Error Performance Analysis of Multiuser CDMA ... - Semantic Scholar
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Error Performance Analysis of Multiuser CDMA Systems with Space-time Coding in Rician Fading Channel Dingli Yang 1, Qiuchan Bai 1, Yulin Zhang 1, Rendong Ji 1, 2, Yazhou Li 1, and Yudong Yang 1 1. Faculty of Electronic and Electrical Engineering, Huaiyin Institute of Technology, Huai‟an 223003, China 2. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Email: [email protected], [email protected], [email protected] , [email protected], [email protected], [email protected]
Abstract—In this paper, the error performance of multiuser CDMA system with space-time coding is studied in Rician fading channel, and the corresponding bit error rate (BER) and symbol error rate (SER) analysis are presented. Based on the performance analysis, a simple and effective multiuser receiver scheme is developed. The scheme has linear decoding complexity when it compares to the existing scheme. Based on the performance analysis, and using mathematical manipulation, the BER and SER of multiuser space-time coded CDMA system are derived, respectively. As a result, accurate closed-form expressions of BER and SER are respectively obtained. With these expressions, the performance of multiuser CDMA system with space-time coding can be evaluated effectively. Computer simulation shows that the developed receiver scheme has almost the same performance as the existing scheme, and the theoretical BER and SER can match the corresponding simulation results well. Index Terms—Space-Time Coding; Code Division Multiple Access (CDMA); Rician Fading; Low Complexity; Bit Error Rate; Symbol Error Rate
I.
INTRODUCTION
Recently, multiple-input and multiple-output (MIMO) technique is well known to offer improvements in bandwidth efficiency along with diversity and coding benefits over wireless fading channels [1-4]. Especially space-time block coding (STBC) in MIMO systems can provide effective diversity for combating fading effect [510], and has received much interests. However, the conventional STBC scheme is only used for single-user environment, and thus it will be not suitable for multiuser scenario in practice. Hence, it is necessary to extend the STBC scheme into multiuser CDMA scenario for practical purposes. Based on different multiuser spacetime coded system models, the receiver schemes are designed in references [11-18] and references therein. In reference [11], the bit error rate (BER) analysis is provided for space-time coded CDMA system with the conventional matched filter receiver, but the analysis method and system model are applicable only to the BPSK modulation and downlink, and does not provide the closed-form BER expression for Rician fading © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.12.3462-3469
channel. The given symbol error rate (SER) expression for space-time codes needs summing from 0 to infinity, whereas the infinity is difficult to decide. In reference [12], a minimum variance linear receiver scheme for multiuser MIMO system is proposed, but the system needs to design and optimize the weighted matrix to suppress the multiuser interference (MUI). As a result, the computational complexity is very complicated. In reference [13], the performance of multiuser CDMA system with transmit diversity is studied, but the analysis is limited in BPSK modulation and two transmit antennas. For multicarrier-CDMA (MC-CDMA) system, a least mean-square based adaptive receiver scheme is proposed for MC-CDMA with Alamouti‟s STBC [14], but the scheme is limited in two transmit antennas and one receive antenna. The performance of space-time coded MC-CDMA system is analyzed in Nakagami fading channel [15], but the analysis is limited in Alamouti‟s STBC and BPSK modulation. Blind space-time multiuser detection schemes are presented in reference [16] for MC-CDMA system. The performance of space-time coded MIMO system with cross-layer design is analyzed in spatially-correlated and Keyhole Nakagami fading channel [17], but the analytical method is applicable only to single user system. Reference [18] gives the effective combination of CDMA system and different space-time codes, and the developed decorrelative receiver scheme can decouple the detection of different users, but the decoding complexity is exponential for each user, which will not benefit practical application. Moreover, the above schemes basically do not provide the error performance analysis, and are limited in Rayleigh fading channel, whereas in practice, the system may experience the Rician fading due to the direct-path propagation. Due to the reason above, the error performance of space-time coded CDMA system in Rician fading channel is studied. Firstly, a multiuser space-time coded CDMA system model is presented, and then a lowcomplexity multiuser receiver scheme is proposed by utilizing maximum ratio combining (MRC) method and orthogonality of space-time coding. The presented spacetime coded CDMA system can achieve effective MUI suppression by using multiuser detection method. After
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decorrelating, each user has linear rather than exponential decoding complexity. According to the performance analysis, and using mathematical calculation, the average BER and SER of the system are derived in detail. As a result, accurate and approximate closed-form expressions of BER and SER are attained for space-time coded CDMA with orthogonal and quasi-orthogonal spreading code, respectively. Simulation results show that the proposed low-complexity scheme can obtain almost the same performance as the existing scheme. Theoretical BER and SER will be in good agreement with the corresponding simulations. Thus, the effectiveness of the theoretical formulae is verified. Note: the superscripts ()T , () , () H are used to stand for the transpose, complex conjugate, and Hermitian transpose, respectively. E{} and I N denote the statistical expectation and identity matrix, respectively. vec( ?) stand for matrix vectorization operator. II.
(2)
where D u is a N T space-time coding matrix of user u, is spreading code, and T 1 Cu (q) Cu [ Cu (1) ,, Cu (Q)] corresponds to T normalized spreading codes of length Q used to spread D u for user u (u 1,2,U ) , here conventional orthogonal WalshHadamard code and quasi-orthogonal Gold code in CDMA system are considered. These different spreading codes for different users are employed as well. Based on the analytical method in [18], obtain the baseband received signal at qth ( q 1,2,,Q ) chip period is obtained as follows U
y(q) u H uVu (q) w(q) , q 1,2,,Q (3) u 1
the channel gain from the nth transmit antenna to the lth receive antenna, which is assumed to be constant over a frame of T symbols and varied from frame to frame. For Rician fading channels, the { h u,ln } are modeled as independent complex Gaussian random variables with respective means mI and mQ for the real and imaginary parts and variance of 0.5 per dimension [11, 19], and
where Hu [h u ]l,n is L N channel matrix of user u . w(q), q 1,,Q is L1 noise vector, whose element {w l (q), l 1,, L, q 1,,Q } are independent, identically distributed (i.i.d ) complex Gaussian random variables with zero-mean and unit-variance. Nu denotes the average signal-to-noise ratio per receive antenna for user u at the receiver during the transmission of spacetime coding matrix Du (which corresponds to Q chip periods), this SNR adopts the definition similar to Ref.[18] for comparison consistency. Substituting (2) into (3), the received signal at qth chip period can be expressed as U
y(q) u H u Du Cu (q) w(q) , q 1,2,,Q (4) u 1
L
| hu ,ln |2
obeys
a
noncentral
chi-square
n 1 l 1
distribution with 2NL degrees of freedom. According to reference [19] and changing variable, the probability density function (pdf) of can be obtained by f ( ) ( / x2 )( NL 1)/ 2 (2 2 ) 1 e( x
2
)/(2 2 )
I NL 1 ( x 2 / 2 ), 0
0.5 , and I v (x) is the vth-order modified Bessel function of the first kind [20-21]. 2
LOW-COMPLEXITY MULTIUSER RECEIVER
In this section, A low-complexity multiuser receiver design for multiuser space-time coded CDMA system is given. For multiuser CDMA system with space-time coding, the block length of space-time code is set equal to Q chip periods. Then according to reference [18], the
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In order to express (4) more compactly, the matrices is defined as: Su u H u Du , S [S1 , S2 ,,SU ] ,
C [C1T ,,CTU ]T Cu [ Cu (1),, Cu (Q)] , , Y [y(1) , ,y (Q)] , and W [w(1) ,, w(Q)] . Thus, (4) is changed to U
(1)
where x2 NL(mI2 mQ2 ) is the noncentrality parameter,
III.
