The design of MIMO precoding matrix at the BS and MIMO receive matrices at the RN and the UTs will be straightforward as we use only local CSI.

At the RN, we have the estimate of

_{H 1}F, and the optimum MIMO receive matrix

_{D R}is obtained from

${\mathit{D}}_{R}=\underset{{\stackrel{\u0304}{\mathit{D}}}_{R}}{min}E\left\{{\u2225{\stackrel{\u0304}{\mathit{D}}}_{R}\left({\mathit{H}}_{1}\mathit{F}\mathit{x}+\frac{{\mathit{n}}_{1}}{{\beta}_{1}}\right)-\mathit{x}\u2225}^{2}\right\}$

(4)

as

${\mathit{D}}_{R}={\mathit{R}}_{x}{\mathit{F}}^{H}{\mathit{H}}_{1}^{H}{\left({\mathit{H}}_{1}\mathit{F}{\mathit{R}}_{x}{\mathit{F}}^{H}{\mathit{H}}_{1}^{H}+\frac{1}{{\beta}_{1}^{2}}{\mathit{R}}_{{n}_{1}}\right)}^{-1},$

(5)

where ${\mathit{R}}_{x}=E\left\{\mathit{x}{\mathit{x}}^{H}\right\}$ denotes the transmit vector correlation matrix, ${\mathit{R}}_{{n}_{1}}=E\left\{{\mathit{n}}_{1}{\mathit{n}}_{1}^{H}\right\}$ denotes the additive noise correlation matrix, and ^{(·)H}denotes conjugate transpose.

Let us define the singular value decomposition (SVD) of the channel matrix

_{H 1} as

${\mathit{H}}_{1}={\mathit{U}}_{1}{\mathit{\Sigma}}_{1}{\mathit{V}}_{1}^{H}.$

(6)

From [

16,

17], we can assume that the matrix

F is in the form:

$\mathit{F}={\mathit{V}}_{1}^{\left(r\right)}\mathit{\Phi},$

(7)

where

${\mathit{V}}_{1}^{\left(r\right)}$ contains the first

*r* columns of the matrix

_{V 1}and

Φ∈

^{ℂ r×r}. Then, from Equation (

5) matrix

_{D R}can be also written as

${\mathit{D}}_{R}={\mathit{\Delta}}_{R}{\mathit{U}}_{1}^{\left(r\right)\phantom{\rule{1em}{0ex}}H},$

(8)

where ${\mathit{\Delta}}_{R}\in {\u2102}^{r\times r}$ and ${\mathit{U}}_{1}^{\left(r\right)\phantom{\rule{1em}{0ex}}H}$ contains the first *r* columns of _{U 1}.

At the BS, we assume we have the estimate of the channel matrix

_{H 1}. MIMO precoding matrix at the BS is derived from the following optimization:

$\begin{array}{c}\mathit{F}=\underset{\stackrel{\u0304}{\mathit{F}}}{min}E\left\{{\u2225{\mathit{D}}_{R}\left({\mathit{H}}_{1}\stackrel{\u0304}{\mathit{F}}\mathit{x}+\frac{{\mathit{n}}_{1}}{{\beta}_{1}}\right)-\mathit{x}\u2225}^{2}\right\},\\ \text{s.t.}{\beta}_{1}^{2}\text{tr}\left(\stackrel{\u0304}{\mathit{F}}{\mathit{R}}_{x}{\stackrel{\u0304}{\mathit{F}}}^{H}\right)={P}_{{\text{T}}_{B}}.\hfill \end{array}$

(9)

The MIMO precoding matrix

F can be obtained in several ways. Using the approach presented in [

16,

17], we can substitute the solution for

_{D R} from (5) in (9) and then find the optimum

F. Another approach is used in [

10]. The matrices

F and

_{D R}are designed iteratively. In this case, we start with some solution for

F, then we calculate

_{D R}, then use this solution to update the matrix

F and so on. Unlike these approaches, in this article, we want to be able to design the spatial processing matrices at the transmitter and at the receiver independently. The transmit MIMO processing matrices are designed assuming only eigenmode decomposition at the receiver, regardless of the actual spatial processing used at the receiver. This is the worst case assumption as only the transmitter would have to deal with the noise and spatial interference. Therefore, the matrix

