IEEE - 20180
Joint Optimization of Precoder and Decoder in Multiuser MIMO Systems with Imperfect Channel State Information(CSI) M.Raja, P. Muthuchidambaranathan Dept. of Electronics and Communication Engineering, National Institute of Technology, Tiruchirappalli, India Email:
[email protected].
[email protected] Abstract-In this paper, we address the problem of joint transceiver optimization for downlink of multiuser multiple input, mUltiple-output systems employing improper constella tions such as binary phase shift-keying (BPSK) and M-ary amplitude shift-keying (M-ASK). Proposed are novel joint linear transceivers that minimize the total mean squared error (TMSE), subject to a total transmit power constraint. The joint linear transceiver designs are carried out for the cases of imperfect channel state information (CSI). The channel model takes into account both transmit and receive correlations as well as the channel estimation error. Simulation results indicate that the performance improvement of
the proposed joint transceiver
designs over the previously-proposed designs.
1. INTRODUCTION
Multiple-input multiple-output (MIMO) system has at tracted considerable attention recently because of the higher system throughput and better spectral efficiency. Multiuser MIMO which can achieve higher sum capacity, effectively improves average and cell edge throughput, appeals enormous attentions recently [1], [2]. The downlink of multiuser MIMO wireless communications is considered in [3], [4], [5] because of the potential for improving reliability and capacity of the system. 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) [6], [7]. Joint linear transceiver design for downlink of MU-MIMO systems is proposed, with an assumption of perfect CSI in [8], [9], [10]. To enable precoding at the transmitter, the channel estimation has to be performed at the receiver, and the channel state information at the receiver (CSIR) is feedback to the transmitter [11]. But the feedback information is always not perfect because of channel estimation error and/or delay. The imperfect CSI is considered and TMSE minimization problem is formulated for both uplink and downlink in [12], [13], [14]. The minimum TMSE design is formulated as a nonconvex optimization problem under a total transmit power constraint, robust adaptive joint transmit filters, receive filter for uplink and downlink of MU-MIMO systems are obtained by solving a nonconvex optimization problem. Iterative algorithm based on the Karush-Kuhn-Tucker (KKT) conditions are used to solve nonconvex optimization problem. In [12], [13] imperfect CSI
considers the impact of channel estimation error and channel correlations at transmitter for uplink, and channel correlation at receiver for downlink. But [15], [16] considers both transmit and receive correlation information. The feedback information is assumed to be error free for the purpose of convenient analysis. In [17] proposed a novel linear transmit precoding strategy for single user and MU-MIMO systems employing improper signal constellations. And it shows that the existing linear precoders are SUboptimum for systems employing improper modulation schemes and their performance can be improved by designing the system with modified cost functions, and by exploitation of the improperness of signal constellation. And also it investigate the design of robust precoders in the presence of a perfect and imperfect CSI. In case of downlink of MU-MIMO system, proposed a precoder based on nullspace of channel transmission matrix is employed to decouple multiuser channels [17]. The conventional MU MIMO transceiver design for downlink under the minimum TMSE criterion exhibits good BER performance for proper modulation schemes, e.g., M-QAM, and M-PSK [12], [13], [14] than improper modulation schemes, e.g., BPSK and M ASK. The improved minimum TMSE design for improper signal constellations was proposed in [17] and shown to give superior BER performance than the conventional design in [12], [13], [14]. Recently [18] proposed a novel joint linear transceiver design under the minimum TMSE criterion, for single user MIMO systems employing improper signal constellations. The joint design is formulated into an optimization problem, the optimum closed-form precoder and decoder are derived under both the scenario of perfect and imperfect CSI. However, to the best of our knowledge, no attention has been paid to the optimum joint linear transceiver design for the MU-MIMO systems which employ improper modulation techniques, ei ther under the perfect CSI or imperfect CSI assumption. To fill this gap, this paper shall examine the problem of joint linear precoding/decoding design for downlink of MU-MIMO system employing improper constellations with imperfect CSI. An improved minimum TMSE transceiver is designed for the case of imperfect 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
ICCCNT - 2012 July 26-28,2012,Coimbatore, India
IEEE - 20180
we use the channel model in [19], channel of the jth user is denoted as where is a spatially = white matrix whose entries are independent and identically distributed (ij.d.) Nc(O, and j = ... , K. The matrices and represent the normalized transmit and receive correlations (i.e., with unit diagonal entries), respectively.Both and are assumed to be full-rank and known to both the transmitter and the receiver. In general feedback information are not perfect due to feed back delays and errors, transmitter can only get an erroneous estimate of the true channel When the channel is spatially correlated, the jth user channel error model can be written as (1)
Hj RU,�Hw,jR¥2, Hw,j 1) 1, RT RR,j RT RR,j Fig. 1.