Vu (q) Du Cu (q)
SYSTEM MODEL
In this paper, a synchronous CDMA communication system with N transmit antennas and L receive antennas and U active users that operates over a flat and quasi-static Rician channel is considered. The multiuser CDMA system employs the space-time block coding schemes (such as conventional full-diversity G 2 , G 3 , H 3 , G 4 , H 4 code [5,8], full-rate X code [7]) to transmit the data. For each user u , the channel gain h u,ln denotes
N
transmitted signal matrix of user u at qth ( q 1,2,,Q ) chip period is
Y Su Cu W SC W
(5)
u 1
Then according to reference [22], obtain the ML estimate of S conditioned on {H u and {Du is obtained as Sˆ YC H (CC H )1 S WC H (CC H )1
(6)
Here, the Moore-Penrose inverse matrix CH (CCH )-1 can be expressed as a multiuser decorrelator [18, 22], and thus the ML estimate Sˆ is an effective output of the decorrelator with the input being the received data Y . By this decorrelator, the multiuser interference is cancelled, and the detection of different users is decoupled. Based
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on the block structure of S , the ML estimate Sˆu of Su can be easily achieved. While for user u , all data information on the transmitted code matrix Du is contained in Su . Hence, the code matrix and corresponding information symbols via the achieved Sˆ u
is evaluated. According to the definition of Su , Assuming that Sˆ [sˆT , sˆT ,..., sˆT ]T , where sˆ is a 1 T row u
u ,1
u ,2
u, L
u ,l
vector. Thus when G 3 code scheme is employed, T=8. Considering that Su has the receiver signal form similar to the conventional STBC in single user scenario [7-8], the simple decoding scheme for the space-time coded CDMA system with G 3 code after performing multiuser decorrelation is obtained by utilizing the MRC method and complex orthogonality of STBC, i.e., dˆu1 arg
L
min || (sˆ
u , l 1 u , l1
l 1
du 1
h
sˆu ,l 2 hu,l 2 sˆu ,l 3 hu,l 3 3
sˆu ,l 5 hu,l1 sˆu,l 6 hu ,l 2 sˆu,l 7 hu ,l 3 2 u | hu ,ln |2 du1 ) ||2 n 1
dˆu 2 arg
L
min || (sˆ l 1
du 2
u , l1 u , l 2
h
u ,l 2 u ,l1
sˆ
h
sˆu ,l 4 hu,l 3 sˆu ,l 3 hu,l 3 3
sˆu,l 5 hu ,l1 sˆu,l 6 hu ,l 2 sˆu,l 7 hu ,l 3 2 u | hu ,ln |2 du 2 ) ||2 n 1
dˆu 3 arg
L
min || (sˆ l 1
du 3
u ,l1 u ,l 3
h
sˆu ,l 3 hu,l1 sˆu ,l 4 hu,l 2 sˆu ,l 5 hu,l 3
dˆu 4 arg
L
min || (sˆu ,l 3 hu,l 3 sˆu,l 4 hu ,l 2 ) ||2
(9)
l 1
du 4
From (7) and (8) as well as (9), it is observed that the developed decoding scheme has linear complexity. For Ref.[18], its receiver decoding schemes with coherent detection (i.e. (44) and (45) in [18]) are shown as follows: 1) For general spreading codes:
Dˆ u arg min{vec H (Sˆu u H u Du ) k1 {du ,1 ,..., d u , p }
vec(Sˆu u H u Du )}
(10)
2) For orthogonal spreading codes: Dˆ u arg min || Sˆu u H u Du ||2F
(11)
{du ,1 ,..., d u , p }
From the above two equations, it can be seen that the decoding scheme in [18] has exponential complexity. Namely, if is a constellation consists of M symbols, the search times that need to obtain the transmitted P symbols is M P . Thus, when M and P become larger, the complexity will be much higher. As a result, the implementation complexity of the system will be increased significantly. While for our scheme, the needed search times are MP only. Based on this, the complexity comparison between the developed scheme and the existing scheme [18] in Table.1 is given. From Table 1, it can be seen that the proposed scheme has lower complexity than the existing scheme.
3
sˆu,l 7 hu ,l1 sˆu,l 8 hu ,l 2 2 u | hu ,ln |2 du 3 ) ||2
TABLE I.
n 1
dˆu 4 arg
L
min || (sˆ l 1
du 4
u ,l 3 u ,l 2
h
sˆu ,l 2 hu,l 3 sˆu ,l 4 hu,l1 sˆu ,l 6 hu,l 3
3
sˆu,l 7 hu ,l 2 sˆu,l 8 hu ,l1 2 u | hu ,ln |2 du 4 ) ||2
(7)
Scheme Existing scheme Developed scheme
COMPARISON OF COMPLEXITY QPSK (M=4, P=2) 16 8
8PSK (M=8, P=2) 64 16
16QAM (M=16, P=3) 4096 48
16QAM (M=16, P=4) 65536 64
n 1
When G2 code [5] is used, T=2, Sˆu [sˆuT,1 , sˆuT,2 ]T , the corresponding simple decoding scheme can be given by dˆu1 arg
L
min || (sˆu ,l1hu,l1 sˆu,l 2 hu ,l 2 ) ||2
du 1
dˆu 2 arg
l 1
L
min || (sˆu ,l1hu,l 2 sˆu,l 2 hu ,l1 ) ||2
(8)
l 1
du 2
Besides, when X code [7] is employed, T=4, ˆ Su [sˆuT,1 , sˆuT,2 , sˆuT,3 , sˆuT,4 ]T , the simple decoding scheme can be given by can be attained as dˆu1 arg
L
min || (sˆ
du 1
dˆu 2 arg
h
u ,l 2 u ,l 2
sˆ
h
2
min || (sˆu ,l1hu,l 2 sˆu,l 2 hu ,l1 ) ||2 l 1 L
min || (sˆu ,l 3 hu,l 2 sˆu,l 4 hu ,l 3 ) ||2
du 3
l 1
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SYSTEM PERFORMANCE ANALYSIS IN RICIAN FADING CHANNEL
In this section, the error performance analysis of spacetime coded CDMA system in Rician fading channel is given, and the closed-form expressions of average BER and SER is derived. Let W WC H (CC H )1 and Wu be the L T submatrix of W consisting of the columns starting from (u-1)T 1 to uT , then W [W ,...,W ] . According to 1
U
reference [18], the covariance matrix of vec( W ) is expressed as E{vec(W )vec H (W )} [(CC H ) 1 ]T I L
) ||
L
du 2
dˆu 3 arg
l 1
u , l1 u , l1
IV.
With Eq.(12), the calculated as follows:
matrix
VW E{W H W H }
VW (CC H )1
(12) is (13)
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Let C be (CC H )-1 , then with (12) and (13), the following equations can be obtained.
VWu E{WuH Wu } C(u 1)T 1:uT ,(u 1)T 1:uT , u 1,,U
(14)
When the orthogonal Walsh-Hadamard codes are used for spreading codes, the matrix C will be identity matrix I TU , and accordingly, VWu is also identity matrix (i.e. I T ). While the quasi-orthogonal Gold codes are employed for spreading codes, C will be a symmetric Toeplitz matrix with first row being [a, u, u,...u] , where (UT-1) is included. Thus, VWu is also a symmetric Toeplitz matrix with first row being [a, u, u,...u] , where (T -1) is included. According to the above analysis, the elements of W are complex Gaussians variables with mean zero. So the elements {wu ,lt } of Wu (u 1,, U) are also complex Gaussians variables with mean zero and variance unit for orthogonal spreading code case, while in the case of quasi-orthogonal spreading code, their covariance is a or . Namely, E{| wu ,lt |2 } a and E{wu ,lt wu,l t } , l l or t t . Using Eq.(6) and the definition of Su , the effective output of the decorrelator of user u can be written as: Sˆu Su Wu u Hu Du Wu , u 1,2,, U .
Pb, q ( ) j erfc( k j )
where M is the constellation size, erfc( ) is the complementary error function, j and k j are constants which depend on M , and the values of the constant sets { j , k j } for MQAM can be found in [19]. Besides, the high-accuracy approximate BER for MPSK with Gray coding over AWGN channel is given as [23] 2
Pb, p ( ) erfc( k j )
where the { j , k j } are given by { (1/ log 2M ,sin 2 ( / M )) , (1/ log2M ,sin 2 (3 / M )) } for MPSK. By using Eqs.(17)-(20), the average BER for multiuser space-time coded CDMA system with MQAM or MPSK modulation is evaluated as follows
Pb j erfc( k j ) f ( )d j G(k j ) (21)
G( )
0
[1 v / Q1 ( 2 NLK , 2 R / u ) 1 2 e a d ] 0
K (mI2 mQ2 ) / (2 2 )
can
be
obtained as f ( ) (
e NLK R / u I NL 1 (2
R
u
)( NL 1)/ 2 (
RNLK
u
0
Using (1) and changing variable, the pdf of in (16)
NLK
( n 1)/ 2 e ( R /
u
a )
e NLK
NL 1
( n 1
I n (2
R ) u NLK
RNLK
u
n2
)d g1 g2
where the equality Qm (a, b) Q1 (a, b) e ( a
2
b2 )/ 2
m 1
(b / a) I (ab) i
i 1
(17)
g1 1 2 / Q1 ( 2 NLK , R / (au )t )et / 2 dt 2
0
2Q1 (v, w) [1 u a / (u a R)]
The cumulative distribution function (cdf) of can be expressed as
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is
i
F ( ) 1 QNL ( 2 NLK , 2R / u ) ,
(23)
utilized. Using change of variables and the results of Appendix I in [24], g1 in Eq.(23) can be expressed as
)( NL 1)/ 2
), 0
(22)
G(a) 1 a / QNL ( 2 NLK , 2R / u ) 1 2 e a d
l 1 n 1
f ( ) erfc( a )d , 0 .
Substituting (17) and (18) into (22) gives
a
(16)
0
N
u | hu ,ln |2 / R u ||
factor
j
where G() denotes the integration in (21) defined as
N
Rician
0
j
l 1 n 1
with
(20)
j 1
u | hu ,ln |2 / R u || Hu ||2F / R u / R L
(19)
j
(15)
A. Orthogonal Walsh-Hadamard Code Case When the orthogonal Walsh-Hadamard code is used for spreading code, the covariance of Wu will be identity matrix, and its elements are i.i.d . complex Gaussian variables with mean zero and variance unit. Based on this, utilizing the orthogonality of space-time block coding, the effective SNR for user u at the receiver is given by L
where Qm ( , ) is the generalized Marcum Q-function [19, 21]. With Eq.(17), the pdf of for Rayleigh fading channel can be obtained by setting K 0 . According to Refs.[19-20, 23], the BER of coherent Mary QAM (MQAM) with Gray coding over an Additive White Gaussian Noise (AWGN) channel is given as
(18)
exp[(v w2 ) / 2]I 0 (vw) 2
where
(24)
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v w
NLK [ R / 2 u a u a( R u a)]
Equation (29) is accurate closed-form expressions of the average SER of multiuser space-time coded CDMA with MQAM in Rician fading channels.
