Precoders

Precoders are signal processing techniques used at the transmitter in multi-antenna wireless systems to pre-adjust transmitted signals based on the channel state information. The goal is to control how signals combine at receivers, suppress interference in multi-user scenarios, and improve spectral efficiency and reliability.

In a MIMO downlink, a precoding matrix W maps data streams s to transmit antennas x = W s. The received signal is y = H x + n, where H is the channel matrix and n is noise. By choosing W according to H, interference among streams or users can be reduced and energy can be steered toward intended receivers. Precoders thus form the wireless link prior to modulation and transmission.

Common linear precoders include zero-forcing (ZF) precoding, which uses the channel inversion to cancel inter-stream interference; MMSE precoding, which regularizes inversion to balance interference suppression with noise amplification; and matched-filter (conjugate) precoding, which maximizes received signal gain. Regularized zero-forcing (RZF) blends ZF and maximum ratio transmission to accommodate noise and CSI imperfections. Non-linear precoding techniques, such as Tomlinson-Harashima precoding and vector perturbation, attempt more sophisticated interference pre-cancellation. Dirty paper coding is a theoretical optimum for broadcast channels but is complex to implement in practice.

Practical considerations include the requirement for accurate channel state information at the transmitter (CSIT); performance degrades with imperfect or outdated CSI. Computational complexity and power constraints (per-antenna and total) also influence design choices, and precoding is typically applied per subcarrier in OFDM systems. Robust designs address channel variation, feedback overhead, and latency.

Applications are widespread in wireless communications, notably in 4G/5G downlink, Wi-Fi MIMO, and other multi-antenna systems, where precoding is a central mechanism for directing energy and managing interference.