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Hybrid Beamforming Full-Duplex Relay in Massive MIMO Systems

  • THEORY AND METHODS OF SIGNAL PROCESSING
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Abstract

In this paper, we consider a point-to-point MIMO system with the amplified-and-forward full-duplex relay that the source can communicate with destination through the relay. The optimization problem for this scenario is non-convex, and finding the hybrid beamforming matrices, is a significant challenge. Here we propose two methods to design hybrid beamforming matrices in a full-duplex relay scenario. In the first method, we separate the transmitter and receiver sides of the relay and formulate the optimization problems. In the second method, we designed the beamforming relay matrices in a joint manner and proposed an algorithm to find a nearly-optimum full-duplex relay to achieve higher performance. Simulation results show that the joint optimization method achieves higher spectral efficiency than the separation method (first method). In addition the impact of the number of RF chains, is carried out by simulation. The simulation demonstrates that the number of antennas in the relay has an optimum value for a fixed number of RF chains.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Babol Noshirvani University of Technology, 47148–71167, Babol, Mazandaran, Iran

    M. Golbabapour & M. Reza Zahabi

Authors
  1. M. Golbabapour
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  2. M. Reza Zahabi
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Correspondence to M. Reza Zahabi.

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Appendices

APPENDIX 1

MSE problem in receiver side of the relay is

$$\mathbb{E}\left[ {\left\| {{\mathbf{s}} - {\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}{{{\mathbf{y}}}_{{\text{R}}}}} \right\|_{2}^{2}} \right]$$
$$ = \mathbb{E}\left[ {{{{\left( {{\mathbf{s}} - {\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}{{{\mathbf{y}}}_{{\text{R}}}}} \right)}}^{H}}\left( {{\mathbf{s}} - {\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}{{{\mathbf{y}}}_{{\text{R}}}}} \right)} \right]$$
$$ = \mathbb{E}\left[ {{\text{tr}}\left( {\left( {{\mathbf{s}} - {\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}{{{\mathbf{y}}}_{{\text{R}}}}} \right){{{\left( {{\mathbf{s}} - {\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}{{{\mathbf{y}}}_{{\text{R}}}}} \right)}}^{H}}} \right)} \right]$$
$$ = {\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{s}}{{{\mathbf{s}}}^{H}}} \right]} \right) - 2R\left\{ {{\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{sy}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R}}1}}}} \right)} \right\}$$
$$ + \,\,{\text{tr}}\left( {{\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R}}1}}}} \right).$$

Since the optimization problem is done on Variables \({{{\mathbf{F}}}_{{{\text{R1}}}}}\) and \({{{\mathbf{W}}}_{{{\text{R1}}}}}\), we can add a phrase independent of these two variables. So we add, \({\text{tr}}\left( {{\mathbf{W}}_{{{\text{MMSE}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{W}}}_{{{\text{MMSE}}}}}} \right) - {\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{s}}{{{\mathbf{s}}}^{H}}} \right]} \right)\), which is independent of the problem variables, and we have:

$$\begin{gathered} f = {\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{s}}{{{\mathbf{s}}}^{H}}} \right]} \right) - 2R\left\{ {{\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{sy}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R}}1}}}} \right)} \right\} \\ + \,\,{\text{tr}}\left( {{\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R}}1}}}} \right) \\ \end{gathered} $$
$$ + \,\,{\text{tr}}\left( {{\mathbf{W}}_{{{\text{MMSE}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{W}}}_{{{\text{MMSE}}}}}} \right) - {\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{s}}{{{\mathbf{s}}}^{H}}} \right]} \right)$$
$$\begin{gathered} = {\text{tr}}\left( {{\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R1}}}}}{{{\mathbf{W}}}_{{{\text{R1}}}}}} \right) \\ - \,\,2R\left\{ {{\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{sy}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R}}1}}}} \right)} \right\} \\ \end{gathered} $$
$$ + \,\,{\text{tr}}\left( {{\mathbf{W}}_{{{\text{MMSE}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{W}}}_{{{\text{MMSE}}}}}} \right).$$

With knowing, \({\mathbf{W}}_{{{\text{MMSE}}}}^{H} = \mathbb{E}\left[ {{\mathbf{sy}}_{{\text{R}}}^{H}} \right]\mathbb{E}{{\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]}^{{ - 1}}}\),

$$\mathbb{E}\left[ {{\mathbf{sy}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R}}1}}} = {\mathbf{W}}_{{{\text{MMSE}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R1}}}}}$$

and

$$\begin{gathered} {\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{sy}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R1}}}}}{{{\mathbf{W}}}_{{{\text{R1}}}}}} \right) \\ = {\text{tr}}\left( {\mathbb{E}\left[ {{\mathbf{sy}}_{{\text{R}}}^{H}} \right]\mathbb{E}{{{\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]}}^{{ - 1}}}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R}}1}}}} \right), \\ \end{gathered} $$

the equation is simplified as follows:

$$\begin{gathered} f = {\text{tr}}\left( {{\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R}}1}}}{{{\mathbf{W}}}_{{{\text{R1}}}}}} \right) \\ - \,\,2R\left\{ {{\text{tr}}\left( {{\mathbf{W}}_{{{\text{MMSE}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{F}}}_{{{\text{R1}}}}}{{{\mathbf{W}}}_{{{\text{R1}}}}}} \right)} \right\} \\ \end{gathered} $$
$$ + \,\,{\text{tr}}\left( {{\mathbf{W}}_{{{\text{MMSE}}}}^{H}\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]{{{\mathbf{W}}}_{{{\text{MMSE}}}}}} \right)$$
$$\begin{gathered} = {\text{tr}}\left( {\left( {{\mathbf{W}}_{{{\text{MMSE}}}}^{H} - {\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}} \right)\mathbb{E}\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]} \right. \\ \left. { \times \,\,{{{\left( {{\mathbf{W}}_{{{\text{MMSE}}}}^{H} - {\mathbf{W}}_{{{\text{R1}}}}^{H}{\mathbf{F}}_{{{\text{R1}}}}^{H}} \right)}}^{H}}} \right) \\ \end{gathered} $$
$$ = \mathbb{E}{{\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]}^{{\frac{1}{2}}}}\left\| {{{{\mathbf{W}}}_{{{\text{MMSE}}}}} - {{{\mathbf{F}}}_{{{\text{R1}}}}}{{{\mathbf{W}}}_{{{\text{R1}}}}}} \right\|_{{\text{F}}}^{2}.$$

So MSE problem on the receiver side of the relay node is:

$$\mathop {{\text{min }}}\limits_{{{{\mathbf{W}}}_{{{\text{R1}}}}},{{\;}}{{{\mathbf{F}}}_{{{\text{R}}1}}}} \mathbb{E}{{\left[ {{{{\mathbf{y}}}_{{\text{R}}}}{\mathbf{y}}_{{\text{R}}}^{H}} \right]}^{{\frac{1}{2}}}}\left\| {{{{\mathbf{W}}}_{{{\text{MMSE}}}}} - {{{\mathbf{F}}}_{{{\text{R1}}}}}{{{\mathbf{W}}}_{{{\text{R1}}}}}} \right\|_{{\text{F}}}^{2}.{{\;}}$$
$${\text{s}}.{\text{t}}.\,\,{{{\mathbf{F}}}_{{{\text{R}}1}}} \in {{\mathcal{F}}_{{{\text{R}}1}}}.$$

APPENDIX 2

The optimization problem in an optimal relay is

$$\mathop {\max ~}\limits_{{\mathbf{G}}~} {\text{lo}}{{{\text{g}}}_{2}}\left| {{{{\mathbf{I}}}_{{{{N}_{{\text{S}}}}}}} + \frac{\rho }{{{{N}_{{\text{S}}}}~}}{{{\mathbf{B}}}^{{ - 1}}}{{{\mathbf{H}}}_{{{\text{RD}}}}}{\mathbf{CG}}{{{\mathbf{H}}}_{{{\text{SR}}}}}{\mathbf{H}}_{{{\text{SR}}}}^{H}{\mathbf{H}}_{{{\text{RD}}}}^{H}} \right|,$$
$${\text{s}}.{\text{t}}.\,\,{\text{tr}}\left( {{\mathbf{CG}}\left( {{{{\mathbf{H}}}_{{{\text{SR}}}}}{{{\mathbf{W}}}_{{\text{S}}}}{\mathbf{W}}_{{\text{S}}}^{H}{\mathbf{H}}_{{{\text{SR}}}}^{H} + \sigma _{{{{{\mathbf{n}}}_{{{\text{SR}}}}}}}^{2}{{{\mathbf{I}}}_{{{{N}_{{\text{R}}}}}}}} \right){{{\mathbf{G}}}^{H}}{{{\mathbf{C}}}^{H}}} \right) \leqslant {{P}_{{\text{R}}}},$$

where \({\mathbf{C}} = {{\left( {{{{\mathbf{I}}}_{{{{M}_{{\text{R}}}}}}} - {{{\mathbf{F}}}_{{{\text{R2}}}}}{{{\mathbf{W}}}_{{\text{R}}}}{{{\mathbf{F}}}_{{{\text{R1}}}}}{{{\mathbf{H}}}_{{{\text{RR}}}}}} \right)}^{{ - 1}}}\) and \({\mathbf{B}} = \sigma _{{{{{\mathbf{n}}}_{{{\text{SR}}}}}}}^{2}{{{\mathbf{H}}}_{{{\text{RD}}}}}{\mathbf{CG}}{{{\mathbf{G}}}^{H}}{{{\mathbf{C}}}^{H}}{\mathbf{H}}_{{{\text{RD}}}}^{H} + \sigma _{{{{{\mathbf{n}}}_{{{\text{RD}}}}}}}^{2}{{{\mathbf{I}}}_{N}}\) is the Covariance matrix of the noise. \({{P}_{{\text{R}}}}\) and \(\rho \) are the relay total power and power coefficient, respectively.

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Golbabapour, M., Reza Zahabi, M. Hybrid Beamforming Full-Duplex Relay in Massive MIMO Systems. J. Commun. Technol. Electron. 68, 141–150 (2023). https://doi.org/10.1134/S1064226923020079

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