On the Performance of MCCDMA Systems withPartial Combining and Multiple Antennas in FadingChannels
Flavio Zabini
DEIS, University of Bologna, ItalyEmail: fzabini@deis.unibo.it
Barbara M. Masini
IEIIT/CNR, University of Bologna, ItalyEmail: barbara.masini@unibo.it
Andrea Conti
ENDIF, University of Ferrara, ItalyEmail: a.conti@ieee.org
Abstract
—In this paper we analytically evaluate the downlinkperformance of a multicarrier code division multiple access(MCCDMA) system by employing partial combining whenmultiple antennas at the transmitter or at the receiver areconsidered. It is known that for single reception/transmission,the partial combining technique depends on a parameter whichcan be optimized as a function of the number of subcarrier, thenumber of active users and the SNR improving the performancewith respect to other classical techniques (such as maximal ratiocombining, equal gain combining, or orthogonality restoringcombining). In this paper we extend the analysis considering alsomultiple antennas aiming at showing how the spatial diversityaffects the performance as a function of the partial combiningparameter.
I. I
NTRODUCTION
Multicarrier code division multiple access (MCCDMA)consists in a combination of orthogonal frequency divisionmultiplexing (OFDM) and spreading. Systems based on MCCDMA are widely considered to counteract the frequencyselectivity of the wireless channel, to avoid intersymbol andintercarrier interference and achieve high spectral efﬁciency.The basic principle of MCCDMA is to spread each datasymbol over several subcarriers and, hence, to efﬁcientlyexploit frequency diversity of the channel [1]. In this work we consider the MCCDMA architecture presented in [2]and [3], where the spreading is performed in the frequencydomain and WalshHadamard (WH) codes are used withspreading factor equal to the number of subcarriers. We focuson the downlink, hence the system is assumed synchronousand different users experience the same channel. However,in spite of this, in frequency selective fading channels, theorthogonality between spreading sequences is corrupted byfading, thus the performance at the receiver side is affectedby multiuserinterference (MUI) [2], [3].
Focusing on linear combining techniques and assuming perfectchannel state information (CSI), many techniques are knownin the literature trying to minimize the bit error probability: maximum ratio combining (MRC) reduces the effectof noise but completely destroys the orthogonality betweenthe sequences (increasing the MUI), whereas orthogonalityrestoring combining (ORC) reduces the effect of MUI butenhances that of noise. It is shown in [4] that partial combining(PC) technique represent a good tradeoff to reduce both theeffect of noise and MUI. In single transmission and receptionsystems (single input single output  SISO  systems), PCconsists in choosing the complex weighs for the
m
th
subcarrier(
m
= 0
,...,M
−
1
) as:
G
m
=
H
∗
m

H
m

1+
β
, β
∈
[
−
1
,
1]
,
(1)where
H
m
is the
m
th
complex channel coefﬁcient and
β
is thePC parameter to be optimized in order to minimize the bit errorprobability (BEP). It is readily understandable that when
β
=0
,
−
1
and
1
, then (1) coincides with EGC, MRC and ORC,respectively. In [4] the optimal value of
β
for SISO systemswas derived as a function of the number of subcarrier
M
, thenumber of active users
N
u
and the mean signaltonoiseratio(SNR) averaged over smallscale fading,
γ
. In this work weconsider multiple antennas at the transmitter or at the receiverside in order to extend the analytical framework to jointlyconsider PC on both subcarriers and branches when multipleantennas are employed and to verify the improvement achievedby adding spatial diversity. For a deep investigation on howantenna diversity affects MCCDMA systems performance,see, at instance, [5]–[7].
II. S
YSTEM
M
ODEL
Following the MCCDMA architecture presented in [4],each datasymbol is copied over all subcarriers and multipliedby the chip assigned to each particular subcarrier. Binaryphase shift keying (BPSK) modulation is assumed (see Fig.1 for details).
