i}|^{2}+{\sigma_{s}^{2}}b+c+{\sigma_{2}^{2}}}, \end{array} $$

(14)

*ϕ*
_{
i
} is the *i*
^{th} diagonal element of *Φ*
_{1}, \(\alpha _{i}=\lambda _{1,i}^{2}\lambda _{2,i}^{2}\), \(\beta _{i}={\sigma _{s}^{2}}\left (\lambda _{1,i}^{2}+\sigma _{h_{1}|\widehat {h}_{1}}^{2}\right)\lambda _{2,i}^{2}+{\sigma _{1}^{2}}\lambda _{2,i}^{2}\), \(\gamma _{i}={\sigma _{s}^{2}}\mid \lambda _{1,i}\mid ^{2}+\)
\({\sigma _{s}^{2}}\sigma _{h_{1}|\widehat {h}_{1}}^{2}+{\sigma _{1}^{2}},\)

$$\begin{array}{*{20}l} &b=\sigma_{h_{2}|\widehat{h}_{2}}^{2}\sum_{i=1}^{N_{s}}\left(\lambda_{1,i}^{2}+\sigma_{h_{1}|\widehat{h}_{1}}^{2}\right)\mid\phi_{i}\mid^{2}, \end{array} $$

(15)

$$\begin{array}{*{20}l} &c=\sigma_{h_{2}|\widehat{h}_{2}}^{2}\sum_{i=1}^{N_{s}}\mid\phi_{i}\mid^{2}, \end{array} $$

(16)

and *λ*
_{1,i
} and *λ*
_{2,i
} denote the *i*
^{th} singular values of \(\overline {\mathbf {H}}_{1}\) and \(\overline {\mathbf {H}}_{2}\), respectively. Details of transformation from (11) to (13) are presented in Appendix B. The optimization problem (13) is easier to solve than (11). Once the optimal *Φ*
_{1} is derived, **Q**
_{opt} and **W**
_{opt} are computed using (12) and (10), respectively.

When \(\sigma _{h_{1}|\widehat {h}_{1}}^{2}=0\) and \(\sigma _{h_{2}|\widehat {h}_{2}}^{2}=0\), *b*=0, *c*=0, \(\beta _{i}={\sigma _{s}^{2}}\lambda _{1,i}^{2}\lambda _{2,i}^{2}+{\sigma _{1}^{2}}\lambda _{2,i}^{2}\), and \(\gamma _{i}={\sigma _{s}^{2}}\mid \lambda _{1,i}\mid ^{2}+{\sigma _{1}^{2}},\). (13) becomes

$$ \max_{\phi_{i}}\sum_{i=1}^{N_{s}}\frac{\lambda_{1,i}^{2}\lambda_{2,i}^{2}\mid\phi_{i}\mid^{2}} {\left({\sigma_{s}^{2}}\lambda_{1,i}^{2}\lambda_{2,i}^{2}+ {\sigma_{1}^{2}}\lambda_{2,i}^{2}\right)\mid\phi_{i}\mid^{2}+{\sigma_{2}^{2}}}, $$

(17a)

$$ \text{subject to} \sum_{i=1}^{M}\left({\sigma_{s}^{2}}\mid\lambda_{1,i}\mid^{2}+{\sigma_{1}^{2}}\right)\mid\phi_{i}\mid^{2}\leq P_{r}, $$

(17b)

which are equivalent to (24) and (25) of [19]. Therefore, the minimum conditional MSE criterion reduces to the MSE criterion when the CSI mismatch vanishes.

### 4.2 Global solution by genetic algorithm

It is observed from (13) that multiplying *ϕ*
_{
i
} by a constant maximizes the relay power and the value of the objective function (13a). Therefore, the optimal *ϕ*
_{
i
} can be obtained while (13b) achieves equality. Based on this observation, the values of chromosomes in genetic algorithm (GA) is optimized within 0 and 1, then multiplied with a constant *α* which is obtained when (13b) achieves equality, i.e.,

$$ {{}{\begin{aligned} \alpha=\frac{P_{r}}{\sum_{i=1}^{M}{\sigma_{s}^{2}}\mid\phi_{i}\mid^{2}\mid{\lambda}_{1,i}\mid^{2}+{\sigma_{s}^{2}}\sigma^{2}_{h_{1}|\widehat{h}_{1}} \mid\phi_{i}\mid^{2}+{\sigma_{1}^{2}}\mid\phi_{i}\mid^{2}}. \end{aligned}}} $$

(18)

Substituting *α*
*ϕ*
_{
i
} into (13a) yields the value of the fitness function of GA.