F is designed in a non-iterative way by assuming at the BS that

_{Δ R}=

_{
I
r
}, where

${\mathit{I}}_{r}\in {R}^{r\times r}$ denotes the identity matrix. At high signal-to-noise ratios (SNRs) this assumption is true. Equation (

9) can be written then as

$\begin{array}{c}\mathit{\Phi}=\underset{\stackrel{\u0304}{\mathit{\Phi}}}{min}E\left\{{\u2225\left({\mathit{\Sigma}}_{1}^{\left(r\right)}\stackrel{\u0304}{\mathit{\Phi}}\mathit{x}+\frac{{\mathit{n}}_{1}^{\prime}}{{\beta}_{1}}\right)-\mathit{x}\u2225}^{2}\right\},\\ \phantom{\rule{0.2em}{0ex}}\text{s.t.}{\beta}_{1}^{2}\text{tr}\left(\stackrel{\u0304}{\mathit{\Phi}}{\mathit{R}}_{x}{\stackrel{\u0304}{\mathit{\Phi}}}^{H}\right)={P}_{{\text{T}}_{B}}\hfill \end{array}$

(10)

where ${\mathit{\Sigma}}_{1}^{\left(r\right)}\in {\u2102}^{r\times r}$ is a diagonal matrix with *r* largest singular values of _{H 1}on the main diagonal and ${\mathit{n}}_{1}^{\prime}={\mathit{U}}_{1}^{\left(r\right)\phantom{\rule{0.2em}{0ex}}H}{\mathit{n}}_{1}$.

Using the method of Lagrangian multipliers, from Equation (

10) it can easily be shown that the optimum

F is in the form of

$\mathit{F}={\mathit{V}}_{1}^{\left(r\right)}{\left({\mathit{\Sigma}}_{1}^{\left(r\right)\phantom{\rule{1em}{0ex}}2}+\frac{\text{tr}\left({\mathit{R}}_{{n}_{1}}^{\prime}\right)}{{P}_{{\text{T}}_{B}}}{\mathit{I}}_{r}\right)}^{-1}{\mathit{\Sigma}}_{1}^{\left(r\right)\phantom{\rule{1em}{0ex}}T}.$

(11)

From Equation (11), it follows that the optimum Φ is diagonal positive definite power-loading matrix. If the elements of the additive noise vector at the input of the RN antenna array are independent and identically distributed (i.i.d.) zero mean complex Gaussian random variables with variance ${\sigma}_{{n}_{1}}^{2}$ then $\text{tr}\left({\mathit{R}}_{{n}_{1}}^{\prime}\right)=r{\sigma}_{{n}_{1}}^{2}$. Then, we only need to feedback the noise variance ${\sigma}_{{n}_{1}}^{2}$ from RN to the BS to design the MIMO precoding matrix F.

Under the assumption that the estimate of

_{H 2,k}F_{
R
}_{
k
}is available at the

*k* th UT, the

*k* th UT MIMO receive matrix is obtained from

$\phantom{\rule{-3.0pt}{0ex}}{\mathit{D}}_{k}=\underset{{\stackrel{\u0304}{\mathit{D}}}_{k}}{min}E\left\{{\u2225{\stackrel{\u0304}{\mathit{D}}}_{k}{\mathit{H}}_{2,k}{\mathit{F}}_{{R}_{k}}{\mathit{x}}_{{R}_{k}}+\frac{1}{{\beta}_{2}}{\stackrel{\u0304}{\mathit{D}}}_{k}{\mathit{n}}_{2,k}-{\mathit{x}}_{{R}_{k}}\u2225}^{2}\right\}$

(12)

as

$\phantom{\rule{-6.0pt}{0ex}}{\mathit{D}}_{k}={\mathit{R}}_{{x}_{R},k}{\mathit{F}}_{{R}_{k}}^{H}{\mathit{H}}_{2,k}^{H}\phantom{\rule{0.3em}{0ex}}\times \phantom{\rule{0.3em}{0ex}}{\left(\phantom{\rule{0.3em}{0ex}}{\mathit{H}}_{2,k}{\mathit{F}}_{{R}_{k}}{\mathit{R}}_{{x}_{R},k}{\mathit{F}}_{{R}_{k}}^{H}{\mathit{H}}_{2,k}^{H}+\frac{{\mathit{R}}_{{n}_{2},k}}{{\beta}_{2}^{2}}\phantom{\rule{0.3em}{0ex}}\right)}^{-1}\phantom{\rule{0.3em}{0ex}},$

