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Procedía Technology 6 (2012) 708 - 715

2nd International Conference on Communication, Computing and Security [ICCCS-2012]

Joint Precoding and Decoding in MU-MIMO Downlink Systems with Perfect Channel State Information (CSI)

M.Rajaa, P. Muthuchidambaranathanb

aDept. of Electronics and Communication Engineering, National Institute of Technology, Tiruchirappalli, India bDept. of Electronics and Communication Engineering, National Institute of Technology, Tiruchirappalli, India

Abstract

In this paper, we propose an joint optimization of precoder and decoder for downlink transmissions in multiuser multiple-input, multiple-output (MU-MIMO) systems with perfect channel state information (CSI). This paper focuses on the scenario when an improper constellation such as binary phase shift-keying (BPSK) or M-ary amplitude shift-keying (M-ASK) is employed. The proposed joint design aims to minimize all the users total mean-squared-error (TMSE) under the constraint of total transmit power. Simulation results show that the proposed joint linear precoder and decoder designs improves BER performance of MU-MIMO downlink systems and verify its effectiveness.

©2012ElsevierLtd...Selectionand/or peer-review under responsibility of the Department of Computer Science & Engineering, National Institute of Technology Rourkela

Keywords: Multiple-input multiple-output (MIMO); Total mean square error (TMSE); Channel state information (CSI); Linear precoding; Diversity; Improper modulations.

1. Introduction

A MIMO system consists of several antenna elements, plus adaptive signal processing, at both transmitter and receiver, and the MIMO systems have bring considerable attentions recently due to their huge amount of improvement in capacity without additional spectrum and transmit power [1][2]. The Multiuser MIMO (MU-MIMO) technique could improve the system performance significantly than traditional single-user MIMO (SU-MIMO) by utilizing the spatial division multiple access (SDMA) technique [3][4]. In the downlink of MU-MIMO systems, the multiple users can be served by using multiple antennas at the base station (BS). Various performance measurements have been considered to obtain a joint transceiver structure for MU-MIMO systems with both uplink and downlink configuration, such as minimum mean-square error (MMSE) from all the data streams, maximum sum capacity and minimum bit error rate (BER) [5][6].

Joint design of precoding and decoding vectors for each transmitting data stream is proposed in [7][8]. This design can provide greater performance, but there is no closed-form solution for precoding and decoding vectors. Since, the

* Corresponding author. Tel.: +91-978-614-4841. E-mail address: raja.sanjeeve@gmail.com

2212-0173 © 2012 Elsevier Ltd...Selection and/or peer-review under responsibility of the Department of Computer Science & Engineering, National Institute of Technology Rourkela doi: 10.1016/j.protcy.2012.10.085

iterative algorithms are always required to calculate the numerical solutions. Joint Transceiver design for downlink MU-MIMO systems with perfect and imperfect CSI is investigated in [9], and it aims to minimize the sum mean-squared-error (MSE) of all user under the constraint of total transmit power. The optimum solution for precoding and decoding vectors are obtained by using iterative algorithm. Joint linear transceiver design for uplink MU-MIMO system with minimum total mean-square error (TMSE) problem has been proposed in [10][11]. Two different types of methods based on iterative algorithms are used [10], to solve this problem. One is based on the associated Karush-Kuhn-Tucker conditions, the other is to solve an equivalent problem, approaching the solution by solving a sequence of semi-definite programming problems. The uplink-downlink duality in sum MSE under imperfect CSI is displayed in [12].

A novel linear transmit precoding MIMO systems employing improper signal constellations are proposed in [13]. And their performance are improved by designing the system with modified cost functions and by exploitation of the improperness of signal constellation. And it achieves a superior performance than the conventional linear and non-linear precoders by utilizing improperness of the signal constellation. And also it investigate the design of robust precoders in the presence of a perfect and imperfect CSI. In case of downlink MU-MIMO system, proposed a precoder based on nullspace of channel transmission matrix is employed to decouple multiuser channels [13]. The conventional MU-MIMO transceiver design for both uplink and downlink under the minimum TMSE criterion exhibits good BER performance for proper modulation schemes, e.g., M-QAM, and M-PSK [9][10][12] than improper modulation schemes, e.g., BPSK and M-ASK. The improved minimum TMSE design for improper signal constellations was proposed in [13] and shown to give superior BER performance than the conventional design in [9][10][12].

