A low complexity semidefinite relaxation for large-scale MIMO detection

Abstract

Many wireless communication problems is based on a convex relaxation of the maximum likelihood problem which further can be cast as binary quadratic programs (BQPs). The two standard relaxation methods that are widely used for solving general BQPs such as spectral methods and semidefinite programming problem (SDP), each have their own advantages and disadvantages. It is widely accepted that small and medium sized SDP problems can be solved efficiently by interior point methods. Albeit, semidefinite relaxation has a tighter bound for large scale problems, but its computational complexity is high. However, Row-by-Row method (RBR) for solving SDPs could be opted for an alternative for large-scale MIMO detection because of low complexity. The present work is a spectral SDP-cut formulation to which the RBR is applied for large-scale MIMO detection. A modified RBR algorithm with tighter bound is presented to specify the efficiency in detecting massive MIMO.

Appendices

Appendix A: Simplification of the $z_r$

The Eq. (18) can be written as

$$\begin{aligned}&z_r+\frac{\lambda }{\rho }\varLambda _r^{-1}z_r =-\frac{1}{2\rho }Q_r^\top c_1\\&\quad \Rightarrow \left( I+\frac{\lambda }{\rho }\varLambda _r^{-1}\right) z_r=-\frac{1}{2\rho }Q_r^\top c_1\\&\quad \Rightarrow \frac{\lambda }{\rho }\varLambda _r^{-1} (\frac{\rho }{\lambda }(\varLambda _r+I)z_r=-\frac{1}{2\rho }Q_r^\top c_1\\&\quad \Rightarrow z_r=-\frac{1}{2\lambda }\left( \frac{\rho }{\lambda }\varLambda _r+I\right) ^{-1} \varLambda _rQ_r^\top c_1 . \end{aligned}$$

Appendix B: Proof of: $D\varLambda _r^{-1}D=\varLambda _r^{-1}D^2$

It is to be noted that D is a diagonal matrix and so also $\varLambda _r$. Therefore, $D\varLambda _r^{-1}D=\varLambda _r^{-1}D^2$.

Appendix C: Proof of: $Q_rD^2Q_r^\top =[(\frac{\rho }{\lambda }B+I)^2]^{-1}$

$$\begin{aligned} Q_rD^2Q_r^\top&\,=Q_r\left( \frac{\rho }{\lambda }\varLambda _r+I\right) ^{-1} \left( \frac{\rho }{\lambda }\varLambda _r+I\right) ^{-1}Q_r^{-1}\\&=\left[ Q_r\left( \frac{\rho }{\lambda }\varLambda _r+I\right) \left( \frac{\rho }{\lambda }\varLambda _r+I\right) Q_r^{-1}\right] ^{-1}\\&=\left[ \frac{\rho ^2}{\lambda ^2}Q_r\varLambda _r^2Q_r^{-1} +\frac{2\rho }{\lambda }Q_r\varLambda _rQ_r^{-1}+I\right] ^{-1}\\&=\left[ \frac{\rho ^2}{\lambda ^2}B^2+\frac{2\rho }{\lambda }B+I\right] ^{-1}\\&=\left[ \left( \frac{\rho }{\lambda }B+I\right) ^2\right] ^{-1}=S^2 \end{aligned}$$

where $S=(\frac{\rho }{\lambda }B+I)^{-1}$.