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4392 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011
BER Analysis of Uplink OFDMA in the Presenceof Carrier Frequency and Timing Offsets
on Rician Fading ChannelsK. Raghunath, Yogendra U. Itankar, A. Chockalingam, Senior Member, IEEE, and
Ranjan K. Mallik, Senior Member, IEEE
Abstract—In orthogonal frequency-division multiple access(OFDMA) on the uplink, the carrier frequency offsets (CFOs)and/or timing offsets (TOs) of other users with respect to a desireduser can cause multiuser interference (MUI). Analytically evaluat-ing the effect of these CFO/TO-induced MUI on the bit error rate(BER) performance is of interest. In this paper, we analyze theBER performance of uplink OFDMA in the presence of CFOs andTOs on Rician fading channels. A multicluster multipath channelmodel that is typical in indoor/ultrawideband and underwateracoustic channels is considered. Analytical BER expressions thatquantify the degradation in BER due to the combined effect ofboth CFOs and TOs in uplink OFDMA with M -state quadratureamplitude modulation (QAM) are derived. Analytical and simula-tion BER results are shown to match very well. The derived BERexpressions are shown to accurately quantify the performancedegradation due to nonzero CFOs and TOs, which can serve asa useful tool in OFDMA system design.
Index Terms—Bit error rate (BER) analysis, carrier frequencyoffset (CFO)/timing offset (TO), multiuser interference (MUI), Ri-cian fading, uplink orthogonal frequency-division multiple access(OFDMA).
I. INTRODUCTION
IN ORTHOGONAL frequency-division multiple access(OFDMA) on the uplink, factors including 1) timing offsets
(TOs) of different users that are caused by path delay differ-ences between different users and imperfect timing synchro-nization and 2) carrier frequency offsets (CFOs) of differentusers that are induced by Doppler effects and/or poor oscillatoralignments can destroy orthogonality among subcarriers atthe receiver and cause multiuser interference (MUI) [1]–[8].In practical wireless OFDMA systems, the detrimental ef-fects of CFOs and TOs are reduced through tight closed-loop
Manuscript received April 20, 2011; revised July 25, 2011; acceptedAugust 30, 2011. Date of publication September 19, 2011; date of currentversion December 9, 2011. This work was supported in part by the DefenceResearch and Development Organisation (DRDO)-IISc Program on AdvancedResearch in Mathematical Engineering. The review of this paper was coor-dinated by Dr. E. K. S. Au. This paper was presented in part at the 2010IEEE International Communications Conference, Cape Town, South Africa,May 23–27.
K. Raghunath is with Semtronics Micro Systems Private Ltd., Bangalore560009, India (e-mail: [email protected]).
Y. U. Itankar and A. Chockalingam are with the Department of ElectricalCommunication Engineering, Indian Institute of Science, Bangalore 560012,India (e-mail: [email protected]; [email protected]).
R. K. Mallik is with the Department of Electrical Engineering, Indian Insti-tute of Technology Delhi, New Delhi 110016, India (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2011.2168250
frequency/timing correction between the mobile transmittersand the base station (BS) receiver. Such close-loop techniquesare expensive in terms of feedback bandwidth and mobiletransmit oscillator cost. Alternatively, the effects of MUI effectsdue to large CFOs and TOs can be countered through the useof interference cancellation techniques at the receiver [1]–[8].In such situations, characterization of the performance degra-dation due to CFO/TO-induced loss of orthogonality becomesimportant. However, we note that analytical characterizationof the bit/symbol error performance of uplink OFDMA in thepresence of CFOs and TOs has not been adequately addressedin the literature. Most bit error rate (BER) evaluations in uplinkOFDMA are based on simulations, e.g., [1]–[8].
In terms of analytical evaluation, an approximate analysisof the signal-to-noise ratio (SNR) degradation and BER of“single-user OFDM” with CFO on additive white Gaussiannoise (AWGN) channels was introduced in [9]. Later, in [10],Santhanam and Tellambura presented an exact BER analysisof single-user OFDM with CFO on AWGN channels. Fur-thermore, making a Gaussian approximation of the intercar-rier interference (ICI), Rugini and Banelli extended the BERanalysis of single-user OFDM to frequency-selective Rayleighand Rician fading with CFO in [11]. However, the analyses in[9]–[11] do not consider TOs. In [12], an approximate averagesignal-to-interference (SIR) analysis for single-user OFDMwith TO alone (assuming zero CFO) was presented. In [13], anapproximate symbol-error-rate analysis of single-user OFDMwith both CFO and TO is presented. However, references[9]–[13] do not consider “multiuser OFDM” on the uplink (i.e.,uplink OFDMA).
In terms of the performance analysis of uplink OFDMA,Raghunath and Chockalingam [7] and Park et al. [14] derivedanalytical expressions for the average SIR at the receiver. In[14], SIR expressions considering only TO (assuming zeroCFO) are derived. In [7], SIR expressions considering bothCFOs and TOs are derived. However, to our knowledge, ananalytical derivation of BER expressions for uplink OFDMAin the presence of both CFO and TO on Rician fading channelshas not been reported. Our contribution in this paper aims to fillthis gap. In particular, we derive analytical BER expressionsthat quantify the degradation in BER due to the combinedeffect of both CFOs and TOs in uplink OFDMA with M -statestate quadrature amplitude modulation (QAM) on Rician fadingchannels using probability density function (pdf) [15] and char-acteristic function (CF) approaches. Another interesting aspect
0018-9545/$26.00 © 2011 IEEE
RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4393
Fig. 1. Multicluster multipath channel model.
of our contribution is that we carry out this BER analysis fora general “multicluster” multipath Rician fading model, whichis typical in indoor/ultrawideband (UWB) and underwateracoustic channels [16]–[20]. Our numerical and simulation re-sults show that the BER expressions derived accurately quantifythe performance degradation due to nonzero CFOs and TOs.
