Performance of regressionbased precoding for multiuser massive MIMOOFDM systems
 Ali Yazdan Panah†^{1}Email author,
 Karthik Yogeeswaran^{1} and
 Yael Maguire^{1}
DOI: 10.1186/s1363401603351
© Panah et al. 2016
Received: 1 July 2015
Accepted: 13 March 2016
Published: 13 April 2016
Abstract
We study the performance of a singlecell massive multipleinput multipleoutput orthogonal frequencydivision multiplexing (MIMOOFDM) system that uses linear precoding to serve multiple users on the same timefrequency resource. To minimize overhead, the channel estimates at the base station are obtained via combtype pilot tones during the training phase of a timedivision duplexing system. Polynomial regression is used to interpolate the channel estimates within each coherence block. We show how such regressors can be designed in an offline fashion without the need to obtain channel statistics at the base station, and we assess the downlink performance over a wide range of system parameters.
Keywords
MIMO OFDM Massive MIMO Least squares Interpolation Channel estimation Zeroforcing Beamforming Precoding1 Introduction

Massive MIMO can increase the throughput and simultaneously improve the radiated energy efficiency via energy focusing.

Massive MIMO can be built with rather inexpensive components by replacing highpower (W) linear amplifiers with lowpower (mW) counterparts.

Massive MIMO can simplify the multipleaccess layer (MAC) by scheduling the users on the entire band without the need for feedback^{1}.
Such benefits largely stem from asymptotic results on random matrix theory that illustrates how the effects of uncorrelated noise and smallscale fading are virtually eliminated (and the required transmitted energy per bit vanishes) as the number of antennas in a MIMO cell grows to infinity.
Massive MIMO systems are also versatile over a wide range of system parameters. For instance, the beamforming gain afforded by using a large number of transmit antennas may be used to overcome the large path loss associated with mmWave links in urban areas [6]. Alternatively, the beamforming gain may be harnessed at VHF/UHF frequencies to provide widecoverage connectivity to rural areas of the world [7]. Given such promises, the practical and theoretical aspects of massive MIMO systems are actively under scrutiny for potential beyond4G wireless communication deployments not only by standardization entities such as the 3rd Generation Partnership Project (3GPP) but also by many industrial base station and device manufacturers worldwide.
Coherent massive MIMO systems require channel state information (CSI) at the base station in order to compute linear precoder filters for the downlink and equalization filters for the uplink. Such systems are typically designed for a timedivision duplexing (TDD) scheme where the uplink and downlink share the transmission bandwidth. This is primarily due to the fact that the CSI may be readily obtained in TDD mode when reciprocity is maintained in the signal path. For example, the base station may estimate the downlink (and uplink) channel using pilot symbols transmitted by the users during an uplink “training phase” [8]. The estimation of CSI is a wellstudied area for MIMO [8], OFDM [9, 10], and MIMOorthogonal frequencydivision multiplexing (OFDM) [11] systems. For multiuser systems, the base station may use the estimated CSI obtained from uplink pilots to construct linear precoders (and equalizers). Fortunately, in the massive MIMO regime, the performance of such filters are known to be close to the optimal schemes. In this context, matchedfilter (MF) and zeroforcing (ZF) are two popular linear filters [12]. The gains due to linear processing must be weighed by the increases in baseband computational complexity as a result of adding more antenna elements at the base station. For instance, MF and ZF equalization are known to have linear and cubic complexity, respectively, in the number of users. This may present a bottleneck given current hardware capabilities; hence, some researchers have devised suboptimal methods with reduced complexity such as the ordering scheme proposed in [13] for MF or the inversionapproximation for ZF proposed in [14]. The accuracy of these linear filters depend on the accuracy of the CSI on which they are obtained from.
Interpolating a reduced set of pilots is a popular method of estimating the CSI across the frequency band in single and multiuser MIMOOFDM system (see, e.g., [9, 10, 15–17] and references therein). In this paper, we study the effects of regressionbased interpolation of CSI and its effects on the accuracy of linear precoding in a downlink massive MIMO system. We propose polynomial regression as a way to interpolate the multiplexed pilots in the uplink into a single channel estimate over a block of bandwidth, i.e., over a coherence block. These regressors may be computed in an offline fashion without any knowledge of the channel. In Section 2, we formulate the problem and propose some notation and in Section 3, we present numeric results. We make concluding remarks in Section 4.