Transmitter design for MU-MIMO downlink
Hj
Hj.
Hj, R1R,/2jHw,jR1T/2 Ej Re,1 /R2,jEw,jRT1 /2. Hw,j Ew,j H - RR,1 /2)Hw,)·RT1 /2 Re,1 /R2,)Ew ·RT1 /2 1 a�e,j Re, R , j [IN j RR � ] a , ; �e, , R 1 Tr(RrEw,)a;/ j Ptr,j Ptr,j A
where and = = The entries of and are independent. Channel estimation is performed based on orthogonal training method, the channel is Fig. 2.
Receiver design for MU-MIMO downlink
where Jomt linear transceiver design in downlink of MU-MIMO systems with imperfect CSI is presented in Section II. The superiority of the proposed joint linear transceiver design over the conventional designs is verified with simulation results in Section III. Finally conclusions are given in Section IV. Notations: Throughout this paper, upper (lower) case bold face letters are for matrices (vectors), ( ) T denotes matrix transpose, ( - ) H stands for matrix conjugate transpose, * means matrix conjugate, is expectation, is Euclidian norm, Tr(·) is the trace operation and is an x identity matrix. II.
DOWNLINK DESIGN IN
-
MU-MIMO WITH IMPERFECT CSI
In this section, we propose a joint transceiver design for the downlink MU-MIMO systems that minimizes the TMSE under a constraint on total base station (BS) transmit power, which is illustrated in Fig. 1 and Fig. 2 for transmitter and receiver correspondingly. And also it is assumed that both the transmitter and receivers have imperfect CSI. A.
Channel Model
and B.
(2)
,J
+
and and is training power of the user j is independent with data and noise vector.
System Model
NT NR,j 1, Bj 1(Bj min(NT,BjNR)) .I I ( ) IN N N B Fj2:=,f=1 Bj.1, NT Bj. Xj Fjsj. E[SjSf] IB,j' E[l xI12] j2::K=1 E[I!FjSj1 2] j2::K=1 Tr(FjFf) PT· E[sjsJ] i= i N1,R,i 1 Yi Hi[ ,\,L..jK=1 FjSj] ni Gi i , Bi NR,i -
E(-)
+
J
In this section, we consider a MU-MIMO system with im perfect CSI because of the time-varying nature of the wireless channel. In that case obtaining the channel information at both the transmitter and receiver can be difficult and also the obtained channel information is not the same as the instantaneous channel information. Because of this reason, the MU-MIMO downlink transceiver design under perfect CSI is no longer optimum for the systems operating with estimated channel information. So we need to design a better transceiver that consider the channel estimation error. This is a main objective of this section.
