LTE

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Published in IET Communications
Received on 2nd May 2010
Revised on 20th April 2011
doi: 10.1049/iet-com.2010.0344

ISSN 1751-8628

Random characteristics of carrier frequency
offset and a joint fading branch correlation
in an asynchronous multi-carrier coded-division
multiple-access system
J.I.-Z. Chen
Department of Electrical Engineering, Dayeh University, 168 University Road, Dasuen, Changhwa 51505, Taiwan
E-mail: jchen@mail.dyu.edu.tw

Abstract: The carrier frequency offset (CFO) is one of the critical parameters for degrading the overall performance of radio
systems modulated with a scheme of multiple carriers. The evaluation of system performance for a multi-carrier coded-
division multiple-access (MC-CDMA) radio system with CFO effect mentioned previously is investigated in this study. Apart
from the aforementioned parameter, the other important parameter referred to as fading branch correlation (FBC) occurring in
the propagating channel is rarely applied in the study of MC-CDMA systems’ performance. For the sake of simplicity,
subcarriers are typically considered independently. However, the randomisation incurred in generating CFO and FBC should
be simultaneously considered to calculate the overall performance. The focus of this work is an aggregate investigation to
determine which parameter, that is, CFO or FBC, primarily dominates the performance of an MC-CDMA system. Assume
that applying the same quantity of CFO (1) and FBC (l ) in serving the simulation, that is, both values of CFO and FBC are
assumed and assigned in the interval of (0.4, 1.0). Moreover, several novel formulas include both CFO and FBC parameters
and many three-dimensional curves are presented and illustrated in this article.

1 Introduction subcarriers is not optimal as opposed to the uncorrelated
case, and a better power-loading scheme was proposed.
It is well known that the phenomenon of carrier frequency To maintain high bandwidth efﬁciency in advance, an MC-
offset (CFO) is mainly caused by the reason of frequency CDMA system is subject to correlated fading for different
mismatch, which can be caused by Doppler shift because of subcarriers because the frequency space between adjacent
the vehicle motion or the frequency differences between the subcarriers of the system is inadequate; thus, corrected
transmitter and the receiver oscillator [1]. In multi-carrier fading usually occurs among the spatially separated receiver
wireless systems, CFO is going to give rise to inter-carrier antennas. The fading branch correlation (FBC) problem that
interference (ICI) which thereby incurs the degradation of occurred in an MC-CDMA system that combines with the
system performance for amulti-carrier coded-division maximum ratio combining (MRC) scheme at the receiver
multiple-access (MC-CDMA) system. In the past, there has ever been investigated and published by several
have been several researches that were focused on the researches. In [6], the calculation of moment-generating
issues of ICI caused by the CFO for the system using function (MGF) was adopted as one mean to analyse the
multi-carrier signalling techniques. Most recently, in [2] the system performance with BER for an MC-CDMA system
authors derived an analytical expression of the error over an independent and a correlated Nakagami-m fading
probability of orthogonal frequency division multiplexing channel. The performance of an MC-CDMA system with
(OFDM) system for CFO. In [3], the authors study is in an MRC diversity working over correlated Nakagami-m
two folds: a probability density function of multiple access fading in a multiple-cell environment was calculated in [7].
interference (MAI) plus background noise was not only In [8] branch correlation over a shadowed fading channel
derived for the asynchronous uplink CDMA system, but the was investigated by implementing both macro- and micro-
methods were also extended to examine the impact of CFO diversity techniques, assuming an MRC at the micro level
on the bit error rate (BER) performance for an MC-CDMA and selection combining (SC) at the macro level,
system over a frequency-selective fading channel. On the respectively. Nevertheless, the aforementioned publications
basis of considering CFO error in different kinds of almost exclusively considered the phenomena of FBC and
propagation fading channels, the evaluation of system CFO independently only; however, in the real world, in the
performance for an OFDM system was held in [4]. The stage of wireless communication systems, the relationship
authors in [5] proved that uniform loading of the active between FBC and CFO for an MC-CDMA system should

