Precoding with CSIT

Maximum ratio transmission (MRT) is the transmit-side analogue of maximal ratio combining. For a single-user MISO channel $y = \mathbf{h}^H\mathbf{x} + n$ with $\mathbf{h} \in \mathbb{C}^{N_t}$, the MRT beamforming vector is:

$$\mathbf{w}_{\text{MRT}} = \frac{\mathbf{h}}{\|\mathbf{h}\|}$$

which transmits $\mathbf{x} = \mathbf{w}_{\text{MRT}} s$ for scalar symbol $s$. This maximises the received SNR:

$$\text{SNR}_{\text{MRT}} = \frac{P\|\mathbf{h}\|^2}{\sigma^2}$$

MRT achieves the full array gain $N_t$ and the full transmit diversity order $N_t$ for a MISO channel.

For a multi-user MISO broadcast channel where user $k$ has channel $\mathbf{h}_k^H \in \mathbb{C}^{1 \times N_t}$, define $\mathbf{H} = [\mathbf{h}_1, \ldots, \mathbf{h}_K]^H$. The ZF precoding matrix is:

$$\mathbf{W}_{\text{ZF}} = \mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}\mathbf{D}$$

where $\mathbf{D}$ is a diagonal power allocation matrix. Each user receives:

$$y_k = \mathbf{h}_k^H \mathbf{w}_k s_k + n_k$$

with zero inter-user interference: $\mathbf{h}_k^H\mathbf{w}_j = 0$ for $j \neq k$.

ZF precoding requires $N_t \geq K$ and completely eliminates multi-user interference at the cost of power efficiency (noise enhancement at the transmitter side).

MMSE precoding (also called regularised ZF precoding) adds a regularisation term analogous to the MMSE receiver:

$$\mathbf{W}_{\text{MMSE}} = \mathbf{H}^{H}\left(\mathbf{H}\mathbf{H}^{H} + \frac{K\sigma^2}{P}\mathbf{I}\right)^{-1}\mathbf{D}$$

This balances inter-user interference suppression with transmit power efficiency. At high SNR, MMSE precoding converges to ZF precoding; at low SNR, it converges to MRT (matched filtering), which is the power-optimal strategy when interference is negligible compared to noise.

Dirty paper coding (DPC) is the information-theoretically optimal nonlinear precoding strategy. Based on Costa's 1983 result, DPC shows that if the transmitter knows the interference $\mathbf{s}$ non-causally (i.e., knows other users' signals), it can pre-subtract the interference without any rate loss:

$$C_{\text{DPC}} = \log_2\!\left(1 + \frac{P}{\sigma^2}\right)$$

the same capacity as if the interference did not exist.

In the MIMO broadcast channel, DPC combined with successive encoding achieves the capacity region (the "dirty paper" region). However, DPC requires impractical complexity for exact implementation. Practical approximations include Tomlinson-Harashima precoding and vector perturbation.