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On the Analysis of MIMO-ZF Receiver Over Fully Correlated MIMO Rayleigh Fading with LMMSE Channel Estimation

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Abstract

For multiple-input multiple-output (MIMO) communication systems, the zero-forcing (ZF) receiver has attracted great research interest in recent years. The purpose of this work is to analyze the effect of channel estimation error on the performance of the MIMO-ZF receiver over spatially correlated Rayleigh fading channels. Unlike previous works, where the channel is assumed with transmit correlation but without receiver correlation, we treat the fully correlated channel cases. Furthermore, we assume that the channel estimation error is a Gaussian and correlated matrix. The distribution of the post-processing signal-to-noise ratios is approximated. An approximation of the average bit error rate (BER) is derived and compared to Monte Carlo simulation results. The proposed BER expression takes into consideration that the post-processing noise in not Gaussian. Numerical results show an excellent agreement between the analytical approximation and Monte Carlo simulation curves.

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Notes

  1. Note that \({\varvec{R}}_{ {{\varvec{H}}}, {{\varvec{H}}}}={\mathbb{E}}[ {\varvec{H}} {\varvec{H}}^H ] \) denotes the covariance of the channel matrix and does not correspond to the conventional definition of covariance \({\mathbb{E}}[ {\text{vec}} ( {\varvec{H}}) {\text{vec}} ( {\varvec{H}} )^H] \).

  2. Since \(({\varvec{A}}\otimes {\varvec{B}})^H={\varvec{A}}^H \otimes {\varvec{B}}^H\).

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Laval University, Quebec City, QC, Canada

    Mohamed Lassaad Ammari & Paul Fortier

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  1. Mohamed Lassaad Ammari
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Correspondence to Mohamed Lassaad Ammari.

Appendix: Proof of the First Inequality in (32)

Appendix: Proof of the First Inequality in (32)

The conditioned covariance of the virtual noise is given by (31)

$$\begin{aligned} {\mathbb{E}}[{\varvec{v}}_{x} {\varvec{v}}_{x}^H| \hat{\varvec{H}},{\varvec{x}}_i ] =({\varvec{x}}_i' \otimes {\varvec{H}}^{\dag } ) {\varvec{U}}_s{\varvec{D}}_s{\varvec{U}_s^H} ({\varvec{x}}_i^{*} \otimes {\varvec{H}}^{\dag H} ) \end{aligned} $$
(47)

Since \({\varvec{U}}_s\) is a unitary matrix, we have \(({\varvec{x}}_i' \otimes {\varvec{H}}^{\dag } ) {\varvec{U}}_s \sim ({\varvec{x}}_i' \otimes {\varvec{H}}^{\dag } )\). Thus, we can write

$$\begin{aligned} {\mathbb{E}}[{\varvec{v}}_{x} {\varvec{v}}_{x}^H| \hat{\varvec{H}},{\varvec{x}}_i ] \sim ({\varvec{x}}_i' \otimes {\varvec{H}}^{\dag } ) {\varvec{D}}_s ({\varvec{x}}_i^{*} \otimes {\varvec{H}}^{\dag H} ) \end{aligned}$$
(48)

We recall that \({\varvec{D}}_s \) is a diagonal matrix whose elements satisfy \(\left[ {\varvec{D}}_s \right] _{ii} \ge \lambda _{\text{min}}^s\). Hence, we have the first inequality of (32)

$$\begin{aligned} {\mathbb{E}}[{\varvec{v}}_{x} {\varvec{v}}_{x}^H| \hat{\varvec{H}},{\varvec{x}}_i ] \ge \lambda _{\text{min}}^s ({\varvec{x}}_i' \otimes {\varvec{H}}^{\dag } ) {\varvec{I}}_{N_t N_r} ({\varvec{x}}_i^{*} \otimes {\varvec{H}}^{\dag H} ) \end{aligned} $$
(49)

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Ammari, M.L., Fortier, P. On the Analysis of MIMO-ZF Receiver Over Fully Correlated MIMO Rayleigh Fading with LMMSE Channel Estimation. Wireless Pers Commun 85, 1025–1042 (2015). https://doi.org/10.1007/s11277-015-2823-2

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