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doi: 10.1049/iet-com.2010.0344 & The Institution of Engineering and Technology 2011

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be taken into account. On the other hand, in fact, the came from the parallel-to-serial converter is given as
correlation between adjacent subcarriers would take place in
most radio environments with the modulation technique in M −1
multiple carriers. The analysis of system performance for an S0 (t) = s(0) PTC (t − vTc ) (1)
v
MC-CDMA system might be over-estimated (optimistic) v=0
owing to the assumption of the independence of fading
subcarriers. It is valuable and interesting to explore the
FBC and CFO effect together with signature sequences. where PTC (t) is defined as the unit amplitude pulse over the
Therefore in this study, the concept of adopting to join the interval of chip time [0, Tc], and
CFO and FBC, which were considered as two key
parameters, for an MC-CDMA system with an MRC 1 M −1 (0)
operating over a correlated Rayleigh fading channel is s(0) =
v j P (v), v = 0, 1, . . . , M − 1 (2)
examined, with the analysis of BER performance. M m=0 m h
The rest of the paper is organised as follows. The system
models of an MC-CDMA system are established in Section 2. where M denotes the number of subcarriers, which supports
In Section 3, the first and second statistical moments are each user operating within a cell, that is, M ¼ ⌊N/K⌋,
illustrated then the BER data of an MC-CDMA system are Ph(v) ¼ exp[j2pmv/M ], v ¼ 0, 1, . . . , M 2 1 represent the
evaluated. In Section 4, the results from computer analysis are phase components in the transmitter. To ensure frequency
presented. There is a brief conclusion drawn in Section 5. non-selectivity in each subchannel in an MC-CDMA
system, it is often necessary to select a sufficiently large
number of subcarriers. Consequently, such selection also
2 MC-CDMA system models causes a reduction in frequency separation between adjacent
subcarriers. In (2), j(0) is defined as
m
In this section, the scenarios of an MC-CDMA system are
described. Consider that the system is equipped with
jm = b(0) a(0) Ci,m ,
(0)
i
(0)
m = 0, 1, . . . , M − 1 (3)
transmitter and receiver models for an activating mobile
unit (referenced user) and illustrated in Fig. 1. A
performance analysis of an MC-CDMA system is first where b(0) [ [−1, 1] denotes the data bits of the referenced
i
(0)
evaluated by taking both FBC and CFO into account. In the user during the ith user’s signal interval, Ci,m represents the
block diagram it is assumed that there exist K simultaneous mth chip of the referenced subscriber during the ith bit
subscribers who are individually given with total N interval and a(0) indicates the fading gain of the referenced
subcarriers provided within a single cell, that is, there are M user. The subscript i will be omitted henceforth. The delay
orthogonal subcarriers shared by K uplink subscribers. A time, ti , denotes that which occurs during each user’s bit
signature sequence chip with a spreading code assumed that interval in an uplink asynchronous system. Moreover, an
has the equivalent spreading factor to provide with the MC-CDMA system experiencing a wide sense stationary
number of subcarriers with length L and apply binary phase uncorrelated scattering frequency-selective fading channel is
shift keying to modulate each of the subcarriers. This considered, wherein each user, k, has an impulse response
configuration is like the technique of OFDM signal h (k)(t; t). Since the subchannel bandwidth is assumed to be
modulation in a direct-sequence spread-spectrum scheme less than the channel coherent bandwidth, the consideration
when the frequency of the subcarrier is 1/Ts Hz, where Ts is of uncorrelated fading characteristics between the
the symbol duration. The inter-symbol interference between subchannels has become invalid. It also means that the FBC
contiguous OFDM symbols is ignorable, whereas the length should exist between subchannels as fD ¼ W/M , ( fD)c ,
of the cyclic prefix considered is longer than the maximum where W is the system bandwidth, and ( fD)c represents the
access delay time, and it is assumed that a perfect phase coherent bandwidth. The statistical model between the nth
correlation can be obtained. Thus, after the zeroth-order and the mth subchannel fading coefficients experienced by
interpolation took place the non-discrete time signal that the kth subscriber can be characterised as the frequency

