MC-CDMA with Quadrature Spreading over. Frequency Selective Fading Channels. Slimane Ben Slimane. Radio Communication Systems. Department of ...
MC-CDMA with Quadrature Spreading over Frequency Selective Fading Channels Slimane Ben Slimane
Radio Communication Systems Department of Signals, Sensors and Systems Royal Institute of Technology 100 44 Stockholm, Sweden Abstract | MultiCarrier CDMA (MC-CDMA) schemes resolve the frequency selectivity in multipath fading channels and have good spectral properties. Spreading the information over many subcarriers adds exibility to the system and a redundancy between subcarriers that can be combined nicely with the derived frequency diversity out from the fading channel. In this paper the performance of a MC-CDMA scheme over frequency selective, slowly fading channels is studied analytically and by computer simulations. The MC-CDMA system uses a BPSK modulation, quadrature (complex) spreading codes, and a simple correlator receiver. Quadrature spreading codes are used to reduce the e ect of the multipath fading channel and restore some of the orthogonality losses between users. The obtained results show considerable performance improvement compared to conventional Orthogonal Frequency Division Multiplexing (OFDM) scheme and MCCDMA schemes that use non-quadrature spreading. The effect of frequency o sets on the system performance is also addressed in this paper.
I. Introduction
N mobile communication systems, the system capacity
Iand performance are limited by the multipath fading chan-
nel. The most common approach in dealing with this channel e ect is by means of diversity where the system tries to take advantage and exploits the multipath channel. MultiCarrier CDMA (MC-CDMA) techniques have such capability and can be eciently used for such applications. The principle of MC-CDMA has been introduced not long ago and these schemes have attracted a lot of research interest since [1-10]. MC-CDMA systems are known to have good spectral properties and are robust against frequencyselective fading. Orthogonal Frequency Division Multiplexing (OFDM) modulation is rst used to reduce Intersymbol Interference (ISI) by using narrowband subcarriers and thus resolving the frequency selectivity of the channel. Orthogonal spreading codes are then used to create redundancy and increase the diversity gain of the system. Orthogonal codes are easily generated using Hadamard matrices [11]. However, the biggest challenge in multicarrier systems is to preserve such orthogonality after the transmitted signal has gone through the channel, since orthogonality loss creates crosstalk between users and introduces an error oor in the system performance that cannot be reduced by just increasing the transmitted power. Many MC-CDMA detectors have been proposed in the lit-
erature and analyzed. The simplest is a conventional correlator detector with Channel State Information (CSI) (called Orthogonality Restoring Correlation (ORC) detector [3]). This technique illuminates the error oor, but at the expense of a noise ampli cation, giving an overall system performance worse than that of conventional OFDM scheme. In reducing this noise ampli cation, a correlator detector with Threshold (TORC) has been studied in [2], [8], and a TORC with multi-stage detection in [10]. These detectors reduce the error oor and their performance depends on the threshold. The optimum receiver for MC-CDMA is based on the Maximum Likelihood Sequence Estimation (MLSE) [4]. However, its complexity increases exponentially with the number of users and can only be used for very small numbers. All detectors that have been proposed and studied consider non-quadrature spreading codes. Quadrature (complex) spreading codes are usually used to combat intentional jamming by forcing the jammer to split its power between the quadrature components of the carrier signal. For MC-CDMA systems, quadrature spreading can be used to reduce the e ect of multipath fading and reduce interference from other users. In this paper a MC-CDMA system that uses quadrature spreading codes is studied. Its performance over frequency selective, slowly fading channels is analyzed analytically and by computer simulations. Quadrature spreading codes are used to reduce interference between users, improve the received signal level, and thus reduce the complexity of the MC-CDMA detector. Very good performance is achieved by just using a simple correlator receiver with and without CSI. In fact, the obtained results are considerably better than that of a conventional OFDM scheme and MC-CDMA schemes with non-quadrature spreading codes. II. System Model
A MC-CDMA scheme that uses a BPSK baseband modulation scheme and a quadrature (complex) spreading sequence is considered in this paper. The system consists of Nu users, each using an orthogonal Hadamard spreading code followed by an OFDM modulator. The transmitter block diagram of user n is shown in Figure 1. The input data is rst modulated in baseband using BPSK, multiplied by the complex spreading sequence, and then transmitted using N
orthogonal subcarriers. As shown in Figure 1, the code on the quadrature side is just a cyclic shift of that of the inphase side. The choice of such an arrangement will become clear in the following. Consider an OFDM block, the transmitted MC-CDMA signal can be written as follows:
x(t) =
Re xn (t)ej(2fc t+)
n=0
= Re where
n
NX u ?1
("NX ?