A. Multiple Transmitting Antennas
In Fig. 1(a) the transmitter block scheme for the case of multiple transmitting antennas is presented: the transmittedsignal referred to the
k
th
user on the
l
th
antenna, can be writtenas:
s
(
l,k
)
(
t
) =
2
E
b
LM
+
∞
i
=
−∞
M
−
1
m
=0
c
(
k
)
m
a
(
k
)
[
i
]
g
(
t
−
iT
b
)
×
cos(2
πf
m
t
+
φ
m
)
,
(2)where
E
b
is the energy per bit,
L
the number of antennas,
M
the number of subcarriers. Index
i
denotes the data index,
m
is the subcarrier index,
c
m
is the
m
th
chip of the WH
9781424416455/08/$25.00 ©2008 IEEE1374
code sequence,
a
(
k
)
[
i
]
is the datasymbol transmitted by user
k
during the
i
th
symbol time,
g
(
t
)
is a rectangular pulsewaveform with duration
[0
,T
]
and unitary energy,
T
b
isthe bittime,
f
m
is the
m
th
subcarrierfrequency and
φ
m
isthe random phase uniformly distributed within
[
−
π,π
[
. Inparticular,
T
b
=
T
+
T
g
is the total OFDM symbol duration,increased with respect to
T
of a timeguard
T
g
(insertedbetween consecutive multicarrier symbols to eliminate theresidual inter symbol interference, ISI, due to the channeldelay spread).We now assume to jointly combine both antennas and subcarriers contributions, by weighting the signal on each antennaand subcarrier with the partial combining coefﬁcient,
G
m,l
,deﬁned as in (1) when jointly considering antenna diversity(see Fig. 1(a) as example):
G
m,l
=
H
∗
m,l

H
m,l

1+
β
, β
∈
[
−
1
,
1]
,
(3)Hence, assuming uncorrelated fading channels, the total diversity degree is in the range
[
M,LM
]
depending on the correlation among the antennas. All users exploit all the subchannels(e.g, spreading factor equal to the number of subcarriers),thus, considering perfect channel state information (CSI) atthe transmitter, the total transmitted signal on the
l
th
antennaresults in:
s
(
l
)
(
t
) =
N
u
−
1
k
=0
s
(
l,k
)
(
t
) =
2
E
b
LM
N
u
−
1
k
=0+
∞
i
=
−∞
M
−
1
m
=0
c
(
k
)
m
×
a
(
k
)
[
i
]
G
m,l
g
(
t
−
iT
b
)cos(2
πf
m
t
+
φ
m
)
,
(4)where
N
u
is the number of active users and, because of the useof orthogonal codes,
N
u
≤
M
. Finally, the total transmittedsignal over
L
antennas is given by:
s
(
t
) =
L
−
1
l
=0
s
(
l
)
(
t
)
.
(5)
B. Multiple Receiving Antennas
Following assumptions made in Section IIA (i.e., uncorrelated fading among the subcarriers and perfect CSI), in theopposite case of single transmission and multiple receivingantennas (see Fig. 1(b) for details), the total transmitted signalcan be written as:
s
(
t
) =
2
E
b
LM
N
u
−
1
k
=0+
∞
i
=
−∞
M
−
1
m
=0
c
(
k
)
m
×
a
(
k
)
[
i
]
g
(
t
−
iT
b
)cos(2
πf
m
t
+
φ
m
)
,
(6)III. D
ECISION
V
ARIABLE
We consider the downlink of a MCCDMA system, henceusers are synchronous and different delays affecting eachsubcarriers are assumed perfectly compensated. We considera frequencydomain channel model with transfer function foreach subchannel,
H
(
f
)
, given by:
H
(
f
)
≃
H
(
f
m
) =
α
m
e
jψ
m
,
(7)where
α
m
and
ψ
m
are the
m
th
amplitude and phase coefﬁcients, respectively. Each
H
(
f
m
)
is considered independentidentically distributed (i.i.d.) complex zeromean Gaussianrandom variable (r.v.) (i.e., each subcarrier experiments ﬂatfading, uncorrelated with the other subcarriers) with variance
σ
2
H
deﬁned as:
E
{
α
2
m,l
}
= 2
σ
2
H
= 1
,
(8)where
E
{·}
is the statistical expectation. Hence, we can write:
M
−
1
m
=0
L
−
1
l
=0
E
{
α
2
m,l
}
=
LM,
∀
l,m.