The rest of this paper is organized as follows: In Section II,the considered system and channel models are introduced. InSection III, the BER analysis for the case of zero CFOs/TOs ispresented. BER analysis in the presence of nonzero CFOs/TOsis presented in Section IV. Results and discussions are pre-sented in Section V. Conclusions are presented in Section VI.
II. SYSTEM MODEL
Consider an uplink OFDMA system with K users, whereeach user communicates with a BS through an independentmulticluster multipath Rician fading channel. We assume thatthere are N subcarriers in the system and that each user is allot-ted M subcarriers such that a subcarrier is allotted to only oneuser. Let Xu = [X(u)
1 X(u)2 · · ·X(u)
M ] denote the current frame
of the uth user consisting of M symbols, where X(u)k , k ∈ Su,
denotes the uth user’s symbol on the kth subcarrier; Su be theset of subcarriers allotted to the uth user; and E[|X(u)
k |2] = 1,where E[.] denotes the expectation operator. Then,
⋃Ku=1 Su =
{0, 1, . . . , N − 1}, and Su
⋂Sv = φ for u �= v. The length
of the cyclic prefix (CP) added is Ng sampling periods andis assumed to be equal to the channel delay spread L − 1normalized by the sampling period (i.e., Ng ≥ L − 1). Afterinverse discrete Fourier transform processing and CP insertionat the transmitter, the time-domain sequence of the uth usercorresponding to the current frame xu
n is given by
xun =
1N
∑k∈Su
Xuk e
j2πnkN , −Ng ≤ n ≤ N − 1. (1)
We consider a multicluster multipath channel model withNc eigenpaths (clusters), as shown in Fig. 1. Such multiclusterchannel models are typical in indoor/UWB channels [16]–[18]and underwater acoustic channels [19], [20]. Each cluster con-sists of a stable dominant component and many nondominantrandomly scattered components. A Rice fading model is usedfor each cluster. The first path of each cluster is the dominantcomponent for that cluster. We define the following parameters
in the channel model: Let Ti, i = 0, 1, . . . , Nc − 1 denote thearrival time of the first path of the ith cluster; Np,i denote thenumber of multipaths in the ith cluster including the dominantpath; and Pi denote the expected power of the ith cluster sothat
∑Nc−1i=0 Pi = 1 and Pi ∝ e−βTi , where β is the exponential
power decay factor. Let Ωpi denote the expected power ofthe pth multipath of the ith cluster, p = 0, 1, . . . , Np,i − 1,and Ki denote the Rice factor of ith cluster. We have Ω0i =PiKi/Ki + 1 as the power of the dominant component ofthe ith cluster,
∑Np,i−1p=1 Ωpi = Pi/Ki + 1 as the power of
the scattered components of the ith cluster, and Ωpi ∝ e−ζip,where ζi is the decay factor for the scattered components ofthe ith cluster β < ζi’s. The clusters are nonoverlapping, i.e.,Ti+1 − Ti ≥ Np,i.
Let hn denote the multicluster multipath channel impulseresponse. hTi
=√
Ω0iejθi is the gain of the dominant com-
ponent, and θi is the phase of the ith cluster’s dominantpath. hTi+1, hTi+2, . . . , hTi+Np,i−1 are the gains of the scat-tered components of the ith cluster, where hTi+p’s, p =1, . . . , Np,i − 1 are modeled as hTi+p ∼ CN (0,Ωpi). The totaldelay spread in the channel is L − 1 = TNc−1 + Np,Nc−1 − 1.The channel impulse response can be written as
hn =Nc−1∑i=0
hn,i
(Ti, θi, Np,i,Ω0i,Ω1i, . . . ,Ω(Np,i−1)i
)︸ ︷︷ ︸Θi
(2)
where hn,i is the ith cluster impulse response, and Θi is theparameter set for the ith cluster. Now, in a multiuser scenario
hun =
Nuc −1∑i=0
hun,i (Θu
i ) (3)
where hun is the impulse response of the uth user’s channel
having Nuc clusters.
III. BIT ERROR RATE ANALYSIS WITH ZERO CARRIER
FREQUENCY OFFSETS AND TIMING OFFSETS
In this section, we consider the analysis in the case of perfectsynchronization, where the CFOs and TOs are zero. The uthuser’s signal at the BS receiver input is given by
sun = xu
n � hun (4)
4394 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011
where � denotes circular convolution. The uth user’s channelcoefficient on the kth subcarrier in frequency domain Hu
k isgiven by
Huk =
L−1∑n=0
hune
−j2πnkN (5)
with E[Huk ]=
∑Nuc−1
i=0
√Ωu
0iejθu
ie−j2πT ui k/N , and var((Hu
k ))=
var((Huk )) = σ2 = 1/2
∑Nuc −1
i=0
∑Nup,i−1
p=1 Ωupi. Define s
Δ=|E[Hu
k ]| and κ = s2/2σ2. The DFT output at the receiver onthe kth subcarrier is
Y uk = Hu
k Xuk + Wu
k (6)
where Wuk is the output noise of variance σ2
n. In the succeedingsubsections, we derive average BER expressions using twoapproaches, i.e., one using the pdf approach (Approach 1) andanother using the CF approach (Approach 2).
A. Approach 1
Here, we derive the BER expression using the pdf approach.The probability of error conditioned on Hu
k , which is denotedby Pe(Hu
k ), is given by
Pe (Huk ) =
1√2π
∞∫√
a|Huk |2
σ2n
e−y2
2 dy (7)
where a = 1 for quadrature phase-shift keying (QPSK) anda = 2 for binary phase-shift keying (BPSK). Unconditioningover the Rician pdf of R = |Hu
k |, we get the unconditional BERexpression as
Pe =1√2π
∞∫0
∞∫√
ar2
σ2n
e−y2
2 dyfR(r)dr (8)
where
fR(r) = e−κ r
σ2e
−r2
2σ2 I0
( rs
σ2
), r ≥ 0 (9)
and I0(rs/σ2) =∑∞
c=0(r2cs2c/(2σ2)2c(c!)2). Changing the
order of integration and after some simplification, the analyticalexpression for the BER can be derived as1
Pe =12
[1 − e−κ
∞∑c=0
s2c
(2σ2)cc!
