RZF

RZF, or Regularized Zero Forcing, is a linear precoding technique used in multi-antenna wireless communications to mitigate inter-user interference while controlling noise amplification. It sits between zero-forcing (ZF) and maximum ratio transmission (MRT) as a compromise that performs well across a range of signal-to-noise ratios and channel conditions.

In downlink multi-user MIMO or multi-antenna broadcast channels, the precoding matrix W for K users and N transmit antennas is commonly expressed as W = H^H (H H^H + α I)^-1, where H is the channel matrix (K users by N transmit antennas) and α ≥ 0 is a regularization parameter. A normalization or power-control factor is often applied to meet a given transmit power constraint. The term (H H^H + α I)^-1 regularizes the inverse of H H^H, reducing sensitivity to ill-conditioned channels and avoiding excessive noise amplification that can occur with pure ZF.

Interpretation and regimes: as α approaches zero, RZF approaches ZF precoding, aggressively canceling interference but potentially amplifying noise. As α becomes large, the method leans toward more uniform beamforming similar to MRT, with reduced interference suppression but greater robustness to noise. Thus, the regularization parameter trades off interference suppression against noise enhancement and is typically chosen based on the operating SNR, channel statistics, and power constraints.

Applications and variants: RZF is widely used for downlink precoding in wireless standards and research on MIMO broadcast channels. It is closely related to MMSE precoding and can be viewed as a Tikhonov-regularized solution. Practical implementations select α to optimize sum rate and reliability under CSIT constraints.