Elsevier

Physical Communication

Volume 10, March 2014, Pages 3-10
Physical Communication

Full length article
Maximizing capacity with trellis exploration aided limited feedback precoder design for multiuser MIMO-MAC

https://doi.org/10.1016/j.phycom.2013.11.004Get rights and content

Abstract

In this paper, a linear trellis based precoding technique is proposed for maximizing the multiuser MIMO capacity in a multiple access channel with inter symbol interference (ISI). We use the trellis exploration algorithm to design a precoding matrix that minimizes the interference from other users by allocating power in the appropriate eigenmodes. Employing a finite number of phases, the precoder matrix for each user is first custom designed at the receiver. Then a bit sequence indicating the phase indices of the optimized precoding matrix elements for the corresponding user is fed back to the transmitter. We show that under this approach, the sum rate capacity achieved is comparable to the optimal sum-rate capacity employing the well-known water filling solution with complete channel knowledge at the transmitter for spatio-temporal vector coding (STVC). Our simulations for various multiuser MIMO cases, show that the per user precoding with limited feedback and equal power allocation strategy achieves desirable capacity gains relative to the eigenbeamforming and Grassmannian precoding.

Introduction

It is well-known that, Precoding in MIMO (multiple-input multiple-output) systems can be used to enhance the system throughput. For multiuser MIMO, multiple access interference (MAI) is a limiting factor in achieving higher throughput. In an i.i.d (independent identically distributed) Rayleigh-fading multiple access MIMO channel, if complete channel state information (CSI) is available at the transmitter, an iterative “water-filling” scheme can be used to optimize the MIMO capacity  [1]. This method designs an optimal user input covariance matrix from any initial point by iterative “water-filling”. For the single user MIMO, channel capacity is achieved by exploiting the asymptotic structure of spatio-temporal vector coding (STVC) in  [2]. This was extended to include the multiuser MIMO scenario in  [3]. However, the application of spatial “waterfilling” in a multiuser environment is limited by: (1) availability of full CSI at the transmitter; (2) quantizing and feeding back channel fading parameters to the transmitter every time the channel changes its state (in the case of feedback based systems). The feedback process has transmission overhead, if the reverse channel is a limited rate channel. Limited feedback linear precoding techniques in these cases are very useful. In  [4], unitary precoders are selected in Grassmannian manifold to optimize the desired performance metrics at the receiver. A few bits specifying the index of the corresponding precoding matrix is sent back to the transmitter. Design of codebook based on transmit signal covariance matrices for use with limited rate feedback channel to increase the sum-capacity in a multiuser MIMO environment is presented in  [5]. These codebooks are known to both transmitters and receivers. The transmitter chooses the best precoder matrix based on the codebook index sent by the receiver on the feedback channel that maximizes the sum rate capacity. Recent works on multiuser linear feedback precoding for closed-loop MIMO-OFDM downlink and MIMO-SC-FDMA uplink scenario are presented in  [6], [7] respectively. A nonlinear precoding scheme minimization of the bit-error-rate in the high signal to noise ratio regime, for the downlink of a multiuser MIMO channel  [8]. However, none of these works consider multiuser interference in the uplink while designing the precoder. Trellis exploration based online precoding techniques have been presented in  [9], [10] to maximize the single user MIMO capacity in non-dispersive channel. These demonstrate non-unitary and unitary approaches of precoder design to improve the SU-MIMO system throughput. Further, the performance is evaluated under the context of flat fading channel scenario only. In this paper, we extend our previous works in  [9], [10] to design precoder F that maximizes multiuser MIMO multiple access (uplink) capacity in a non-cooperative broadband frequency selective and narrowband flat fading transmission channel environments. We assume that the total transmit power constraint exists for each user in this environment with equal power for each user. Our proposed scheme uses the eigendecomposition of a user interference matrix at the receiver to generate the eigenvectors corresponding to maximum signal to interference and noise ratio for each user. The trellis approach aims to match the direction of eigenvectors of the generated precoder matrix with eigenvectors of the user interference matrix. This ensures that user transmissions are along the proper “eigenmodes”. Given that the trellis exploration algorithm only operates on a finite set of phases, optimal precoders can first be custom-designed at the receiver, and then a small number of bits indicating phase indices of the precoder elements is fed back to reconstruct F at the transmitter. Simulation results show that the ergodic sum-rate capacity of the proposed trellis exploration based multiuser precoding is comparable to the optimal “waterfilling” approach that requires CSI at the transmitter. The sum rate performance improvement comes at the cost of increased receiver complexity due to multiple trellis searches. Additionally, we show that the sum capacity improvement with our proposed approach is robust with noisy feedback. We also provide a sum rate capacity comparison with eigenbeamforming and Grassmannian codebook  [5] based linear precoding in a non-dispersive multiuser MIMO channel.

Our results demonstrate that: (1) with additional receiver complexity, the trellis precoding improves the multiuser MIMO dispersive channel capacity even with equal user power allocation; (2) the number of feedback bits required per user per transmission relative to the quantized CSI bits per user per transmission decreases with the increase in the order (MtMr) of MIMO system; (3) the trellis based precoder achieves higher sum capacity than eigenbeamforming and Grassmannian codebook based precoding in MU-MIMO MAC, and (5) storage requirement for trellis based precoding approach is much less than the codebook based precoding and does not increase with the number of users in the system.