√2aσ2
σ2n + aσ2
·(
c∑l=0
(2l)!(l!)2
(σ2
n
σ2n + aσ2
)l(12
)(2l+0.5))]
. (10)
1The derivation of (10) is given in the Appendix. Similar derivation steps willbe used in the analysis with CFOs and TOs in Section IV as well.
B. Approach 2
Here, we derive the BER expression using the CF approach.The probability of error conditioned on instantaneous SNR γ =|Hu
k |2/σ2n, which is denoted by Pe(γ), is given by
Pe(γ) = Q (√
aγ) =1π
π2∫
0
exp(
−aγ
2 sin2 φ
)dφ (11)
where γ is a noncentral chi-square distributed random variablewith 2◦ of freedom, and the second step in the precedingdiscussion is by Craig’s formula [21]. The CF of γ is given by
ψγ(jω) =1(
1 − jωγ01+κ
) exp
(jωκγ01+κ
1 − jωγ01+κ
)(12)
where γ0 is the average SNR given by γ0 = (s2 + 2σ2)/σ2n.
Now, unconditioning over the random variable γ, we get theunconditional BER expression as
Pe =1π
π2∫
0
ψγ
(−a
2 sin2 φ
)dφ
=1π
π2∫
0
sin2 φ(sin2 φ + aγ0
2(1+κ)
) exp
⎧⎨⎩−κaγ02(1+κ)(
sin2 φ + aγ02(1+κ)
)⎫⎬⎭ dφ.
(13)
Let ηΔ= (aγ0/2(1 + κ))/ sin2 φ + (aγ0/2(1 + κ)); then
Pe =1π
π2∫
0
(1 − η) exp(−κη)dφ. (14)
Using Taylor series expansion, the integrand can be written as
(1 − η) exp(−κη) =(
1 − κη +κ2η2
2!− κ3η3
3!· · ·)
− η
(1 − κη +
κ2η2
2!− κ3η3
3!· · ·)
= 1 − η(κ + 1) + η2
(κ2
2!+
κ
1!
)
− η3
(κ3
3!+
κ2
2!
)· · ·
= 1 +∞∑
c=1
(−η)c
(κc
c!+
κ(c−1)
(c − 1)!
). (15)
Substituting (15) in (14) and exchanging the integration andsummation, we get
Pe =12
+∞∑
c=1
(−1)c
(κc
c!+
κc−1
(c − 1)!
)1π
π2∫
0
ηcdφ. (16)
RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4395
The inner integral evaluates as
1π
π2∫
0
ηcdφ =1
2(1 + 2(1+κ)
aγ0
)c− 12
·c−1∑l=0
(c − 1
l
)(2l
l
)(1 + κ
2aγ0
)l
, for c ≥ 1. (17)
Finally, we get
Pe =12
⎡⎢⎣1 +∞∑
c=1
(−1)c(1 + 2(1+κ)
aγ0
)c− 12
(κc
c!+
κc−1
(c − 1)!
)
·c−1∑l=0
(c − 1
l
)(2l
l
)(1 + κ
2aγ0
)l
⎤⎥⎦. (18)
Although the final BER expressions (10) and (18) contain aninfinite sum, only the first few terms are significant as c! rapidlyincreases with increase in c. In addition, when s is very smalland tends to zero, κ also tends to zero, making γ0 =2σ2/σ2
n,and
Pe =12
(1 −
√aσ2
σ2n + aσ2
)=
12
(1 −
√a2γ0
1 + a2γ0
)(19)
which is the well-known BER expression in Rayleigh fadingfor BPSK (a = 2) and QPSK (a = 1) [22].
IV. BIT ERROR RATE ANALYSIS WITH NONZERO CARRIER
FREQUENCY OFFSETS AND TIMING OFFSETS
In this section, we consider the case of imperfect synchro-nization, where both the CFOs and TOs are nonzero. Let εu,u = 1, 2, . . . ,K denote the uth user’s residual CFO normal-ized by the subcarrier spacing |εu| ≤ 0.5,∀u, and let μu, u =1, 2, . . . ,K denote the uth user’s residual TO in the number ofsampling periods at the receiver. The DFT output on the kthcarrier of the uth user at the receiver in the presence of CFOsand TOs can be written in the form
Y uk = Hu
k,kXuk +
∑q∈Suq �=k
Huk,qX
uq +
u∑q∈Su
Hu,Ik,q Xu,I
q
︸ ︷︷ ︸self interference (SI)
+K∑
v=1,v �=u
∑q∈Sv
Hvk,qX
vq + Hv,I
k,q Xv,Iq
︸ ︷︷ ︸MUI
+Wuk (20)
where Xuq and Xu,I
q are the symbols from the current andinterfering frames, respectively, of the uth user, and Wu
k is theoutput noise of variance σ2
n. If the TO is −ve, the interferingframe will be the previous frame; if the TO is +ve, the inter-fering frame will be the next frame. Coefficients Hk,q’s dependon the CFO and TO values. To write the expressions for these
coefficients for Ng = L − 1, we need to consider four differentcases of TOs, which are referred to as cases a–d of TOs, where0 < −μu ≤ Ng for case a, −μu > Ng for case b, 0 < μu < Lfor case c, and μu ≥ L for case d. Using l to denote the pathindex and defining
Γu,lqk (n1, n2)
Δ=1N
n2∑n=n1
ej2πn(q−k+εu)
N (21)
Γu,lqk (n1, n2) = 0 for n1 > n2, the expressions for Hu
k,q fordifferent cases of TOs can be written as
Huk,q = e
j2πμuqN
L−1∑l=0
hul e
−j2πlqN Γu,l
qk (nα1 , nα2) (22)
where (nα1 , nα2) corresponding to different cases a–d aregiven by
(na1 , na2) ={
(0, N − 1), for l ≤ Ng + μu
(l − μu − Ng, N − 1), for l > Ng + μu
(23)
(nb1 , nb2) = (l − μu − Ng, N − 1), ∀l (24)
(nc1 , nc2) ={
(0, N − 1 − μu + l), for 0 ≤ l ≤ μu − 1(0, N − 1), for l ≥ μ
(25)
(nd1 , nd2) = (0, N − 1 + l − μu) ∀l. (26)
