Chapter Summary

Chapter 6 Summary: Linear Precoding

Key Points

• 1.

  MRT Precoding. Maximum ratio transmission sets $\mathbf{v}_{k} = \mathbf{h}_k/\|\mathbf{h}_k\|$, maximising the SNR to each user via the Cauchy--Schwarz inequality. MRT delivers full array gain ($N_t$-fold SNR increase) but is interference-limited when $K$ is not negligible relative to $N_t$. In the massive MIMO regime, favorable propagation makes MRT asymptotically optimal.

• 2.

  ZF Precoding. Zero-forcing projects each precoding vector into the null space of all other users' channels, eliminating multi-user interference completely. The cost is a power penalty: the effective array gain drops from $N_t$ (MRT) to $N_t - K$ (ZF), reflecting the degrees of freedom consumed by interference nulling. ZF becomes ill-conditioned when $K \to N_t$.

• 3.

  RZF/MMSE Precoding. Regularized zero-forcing adds $\alpha\mathbf{I}$ to the Gram matrix: $\mathbf{W} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H} + \alpha\mathbf{I})^{-1}$. With optimal $\alpha^{\star} = K\sigma^2/P_t$, RZF achieves the best tradeoff between noise amplification and residual interference at any SNR and loading. It smoothly bridges MRT ($\alpha \to \infty$) and ZF ($\alpha \to 0$).

• 4.

  Per-Antenna Power Constraints. Practical systems have individual power amplifiers per antenna element. The per-antenna constraint makes precoder design a convex optimisation problem solvable via uplink-downlink duality with antenna-dependent noise. The rate loss relative to sum power is typically small but must be accounted for.

• 5.

  Gap to DPC Capacity. The MIMO broadcast channel capacity is achieved by dirty-paper coding, a nonlinear technique that is computationally intractable. RZF precoding captures over 95% of the DPC capacity at $N_t/K \geq 4$, and the gap vanishes in the massive regime. This small gap justifies the universal adoption of linear precoding in 4G/5G systems.

• 6.

  Engineering Perspective. The choice among MRT, ZF, and RZF depends on the antenna-to-user ratio and SNR regime. RZF is the practical default in 5G NR, with the regularization parameter adapted to the estimated noise level and channel conditions. Per-antenna constraints and computational complexity are the binding practical considerations.