161, Gajeong-dong, Yuseong-gu, Daejeon, 305-700, Korea. E-mail: {topsop ... correlation and cross-correlation functions within certain vicinity of the zero shifts ...
LS Codes Assisted Multipath Channel Interference Canceller for MIMO-OFDM Systems Wonsop Kim, Jae Joon Park, Hyun Kyu Chung
Jongsub Cha, Hyuckjae Lee
Mobile Telecommunication Research Laboratory Electronics and Telecommunications Research Institute 161, Gajeong-dong, Yuseong-gu, Daejeon, 305-700, Korea E-mail: {topsop, jjpark, hkchung}@etri.re.kr
School of Engineering Information and Communications University 103-6, Munji-dong, Yuseong-gu, Daejeon, 305-732, Korea E-mail: {s1009, hjlee}@icu.ac.kr
Abstract—MIMO-OFDM system is a promising way to achieve high data-rate wireless transmission. In this paper, Loosely Synchronous (LS) codes assisted Multipath Channel Interference Canceller (MCIC) is proposed for the channel estimation of MIMO-OFDM systems. Since LS codes have perfect autocorrelation and cross-correlation functions within certain vicinity of the zero shifts, LS codes assisted MCIC can substantially reduce multipath channel interference. Simulation results show that the MCIC estimator outperforms least square estimator and has slightly inferior performance as compared with linear minimum mean square error (LMMSE) estimator. Moreover the MCIC estimator has the advantage of not requiring any information about the channel statistics like auto-correlation of channels. Keywords- channel estimation, LS code, MCIC, MIMO-OFDM systems, multipath channel.
I. INTRODUCTION High data-rate transmission techniques in mobile communication systems have gained considerable interest in recent years. For instance, there are Multiple-Input MultipleOutput (MIMO) systems, Orthogonal Frequency Division Multiplexing (OFDM) systems, and MIMO-OFDM systems. MIMO systems have shown their potential by providing high data-rate and diversity gain obtained through spatial multiplexing system commonly referred to as Bell Lab Layered Space-Time (BLAST) and Space-Time Block Coding (STBC) [1] respectively.
applied directly for channel estimation because signals from other transmission antennas act as interference. LS codes [5], one of the smart codes based on Golay complementary pairs (GCP) [4], have the zero correlation zone (ZCZ) or Interference Free Window (IFW). It is proposed that the multiple antennas and multipath interference can be significantly reduced by utilizing the ZCZ or IFW. In this paper, we propose the MCIC for the channel estimation of MIMO-OFDM systems by using LS codes as training sequences. Since LS codes have perfect autocorrelation and cross-correlation functions within certain vicinity of the zero shifts, the MCIC can substantially reduce multipath channel interference by using these correlation properties of LS codes. This paper is organized as follows. In section II, a system model of MIMO-OFDM systems is described. In section III, properties of LS codes are described. In section IV, the MCIC is described. Finally, Section V and VI give simulation results and conclusions respectively. II.
MIMO-OFDM systems with k synchronous Alamouti system [2] blocks at the transmitters are shown in Fig. 1. The output over two consecutive periods at the j th receive antenna, Y1, j (k) and Y2, j (k) , for k =1,2,…,N, can be written as; LScodes generator
Since OFDM is commonly used for high data-rate wireless communication due to its inherent error susceptibility, the combination of OFDM with MIMO systems is regarded as one of the most promising schemes for the next generation high speed wireless communication. However, it is impossible to know perfect channel characteristics at the receiver in terms of realistic aspect. Thus, reliable channel estimations are needed to take the advantages of the MIMO-OFDM systems. For the OFDM systems with Single-Input Single Output (SISO), various channel estimation techniques such as least square [3] and linear minimum mean square error (LMMSE) [3] estimators was introduced to estimate channel. In MIMOOFDM systems, on the other hand, these techniques can not be
1550-2252/$25.00 ©2007 IEEE
SYSTEM MODEL
Space-Time Block Encoder ∗ −x2,1 x1,1 ∗ x x 1,1 2,1
IFFT IFFT
IFFT IFFT
SK MBP od Mod.