It is noted that, in cases a and b, interference is only due to theprevious frame, and in cases c and d, interference is only dueto the next frame. Based on this observation, the expressionsfor Hu,I
k,q ’s for cases a and b can be written as
Hu,Ik,q =e
j2π(μu+Ng)q
N
L−1∑l=Ng+μu+1
hul e
−j2πlqN Γu,l
qk (0, nα1−1) (27)
where nα1 in (27) is na1 for case a and nb1 for case b. Forcase a, paths l ≤ Ng + μu do not contribute to previous frameinterference, which corresponds to Γu,l
qk (0, nα1 − 1) = 0 forthese paths. For case b, all paths contribute to previous frameinterference. Thus, Γu,l
qk �= 0 for all paths. Thus, Hu,Iqk for cases
a and b can be written as
Hu,Ik,q = e
j2π(μu+Ng)q
N
L−1∑l=0
hul e
−j2πlqN Γu,l
qk (0, nα1 − 1). (28)
Likewise, the expressions for Hu,Ik,q s for cases c and d can be
written as
Hu,Ik,q =e
−j2π(Ng−μu)q
N
μu−1∑l=0
hul e
−j2πlqN Γu,l
qk (nα2 +1, N−1) (29)
where nα2 in (29) is nc2 for case c and nd2 for case d.For case c, paths l ≥ μu do not contribute to next frameinterference, which corresponds to Γu,l
qk (nα2 + 1, N − 1) = 0for these paths. For case d, all paths contribute to next frame
4396 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011
interference. Thus, Γu,lqk �= 0 for all paths. Thus, Hu,I
qk for casesc and d can be written as
Hu,Ik,q =e
−j2π(Ng−μu)q
N
L−1∑l=0
hul e
−j2πlqN Γu,l
qk (nα2 +1, N−1). (30)
We note that, due to the combined effect of CFOs and TOs,two conditions hold.
1) The means of different coefficients are given by
sukq
Δ= E[Hu
k,q
]= e
j2πμuqn
L−1∑l=0
E [hul ] e
−j2πlqN Γu,l
qk (nα1 , nα2).
(31)
The first path of each cluster is dominant nonrandom2
and contributes to the mean. Thus, E[hul ] �= 0 for l ∈
{Tu0 , Tu
1 , . . . , TuNu
c −1}, and
sukq =e
j2πμuqN
Nuc−1∑
i=0
√Ωu
0iejθu
i e−j2πT u
iq
N Γu,T ui
qk (nα1 , nα2). (32)
Similarly
su,Ikq
Δ= E
[Hu,I
k,q
]
=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
ej2π(μu+Ng)q
N
Nuc −1∑i=0
√Ωu
0iejθu
i e−j2πT u
iq
N
·Γu,T ui
qk (0, nα1 − 1), for μu < 0
e−j2π(Ng−μu)q
N
Nuc −1∑i=0
√Ωu
0iejθu
i e−j2πT u
iq
N
·Γu,T ui
qk (nα2 + 1, N − 1), for μu > 0
.
(33)
2) The coefficients of any given user u, (i.e., Huk,qs) are
correlated, whereas the coefficients of any two differentusers (i.e., Hu
k,qs and Hvk,qs, u �= v) are uncorrelated.
Computation of the exact BER would involve M -fold inte-gral in the case of the system with only CFOs and 2M -foldintegral for the system with both CFOs and TOs (where M isthe number of subcarriers allotted to each user). To reduce thiscomputational complexity, we proceed to obtain an analyticalexpression for the BER using three steps.
1) Since Huk,qs are correlated, we obtain an estimate of Hu
k,q
and Hu,Ik,q , in terms of Hu
k,k.2) Obtain expressions for the variances of SI/MUI and the
SINR conditioned on |Huk,k|.
3) Obtain an expression for the BER conditioned on |Huk,k|
by assuming the estimation errors in Huk,qs and Hu,I
k,q s tobe Gaussian, and uncondition it to obtain unconditionalBER.
Step 1 To obtain an estimate of Huk,q in terms of Hu
k,k, weuse the fact that, if two nonzero mean complex Gaussianrandom variables X and Y having means mx and my ,
2The analysis is valid for cases where the first path is not the dominantcomponent for that cluster. The index corresponding to the first path (i.e.,index zero) in Ω in (32) and (33) must be suitably changed to the index ofthe dominant path.
respectively, are correlated, an estimate of one variable(e.g., Y ) can be obtained, in terms of the other variable,as [23]
Y =CX,Y
σ2X
X +(
my − CX,Y
σ2X
mX
)(34)
with an estimation error EY =Y − Y of variance σ2Y −
C2X,Y /σ2
X , where CX,Y is the covariance of X and Y , andσ2
X and σ2Y are the variances of X and Y , respectively.
Using this, we can write all Hk,qs in (20) in terms of Hk,k,to get
Y uk = Hu
k,kXuk
+∑q∈Suq �=k
(Cu
k,q
(σuk )2
Huk,k+
(su
kq−Cu
k,q
(σuk )2
sukk
)+Eu
q
)Xu
q
+∑q∈Su
(Cu,I
k,q
(σuk )2
Huk,k+
(su,I
kq −Cu,I
k,q
(σuk )2
sukk
)+Eu,I
q
)Xu,I
q
+K∑
v=1,v �=u
∑q∈Sv
Hvk,qX
vq + Hv,I
k,q Xv,Iq + Wu
k (35)
where Cuk,q = E[(Hu
k,k − sukk)(Hu
k,q − sukq)
∗], Cu,Ik,q =
E[(Huk,k−su
kk)(Hu,Ik,q − su,I
kq )∗], (σuk )2 =E[(Hu
k,k−sukk)
(Huk,k−su
kk)∗], and (.)∗ denotes the conjugate operation.