,
R u a NLK [ R / 2 u a u a( R u a)] R u a
.
According to Eq.(23), g 2 is written as: a
g2
0
NL 1
n 1
( n 1)/ 2 e ( R /
u
a )
e NLK
NL 1 n 1
RNLK
I n (2
R
( NLK ) u
n/ 2
u
)d
exp( NLK )
RLKN 1 (n 1/ 2) R n (au )1/ 2 F (n ;n 1; ) (25) n 1/ 2 1 au R 2 (n 1) (au R)
The above derivation utilizes Eq.(6.643), Eq.(9.220) and Eq.( 9.215) in reference [21] . With Eqs.(24) and (25), G(a) in (21) can be expressed as G(a) 2Q1 (v, w) [1 au / (au R)] exp[(v 2 w2 ) / 2]I 0 (vw) NL 1
n 1
exp( NLK )
RLKN 1 (n 1/ 2) R (au ) F (n ;n 1; ) (26) n 1/ 2 1 au R 2 (n 1) (au R) n
Pb j {2Q1 (v, w) [1 k j u / (k j u R)] j
exp[(v 2 w2 ) / 2]I 0 (vw)
n 1
exp( NLK )
same error probability. So just one of decision metrics and the corresponding effective SNR is analyzed. Without loss of generality, symbol d u,1 is considered, then with (7) and (15), the corresponding decision metrics is L
z 2 u (| hu ,l1 |2 | hu ,l 2 |2 | hu ,l 3 |2 )du1 w
1/ 2
Eq.(27) is an accurate BER expression for multiuser space-time coded CDMA system with MQAM, and it is also a high-accuracy approximate BER expression for multiuser space-time coded CDMA system with MPSK modulation, which is shown to match the simulation well. According to Refs. [19-20], utilizing (17) and (26), a closed-form approximate expression of average SER of multiuser space-time coded CDMA system with MPSK modulation is obtained as:
(30)
l 1
where L
w wu ,l1hu,l1 wu ,l 2 hu,l 2 wu ,l 3 hu,l 3 wu*,l 5 hu ,l1 l 1
* u ,l 6 u ,l 2
w
h
wu*,l 7 hu ,l 3 is an equivalent noise. According to
the previous analysis, it will be a complex Gaussian random variable with mean zero, and the variance is L
3
l 1
n 1
var{w} 2a[ | hu ,ln |2 (hu*,l1hu ,l 2 hu*,l 2 hu ,l1
hu*,l1hu ,l 3 hu*,l 3 hu ,l1 hu*,l 3 hu ,l 2 hu*,l 2 hu ,l 3 ) ]
(n 1/ 2) R (k j u ) 1 RLKN F (n ;n 1; )} (27) n 1/ 2 1 (n 1) (k j u R) 2 k j u R n
according to (7) is evaluated. Due to symmetry considerations, the symbols d u,1 , d u,2 , d u,3 , d u,4 have the
1/ 2
where v and w are defined in (24) and F 1 (x,y;z) is the confluent hypergeometric function [21]. Substituting (26) into (21) yields
NL 1
B. Quasi-orthogonal Gold Code Case In this subsection, the BER and SER performance of the system is studied when quasi-orthogonal Gold code is used for spreading code. Under this scenario, the covariance of Wu will be not identity matrix. For the simplicity of analysis, the G3 code is taken as an example to analyze the corresponding system performance. When G3 code is employed, the decision metrics for the detection of the transmitted symbols {d u,p , p 1,,4}
L
3
2(a ) | hu ,ln | 2
(31)
l 1 n 1
where
the
hu ,ln hu*,lm hu ,lm hu*,ln
inequality
| hu,ln |2 | hu,lm |2 is utilized. Thus the lower bound of
effective
SNR
l R [ u / (a )] || Hu || R ul || Hu || obtained by using (30) and (31). 1
2 F
1
ul u / (a )
2 F
by can
be (32)
It is well known that the CDMA system performance with quasi-orthogonal spreading code is worse than that Ps , p f ( ) erfc( )sin( / M ))d G(sin 2 ( / M )) (28) with orthogonal spreading code. Namely, the BER of the 0 former is higher than that of the latter, and the corresponding effective SNR (denoted by no ) is lower Similarly, the average SER of the system with MQAM than the latter o (i.e. in (16), is effective SNR for the is evaluated as orthogonal spreading code case). Hence, o can be Ps , q 1 [1 (1 1/ M )G(1.5 / ( M 1))]2 (29) regarded as the upper bound of no . Thus: l no o . Moreover, is small and a is slightly larger than 1 in general, and thus l in (32) is indeed lower than o in
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(16). Hence, the upper bound and lower bound of no exist. To obtain the approximate BER or SER expression, the mean value between the upper bound and lower bound of no is taken as its approximate value, that is,
no ( l o ) / 2 R1 || Hu ||2F (ul u ) / 2 R1 || Hu ||2F no
(33)
is regarded as an approximate value of no , and
no (ul u ) / 2 [1 (a )1 ]u / 2
is an approximate value of effective SNR accordingly. By substituting u with no , and utilizing (27), the closedform expression of average BER of the multiuser spacetime coded CDMA system with quasi-orthogonal spreading code can be given by Pb j {2Q1 (v, w) [1 k j no / (k j no R)] j
exp[(v 2 w2 ) / 2]I 0 ( ) NL 1
n 1
exp( NLK )
n 1/ 2 (n 1/ 2) R (k j no ) 1 RLKN F (n ;n 1; )} (34) n 1/ 2 1 (n 1) (k j no R) 2 k j no R
By substituting u with no into (26), and then using (28) and (29), the closed-form approximate expressions of average SER of the multiuser space-time coded CDMA system with MPSK and MQAM for quasiorthogonal spreading code case is obtained as follows. Ps , p G (sin 2 ( / M ))
(35)
and Ps , q 1 [1 (1 1/ M )G (1.5 / ( M 1))]2 (36)
where : G (a) 2Q1 (v, w) [1 ano / (ano R)] exp[(v 2 w2 ) / 2]I 0 (vw)
exp( NLK )
mapping of the bits to symbol is employed. The MonteCarlo method is employed for simulation. For different STBCs, the different modulation modes to maintain the same the transmission rate is employed. 6 active users are considered in the system, and conventional Gold codes (P=63) and Walsh-Hadamard (W-H) code (P=64) are used for spreading code, respectively. The simulation results are shown in Fig.1-Fig.4. In these figures, „G2CDMA‟, „G3-CDMA‟, „H3-CDMA‟, „G4-CDMA‟ and „H4-CDMA‟ denote the CDMA system with G2, G3, H3, G4 and H4 code, respectively. „scheme 1‟ and „scheme 2‟ represent the existing decoding scheme in [18] and our improved scheme, respectively. The average BER and SER are obtained by averaging over 107 Monte-Carlo realizations, and thus the results can be accurate enough to reflect the actual values. Fig. 1 shows the BER versus SNR for different spacetime coded CDMA systems with the Rician factor K=0, 5dB, where single receive antenna (1Rx) is considered, and Gold code is used for spreading code. For G2-CDMA, 8PSK modulation is employed, while for H3-CDMA, 16QAM is used instead. Thus, the overall transmission rate is 3 bit/s/Hz. From Fig. 1, it can be seen that H3CDMA performs better than G2-CDMA due to its larger diversity gain. Moreover, it is observed that multiuser space-time coded CDMA system with the developed scheme 2 has almost the same performance as multiuser space-time coded CDMA system with the existing scheme 1 due to better approximation and full utilization of complex orthogonality of space-time block coding, but the implement complexity of scheme 2 is much lower than scheme 1 because of the linear decoding. It means that our scheme 2 is valid and makes a good tradeoff between performance and complexity. Besides, the average BER in Rician fading channel is obviously lower than that in Rayleigh fading channel ( K 0 ) due to the presence of direct path. In the following simulation, the scheme 2 will be employed for the system evaluation due to its simplicity.
RLKN 1 (n 1/ 2) R n (ano )1/ 2 F (n ;n 1; )} n 1/ 2 1 ano R 2 (n 1) (ano R) n 1 Based on the above analysis, the closed-form expression of average BER/SER of multiuser CDMA system with other space-time codes (such as H 3 , H 4 , G4 , G2 , X code) can be easily obtained.
0
10
NL 1
-1
10
K=0
-2
BER
10
-3
10
V.
SIMULATION RESULTS AND THEORETICAL EVALUATION
In this section, the effectiveness of the developed scheme and the derived theoretical expressions by computer simulation for different space-time coded CDMA systems are evaluated. The G2 , G3 , H 3 , G4 , H 4 codes are used for evaluation. In simulation, the channel is assumed to be quasi-static flat Rician fading. Every data frame includes 480 information bits, and Gray
© 2014 ACADEMY PUBLISHER
K=5dB
G2-CDMA w ith 8PSK and scheme 1
-4
10
G2-CDMA w ith 8PSK and scheme 2 H3-CDMA w ith 16QAM and scheme 1 H3-CDMA w ith 16QAM and scheme 2
-5
10
0
5
10 average SNR (dB)
15
20
Figure 1. BER versus SNR for space-time coded CDMA systems with one receive antenna.