Fig. 1 MC-CDMA system block diagram
a Transmitter
b Receiver

2552 IET Commun., 2011, Vol. 5, Iss. 17, pp. 2551–2557
& The Institution of Engineering and Technology 2011 doi: 10.1049/iet-com.2010.0344

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correlation function and is written as and ICFO denote MAI and CFO, respectively. The ICI
caused by the CFO phenomena may significantly
∗
F(k) (fn , fm ) = E[H (k) (fn ; t)H (k) (fm ; t)] (4) deteriorate the system performance. It is well known that
the phase noise is equivalent to a random phase modulation
where H (k)( f; t) denotes the Fourier transform of h (k)(t; t), of the carrier, and it inherently appears in the oscillator of
and the superscript ∗ indicates complex conjugate. Provided both the transmitter and the receiver. However, the small
that the assumption of fading channel is characteristic of the amount of phase noise is considered in this paper during
Rayleigh fading process among all M subcarriers, the one MC-CDMA symbol. This reason results in the case
covariance matrix for the user k can be described as with phase noise only at the end of the receiver. As the
CFO happens to the vth subcarrier of a referenced user
∗ operating in an MC-CDMA system within a single cell, it is
RF = E[H (k) (t)H (k) (t)] (5) able to be shown and easily expressed with a geometric
series as [9]
After the signal is impaired by the CFO, which arrives at the
receiver input of an asynchronous MC-CDMA system during M −1
a one-bit interval, it can be written as exp[j2p(v + 1(k) )q/M ]
ICFO,v =
q=0
M
K−1 (10)
r(t) = g(k) pTC (t − tk )
v (6) sin[p(v + 1(k) )] exp[jp(M − 1)(v + 1(k) )/M ]
k=0 =
M sin[p(v + 1(k) )]
where tk ¼ vTc + tk, k ¼ 0, 1, . . . , K 2 1 and
where 1(k) has been defined in (7). Now the aim of this study
is to focus on the discussion on joint impact of CFO and FBC
1 M −1 M −1 (k)
g(k) =
v j P (v + w′ ) (7) in an asynchronous MC-CDMA system. However, it is worth
M v=0 m=0 m h exploring the relationship between CFO and FBC themselves.
As previously mentioned, to preserve the effectiveness of the
where j(k) has been shown in (3) for the zeroth user
m
assumption of uncorrelated fading characteristics between
sub-channels, it is necessary to maintain the condition of
√ fD , (fD)coh. On the other hand, when the inverse happens,
Ph (v + w′ ) = exp j2p mv + 1(k) /M that is, when the condition fD . ( fD)c is valid, the
subchannels suffer from significant correlation. Therefore if
is the phase generated at the receiver, where 1(k) denotes the the event of CFO outcome is considered as a random
normalised frequency offset owing to the frequency process, then the probability of the event to generate CFO
mismatched between the transmitter and the receiver and is with the former condition, fD , ( fD)c , will become larger
defined as 1(k) = f0(k) /fD , where f0(k) is the frequency offset than that of the latter condition, fD . ( fD)c . This is because
of the kth user, and the space between subcarriers of each of the frequency separation, fD ¼ W/M, decreasing in the
user is defined as fD ¼ (MTC)21. When the CFO is involved former case. Mathematically, express this event as
in the received signal for the referenced subscriber,
subsequently it can be obtained by the v-th FFT (fast Prob{fo |fD , (fD )c } . Prob{fo |fD . (fD )c } (11)
Fourier transform) input corresponding to the MAI
(multiple access interference) without noise, expressed as where Prob{.} indicates the probability of an event, fo has
(v+1)Tc
been defined in (7). Accordingly, the function between FBC
1 and CFO can be determined as
(g(0) )′ =
v g (0)P(t − vTc ) dt
vTc Tc v
1 1 − exp[−j2p(i − j − 1(k) )]
= b(k) a(k) Ci,m Ph (v + w′ )
(k) l(k) = (12)
M i ij
j2p(i − j − 1(k) )
k v m

