Improving Expectation Propagation with Lattice Reduction for Massive MIMO Detection

Senjie Zhang＊, Shi Jin, Chao-Kai Wen, Zhiqiang He

1 Key Laboratory of Universal Wireless Communications, Ministry of Education,Beijing University of Posts and Telecommunications, Beijing 100876, China

2 National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

3 Institute of Communications Engineering, National Sun Yat-sen University, Taiwan, China

Abstract: Efficient massive MIMO detection for practical deployment, which is with spatially correlated channel and high-order modulation, is a challenging topic for the fifth generation mobile communication (5G). In this paper, we propose a lattice reduction aided expectation propagation (LRA-EP) algorithm for massive MIMO detection. LRA-EP applies expectation propagation in lattice reduced MIMO system to approach the distribution of lattice reduced constellation point by iterative refinement on its parameters (mean and covariance). The parameter refinement is based on the lattice reduced, well-conditioned MIMO channel. Numerical result shows that LRA-EP outperforms classic EP based MIMO detection(EPD) with 5～7dB in terms of required signalto-noise ratio (SNR) for 1% packet error rate in spatially correlated channel for 256-QAM.We also show that LRA-EP has lower computation complexity than EPD.

Keywords: MIMO detection; lattice reduction; expectation propagation; massive MIMO

I. INTRODUCTION

Massive multiple-input and multiple-output(massive MIMO or large-scale MIMO) is a key technology for the fifth generation mobile communication (5G) [1, 2]. The efficient detections in massive MIMO draw lots of attention [3,13,14]. In various MIMO detection schemes, expectation propagation (EP) based MIMO detection is superior with near optimum performance, high flexibility and simple implementation. EP is a powerful technique in Bayesian machine learning [4] which is applied to MIMO detection by constructing a Gaussian approximation to the posterior distribution of the transmitted symbol vector [5, 6].EP based MIMO detection (EPD) approaches near optimum performance in low spatial correlated channel [7]. EPD utilizes classic linear minimum mean square error (MMSE) filtering iteratively for message refinement. This leads to high flexibility and simple implementation.However, MMSE based message refinement degrades in spatially correlation channel due to noise amplification. Unfortunately many practical massive MIMO channel is spatially correlated according to 3GPP 3D channel model [8]. The performance gain of EPD over MMSE detection reduces significantly in 3GPP 3D channel.

The authors proposed lattice reduction aided expectation propagation (LRA-EP) for massive MIMO detection in this paper.

In this paper, we propose lattice reduction aided expectation propagation (LRA-EP) to improve performance in spatially correlated channel. With lattice reduction [9], the spatially correlated channel is converted to near-independent channel for MIMO detection. Consequently the noise amplification effect in MMSE filtering becomes less. Better performance in MIMO detection by applying MMSE detection and sphere decoder in lattice reduced MIMO system were reported in [10]and [11]. Its application in massive MIMO was proposed [12]. LRA-EP applies EP in lattice reduced MIMO system which results in better parameter refinement (based on MMSE filtering) and overall detection performance.Numerical result shows that LRA-EP outperforms EPD with 5～7dB in terms of required signal-to-noise ratio (SNR) for 1% packet error rate (PER) in 3GPP 3D channel.

This paper is organized as follows. In Section II, we briefly review the lattice reduced MIMO system model. In Section III, LRA-EP is proposed. The performance and complexity is evaluated in Section IV. Finally, in Section V, the concluding remarks are given.

II. SYSTEM MODEL WITH LATTICE REDUCTION

We consider an uplink massive MIMO system with K users as follows:

where r is an Nr×1 complex vector for received signals,and skare complex channel coefficients matrix and transmitted data vector for k-th user, G(c)is an Nr× Ntcomplex channel matrix for the overall virtual MIMO system, s is an Nt×1 complex vector for transmitted symbols, and ω is an Nr×1 complex Gaussian vector for noise with mean 0 and covariance E{ω⋅ωH}=σ2I, where E{ } denotes expectation.

To facilitate computation and reduce complexity, we use an equivalent real-domain system model as:

With lattice reduction, the system model can be reformulated as:

where H=G ⋅T and z=T−1⋅x. Here T and T−1are both integer unimodular matrix. Lenstra-Lenstra-Lovasz (LLL) algorithm can be used to obtain T and T. LLL algorithm makes H more orthogonal than G.

The application of lattice reduction is at cost of using a transformed constellation.Since xk∈Ωxis finite, zkis also finite. The original (X-axis) real-domain constellation Ωxis transformed to the Z-axis real-domain constellation Ωz(k). Ωz(k)can be obtained with simple computation [9] .It should be noted that unlike regular constellation Ωxwhich is common for all streams, Ωz(k)differs for different stream (i.e. different k).

III. LRA-EP ALGORITHM

LRA-EP is based on the lattice reduced MIMO system model in (3). It applies EP framework to lattice reduced system to findp(z), the probability of Z-axis constellation point.

In LRA-EP, the desired distribution p(z)is approximated by a conditional distribu-tion q(z|γ,Λ) within an exponential family.EP provides a framework to find parameters(γ,Λ) with moment matching condition iteratively. The closeness is in terms of Kullback-Leibler divergence.

We choose

where t is iteration index, N(a;b,C) is multivariate Gaussian distribution and

N(z;μ(t),Σ(t)), the extrinsic mean of covariance of z (noted ascan be obtained. Next, the posterior mean of covariance of z (noted asis obtained according toN) and

Based on the EP principle, parameters(γ,Λ) can be iteratively refined with satisfying moment matching condition as:

Algorithm.1 summarizes the detail procedure of LRA-EP.

