978-1-4673-1008-6/12/$31.00 2012 IEEE. Precoding for MIMO Systems with Grassmannian codebooks Imen Turki MEDIATRON Higher School of Communication Ariana, Tunisia turki.imene@gmail.com Ines Kammoun LETI Departement National School of Engineering Sfax, Tunisia ines.kammoun@isecs.rnu.tn Mohamed Siala MEDIATRON Higher School of Communication Ariana, Tunisia mohamed.siala@supcom.rnu.tn Abstract—In this paper, we investigate limited feedback pre- coding for two single-user MIMO systems. The first one employs spatial diversity at the transmitter and Maximum Ratio Combin- ing at the receiver. The second one uses spatial multiplexing at the transmitter and V-BLAST signal processing at the receiver. For both systems, the precoder is selected at the receiver among a codebook and its index is conveyed to the transmitter through an error free feedback channel. For the first system, we consider two Grassmannian precoding codebooks and show through simulation results that a small number of feedback bits can reach the theoretical performance. For the case of a single-user system based on spatial multiplexing, we propose a novel criterion for selecting among codebook matrices the optimum precoding matrix at the receiver which maximizes the minimum signal to interference and noise ratio over the transmitted substreams. Simulation results show that our proposed selection criterion outperforms three other criteria introduced in the literature for the spatial muliplexing MIMO system. They also show a significant enhancement over the unprecoded spatial multiplexing system. Keywords: MIMO systems, precoding, spatial diversity, spatial multiplexing, codebook, Grassmann space. I. INTRODUCTION The multi-antenna systems are used in different ways and for different purposes, namely spatial diversity, spatial mul- tiplexing and transmit and/or receive beamforming. Spatial diversity employs multiple spatially uncorrelated antennas to provide diversity gain and increase the reliability of wireless links. Spatial multiplexing consists on demultiplexing the data stream into multiple substreams that are sent over different antennas to achieve high spectral efficiencies. Beamforming can increase the signal to interference and noise ratio in one direction and decrease the effects of interference. When the channel is perfectly known at the transmitter, linear precoding [1] can be considered. However, when the transmitter does not have perfect channel state information, two approaches can be envisaged: channel quantization [2], [3] and limited feedback signal design [4], [5]. In our study, we consider the limited feedback signal design. The basic idea of this approach is to design the signal at the receiver rather than at the transmitter. The receiver is then called to sent back the gains to be applied on the transmit antennas to the transmitter. To reduce the signaling overhead on the uplink, one solution is to design in advance a finite codebook known to both the transmitter and the receiver. Thus, the receiver will convey the index of the chosen codebook element back to the transmitter according to the channel state. For the case of a single-user system based on spatial multi- plexing, three criteria have been proposed respectively in [6], [7] and [8] for selecting the precoding matrix. These criteria are based respectively on minimizing the error probability, minimizing a function of the minimum mean squared error (MSE) matrix and maximizing the mutual information. In [5], authors show that the precoder codebook design method for maximizing the average effective channel power is related to packing subspaces on the Grassmann manifold with largest minimum chordal distance. In this paper, we consider, first, a precoded spatial diver- sity MIMO system with transmit beamforming and receive combining. We present the codebook design problem which is equivalent to packing one-dimensional subspaces known as Grassmannian line packing. We illustrate the performance of this system for two possible codebooks designed on the Grassmann manifold. Then, we consider a precoded spatial multiplexing MIMO system with a V-BLAST receiver [9]. At the transmitter side, a data stream is demultiplexed into multiple substreams that are premultiplied by a precoding matrix before being sent over different antennas. The V-BLAST receiver detects the strongest substream (in the sense of signal to noise ratio), then cancels the effect of this strongest substream from the received signal, and then proceeds to detect the strongest of the remaining substreams, and so on. This receiver can be based on the Zero Forcing (ZF) or the Minimum Mean Square Error (MMSE) criteria. We propose a novel selection criterion of the precoding matrix at the receiver based on maximizing the Signal to Noise Ratio (SNR) over the transmitted substreams with a successive cancellation of interference of each detected subtream for the ZF criterion. For the MMSE criterion, the minimum Signal to Interference and Noise Ratio (SINR) is considered because of the residual interference inherent to the MMSE criterion. For presentation simplicity, we legitimize a single notation throughput the paper, namely the SINR. For an error probability performance evaluation of the system using our proposed criterion, we adopt two precoder codebooks which are designed on the Grassmannian using the methods