Improved Minimum Total MSE Transceiver Design With Imperfect CSI at Both Ends of a MIMO Link Kamalakar.Sanka, M.Raja and P.Muthuchidambaranathan Department of Electronics and Communication Engineering National Institute of Technology Tiruchirappalli, India.
[email protected],
[email protected],
[email protected] Abstract—In this paper, we propose a novel joint linear transceiver design for single-user multiple-input, multipleoutput (SU-MIMO) systems employing improper signal constellations. In particular, improved minimum mean square error (MMSE) transceiver is designed based on modified cost functions, with channel mean as well as both transmit and receive correlation information at both ends. The joint design is formulated into an optimization problem. The optimum closed-form precoder and decoder are derived. Compared to the case with perfect channel state information (CSI), linear filters are added at both ends to balance the suppression of channel noise and the noise from imperfect channel estimation. The superiority of the proposed joint linear transceiver design over the conventional solutions is verified by simulation results. Index Terms—Channel state information (CSI), meansquare error (MSE), multiple-input multiple-output (MIMO), linear precoding, spatial multiplexing.
I.
INTRODUCTION
Multiple Input Multiple Output (MIMO) systems have been extensively studied over the last decade [1]. The capacity analysis of MIMO systems has shown significant gains over single-input single-output (SISO) systems [2]. Depending on the channel condition, the first generation MIMO technique aims at achieving a higher data rate, such as spatial multiplexing [3], or a higher diversity, such as space-time coding [4]. These techniques do not require the knowledge of channel state information (CSI) at the transmitter (CSIT). Joint transceiver design requires CSI at both transmitter (CSIT) and receiver (CSIR). In coherent communications, CSIR is obtained by channel estimation, is transferred to the transmitter side [5]. Thus, usually CSIT is imperfect, due to channel estimation error and/or limitations of the feedback link. Compared to Non-Linear designs, linear designs are less complex, especially for mobile station. A joint transceiver design under MSE-based, signal-to-interferenceplus-noise-ratio (SINR)-based or BER-based criteria have been studied in [6]. The minimum total MSE based transceiver design gives convenient analysis, balances interference and noise suppression, widely used in singleuser, multi-user MIMO. Under different assumption of CSI, MMSE based joint transceiver designs have been studied in [6]-[10]. In [6]-[7], the MMSE based joint transceivers were designed under perfect CSI at both ends. Later imperfect CSI ___________________________________ 978-1-4244 -8679-3/11/$26.00 ©2011 IEEE 23
was considered. In [10, Ch.7], the same imperfect CSI is assumed at both ends, but the channel correlation has not been considered. In [11] the same imperfect CSI is assumed at both ends, with both transmit and receive correlation. In this paper, we propose a novel joint linear transceiver design to minimize modified cost functions with same imperfect CSI at both ends as well as both transmit and receive correlation information at both ends of a single-user MIMO link. Here the CSIR is obtained by channel estimation, is transferred to the transmitter side. To simplify the analysis, we assume that the feedback is error-free and instantaneous, as in [8] and [12, Sec.VI], which implies that the CSIT is the same as CSIR. The major contributions of this paper are summarized as follows. x We show that the existing joint linear transceiver designs 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. The proposed joint linear transceiver designs are compared to the existing joint linear transceiver designs in single-user MIMO (SU-MIMO) systems, and are shown to have superior performance without any loss of spectrum efficiency. x With perfect CSI at both ends as well as both transmit and receive correlation, based on modified cost function, improved minimum total MSE transceiver designed under a total transmit power constraint. x The joint linear transceiver design is extended to more general case with the same imperfect CSI at both ends, as well as both transmit and receive correlation. The rest of the paper is organized as follows. The improved joint linear transceiver design with transmitter and receiver correlation under perfect CSI is introduced in Section II for SU-MIMO systems. The issue of robust transceiver design with imperfect channel state information is addressed in Section III. The superiority of the proposed joint linear transceiver design over the conventional
Fig.1 Block diagram of the transceiver design for SU-MIMO systems.
solutions are verified by simulation results in Section IV and conclusions are given in Section V. Notation: Throughout this paper, Upper (lower) case boldface letters are for matrices (vectors), T denotes matrix
transpose,
matrix
H
matrix conjugate, *
conjugate
transpose,
E[] expectation, Euclidian norm,
Tr() trace operation and I N an NxN identity matrix.