Recently [14] proposed TMSE based optimum joint linear transceiver design for the SU-MIMO systems which employs improper modulation techniques. In this paper, we propose joint linear precoding and decoding design for downlink MU-MIMO systems employing improper constellations. An improved minimum TMSE transceiver is designed for the case of perfect CSI and used to develop an iterative design procedure for the optimum precoding and decoding matrices.

The rest of the paper is organized as follows. The proposed joint linear design of precoder and decoder in downlink MU-MIMO systems is presented in Section 2. The superiority of the proposed joint linear precoder and decoder designs over the conventional designs is verified with simulation results in Section 3. Finally conclusions are given in Section 4.

Notations: Throughout this paper, upper (lower) case boldface letters are for matrices (vectors), (■)T denotes matrix transpose, ()H stands for matrix conjugate transpose, (■)* means matrix conjugate, E(■) is expectation, y y is Euclidian norm, Tr( ) is the trace operation and IN is an N x N identity matrix.

2. joint linear precoder and decoder design in downlink MU-MIMO systems

It Consider the downlink of a MU-MIMO system with NT transmit antennas at the base station, and K users, each with NR,j receive antennas, where j = 1,...,K. Suppose the user j has Bj data streams which is denoted by Bj x 1(Bj < min(NT, Nr)) and the total number of substreams are B = Bj. Linear precoder of the user j at BS is denoted as Fj, j = 1,...,K with matrix size NT x Bj. Data vectors are assumed to have the same statistics, output form jth precoders are represented as

x j = Fj s j (1)

H, i = 1,...,K denotes the downlink channel matrix of user i. The data symbols are assumed to be uncorrelated and have zero mean and unit energy, i.e., E[sjsj] = IB j. The signal after the precoders are satisfies the following total transmit power constraint:

E [||x||2] = |e [||f js j ||2] = £ Tr(F j Fj) = ft. (2)

j=1 j=1

we consider the transceiver design for the downlink MU-MIMO systems using an improper modulations (for which E [sjsj] = 0) such as BPSK and 4-ASK. Precoded signals are transmitted across a slowly-varying flat Rayleigh fading

Modulator s 1 F1 x 1

G1 S1 -> Re () S1 -> Demodulator

Modulator s 2 -*■ F2 y NR.2 . Li G 2 !.2 -> Re () >2 -> Demodulator

; • i k . N T H ^V N, k . VL

Modulator s K F x K _

G sK Re () sK Demodulator

Fig. 1. Transmitter and Receiver design for downlink MU-MIMO systems

channels. The signal received at the receive antennas of user i is given by [9][12].

yi = H

The received vector is fed to the decoder Gi , i = 1,...,K which is a Bi x NRi matrix. Then the resultant vector from output of decoder is:

ri = GiHi

+ Gi-ni

where the NT i x 1 vector n represents spatially and temporally additive white Gaussian noise (AWGN) of zero mean and variance o2. The conventional downlink transceiver problem is formulated as minimizing the TMSE under the total transmit power constraint specified by (2):

£i = E[Hr - Sill2] = E

- K 2]

GiHi I F j s j + Gi-n-- — s-

- LJ=1 -

This design criterion is optimum for the systems with proper modulations E [sjsj] = 0 such as M-QAM and M-PSK. However, for the improper modulation schemes (BPSK and M-ASK) considered in this paper, same conventional optimization approach fails to provide a optimum performance due the a complex-values filter output. The decision in a system employing improper constellation is based on only real part of the output. The same strategy in [14] is extended to the case of downlink MU-MIMO systems. We proposed a new joint linear precoder and decoder design in downlink MU-MIMO systems based on minimizing the TMSE under total transmit power constraint. And the newly defined error vector is,

e = ri - Si

where ri = ^GiHi follows:

jF J s J

+ Gin . With the newly defined error vector, the TMSE can be computed as

E[||e||2] = E[Hr - Sill2]

