Accurate approximation of ZF Massive MIMO channel rate with a finite antenna over ricean fading channel

In a typical downlink massive multiple-input and multiple-output (massive MIMO) system, the available degree of freedom (DoF) and multiplexing gain improve the channel rate. However, while a finite base station (BS) antenna limits the DoF, the line of sight (LoS) component decreases the multiplexing gain as a result of reducing the rank of the Ricean channel matrix. This paper provides the simple and tight approximations of the massive MIMO channel rate using the zero-forcing (ZF) precoding over the Ricean fading channel and with a finite antenna. We derive an accurate approximation of the ZF Ricean channel covariance matrix by exploiting the arbitrary-rank channel mean. This allows the ZF precoding to focus on the nonzero eigenmode of the correlation matrix in order to improve the channel gain. Subsequently, we deduce the first-, second- and third-eigenvalue moments. We then obtain an asymptotic series expansion in the closed-form expressions of the ZF achievable rates. Numerical results confirm the accuracy of the approximations, as the approximated ZF achievable rates converge to the ideal channel rate. In particular, the convergence shows a dependence on the Ricean factor and effective signal-to-noise ratio (SNR), while a finite BS antenna in ZF massive MIMO channel offers the required diversity order.