Robust Non-linear Precoder for Multiuser MISO Systems Based on Delay and Channel Quantization

Abstract

In multiuser multiple-input single-output (MISO) systems, non-linear precoder is able to achieve the theoretical sum capacity of downlink channel with perfect channel state information (CSI). However, the perfect CSI is not available at the transmitter in practical system, especially in frequency division duplex (FDD) system where the imperfect CSI is the delayed, quantized channel direction information relayed back from the receiver through a dedicated feedback channel. So the performance of conventional non-linear precoder degrades significantly. In this paper, a robust non-linear Tomlinson–Harashima precoding (THP) based on sum mean squared error (SMSE) minimization for the downlink of multiuser MISO FDD systems is proposed. The proposed precoder is robust to the channel uncertainties arising from channel delay and quantization error. Furthermore, an improved non-linear THP with channel magnitude information (CMI) consideration is introduced to compensate the instantaneous CMI shortage at the transmitter. Additionally, the computational complexity of both proposed precoders can be reduced remarkably by Cholesky factorization with symmetric permutation. Simulation results demonstrate the improvement in bit error ratio performance and illustrate the SMSE performance of the proposed algorithms compared with conventional THP with perfect CSI in the literature.

Appendix

Here we give a detailed derivation process of the expression (37).

$$\begin{aligned} SMSE_{THP}&= \mathbb{E }_{\mathbf{{H}}_n |{\hat{\mathbf{H}}}_{n - 1} } \left\{ \left\| {\beta _n^{ - 1} \mathbf{{H}}_n \mathbf{{F}}_n {\tilde{\mathbf{a}}}_n + \beta _n^{ - 1} \mathbf{{n}}_n - \mathbf{B}_n {\tilde{\mathbf{a}}}_n } \right\| ^2 \right\} \nonumber \\&= \mathbb{E }_{\mathbf{{H}}_n |{\hat{\mathbf{H}}}_{n - 1} } \left\{ \left\| {\beta _n^{ - 1} \mathbf{{H}}_n \mathbf{{F}}_n {\tilde{\mathbf{a}}}_n - \mathbf{B}_n {\tilde{\mathbf{a}}}_n } \right\| \right\} + M\beta _n^{ - 2} \sigma ^2 \end{aligned}$$

(61)

Inserting (7) and (28) into (60), we get

$$\begin{aligned}&SMSE_{THP} \nonumber \\&\quad = \mathbb{E }_{\mathbf{{H}}_n |{\hat{\mathbf{H}}}_{n - 1} } \left\{ \left\| {\rho \alpha \mathbf{{H}}_n {\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n \mathbf{B}_n {\tilde{\mathbf{a}}}_n - \mathbf{B}_n {\tilde{\mathbf{a}}}_n } \right\| \right\} + M\beta _n^{ - 2} \sigma ^2 \nonumber \\&\quad = \mathbb{E }_{\mathbf{{H}}_n |{\hat{\mathbf{H}}}_{n - 1} } \{ \text{ tr }[(\rho \alpha (\mathbf{{A}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1} + \mathbf{{C}}_{n - 1} \mathbf{{S}}_{n - 1} + \mathbf{{E}}_n ){\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n - \mathbf{{I}}_M ) \nonumber \\&\qquad (\rho \alpha (\mathbf{{A}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1} + \mathbf{{C}}_{n - 1} \mathbf{{S}}_{n - 1} + \mathbf{{E}}_n ){\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n - \mathbf{{I}}_M )^H \mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H \} + M\beta _n^{ - 2} \sigma ^2 \end{aligned}$$

(62)

Note that $\mathbf{{S}}_{n - 1}$ is distributed in the null space of ${\hat{\mathbf{H}}}_{n - 1}$, so (61) change into

$$\begin{aligned}&SMSE_{THP} \nonumber \\&\quad =\mathbb{E }_{\mathbf{{H}}_n |{\hat{\mathbf{H}}}_{n - 1} } \{\text{ tr }[(\rho \alpha (\mathbf{{A}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1} + \mathbf{{E}}_n ){\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n - \mathbf{{I}}_M )\nonumber \\&\qquad (\rho \alpha (\mathbf{{A}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1} + \mathbf{{E}}_n ){\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n - \mathbf{{I}}_M )^H \mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H \} + M\beta _n^{ - 2} \sigma ^2 \nonumber \\&\quad {\mathop {=}\limits ^a} \text{ tr }((\rho ^4 \alpha ^4 \mathbf{Q}_n^H {\hat{\mathbf{H}}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1}^H {\hat{\mathbf{H}}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n - (\rho \alpha )^2 {\hat{\mathbf{H}}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n - (\rho \alpha )^2 \mathbf{Q}_n^H {\hat{\mathbf{H}}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1}^H \nonumber \\&\qquad + (\rho \alpha )^2 \varepsilon _e^2 \mathbf{Q}_n^H {\hat{\mathbf{H}}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n + \mathbf{{I}}_M )\mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H ) + M\beta _n^{ - 2} \sigma ^2 \nonumber \\&\quad =\text{ tr }[((\rho \alpha )^2 {\hat{\mathbf{H}}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1}^H - \mathbf{Q}_n^{ - 1} )\mathbf{Q}_n \mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H \mathbf{Q}_n^H ((\rho \alpha )^2 {\hat{\mathbf{H}}}_{n - 1} {\hat{\mathbf{H}}}_{n - 1}^H - \mathbf{Q}_n^{ - 1} )] \nonumber \\&\qquad + \varepsilon _e^2 \rho ^2 \alpha ^2 \text{ tr }({\hat{\mathbf{H}}}_{n - 1}^H \mathbf{{QB}}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H \mathbf{Q}^H {\hat{\mathbf{H}}}_{n - 1} ) + M\beta _n^{ - 2} \sigma ^2 \end{aligned}$$

(63)

where (a) holds because $\mathbf E _n$ is independent of $\mathbf{A }_{n-1}$ and $\mathbb{E } \{\mathbf{E }_{n}\}=\mathbf{0}$, it also follows from (8) and (9). From the expressions of $\mathbf{Q}_n$ and $\varOmega _n, SMSE_{THP}$ can be written as

$$\begin{aligned}&SMSE_{THP} \nonumber \\&\quad = \text{ tr }[{{\varOmega }}_n \mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H {{\varOmega }}_n ] + \varepsilon _e^2 \rho ^2 \alpha ^2 \text{ tr }({\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n \mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H \mathbf{Q}_n^H {\hat{\mathbf{H}}}_{n - 1} ) + M \beta _n^{ - 2} \sigma ^2 \end{aligned}$$

(64)

For analyzing the $SMSE_{THP}$ performance conveniently, utilizing (36), the final $SMSE_{THP}$ expression is

$$\begin{aligned} SMSE_{THP} = \text{ tr }[{{\varOmega }}_n \mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H {{\varOmega }}_n ] + (\varepsilon _e^2 \rho ^2 \alpha ^2 + \gamma ^{ - 1} \rho ^2 \alpha ^2 )\text{ tr }({\hat{\mathbf{H}}}_{n - 1}^H \mathbf{Q}_n \mathbf{B}_n \mathbf{R}_{{\tilde{\mathbf{a}}}_n } \mathbf{B}_n^H \mathbf{Q}_n^H {\hat{\mathbf{H}}}_{n - 1} ) \end{aligned}$$