Multi-objective optimization design for multi-source multicasting MIMO AF relay systems

Abstract

In this paper, we consider a multi-source multicasting two-hop multi-input multi-output amplify-forward (AF) relay system, where multiple source nodes multicast their own messages to a group of receivers with the cooperation of an AF relay node. In particular, we aim at minimizing the transmission power consumption and the mean-squared error (MSE) of the receiver estimated signal simultaneously. However, the two objectives are coupled and even conflicting. In view of this, a multi-objective optimization (MOO) framework is adopted to achieve the trade-off between the two objectives. The formulated MOO problem (MOOP) takes into account the constraints of the MSE upper bound and the maximum transmission power budget. Since the MOOP is non-convex and hard to tackle, we propose a resource allocation algorithm by exploiting the weighted Tchebycheff approach and the optimal structure of the relay precoding matrix. Simulation results not only demonstrate the effectiveness of the proposed algorithm, but also unveil an important trade-off between the total power consumption and the MSE at receivers.

Appendix A

MSE matrix (12) can be reformulated as:

$$ MSE_{k} = {\text{tr}}\left\{ {\left[ {{\mathbf{I}}_{{N_{b} }} + {\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{Q}}^{H} {\mathbf{G}}_{k}^{H} ({\mathbf{G}}_{k} {\mathbf{QQ}}^{H} {\mathbf{G}}_{k}^{H} + {\mathbf{I}}_{{N_{d} }} )^{ - 1} {\mathbf{G}}_{k} {\mathbf{QHF}}} \right]^{ - 1} } \right\} $$

(30a)

$$ = {\text{tr}}\left\{ {\left[ {{\mathbf{I}}_{{N_{b} }} - {\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{Q}}^{H} {\mathbf{G}}_{k}^{H} ({\mathbf{G}}_{k} {\mathbf{QHFF}}^{H} {\mathbf{H}}^{H} {\mathbf{Q}}^{H} {\mathbf{G}}_{k}^{H} + ({\mathbf{G}}_{k} {\mathbf{QQ}}^{H} {\mathbf{G}}_{k}^{H} + {\mathbf{I}}_{{N_{d} }} ))^{ - 1} {\mathbf{G}}_{k} {\mathbf{QHF}}} \right]} \right\} $$

(30b)

$$ = {\text{tr}}\left\{ {{\mathbf{I}}_{{N_{b} }} - {\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{Q}}^{H} {\mathbf{G}}_{k}^{H} ({\mathbf{G}}_{k} {\mathbf{Q}}({\mathbf{HFF}}^{H} {\mathbf{H}}^{H} + {\mathbf{I}}_{{N_{r} }} ){\mathbf{Q}}^{H} {\mathbf{G}}_{k}^{H} + {\mathbf{I}}_{{N_{d} }} )^{ - 1} {\mathbf{G}}_{k} {\mathbf{QHF}}} \right\} $$

(30c)

$$ = {\text{tr}}\left\{ {{\mathbf{I}}_{{N_{b} }} - {\mathbf{F}}^{H} {\mathbf{H}}^{H} ({\mathbf{U}}^{ - 1} - ({\mathbf{UQ}}^{H} {\mathbf{G}}_{k}^{H} {\mathbf{G}}_{k} {\mathbf{QU}} + {\mathbf{U}})^{ - 1} ){\mathbf{HF}}} \right\} $$

(30d)

$$ = {\text{tr(}}{\mathbf{I}}_{{N_{b} }} - {\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{U}}^{ - 1} {\mathbf{HF}}) + {\text{tr}}({\mathbf{F}}^{H} {\mathbf{H}}^{H} ({\mathbf{UQ}}^{H} {\mathbf{G}}_{k}^{H} {\mathbf{G}}_{k} {\mathbf{QU}} + {\mathbf{U}})^{ - 1} {\mathbf{HF}}), $$

(30e)

where ${\mathbf{U}} = {\mathbf{HFF}}^{H} {\mathbf{H}}^{H} + {\mathbf{I}}_{{N_{r} }}$.

In the above transformations, the inversion lemma is applied to obtain formula (30b) and (30c), while the matrix equality ${\mathbf{B}}^{H} ({\mathbf{BCB}}^{H} + {\mathbf{I}})^{ - 1} {\mathbf{B}} = {\mathbf{C}}^{ - 1} - ({\mathbf{CB}}^{H} {\mathbf{BC}} + {\mathbf{C}})^{ - 1}$ is used to obtain formula (30d).

The first term in the right side of (30e) can be transformed by utilizing the matrix inversion lemma as:

$$ \begin{aligned} & {{\text{tr}}({\mathbf{I}}_{{N_{b} }} - {\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{U}}^{ - 1} {\mathbf{HF}})} \\ & {\quad = {\text{tr}}\left\{ {{\mathbf{I}}_{{N_{b} }} - {\mathbf{F}}^{H} {\mathbf{H}}^{H} ({\mathbf{HFF}}^{H} {\mathbf{H}}^{H} + {\mathbf{I}}_{{N_{r} }} )^{ - 1} {\mathbf{HF}}} \right\}} \\ & { \quad= {\text{tr}}\left\{ {\left[ {{\mathbf{I}}_{{N_{b} }} + {\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{HF}}} \right]^{ - 1} } \right\}.} \\ \end{aligned} $$

(31)

Applying the relay structure in (16), the matrix inversion lemma and the matrix equality ${\mathbf{B}}^{H} ({\mathbf{BCB}}^{H} + {\mathbf{I}})^{ - 1} {\mathbf{B}} = {\mathbf{C}}^{ - 1} - ({\mathbf{CB}}^{H} {\mathbf{BC}} + {\mathbf{C}})^{ - 1}$ once again into the second term in the right side of (30e), then we can get the transformation as:

$$ \begin{aligned} & {{\text{tr}}({\mathbf{F}}^{H} {\mathbf{H}}^{H} ({\mathbf{UQ}}^{H} {\mathbf{G}}_{k}^{H} {\mathbf{G}}_{k} {\mathbf{QU}} + {\mathbf{U}})^{ - 1} {\mathbf{HF}})} \\ & {\quad = {\text{tr}}\left\{ {{\mathbf{F}}^{H} {\mathbf{H}}^{H} ({\mathbf{HFT}}^{H} {\mathbf{G}}_{k}^{H} {\mathbf{G}}_{k} {\mathbf{TF}}^{H} {\mathbf{H}}^{H} + {\mathbf{U}})^{ - 1} {\mathbf{HF}}} \right\}} \\ & {\quad = {\text{tr}}\left\{ {{\mathbf{F}}^{H} {\mathbf{H}}^{H} ({\mathbf{U}}^{ - 1} - {\mathbf{U}}^{ - 1} {\mathbf{HF}}({\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{U}}^{ - 1} {\mathbf{HF}} + ({\mathbf{T}}^{H} {\mathbf{G}}_{k}^{H} {\mathbf{G}}_{k} {\mathbf{T}})^{ - 1} )^{ - 1} ){\mathbf{HF}}} \right\}} \\ & {\quad = {\text{tr}}\left\{ {\left[ {({\mathbf{F}}^{H} {\mathbf{H}}^{H} {\mathbf{U}}^{ - 1} {\mathbf{HF}})^{ - 1} + {\mathbf{T}}^{H} {\mathbf{G}}_{k}^{H} {\mathbf{G}}_{k} {\mathbf{T}}^{ - 1} } \right]} \right\},\quad \forall k.} \\ \end{aligned} $$

(32)

Therefore, we can get MSE formulation (17).