Space-Time Block Encoder
−x2,∗k x1,k ∗ x1,k x2,k
C H A N N E L
FFT FFT
FFT FFT
Space-Time BlockDecoder
BPSK Dem od Mod.
FFT FFT
IFFT IFFT
IFFT IFFT
Channel Estimation
Figure 1. A simplified block diagram for a MIMO-OFDM system with LS codes generator
1465
k
∑ {H
i 1, j
}
( k ) X 1, i ( k ) + H 2i , j ( k ) X 2 , i ( k ) + W1, j ( k ),
i =1
Y2 , j ( k ) =
k
∑ {H
i 2, j
Auto-correlation
(1)
Correlation value
Y1, j ( k ) =
}
( k ) X 1,∗ i ( k ) − H 1,i j ( k ) X 2∗, i ( k ) + W 2 , j ( k ),
i =1
where X1,i(k), X2,i (k) are transmitted from the i block simultaneously during the first symbol period, − X ∗2,i ( k ), X 1,∗i (k ) are transmitted from the i th block during the second symbol period, H1,i j and H 2,i j denote the channel frequency responses of the i th transmission block and the j th receive antenna respectively, W1, j ( k ) and W2, j ( k ) represent the frequency response of Additive White Gaussian Noise (AWGN). We also define following equations. th
H i (k ) H 2,i j (k ) T Wj (k ) = W1, j (k ), W2,∗ j (k ) , Hij (k ) = i1, j∗ . i ∗ H2, j (k ) − H1, j (k )
III.
- 2 0
0
(3)
(4)
A. Properties of uncorrelated LS codes in interference free window LS codes are defined as the combination of C and S subsequences, a Golay complementary pair, with zeros inserted to avoid overlapping between the two subsequences. If (C0 , S0 ) and (C1 , S1 ) are both Golay pairs of LS codes, we say that two LS codes are a mate. Fig. 2 shows a simple example of LS codes whose components are equal to ±1 or 0 . As a result of inserted zeros, LS codes have features that aperiodic autocorrelation sidelobes and cross-correlations are zero within IFW W0 . Fig. 3 shows auto-correlation and cross-correlation properties of a mate with length N = 64 and IFW W0 = 32 .
0
0
W0
- 0 . 5 - 1
- 2
0
IFW
0
Code length
B. Properties of correlated LS codes in interference free windows The main purpose of zeros insertion of the LS codes is to avoid the sequences C0 and C1 overlapping with the sequences S0 and S1 . Note that it is also necessary to insert enough guard intervals between sequences with length longer than the maximum delay of the multipath channel. Under the assumption that W0 / 2 is the maximum delay of the multipath channel without decreasing the energy efficiency suggested in [5], we define that the length of IFW W0 is equal to N p − 1 , where N p is length of a complementary pair. As shown in Fig. 4, using Golay codes of N p =16 which are generated by using Hadamard matrix [5], the properties of possible IFWs are as follows; c0cc 1 0 − c1 s0s1s0 − s1 c0c1 s0s1 c0c1 − c0c1 s0s1 − s0s1 c − c s − s → c −c c c s0 − s1s0s1 0 1 0 1 0 1 0 1 c −c − c −c s − s − s − s c0 s0 0 1 0 1 0 1 0 1 c s → cc 1 1 1 0c1 − c0 s1s0s1 − s0 c c s s cc − cc s s − s s 1 0 1 0 1 0 1 0 1 0 1 0 → c1 − c0 s1 − s0 c1 −c0cc s1 − s0s1s0 1 0 c −c − c − c s − s − s − s 1 0 1 0 1 0 1 0
set 1. set 2.
mate, W0 IFW W0 IFW 2
mate
W0
4
set 3. mate
IFW
set 4.
mate
(a) Three possible IFW properties using complementary pairs of Golay codes generated by Hadamard matrix
S IF W
2
zeros
2
0 . 5
IF W
C
0
1
1
zero s
2
Figure 3. Correlation properties of a LS code mate
LOOSELY SYNCHRONOUS CODES
C
W 0IFW
(b) Cross-correlation property
Equation (3) can be reduced to the following form; Y(k ) = H (k )X(k ) + W (k ).