In Fig. 2, the theoretical average BER/SER and simulation results of different space-time coded CDMA systems with 1Rx and orthogonal W-H code are given.
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The space-time codes G2 and G3 are considered, and the Rician factor K=6dB. Regarding modulation, 4QAM is employed for G2-CDMA, whereas 16QAM is used for G3-CDMA, resulting in the overall transmission rate of 2bit/s/Hz. The (27) and (29) are employed for computing the theoretical BER and SER of the system, respectively. It is found that the theoretical BER and SER are in good agreement with the simulated values. Moreover, G3CDMA system outperforms G2-CDMA system due to greater diversity. The above results show that the derived theoretical formulae for space-time coded CDMA system with orthogonal W-H code are valid for performance evaluation.
mathematical derivation, accurate and approximate closed-form expressions of BER and SER are obtained, respectively. With these expressions, the error performance of space-time coded CDMA can be effectively assessed. Simulation results show that the derived theoretical BER and SER expressions are in good agreement with the corresponding simulation results. The developed receiver scheme can obtain almost the same performance as the existing scheme, and it has lower complexity than the latter. 0
10
G2-CDMA with 8PSK (theory) G2-CDMA with 8PSK (simu.) H4-CDMA with 16QAM (theory) H4-CDMA with 16QAM (simu.)
-2
10 0
BER/SER
10
G2-CDMA with 4QAM (theory) G2-CDMA with 4QAM (simu.) G3-CDMA with 16QAM (theory)
10 BER/SER
BER
-4
10
SER
G3-CDMA with 16QAM (simu.)
-2
-6
10 -4
10
SER
0
5
10
15
average SNR (dB)
Figure 3. BER/SER versus SNR for space-time coded CDMA systems with two receive antennas (K=6dB)
-6
10
BER
0
5
10 average SNR (dB)
15
20
Figure 2. BER/SER versus SNR for space-time coded CDMA systems with one receive antenna (K=6dB)
In Fig. 3, it plots the theoretical average BER and simulation results of space-time coded CDMA systems with Gold code and two receive antenna, where G2 and H 4 code are considered, and Rician factor K=6 dB. For G2-CDMA, 8PSK modulation is used, while for H4CDMA, 16QAM modulation is employed. Thus, the overall transmission rate is 3 bit/s/Hz. The (34) is employed for the theoretical BER calculation of the system. The SER performance expressions (35) and (36) are used for MPSK and MQAM, respectively. From Fig.3, it can be seen that the theoretical BER and SER are very close to the corresponding simulation results. Besides, it is found that H4-CDMA system outperforms G2-CDMA system due to high diversity gain. The above results show the derived BER and SER expressions for space-time coded CDMA system with Gold code are also effective for performance evaluation. VI.
CONCLUSIONS
The error performance of multiuser space-time coded CDMA systems over Rician fading channel is investigated. A simple and effective multiuser receiver scheme is developed for space-time coded CDMA systems. The scheme can effectively suppress MUI via multiuser detection method, and greatly reduce the high decoding complexity of the existing scheme. According to the performance analysis of the system, and using the © 2014 ACADEMY PUBLISHER
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewer for his valuable comments. This research was supported in part a grant from the national natural science foundation of China(GrantNo. 61203056), Foundation of Huaian industrial projects (GrantNo. HAG2013064), and foundation of Huaiyin Institute of Technology (GrantNo. HGB1202). REFERENCES [1] Chae, C.-B., Forenza, A., W.Heath, R., McKay, M. and Collings, I., Adaptive MIMO transmission techniques for broadband wireless communication systems. IEEE Commun. Magazine, 48 (2010), 112-118. [2] X. Hu, Z. Chen, and F. Yin, "Impulsive Noise Cancellation for MIMO Power Line Communications," Journal of Communications, vol. 9, no. 3, pp. 241-247, 2014. [3] Siam, M.Z., Krunz, M., An overview of MIMO-oriented channel access in wireless networks. IEEE Wireless Communications, 15(2008), 63-69. [4] X. Shao and C. H. Slump, "Opportunistic Error Correction for MIMO-OFDM: From Theory to Practice," Journal of Communications, vol. 8, no. 9, pp. 540-549, 2013. [5] Tarokh, V., Jafarkhani, H. and Calderbank, A.R., Spacetime block codes from orthogonal designs. IEEE Trans. Inform. Theory, 45 (1999), 1456-1467. [6] M.M. KAMRUZ ZAMAN and Li Hao, "Performance of Turbo-SISO, Turbo-SIMO, Turbo-MISO and TurboMIMO system using STBC," Journal of Communications, vol. 6, no.8, pp.633-639, 2011. [7] Yu Xiangbin, Xu Dazhuan and Bi Guangguo, Full-rate complex orthogonal space-time block code for multiple antennas. Wireless Pers. Commun., 40 (2007), 81-89.
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[8] Tarokh, V., Jafarkhani, H. and Calderbank, A.R., Spacetime block coding for wireless communications: performance results. IEEE J. Select. Areas Commun. 17(1999), 451-460. [9] Larsson, E.G. and Stoica, P., Space-time block coding for wireless communications. Cambridge University Press, 2003. [10] J. Gao and Y. Bai, "Quasi-Orthogonal Space-Time Block Code with Givens Rotation for SC-FDE Transmission in Frequency Selective Fading Channels," Journal of Communications, vol. 8, no. 12, pp. 832-838, 2013. [11] Zhang, H. Gulliver, T.A., Capacity and error probability analysis for orthogonal space-time block codes over fading channels. IEEE Trans. Wireless Commun., 4 (2005) 808819. [12] Shabazpanahi, S., Beheshti, M., Gershmann, A.B., et al., Miniumum variance linear receivers for multiaccess MIMO wireless systems with space-time block coding. IEEE Trans. Signal Process. 52 (2004), 3306-3312. [13] Sacramento, A. and Hamouda, W., Performance of multiuser-coded CDMA systems with transmit diversity over Nakagami-m fading channels. IEEE Trans.Veh.Technol., 58 (2009), 2279-2287. [14] Seo, B., Ahn, W.-G., Jeong, C. and Kim, H.-M., Fast convergent LMS adaptive receiver for MC-CDMA systems with space-time block coding. IEEE Communication Lett., 14 (2010), 737-739. [15] Z. Li and M. Latva-aho, “Performance of space-time block coded MC-CDMA in Nakagami fading channels,” IEE Electronics Letters, vol. 39, no. 2, pp. 222-224, 2003. [16] Zhang Xiaofei, Feng Gaopeng, Gao Xin, Xu Dazhuan, Blind multiuser detection for MC-CDMA with antenna array. Computers and Electrical Engineering, 36 (2010), 160-168. [17] Qi, S. Aissa, A. Maaref, "Cross-Layer Design for MIMO Orthogonal STBC Systems Over Spatially-Correlated and Keyhole Nakagami-m Fading Channels," Wiley Journal Wireless Communications and Mobile Computing, in press; published online Aug. 2009. [18] Li, H. and Li, J., Differential and coherent decorrelating multiuser receivers for space-time coded CDMA systems. IEEE Trans. Signal Process., 50 (2002), 2529-2536. [19] Proakis, J.G., Digital communications, 5th ed. New York: McGraw-Hill, 2007. [20] Simon, M.K. and Alouini, M.S., Digital communication over fading channels: a unified approach to performance analysis. New York: Wiley, 2000. [21] Gradshteyn, I.S. and Ryzhik, I.M., Table of integrals, series, and products. 7th ed. San Diego, CA: Academic, 2007.
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[22] Verdu, S., Multiuser detection. Cambridge, U.K: Cambridge university press, 1998. [23] Lu, J., Letaief, K.B., Chuang, J.C.-I., et al, M-PSK and MQAM BER computation using signal-space concepts, IEEE Trans. on Commun., 47 (1999), 181-184. [24] Lindsey, W.C., Error probabilities for Rician fading multichannel reception of binary and n-ary signals. IEEE Trans. Inf. Theory, 10 (1964), 339-350.
Dingli Yang received his M.S. degree in electronic and information engineering from Southeast University, China, in June 2006. His current research interest includes communication, digital signal processing, image processing.
Qiuchan Bai received his M.S. degree in electronic and information engineering from Northwestern polytechnical University, China, in June 2006.His current research interest includes image processing, and pattern recognition. Yulin Zhang received his Ph.D. degree in communication and control engineering, from Jiangnan University, China, in 2010. His current research interest includes communication, pattern recognition. Rendong Ji received his M.S. degree in electronic and information engineering from Qufu Normal University, China, in June 2006. His current research interest includes digital signal processing. Now he is pursuing a Ph.D in Nanjing University of Aeronautics and Astronautics. Yazhou Li received his M.S. degree in electronic and information engineering from Chongqing University, China, in June 2008. His current research interest includes communication, image processing. Yudong Yang received his M.S. degree in school of information science and engineering from Southeast University, China, in 2004, and received his Ph.D. degree in School of Electronic and Optical Engineer from Nanjing University of Science & Technology, China, in 2012. His current research interest includes communication, pattern recognition.
