Optimal Tx-BF for MIMO SC-FDE Systems

Transmit beamforming (Tx-BF) for multiple-input multiple-output (MIMO) channels is an effective means to improve system performance. In frequency-selective channels, Tx-BF can be implemented in combination with single-carrier frequency-domain equalization (SC-FDE) to combat inter-symbol interference. In this paper, we consider the optimal design of the Tx-BF matrix for a MIMO SC-FDE system employing a linear minimum mean square error (MSE) receiver. We formulate the Tx-BF optimization problem as the minimization of a general function of the stream MSEs, subject to a transmit power constraint. The optimal structure of the Tx-BF matrix is obtained in closed form and an efficient algorithm is proposed for computing the optimal power allocation. Our simulation results validate the excellent performance of the proposed scheme in terms of uncoded bit-error rate and achievable bit rate.

I. Introduction

Multiple-Input multiple-output (MIMO) systems are a promising technology to improve the spectrum efficiency and/or error performance of wireless networks. In order to fully exploit the benefits of the multiple antennas and the available channel state information at the transmitter (CSIT), appropriate MIMO transmit beamforming (Tx-BF) schemes are required. Optimal MIMO Tx-BF designs for flat fading channels and frequency selective fading channels in combination with orthogonal frequency-division multiplexing (OFDM) have been well studied in the literature, cf. [1]–[3]. However, the design methodology used in [1]–[3] is not directly applicable to MIMO systems employing single carrier frequency-domain equalization (SC-FDE) [4], where a block circular matrix structure is imposed on the equalization matrix to enable efficient frequency domain implementation. In particular, for MIMO SC-FDE systems, the system performance metrics depend on the mean square errors (MSEs) of the spatial data streams in the time domain, instead of the subcarrier MSEs in the frequency domain as is the case for OFDM systems. Tx-BF design for SC-FDE systems has been investigated in several works. For example, adopting the arithmetic MSE (AMSE) as the performance metric, [5] proposed optimal Tx-BF for a MIMO SC-FDE system with both linear and decision feedback equalization. However, obtaining the optimal Tx-BF matrix design directly minimizing the bit error rate (BER) or maximizing the achievable bit rate (ABR) is much more challenging since, unlike the AMSE, these performance metrics are nonlinear functions of the data stream MSEs. In [6], the authors provide a first attempt to minimize the BER of a multiple-input single-output (MISO) SC-FDE system. However, since only one data stream is transmitted, minimizing the BER is equivalent to minimizing the AMSE for MISO systems. To the best of the author's knowledge, Tx-BF design for MIMO SC-FDE systems with general non AMSE based objective functions has not been studied in the literature yet.