- 1
Cross-correlation
(2)
The received signals from all receive antennas at the k th tone over two consecutive OFDM block periods can be written in a matrix form as Y1 (k ) H11 (k ) H12 (k ) " H1k (k ) X1 (k ) W1 (k ) 1 k 2 Y2 (k ) = H2 (k ) H2 (k ) " H2 (k ) X2 (k ) + W2 (k ) . # # # # # # # 1 k 2 ( k ) ( k ) Y X W ( k ) ( k ) ( k ) H H " H k m (k ) m m m m
0 - 0 . 5
(a) Auto-correlation property
T
T
0 . 5
Code length
Correlation value
T
Yj (k ) = Y1, j (k ), Y (k ) , Xi (k ) = X1, i (k ), X 2, i (k ) , ∗ 2, j
1
S
1 4
Figure 2. Formation of LS codes
tim e
t im e
t im e IF W
(b) Three possible IFWs Figure 4. Three possible cross-correlation properties
1466
- 0
Correlation value
0
- 0
Correlation value
0
- 0
Correlation value
2 0
Ru1 , v1
1
. 5
W 0 IF W 2
- 6
0
- 4
0
- 2
0
. 5
0
Code length
2
0
4
0
6
0
W0
4
20
Rv1 , u1
1
-1
0
20
-1
R v1 , u2
1
-1
-20
0
W 0 IF W 2
-0.5 20
W 0 IF W 2
-1
-20
0
20
-20
0
20
-20
0
0
W 0 IFW 2
-0.5
20
-1
Rv 2 , u 2
1 0.5
0
W 0 IFW 2
-0.5 -1
R v2 , u1
1 0.5
0
0
-0.5
-20
0
W 0 IF W 2
-0.5 20
-1
-20
0
20
Code length - 6
0
- 4 0
- 2 0
0
C ode leng th
2 0
4 0
6
(a) Cross-correlation Ra ,b of each LS code in group 1, ( u1 , u2 , v1 , v2 )
0
0
W0
8 - 6
0
- 4 0
- 2
0
0
C ode length
2
0
4
0
Ru1 , − v1
1 0.5
0.5
0.5
0
0
0
W 0 IFW 2
-0.5 -1
-20
0
W0
20
-1
R − v1 , u1
1
- 1
0
0
C ode length
-20
0
20
-1
R − u2 , v 2
1 0.5 0
W 0 IFW 2
-0.5 -20
0
W0 IFW 2
-0.5 20
W 0 IFW 2 -20
0
-1
-20
0
20
-1
-20
0
0
W 0 IFW 2
-0.5 20
-1
Rv2 , − u2
1 0.5
0
W 0 IFW 2
-0.5
20
R v2 , u1
1 0.5
0
0 -0 .5 -1
R − v1 , − u2
1 0.5
0 .5
-20
0
W 0 IF W 2
-0 .5 20
-1
-2 0
0
20
Code length
16 0
W0 IFW 2
-0.5
R− u2 , − v1
1
Code length
0
- 2
Ru1 ,v 2
1
6 0
. 5
. 5
-20
0
W 0 IFW 2
-0.5
0.5
0.5
. 5
. 5
0
0
W 0 IFW 2
-0.5
Ru 2 , v 2
1 0.5
Code length
0
. 5
-20
R u 2 , v1
1 0.5
0
0
-1
Ru1 , v 2
1 0.5
0.5
-0.5
Correlation value
Correlation value
0
W0
Correlation value
- 0
. 5
Correlation value
Correlation value
0
1
0
2
(b) Cross-correlation
0
Ra ,b of each LS code in group 2, ( u1, −u2 , −v1, v2 )
Figure 6. Cross-correlation of each LS code in groups
Figure 5. Four weakly correlated cross-correlations
III.B.1) Set 1, 2, 3 and 4 are mates, respectively. III.B.2) If set 1 is initial mate, IFW between set 1 and 2 is W0 / 2 . III.B.3) If set 1 is initial mate, IFW between set 1 and 3 is W0 / 4 . III.B.4) If set 1 is initial mate, IFW between set 1 and 4 is W0 / 4 . III.B.5) If set 2, 3 or 4 are initial mates respectively, properties of IFWs among three other mates are the same as the set 1, the initial mate.