JOURNAL OF NETWORKS, VOL. 9, NO. 12, DECEMBER 2014
Error Performance Analysis of Multiuser CDMA Systems with Space-time Coding in Rician Fading Channel Dingli Yang 1, Qiuchan Bai 1, Yulin Zhang 1, Rendong Ji 1, 2, Yazhou Li 1, and Yudong Yang 1 1. Faculty of Electronic and Electrical Engineering, Huaiyin Institute of Technology, Huai‟an 223003, China 2. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Email: [email protected], [email protected], [email protected] , [email protected], [email protected], [email protected]
Abstract—In this paper, the error performance of multiuser CDMA system with space-time coding is studied in Rician fading channel, and the corresponding bit error rate (BER) and symbol error rate (SER) analysis are presented. Based on the performance analysis, a simple and effective multiuser receiver scheme is developed. The scheme has linear decoding complexity when it compares to the existing scheme. Based on the performance analysis, and using mathematical manipulation, the BER and SER of multiuser space-time coded CDMA system are derived, respectively. As a result, accurate closed-form expressions of BER and SER are respectively obtained. With these expressions, the performance of multiuser CDMA system with space-time coding can be evaluated effectively. Computer simulation shows that the developed receiver scheme has almost the same performance as the existing scheme, and the theoretical BER and SER can match the corresponding simulation results well. Index Terms—Space-Time Coding; Code Division Multiple Access (CDMA); Rician Fading; Low Complexity; Bit Error Rate; Symbol Error Rate
I.
INTRODUCTION
Recently, multiple-input and multiple-output (MIMO) technique is well known to offer improvements in bandwidth efficiency along with diversity and coding benefits over wireless fading channels [1-4]. Especially space-time block coding (STBC) in MIMO systems can provide effective diversity for combating fading effect [510], and has received much interests. However, the conventional STBC scheme is only used for single-user environment, and thus it will be not suitable for multiuser scenario in practice. Hence, it is necessary to extend the STBC scheme into multiuser CDMA scenario for practical purposes. Based on different multiuser spacetime coded system models, the receiver schemes are designed in references [11-18] and references therein. In reference [11], the bit error rate (BER) analysis is provided for space-time coded CDMA system with the conventional matched filter receiver, but the analysis method and system model are applicable only to the BPSK modulation and downlink, and does not provide the closed-form BER expression for Rician fading © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.12.3462-3469
channel. The given symbol error rate (SER) expression for space-time codes needs summing from 0 to infinity, whereas the infinity is difficult to decide. In reference [12], a minimum variance linear receiver scheme for multiuser MIMO system is proposed, but the system needs to design and optimize the weighted matrix to suppress the multiuser interference (MUI). As a result, the computational complexity is very complicated. In reference [13], the performance of multiuser CDMA system with transmit diversity is studied, but the analysis is limited in BPSK modulation and two transmit antennas. For multicarrier-CDMA (MC-CDMA) system, a least mean-square based adaptive receiver scheme is proposed for MC-CDMA with Alamouti‟s STBC [14], but the scheme is limited in two transmit antennas and one receive antenna. The performance of space-time coded MC-CDMA system is analyzed in Nakagami fading channel [15], but the analysis is limited in Alamouti‟s STBC and BPSK modulation. Blind space-time multiuser detection schemes are presented in reference [16] for MC-CDMA system. The performance of space-time coded MIMO system with cross-layer design is analyzed in spatially-correlated and Keyhole Nakagami fading channel [17], but the analytical method is applicable only to single user system. Reference [18] gives the effective combination of CDMA system and different space-time codes, and the developed decorrelative receiver scheme can decouple the detection of different users, but the decoding complexity is exponential for each user, which will not benefit practical application. Moreover, the above schemes basically do not provide the error performance analysis, and are limited in Rayleigh fading channel, whereas in practice, the system may experience the Rician fading due to the direct-path propagation. Due to the reason above, the error performance of space-time coded CDMA system in Rician fading channel is studied. Firstly, a multiuser space-time coded CDMA system model is presented, and then a lowcomplexity multiuser receiver scheme is proposed by utilizing maximum ratio combining (MRC) method and orthogonality of space-time coding. The presented spacetime coded CDMA system can achieve effective MUI suppression by using multiuser detection method. After
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decorrelating, each user has linear rather than exponential decoding complexity. According to the performance analysis, and using mathematical calculation, the average BER and SER of the system are derived in detail. As a result, accurate and approximate closed-form expressions of BER and SER are attained for space-time coded CDMA with orthogonal and quasi-orthogonal spreading code, respectively. Simulation results show that the proposed low-complexity scheme can obtain almost the same performance as the existing scheme. Theoretical BER and SER will be in good agreement with the corresponding simulations. Thus, the effectiveness of the theoretical formulae is verified. Note: the superscripts ()T , () , () H are used to stand for the transpose, complex conjugate, and Hermitian transpose, respectively. E{} and I N denote the statistical expectation and identity matrix, respectively. vec( ?) stand for matrix vectorization operator. II.
(2)
where D u is a N T space-time coding matrix of user u, is spreading code, and T 1 Cu (q) Cu [ Cu (1) ,, Cu (Q)] corresponds to T normalized spreading codes of length Q used to spread D u for user u (u 1,2,U ) , here conventional orthogonal WalshHadamard code and quasi-orthogonal Gold code in CDMA system are considered. These different spreading codes for different users are employed as well. Based on the analytical method in [18], obtain the baseband received signal at qth ( q 1,2,,Q ) chip period is obtained as follows U
y(q) u H uVu (q) w(q) , q 1,2,,Q (3) u 1
the channel gain from the nth transmit antenna to the lth receive antenna, which is assumed to be constant over a frame of T symbols and varied from frame to frame. For Rician fading channels, the { h u,ln } are modeled as independent complex Gaussian random variables with respective means mI and mQ for the real and imaginary parts and variance of 0.5 per dimension [11, 19], and
where Hu [h u ]l,n is L N channel matrix of user u . w(q), q 1,,Q is L1 noise vector, whose element {w l (q), l 1,, L, q 1,,Q } are independent, identically distributed (i.i.d ) complex Gaussian random variables with zero-mean and unit-variance. Nu denotes the average signal-to-noise ratio per receive antenna for user u at the receiver during the transmission of spacetime coding matrix Du (which corresponds to Q chip periods), this SNR adopts the definition similar to Ref.[18] for comparison consistency. Substituting (2) into (3), the received signal at qth chip period can be expressed as U
y(q) u H u Du Cu (q) w(q) , q 1,2,,Q (4) u 1
L
| hu ,ln |2
obeys
a
noncentral
chi-square
n 1 l 1
distribution with 2NL degrees of freedom. According to reference [19] and changing variable, the probability density function (pdf) of can be obtained by f ( ) ( / x2 )( NL 1)/ 2 (2 2 ) 1 e( x
2
)/(2 2 )
I NL 1 ( x 2 / 2 ), 0
0.5 , and I v (x) is the vth-order modified Bessel function of the first kind [20-21]. 2
LOW-COMPLEXITY MULTIUSER RECEIVER
In this section, A low-complexity multiuser receiver design for multiuser space-time coded CDMA system is given. For multiuser CDMA system with space-time coding, the block length of space-time code is set equal to Q chip periods. Then according to reference [18], the
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In order to express (4) more compactly, the matrices is defined as: Su u H u Du , S [S1 , S2 ,,SU ] ,
C [C1T ,,CTU ]T Cu [ Cu (1),, Cu (Q)] , , Y [y(1) , ,y (Q)] , and W [w(1) ,, w(Q)] . Thus, (4) is changed to U
(1)
where x2 NL(mI2 mQ2 ) is the noncentrality parameter,
III.
