Joint Source and Relay Precoder Design in Amplify-and-Forward MIMO Relay Systems with Direct Link

Abstract

This paper investigates the precoder design problem in a two-hop amplify-and-forward multiple-input-multiple-output relay system. Many previous works on this problem are based on the minimum mean-square error criterion and the presence of a direct link between the source and the destination is ignored. In this paper, we propose a new method for joint source and relay precoder design based on maximizing the mutual information between the source and the destination, taking both the relay link and the direct link into account. In contrast to previous works, which consider the transmit power constraints of the source and the relay independently, we assume a total power constraint on the sum transmit power of the source and the relay instead to study also the optimal power distribution over the two nodes. A constrained optimization problem with respect to the unknown source precoder matrix and relay precoder matrix is then formulated, which is nonconvex and very difficult to solve directly. We propose a structural constraint on the precoders by analyzing the structure of the problem and referring to related works. With the proposed precoders’ structure and by applying the Hadamard’s inequality, the original problem is simplified from a matrix-valued problem to a scalar-valued one. However, the new scalar-valued problem is still nonconvex and we manage to convert it into two subproblems and solve it in an iterative fashion. By using the Karash–Kuhn–Tucker (KKT) conditions, we give out the closed-form solutions to the subprobelms. Simulation results demonstrate that the proposed design method converges rapidly and significantly outperforms the existing methods.

Appendices

Appendix A: Proof of (10)

$$\begin{aligned} \mathbf{{I}}({\mathbf{{y}}_D},{} \,\mathbf{{s}})&= \frac{1}{2}{\log _2}\left| \mathbf{{I}} + \mathbf{{H}}{\mathbf{{R}}_S}{\mathbf{{H}}^H}\mathbf{{R}}_Z^{ - 1}\right| \nonumber \\&= \frac{1}{2}{\log _2}\left| \mathbf{{I}} + \sigma _S^2\mathbf{{H}}{\mathbf{{H}}^H}\mathbf{{R}}_Z^{ - 1}\right| \nonumber \\&= \frac{1}{2}{\log _2}\left| \mathbf{{I}} + \sigma _S^2\left[ {\begin{array}{cc} {{\mathbf{{H}}_{SD}}{\mathbf{{F}}_S}} \\ {{\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}{\mathbf{{H}}_{SR}}{\mathbf{{F}}_S}} \\ \end{array}} \right] \left[ {\begin{array}{cc} {{{({\mathbf{{H}}_{SD}}{\mathbf{{F}}_S})}^H}} &{} {{{({\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}{\mathbf{{H}}_{SR}}{\mathbf{{F}}_S})}^H}} \\ \end{array}} \right] \right. \nonumber \\&\left. \times {\left[ {\begin{array}{cc} {{\mathbf{{R}}_{Z0}}} &{} \mathbf{{0}} \\ \mathbf{{0}} &{} {{\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}{\mathbf{{R}}_{Z1}}\mathbf{{F}}_R^H\mathbf{{H}}_{RD}^H + {\mathbf{{R}}_{Z2}}} \\ \end{array}} \right] ^{ - 1}}\right| \nonumber \\&= \frac{1}{2}{\log _2}\left| \mathbf{{I}} + \sigma _S^2\left[ {\begin{array}{cc} {{{({\mathbf{{H}}_{SD}}{\mathbf{{F}}_S})}^H}} &{} {{{({\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}{\mathbf{{H}}_{SR}}{\mathbf{{F}}_S})}^H}} \\ \end{array}} \right] \right. \nonumber \\&\left. \times \left[ {\begin{array}{cc} {\mathbf{{R}}_{Z0}^{ - 1}} &{} \mathbf{{0}} \\ \mathbf{{0}} &{} {{{({\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}{\mathbf{{R}}_{Z1}}\mathbf{{F}}_R^H\mathbf{{H}}_{RD}^H + {\mathbf{{R}}_{Z2}})}^{ - 1}}} \\ \end{array}} \right] \left[ {\begin{array}{cc} {{\mathbf{{H}}_{SD}}{\mathbf{{F}}_S}} \\ {{\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}{\mathbf{{H}}_{SR}}{\mathbf{{F}}_S}} \\ \end{array}} \right] \right| \nonumber \\&= \frac{1}{2}{\log _2}\left| \mathbf{{I}} + \frac{{\sigma _S^2}}{{\sigma _{Z0}^2}}\mathbf{{F}}_S^H\mathbf{{H}}_{SD}^H{\mathbf{{H}}_{SD}}{\mathbf{{F}}_S}\right. \nonumber \\&\left. + \frac{{\sigma _S^2}}{{\sigma _{Z2}^2}}\mathbf{{F}}_S^H\mathbf{{H}}_{SR}^H\mathbf{{F}}_R^H\mathbf{{H}}_{RD}^H{\left( \frac{{\sigma _{Z1}^2}}{{\sigma _{Z2}^2}}{\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}\mathbf{{F}}_R^H\mathbf{{H}}_{RD}^H + \mathbf{{I}}\right) ^{ - 1}}{\mathbf{{H}}_{RD}}{\mathbf{{F}}_R}{\mathbf{{H}}_{SR}}{\mathbf{{F}}_S}\right| \qquad \end{aligned}$$