IV.A.4) W0 / 2 is assumed to be the maximum delay of the channel. IV.A.5) Guard interval of OFDM systems is about a quarter of an OFDM symbol duration. IV.A.6) Since the locations and values of correlations among four LS codes are known in advance, interference cancellation method can be used. IV.A.7) To estimate the channel, transmission antennas must transmit at least two different groups, group 1 and group 2.
For the mates having length N = 128 and IFW W0 = 64 , Fig. 5 shows four weakly correlated cross-correlations
Here we consider the MIMO-OFDM system as shown in Fig. 1. Suppose that four transmission antennas simultaneously transmit group 1 at a given symbol period. After transmitting group 1 from the first four antennas, another four antennas also transmit group 1, and repeat transmission of the group 1 for the rest of transmission antennas. After transmitting group 1 for all antennas, transmission antennas also transmit group 2 like the method of group 1 transmission for all antennas. At the receiver, we can estimate the channel with Double Space Time Transmit Diversity (DSTTD) [8] blocks. If we restrict ourselves to a DSTTD-OFDM system with four transmission antennas and two receive antennas, the time domain received signals at the first and second receive antenna can be represented respectively as;
IV.
LS CODES ASSISTED MULTIPATH CHANNEL INTERFERENCE CANCELLER
A. The MCIC using weakly correlated LS codes For the mates having length N = 128 and IFW W0 = 64 , Fig. 6 shows each cross-correlation Ra ,b of four weakly correlated LS codes in group 1, (u1 , u2 , v1 , v2 ) , and group 2, (u1 , −u2 , −v1 , v2 ) , respectively. Based on the following seven properties, we propose the MCIC for MIMO-OFDM systems using four LS codes as training sequences. IV.A.1) The length of IFW W0 is equal to N p − 1 , where N p is length of a complementary pair. IV.A.2) A mate of LS codes has maximum IFW W0 . IV.A.3) ( p1 , p2 ) and ( z1 , z2 ) are mates of LS codes respectively, and they are weakly correlated with IFW W0 / 2 .