Section snippets

Channel and limited feedback system model

The input output relationship of a multiuser dispersive SISO channel can be written in simplified matrix form as y=i=1MHixi+n where the channel matrix Hi is given by

Hi=[h0i00hνih0i00hνih0i0hνih0i00hνi]. Here, {hmi}m=0ν are channel coefficients, and ν is the channel memory due to multipath. In the case of space–time vector coding (STVC), each size-N blocks of input symbols is padded with ν zeros. The input xi and output y vectors are given by [x0ix1ixN1i]T and [y0y1yN+ν1]T,

Capacity analysis and proposed approach

The normalized sum-rate capacity for multiuser Gaussian vector channel can be obtained from  [3] as

C=tr(R)i=EiMax1N+νlog2det(Rn+i=1MHiRiHiH)det(Rn) where {.}H denotes the Hermitian operation. With si=Fixi, the input covariance matrix for the i-th user Ri can be given as Ri=E{Fixi(Fixi)H}=FiΣiFiH. Rn=E{nnH} is the noise covariance matrix. Assuming equal power allocation for each user, i.e.,  Σi=E{xixiH}=EiNMtIMt, the capacity equation given in  (4) can be simplified as C=tr(R)i=EiMax1N+νlog2det(

Trellis exploration algorithm

In this section, we describe the use of the trellis exploration algorithm to determine an optimal precoding matrix {F}i=1M to accomplish the goal presented in Section  3. Specifically, each Fi corresponds to Fi=(F1,1iF1,2iF1,NMtiF2,1iF2,2iF2,NMtiFNMt,1iFNMt,2iFNMt,NMti) where, Fm,pi denotes a single phase value that is chosen from a set of L candidate phases. The candidate phases φ1, , φL for the i-th user are taken from a uniformly distributed discrete phase space, scaled according to

Simulation results

First, we consider a block length N of 10 and 5 users in the MIMO multiple access dispersive channel. The capacity of the multiuser STVC system with complete CSI is compared with our trellis based STVC system for 2×2 and 3×3 MIMO multiuser cases. The channel per user is assumed to be i.i.d Rayleigh fading with N blocks of independent data vectors per user per transmission. For each user, the channel has 4 independent Rayleigh fading paths with the average power Sx on each path given by 0, 0.25

Conclusion

In this paper, we develop a novel trellis exploration based linear precoder design strategy to enhance the system throughput in MIMO multiple access ISI channels. The precoding matrix for each user is custom designed at the receiver as receiver has the complete channel information. Specifically, the second norm of the difference between the eigenvectors of the precoder matrix and the user interference matrix is used by the trellis exploration algorithm as the metric to minimize. After

Sayak Bose received the B.Tech. degree from Kalyani University, India, in 2003 and the M.S. degree from the University of Kansas, Lawrence, in 2009. Currently, he is pursuing the Ph.D. degree from Kansas State University, Manhattan. He worked as an Assistant Systems Engineer at Tata Consultancy Services, India, for over three years. His research interests include communication theory and systems, MIMO signal processing, and optimized reconfiguration methodologies.

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Sayak Bose received the B.Tech. degree from Kalyani University, India, in 2003 and the M.S. degree from the University of Kansas, Lawrence, in 2009. Currently, he is pursuing the Ph.D. degree from Kansas State University, Manhattan. He worked as an Assistant Systems Engineer at Tata Consultancy Services, India, for over three years. His research interests include communication theory and systems, MIMO signal processing, and optimized reconfiguration methodologies.

Balasubramaniam (Bala) Natarajan received his B.E and Ph.D. degrees, both in electrical engineering from Birla Institute of Technology and Science, Pilani, India and Colorado State University, Fort Collins, CO, in 1997 and 2002, respectively.

Since fall 2002, he has been a faculty member in the Department of Electrical and Computer Engineering at Kansas State University, where he is currently an Associate Professor and the director of the WiCom (Wireless Communication) and Information Processing Research Group. He was also involved in telecommunications research at Daimler Benz Research Center, Bangalore, India, in 1997. His research interests include spread spectrum communications, multi-carrier CDMA and OFDM, multiuser detection, cognitive radio networks, sensor signal processing, distributed detection and estimation, and antenna array processing.

Dr. Nataro has numerous journal and refereed conference publications in the wireless communications and signal processing arena, has published a book entitled Multi-carrier Technologies for Wireless Communications (Kluwer Academic Publishers, 2002) and holds a patent on customized spreading sequence design algorithm for CDMA systems. He also has two pending patents in adaptive/customized precoder design for MIMO systems.

Dalin Zhu received his Bachelor and Master degrees from Beijing University of Posts and Telecommunications (BUPT), Beijing, China and Kansas State University, US in 2007 and 2009, respectively. Now, he is working as a wireless research staff member and project manager at NEC Laboratories China (NLC). His research interests include communication theory, control theory and efficient optimization methodologies. Currently, he is working on dynamic resource allocation and baseband pooling under the framework of centralized RAN. He is an IEEE member and one of NEC’s delegates in 3GPP.

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