- ( K \ 2]

E ft GiHi IF js j + G-n J — s-

- V Lj=1 J -

Tr|e| a + 0.5^Gni + G*n*^ — s- ß + 0.5^nfGf + nfG^ -sf ||

(8) (9)

where a = 0.5 G,H

.5^ G,H,

jF j s j

K Ï7*c* j=1 FjSj

and ß = 0.5

IK=1 j Fj

Hf Gf+

11l sfFT

'j=i*j * j Ia, £ [n,nf

From the assumptions on the statistics of the channel, noise and data, one has E [s,sf ] = E[s,sf c2(INr and E[n,] = E [n,nf] = E[n*nf] = 0. Using these facts and after some manipulations (10) can be simplified to

E[||e||2] = Y+ 8 + X + ^ - 0.^G,H,F, + G*H*F; + FfHfGf + FjTHfG^

+IBi + 0.25c2 ^ G,Gf + G*Gf

where 7 = G,H,-and Z = G*H*

j F j Ff

Hf Gf ,5 = G,H,

j F j F

Hf Gf, X = G*H*

I=1 f*fk

Hf Gf,

2K=1 F* Ff

Hf Gf.

The main objective of downlink MU-MIMO systems design is to find a pair F, and G, to minimize E[||e||2] subject to the total BS transmit power constraint. That is, the improved TMSE design for downlink MU-MIMO systems employing improper modulations is expressed as

{(Fj ,Gj j

E [||e||2] s.t.

I Tr (F j j < Py. j=1

Here we form the Lagrangian to find the solution for the problem in (12)

V = E [||e||2]+^

I^(W)] -pr)

where ^ is the Lagrange multiplier. By substituting (11) into (13) and then taking the derivatives of n with respect to f, and Gi, the associated Karush-Kuhn-Tucker (KKT) conditions can be obtained and given in the following.

First, the value of ¿r- can be found by using the cyclic property of the trace function. Setting ¿r- 0, 1 < z < k and taking the complex conjugates of both sides and it yields

Q + W + a^Gz-

2Ff Hf

where Q

IK=1 F jFf il

Hf, and W = G*H*

IK=1F* Ff

Similarly, setting = 0 and taking the complex conjugates of both sides and it yields

0 + A + 2ßFz

2Hf Gf

where 0

IK=1 Hf Gf G j H j

Fz, and A

IK=1 Hf Gf G*H*

Next, by post-multiplying both sides of (14) by Gf and taking Xf=i on both sides, one obtains

I{QGf + WGf + an2GzGf} = 2 iFfHfGf

Likewise, pre-multiplying both sides of (15) by Ff and taking XK=1 on both sides, produces

I {Ff 0 + Ff A + Ff 2jUFz} = 2 I Ff Hf G

z=1 Lz=1

It then follows from (16) and (17) that:

K {QGf + WGf + olGzGf } = K {Ff 0 + Ff A + Ff 2^Fz}

z=1 z=1

Then, by taking the traces of both sides of (18) one has:

I Tr (GzGf)

As in [13], an iterative procedure is developed for to find a optimum solution for Fz and Gz, by using (14), (15) and (19). First, define

I Fj Ff

I f; Ff

Gz,Re + jGz,Im Az,Re + JAz,Im

Bz,Re + JBz,Im Cz,Re + Cz,Im

Then Cz Re and Cz Im can be expressed using (14), in a vector form as

2FzfHzf

Az,Re + Bz,Re + D Az,Im + Bz,Im

Bz,Im — Az,Im Az,Re + Bz,Re — D

[ OzJR ^pm where D = OnlNRz. The above expression also implies that

[ ^Re Gz,Im ]