1 1 2 y1,1 (t) = h1,1 (τ , t ) ∗ p1 (t ) + h2,1 (τ , t ) ∗ p2 (t) + h1,12 (τ , t ) ∗ z1(t ) + h2,1 (τ , t ) ∗ z2 (t ) + w1,1 (t )
(5)
1 2 2 (τ , t ) ∗ p1 (t ) + h12,2 (τ , t) ∗ p2 (t ) + h1,2 (τ , t ) ∗ z1(t) + h2,2 (τ , t ) ∗ z2 (t) + w1,2 (t) y1,2 (t) = h1,2
where “ ∗ ” denotes the convolution, ( p1 (t ), p2 (t )) and ( z1 (t ), z 2 (t )) are mates at time t respectively, each has length N , ( p1 (t ), p2 (t )) and ( z1 (t ), z2 (t )) are weakly correlated with IFW W0 / 2 , both h1,i j (τ , t ) and h2,i j (τ , t ) are time domain
1467
response of channel for i, j = 1,2 , w1,1 and w1,2 are AWGN. At the receiver, we can get the channel parameters by taking respective cross correlation between received signals and LS codes. The channel parameters can be written as follow; 1 1 h1,1 = Ry1,1 , p1 (m) = h1,1 + Rh1
2,1 ∗ p2 , p1
1 1,2
1 1,2
(m) + Rh2 ∗z , p (m) + Rh2 1,1
h = Ry1,2 , p1 (m) = h + Rh1
2,2 ∗ p2 , p1
1
(m) + Rh2
1,2 ∗ z1 , p1
1 1 h2,1 = Ry1,1 , p2 (m) = h2,1 + Rh1 ∗ p , p (m) + Rh2 1
1,1
2,1 ∗ z 2 , p1
1
2,1 ∗ z2 , p2
2
1 1 h2,2 = Ry1,2 , p2 (m) = h2,2 + Rh1
1,2 ∗ p1 , p2
2,2 ∗ z 2 , p1
1,1
2,2 ∗ z2 , p2
1,1
1
1
2,1
1,2 ∗ p1 , z1
2
2,2 ∗ z 2 , z1
h = Ry1,1 , z2 ( m) = h + Rh1 ∗ p , z (m) + Rh1 2 2,1
2 2,1
1,1
2 2 h2,2 = Ry1,2 , z2 (m) = h2,2 + Rh1
1
1
(m) + Rh2
2
1,2 ∗ p1 , z2
2,1 ∗ p2 , z2
(m) + Rh1
1
2
(m) + Rh2
1,2 ∗ z1 , p2
2,1
W0
(m) + Rw1,2 , p2 (m)
W0
4
2,2 ∗ p2 , z1
(m) + Rw1,2 , z1 (m)
1
1
2 1,i
1,i
1 1,i
1 2,1
1 1,1
, v1
4
IFW
1 ′′ h 2
1
2
(10)
2 2 2 2 2 2 ′1 + h2,1 ′1 + E7 , hˆ42 = hˆ1,2 ′1 + h2,2 ′1 + E8 hˆ32 = hˆ1,12 + hˆ2,1 = h1,12 + h2,1 − h1,1 + hˆ2,2 = h1,2 + h2,2 − h1,2
1,i
(9)
1,i
2
1 1 1 1 1 1 1 1 1 1 1 h1,11 = h1,1 − hˆ3′2 − hˆ1′′2, h1,2 = h1,2 − hˆ4′2 − hˆ2′ 2 , h2,1 = h2,1 + hˆ3′2 − hˆ1′′2 2 2 2 2 2 2 1 ˆ 2 1 ˆ 2 2 2 1 ˆ1 1 ˆ 1 2 2 1 ˆ 1 1 ˆ 1 1 1 h2,2 = h2,2 + h4′ − h2′ , h1,1 = h1,1 − h3′ − h1′′ , h1,2 = h1,2 − h4′ − h2′′ 2 2 2 2 2 2 1 1 1 1 2 2 2 2 h2,1 = h2,1 + hˆ3′1 − hˆ1′1, h2,2 = h2,2 + hˆ4′1 − hˆ2′1 2 2 2 2
(11)
Applying (8) to (9), we can get the second reestimated channels. We can also describe the second part of the MCIC as follow; 1 1 1 1 1 1 1 1 1 1 1 h1,11 = hˆ1,1 + h1′2 + h3′′2 , h1,2 = hˆ1,2 + h2′2 + h4′ 2 , h2,1 = hˆ2,1 + h1′2 − h3′′2 2 2 2 2 2 2 1 2 1 2 2 ˆ2 1 1 1 1 2 ˆ2 1 1 1 1 1 1 ˆ ′ ′ ′ ′ ′ h2,2 = h2,2 + h2 − h4 , h1,1 = h1,1 + h1 + h3 , h1,2 = h1,2 + h2 + h4′′ 2 2 2 2 2 2 1 1 1 1 2 ˆ2 1 1 1 1 2 2 ˆ h2,1 = h2,1 + h1′ − h3′ , h2,2 = h2,2 + h2′ − h4′ 2 2 2 2