in [14], i.e., H R 1R/ 2 H w R 1T/ 2 , where H w is a spatially white matrix whose entries are independent and identically distributed(i.i.d.) N ( 0,1). The matrices R T and R R c
represent normalized transmit and receive correlation (i.e., with unit diagonal entries), respectively. Both R T and R R are assumed to be full-rank. The conventional precoder and decoder are derived by minimizing the minimum square error (MSE). 2 E[ ~s s ]
II. IMPROVED TRANSCEIVER DESIGN FOR SU-MIMO SYSTEMS WITH PERFECT CSI In this section, we consider the SU-MIMO system, which is shown in Fig.1. It is assumed that both the transmitter and receiver have perfect CSI as well as transmitter and receiver correlation information. The proposed design will be extended to imperfect CSI in Sections III. 1t It is assumed that antennas are used at the 1 transmitter and r antennas are used at the receiver. The information symbols to be sent are denoted by a % x1 vector s
T
[ V V % ]
where the number of data streams
the precoder satisfies the total power constraint 2
E[ x ]
E[ Fs
2
]
H
Tr (FF )
PT
PT
such that (1)
T
which E[ss ] 0 ). However, for the improper modulation schemes considered in this paper, such as M-ary ASK, T
E[ss ]
I % ), the design criterion OQPSK (for which expressed by (3) is suboptimum. We propose a new joint linear transceiver scheme based on an error criterion defined by e
The reason for the above modification is that the conventional optimization approach expressed by (3) yields a complex valued filter output. However, only the real part of this output is relevant for the decision in a system with an improper constellation. Thus, minimization of the modified cost function in (4) will result in a better estimator. The modified MSE function can be written as follows: E[ e ]
2
Tr ( E [ sˆ s ])
that PT ,
E[ e
2
]
GHFs Gn
(2)
where n is the spatially and temporally white additive Gaussian noise 1 r x1 vector with zero mean and covariance matrix , i.e.,
n~
N
( 0, V n I 1 ). We 2
c
r
use the channel model
24
(5)
is minimized subject to a total power constraint
i.e., min
y
~ s
2
Tr ( E[ Re(GHFs Gn) s ])
Our goal is to find a pair of appropriate F and G , such
matrix H . At the receiver, the 1 r x1 received signal vector Hx n is fed into decoder G (is %x1 r matrix), then the % x1 resultant vector is
(4)
Re(GHFs Gn) s
With transmit power constraint specified by (1).
2
We consider an uncoded MIMO system using improper modulation (for which E[ss T ] z 0 ), such as M-ary amplitude shift keying (ASK), offset quadrature phase shift keying (OQPSK) [13], etc. After the precoder, the data vector x is transmitted across the slowly varying, flat, Rayleigh fading MIMO channel, described by the 1 t x1 r
(3)
With transmit power constraint specified by (1). This design criterion is optimum for systems with proper modulations, such as M-QAM and M-PSK (for
% (d min( 1 t , 1 r ))
is chosen and fixed. The transmitted 1 x% precoding symbol vector s is precoded using the t 1 x 1 matrix F , i.e., x Fs is a t vector. The data symbols are assumed to be uncorrelated and to have zero mean and unit 2 E[ss H ] I % identical energy V s 1 , i.e., The signal after
2
E[ (GHFs Gn ) s ]
F, G
E[ e
2
]
(6)
H
Tr (FF ) d PT
The associated Lagrangian is K
E[ e
2
] P (Tr( FF
H
) PT )
(7)
Using our assumptions on the statistics of the channel, noise, and data, with some manipulations, we can simplify (7) as K
ˆ is the estimated channel matrix (i.e., the channel where H 1/ 2
R e ,R E w R T
is the channel estimation error
matrix.
H H H T T T * * * H H H Tr{0.25(GHFF H G GHFF H G G H F F H G
In contrast to the perfect CSI case where the instantaneous MSE is considered, we need to formulate the problem with the average MSE, (i.e., the expected value of MSE where the expectation is taken with respect to H ).
* * * T T T G H F F H G )
* * * H H H T T T 0.5(GHF G H F F H G F H G )
Equation (9) becomes
2 H H * T I 0.25V .(GG G G )} P (Tr(FF ) PT )
%
1/ 2
mean), and E
(8) *ˆ* * H ˆ H 2 H ˆ FF H H ˆ H R G (H e , R V Tr ( R T FF )) G H F F H
n
ce
where P is the Lagrange multiplier. By taking the derivatives of K with respect to F and G [15] with the power constraint, the associated conditions can be obtained as follows:
H H 2 * * * H H GHFF H G H F F H V G n
H
H
H
H
H
H ˆH 2F H
G
H
2H G
(13)
Equation (10) becomes ˆ H G H GH ˆ R V 2 Tr ( R G H G ))F H ˆ H G H G *H ˆ *F * (H T e,R
(9)
ce
H
ˆ G 2H
H
(14)
(10)
From the trace of post-multiplying both sides of (13)
From the trace of post-multiplying both sides of (9)
by G and the trace of pre-multiplying both sides of (14)
H
by G and the trace of pre-multiplying both sides of (10)
H
H
by F , we obtain
H
P
by F , we obtain 2
P
H
V Tr ( GG ) / 2 PT
(11)
n
HFF H