1
m=0
sm ej2 mT t
#
o
ej(2fc t+)
r
NX u ?1 sm = 2ETb (cm;n + jcN ?1?m;n)an ; n=0
)
(2)
N
and an = 1, with equal probability, is the information from user n during the considered OFDM block. Transmitted through the channel, the MC-CDMA signal is attenuated by the multipath fading channel and also affected by the AWGN channel. The mobile radio channel is modeled as a tapped delay line with impulse response
h(t) =
PX ?1 i=0
i (t ? i )
(3)
where i is a Rayleigh random variable tap weight, i is the relative time delay of the ith path, and P is the number of paths. At the receive end, the received signal is rst downconverted to baseband using a local oscillator (fl = fc ) as shown in Figure 2, correlated at each subcarrier frequency, and then sampled at the block rate. For a slowly varying frequency selective fading channel and an OFDM guard interval larger that the maximum delay spread of the channel, the received sample at the mth subcarrier is free of intersymbol interference and is given by ZT 1 r = p r(t)e?j2 mT t dt m
T
0
p
(4) = hm ej m Tsm + zm where sm is as de ned in (2), zm is a complex Gaussian random variable with zero mean, and
hm ej m =
PX ?1 i=0
i e?j2 i (fc + mT )
(5)
is a complex Gaussian random variable representing the channel attenuation factor at subcarrier m, which is regarded as constant over one block interval. The hm 's are
e
c1,n+jcN-2,n
e jw1
t
an
BPSK
multiplex
(1)
is the transmitted signal at subcarrier m, cm;n are elements of a Hadamard orthogonal code of length N , with jc j = p1 ; m;n
input data
jw0 t
c0,n+jcN-1,n
cN-2,n+jc1,n
e jwN-2 t
cN-1,n+jc0,n
e jwN-1 t
xn(t)
Fig. 1. Transmitter block diagram for one user of a MC-CDMA that uses quadrature spreading. cos (wl t) r(t)
local oscillator
user 1 decorrelator receiver
phase rotation
user separation & detection user N
sin (wl t)
Fig. 2. Multicarrier CDMA receiver block diagram.
identically distributed Rayleigh random variables with probability density function (pdf)
fh (x) = 2xe?x2 ; x 0;
(6) and the m 's are identically uniformly distributed over [0 2[. For a suciently slow fading channel, the phase shift m can be estimated without error. Thus, considering the output at all subcarriers and after phaseTrotations, we obtain a received vector r = [r0 ; r1 ; ; rN ?1 ] , given by the following
r
r = E2b HCa + z
(7)
where H = diagfh0; h1 ; ; hN ?1 g, is a diagonal matrix, 3 2 c + jc 0;0 N ?1;0 c0;N ?1 + jcN ?1;N ?1 75 ; (8) .. .. ... C = 64 . . cN ?1;0 + jc0;0 cN ?1;N ?1 + jc0;N ?1 is an N N square matrix, a is the data information vector from all users, and z is a vector of complex Gaussian random variables with zero mean and variance N0 =2. A. The MC-CDMA Detector Consider the following two elements of the received vector in (7),
r
NX u ?1 (cm;k + jcN ?1?m;k ) ak + zm ; rm = hm E2b k=0
r
NX u ?1 rN ?1?m = hN ?1?m E2b (cN ?1?m;k + jcm;k ) ak k=0
The conditional bit error probability of the system for a given Channel State Information (CSI) vector, h0 = (h00 ; h01; ; h0N=2?1 ), can be written as
+zN ?1?m;
0v u Eb P (errorjh0 ) = 12 erfc B t 4N PN=N? @u
and de ne two new element vm and vN ?1?m as follows:
vm = Re frm g + Im frN ?1?m g vN ?1?m = Im frm g + Re frN ?1?m g :
(9)
By doing the above combination for all values of m a new vector v = [v0 ; v1 ; ; vN ?1 ]T is obtained with
p
v = Eb H0C0a + z0 ;
where with
0
1
2 1
2 1
hN ?1?m ; m = 0; 1; ; N=2 ? 1; h0m = hm + p 2
1
0
p ? fh (x) = 2xe?2x2 + 2x2 ? 1 e?x2 erf(x); 0
erf() is the error function,
2 c ; 6 . 0 . C =4 .