Remark: Without lattice reduction, i.e.T=I, LRA-EP rollbacks to EPD. When the iteration number equals 1, LRA-EP is identical with lattice reduction aided MMSE detector.

IV. NUMERICAL RESULTS AND DISCUSSION

We compared LRA-EP with classic MMSE detector and EPD by means of Monte Carlo simulations. We choose the iteration number of LRA-EP and EP to be 5.

Algorithm 1. Procedure of LRA-EP.Pre-processing:1: Get T and T from G with Lenstra-Lenstra-Lovasz (LLL) algorithm 2: Compute A=HH ⋅H and y′=HH ⋅y Initialization:3: Set γ(0)=02Nt×1 and NITER Iteration:4: for NITER do (NITER is the number of iteration)() 2 (1)t t■■■■■■Σ ■σ− −■−1 5:A Λ ■μ Σ y′γ() () 2 (1)t t t= ⋅+= ⋅ ⋅+(σ− −diag( ))■z() ()A B A B t t ⋅ ⋅=diag ■μ Σ γ)−■t→() 1 () (1)dd(t t−→)■■■■− ■6:)−1■where dd diagdiag■■■v 1 () (1)(t )A Bt dd− − ■(v■■■■(■ →=diag Σt )−diag (Λ ■■−1(·= ·)■ ■■( ())−1■() () ()■■■■■v z t =E ,(k)7:B → →t t v A B A B }according to N(; )zz B() () ()t =Cov ,{zz{t t zz() v()AB AB tt→ , → and Ωz A B A B→ v →}■γ() 1 1 t= −diag )}() () () ()t t t t AB A B−(B B )(■ ■8:→→■⋅Λ■() 1 1 t= −{z v z v diag⋅dd dd{dd dd−−(v v() ()t −(A→B)t B )}9: end for Output:10: Compute the mean and variance of z()AB → with relationship x=Tz 11: Obtain log-likelihood ratio (LLR) from the mean and variance of x t→ from z()A B t→ and v()A B t

To verify the convergence of iteration,high-order modulation (256-QAM) is used.For 256-QAM, EPD uses regular constellation with size=16 and LRA-EP uses transformed constellation with size=9. For the channel coding, we use 3GPP LTE Turbo code of rate 3/4 and length 3392.

In simulation, the 3D channel model proposed by 3GPP [8] is used. It defines a 2D planar antenna array, where antenna elements are uniformly placed (0.5λ) in both vertical and horizontal direction. The 2D planar array has Nr= M ×N ×P antenna elements, where N is the number of columns, M is the number of antenna elements with the same polariza-tion in each column and P is the number of polarization. According to [8], in simulation the base station uses a 2D planar antenna array with Nr= M × N ×P =4 ×4 ×2 =32 for uplink massive MIMO detection. This antenna configuration leads to a spatially correlation uplink channel.

Fig. 1. Receiver performance in 3GPP 3D channel, SU 32x8, 256-QAM.

Fig. 2. Receiver performance in 3GPP 3D channel, MU 32x12, 256-QAM.

Both single-user MIMO (SU-MIMO) and multiuser MIMO (MU-MIMO) are evaluated. In SU-MIMO, a user equipment (K=1)with 8 antennas transmits data to base station and a 32×8 MIMO system is established. In MU-MIMO, two user equipment (K=2, each with 6 antennas) transmit data to base station and a 32×12 virtual MIMO system is established.

Fig.1 and 2 show the PER-SNR curves of LRA-EP, EPD and MMSE detector for SU-MIMO and MU-MIMO, respectively.

As figure 1 shown, MMSE detector lose its optimum performance for massive MIMO in spatially correlated channel. It is contrary to MMSE detector’s behavior in independent and identically distributed Rayleigh channel. This observation indicates the impact of channel correlation. The channel correlation also degrades EPD’s performance. The slope of EPD’s PER-SNR curve is same as MMSE.Benefit from lattice reduced channel, the correlation reduces and LRA-EP has steeper curve than EPD and MMSE detector. For SU-MIMO, LRA-EP outperforms EPD with 7dB in terms of required SNR for 1% PER. It outperforms MMSE detector with 10dB.

For MU-MIMO, the performance gain of LRA-EP over EPD and MMSE detector is 5dB and 15dB. Comparing figure 1 and figure 2, we observe that LRA-EP’s gain over EPD reduces by 2dB when switching from SU-MIMO to MU-MIMO. It is due to the fact that two user equipment in MU-MIMO are seldom to locate closely. It means the spatial correlation for MU-MIMO is lower than SU-MIMO.Consequently EPD is closer to optimum detector in MU-MIMO.

Comparing LRA-EP and EPD, the computation procedure is identical except line 1,7 and 10 of Algorithm.1. For line 1, the extra complexity, e.g. LLL algorithm and determination of, can be shared among the subcarriers inside coherent bandwidth. Benefited fromLRA-EP requires lower complexity than EP for line 7 of Algorithm.1 which is with heavy computation for exponential function. Line 10 of Algorithm.1 requires negligible complexity compared to other steps.The overall computation complexity of LRAEP is lower than EPD.

V. CONCLUSION

In this paper, we proposed lattice reduction aided expectation propagation (LRA-EP) for massive MIMO detection. Utilizing the lattice reduced channel with better condition, LRAEP obtains better parameters refinement and detection performance in spatially correlated channel. Numerical result shows that LRA-EP outperforms EPD with 5～7dB in terms of required SNR for 1% packet error rate in spatially correlated 3GPP 3D channel for 256-QAM.LRA-EP has lower computation complexity than EPD.