Since obtaining channel information at the receiver and the transmitter can be difficult due to channel dynamics, CSI is usually not accurate instantaneous channel information. For the single-user MIMO system considered in this work, we assume that the channel experiences both transmit correlation and receiver correlation which are known to both the transmitter and the receiver. Let power, V
denote
PT r 2 ce
2
1
n
T
the
V Tr (R ) / PTr , R e , R
entries of E w are i.i.d. N described by [11] as H
c
( 0, V
total
[ I 1 V r
2 ce
2 ce
). Then
1 1 RR ]
H
H
H
H
(15)
n
* * H H G r M G i , H F F H
R e , R V Tr ( R T FF 2
ce
H
)
B r M B i ,
A r M A i
H
and 2F H C r M Ci . Then C r and C i can be expressed using (13), in vector form as
>Cr
training
Ci
@ >G r
ªA B V 2 I r r 1r n G i @ « « B A i i «¬
º » (16) 2 » Ar Br V I1 r» n ¼ Ai Bi
G r and G i are derived as
and the
the CSI model is
ˆ E H
2
V Tr ( GG ) / 2 PT
Let us denote G
III. IMPROVED ROBUST TRANSCEIVER DESIGN FOR SU-MIMO SYSTEMS WITH IMPERFECT CSI
2 n
2P F
* * * H G GHF H G G H F 2 P F H
H
2F H
V
(12)
>G r
Gi
@ >C r
ªA B V 2 I r r 1r n Ci @ « « B A i i «¬
Let us denote F
25
º » 2 » Ar Br V I1 r » n ¼
1
Ai Bi
H H * * Fr M Fi , H G G H
(17)
Q r M Q i ,
H 2 Tr ( R H H G H GH R T V ce e,R G G ) H
2H G
H
Pr M Pi
and
R r M R i . Then R r and R i can be expressed
using (14), in vector form as
ªR r º ªPr «R » « ¬ i¼ ¬
Q r 2 PI 1 Pi Q i
º ªFr º »« » 2 PI 1 F t ¼ ¬ i¼
Pr Q r
(18)
And Fi (Fi 1 ) denotes F ith ((i-1)th) iteration. IV.
Fr and Fi are derived as
ªFr º ªPr Q r 2 PI 1 t «F » « P Q i i ¬ i¼ ¬
1) Upper %x% sub matrix of Fo is chosen to be a scaled identity and to satisfy the power constraints with equality 2) All other remaining entries of Fo are zero.
Q i Pi
t
where constraint for the chosen value of Fo
Let 1 t 1 r 4. The exponential model is used for both transmit and receive correlation [16], [17]. Specifically, the transmit correlation model is given by:
1
º ªR r º » « » Pr Q r 2 PI 1 ¬R i ¼ t ¼ Q i Pi
SIMULATION RESULTS
(19)
From equations (13) & (14), it is observed that transceiver design is a closed form solution. In this paper, precoder and decoder are solved by iteration structure, which is given in Fig.2.
(R T )
LM
U
i j T
for L M
{1, 2...1 T } . The receive correlation
U matrix R R is similarly defined with the exception that T is U replaced by R and that the indices range from 1 to 1 r .
Fig.3 shows the offset QPSK modulation scheme with improved joint transceiver design algorithm under perfect CSI (i.e., no channel estimation error different U
0.0, U
0.0; U
0.5, U
0.0;
U
(V
2 ce
0.5, U
0)
) with correlations 0.5
T R R T R ( T ) and % 4 , 3 different no. of data streams ( ). In Fig.3, effects of transmit and receive correlation are observed under perfect CSI and it is clear that reducing the number of data streams % introduces diversity and thus compensates for the loss caused by channel correlation.
Fig.4 shows the performance comparison of conventional MMSE based joint transceiver design algorithm [11] and improved joint transceiver design algorithm under perfect CSI with different modulation U
0.5, U
0.5
R schemes, correlations ( T ) and no. of data % 4 streams ( ). In Fig.4, the superiority of the proposed joint linear transceiver design over the conventional solutions are verified.
Fig.2 Iteration structure for solving precoder and decoder
Fig.3 offset QPSK with perfect CSI
26
conventional transceiver design. This work can be extended to solve the closed form solution without iteration structure and then it can be applied to multi-user MIMO systems. REFERENCES [1]
[2] [3]
[4] Fig.4 Performance comparison with perfect CSI [5]
[6]
[7]
[8]
[9]
Fig.5 Performance comparison with Imperfect CSI
Fig.5 shows the performance comparison of conventional MMSE based joint transceiver design algorithm [11] and improved joint transceiver design algorithm under imperfect CSI (
V
2 ce
z0
with
PT r / V
2 n
26.016 dB
U
),
with 0.5, U
different
[11]
[12]
0. 5
R modulation schemes, correlations ( T ) and % 4 no. of data streams ( ). In Fig.5, the superiority of the proposed joint linear transceiver design over the conventional solutions are verified.
V.
[10]
CONCLUSIONS
In this paper, the proposed joint linear transceiver design for SU-MIMO systems employing improper signal constellations. The improved MMSE transceiver was designed based on modified cost functions, with both transmit and receive correlation information at both ends under perfect CSI. The transceiver design was a closed form solution and was solved by using iteration structure. The robust transceiver design was also discussed with same imperfect CSI at both ends. The simulation results showed the proposed joint linear transceiver design achieved the better performance in average bit error rate over the
27
[13] [14]
[15] [16]
[17]
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