Semi-blind maximum-likelihood joint channel/data estimation for correlated channels in multiuser MIMO networks

Abstract

The aim of this paper is to investigate receiver techniques for maximum likelihood (ML) joint channel/data estimation in flat fading multiple-input multiple-output (MIMO) channels, that are both (i) data efficient and (ii) computationally attractive. The performance of iterative least squares (LS) for channel estimation combined with sphere decoding (SD) for data detection is examined for block fading channels, demonstrating the data efficiency provided by the semi-blind approach. The case of continuous fading channels is addressed with the aid of recursive least squares (RLS). The observed relative robustness of the ML solution to channel variations is exploited in deriving a block QR-based RLS-SD scheme, which allows significant complexity savings with little or no performance loss. The effects on the algorithms’ performance of the existence of spatially correlated fading and line-of-sight paths are also studied. For the multi-user MIMO scenario, the gains from exploiting temporal/spatial interference color are assessed. The optimal training sequence for ML channel estimation in the presence of co-channel interference (CCI) is also derived and shown to result in better channel estimation/faster convergence. The reported simulation results demonstrate the effectiveness, in terms of both data efficiency and performance gain, of the investigated schemes under realistic fading conditions.

Introduction

High throughput data communication systems require high quality channel estimation at the receiver in order to provide reliable data detection, such as that performed by maximum likelihood (ML) techniques. The channel estimation task is especially challenging in time varying channels, such as the ones often arising in wireless communication links. The time-varying nature of the channel typically requires the use of frequent channel re-training, which in turn increases the data overhead due to training signals, thus reducing the system's overall spectral efficiency.

The problem of channel estimation becomes even more challenging in multiple antenna wireless links (such as multiple input multiple output (MIMO) channels [34]) due to the larger number of channel parameters that need to be estimated. The required training could take away a substantial part of the spectral efficiency gain provided by MIMO systems [32], [16]. A more demanding situation results if colored interference is also present, as it is the case in a multi-user environment [2].

Blind estimation techniques [46] completely eliminate the training overhead, albeit, in general, at the cost of slower convergence and/or lower estimation accuracy. Furthermore, channel phase and ordering indeterminacies are introduced. A more practical approach, lying somewhere between the training-based and blind approaches, is that of semi-blind estimation [8]. It can perform better than either of them, as it provides a means of effectively increasing the training length through the exploitation of the information residing in the unknown part of the signal [24]. Moreover, semi-blind techniques have been shown to exhibit robust performance in the presence of asynchronous interference and other non-idealities (see, e.g. [26]).

Joint estimation of the channel and the data can be considerably more effective than performing the data detection only on the basis of what a short training period can tell about the channel. Most of the algorithms for joint channel/data estimation that have appeared in the literature are based on the iterative approach of alternately estimating the data considering the channel as being known and vice versa, until convergence is achieved. Such works include [43], where the problem was first studied theoretically and the basic algorithmic schemes were developed, and [30], [39], [36], [23], [19], [56], [58], [6], [52], [53], [54]. Formulations of the alternating optimization scheme in the expectation–maximization (EM) framework have also been reported; see, e.g. [11], [33], [42], [51]. Recently, this problem was also casted in information geometric terms [60]. Genetic optimization was employed in [1]. Multi-user interference was taken into account in [13], [27], [25], [49], [10].

ML detection for MIMO is known to offer substantial diversity gain over both linear and decision-feedback based methods [61]. However, its computational requirements increase exponentially with the input signal dimensionality and can easily become prohibitive, even for a moderate size system. Sphere decoding (SD) (e.g. [12]) provides a means of reducing the (expected) complexity of the ML receiver,¹ especially in high signal-to-noise ratios (SNR).

The aim of this paper is to investigate receiver techniques for ML joint channel/data estimation in flat fading MIMO channels that are both (i) data efficient and (ii) computationally attractive. In the block fading case, an alternating least squares (ALS) scheme, which relies on SD for data detection, is examined and its superiority over SD based solely on training data is demonstrated, especially for very short training periods. For continuous fading channels, the observed relative robustness of the ML solution to channel variations is exploited in deriving a block QR-based RLS-SD scheme, which is shown to allow significant complexity savings with little or no performance loss. Results are provided for both Ricean and spatially white and correlated Rayleigh fading channels. The scenario of a multi-user MIMO network, resulting in spatially and temporally colored co-channel interference (CCI), is also studied, through appropriate extensions of the estimation/detection algorithms. Moreover, the optimal training sequence for ML channel estimation in the presence of CCI (in the sense of minimum channel mean squared error (MSE)) is derived. Simulation results are reported that confirm the effectiveness of the studied schemes in realistic environments.

The rest of the paper is organized as follows. Section 2 describes the signal and channel model. The single-user scenario is studied in Section 3, for both the cases of block and continuous channel fading. Section 4 is concerned with the multi-user case. Simulation results are presented in Section 5. Section 6 concludes the paper.

Notation. In the following, $(·{)}^T$, $(·{)}^H$, and $(·{)}^{†}$ denote transpose, complex conjugate transpose and Moore–Penrose pseudoinverse of a matrix, respectively. $\parallel ·\parallel$ is the Frobenius norm. The squared Frobenius norm of a matrix $\mathit{A}$ weighted with a positive definite matrix $\mathit{B}$, i.e., $\mathrm{trace}({\mathit{A}}_{}^H\mathit{BA})$, is denoted by $\parallel \mathit{A}{\parallel }_{\mathit{B}}^2$. The expectation and matrix trace operators will be denoted by $E(·)$ and $\mathrm{tr}(·)$, respectively. $\otimes$ is the (left) Kronecker product. ${\mathit{I}}_m^{}$ is the m th-order identity. Complex conjugation is denoted by ${}^{\star }$. The vectorization operator, which places the columns of a matrix in a vector on top of each other, will be denoted by $\mathrm{vec}$.