Limited-Feedback Low-Encoding Complexity Precoder Design for Downlink of FDD Multi-User Massive MIMO Systems

We investigate a limited feedback precoder based on symbol pairwise error probability (PEP) for a block-faded <inline-formula> <tex-math notation="LaTeX">$K \times n_{t}$ </tex-math></inline-formula> downlink multiple-input multiple-output (MIMO) channel. In the considered system, <inline-formula> <tex-math notation="LaTeX">$K=\lfloor n_{t}^{\alpha }\rfloor $ </tex-math></inline-formula> single-antenna users feedback quantized channel state information to the <inline-formula> <tex-math notation="LaTeX">$n_{t}$ </tex-math></inline-formula>-antenna transmitter using <inline-formula> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula> bits per-transmit-antenna per user. We analytically show that for <inline-formula> <tex-math notation="LaTeX">$\alpha <({1}/{2})$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$B \geq 1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n_{t}\rightarrow \infty $ </tex-math></inline-formula>, both symbol PEP and achievable rate of each of the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> downlink users almost surely converge to the symbol PEP and achievable rate of <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> parallel additive white Gaussian noise (AWGN) channels, respectively. We show that the encoding complexity of the precoder is <inline-formula> <tex-math notation="LaTeX">$O(n_{t}K)$ </tex-math></inline-formula>. We also show that if channel coefficients estimated by the user are corrupted by AWGN noise, the symbol PEP and achievable rate of each user almost surely converge to the symbol PEP and achievable rate in a <italic>scaled</italic> AWGN channel with <inline-formula> <tex-math notation="LaTeX">$B>1$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$n_{t}\rightarrow \infty $ </tex-math></inline-formula>. For correlated channels, we derive a condition, which enables the proposed precoder almost surely to cancel multi-user interference for large <inline-formula> <tex-math notation="LaTeX">$n_{t}$ </tex-math></inline-formula> values. Finally, we numerically compare the bit error rate, encoding complexity, and per-user achievable rate of the proposed scheme with the existing designs.