v = 0, 1, . . . , M − 1 (8) where i and j, i, j ¼ 0, . . . , M 2 1 represent different
subchannel, respectively, and 1(k) is the normalised CFO of
The received signal expressed in the previous equation is user k. The numerical analysis of the relationship between
going to be passed into a block with FFT function as CFO and FBC is shown in Fig. 2. Different subchannel
shown in Fig. 1. After the complex-valued M samples are numbers are indexed by i and j, that is, i, j ¼ 0, . . . , M 2 1.
sampled within one OFDM symbol at the time instant In case in the same subchannel situation, (i 2 j) ¼ 0, it is to
tn ¼ iTs/M, n ¼ 0, 1, . . . , M 2 1, of (g(0) )′ accompanied by
v clarify the fact that the CFO will increase to follow the
the additive white Gaussian noise (AWGN), the waveform FBC decrease, since the separation now is zero. However,
at the output of the FFT block can be determined by this situation is trivial because nothing could occur in a
state which remains in the same sub-channel. Once, the
M −1
2pvn separation becomes non-zero between subchannels, it is
h(0) = (g(0) (tn ))′ =
n v (g(0) )′ exp −j
v + Nn worth noting that the results will change inversely. For
v=0
M
instance, the results from the case of (i 2 j) ¼ 1 are totally
= Dn + IMAI + ICFO + Nn (9) different in that of the case of (i 2 j) ¼ 0. Moreover, the
corresponding results from the assumption with (i 2 j) ¼ 1
where the last component, Nn , expresses the contribution of to (i 2 j) ¼ 4 are also illustrated for comparison in the
passing the AWGN into the FFT block, Dn is the desired pictorial of Fig. 2. It is reasonable to claim that the larger
signal component of the referenced subscriber, and IMAI the separation between different subchannels, the lesser the


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multinomial. Traditionally, it has been somewhat
troublesome to directly obtain the jpdf of L corrected
variables with a Rayleigh distribution. The jpdf, however,
can be obtained by the methods of MGF and/or
characteristic function (CF). It has been shown that MGF or
CF approach offers another choice.

3 Calculation of moments and BER
In this subsection the effects of CFO and FBC are assumed to
be coexisting in an MC-CDMA system for the reasons
aforementioned. For comparison, an analysis of the system
performance of an MC-CDMA system without the
correction by a phase-locked loop is implemented. To
compute the signal-to-noise ratio (SNR) for the received
signal at the output of the decision maker, the expectation
(first moment) of the desired signal, the variance (second
Fig. 2 Plots of relationship between CFO and FBC with different moment) of interference and AWGN should be calculated
subchannels separation first. The statistical calculation of an MC-CDMA system
within single-cell environment is analysed as follows. The
branch number of the received signal is set as l ¼ 0, 1, . . . ,
correlation exists in it. The fact of (12) has been proved and
L 2 1. Thereafter, the error probability with coherent
shown in the results of Fig. 2.
technical demodulation conditioned on an instantaneous
So far, in addition to determining the result of the signals
SNR for an asynchronous MC-CDMA system working in
propagating between the transmitter and the receiver of an
the environment simultaneously accompanying the CFO
MC-CDMA system, in order to complete the analysis
and FBC is given as [11]
involving the FBC parameter now to determine the
correlated-Rayleigh channel model of the small-term
channel is necessary. Except for considering that the Pr (error|al , l = 0, 1, . . . , L − 1)
⎛ ⎞
propagation channel is with multipath delay and the √ 2
( js ) ⎠
received signal of different users is independent of each =Q SNR = Q⎝ (15)
other, the received path number is assumed to be equal to (s2 )
T
the number of subcarriers too. The fading path gain ai ,
i ¼ 0, 1, . . . , M 2 1, are characterised as Rayleigh where ( js)2 and (s2 ) represent the first moment of the desired
T
distribution. Since the fading branch correlation is discussed signal and the second moment of all the interference,
in this study, the independence of the receiving branch respectively, the later term that includes three terms
cannot be maintained. The joint probability density function as shown in (9), and Q(.) is the well known Macuamm
(jpdf) of assuming the correlated channel proposed in [10] Q-function, which can be alternately expressed as [12]
is adopted as a specified formula in analytical calculation.
Apart from the fact that the way to algebraically deal with (p/2)
the fading correlated channel in each received branch is a 1 2
/2 sin2 u)
Q(t) = e(−t du (16)
difficulty matter, a novel method by using the generalised p 0
Laguerre polynomial to expand the jpdf can be derived and
obtained as After the desired signal and the total instant of interference are
determined, the average error probability for an MC-CDMA
fa0 ,...,a(L−1) (a0 , . . . , a(L−1) ) system in correlated Rayleigh fading channels can be
obtained by averaging Pr(error|al , l ¼ 0, 1, . . . , L 2 1) over
L
ai L
a2 1
(1/2)n L variates with the jpdf shown in (13) and is written as
= exp − i
×
i=1
s2
i i=1
2s2 i n=0
n! L
1 1 1
Lg (a′j /(2s′2 ),
j 1) Lg (a′j /(2s′2 ),
j 1) Pav = ··· Pr (error|al , l = 0, 1, . . . , L − 1)
× Cij h × + ···
i,j
1 1 0 0 0