Vu (q) Du Cu (q)
SYSTEM MODEL
In this paper, a synchronous CDMA communication system with N transmit antennas and L receive antennas and U active users that operates over a flat and quasi-static Rician channel is considered. The multiuser CDMA system employs the space-time block coding schemes (such as conventional full-diversity G 2 , G 3 , H 3 , G 4 , H 4 code [5,8], full-rate X code [7]) to transmit the data. For each user u , the channel gain h u,ln denotes
N
transmitted signal matrix of user u at qth ( q 1,2,,Q ) chip period is
Y Su Cu W SC W
(5)
u 1
Then according to reference [22], obtain the ML estimate of S conditioned on {H u and {Du is obtained as Sˆ YC H (CC H )1 S WC H (CC H )1
(6)
Here, the Moore-Penrose inverse matrix CH (CCH )-1 can be expressed as a multiuser decorrelator [18, 22], and thus the ML estimate Sˆ is an effective output of the decorrelator with the input being the received data Y . By this decorrelator, the multiuser interference is cancelled, and the detection of different users is decoupled. Based
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on the block structure of S , the ML estimate Sˆu of Su can be easily achieved. While for user u , all data information on the transmitted code matrix Du is contained in Su . Hence, the code matrix and corresponding information symbols via the achieved Sˆ u
is evaluated. According to the definition of Su , Assuming that Sˆ [sˆT , sˆT ,..., sˆT ]T , where sˆ is a 1 T row u
u ,1
u ,2
u, L
u ,l
vector. Thus when G 3 code scheme is employed, T=8. Considering that Su has the receiver signal form similar to the conventional STBC in single user scenario [7-8], the simple decoding scheme for the space-time coded CDMA system with G 3 code after performing multiuser decorrelation is obtained by utilizing the MRC method and complex orthogonality of STBC, i.e., dˆu1 arg
L
min || (sˆ
u , l 1 u , l1
l 1
du 1
h
sˆu ,l 2 hu,l 2 sˆu ,l 3 hu,l 3 3
sˆu ,l 5 hu,l1 sˆu,l 6 hu ,l 2 sˆu,l 7 hu ,l 3 2 u | hu ,ln |2 du1 ) ||2 n 1
dˆu 2 arg
L
min || (sˆ l 1
du 2
u , l1 u , l 2
h
u ,l 2 u ,l1
sˆ
h
sˆu ,l 4 hu,l 3 sˆu ,l 3 hu,l 3 3
sˆu,l 5 hu ,l1 sˆu,l 6 hu ,l 2 sˆu,l 7 hu ,l 3 2 u | hu ,ln |2 du 2 ) ||2 n 1
dˆu 3 arg
L
min || (sˆ l 1
du 3
u ,l1 u ,l 3
h
sˆu ,l 3 hu,l1 sˆu ,l 4 hu,l 2 sˆu ,l 5 hu,l 3
dˆu 4 arg
L
min || (sˆu ,l 3 hu,l 3 sˆu,l 4 hu ,l 2 ) ||2
(9)
l 1
du 4
From (7) and (8) as well as (9), it is observed that the developed decoding scheme has linear complexity. For Ref.[18], its receiver decoding schemes with coherent detection (i.e. (44) and (45) in [18]) are shown as follows: 1) For general spreading codes:
Dˆ u arg min{vec H (Sˆu u H u Du ) k1 {du ,1 ,..., d u , p }
vec(Sˆu u H u Du )}
(10)
2) For orthogonal spreading codes: Dˆ u arg min || Sˆu u H u Du ||2F
(11)
{du ,1 ,..., d u , p }
From the above two equations, it can be seen that the decoding scheme in [18] has exponential complexity. Namely, if is a constellation consists of M symbols, the search times that need to obtain the transmitted P symbols is M P . Thus, when M and P become larger, the complexity will be much higher. As a result, the implementation complexity of the system will be increased significantly. While for our scheme, the needed search times are MP only. Based on this, the complexity comparison between the developed scheme and the existing scheme [18] in Table.1 is given. From Table 1, it can be seen that the proposed scheme has lower complexity than the existing scheme.
3
sˆu,l 7 hu ,l1 sˆu,l 8 hu ,l 2 2 u | hu ,ln |2 du 3 ) ||2
TABLE I.
n 1
dˆu 4 arg
L
min || (sˆ l 1
du 4
u ,l 3 u ,l 2
h
sˆu ,l 2 hu,l 3 sˆu ,l 4 hu,l1 sˆu ,l 6 hu,l 3
3
sˆu,l 7 hu ,l 2 sˆu,l 8 hu ,l1 2 u | hu ,ln |2 du 4 ) ||2
(7)
Scheme Existing scheme Developed scheme
COMPARISON OF COMPLEXITY QPSK (M=4, P=2) 16 8
8PSK (M=8, P=2) 64 16
16QAM (M=16, P=3) 4096 48
16QAM (M=16, P=4) 65536 64
n 1
When G2 code [5] is used, T=2, Sˆu [sˆuT,1 , sˆuT,2 ]T , the corresponding simple decoding scheme can be given by dˆu1 arg
L
min || (sˆu ,l1hu,l1 sˆu,l 2 hu ,l 2 ) ||2
du 1
dˆu 2 arg
l 1
L
min || (sˆu ,l1hu,l 2 sˆu,l 2 hu ,l1 ) ||2
(8)
l 1
du 2
Besides, when X code [7] is employed, T=4, ˆ Su [sˆuT,1 , sˆuT,2 , sˆuT,3 , sˆuT,4 ]T , the simple decoding scheme can be given by can be attained as dˆu1 arg
L
min || (sˆ
du 1
dˆu 2 arg
h
u ,l 2 u ,l 2
sˆ
h
2
min || (sˆu ,l1hu,l 2 sˆu,l 2 hu ,l1 ) ||2 l 1 L
min || (sˆu ,l 3 hu,l 2 sˆu,l 4 hu ,l 3 ) ||2
du 3
l 1
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SYSTEM PERFORMANCE ANALYSIS IN RICIAN FADING CHANNEL
In this section, the error performance analysis of spacetime coded CDMA system in Rician fading channel is given, and the closed-form expressions of average BER and SER is derived. Let W WC H (CC H )1 and Wu be the L T submatrix of W consisting of the columns starting from (u-1)T 1 to uT , then W [W ,...,W ] . According to 1
U
reference [18], the covariance matrix of vec( W ) is expressed as E{vec(W )vec H (W )} [(CC H ) 1 ]T I L
) ||
L
du 2
dˆu 3 arg
l 1
u , l1 u , l1
IV.
With Eq.(12), the calculated as follows:
matrix
VW E{W H W H }
VW (CC H )1
(12) is (13)
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Let C be (CC H )-1 , then with (12) and (13), the following equations can be obtained.
VWu E{WuH Wu } C(u 1)T 1:uT ,(u 1)T 1:uT , u 1,,U
(14)
When the orthogonal Walsh-Hadamard codes are used for spreading codes, the matrix C will be identity matrix I TU , and accordingly, VWu is also identity matrix (i.e. I T ). While the quasi-orthogonal Gold codes are employed for spreading codes, C will be a symmetric Toeplitz matrix with first row being [a, u, u,...u] , where (UT-1) is included. Thus, VWu is also a symmetric Toeplitz matrix with first row being [a, u, u,...u] , where (T -1) is included. According to the above analysis, the elements of W are complex Gaussians variables with mean zero. So the elements {wu ,lt } of Wu (u 1,, U) are also complex Gaussians variables with mean zero and variance unit for orthogonal spreading code case, while in the case of quasi-orthogonal spreading code, their covariance is a or . Namely, E{| wu ,lt |2 } a and E{wu ,lt wu,l t } , l l or t t . Using Eq.(6) and the definition of Su , the effective output of the decorrelator of user u can be written as: Sˆu Su Wu u Hu Du Wu , u 1,2,, U .
Pb, q ( ) j erfc( k j )
where M is the constellation size, erfc( ) is the complementary error function, j and k j are constants which depend on M , and the values of the constant sets { j , k j } for MQAM can be found in [19]. Besides, the high-accuracy approximate BER for MPSK with Gray coding over AWGN channel is given as [23] 2
Pb, p ( ) erfc( k j )
where the { j , k j } are given by { (1/ log 2M ,sin 2 ( / M )) , (1/ log2M ,sin 2 (3 / M )) } for MPSK. By using Eqs.(17)-(20), the average BER for multiuser space-time coded CDMA system with MQAM or MPSK modulation is evaluated as follows
Pb j erfc( k j ) f ( )d j G(k j ) (21)
G( )
0
[1 v / Q1 ( 2 NLK , 2 R / u ) 1 2 e a d ] 0
K (mI2 mQ2 ) / (2 2 )
can
be
obtained as f ( ) (
e NLK R / u I NL 1 (2
R
u
)( NL 1)/ 2 (
RNLK
u
0
Using (1) and changing variable, the pdf of in (16)
NLK
( n 1)/ 2 e ( R /
u
a )
e NLK
NL 1
( n 1
I n (2
R ) u NLK
RNLK
u
n2
)d g1 g2
where the equality Qm (a, b) Q1 (a, b) e ( a
2
b2 )/ 2
m 1
(b / a) I (ab) i
i 1
(17)
g1 1 2 / Q1 ( 2 NLK , R / (au )t )et / 2 dt 2
0
2Q1 (v, w) [1 u a / (u a R)]
The cumulative distribution function (cdf) of can be expressed as
© 2014 ACADEMY PUBLISHER
is
i
F ( ) 1 QNL ( 2 NLK , 2R / u ) ,
(23)
utilized. Using change of variables and the results of Appendix I in [24], g1 in Eq.(23) can be expressed as
)( NL 1)/ 2
), 0
(22)
G(a) 1 a / QNL ( 2 NLK , 2R / u ) 1 2 e a d
l 1 n 1
f ( ) erfc( a )d , 0 .
Substituting (17) and (18) into (22) gives
a
(16)
0
N
u | hu ,ln |2 / R u ||
factor
j
where G() denotes the integration in (21) defined as
N
Rician
0
j
l 1 n 1
with
(20)
j 1
u | hu ,ln |2 / R u || Hu ||2F / R u / R L
(19)
j
(15)
A. Orthogonal Walsh-Hadamard Code Case When the orthogonal Walsh-Hadamard code is used for spreading code, the covariance of Wu will be identity matrix, and its elements are i.i.d . complex Gaussian variables with mean zero and variance unit. Based on this, utilizing the orthogonality of space-time block coding, the effective SNR for user u at the receiver is given by L
where Qm ( , ) is the generalized Marcum Q-function [19, 21]. With Eq.(17), the pdf of for Rayleigh fading channel can be obtained by setting K 0 . According to Refs.[19-20, 23], the BER of coherent Mary QAM (MQAM) with Gray coding over an Additive White Gaussian Noise (AWGN) channel is given as
(18)
exp[(v w2 ) / 2]I 0 (vw) 2
where
(24)
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v w
NLK [ R / 2 u a u a( R u a)]
Equation (29) is accurate closed-form expressions of the average SER of multiuser space-time coded CDMA with MQAM in Rician fading channels.