(8)
transmit (u1 , −u2 , −v1 , v2 ) , group 2, (6) can be represented by following methods;
1
(7), we can get the first reestimated channels. We can describe the first part of the MCIC as follow;
(7)
2
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 ′ + h2,1 ′ − h1,1 ′ − h2,1 ′′ + D1, hˆ1,2 ′ + h2,2 ′ − h1,2 ′ − h2,2 ′′ + D2 hˆ1,1 = h1,1 − h1,1 = h1,2 − h1,2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 ′ + h2,1 ′ + h1,1 ′ + h2,1 ′ + D3, hˆ2,2 ′ + h2,2 ′ + h1,2 ′ + h2,2 ′′ + D4 = h2,1 − h1,1 = h2,2 − h1,2 hˆ2,1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 ′ + h2,1 ′ − h1,1′ − h2,1 ′ + D5, hˆ1,2 ′ + h2,2 ′ − h1,2 ′ − h2,2 ′ + D6 hˆ1,12 = h1,12 − h1,1 = h1,2 − h1,2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 ′ + h2,2 ′′ + D8 ′ + h2,1 ′ + h1,1′ + h2,1 ′ + D7, hˆ2,2 ′ + h2,2 ′ + h1,2 hˆ2,1 = h2,1 − h1,1 = h2,2 − h1,2 2 2 2 2 2 2 2 2
1
for i = 1,2, k = 1,..., 4, Ek = Rw2 ,u ( m) ± Rw2 , −u ( m) 1,i 1 1,i 2 Ek = Rw2 ,− v (m) ± Rw2 , v ( m) for i = 1,2, k = 5,...,8 . Applying (10) to
for i = 1,2, k = 1,...,4, Bk = Rw1 ,u ( m) ± Rw1 ,u ( m) 1,i 1 1,i 2 (m) ± Rw1 ,v (m) for i = 1,2, k = 5,...,8. When four antennas 1,i
W0
2 2 2 2 2 2 ′′1 − h2,1 ′′1 + E5, hˆ22 = hˆ1,2 ′ 1 − h2,2 ′ 1 + E6 = h1,12 − h2,1 − h1,1 − hˆ2,2 = h1,2 − h2,2 − h1,2 hˆ12 = hˆ1,12 − hˆ2,1
2 2 2 2 2 2 ′′1 − h2,1 ′ 1 + B7 , h42 = h1,2 ′ 1 − h2,2 ′′1 + B8 h32 = h1,12 + h2,1 = h1,12 + h2,1 + h1,1 + h2,2 = h1,2 + h2,2 + h1,2
1,i
IFW
delay
2
1 1 1 1 1 1 1 1 ′2 + h2,1 ′2 + E3 , hˆ41 = hˆ1,2 ′2 + h2,2 ′2 + E4 + hˆ2,1 = h1,1 + h2,1 − h1,1 + hˆ2,2 = h1,2 + h2,2 − h1,2 hˆ31 = hˆ1,1
1 2,1
2 2 2 2 2 2 ′1 + h2,1 ′1 + B5, h22 = h1,2 ′1 + h2,2 ′1 + B6 h12 = h1,12 − h2,1 = h1,12 − h2,1 + h1,1 − h2,2 = h1,2 − h2,2 + h1,2
Bk = Rw1
4
W0
where
j
1 1 1 1 1 1 1 1 ′′2 − h2,1 ′ 2 + B3 , h41 = h1,2 ′ 2 − h2,2 ′′2 + B4 h31 = h1,1 + h2,1 = h1,1 + h2,1 + h1,1 + h2,2 = h1,2 + h2,2 + h1,2
where
W0
4
1 1 1 1 1 1 1 1 ′′2 − h2,1 ′ 2 + E1, hˆ21 = hˆ1,2 ′′2 − h2,2 ′′2 + E2 hˆ11 = hˆ1,1 − hˆ2,1 = h1,1 − h2,1 − h1,1 − hˆ2,2 = h1,2 − h2,2 − h1,2
1 1 1 1 ′2 + h2,1 ′2 + B1, h21 = h1,2 ′2 + h2,2 ′2 + B2 h = h − h = h − h + h1,1 − h2,2 = h1,2 − h2,2 + h1,2 1 1,1