n
× fa0 ,...,a(L−1) (a0 , . . . , a(L−1) ) da0 da1 · · · daL−1 (17)
Lg (a′j /(2s′2 ), 1) Lg (a′2L /(2s′2 ), 1)
j 2L
+ C12..(2L) h ×
1 1 Hereafter, the branch number over a small-scale fading
channel is assumed equivalent to the subcarrier number.
(13) Once the jpdf is determined, the average error probability
can be evaluated by involving L-fold integration.
where Lg(v, w) is the generalised Laguerre polynomial of Furthermore, the second moment of the AWGN component
degree g, it is defined as [8] within a cellular environment without any other interference
can be determined as
(d/dv)g [vg P(v)]
Lg (v, w) = L(w−1) (v) × g! =
g (14)
P(v) C−1
MNn,0 (0)
AWGN ) =
(s2 AWGN )] =
[(s2 V (18)
and { · · · }n in (13) is a symbol of the nth power of a c=0
4Tb i


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where C denotes the number of cells. However, the second required to evaluate the system performance is the main
moment is not involved in the previous equation (i.e. reason for which we replace (12) with (20).
C ¼ 1), since the macrodiversity is not considered in this For all K interfering users the amount of AWGN including
study. Moreover, V(0) = E[(a(0) )2 ], i ¼ 0, 1, . . . , M 2 1,
i i the CFO component accompanied with all the subcarriers
represents the average power of the ith path for the zeroth can be calculated as IAWGN = SK−1 ICFO (s2
k=0
2
AWGN ), where
M −1
user (referenced user). To consider the correlation factor in ICFO = Sg=0 (ICFO,g ) . Next, the second moment of the
2 2

the analysis of an asynchronous MC-CDMA system, the MAI plus the ICI for the referenced user, caused by the
correlation coefficient should take an account of the other interfering user conditioned on the fading gain over
assumption in which the total average power becomes frequency-selective fading channel with a correlated
Rayleigh distributed, can be obtained as
⎡ ⎤
L−1 2 L−1
E⎣ a(k)
l
⎦ = 2N (V(k) )
l 1 K M

l=0 l=0 (MAI+ICI) =
s2 ICFO,g (a(k) )2 V(k) − (a(0) )2 V(0) ICFO,0
2
g g 0 0
3n k=0 g=0