(46)

Appendix B: The Algorithm for Determining $\mu$ in (44)

To determine $\mu$ efficiently, we first give out an upper bound and a lower bound of $\mu$ denoted as ${\mu _{U0}}$ and ${\mu _{L0}}$, respectively. Substituting (44) into (45), we have

$$\begin{aligned} {\sum \limits _{i = 1}^4 {\frac{1}{{\varLambda _{2i}^2}}\left( {\mu \sqrt{\frac{{\varLambda _{1i}^2\varLambda _{2i}^2}}{{\varLambda _{1i}^2 + 1}}} - 1} \right) } ^ + } = P/2 \end{aligned}$$

(47)

Since

$$\begin{aligned} {\sum \limits _{i = 1}^4 {\frac{1}{{\varLambda _{2i}^2}}\left( {\mu \sqrt{\frac{{\varLambda _{1i}^2\varLambda _{2i}^2}}{{\varLambda _{1i}^2 + 1}}} - 1} \right) } ^ + }&\ge \sum \limits _{i = 1}^4 {\frac{1}{{\varLambda _{2i}^2}}\left( {\mu \sqrt{\frac{{\varLambda _{1i}^2\varLambda _{2i}^2}}{{\varLambda _{1i}^2 + 1}}} - 1} \right) }\nonumber \\&\ge \sum \limits _{i = 1}^4 {\frac{1}{{\varLambda _{2i}^2}}} \left[ {\mu \mathop {\min }\limits _i \left( \sqrt{\frac{{\varLambda _{1i}^2\varLambda _{2i}^2}}{{\varLambda _{1i}^2 + 1}}} \right) - 1} \right] \end{aligned}$$

(48)

let

$$\begin{aligned} \Bigg [\mu \underbrace{\mathop {\min }\limits _i \left( \sqrt{\frac{{\varLambda _{1i}^2\varLambda _{2i}^2}}{{\varLambda _{1i}^2 + 1}}} \right) }_{: = a} - 1\Bigg ]\underbrace{\sum \limits _{i = 1}^4 {\frac{1}{{\varLambda _{2i}^2}}} }_{: = b} = P/2 \end{aligned}$$

(49)

we have

$$\begin{aligned} {\mu _{U0}} = \left( \frac{P}{{2b}} + 1\right) /a \end{aligned}$$

(50)

On the other hand, in order to satisfy (47), at least we should have

$$\begin{aligned} \mu \mathop {\max }\limits _i \left( \sqrt{\frac{{\varLambda _{1i}^2\varLambda _{2i}^2}}{{\varLambda _{1i}^2 + 1}}} \right) - 1 > 0 \end{aligned}$$

(51)

so we can choose

$$\begin{aligned} {\mu _{L0}} = 1/\mathop {\max }\limits _i \left( \sqrt{\frac{{\varLambda _{1i}^2\varLambda _{2i}^2}}{{\varLambda _{1i}^2 + 1}}} \right) \end{aligned}$$

(52)

With this two initial value, we have the following algorithm to determine $\mu$, where $\epsilon$ is a predetermined convergence precision.