W0
0
equations;
where h1,′i j and h2,′i j are part of the each channel from W0 / 4 to W0 / 2 delay and have bad effects on part of the h1,i j and h2,i j from zero to W0 / 4 delay for i, j = 1,2 , h1,′′ij and h2,′′i j are part of the each channel from zero to W0 / 4 delay and have bad effects on part of the h1,i j and h2,i j from W0 / 4 to W0 / 2 delay for i, j = 1,2 , k = 1,...,4, for Ak = Rw ,u (m) for i, j = 1, 2, Ak = Rw ,v (m) i, j = 1, 2, k = 5,...,8 . Fig. 7 shows the meanings of h′ and h′′ . We also define following equations; 1 1
1 ′ h 2
IFW
2 1,i
2
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 ′ + h2,1 ′ + h1,1 ′′ − h2,1 ′′ + A1, h1,2 ′ + h2,2 ′ + h1,2 ′′ − h2,2 ′′ + A2 h1,11 = h1,1 + h1,1 = h1,2 + h1,2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 ′ + h1,1 ′ − h2,1 ′′ + A3 , h2,2 ′ − h2,2 ′ + h1,2 ′′ − h2,2 ′′ + A4 h2,1 = h2,1 − h1,1′2 − h2,1 = h2,2 − h1,2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 ′ + h2,1 ′ + h1,1′ − h2,1 ′ + A5 , h1,2 ′ + h2,2 ′ + h1,2 ′′ − h2,2 ′ + A6 h1,12 = h1,12 + h1,1 = h1,2 + h1,2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 ′ − h2,2 ′′ + A8 ′ + h1,1′ − h2,1 ′ + A7 , h2,2 ′ − h2,2 ′ + h1,2 h2,1 = h2,1 − h1,1′1 − h2,1 = h2,2 − h1,2 2 2 2 2 2 2 2 2
j
4
4
( m) for i = 1,2, k = 1,2, Dk = Rw2 , −u ( m) for ,u 1,i 2 i = 1,2, k = 3,4, for i = 1,2, k = 5,6, Dk = Rw ,− v ( m) Dk = Rw2 , v (m) for i = 1,2, k = 7,8 . We also define following
where Dk = Rw
2
(m) + Rh2 ∗ z , z (m) + Rw1,2 , z2 (m) 1,2
W0
W0
delay
Figure 7. The meanings of h′ and h′′
where Ra, a (m) is the aperiodic auto-correlation of a and a , Ra, b (m) is the aperiodic cross-correlation of a and b . Because of the two properties, IV.A.3) and IV.A.4), we can not clearly eliminate the multipath channel interference. When four antennas transmit (u1 , u2 , v1 , v2 ) , group 1, (6) can be represented by following methods;
1 1,i
IFW
2
(6)
(m) + Rh2 ∗ z , z (m) + Rw1,1 , z2 (m) 1,1
W0
4
1 ′′ h 2
1 h′ 2
1
(m) + Rh1
2,2 ∗ p2 , z2
W0
( m) + Rw1,1 , z1 (m) ∗p ,z 2
Channel value
h
0
4
(m) + Rw1,2 , p1 (m)
(m) + Rh2 ∗ z , p (m) + Rw1,1 , p2 (m)
2 2 h1,1 = Ry1,1 , z1 (m) = h1,1 + Rh1 ∗ p , z ( m) + Rh2 ∗ z , z (m) + Rh1 2 2 h1,2 = Ry1,2 , z1 ( m) = h1,2 + Rh1
(m) + Rw1,1 , p1 (m)
(m) + Rh2
(m) + Rh2
Channel value
V.