(13)

where ${\mathit{R}}_{{x}_{R},k}=E\left\{{\mathit{x}}_{{R}_{k}}{\mathit{x}}_{{R}_{k}}^{H}\right\}$ denotes the *k* th UT’s RN transmit vector correlation matrix, and ${\mathit{R}}_{{n}_{2},k}=E\left\{{\mathit{n}}_{2,k}{\mathit{n}}_{2,k}^{H}\right\}$ denotes the correlation matrix of the additive noise at the input of *k* th UT antenna array.

Our goal is to use as much as possible of the available users’ spatial resources and at the same time minimize the MU interference (MUI) between different users. Let us consider the MSE at the UTs:

${\text{mse}}_{\mathrm{UT}}=E\left\{{\u2225\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}\left({\mathit{x}}_{R}+{\mathit{n}}_{R}\right)+\frac{1}{{\beta}_{2}}\mathit{D}{\mathit{n}}_{2}-{\mathit{x}}_{R}\u2225}^{2}\right\},$

(14)

where

${\mathit{x}}_{R}={\mathit{D}}_{R}{\mathit{H}}_{1}\mathit{F}\mathit{x},\phantom{\rule{1em}{0ex}}{\mathit{n}}_{R}=\frac{1}{{\beta}_{1}}{\mathit{D}}_{R}{\mathit{n}}_{1}$

(15)

and

$\begin{array}{cc}{\mathit{x}}_{R}& ={\left[{\mathit{x}}_{{R}_{1}}^{T}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{\mathit{x}}_{{R}_{K}}^{T}\right]}^{T},\\ {\mathit{n}}_{R}& ={\left[{\mathit{n}}_{{R}_{1}}^{T}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{\mathit{n}}_{{R}_{K}}^{T}\right]}^{T}.\end{array}$

(16)

We can rewrite this equation as

$\begin{array}{c}{\text{mse}}_{\text{UT}}=E\left\{\phantom{\rule{0.3em}{0ex}}\u2225\overline{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}\left({\mathit{x}}_{R}+{\mathit{n}}_{R}\right)+\stackrel{~}{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}\left({\mathit{x}}_{R}+{\mathit{n}}_{R}\right)\right.\\ \phantom{\rule{1.05em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{3em}{0ex}}\left.{+\frac{1}{{\beta}_{2}}\mathit{D}{\mathit{n}}_{2}-{\mathit{x}}_{R}\u2225}^{2}\right\}.\hfill \end{array}$

(17)

Matrix

$\overline{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}$ is a block diagonal matrix with matrices

_{
D
k
}_{H 2,k}F_{
R
}_{
k
} on the main diagonal. Matrix

$\stackrel{~}{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}$ is given by

$\stackrel{~}{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}=\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}-\overline{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}$

(18)

and represents the MUI.

In order to design the MU MIMO precoding matrix at the RN we have to meet two contradictory requirements. First, we need to minimize the co-channel interference between different users by reducing the overlap of the row spaces spanned by the effective channel matrices of different users. However, to maximize the spatial processing gains we need to use as much as possible of the available UTs’ channel row vector subspaces. Therefore, we factor the MU MIMO precoding matrix at the RN as

${\mathit{F}}_{{R}_{k}}={\mathit{F}}_{{R}_{a},k}{\mathit{F}}_{{R}_{b},k}{\mathit{F}}_{{R}_{c},k},$

(19)

where the matrix F_{R}_{a,k}is used to minimize the MUI from the *k* th UT to the co-channel UTs, matrix F_{R}_{b,k}is used to maximize the received power of the *k* th UT and the matrix F_{R}_{c,k} is used to optimize the *k* th UT performance according to a specific criterion.