Consider a downlink MU-MIMO where BS with trans mit antennas transmits to K users each equipped with receive antennas simultaneously over the same physical re sources, where j = ... , K. Suppose the user j has data streams which is denoted by x :s; and the total number of substreams are = Linear precoder of the user j at BS is denoted as j = ... , K with matrix size x Data vectors are assumed to have the same statistics, output form lh precoders are represented as = The data symbols are assumed to be uncorrelated and have zero mean and unit energy, i.e., = The signal after the precoder satisfies the following total transmit power constraint: =
=
=
(3)
In this section, we consider the downlink MU-MIMO sys tems using an improper modulations (for which 0) such as BPSK and 4-ASK. Precoded signals are transmitted across a slowly-varying flat Rayleigh fading channels. The signal received at the antennas of user is given by [13], [14], . . CDL) + (DL) Wit . h matrix . size x = And the received vector is fed to the decoder = ... , K which is a x matrix. Then the resultant vector from output of decoder is:
ICCCNT - 2012 July 26-28,2012,Coimbatore, India
(4)
IEEE - 20180
NT,i
)
� where the x 1 vector ll DL represents spatially and tem porally additive white Gaussian noise (AWGN) of zero mean ) . and variance (0"�DL)2 The conventional downlink transceiver problem is formulated as minimizing the TMSE under the total transmit power constraint specified by (3):
facts and after some manipulations (9) can be simplified to
Tr{ O.25GiHi [t, FjFf] H{iG{i +O.25GiHiA [ )=1� FjFjT] AHiTGiT +o.25GiRe,R,iG{i XO"�e, i ) ) (0" +O.25GiG{i �DL 2 - O.5GiHiFi +O.25G;H; [ )=1� F;Ff] H{iG{i +O.25G;H; [ )=1� F;FJ]H;G; +O.25G;R:,R,iG;X) *O"�e,i+ +O.25G;G;(0"�DL)2 - O.5G;H;F; - O.5FHAH i Hi GHi - O.5FiTAHiTGiT } X 2::f=1 Tr (RTFjFf). (Gi) E[lle(DL)112] E[lle(DL)112]
�
E[llr�DL) - si112] E
[I GiHi [t, FjSj] +
(5)
Gill�DL)
- Si In
(6)
K
For the case of proper modulation E[sjsJl = 0 such as M QAM and M-PSK, the conventional transceiver design based on (6) is optimum. But for the case of improper modulations, same conventional optimization approach fails to provide a optimum performance. This is because of conventional op timization method produce a complex-values filter output. The decision in a system employing improper constellation is based on only real part of the output. And it was pointed out earlier in [18], in which a novel linear transceiver strategy for SU-MIMO systems employing improper constellations is also proposed. In this paper, the same strategy in [18] is extended to the case of both downlink and uplink 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 for the downlink MU-MIMO systems with improper constellations is modified from conventional approach and it is defined as follows:
(7)
)
� (Gi (Hi+ Ei) [2::;=1 FjsjL+
)
Gill�DL where r�DL = is resultant vector after the decoder. The T SE function for joint transceiver design can be evaluated for improper modulation as follows
)
(8)
) substitute the value of r�DL in (8), we get
K
IBi
10)
where = The main objective of downlink MU-MIMO systems design is to find a pair pre coding matrix (Fi) and decoding matrix to minimize 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 K
j=l Tr (FjFf) PT. �
7](DL)
=
E[lle(DL)) 112]
Gi
Fi,
Gz (AHzfHAHz + Re,R,zO"ce,2 zRT1>) +Gz*Hz*f'.H. zH+ (0"(DL))2G 2FHHH ( + RTO"�e,zRe,R,z ) Fz +AF;+ 2p,(DL)Fz 2H�G� 2::f=1 FjFf, f'.. AH2::f=H1 FjAFf 1> H )*,A* 2::j=l Hj Gj GjHj, 2::j=l ATr(FjF A j H H H 2::j=l Hj Gj GjHj, 2::j=l Tr(Gj Gj). Z =
J
z
z
where
f
=
=
K
e
=
ICCCNT - 2012 July 26-28,2012,Coimbatore, India
,
(14) =
K
K
and III = pression of the Lagrange multiplier is,
O.
(13)
III
=
From the assumptions on the statIstIcs of the chan nel, noise and data, one has E[Sis{i] = E[SiS; = ) ) ) , (ll,(DL)H] = ( (DL)21 NT and E[ll,(DL] = I B" E[ll(DL E[ll�DL\ll�DL)) T] = E[(ll�DL))*(ll�DL))H] = Using these
O"n"
K
(12) where p,(DL) is the Lagrange multiplier. By substituting (10) in (12) and taking the derivatives of 7] with respect to and it can be shown that the associated Karush-Kuhn-Tucker (KKT) conditions are as follows:
K
.
+ p,(DL) ([ � Tr (FjFf) ] - PT)
e
(9)
(11)
Here we form the Lagrangian to find the solution for the problem in (11)
n
E [II � (Gi (Hi+Ei) [t,FjSj] + Gill�DL) ) - Si In
:s;
The ex-
(15)
IEEE - 20180
As in [17], an iterative procedure is developed for to find a optimum solution for Fz and Gz, by using (13), (14) and (??).