(9)Each
α
m,l
=

H
m,l

is assumed Rayleigh distributed, i.i.d.for each subcarrier and each antennas. In the following, thedecision variable for both multiple transmission and singlereception and single transmission and multiple reception isderived.
A. Multiple Transmitting Antennas
In this case, the received signal can be written as:
r
(
t
) =
2
E
b
LM
N
u
−
1
k
=0+
∞
i
=
−∞
L
−
1
l
=0
M
−
1
m
=0
G
m,l
α
m,l
c
(
k
)
m
a
(
k
)
[
i
]
×
g
′
(
t
−
iT
b
)cos(2
πf
m
t
+
ϑ
m
φ
m
+
ψ
m
) +
n
(
t
)
,
(10)where
g
′
(
t
)
is the response of the channel to
g
(
t
)
with unitaryenergy and duration
T
′
T
+
T
d
, being
T
d
≤
T
g
the timedelay,
n
(
t
)
is the additive white Gaussian noise with twosidepower spectral density (PSD)
N
0
/
2
and
ϑ
m
φ
m
+
ψ
m
.Note that, since
ϑ
m
can be considered uniformly distributedin
[
−
π,π
[
, we can consider
∠
H
(
f
m
)
distributed as
ϑ
m
inthe following. The receiver performs the correlation of thereceived signal with
c
(
n
)
m
√
2cos(2
πf
m
t
+
ϑ
m
)
, thus, after somealgebra, the decision variable results in:
v
(
n
)
=
E
b
δ
d
LM
M
−
1
m
=0
L
−
1
l
=0
α
m,l
g
m,l
a
(
n
)
+
E
b
δ
d
LM
M
−
1
m
=0
L
−
1
l
=0
N
u
−
1
k
=0
,k
=
n
α
m,l
g
m,l
c
(
n
)
m
c
(
k
)
m
a
(
k
)
+
M
−
1
m
=0
n
m
(11)where
δ
d
= 1
/
(1+
T
d
/T
)
represents the loss of energy causedby timespreading and the gains
g
m,l
are normalized as:
g
m,l
=
Gα
−
βm,l
,
(12)being
G
=
LM
M
−
1
m
=0
L
−
1
l
=0
α
−
2
βm,l
.
(13)
1375
Multiplying each term of (11) by
G
, we can write:
v
(
n
)
=
U
E
b
δ
d
LM
M
−
1
m
=0
L
−
1
l
=0
α
1
−
βm,l
a
(
n
)
+
I
E
b
δ
d
LM
M
−
1
m
=0
L
−
1
l
=0
N
u
−
1
k
=0
,k
=
n
α
1
−
βm,l
c
(
n
)
m
c
(
k
)
m
a
(
k
)
+
N
1
LM
M
−
1
m
=0
L
−
1
l
=0
α
−
2
βm,lM
−
1
m
=0
n
m
.
(14)Assuming to have a number of subcarriers sufﬁciently highto justify the use of the law of large numbers (LLN) andexploiting the independence among the subcarriers, by applingthe Kolmogorov’s Law, we can write:
1
M
M
−
1
m
=0
L
−
1
l
=0
α
1
−
βm,l
=
E
L
−
1
l
=0
α
1
−
βm,l
,
(15)
1
M
M
−
1
m
=0
L
−
1
l
=0
α
−
2
βm,l
=
E
L
−
1
l
=0
α
−
2
βm,l
.