00
...
c0;N ?1 .. .
cN ?1;0 cN ?1;N ?1
3 75 ;
(12)
is the N N square matrix whose columns are the spreading sequences, and z0 is a vector of Gaussian random variables with zero mean and variance N0 . Using the above vector, the correlator detector removes the interference between di erent information symbols by multiplying the above vector with the inverse of H0 ,
p
0N= ? min @ X
h02
B. Performance Analysis For a given user n, the received sample at a given block interval can be written as 2?1 p N=X cn;i zi0 + cn;N ?1?i zN0 ?1?j yn = an Eb + h0i i=0
2 1
N=2
1? 1 A h0 : min h0i 1
i=0
(14)
2
2
The above inequality can be used to write two bounds (upper and lower) on the average bit error probability of the system. The lower limit in (14) is obtained only when all h0i 's have the same value. Since the probability of such an event is very close to zero for large values of N , the corresponding bound (upper bound) is expected to be loose and is given here for illustration only. Using (14) we can write
PL P b P U ; where
PL =
Z
11
+
2 erfc
0
1
of two fading amplitudes, the noise enhancement will be very small in this case. The probability of having a noise ampli cation is only about 11% as compared to 63% for a MCCDMA system that uses non-quadrature spreading. The receiver can then use the signals from all subcarriers and full orthogonality restoring is obtained. Notice that a TORC receiver can also be used here but it does not appear necessary since as just mentioned there is hardly noise enhancement. User separation is then achieved by multiplying the vector y by the matrix C0T followed by a hard-decision detector for each user.
(13)
we can write the following inequality
y = EbH0? H0C0a + H0? z0 : Since the elements of the diagonal matrix H0 are the sum 1
0
h0min = min h00 ; h01 ; ; h0N=2?1 ;
(11)
having as pdf
1
hi 2
where erfc() is the complementary error function. De ning h0min as
(10)
H0 = diagfh0 ; h0 ; ; h0N= ? ; h0N= ? ; ; h0 ; h0 g
2 1
i=0
0
1 CA ;
PU =
Z
11
+ 0
2 erfc
!
r
Nx 4ENb fhmin (x)dx; 0 2
r E ! x b fh 2
(15)
0
2N0
min (x)dx:
(16)
0
The function fhmin () is the pdf of the random variable h0min and is given by 0
d he?2x2 + pxe?x2 erf(x)i N2 ; x 0: (17) fhmin (x) = ? dx 0
The two bounds can then be written as a function of a single integral
q
PL = 12 ? q N2 NEb0 IN ; PU = 21 ? NEb0 IN ; where
IN = p1
2
Z
1
+ 0
h p
e?x2 N (1+Eb =4N0 ) 1 + xex2 erf(x)
(18)
iN 2
dx:
C. Impact of Frequency Errors on the Performance In multicarrier systems, frequency asynchronism destroys the orthogonality between subcarriers and therefore attenuates the signal and introduces ISI between symbols of an OFDM block. To get rid of both attenuation and interference frequency o sets should be compensated for before the correlator receiver. This may not always be possible especially for a system with narrowband subcarriers and over time varying channels. In this section we look at the effect of frequency o sets on the performance of MC-CDMA systems when quadrature spreading are used. Assume that the local oscillator frequency is given by
fl = fc + fD ; where fD is a constant frequency o set with jfD T j 1. The received sample at the mth subcarrier, given in (4), becomes
p
rm = ej(fD T +D ) sinc(fD T )hmej m Tsm NX ?1 p j i Tsi ej(fD T +D ) sinc(fD T ) hi ei?m + zm ; + 1+ fD T
i=0
i6=m
where D is the phase of fD and sinc(x) = sin(x)=(x). We notice that due to the frequency o set, each symbol is rotated by an angle of fD T + D . The above expression also shows that all symbols within a block are attenuated and interfere with each other. For a constant frequency o set, the phase rotation is the same for all symbols and can be estimated. In the sequel it is assumed that the phase rotator is able to estimate this phase at each subcarrier without error. The output of the correlator receiver from all subcarriers then becomes
r
r = sinc (fD T ) E2b FHCa + z