Gz,Re + jGz,Im (16)
(DL) + 'A(DL) (17) Az,Re J z,Im L) (D (DL) (18) Bz,Re + J'Bz,Im DL) + dDL) (19) dz,Re z,Im
Then Gz,Re and Gz,Im can be expressed as
[ dz,Re DL) dDL) z,Im Bz,Re [ Az,Re Bz,Im - Az,Im
Gz,Re Gz,Im 1
V
=
Az,Im + Bz,Im Az,Re + Bz,Re - V
+V
+
where
=
(T�DL) )2INR,z' Likewise, define Fz,Re
Fz 8
+
+
] - �20) (21) (22) (23) (24)
jFz,Im
(DL) + 'p(DL) pz,Re J z,Im L) (D (DL) Qz,Re + J'Qz,Im (DL) + R(DL) R
RT(T�e,zRe,R,z W A
H 2Hz GzH A
z,Re
z,Im
Then
Fz,Re Fz,Im
]
=
pz,Re + Qz,Re + E Qz,Im - pz,Im pz,Im + Qz,Im pz,Re + Qz,Re - E
[ Rz,Re ] Rz,Im
r
l (25)
where E = 2j.l(DL)INT' Based on the above expressions, the optimum precoder and decoder can be solved by an iteration procedure as outlined in following algorithm 1) Initialize 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, 2) For Z = 1 ... K, find the value of Gz using (20), 3) Find the value of j.l(DL) using (15), 4) For z = 1 ... K, find Fz using (25),
{ [ L�=l FzF� ] } PT, { L�=l FzF� ] } PT, { L�=l (F� - F�-l) (F� - F�-l) H ] }
5) IfTr
that Tr
6) If
then do the scaling such
>
=
else go to next step
Tr'
1 ,2 , ... , NR,l and RT(i, j) = p�-jl for i,j = 1 ,2 , ... , NT. For all simulation results reported in this section, the numbers of transmit and receive antennas are set to be NT = 6 ,NR,l = NR,2 = NR,3 = 2. For Fig. 4 the numbers of data streams are B = 1 or B = 2 or B = 3, all other figures consider B = 2. In P all figures, the signal-to-noise ratio is defined as SNR = I. To have a fair comparison with the results reported in [14], P the SNR in training phase is set to SNRtr = �r = 2 6.01 6 an dB. Fig. 3 displays the performance comparisons of the conven tional joint transceiver design for downlink of MU-MIMO in [14] and the proposed joint transceiver design for downlink of MU-MIMO under imperfect CSI. And the transmit and receive correlation parameters are assumed as PT = PR,l = PR,2 = PR,3 = 0.5 for both BPSK and 4-ASK. As can be seen from the figure, the proposed joint linear transceiver leads to a very large performance improvement, especially for BPSK modulation (an SNR improvement of about 1 0 dB is observed P for BER of 1 0-3). Note that, with SNRtr = �r = 2 6.01 6 dB an and PT = 0.5, one has (T�e = 0.01 5. Fig. 4 shows the diversity performance of the downlink for imperfect CSI with BPSK modulation. The motivation of Fig. 4 is to show that, the proposed transceiver design is enjoys further gain, if the number of data streams is reduced from B = 2 to B = 1. The purpose of Fig. 5 is to examines the effect of channel correlations on the downlink system BER performance under imperfect CSI. For this figure, BPSK mod ulation is employed with the number of data streams B = 2. Various sets of transmit/receive correlations considered are {PT = 0. 9 , PR,l = PR,2 = PR,3 = 0. 9}; {PT = 0. 9 , PR,l = PR,2 = PR,3 = 0.5}; {PT = 0.5 , PR,l = PR,2 = PR,3 = 0.5}; and {PT = 0.5 , PR,l = PR,2 = PR,3 = 0.5}. With P SNRtr = �r = 2 6.01 6 dB the parameter (T�e is 0.01 5 and an 0.073 9 for PT = 0.5 and PT = 0. 9, respectively. In general, Fig. 5 shows that higher values of the transmit and receive correlations lead to a bigger performance loss. IV. CONCLUSIONS This paper addressed the designs of joint linear transceivers with improper constellations for downlink of MU-MIMO systems . Specifically, it designed a joint linear transceiver for downlink of MU-MIMO systems under imperfect CSI assump tion and with improper modulations. In this the transceiver designs are accomplished with an iterative procedure. Per formance advantage of the proposed designs in terms of the system's BER was thoroughly demonstrated with simulation results.