Matrix

F_{R}_{
a
}is obtained from Equation (

17) using the following optimization:

$\begin{array}{c}{\mathit{F}}_{{R}_{a}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a}}}{min}E\left\{{\u2225\stackrel{~}{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}\left({\mathit{x}}_{R}+{\mathit{n}}_{R}\right)+\frac{1}{{\beta}_{2}}\mathit{D}{\mathit{n}}_{2}\u2225}^{2}\right\},\\ \text{s.t.}{\beta}_{2}^{2}\text{tr}\left({\mathit{F}}_{R}\left({\mathit{R}}_{{x}_{R}}+{\mathit{R}}_{{n}_{R}}\right){\mathit{F}}_{R}^{H}\right)={P}_{{\text{T}}_{R}}\hfill \end{array}$

(20)

assuming matrices

_{
D
k
},

F_{R}_{b,k}, and

F_{R}_{c,k} are unitary,

_{
r
k
}=rank(

_{H 2,k}·

_{H 1}) and without the loss of generality that the elements of vectors

_{x R}and

_{n R} are i.i.d. zero mean unit variance random variables. These assumptions correspond to the initial requirement that all UTs use as much as possible of the available subspace for communication. Equation (

20) can be written as

$\begin{array}{c}{\mathit{F}}_{{R}_{a}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a}}}{min}\sum _{k=1}^{K}\text{tr}\left({\stackrel{~}{\mathit{H}}}_{2,k}^{H}{\stackrel{~}{\mathit{H}}}_{2,k}{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a},k}{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a},k}^{H}\right.\\ \phantom{\rule{12.05em}{0ex}}\left.+\frac{\text{tr}\left({\mathit{R}}_{{n}_{2}}\right)}{{P}_{{\text{T}}_{R}}}{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a},k}{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a},k}^{H}\right)\\ \phantom{\rule{2.05em}{0ex}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a}}}{min}\sum _{k=1}^{K}\text{tr}\left({\stackrel{\u0304}{\mathit{F}}}_{{R}_{a},k}{\stackrel{\u0304}{\mathit{F}}}_{{R}_{a},k}^{H}\right.\hfill \\ \phantom{\rule{12.05em}{0ex}}\left.\times \left({\stackrel{~}{\mathit{H}}}_{2,k}^{H}{\stackrel{~}{\mathit{H}}}_{2,k}+\frac{\text{tr}\left({\mathit{R}}_{{n}_{2}}\right)}{{P}_{{\text{T}}_{R}}}{\mathit{I}}_{{M}_{R}}\right)\right)\end{array}$

(21)

The joint co-channel UTs channel matrix

${\stackrel{~}{\mathit{H}}}_{2,k}\in {\u2102}^{({M}_{U}-{M}_{{U}_{k}})\times {M}_{R}}$ is defined as

${\stackrel{~}{\mathit{H}}}_{2,k}=\left[\begin{array}{l}{\mathit{H}}_{2,1}\\ \vdots \\ {\mathit{H}}_{2,(k-1)}\\ {\mathit{H}}_{2,(k+1)}\\ \vdots \\ {\mathit{H}}_{2,K}\end{array}\right]\phantom{\rule{0.25em}{0ex}}.$

(22)

Let us define the SVD of

${\stackrel{~}{\mathit{H}}}_{2,k}$ as

${\stackrel{~}{\mathit{H}}}_{2,k}={\stackrel{~}{\mathit{U}}}_{2,k}{\stackrel{~}{\mathit{\Sigma}}}_{2,k}{\stackrel{~}{\mathit{V}}}_{2,k}^{H},$

(23)

then the non-trivial solution for

F_{R}_{
a
}in Equation (

21) is given by [

17]

${\mathit{F}}_{{R}_{a},k}={\stackrel{~}{\mathit{V}}}_{2,k}{\left({\stackrel{~}{\mathit{\Sigma}}}_{2,k}^{T}{\stackrel{~}{\mathit{\Sigma}}}_{2,k}+\frac{\text{tr}\left({\mathit{R}}_{{n}_{2}}\right)}{{P}_{{\text{T}}_{R}}}{\mathit{I}}_{{M}_{R}}\right)}^{-1/2}$

(24)

Matrix

F_{R}_{
b
} is obtained from

${\mathit{F}}_{{R}_{b}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{b}}}{max}E\left\{{\u2225\overline{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}\left({\mathit{x}}_{R}+{\mathit{n}}_{R}\right)\u2225}^{2}\right\}$

(25)

assuming matrices

_{
D
k
}and

F_{R}_{c,k} are unitary and

_{
r
k
}=rank(

_{H 2,k}·

_{H 1}). Again, without the loss of generality we can assume that the elements of vectors