### 4.3 Relaxed solution by water filling strategy

It is observed from (13a) that the terms of *b* and *c* contain *ϕ*
_{
i
},*i*=1,…,*M*, which prevent deriving an analytical solution to (13). To avoid high computational loads of using the global searching algorithm, a relaxed version of (13) is proposed here.

From (13a), it is noted that increasing the values of *b* and *c* reduces the value of *J*(*Φ*
_{1}), which means

$$\begin{array}{@{}rcl@{}} J(\boldsymbol{\Phi}_{1})\geq J(\boldsymbol{\Phi}_{1})_{\text{max}}, \end{array} $$

(19)

where *J*(*Φ*
_{1})_{max} is computed from (13a) using *b*
_{max} and *c*
_{max}. Here, *b*
_{max} and *c*
_{max} denote the maximum values of *b* and *c*, respectively. From the relay power constraint (13b), the possible values of *b*
_{max} and *c*
_{max} are straightforward to derive and are given below:

$$\begin{array}{@{}rcl@{}} b_{\text{max}}=\sum_{i=1}^{M}\frac{P_{r}\sigma^{2}_{h_{2}|\widehat{h}_{2}}\left(\lambda_{1,i}^{2}+\sigma^{2}_{h_{1}|\widehat{h}_{1}}\right)} {{\sigma_{s}^{2}}\mid{\lambda}_{1,i}\mid^{2}+{\sigma_{s}^{2}}\sigma^{2}_{h_{1}|\widehat{h}_{1}}+{\sigma_{1}^{2}}}, \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} c_{\text{max}}=\sum_{i=1}^{M}\frac{P_{r}\sigma^{2}_{h_{2}|\widehat{h}_{2}}} {{\sigma_{s}^{2}}\mid{\lambda}_{1,i}\mid^{2}+{\sigma_{s}^{2}}\sigma^{2}_{h_{1}|\widehat{h}_{1}}+{\sigma_{1}^{2}}}. \end{array} $$

(21)

Substituting (20) and (21) into (13), and using the Lagrange multiplier technique, the solution of the relaxed version of (13) is given by

$$ {{}{\begin{aligned} \mid\phi_{i}\mid^{2}=\frac{1}{{\sigma_{s}^{2}}\mid{\lambda}_{2,i}\mid^{2}\left(\mid{\lambda}_{1,i}\mid^{2}+\sigma^{2}_{h_{1}|\widehat{h}_{1}}\right) +{\sigma_{1}^{2}}\mid{\lambda}_{2,i}\mid^{2}} \\ \cdot\left(\sqrt{\frac{{\sigma_{s}^{2}}\mid{\lambda}_{1,i}\mid^{2}\mid{\lambda}_{2,i}\mid^{2}\sigma_{2,\text{max}}^{2} }{\mu\left({\sigma_{s}^{2}}\mid{\lambda}_{1,i}\mid^{2}+{\sigma_{s}^{2}}\sigma^{2}_{h_{1}|\widehat{h}_{1}}+{\sigma_{1}^{2}}\right)}}-\sigma_{2,\text{max}}^{2}\right)^{+}, \forall i, \end{aligned}}} $$

(22)

where

$$\begin{array}{@{}rcl@{}} \sigma_{2,\text{max}}^{2}={\sigma_{s}^{2}}{b}_{\text{max}}+{\sigma_{1}^{2}}{c}_{\text{max}}+{\sigma_{2}^{2}}, \end{array} $$

(23)

and (*x*)^{+}=max(*x*,0), *μ* is the Lagrange constant which should be chosen such that (13b) is satisfied.