(16)Since the channel coefﬁcients are Rayleigh distributed:
E
α
1
−
βm,l
= (2
σ
2
H
)
1
−
β
2
Γ
3
−
β
2
,
∀
m, l
(17)
E
α
−
2
βm,l
= (2
σ
2
H
)
−
β
Γ(1
−
β
)
,
∀
m, l,
(18)being
Γ(
·
)
the EulerGamma function. By substituting (17)and (18) in (15) and (16), respectively, and exchanging thesum operation with the statistical expectation, we have:
1
M
M
−
1
m
=0
L
−
1
l
=0
α
1
−
βm,l
=
L
(2
σ
2
H
)
1
−
β
2
Γ
3
−
β
2
(19)
1
M
M
−
1
m
=0
L
−
1
l
=0
α
−
2
βm,l
=
L
(2
σ
2
H
)
−
β
Γ(1
−
β
)
.
(20)Therefore, we can write:
U
=
E
b
δ
d
LM
(2
σ
2
H
)
1
−
β
2
Γ
3
−
β
2
,
(21)
N
=
(2
σ
2
H
)
−
β
Γ(1
−
β
)
M
−
1
m
=0
n
m
.
(22)Since
n
m
are independent Gaussian zeromean random variables with variance
N
0
/
2
, the noise term
N
is gaussiandistributed as:
N
∼N
0
,σ
2
N
,
(23)where
σ
2
N
(2
σ
2
H
)
−
β
Γ(1
−
β
)
MN
0
2
.
(24)For what concern the interference term
I
, we can concatenatethe sums in
m
and
l
in one vector of dimension
LM
and,by exploiting the independence between the coefﬁcients andapplying the methodology proposed in [4], we ﬁnd out:
I
∼N
0
,σ
2
I
(25)where
σ
2
I
E
b
δ
d
(
N
u
−
1)
ζ
β
(
α
)
(26)with
ζ
β
(
α
) = (2
σ
2
H
)
1
−
β
Γ(2
−
β
)
−
Γ
2
3
−
β
2
.
(27)
B. Multiple Receiving Antennas
With multiple receiving antennas, the contributions of eachantenna have to be weighted with the complex coefﬁcients
G
m,l
which in multiple transmission case has been applied atthe transmitter side, as below:
r
(
t
) =
L
−
1
l
=0
G
l,m
r
(
l
)
(28)being
r
(
l
)
(
t
)
the signal received on the
l
th
antenna. Hence, wecan write:
r
(
t
) =
2
E
b
LM
N
u
−
1
k
=0+
∞
i
=
−∞
L
−
1
l
=0
M
−
1
m
=0
G
m,l
α
m,l
c
(
k
)
m
a
(
k
)
[
i
]
×
g
′
(
t
−
iT
b
)cos(2
πf
m
t
+
ϑ
m
φ
m
+
ψ
m
) +
L
−
1
l
=0
n
l
(
t
)
,
(29)being
n
l
(
t
)
the noise at the
l
th antenna supposed to beGaussian with twoside power spectral density (PSD)
N
0
/
2
.For what concern the evaluation of the decision variable, theprocedure is similar to the one presented in [4], except for thefurther degree of freedom given by spatial diversity. Hence,the statistical distribution of
U N
and
I
results in:
U
∼ N
E
b
δ
d
ML
E
{
α
1
−
βl
}
,σ
2
U
N
∼ N
0
,σ
2
N
=
MLN
0
2 (2
σ
2
H
)
−
β
Γ(1
−
β
)
I
∼ N
0
,σ
2
I
=
E
b
δ
d
L
(
N
u
−
1)
ζ
β
(
α
)
.
IV. P
ERFORMANCE
E
VALUATION
In this section we derive the BEP and the optimum PCparameter,
β
, deﬁned as the value of
β
minimizing the BEP.