Step 2 Now, defining (σuq )2 Δ= E[(Hu
k,q − sukq)(H
uk,q − su
kq)∗],
(σEuq)2 Δ= (σu
q )2 − |Cuk,q|2/(σu
k )2, and (σuq,I)
2 Δ=E[(Hu,I
k,q − su,Ikq )(Hu
k,q − su,Ikq )∗], the total variance of
all the terms that are interference to the uth user’s symbolon the kth subcarrier in (35), conditioned on Hu
k,k, isobtained as
σ2I|Hk,k
=|Huk,k|2
⎛⎜⎝∑q∈Suq �=k
∣∣∣Cuk,q
∣∣∣2(σu
k )4+∑q∈Su
∣∣∣Cu,Ik,q
∣∣∣2(σu
k )4
⎞⎟⎠︸ ︷︷ ︸
Δ=A
+∑q∈Su
q�=k
(σu
q
)2−∣∣∣Cu
k,q
∣∣∣2(σu
k )2+∑q∈Su
(σu,I
q
)2−∣∣∣Cu,I
k,q
∣∣∣2(σu
k )2︸ ︷︷ ︸Δ=B1
+∑q∈Suq �=k
∣∣∣∣∣∣sukq −
Cuk,q
(σuk )2
sukk
∣∣∣∣∣∣2+∑q∈Su
∣∣∣∣∣∣ su,Ikq −
Cu,Ik,q
(σuk )2
sukk
∣∣∣∣∣∣2
︸ ︷︷ ︸Δ=B2
+K∑
v=1,v �=u
∑q∈Sv
(σv
q
)2+(σvq,I
)2︸ ︷︷ ︸
Δ=B3
+K∑
v=1,v �=u
∑q∈Sv
∣∣svkq
∣∣2+∣∣∣sv,I
kq
∣∣∣2︸ ︷︷ ︸
Δ=B4
.
(36)
RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4397
TABLE IEXPRESSIONS FOR A AND B1 FOR DIFFERENT TIME OFFSET CASES a–d
TABLE IIEXPRESSIONS FOR B3 FOR DIFFERENT TIME OFFSET CASES a–d
Assuming that, among K users in the system, Kλ usersbelong to case λ, λ∈{a, b, c, d}, the expressions for termsA and B1 in (36) for different TO cases are given in
Table I, where we have (37)–(39), shown at the bottomof the page. Similarly, the expressions for the term B3 in(36) for the different TO cases are given in Table II. Now,
(σuk )2 =
Nuc −1∑i=0
Nup,i−1∑p=1
Ωupi
∣∣∣∣Γu,(T ui +p)
kk (nλ1 , nλ2)∣∣∣∣2 (37)
(σu
q
)2 =Nu
c −1∑i=0
Nup,i−1∑p=1
Ωupi
∣∣∣∣Γu,(T ui +p)
qk (nλ1 , nλ2)∣∣∣∣2 (38)
(σu
q,I
)2 =
⎧⎪⎪⎪⎨⎪⎪⎪⎩Nu
c −1∑i=0
Nup,i−1∑p=1
Ωupi
∣∣∣∣Γu,(T ui +p)
qk (0, nλ1 − 1)∣∣∣∣2 for μu < 0
Nuc −1∑i=0
Nup,i−1∑p=1
Ωupi
∣∣∣∣Γu,(T ui +p)
qk (nλ2 + 1, N − 1)∣∣∣∣2 , for μu > 0
(39)
4398 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011
defining B=B1+B2+B3+B4+σ2n, the SINR at the kth
subcarrier of the uth user conditioned on Huk,k, which was
denoted by γHuk,k
, is given by
γHuk,k
=
∣∣∣Huk,k
∣∣∣2A∣∣∣Hu
k,k
∣∣∣2 + B. (40)
Here, E[Huk,k] = su
kk, and the variance of the real andimaginary parts of Hu
k,k, which was denoted by σ2e , is
1/2(σuk )2. Let κ′ = |su
kk|2/2σ2e .
Step 3 Now, assuming the estimation errors Euq and Eu,I
q tobe Gaussian, we derive average BER expressions usingtwo approaches, one using the pdf approach (Approach 1)and another using the CF approach (Approach 2) in thesucceeding two sections. Approaches 1 and 2 will give thesame result since both of them do the unconditioning in thisstep. These are two different analytical methods to solvethe unconditioning. CF approach 2 is made possible by theuse of finite integral (Craig’s formula) of the conditionalBER and Taylor’s series expansion.
A. Approach 1
Here, we carry out the unconditioning in Step 3 us-ing the pdf approach. The conditional BER, which wasdenoted by Pe(γHu
k,k), can be written as Pe(γHu
k,k) =
1/√
2π∫∞√
aγHuk,k
e−y2/2dy. Unconditioning over the Rician
pdf of R = |Huk,k|, we get an unconditional BER expression as
Pe =1√2π
∞∫0
∞∫√
aγHuk,k
e−y2
2 dyfR(r)dr. (41)
Following the derivation steps similar to those given in theAppendix, (41) can be simplified as
Pe = 1 − e−κ′∞∑
c=0
(sukk)2c
(2σ2e)c c!
c∑l=0
1l!
(b
2σ2e
)l
Il (42)
where
Il =1√2π
∞∫0
y2l
(a −Ay2)le−y2
(12+ B
2σ2e(a−Ay2)
)dy. (43)
The integral in (43) can be evaluated using Simpson’s rule. Wesee that this has infinite discontinuity at y =
√a/A, and there-
fore it is enough to evaluate this integral from 0 to �√
a/A�.