where F is an N N square matrix representing the e ect of the frequency o set and is given by F = [fij ] ; (19) where
fij = f TfD+Tj ? i ej( j ? i ) : D
(20)
It is easy to verify that for small values of fD T , the matrix F becomes the sum of an identity matrix and a skewHermitian matrix. This indicates that if the same information is transmitted at subcarriers m and N ? 1 ? m, the two received signals will experience the same ISI but with opposite signs (assuming an ideal channel, i.e., hi = 1 and i = 0). Therefore, adding these two signals will cancel the ISI term and reduces the e ect of frequency o set errors to just an attenuation of the signal which is similar to the case of a single carrier system. For a multicarrier CDMA system, this same results are obtained when arranging the quadrature spreading codes as in (2) and combining the received signals as described in (9).
Carrying out the above mentioned combination, the vector
v of (10) becomes p v = sinc (fD T ) Eb F0C0a + z0; where C0 , a, z0 are as de ned in (10), and h i F0 = fij 0
with
(21) (22)
j ? i ? =4) fij = fD T hj cos( fD T + j ? i h cos( N ? 1?j N ? 1?j ? N ?1?i + =4) : (23) + fD T ? j + i 0
It is observed that when there is no fading (AWGN only) and for small frequency o sets (1 + fDnT fDnT ), the matrix F0 reduces to an identity matrix and all interference due to a frequency o set is cancelled. However, under multipath fading conditions some of this interference will remain. The system performance will then depend on the phase variations of the multipath fading channel. As most of the interference comes from the two neighbouring subcarriers, the matrix F0 converges to a diagonal matrix for slowly varying channels. III. Numerical Results A MC-CDMA scheme with Nu = N = 64 (assumed equal
to the total number of the OFDM subcarriers) users is considered in this section. The system uses BPSK modulation with ideal coherent detection. The performance is evaluated for a full system and is compared to that of a regular OFDM and also to a MC-CDMA with non-quadrature spreading. Figure 3 gives simulation results for the system average bit error probability as a function of Eb =N0 . Compared to regular OFDM, we notice a considerable performance improvement when CSI is used (orthogonality restoring) and comparable performance for the case of no CSI. When no CSI is used, code orthogonality is partially restored and half of the interference from other users is cancelled. Also given in Figure 3 is a lower bound on the average bit error probability using (15). It is observed that this bound is very tight for high SNRs and can be used as good estimate and avoid lengthy simulations. Figure 4 compares the performance of the system to that of a MC-CDMA that uses a non-quadrature spreading code. Both systems use a correlator receiver. Notice that the error
oor when quadrature spreading is used is only 3:5 10?3 as compared to 2:7 10?2 for the case of non-quadrature spreading. When a non-quadrature spreading is used, the performance is always worse than that of a regular OFDM scheme and this is due to the noise enhancement caused by the channel coecients. This statement can be easily veri ed by computing a lower bound on the bit error probability using (15). That is,
Pb
Z
0
11
+
2 erfc
!
r
Eb f (x)dx; Nx N hmin 0 2
(24)
MC−CDMA with quadrature spreading, N=64
0
10
−1
−1
10
10
−2
−2
−3
10 Bit Error Probability
Bit Error Probability
10
10
Performance of MC−CDMA, N=64
0
10
lower bound
−4
10
with CSI
−3
10
−4
10
with CSI −5
−5
10
10 OFDM MC−CDMA without CSI MC−CDMA with CSI
regular OFDM MC−CDMA with quadrature spreading MC−CDMA with non−quadrature spreading
−6
10
−6
4
6
8
10
12 Eb/No, dB
14
16
18
20
10
4
6
8
10
12 Eb/No, dB
14
16
18
20
Fig. 3. Performance of MC-CDMA system with quadrature spreading over frequency-selective, slowly fading channels, full interleaving is assumed and perfect CSI.