_{x R} and

_{n R}are i.i.d. zero mean unit variance random variables. Equation (

25) is rewritten as

${\mathit{F}}_{{R}_{b}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{b}}}{max}\sum _{k=1}^{K}\mathrm{tr}\left({\stackrel{\u0304}{\mathit{F}}}_{{R}_{b},k}^{H}{\mathit{F}}_{{R}_{a},k}^{H}{\mathit{H}}_{2,k}^{H}{\mathit{H}}_{2,k}{\mathit{F}}_{{R}_{a},k}{\stackrel{\u0304}{\mathit{F}}}_{{R}_{b},k}\right)\phantom{\rule{0.25em}{0ex}}.$

(26)

The non-trivial solution of (26) is given by

${\mathit{F}}_{{R}_{b},k}={\left({\mathit{H}}_{2,k}{\mathit{F}}_{{R}_{a},k}\right)}^{H}={\mathit{F}}_{{R}_{a},k}^{H}{\mathit{H}}_{2,k}^{H}.$

(27)

Finally, we can design the optimum matrix

F_{R}_{c,k}according to a specific optimization criterion. In our case, we use the MMSE criterion so the optimum

F_{R}_{c,k} is obtained from

$\phantom{\rule{-6.0pt}{0ex}}\begin{array}{c}{\mathit{F}}_{{R}_{c}}\phantom{\rule{0.3em}{0ex}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{c}}}{min}E\left\{\phantom{\rule{0.3em}{0ex}}{\u2225\overline{\mathit{D}{\mathit{H}}_{2}{\mathit{F}}_{R}}\left({\mathit{x}}_{R}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\mathit{n}}_{R}\right)+\frac{1}{{\beta}_{2}}\mathit{D}{\mathit{n}}_{2}-{\mathit{x}}_{R}\u2225}^{2}\right\},\\ \text{s.t.}\phantom{\rule{0.5em}{0ex}}{\beta}_{2}^{2}\text{tr}\left({\mathit{F}}_{R}\left({\mathit{R}}_{{x}_{R}}+{\mathit{R}}_{{n}_{R}}\right){\mathit{F}}_{R}^{H}\right)={P}_{{\text{T}}_{R}}\hfill \end{array}$

(28)

assuming the MU MIMO channel is decomposed into the set of parallel SU MIMO channels using matrices

F_{R}_{a,k}. Let us define the SVD of

_{H 2,k}F_{R}_{a,k}F_{R}_{b,k}as

${\mathit{H}}_{2,k}{\mathit{F}}_{{R}_{a},k}{\mathit{F}}_{{R}_{b},k}={\mathit{U}}_{{R}_{k}}{\mathit{\Sigma}}_{{R}_{k}}{\mathit{V}}_{{R}_{k}}^{H}.$

(29)

Again, we can assume in the worst case scenario that at the UTs we perform only eigenmode decomposition of the effective UTs’ channel matrices, i.e.,

${\mathit{D}}_{k}={\mathit{U}}_{{R}_{k}}^{\left({r}_{k}\right)\phantom{\rule{0.2em}{0ex}}H}$. We can rewrite Equation (