B. Approach 2
Here, we uncondition the BER using the CF approach. UsingCraig’s formula, the conditional BER, which was denoted byPe(γHu
k,k), can be written as Pe(γHu
k,k)=1/π
∫ π/2
0 exp(−aγHuk,k
/
2 sin2 φ)dφ. Unconditioning over the r.v, Z = γHuk,k
, we get
Pe =1π
π2∫
0
EZ
[exp
(−az
2 sin2 φ
)]dφ. (44)
Using Taylor series expansion for the exponential in (44)
Pe =1π
π2∫
0
EZ
[1 −
(az
2 sin2 φ
)+
12!
(az
2 sin2 φ
)2
− 13!
(az
2 sin2 φ
)3
+ · · ·]
dφ
=12
+1π
π2∫
0
[ ∞∑l=1
(−a)lEZ(zl)
2ll! sin2l φ
]dφ. (45)
Define gΔ= |Hu
k,k|2. Then, z = g/Ag + B. For 0 ≤ g < B/A
zl =gl
Bl
(1+
Ag
B
)−l
=∞∑
k=0
(−1)k
(l+k−1
k
)Akgk+l
Bk+l. (46)
For B/A ≤ g < ∞
zl =1Al
(1+
BAg
)−l
=∞∑
k=0
(−1)k
(l+k−1
k
)Bkg−k
Ak+l. (47)
Using (46) and (47)
EZ [zl] =∞∑
k=0
(−1)k l + k − 1k
⎡⎢⎣ Ak
Bk+l
BA∫
0
gk+lfG(g)dg
+Bk
Ak+l
∞∫BA
g−kfG(g)dg
⎤⎥⎦ . (48)
The pdf of G is given by fG(g)=(e−κ′/2σ2
e)e−g/2σ2
eI0(√
2κ′g/σ2e),
where I0(√
2κ′g/σ2e) =
∑∞c=0(g
c/(c!)2)(κ′/2σ2e)c. Therefore
BA∫
0
gk+lfG(g)dg =e−κ′
2σ2e
∞∑c=0
1(c!)2
(κ′
2σ2e
)c
·
BA∫
0
e
(−g
2σ2e
)g(k+l+c)dg. (49)
The inner integral in (49) evaluates as
BA∫
0
e( −g
2σ2e)g(k+l+c)dg =
(2σ2
e
)(k+l+c+1)
×
⎡⎣(k + l + c)! − e−B
2Aσ2e
(k+l+c)∑n=0
(k + l + c)!n!
(B
2Aσ2e
)n⎤⎦ .
(50)
For typical values of B/A, the approximate value of momentsin (48) can be obtained by evaluating the first integral only.
RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4399
Combining (45), and (48)–(50), we get
Pe =12
+e−κ′
π
π2∫
0
[ ∞∑l=1
(−a)l
2ll! sin2l φ
∞∑k=0
(−1)k
(l + k − 1
k
)
·(2σ2
e
)(k+l)(
Ak
Bk+l
) ∞∑c=0
κ′c(k + l + c)!(c!)2
·
⎛⎝1 − e−B
2Aσ2e
(k+l+c)∑n=0
(B
2Aσ2e
)n
n!
⎞⎠]dφ.
(51)
For the special case of zero CFO and TO for the desired userand nonzero CFOs and TOs for other users’ CFO and TOs,there will not be any SI (thus, no Gaussian approximation ofestimation error is needed), and only MUI occurs. For this case,A = 0, B = B3 + B4 + σ2
n. The BER in this case is given by(10) or (18), with σ2
n replaced by B3 + B4 + σ2n, which is an
exact closed-form BER expression. For M-QAM (M > 4),the conditional BER is of the form
Pe
(γHu
k,k
)=
√M−1∑j=1
cjQ(√
djγHuk,k
). (52)
For example, for 16-QAM, the expression for conditional BERis [24]
Pe
(γHu
k,k
)=
34Q
(√γHu
k,k
5
)+
12Q
⎛⎝√9γHuk,k
5
⎞⎠−1
4Q(√
5γHuk,k
). (53)
The unconditional BER can be obtained by averaging out eachterm using either the pdf or the CF approach.
V. RESULTS AND DISCUSSIONS
We numerically evaluated the analytical BER performanceand compared them with the simulated performance. In com-puting (51), the number of moments summed in the infinitesum is 100 (i.e., l = 1 to 100). The number of terms summedin the infinite sums with indices k and c is 50. In Fig. 2, weplot the analytical and simulated BER in perfectly synchronized(i.e., zero CFO/TO, εu = μu = 0 for all users) uplink OFDMAwith QPSK, N = 64, K = 4, M = 16 in single-cluster (Nc =1) multipath Rician fading channel with Np,0 = L = 10, andexponential delay profile factor ζ0 = 0.25. Rice factors of 0, 5,and 10 dB are considered. Similar performance plots for a two-cluster model (with parameters Nc = 2, L = 20, β = 0.083,Np,0 = 10, Np1 = 5, and ζ0 = ζ1 = 0.25) are shown in Fig. 3.In this considered case of zero CFOs/TOs, the system is notaffected by interference, and the analysis becomes exact with(10), giving the exact BER. This can be verified by the veryclose match between the analytical and simulated BER plots inFigs. 2 and 3 for various values of the Rice factor.
Fig. 2. BER performance of uplink OFDMA with zero CFOs and TOs forthe single-cluster channel for different values of Rice factors and QPSK. N =64, K = 4, M = 16, L = 10, Np,0 = 10, ζ0 = 0.25, and K0 = 0, 5, 10 dB.Analysis versus simulation.
Fig. 3. BER performance of uplink OFDMA with zero CFOs and TOs for thetwo-cluster channel for different values of Rice factors and QPSK. N = 64,K = 4, M = 16, L = 20, β = 0.083, Np,0 = 10, Np,1 = 5, ζ0 = ζ1 =0.25, and K0 = K1 = 0, 5, and 10 dB. Analysis versus simulation.