Notation: Bold uppercase and lowercase letters represent matrices and vectors, respectively. X ^{∗}, X ^{T}, X ^{H}, X ^{−1}, and X ^{+} denote conjugate, transpose, conjugatetranspose, matrix inverse, and MoorePenrose inverse of a matrix X, respectively.
2 System model
We consider a linearly precoded MUMIMOOFDM system over N subcarriers with M antennas at the base station serving K singleantenna users. The system operates under a hardwarecalibrated time division duplexing (TDD) scheme over a wireless channel with a coherence time of T _{ c } seconds. This allows simultaneous uplink (users to base station) and downlink (base station to users) transmissions across a common frequency band.
2.1 Uplink pilot phase (training)
2.1.1 Least squares channel estimation
Here, t ^{′} and n ^{′} denote subsets of OFDM symbols and subcarriers, respectively, in which user k has transmitted a pilot tone within the rth RB. Let L denote the total number of pilot tones per user per RB.^{2} For example, in Fig. 1 for user 1, we have n ^{′}={1,5,9},t ^{′}={1}, and for user 2, we have n ^{′}={2,6,10},t={1}, and for user 24, we have n ^{′}={4,8,12},t ^{′}={6}, etc. In this case, for any user, we have L=3.
2.2 Regressive interpolation
Finally, it should be noted that while suboptimal by design, the polynomial interpolation method described above may present some advantages compared to the wellknown linear minimum mean square error (LMMSE) channel interpolators of [9, 10]. For example, the polynomial interpolators are both channel model and channel signaltonoise ratio (SNR) independent. Moreover, the perRBbased processing nature of the polynomial interpolation method may lead to computational savings since for N total subcarriers and N ^{′} subcarriers per RB, the LMMSE method requires inversion of complexvalued matrices of size \(\frac {N}{N'}L\), while the polynomial interpolators require inversion of realvalued Vandermonde matrices of size p where \(p<L\le \frac {N}{N'}L\).
2.3 Downlink precoding
where the elements of \(\widehat {\mathbf {C}}[\!n]\) are obtained using polynomial regression via (11).
3 Numeric results
In this section, we assess the performance of the regressionbased linear precoding described in Section 2 using a system level simulator with Monte Carlo simulations. We consider a singlecell multiuser MIMOOFDM system with N=256 subcarriers of which 180 subcarriers are used for data and control signals. Each RB consists of 12 contiguous subcarriers. The base station serves K=24 users using M≥K antennas. The channel between each base station antenna and each user is modeled as a tappeddelay line with an effective delay spread of τ _{rms}. The UL pilot transmission phase consists of 6 OFDM symbols with QPSK pilot symbols multiplexed for 24 users as in Fig. 1. The pilot phase is followed by DL data transmissions with QPSK symbols. The DL transmit powers, path loss, link budgets, and noise variance are such that the SNR for each user is identical.
The channel frequency response estimates are computed using (8) using polynomial regressors of the order p=0,1,2. The interpolation vectors γ _{ k }[n] may be computed offline and selected from the rows of base matrices \(\boldsymbol {\Gamma }_{k}^{<p>} \), where the subscripts denote the user indices corresponding to Fig. 1. We elaborate on this idea using an example below.
Example 1.
3.1 Performance vs. SNR: interpolation accuracy
3.2 Performance vs. M: the massive MIMO effect
4 Conclusions
In this correspondence, we assessed the performance of regressionbased linear precoding in the downlink of a multiuser massive MIMOOFDM system. Simple linear polynomial regressors were used to reduce multiple channel estimates over the resource blocks. These regressors do not depend on the channel statistics and can be computed in an offline manner. Simulations showed that for practical SNR ranges, the performance of the proposed methods are close to the genieaided system, even for loworder regression selections. Moreover, the order of the regressor vectors may be adapted to the channel conditions to obtain optimal performance.
5 Endnotes
^{1} This is true for timedivision duplexing (TDD) where channel reciprocity holds.
^{2} Assumed to be equal for all users over any RB.
^{3} Otherwise, the ZF precoder matrix in (13) does not exist.
Declarations
Acknowledgements
The authors would like to thank members of the Connectivity Lab at Facebook for their valuable input during the course of this project.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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