,
R u a NLK [ R / 2 u a u a( R u a)] R u a
.
According to Eq.(23), g 2 is written as: a
g2
0
NL 1
n 1
( n 1)/ 2 e ( R /
u
a )
e NLK
NL 1 n 1
RNLK
I n (2
R
( NLK ) u
n/ 2
u
)d
exp( NLK )
RLKN 1 (n 1/ 2) R n (au )1/ 2 F (n ;n 1; ) (25) n 1/ 2 1 au R 2 (n 1) (au R)
The above derivation utilizes Eq.(6.643), Eq.(9.220) and Eq.( 9.215) in reference [21] . With Eqs.(24) and (25), G(a) in (21) can be expressed as G(a) 2Q1 (v, w) [1 au / (au R)] exp[(v 2 w2 ) / 2]I 0 (vw) NL 1
n 1
exp( NLK )
RLKN 1 (n 1/ 2) R (au ) F (n ;n 1; ) (26) n 1/ 2 1 au R 2 (n 1) (au R) n
Pb j {2Q1 (v, w) [1 k j u / (k j u R)] j
exp[(v 2 w2 ) / 2]I 0 (vw)
n 1
exp( NLK )
same error probability. So just one of decision metrics and the corresponding effective SNR is analyzed. Without loss of generality, symbol d u,1 is considered, then with (7) and (15), the corresponding decision metrics is L
z 2 u (| hu ,l1 |2 | hu ,l 2 |2 | hu ,l 3 |2 )du1 w
1/ 2
Eq.(27) is an accurate BER expression for multiuser space-time coded CDMA system with MQAM, and it is also a high-accuracy approximate BER expression for multiuser space-time coded CDMA system with MPSK modulation, which is shown to match the simulation well. According to Refs. [19-20], utilizing (17) and (26), a closed-form approximate expression of average SER of multiuser space-time coded CDMA system with MPSK modulation is obtained as:
(30)
l 1
where L
w wu ,l1hu,l1 wu ,l 2 hu,l 2 wu ,l 3 hu,l 3 wu*,l 5 hu ,l1 l 1
* u ,l 6 u ,l 2
w
h
wu*,l 7 hu ,l 3 is an equivalent noise. According to
the previous analysis, it will be a complex Gaussian random variable with mean zero, and the variance is L
3
l 1
n 1
var{w} 2a[ | hu ,ln |2 (hu*,l1hu ,l 2 hu*,l 2 hu ,l1
hu*,l1hu ,l 3 hu*,l 3 hu ,l1 hu*,l 3 hu ,l 2 hu*,l 2 hu ,l 3 ) ]
(n 1/ 2) R (k j u ) 1 RLKN F (n ;n 1; )} (27) n 1/ 2 1 (n 1) (k j u R) 2 k j u R n
according to (7) is evaluated. Due to symmetry considerations, the symbols d u,1 , d u,2 , d u,3 , d u,4 have the
1/ 2
where v and w are defined in (24) and F 1 (x,y;z) is the confluent hypergeometric function [21]. Substituting (26) into (21) yields
NL 1
B. Quasi-orthogonal Gold Code Case In this subsection, the BER and SER performance of the system is studied when quasi-orthogonal Gold code is used for spreading code. Under this scenario, the covariance of Wu will be not identity matrix. For the simplicity of analysis, the G3 code is taken as an example to analyze the corresponding system performance. When G3 code is employed, the decision metrics for the detection of the transmitted symbols {d u,p , p 1,,4}
L
3
2(a ) | hu ,ln | 2
(31)
l 1 n 1
where
the
hu ,ln hu*,lm hu ,lm hu*,ln
inequality
| hu,ln |2 | hu,lm |2 is utilized. Thus the lower bound of
effective
SNR
l R [ u / (a )] || Hu || R ul || Hu || obtained by using (30) and (31). 1
2 F
1
ul u / (a )
2 F
by can
be (32)
It is well known that the CDMA system performance with quasi-orthogonal spreading code is worse than that Ps , p f ( ) erfc( )sin( / M ))d G(sin 2 ( / M )) (28) with orthogonal spreading code. Namely, the BER of the 0 former is higher than that of the latter, and the corresponding effective SNR (denoted by no ) is lower Similarly, the average SER of the system with MQAM than the latter o (i.e. in (16), is effective SNR for the is evaluated as orthogonal spreading code case). Hence, o can be Ps , q 1 [1 (1 1/ M )G(1.5 / ( M 1))]2 (29) regarded as the upper bound of no . Thus: l no o . Moreover, is small and a is slightly larger than 1 in general, and thus l in (32) is indeed lower than o in
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(16). Hence, the upper bound and lower bound of no exist. To obtain the approximate BER or SER expression, the mean value between the upper bound and lower bound of no is taken as its approximate value, that is,
no ( l o ) / 2 R1 || Hu ||2F (ul u ) / 2 R1 || Hu ||2F no
(33)
is regarded as an approximate value of no , and
no (ul u ) / 2 [1 (a )1 ]u / 2
is an approximate value of effective SNR accordingly. By substituting u with no , and utilizing (27), the closedform expression of average BER of the multiuser spacetime coded CDMA system with quasi-orthogonal spreading code can be given by Pb j {2Q1 (v, w) [1 k j no / (k j no R)] j
exp[(v 2 w2 ) / 2]I 0 ( ) NL 1
n 1
exp( NLK )
n 1/ 2 (n 1/ 2) R (k j no ) 1 RLKN F (n ;n 1; )} (34) n 1/ 2 1 (n 1) (k j no R) 2 k j no R
By substituting u with no into (26), and then using (28) and (29), the closed-form approximate expressions of average SER of the multiuser space-time coded CDMA system with MPSK and MQAM for quasiorthogonal spreading code case is obtained as follows. Ps , p G (sin 2 ( / M ))
(35)
and Ps , q 1 [1 (1 1/ M )G (1.5 / ( M 1))]2 (36)
where : G (a) 2Q1 (v, w) [1 ano / (ano R)] exp[(v 2 w2 ) / 2]I 0 (vw)
exp( NLK )
mapping of the bits to symbol is employed. The MonteCarlo method is employed for simulation. For different STBCs, the different modulation modes to maintain the same the transmission rate is employed. 6 active users are considered in the system, and conventional Gold codes (P=63) and Walsh-Hadamard (W-H) code (P=64) are used for spreading code, respectively. The simulation results are shown in Fig.1-Fig.4. In these figures, „G2CDMA‟, „G3-CDMA‟, „H3-CDMA‟, „G4-CDMA‟ and „H4-CDMA‟ denote the CDMA system with G2, G3, H3, G4 and H4 code, respectively. „scheme 1‟ and „scheme 2‟ represent the existing decoding scheme in [18] and our improved scheme, respectively. The average BER and SER are obtained by averaging over 107 Monte-Carlo realizations, and thus the results can be accurate enough to reflect the actual values. Fig. 1 shows the BER versus SNR for different spacetime coded CDMA systems with the Rician factor K=0, 5dB, where single receive antenna (1Rx) is considered, and Gold code is used for spreading code. For G2-CDMA, 8PSK modulation is employed, while for H3-CDMA, 16QAM is used instead. Thus, the overall transmission rate is 3 bit/s/Hz. From Fig. 1, it can be seen that H3CDMA performs better than G2-CDMA due to its larger diversity gain. Moreover, it is observed that multiuser space-time coded CDMA system with the developed scheme 2 has almost the same performance as multiuser space-time coded CDMA system with the existing scheme 1 due to better approximation and full utilization of complex orthogonality of space-time block coding, but the implement complexity of scheme 2 is much lower than scheme 1 because of the linear decoding. It means that our scheme 2 is valid and makes a good tradeoff between performance and complexity. Besides, the average BER in Rician fading channel is obviously lower than that in Rayleigh fading channel ( K 0 ) due to the presence of direct path. In the following simulation, the scheme 2 will be employed for the system evaluation due to its simplicity.
RLKN 1 (n 1/ 2) R n (ano )1/ 2 F (n ;n 1; )} n 1/ 2 1 ano R 2 (n 1) (ano R) n 1 Based on the above analysis, the closed-form expression of average BER/SER of multiuser CDMA system with other space-time codes (such as H 3 , H 4 , G4 , G2 , X code) can be easily obtained.
0
10
NL 1
-1
10
K=0
-2
BER
10
-3
10
V.
SIMULATION RESULTS AND THEORETICAL EVALUATION
In this section, the effectiveness of the developed scheme and the derived theoretical expressions by computer simulation for different space-time coded CDMA systems are evaluated. The G2 , G3 , H 3 , G4 , H 4 codes are used for evaluation. In simulation, the channel is assumed to be quasi-static flat Rician fading. Every data frame includes 480 information bits, and Gray
© 2014 ACADEMY PUBLISHER
K=5dB
G2-CDMA w ith 8PSK and scheme 1
-4
10
G2-CDMA w ith 8PSK and scheme 2 H3-CDMA w ith 16QAM and scheme 1 H3-CDMA w ith 16QAM and scheme 2
-5
10
0
5
10 average SNR (dB)
15
20
Figure 1. BER versus SNR for space-time coded CDMA systems with one receive antenna.