The BER performance in the presence of nonzero CFOs andTOs for QPSK modulation are plotted in Figs. 4 and 5. Fig. 4is for the single-cluster model, and Fig. 5 is for the two-clustermodel for the same system and channel parameters as inFigs. 2 and 3, respectively. The nonzero CFO and TO valuesof different users are [ε1, ε2, ε3, ε4] = [0.1, 0.2,−0.15,−0.3]and [μ1, μ2, μ3, μ4] = [−1,−5, 1, 5]. From Figs. 4 and 5,we see that there is close agreement between the analyticaland simulated BERs. This indicates that the approximationmade to handle the correlation in the channel coefficients ofsubcarriers of the same user is quite effective. The error floorsseen in Figs. 4 and 5 can be analytically inferred from (40),where the term B in the denominator is a sum of interferencevariances B1 to B4 in (36) and the noise variance σ2
n, i.e.,B = B1 + B2 + B3 + B4 + σ2
n, and A in the denominator isalso a function of interference variances. Thus, even if σn = 0(i.e., infinite SNR), the interference variances will leave aresidual floor in the error performance.
4400 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011
Fig. 4. BER performance of uplink OFDMA with nonzero CFOs andTOs for the single-cluster channel for different values of Rice fac-tors and QPSK. N = 64, K = 4, M = 16, L = 10, Np,0 = 10, ζ0 =0.25, K0 = 0, 5, 10 dB, [ε1, ε2, ε3, ε4] = [0.1, 0.2,−0.15,−0.3], and[μ1, μ2, μ3, μ4] = [−1,−5, 1, 5]. Analysis versus simulation.
Fig. 5. BER performance of uplink OFDMA with nonzero CFOs and TOs forthe two-cluster channel for different values of Rice factors and QPSK. N = 64,K = 4, M = 16, L = 20, β = 0.083, Np,0 = 10, Np,1 = 5, ζ0 = ζ1 =0.25, K0 = K1 = 0, 5, 10 dB, [ε1, ε2, ε3, ε4] = [0.1, 0.2,−0.15,−0.3],and [μ1, μ2, μ3, μ4] = [−1,−5, 1, 5]. Analysis versus simulation.
In Figs. 6 and 7, the BER plots for the case of 16-QAM areplotted. Fig. 6 is for the single-cluster model, and Fig. 7 is forthe double-cluster model. Here again, analytical and simulationresults closely match, validating the analysis. As expected,nonzero CFOs and TOs cause the error floors observed in theperformance plots. Investigation of low-complexity interfer-ence cancellation/equalization algorithms to reduce these errorfloors is an interesting topic. In particular, message-passing al-gorithms on graphical models are quite promising and attractivein such systems.
VI. CONCLUSION
We have analyzed the BER performance of uplink OFDMAon Rician fading channels in the presence of nonzero CFOsand TOs. We have considered a multicluster multipath channel
Fig. 6. BER performance of uplink OFDMA with nonzero CFOs andTOs for the single-cluster channel for different values of Rice fac-tors and 16-QAM. N = 64, K = 4, M = 16, L = 10, Np,0 = 10,ζ0 = 0.25, K0 = 0, 5, 10 dB, [ε1, ε2, ε3, ε4] = [0.1, 0.2,−0.15,−0.3], and[μ1, μ2, μ3, μ4] = [−1,−5, 1, 5]. Analysis versus simulation.
Fig. 7. BER performance of uplink OFDMA with nonzero CFOs and TOsfor the two-cluster channel for different values of Rice factors and 16-QAM.N =64, K =4, M =16, L=20, β = 0.083, Np,0 =10, Np,1 =5, ζ0 =ζ1 =0.25, K0 =K1 =0, 5, 10 dB, [ε1, ε2, ε3, ε4]=[0.1, 0.2,−0.15,−0.3], and[μ1, μ2, μ3, μ4]=[−1,−5, 1, 5]. Analysis versus simulation.
model that is typical in indoor/UWB and underwater acousticchannels. For this general multicluster Rician channel model,we have derived analytical BER expressions using pdf andCF approaches. Analytical results have been shown to closelymatch with simulation results. The derived expressions havebeen shown to accurately quantify the degradation due tononzero CFOs and TOs, which can serve as a useful tool inOFDMA system design. In particular, such analytical character-ization of performance loss due to nonzero CFOs and TOs canpoint to the level of receiver signal processing sophisticationneeded to substantially recover the lost performance. Suitablelow-complexity detection/equalization algorithms (includingalgorithms based on message passing on graphical models) thatcan alleviate performance loss due to nonzero CFOs and TOscan be investigated.
RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4401
APPENDIX
DERIVATION OF (10)
Substituting the expression for I0(.) in (9), and (9) in (8) andchanging the order of integration, we can write
Pe =e−κ
√2π
∞∑c=0
s2c
(2σ2)2c(c!)2
·∞∫
0
√σ2
ny2
a∫0
r2c+1
σ2e
−r2
2σ2 dre−y2
2 dy. (54)
Substituting r2/2σ2 = t, (54) becomes
Pe =e−κ
√2π
∞∑c=0
s2c
(2σ2)c(c!)2
∞∫0
σ2ny2
2aσ2∫0
tce−tdte−y2
2 dy. (55)
With U = σ2ny2/2aσ2, the inner integral in (55) evaluates as
U∫0
tce−tdt = c! − e−Uc∑
l=0
c!l!
U l. (56)
Substituting (56) in (55), we get
Pe =e−κ
√2π
∞∑c=0
s2c
(2σ2)c(c!)
∞∫0
e−y2
2 dy
− e−κ
√2π
∞∑c=0
s2c
(2σ2)c(c!)
c∑l=0
1l!
