Per-Antenna Power Constrained Transceiver Design for MIMO Multisource and Multidestination Amplify-and-Forward Relay Systems

Abstract

This study addresses the transceiver design method for multiple-input multiple-output multisource multidestination amplify-and-forward relay systems. The source, relay, and destination filters are designed to minimize the sum mean-squared-error under the per-antenna power constraints at the source and relay nodes. The joint optimization is challenging due to its non-convexity and multiple power constraints. To resolve these difficulties, we propose a source and relay filter design method and present an alternating algorithm that is based on the block-coordinate descent method. Specifically, by introducing transmit-centric modified MSEs and reformulating the problem, the non-convex problem is transformed into tractable forms with multiple power constraints. The multiple power constraints are then adjusted by semidefinite programming of the source filters and a 1-D line search method for the relay transceiver. Simulation results demonstrate the effectiveness of the proposed schemes under the per-antenna power constraints that are compared with the conventional method under the sum power constraints.

Appendix

1.1 Derivation of the modified sum-MSE

The full expression for the sum-MSE (6) is as follows:

$$\begin{aligned} \mathbf{MSE }_{\varSigma }&= \sum _{k=1}^{K} \mathbf{MSE }_k \nonumber \\&=\sum _{k=1}^{K} \Bigg \lbrace \text {Tr}\Big ( {\mathbf{D}}_k{\mathbf{G}}_k{\mathbf{B}}{\mathbf{H}}_k{\mathbf{A}}_k {\mathbf{A}}_k^{H}{\mathbf{H}}_k^{H}{\mathbf{B}}^{H}{\mathbf{G}}_k^{H}{\mathbf{D}}_k^{H} \nonumber \\&\quad -{\mathbf{D}}_k{\mathbf{G}}_k{\mathbf{B}}{\mathbf{H}}_k{\mathbf{A}}_k -{\mathbf{A}}_k^{H}{\mathbf{H}}_k^{H}{\mathbf{B}}^{H}{\mathbf{G}}_k^{H}{\mathbf{D}}_k^{H} +{\mathbf{I}}_{N_s} \nonumber \\&\quad +\sigma _{n_r}^2 {\mathbf{D}}_k{\mathbf{G}}_k{\mathbf{B}}{\mathbf{B}}^{H}{\mathbf{G}}_k^{H}{\mathbf{D}}_k^{H} +\sigma _{n_k}^2 {\mathbf{D}}_k {\mathbf{D}}_k^H \Big ) \nonumber \\&\quad + \sum _{i \ne k }^{K} \text {Tr}\Big ( {\mathbf{D}}_k{\mathbf{G}}_k{\mathbf{B}}{\mathbf{H}}_i{\mathbf{A}}_i {\mathbf{A}}_i^{H}{\mathbf{H}}_i^{H}{\mathbf{B}}^{H}{\mathbf{G}}_k^{H}{\mathbf{D}}_k^{H} \Big ) \Bigg \rbrace \end{aligned}$$

(23a)

$$\begin{aligned}&=\sum _{k=1}^{K} \mathbf{MSE }_k^{\text {(MOD)}} . \end{aligned}$$

(23b)

By the simple notation exchanges; i to k, and k to i, in the summation of the inter-user interference terms of (23a), equality (23b) can be obtained.