A. Multiple Transmitting Antennas
Following the previously mentioned assumptions, the BEPcan be evaluated as:
P
b
= 12
erfc
U
2(
σ
2
I
+
σ
2
N
)
= 12
erfc
√
L
·
SNIR
,
whereSNIR
=
γ
Γ
2
3
−
β
2
2
N
u
−
1
M
γ
Γ(2
−
β
)
−
Γ
2
3
−
β
2
+ Γ(1
−
β
)
,
1376
represents the signaltonoise plus interference ratio for thesingle transmission/reception scenario described in [4] and
γ
= 2
σ
2
H
E
b
δ
d
N
0
,
(30)is the mean SNR averaged over smallscale fading. Note that,being SNIR equal to the single transmission/reception case,the optimum value of
β
minimizing the BEP does not changewith multiple transmitting antennas, but the performance isimproved by the spatial diversity.
B. Multiple Receiving Antennas
The unconditioned BEP with
L
receiving antennas is givenby:
P
b
≃
12
erfc
L
−
1
l
=0
E
b
δ
d
N
0
L
(2
σ
2
l
)
1
−
β
2
Γ
3
−
β
2
(31)
×
1
L
−
1
l
=0
E
b
δ
d
N
0
L
2
N
u
−
1
M
σ
2
I
(
β,l
) + (2
σ
2
l
)
−
β
Γ[1
−
β
]
being
σ
2
I
(
β,l
) = (2
σ
2
l
)
1
−
β
Γ[2
−
β
]
−
Γ
2
3
−
β
2
(32)
+
L
−
1
l
′
=0
,l
′
=
l
σ
2
l,l
′
(
β
)
−
12(2
σ
2
l
)
1
−
β
2
(2
σ
2
l
′
)
1
−
β
2
Γ
2
3
−
β
2
,
where, due to the independence among the subcarriers,
σ
2
l,l
′
(
β
) = 0
.
5
·
E
{
α
1
−
βm,l
α
1
−
βm,l
′
}
,
∀
m
. In case of uncorrelatedantennas, the second term in (32) is zero, thus (31) reduces tothe following expression:
P
b
≃
12
erfc
√
SNIR
,
(33)whereSNIR
=Γ
2
3
−
β
2
γ
2
N
u
−
1
M
Γ(2
−
β
)
−
Γ
2
3
−
β
2
γ L
+ Γ(1
−
β
)
.
(34)It is immediate to verify that when
L
= 1
, (33) reduces to theexpression found in [4]. Note that (32) is general, thus validalso for correlated fading among the antennas.We now aim at evaluation the optimum value of
β
whichminimizes the BEP. It can be found out that for spatialuncorrelated antennas:
β
(opt)
= argmin

β
{
P
b
}
= argmax

β
{
SNIR
}
.
(35)By deﬁning the parameter
ξ .
= 2
γ N
u
−
1
M ,
(36)we ﬁnd that (36) leads to:
ξ
=
L
1Ψ
3
−
β
2
−
Ψ(1
−
β
)+
β
−
1
−
1
,
(37)where
Ψ(
x
)
is the logarithmic derivative of the Euler Gammafunction. Note that
ξ
is a monotonic function of
β
, thusinvertible, and it conﬁrms that if the system is noiselimitedMRC is optimum, while if the system is interference limited,a choice close to ORC is required.Is also possible to observe that for high values of mean SNR,
γ
, the second term at the denominator of (30) is negligible,thus, for interferencelimited systems the SNIR evaluatedfor multiple transmissions agrees with the one of multiplereceptions.V. N
UMERICAL
R
ESULTS
The mean BEP as a function of
β
for various SNRs,
γ
,and diversity degrees,
L
, at the transmitter and at the receiveris shown in Fig. 2 in fully loaded system conditions (
N
u
=
M
= 1024
). In Fig. 2(a) the improvement on the BEP given byspatial diversity at the transmitter can be appreciated especiallyincreasing the mean SNR,
γ
. It is also noticeable that theperformance strongly depends on
β
. Same conclusions can bedrawn by observing Fig. 2(b), where the BEP vs.