In Fig. 2, the theoretical average BER/SER and simulation results of different space-time coded CDMA systems with 1Rx and orthogonal W-H code are given.
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The space-time codes G2 and G3 are considered, and the Rician factor K=6dB. Regarding modulation, 4QAM is employed for G2-CDMA, whereas 16QAM is used for G3-CDMA, resulting in the overall transmission rate of 2bit/s/Hz. The (27) and (29) are employed for computing the theoretical BER and SER of the system, respectively. It is found that the theoretical BER and SER are in good agreement with the simulated values. Moreover, G3CDMA system outperforms G2-CDMA system due to greater diversity. The above results show that the derived theoretical formulae for space-time coded CDMA system with orthogonal W-H code are valid for performance evaluation.
mathematical derivation, accurate and approximate closed-form expressions of BER and SER are obtained, respectively. With these expressions, the error performance of space-time coded CDMA can be effectively assessed. Simulation results show that the derived theoretical BER and SER expressions are in good agreement with the corresponding simulation results. The developed receiver scheme can obtain almost the same performance as the existing scheme, and it has lower complexity than the latter. 0
10
G2-CDMA with 8PSK (theory) G2-CDMA with 8PSK (simu.) H4-CDMA with 16QAM (theory) H4-CDMA with 16QAM (simu.)
-2
10 0
BER/SER
10
G2-CDMA with 4QAM (theory) G2-CDMA with 4QAM (simu.) G3-CDMA with 16QAM (theory)
10 BER/SER
BER
-4
10
SER
G3-CDMA with 16QAM (simu.)
-2
-6
10 -4
10
SER
0
5
10
15
average SNR (dB)
Figure 3. BER/SER versus SNR for space-time coded CDMA systems with two receive antennas (K=6dB)
-6
10
BER
0
5
10 average SNR (dB)
15
20
Figure 2. BER/SER versus SNR for space-time coded CDMA systems with one receive antenna (K=6dB)
In Fig. 3, it plots the theoretical average BER and simulation results of space-time coded CDMA systems with Gold code and two receive antenna, where G2 and H 4 code are considered, and Rician factor K=6 dB. For G2-CDMA, 8PSK modulation is used, while for H4CDMA, 16QAM modulation is employed. Thus, the overall transmission rate is 3 bit/s/Hz. The (34) is employed for the theoretical BER calculation of the system. The SER performance expressions (35) and (36) are used for MPSK and MQAM, respectively. From Fig.3, it can be seen that the theoretical BER and SER are very close to the corresponding simulation results. Besides, it is found that H4-CDMA system outperforms G2-CDMA system due to high diversity gain. The above results show the derived BER and SER expressions for space-time coded CDMA system with Gold code are also effective for performance evaluation. VI.
CONCLUSIONS
The error performance of multiuser space-time coded CDMA systems over Rician fading channel is investigated. A simple and effective multiuser receiver scheme is developed for space-time coded CDMA systems. The scheme can effectively suppress MUI via multiuser detection method, and greatly reduce the high decoding complexity of the existing scheme. According to the performance analysis of the system, and using the © 2014 ACADEMY PUBLISHER
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewer for his valuable comments. This research was supported in part a grant from the national natural science foundation of China(GrantNo. 61203056), Foundation of Huaian industrial projects (GrantNo. HAG2013064), and foundation of Huaiyin Institute of Technology (GrantNo. HGB1202). REFERENCES [1] Chae, C.-B., Forenza, A., W.Heath, R., McKay, M. and Collings, I., Adaptive MIMO transmission techniques for broadband wireless communication systems. IEEE Commun. Magazine, 48 (2010), 112-118. [2] X. Hu, Z. Chen, and F. Yin, "Impulsive Noise Cancellation for MIMO Power Line Communications," Journal of Communications, vol. 9, no. 3, pp. 241-247, 2014. [3] Siam, M.Z., Krunz, M., An overview of MIMO-oriented channel access in wireless networks. IEEE Wireless Communications, 15(2008), 63-69. [4] X. Shao and C. H. Slump, "Opportunistic Error Correction for MIMO-OFDM: From Theory to Practice," Journal of Communications, vol. 8, no. 9, pp. 540-549, 2013. [5] Tarokh, V., Jafarkhani, H. and Calderbank, A.R., Spacetime block codes from orthogonal designs. IEEE Trans. Inform. Theory, 45 (1999), 1456-1467. [6] M.M. KAMRUZ ZAMAN and Li Hao, "Performance of Turbo-SISO, Turbo-SIMO, Turbo-MISO and TurboMIMO system using STBC," Journal of Communications, vol. 6, no.8, pp.633-639, 2011. [7] Yu Xiangbin, Xu Dazhuan and Bi Guangguo, Full-rate complex orthogonal space-time block code for multiple antennas. Wireless Pers. Commun., 40 (2007), 81-89.
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[8] Tarokh, V., Jafarkhani, H. and Calderbank, A.R., Spacetime block coding for wireless communications: performance results. IEEE J. Select. Areas Commun. 17(1999), 451-460. [9] Larsson, E.G. and Stoica, P., Space-time block coding for wireless communications. Cambridge University Press, 2003. [10] J. Gao and Y. Bai, "Quasi-Orthogonal Space-Time Block Code with Givens Rotation for SC-FDE Transmission in Frequency Selective Fading Channels," Journal of Communications, vol. 8, no. 12, pp. 832-838, 2013. [11] Zhang, H. Gulliver, T.A., Capacity and error probability analysis for orthogonal space-time block codes over fading channels. IEEE Trans. Wireless Commun., 4 (2005) 808819. [12] Shabazpanahi, S., Beheshti, M., Gershmann, A.B., et al., Miniumum variance linear receivers for multiaccess MIMO wireless systems with space-time block coding. IEEE Trans. Signal Process. 52 (2004), 3306-3312. [13] Sacramento, A. and Hamouda, W., Performance of multiuser-coded CDMA systems with transmit diversity over Nakagami-m fading channels. IEEE Trans.Veh.Technol., 58 (2009), 2279-2287. [14] Seo, B., Ahn, W.-G., Jeong, C. and Kim, H.-M., Fast convergent LMS adaptive receiver for MC-CDMA systems with space-time block coding. IEEE Communication Lett., 14 (2010), 737-739. [15] Z. Li and M. Latva-aho, “Performance of space-time block coded MC-CDMA in Nakagami fading channels,” IEE Electronics Letters, vol. 39, no. 2, pp. 222-224, 2003. [16] Zhang Xiaofei, Feng Gaopeng, Gao Xin, Xu Dazhuan, Blind multiuser detection for MC-CDMA with antenna array. Computers and Electrical Engineering, 36 (2010), 160-168. [17] Qi, S. Aissa, A. Maaref, "Cross-Layer Design for MIMO Orthogonal STBC Systems Over Spatially-Correlated and Keyhole Nakagami-m Fading Channels," Wiley Journal Wireless Communications and Mobile Computing, in press; published online Aug. 2009. [18] Li, H. and Li, J., Differential and coherent decorrelating multiuser receivers for space-time coded CDMA systems. IEEE Trans. Signal Process., 50 (2002), 2529-2536. [19] Proakis, J.G., Digital communications, 5th ed. New York: McGraw-Hill, 2007. [20] Simon, M.K. and Alouini, M.S., Digital communication over fading channels: a unified approach to performance analysis. New York: Wiley, 2000. [21] Gradshteyn, I.S. and Ryzhik, I.M., Table of integrals, series, and products. 7th ed. San Diego, CA: Academic, 2007.
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[22] Verdu, S., Multiuser detection. Cambridge, U.K: Cambridge university press, 1998. [23] Lu, J., Letaief, K.B., Chuang, J.C.-I., et al, M-PSK and MQAM BER computation using signal-space concepts, IEEE Trans. on Commun., 47 (1999), 181-184. [24] Lindsey, W.C., Error probabilities for Rician fading multichannel reception of binary and n-ary signals. IEEE Trans. Inf. Theory, 10 (1964), 339-350.
Dingli Yang received his M.S. degree in electronic and information engineering from Southeast University, China, in June 2006. His current research interest includes communication, digital signal processing, image processing.
Qiuchan Bai received his M.S. degree in electronic and information engineering from Northwestern polytechnical University, China, in June 2006.His current research interest includes image processing, and pattern recognition. Yulin Zhang received his Ph.D. degree in communication and control engineering, from Jiangnan University, China, in 2010. His current research interest includes communication, pattern recognition. Rendong Ji received his M.S. degree in electronic and information engineering from Qufu Normal University, China, in June 2006. His current research interest includes digital signal processing. Now he is pursuing a Ph.D in Nanjing University of Aeronautics and Astronautics. Yazhou Li received his M.S. degree in electronic and information engineering from Chongqing University, China, in June 2008. His current research interest includes communication, image processing. Yudong Yang received his M.S. degree in school of information science and engineering from Southeast University, China, in 2004, and received his Ph.D. degree in School of Electronic and Optical Engineer from Nanjing University of Science & Technology, China, in 2012. His current research interest includes communication, pattern recognition.