∞∫0
U le−y2
2 dy. (57)
Noting that 1/√
2π∫∞0 e−y2/2dy = 1/2 and κ = s2/2σ2, we
have∑∞
c=0(s2c/(2σ2)c(c!)) = eκ. With this, the first term in
(57) evaluates to 1/2, and hence
Pe =12− e−κ
∞∑c=0
s2c
(2σ2)c(c!)
c∑l=0
1l!
(σ2
n
2aσ2
)l
Il (58)
where Il = 1/√
2π∫∞0 y2le−y2/2(1 + σ2
n/aσ2)dy. With bΔ=
1/2(1 + σ2n/aσ2), Il can be written as
Il =0.5√2π
∞∫−∞
y2le−by2dy. (59)
Using the notation F (.) to denote the Fourier transform, wehave
∞∫−∞
y2le−by2dy =
1bl
d2lF (e−t2)df2l
∣∣∣∣∣f=0
. (60)
Evaluating (59) and (60) evaluates to
Il =(
12
)(2l+1.5)( 2aσ2
aσ2 + σ2n
)(l+0.5) (2l)!l!
. (61)
Substituting (61) in (58), we get (10).
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4402 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011
K. Raghunath received the B.Tech. degree in elec-tronics and communication engineering from SriVenkateswara University, Tirupati, India, the M.E.degree in communication systems engineering fromP. S. G. College of Technology, Coimbatore, In-dia, and the Ph.D. degree in electrical communica-tion engineering from Indian Institute of Science,Bangalore, India.
He was a Faculty Member in engineering collegesfor about 18 years. He is currently with SemtronicsMicro Systems Private Ltd., Bangalore, as Mentor
R&D. His current research interests include the implementation of long-term-evolution systems.
Yogendra U. Itankar received the B.Tech. degree inelectronics and communication engineering from theInternational Institute of Information Technology,Hyderabad, India, in 2009. He is currently workingtoward the M.Sc.(Engg.) degree in electrical andcommunication engineering with the Department ofElectrical Communication Engineering, Indian Insti-tute of Science, Bangalore, India.
His current research interests are low-complexityequalization in large-dimension multiple-input–multiple-output intersignal-interference channels
and underwater acoustic communications.
A. Chockalingam (S’92–M’93–SM’98) was born inTamil Nadu, India. He received the B.E. (Honors) de-gree in electronics and communication engineeringfrom the P. S. G. College of Technology, Coimbatore,India, in 1984, the M.Tech. degree (with special-ization in satellite communications) from the IndianInstitute of Technology, Kharagpur, India, in 1985and the Ph.D. degree in electrical communication en-gineering from the Indian Institute of Science (IISc),Bangalore, India, in 1993.
During 1986 to 1993, he was with the Trans-mission R&D Division, Indian Telephone Industries Ltd., Bangalore. FromDecember 1993 to May 1996, he was a Postdoctoral Fellow and an AssistantProject Scientist with the Department of Electrical and Computer Engineering,University of California, San Diego. From May 1996 to December 1998, heserved Qualcomm, Inc., San Diego, as a Staff Engineer/Manager in the systemsengineering group. In December 1998, he joined the faculty of the Departmentof Electrical Communication Engineering, IISc, Bangalore, where he is aProfessor, working in the area of wireless communications and networking.
Dr. Chockalingam served as an Associate Editor for the IEEE TRANSAC-TIONS ON VEHICULAR TECHNOLOGY from May 2003 to April 2007. Hecurrently serves as an Editor for the IEEE TRANSACTIONS ON WIRELESS
COMMUNICATIONS. He served as a Guest Editor for the IEEE JOURNAL ON
SELECTED AREAS IN COMMUNICATIONS (Special Issue on Multiuser Detec-tion for Advanced Communication Systems and Networks). Currently, he servesas a Guest Editor for the IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL
PROCESSING (Special Issue on Soft Detection on Wireless Transmission). He isa Fellow of the Institution of Electronics and Telecommunication Engineers andthe Indian National Academy of Engineering. He received the SwarnajayantiFellowship from the Department of Science and Technology, Government ofIndia.
Ranjan K. Mallik (S’88–M’93–SM’02) receivedthe B.Tech. degree from the Indian Institute of Tech-nology, Kanpur, in 1987 and the M.S. and Ph.D.degrees from the University of Southern California,Los Angeles, in 1988 and 1992, respectively, all inelectrical engineering.
From August 1992 to November 1994, he was aScientist with the Defence Electronics Research Lab-oratory, Hyderabad, India, working on missile andelectronic warfare projects. From November 1994 toJanuary 1996, he was a Faculty Member with the
Department of Electronics and Electrical Communication Engineering, IndianInstitute of Technology. From January 1996 to December 1998, he was withthe faculty of the Department of Electronics and Communication Engineering,Indian Institute of Technology, Guwahati. Since December 1998, he has beenwith the faculty of the Department of Electrical Engineering, Indian Instituteof Technology, Delhi, where he is currently a Professor. His research interestsare diversity, combining and channel modeling for wireless communications,space-time systems, cooperative communications, multiple-access systems,difference equations, and linear algebra.
Dr. Mallik is a member of Eta Kappa Nu; the IEEE Communications,Information Theory, and Vehicular Technology Societies; the American Math-ematical Society; and the International Linear Algebra Society. He is a Fellowof the Indian National Academy of Engineering, the Indian National ScienceAcademy, The National Academy of Sciences, India, Allahabad, The Institutionof Engineering and Technology, U.K., and The Institution of Electronics andTelecommunication Engineers, India, and a Life Member of the Indian Societyfor Technical Education. He is an Associate Member of The Institution ofEngineers (India). He is an Area Editor for the IEEE TRANSACTIONS ON
WIRELESS COMMUNICATIONS and an Editor for the IEEE TRANSACTIONS
ON COMMUNICATIONS. He is a recipient of the Hari Om Ashram PreritDr. Vikram Sarabhai Research Award in the field of electronics, telematics,informatics, and automation and the Shanti Swarup Bhatnagar Prize in Engi-neering Sciences.
