New Mean-Squared-Error Filter Design Method for Multiple-Input Multiple-Output Amplify-and-Forward Relay Systems with a Non-negligible Direct Link

Abstract

In this paper, we propose a new mean-squared-error (MSE) minimizing transceiver design method for multiple-input multiple-output amplify-and-forward relay systems with a non-negligible source-to-destination direct link. In earlier works, the direct link has not been fully exploited nor have optimal solutions been presented in analytical forms. Motivated by these weaknesses, we investigated a new source-relay-destination filter design method. To address the difficulties resulting from multiple power constraints and the power allocation between two precoders at the source node, we formulated the MSE minimization problem by introducing a regularizing factor. From the Karush–Kuhn–Tucker conditions of the individual transceiver optimization problem, semi-closed form filter solutions for source, relay, and destination were derived. Then, resorting to the proposed iterative joint optimizing algorithm, a local optimal point was accessible. Through numerical simulations, the efficacy of the proposed method was illustrated, and it was observed that only a few iterations the MSE performances of the proposed method surpass those of conventional schemes.

Appendix: The monotonicity of function \(p({{\bar{\mu }}}_1)\)

Substituting ${\tilde{\mathbf{A}}}_1({{\bar{\mu }}}_1)$ in the function $p({{\bar{\mu }}}_1)$ with ${\tilde{\mathbf{A}}}_1$ in case 3, the differentiation of $p({{\bar{\mu }}}_1)$ with respect to ${{\bar{\mu }}}_1$ is written by the following:

$$\begin{aligned} \frac{dp({{\bar{\mu }}}_1)}{d{{\bar{\mu }}}_1}&= -2 P_s \text {Tr}\big ( {\varvec{\varSigma }}_A^H{\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-1}{\bar{\mathbf{U}}}_A^H{\varvec{\varSigma }}_A{\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-1}{\bar{\mathbf{U}}}_A^H \nonumber \\&\quad \times {\mathbf{L}}_A{\mathbf{L}}_A^H{\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-1}{\bar{\mathbf{U}}}_A^H \big ) \end{aligned}$$

(26)

$$\begin{aligned}&= -2 P_s \text {Tr}({\mathbf{Q}}_{A}{\mathbf{Q}}_{A}^H) \le 0 \end{aligned}$$

(27)

where ${\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-1}{\bar{\mathbf{U}}}_A^H=\text {SVD}({\varvec{\varOmega }}_1 {\varvec{\varOmega }}_2 + {{\bar{\mu }}}_1{\mathbf{I}}_{N_s}+\frac{\text {Tr}({\varvec{\varPsi }})-{{\bar{\mu }}}_1 P_s}{P_{r,A}}{\mathbf{H}}^H{\mathbf{B}}^H{\mathbf{B}}{\mathbf{H}})$, ${\varvec{\varSigma }}_A = {\mathbf{I}}_{N_s}-\frac{P_s}{P_{r,A}}{\mathbf{H}}^H{\mathbf{B}}^H{\mathbf{B}}{\mathbf{H}}$, and ${\mathbf{L}}_A={\varvec{\varOmega }}_1({\mathbf{I}}_{N_s}-{\mathbf{D}}_2{\mathbf{C}}_2{\tilde{\mathbf{A}}}_2)$. (27) is obtained by substituting ${\mathbf{L}}_A^H{\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-1}{\bar{\mathbf{U}}}_A^H{\varvec{\varSigma }}_A^H{\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-\frac{1}{2}}$ with ${\mathbf{Q}}_{A}$ which is defined by ${\mathbf{Q}}_{A}={\mathbf{L}}_A^H{\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-1}{\bar{\mathbf{U}}}_A^H{\varvec{\varSigma }}_A^H{\bar{\mathbf{U}}}_A{\bar{\varvec{\varLambda }}}_A^{-\frac{1}{2}}$. Thus, the function $p({{\bar{\mu }}}_1)$ is monotonically decreasing.