Performance of Multi-user MIMO Precoding with Limited Feedback over Measured Channels Florian Kaltenberger, David Gesbert, Raymond Knopp Marios Kountouris Eurecom Wireless Networking and Communications Group 2229, Route des Cretes - B.P. 193 The University of Texas at Austin 06904 Sophia Antipolis, France Austin, TX 78712, USA Abstract—In multi-user multiple-input multiple-output (MU- vector quantization (CVQ) along with linear precoding at the MIMO) systems, channel state information at the transmitter transmitter [5], [6]. (CSIT) allows for multi-user spatial multiplexing and thus in- In this paper we use real channel measurements to study creases the system throughput. We assume that CSIT is obtained by means of a finite-rate feedback channel through channel the effect of CVQ on the sum rate of such a MU-MIMO vector quantization (CVQ) at the receiver. In this paper we use system. We consider CVQ using a Fourier codebook, a random real channel measurements to study the effect of CVQ on the codebook and a random codebook exploiting the second order sum rate of a MU-MIMO system employing linear precoding. statistics of the channel. We compute the achievable sum- The measurement data has been acquired using Eurecom’s rate of these schemes using a standard linear MU-MIMO MIMO Openair Sounder (EMOS). The EMOS can perform real- time MIMO channel measurements synchronously over multiple precoder such as zero-forcing (ZF) and regularized inversion users. We consider CVQ using a Fourier codebook, a random (also called MMSE precoder) [7]. For comparison we also codebook and a random codebook exploiting the second order study the performance of MU-MIMO linear precoding based statistics of the channel. For comparison, we also show the on full CSIT and a single-user multiple-input single-output capacity of a single-user system using time division multiple (MISO) time division multiple access (TDMA) system with access (TDMA) with no CSIT at all. The results show that the Fourier codebook shows very poor performance in the measured full CSIT as well as no CSIT. channels. Random codebooks—although suboptimal—provide a MU-MIMO channel measurements have been obtained us- much better performance in the measured channels. ing Eurecom’s MIMO Openair Sounder (EMOS) [8]. The EMOS can perform real-time channel measurements syn- I. INTRODUCTION chronously over multiple users moving at vehicular speed. The We study the downlink (broadcast) channel of a wideband measured channels are stored to disk for offline analysis. For multi-user multiple-input multiple-output (MU-MIMO) system this paper, we have used four transmit antennas and four users with multiple antennas at the base-station (BS) and possibly with one antennas each. The channel measurements have been multiple antennas at the user equipment (UE). Information used in [9] to evaluate the capacity of linear multi-user MIMO theory reveals that if there is full channel state information precoding schemes assuming a perfect feedback channel. The at the transmitter (CSIT) and the receiver (CSIR), the opti- spatial correlation of the measured channels has been studied mum transmit strategy for the MU-MIMO broadcast channel in [10] and the results will be used in this paper to generate involves a theoretical pre-interference cancellation technique the random codebooks. known as dirty paper coding (DPC) combined with an implicit To the best of our knowledge, no such comparison based user scheduling and power loading algorithm [1], [2]. on real MU channel measurements has been reported. Real While the assumption of good CSIR can be justified by the indoor channel measurements have been used in [11] for the ability of the receiver to estimate the channel using training evaluation of the proposed MU-MIMO scheme. Real outdoor symbols on the downlink, perfect CSIT is almost impossible channel measurements have been used in [12] to study limited to achieve. However, it is possible to obtain partial CSIT by feedback. However, the channel measurements were obtained means of a limited feedback channel as envisioned in, for with one receiver at different times and not synchronously as example, 3GPP long term evolution (LTE). For a state of the in our measurements. art on this topic, we refer the interested reader to [3], [4]. Paper Organization: We introduce the signal model in Section II. The investigated channel vector quantization and There are many possible choices of what information to MU-MIMO linear precoding schemes are given in III. In feed back and an optimal solution has yet to be found. Also, Section IV we describe the EMOS in some more detail and given a certain