β
is plottedconsidering multiple receiving antennas. From a comparisonbetween the two ﬁgures, it appears that the optimum value of
β
is in the range [0.4, 0.7] and, on equal diversity degrees,multiple transmission performs better than multiple reception.
1
This is caused by summation of noise on all the antennasbranches when multiple reception is assumed.When multiple antennas at the receiver are considered, Fig.3 shows the mean BEP as a function of the mean SNR
γ
in[dB] for
β
= 0
.
5
and various diversity degrees,
L
. Fully loaded(
N
u
=
M
= 1024
) and half loaded (
N
u
=
M/
2
) systemconditions are considered. It is possible to observe that, byﬁxing a target BEP,
P
b
= 10
−
3
, it can be achieved only if thesystem is half loaded with single reception, whereas a fullyloaded system can be considered by adopting two antennas atthe receiver.A
CKNOWLEDGMENT
The authors would like to thank Prof. Oreste Andrisano forsupporting this research activity and Prof. Larry Greensteinfor motivating this work.R
EFERENCES[1] I. Cosovic amd S. Kaiser, “A Uniﬁed Analysis of Diversity Exploitation inMulticarrier CDMA”, IEEE Transactions on Vehicular Technology, Vol.56, No. 4, July 2007, Page(s): 2051  2062.[2] N. Yee, J.P. Linnartz and G. Fettweis, ”MultiCarrierCDMA in indoorwireless networks”, in Conference Proceedings PIMRC ’93, Yokohama,Sept, 1993. p 109113[3] N. Yee and J.P. Linnartz, ”BER of multicarrier CDMA in an indoorRician fading channel”, Conference Record of The TwentySeventhAsilomar Conference on Signals, Systems and Computers, 1993., 13Nov. 1993 Pages:426  430 vol.1[4] A. Conti, B. Masini, F. Zabini and O. Andrisano, “On the downlink Performance of MultiCarrier CDMA Systems with Partial Equalization”,IEEE Transactions on Wireless Communications, Vol. 6, Issue 1, Jan.2007 Page(s):230  239.
1
Note that, being in the downlink, multiple transmission can be easilyimplemented at the base station.
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(a) Multiple transmission and single reception. (b) Single transmission and multiple reception.Fig. 1. Multiple transmission and multiple reception block schemes.
−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
β
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
P
b
γ
=5dB
γ
=10dB
γ
=15dBL=1L=2L=4
(a) Multiple transmission and single reception.
−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
β
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
P
b
L=1L=2L=4
γ
=5dB
γ
=10dB
γ
=15dB
(b) Single transmission and multiple reception.Fig. 2. Mean BEP as a function of
β
for different SNRs,
γ
, and diversity degrees,
L
.
0 5 10 15
γ
[dB]10
−4
10
−3
10
−2
10
−1
10
0
P
b
L=1, N
u
=ML=2, N
u
=ML=4, N
u
=ML=1, N
u
=M/2L=2, N
u
=M/2L=4, N
u
=M/2
Fig. 3. Mean BEP as a function of mean SNR for
β
= 0
.
5
, various diversitydegrees,
L
, at the receiver side and fully or half loaded conditions.[5] J. Tang, X. Zhang, “Transmit selection diversity with maximalratiocombining for multicarrier DSCDMA wireless networks over Nakagamim fading channels”, Selected Areas in Communications, IEEE Journal on,Volume 24, Issue 1, Jan. 2006 Page(s):104  112 .[6] LL. Yang, L. Hanzo, “Performance of broadband multicarrier DSCDMA using spacetime spreadingassisted transmit diversity”, WirelessCommunications, IEEE Transactions on, Volume 4, Issue 3, May 2005Page(s):885  894.[7] Y. Zhang, L. B. Milstein, P. H. Siegel, “Tradeoff between diversitygain and interference suppression in a MIMO MCCDMA system”,Communications, IEEE Transactions on Volume 53, Issue 4, April 2005Page(s):623  631.
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