28) as

$\phantom{\rule{-17.0pt}{0ex}}\begin{array}{c}{\mathit{F}}_{{R}_{a}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{c}}}{min}\sum _{k=1}^{K}E\left\{\u2225{\mathit{D}}_{k}{\mathit{H}}_{2,k}{\mathit{F}}_{{R}_{k}}\left({\mathit{x}}_{{R}_{k}}+{\mathit{n}}_{{R}_{k}}\right)\right.\\ \phantom{\rule{8em}{0ex}}\left.{+\frac{1}{{\beta}_{2}}{\mathit{D}}_{k}{\mathit{n}}_{2,k}-{\mathit{x}}_{{R}_{k}}\u2225}^{2}\right\}\\ =\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{c}}}{min}\sum _{k=1}^{K}E\left\{\u2225{\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)}{\mathit{\Phi}}_{{R}_{k}}\left({\mathit{x}}_{{R}_{k}}+{\mathit{n}}_{{R}_{k}}\right)\right.\\ \phantom{\rule{8em}{0ex}}\left.{+\frac{1}{{\beta}_{2}}{\mathit{n}}_{2,k}^{\prime}-{\mathit{x}}_{{R}_{k}}\u2225}^{2}\right\}\\ \phantom{\rule{7.05em}{0ex}}=\underset{{\stackrel{\u0304}{\mathit{F}}}_{{R}_{c}}}{min}\sum _{k=1}^{K}\phantom{\rule{0.3em}{0ex}}\text{tr}\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}{\mathit{R}}_{{x}_{R},k}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.6em}{0ex}}{\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)}{\mathit{\Phi}}_{{R}_{k}}\phantom{\rule{0.3em}{0ex}}\left({\mathit{R}}_{{x}_{R},k}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\mathit{R}}_{{n}_{R},k}\right)\phantom{\rule{0.3em}{0ex}}{\mathit{\Phi}}_{{R}_{k}}^{H}\phantom{\rule{0.3em}{0ex}}{\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)H}\right.\hfill \\ \phantom{\rule{17em}{0ex}}-{\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)}{\mathit{\Phi}}_{{R}_{k}}{\mathit{R}}_{{x}_{R},k}-{\mathit{R}}_{{x}_{R},k}{\mathit{\Phi}}_{{R}_{k}}^{H}{\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)\phantom{\rule{1em}{0ex}}H}\\ \phantom{\rule{17em}{0ex}}\left.+\frac{\text{tr}\left({\mathit{R}}_{{n}_{2},k}^{\prime}\right)}{{P}_{{\text{T}}_{R}}}{\mathit{\Phi}}_{{R}_{k}}\left({\mathit{R}}_{{x}_{R},k}+{\mathit{R}}_{{n}_{R},k}\right){\mathit{\Phi}}_{{R}_{k}}^{H}\right]\phantom{\rule{0.25em}{0ex}},\end{array}$

(30)

where we have assumed that the optimum

F_{R}_{
c
}is in the form [

17]

${\mathit{F}}_{{R}_{c},k}={\mathit{V}}_{{R}_{k}}^{\left({r}_{k}\right)}{\mathit{\Phi}}_{{R}_{k}}$

(31)

and

${\mathit{n}}_{2,k}^{\prime}={\mathit{U}}_{{R}_{k}}^{\left({r}_{k}\right)\phantom{\rule{0.2em}{0ex}}H}{\mathit{n}}_{2,k}$. After setting the derivative of (30) to zero, we have

$\begin{array}{c}\left({\mathit{R}}_{{x}_{R},k}+{\mathit{R}}_{{n}_{R},k}\right){\mathit{\Phi}}_{{R}_{k}}^{H}\left({\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)\phantom{\rule{1em}{0ex}}H}{\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)}+\frac{\text{tr}\left({\mathit{R}}_{{n}_{2},k}^{\prime}\right)}{{P}_{{\text{T}}_{R}}}{\mathit{I}}_{{r}_{k}}\right)\\ \phantom{\rule{1em}{0ex}}-{\mathit{R}}_{{x}_{R},k}{\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)}=\mathbf{0}.\hfill \end{array}$

(32)

From Equation (

32) we have

$\begin{array}{c}{\mathit{F}}_{{R}_{a}}={\mathit{V}}_{{R}_{k}}^{\left({r}_{k}\right)}{\left({\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)\phantom{\rule{0.2em}{0ex}}2}+\frac{\text{tr}\left({\mathit{R}}_{{n}_{2},k}^{\prime}\right)}{{P}_{{\text{T}}_{R}}}{\mathit{I}}_{{r}_{k}}\right)}^{-1}\\ \phantom{\rule{1em}{0ex}}\times {\mathit{\Sigma}}_{{R}_{k}}^{\left({r}_{k}\right)}{\mathit{R}}_{{x}_{R},k}{\left({\mathit{R}}_{{x}_{R},k}+{\mathit{R}}_{{n}_{R},k}\right)}^{-1}.\end{array}$

(33)

Finally, the parameter

_{β 2} is chosen such to set the total transmit power at the RN to

*P*_{T}_{R}:

${\beta}_{2}^{2}\text{tr}\left[{\mathit{F}}_{R}\left({\mathit{R}}_{{x}_{R}}+{\mathit{R}}_{{n}_{R}}\right)\phantom{\rule{0.2em}{0ex}}{\mathit{F}}_{R}^{H}\right]={P}_{{\text{T}}_{R}}.$

(34)