feedback strategy, it is not trivial how to explain how the channel measurements are performed. In use this feedback at the transmitter in an optimal way. One Section V the measurement campaign is described and results solution that has also been proposed for 3GPP LTE is channel are discussed. We finally give conclusions in Section VI. This research was supported by the project PACAM with SFR, the EC Notation: Column vectors and matrices are denoted by a under FP7 Network of Excellence project NEWCOM++ and Eurecom. and A respectively. IM is the identity matrix of size M and 0M is an M-dimensional vector of zeros. The Euclidean (ℓ2) the channel vector norm hk as CQI. Note that this choice of norm of a vector a is denoted by a . E denotes expectation, CQI is not suitable for multi-userk k scheduling, since it does not and (m, C) denotes a multivariatek k proper complex normal take the multi-user interference and the quantization error into distributionCN with mean vector m and covariance matrix C. account. However, in this paper we are only interested in the precoder design and do not consider scheduling. Moreover, we II. SYSTEM MODEL assume that the channel vector norm is not quantized, since we We consider a multi-user, multi-antenna downlink channel are only interested in the ability of the codebook to capture the in which a BS equipped with M antennas communicates with spatial properties of the channel. The codebook is designed K M terminals, each equipped with one antenna. The off-line and there are several possibilities. In thisC paper we ≤ received signal y C of the k-th user at time m and consider the following three designs: a Fourier codebook, a k,m,q ∈ frequency q is mathematically described as random codebook and a correlated random codebook. T 1) Fourier Codebook: The Fourier codebook is obtained by yk,m,q = hk,m,qxm,q + nk,m,q for k = 1,...,K (1) defining ci as the top M rows of the i-th column of the DFT M matrix of size , i.e., where hk,m,q C represents the k-th user channel response C ∈ M at time m and frequency q, xm,q C is the vector of trans- 1 −2πji/C −2πji(M−1)/C T ∈ ci = [1, e ,...,e ] . (4) mitted symbols at time m and frequency q, and nk,m,q C √ ∈ M is i.i.d. circularly symmetric additive complex Gaussian noise The Fourier codebook lookup (3) can be implemented effi- 2 with zero mean and variance σ , k. ciently (in terms of memory and computation) by means of Equation (1) can also be written∀ in matrix notation by T an inverse fast Fourier transform (IFFT). Also, the codebook defining Hm,q = [h1,m,q . hK,m,q] and the vectors ym,q does not need to be stored at the transmitter, as it can be easily and nm,q accordingly: reconstructed. 2) Random Codebook: The quantization vectors of the ym,q = Hm,qxm,q + nm,q. (2) random codebook are drawn from an i.i.d. complex Gaussian We assume that each of the receivers has perfect and instan- distribution on the M-dimensional unit sphere, i. e., c i ∈ taneous knowledge of its own channel. Further we assume (0M , IM ) and ci = 1. a zero-delay error-free finite-rate feedback channel with a CN3) Correlated Randomk k Codebook: The quantization vectors resolution of B bits for each subcarrier q and time m. Note, of the correlated random codebook are drawn from com- that we do not exploit time or frequency correlation in the plex Gaussian distribution on the M-dimensional unit sphere, feedback encoder. The transmitter is subject to an average whose covariance matrix matches the transmit correlation power constraint, i. e., E xH x P , which implies that matrix of the channel, i. e., c (0 , R ) and c = 1. { m,q m,q} ≤ i M Tx i the total transmit power is not dependent on the number of The transmit correlation matrix∈ CNis defined as k k transmit antennas. For notation convenience, in the following R = E HH H . (5) sections we drop the time and frequency indices. Tx { } In this paper we estimate RTx from the measurements by III. LINEAR PRECODING WITH LIMITED FEEDBACK taking the mean of HH H over all frequencies q and all frames In this section we present a MU-MIMO scheme that uses m in one measurement [10]. linear precoding based on quantized channel feedback. This Except for the Fourier codebook, we assume that each scheme has also been proposed for UMTS-LTE [5]. Although users has a different and independently generated quantization nonlinear precoding schemes can achieve a better sum rate codebook as in [14]. than linear precoding, they often exhibit more complexity B. Linear Precoding and a lack of robustness with respect to imperfect CSIT. C Also, it has been shown in [13], that under certain conditions Let sk denote the k-th user transmit symbol.