(12)
SIMULATION RESULTS
We consider a DSTTD-OFDM system with four transmission and two receive antennas. The parameters of the simulation are as follow. The entire channel bandwidth W , 20MHz, is divided into 128 subchannels N s . A BPSK signal constellation is used. The OFDM symbol duration Ts is 8 µ s . An additional 1.6 µ s guard interval Tg is used to provide protection from intersymbol interference. Two channel models are used, namely, Channel A with τ rms = 50 ns, and Channel B with τ rms = 150 ns. Both channels assume an exponentially decaying power delay profile with 11multipaths and 31 multipaths respectively, which are independently generated using Jake’s model. We have used Monte-Carlo simulations to
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generate channel auto-correlation matrix for LMMSE estimation. To suppress the error propagation, training symbols are periodically inserted in the data stream after every 20 OFDM blocks. Table 1 summarizes both training symbols per transmission antenna and total training periods of the channel estimators. The performance of the system is measured in terms of BER versus Eb / N0 for a minimum mean square error equalizer based on the estimated channel. Fig. 8 (a) and (b) show the comparative BER performance for Least Square estimation, LMMSE estimation, and the MCIC estimation. From Fig. 8 (a) at the target BER of 10−3 , we can see a about 2.3dB gain and 0.2dB loss in Eb / N0 for the MCIC estimation over Least Square and LMMSE estimations respectively. From Fig. 8 (b) at the target BER of 10−3 , we can also see a about 2dB gain and 0.2dB loss in Eb / N0 for the MCIC estimation over Least Square and LMMSE estimations respectively. From Fig. 8 (a) and (b), The MCIC estimator performs within 1 dB limit of the system with known channel. Fig. 8 (a) and (b) show that the MCIC estimator substantially reduces multipath channel interference in the channel with relatively small or large delay spread.
TABLE I.
TRAINING SYMBOLS PER TRANSMISSION ANTENNA AND TOTAL TRAINING PERIODS
Estimators Training symbols per transmission antenna
Least square
LMMSE
MCIC
1 Ns
1 Ns
4 Ns
Total training periods
4 Ts
4 Ts
4 Ts
VI. CONCLUSIONS In this paper, LS codes assisted MCIC is proposed for MIMO-OFDM systems. The proposed MCIC uses both weakly correlated LS codes and interference cancellation method in the presence of channel with relatively small or large delay spread. The simulation results show that the MCIC estimator outperforms least square estimator, although it has slightly inferior performance as compared with LMMSE estimator. Moreover the MCIC estimator has the advantage of not requiring any information about the channel statistics like autocorrelation of channels. As a result, the proposed MCIC estimator is more efficient for reducing multipath channel interference than least square and LMMSE estimators in the channel with relatively small or large delay spread. REFERENCES
10
10
[1]
M MSE linear detec tion
0
Least Square-est., L=11 MCIC-es t., L=11 LMMSE-es t., L=11 perfect c hannel, L=11
[2]
-1
BER
[3] 10
10
-2
[4]
-3
[5] 10
-4
0
2
4
6
8
10 12 Eb/No[dB]
14
16
18
20
[6]
(a) The MCIC, channel A 10
Bit Error Rate
10
10
10
10
[7]
0
Least S quare-est., L=31 MCIC-es t., L=31 LMMS E-es t., L=31 perfect c hannel, L=31
-1
[8]
-2
[9] -3
-4
0
2
4
6
8
10 12 E b/No[dB]
14
16
18
20
(b) The MCIC, channel B Figure 8. BER performance of Proposed MCIC for 4 × 2 DSTTD-OFDM system
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