Fig. 4. Performance of MC-CDMA systems over frequency-selective, slowly fading channels with quadrature spreading and nonquadrature spreading, full interleaving is assumed and perfect CSI.
where
complicated receivers such as an MLSE receiver. The e ect of frequency o sets on the system performance has been addressed in this paper and analized. It has been shown that the performance is dependent on the phase variations of the fading channel. For slowly varying channels, quadrature spreading can help reduce the e ect of frequency o sets.
fhmin (x) = N [1 ? Fh (x)]N ?1 fh (x) 2 = 2Nxe?Nx ; x 0;
(25)
with fh (x) as given in (6). Carrying out the integration in (24), the lower bound reduces to
" s
#
0 ; Pb 12 1 ? 1 +EbE=N=N b 0
(26)
which is the bit error rate of the regular OFDM scheme and is completely independent of N . IV. Conclusions
In this paper, we have studied the performance of a multicarrier CDMA system over frequency selective fading channels. This system uses BPSK modulation, quadrature spreading codes, and a simple correlator receiver with and without Channel State Information (CSI). The two spreading codes of each user were arranged in way that helped restore some of the orthogonality losses between codes caused by the frequency selectivity of the channel. This consisted of using one code on the inphase side and a cyclic shift of the same code on the quadrature side. As a result, the interference from other users has been reduced and the performance of the system has been considerably improved. The system performance has been examined analytically and by computer simulations. The obtained results showed that a MC-CDMA system with quadrature spreading performs much better than a conventional OFDM system and MC-CDMA systems with non-quadrature spreading. This performance improvement is obtained without the need for
References [1] K. Fazel, S. Kaiser, and M. Schnell, \A exible and high performance mobile communications system based on orthogonal multicarrier SSMA," Wireless Personal Communications 2, pp. 121144, 1995. [2] T. Mueller, K. Brueninghaus, and H. Rohling, \Performance of coherent OFDM-CDMA for Broadband mobile communications," Wireless Personal Communications 2, pp. 295-305, 1996. [3] N. Yee and J.-P. Linnartz, \Controlled equalization of multicarrier CDMA in an indoor Rician fading channel," Proc. IEEE VTC '94, pp. 1665-1669, Stockholm 1994. [4] K. Fazel, \Performance of CDMA/OFDM for mobile communication systems," ICUPC '93, pp. 975-79, 1993. [5] N. Yee, J.-P. Linnartz, and G. Fetteweis, \Multi-carrier CDMA in indoor wireless networks," IEICE Trans. Commun., Japan, vol. E77-B, pp. 900-904, July 1994. [6] L. Vanderdorpe, \Multitone spread spectrum multiple access communications system in a multipath Rician fading channel," IEEE Trans. Veh. Techn., vol. 44, NO. 2, pp. 327-337, May 1995. [7] E. A. Sourour and M. Nakagawa, \Performance of orthogonal multicarrier CDMA in a multipath fading channel," IEEE Trans. Commun., vol. 44, NO. 3, March 1996. [8] N. Yee and J. Linnartz, \Controlled equalization of MC-CDMA in an indoor Rician fading channel," VTC '94, pp. 1665-69, 1994. [9] L. J. Cimini, \Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing," IEEE Trans. Commun., vol. COM-33, no. 7, July 1985. [10] D. N. Kalofonos and J. G. Proakis, \Performance of the multistage detector for a MC-CDMA system in a Rayleigh fading channel," Globecom '96, pp. 1784-88, 18-22 November 1996, London, UK. [11] J. G. Proakis, Digital Communications, McGraw Hill, Third Edition, 1995.