(21)
+ 2N (N − 1)G2 (1.5) × (V(k) )1/2
i
i=j
where a(k) is one of the fading path gain of the referenced
g
×(V(k) )1/2 2 F1
1 1
− , − ; 1; lij user. Thus, the total second moment, s2 , can be obtained
T
j
2 2 by combining (18) with (19) and expressed as
(19)
K M
1
where a quasi-Gaussian correlation model of an equally s2 =
T ICFO,g (a(k) )2 V(k) − (a(0) )2 V(0) ICFO,0
2
g 0
3n k=0 g=0
spaced linear array with an arbitrary correlation coefficient,
lij , is adopted, being given as [8] M −1
MN0 (0)
+ 2
ICFO,g V (22)
g=0
4Tb g
lij = exp[−0.5h(i − j)2 (d)2 ], i, j = 0, . . . , L − 1 (20)
An assessment of the average BER performance of an MC-
where h 21.4 is a coefficient chosen from setting the CDMA system when both the CFO and FBC parameters
correlation model equal to the Bessel correlation model are considered can be obtained by combining (18) and (21)
with a 23 dB point [8], and d ¼ d/l is the normalised with (22). Certainly, such assessment requires a calculation
distance between two neighbouring branches, where d is the of the second moment of the total instant of interferences.
separation between transmitter and receiver, and l denotes As it is so commonly presented, the derivation of signal
the wavelength of carrier frequency. The parameter d is power will not be shown again in this paper. On the basis
applied to determine the threshold level of correlation. The of the novel derived results, the phenomenon of correlation
assigned values of d are arranged in the interval of (0.4, 1), exists between different branches and CFO caused by the
in which d ¼ 0.4 and d ¼ 1 represent two extreme inconsistence of oscillating frequency will be validated and
conditions, that is, fully correlated and uncorrelated illustrated by means of numerical analysis in the following
branches, respectively. The parameter of wavelength section.

Fig. 3 Plots of BER against different CFO (1) and FBC (d) values


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4 Numerical results case the Eb/N0 and subcarrier number are considered as the
values with 5 dB and M ¼ 64, respectively, the system
The previously derived algebraic results of system BER will always stay at about 1025 after the value of CFO
performance of an MC-CDMA system simultaneously arrives at 1 ¼ 0.42 and whatever the FBC values are. It is
including CFO and FBC are numerically evaluated in this known that the larger the value is for the CFO parameter,
section. In order to figure out the comparison results from the more the degradation of the system performance is.
involving both FBC and CFO at the same time, the figures However, the effect of FBC to system performance is not
are shown with many 3D curves in Figs. 3 – 5. The vertical same as that of CFO, that is, the performance of an MC-
axis represents the different bit error probability in Fig. 3; CDMA system will become superior when the value of the
however, the other two axes (X and Y ) correspondingly FBC increases, since most of the FBC is specifically
express the CFO and FBC with distinct values. Both values determined by the distance between the correlated branches.
of CFO and FBC are assumed and assigned in the interval Moreover, the other important point of view worth noting is
of (0.4, 1), that is, 0.4 ≤ (1, d) ≤ 1 [13]. The system BER the comparison between the results from the effect of CFO
performance of an MC-CDMA system will remain stable and FBC. It is easy to seize the fact that the system BER
after some fixed values of CFO and FBC. For instance, in will stay at about 1021 and 1022 corresponding to the

Fig. 4 Plots of BER against CFO (1) and FBC (d), corresponding to different values of fading parameters

Fig. 5 Plots of BER against CFO (1) and FBC (d) corresponding to different values of user numbers


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values of CFO and FBC, which are set as the same value 0.66. 6 Acknowledgments
Thus, the system performance of an MC-CDMA system is
definitely deeply dominated by the factor of CFO. The The author would like to thank the anonymous reviewers and the
mentioned facts can also be understood in the compared editor for their helpful comments that considerably improved the
results from Figs. 3 and 4 where the planes of BER are quality of this paper. Appreciation is also expressed to Dr.
opposed to the parameters of CFO and FBC. Furthermore, Cheryl J. Rutledge (retired Associate Professor of English,
the results from numerical calculation of subcarrier with Dayeh University) for her editorial assistance.
different values, m ¼ 64, 128 and 512 are illustrated as
different layers shown in Fig. 4. Hence, it is naturally
reasonable to describe that the larger the number of 7 References
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research, for example, the design and implementation of an wireless communications’, IEEE Trans. Commun., 2003, 51,
adaptive algorithm for fair adjustment. pp. 1745– 1748


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