Similarly, define

+ Bz,Re + D

z,Re + Bz,Re Bz,Im Az,Im

Az,Im + Bz,Im

+ Bz,Re — D

z,Re + Dz, Re

I Hf Gf G JH

Fz,Re + JFz,Im Pz,Re + JPz, Im

I Hf Gf g h;

Qz,Re + jQz,Im Rz,Re + Rz,Im

Then Rz Re and Rz Im can be expressed using (15), in a vector form as

2Hf Gf

Pz,Re + Qz,Re + E Qz,Im — Pz,Im Pz,Im + Qz,Im Pz,ReQz,Re — E

z,Re z, Im

where E = 2ßINT. Equivalently,

Pz,Re + Qz,Re + E Qz,Im — Pz,Im Pz,Im + Qz,Im Pz,ReQz,Re — E

— 1 Rz,Re

(20) (21)

Based on the above expressions, the optimum precoder and decoder can be solved by an iteration procedure as illustrated in Fig. 2, where Fz denotes Fz in the ith iteration and Fz, z = 1 ■■■K. and Fz is chosen such that the Bz x Bz upper sub-matrix of Fz is a scaled identity matrix (which satisfies the power constraint with equality), while all the other remaining entries of Fz are zero,

Fig. 2. Iterative procedure for solving the optimum precoder and decoder.

3. NUMERICAL RESULTS

In this section, we present simulation results to validate and investigate our proposed precoding and decoding designs in MU-MIMO downlink with perfect CSI, in terms of bit error rate (BER) as developed in Section 2. For all results, we assume that the numbers of transmit antenna at BS is NT = 6 , the number of receive antenna at each user is NR1 = Nr2 = Nr3 = 2, the number of data streams are B = 2, and the channel between base station and each user is a rayleigh fading channel. In both figures, the signal-to-noise ratio is defined as SNR = A- and the BER cures of user 1 are displayed. The proposed joint linear transceiver design in MU-MIMO systems for downlink with perfect CSI is compared with the previously-designed joint linear transceiver strategy in [9], but without taking into account specific property of improper modulations.

In Fig. 3, system BER versus SNR is plotted when the numbers of data stream is B= 2. It displays the performance comparisons of the conventional joint transceiver design for downlink MU-MIMO systems in [9] and the proposed joint transceiver design in downlink MU-MIMO systems under perfect CSI for both BPSK and 4-ASK. As shown in Fig. 3, the proposed joint linear precoding and decoding designs leads to a very large performance improvement, especially for BPSK modulation (an SNR improvement of about 15 dB is observed for BER of 10-3). It should be emphasized that the proposed designs under comparison take into account the one-dimensional property of improper modulations. It is clear from the figure that a significant performance improvement is achieved by performing proposed joint precoding and decoding. With the assumption of perfect CSI, it can be seen that the BER performance curves improve exponentially with SNR.

The purpose of Fig. 4 is to examines the effect of diversity on the downlink MU-MIMO system BER performance under perfect CSI and BPSK modulation.The proposed joint prcoding and decoding design is enjoys further gain, if the number of data streams is reduced from B = 2 to B = 1.

4. CONCLUSIONS

In this paper, an improved joint linear transceivers design with improper constellations for MU-MIMO downlink systems with perfect CSI is proposed. The joint linear precoder and decoder designs are formulated into an optimization problem. The optimum closed-form precoder and decoder are derived by solving that optimization problem with an iterative procedure. Simulation results indicate that our proposed design considerably outperform the traditional cases. The proposed algorithm can also be extended into the case of uplink MU-MIMO system and also with imperfect CSI.

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Fig. 3. Performance comparison of the conventional downlink transceiver and proposed downlink transceiver with perfect CSI, for BPSK and 4-ASK. NT = 6, NR1 = Nr,2 = Nr,3 = 2, B = 2.

SNR = PT/a'n (dB)

Fig. 4. Diversity Performance of the proposed downlink transceiver with perfect CSI, for BPSK. Nt = 6, Nr,1 = Nr,2 = Nr,3 = 2, B = 1,2.