BER Analysis of Uplink OFDMA in the Presence - Electrical ...

4392 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011BER Analysis of Uplink OFDMA in the Presenceof Carrier Frequency and Timing Offsetson Rician Fading ChannelsK. Raghunath, Yogendra U. Itankar, A. Chockalingam, Senior Member, IEEE, andRanjan K. Mallik, Senior Member, IEEEAbstract—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 evaluatingthe 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 simulationBER 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), Ricianfading, uplink orthogonal frequency-division multiple access(OFDMA).I. INTRODUCTIONIN ORTHOGONAL frequency-division multiple access(OFDMA) on the uplink, factors including 1) timing offsets(TOs) of different users that are caused by path delay differencesbetween different users and imperfect timing synchronizationand 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 effectsof CFOs and TOs are reduced through tight closed-loopManuscript 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 coordinatedby 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: kn_raghu@yahoo.com).Y. U. Itankar and A. Chockalingam are with the Department of ElectricalCommunication Engineering, Indian Institute of Science, Bangalore 560012,India (e-mail: yogendra@ece.iisc.ernet.in; achockal@ece.iisc.ernet.in).R. K. Mallik is with the Department of Electrical Engineering, Indian Instituteof Technology Delhi, New Delhi 110016, India (e-mail: rkmallik@ee.iitd.ernet.in).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.2011.2168250frequency/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 degradationdue 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. Furthermore,making a Gaussian approximation of the intercarrierinterference (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 characteristicfunction (CF) approaches. Another interesting aspect0018-9545/$26.00 © 2011 IEEE

RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4393Fig. 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 resultsshow 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 presentedin Section V. Conclusions are presented in Section VI.II. SYSTEM MODELConsider 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 allottedM subcarriers such that a subcarrier is allotted to only oneuser. Let X u =[X (u)1 X (u)2 ···X (u)M] denote the current frameof the uth user consisting of M symbols, where X (u)k , k ∈ S u,denotes the uth user’s symbol on the kth subcarrier; S u be theset of subcarriers allotted to the uth user; and E[|X (u)k |2 ]=1,where E[.] denotes the expectation operator. Then, ⋃ K⋂ u=1 S u ={0, 1,...,N − 1}, and S u Sv = φ for u ≠ v. The lengthof the cyclic prefix (CP) added is N g sampling periods andis assumed to be equal to the channel delay spread L − 1normalized by the sampling period (i.e., N g ≥ L − 1). Afterinverse discrete Fourier transform processing and CP insertionat the transmitter, the time-domain sequence of the uth usercorresponding to the current frame x u n is given byx u n = 1 ∑Xk u e j2πnkN , −Ng ≤ n ≤ N − 1. (1)Nk∈S uWe consider a multicluster multipath channel model withN c 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 consistsof 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 parametersin the channel model: Let T i , i =0, 1,...,N c − 1 denote thearrival time of the first path of the ith cluster; N p,i denote thenumber of multipaths in the ith cluster including the dominantpath; and P i denote the expected power of the ith cluster sothat ∑ N c −1i=0P i =1and P i ∝ e −βT i, where β is the exponentialpower decay factor. Let Ω pi denote the expected power ofthe pth multipath of the ith cluster, p =0, 1,...,N p,i − 1,and K i denote the Rice factor of ith cluster. We have Ω 0i =P i K i /K i +1 as the power of the dominant component of∑ Np,i −1p=1 Ω pi = P i /K i +1 as the power ofthe ith cluster,the scattered components of the ith cluster, and Ω pi ∝ e −ζip ,where ζ i is the decay factor for the scattered components ofthe ith cluster β

4394 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011where ⋆ denotes circular convolution. The uth user’s channelcoefficient on the kth subcarrier in frequency domain Hku isgiven byL−1∑Hk u = h u ne −j2πnkN (5)n=0with E[Hk u]=∑ Nc u −1√ i=0 Ωu0i e jθu ie −j2πT i uk/N , and var(R(Hk u))=var(I(Hk u)) = σ2 =1/2 ∑ Nc u −1 ∑Np,i u −1i=0 p=1 Ω u pi . Define s =Δ|E[Hk u]| and κ = s2 /2σ 2 . The DFT output at the receiver onthe kth subcarrier isY uk = H u k X u k + W u k (6)where W u k 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 1Here, we derive the BER expression using the pdf approach.The probability of error conditioned on Hk u , which is denotedby P e (Hk u ), is given byP e (Hk u )= √ 12π √∫∞a|H u k| 2σ 2 ne −y22 dy (7)where a =1 for quadrature phase-shift keying (QPSK) anda =2for binary phase-shift keying (BPSK). Unconditioningover the Rician pdf of R = |Hk u |, we get the unconditional BERexpression aswhereP e = √ 1 ∫∞2π0∫ ∞√ar 2σ 2 ne −y22 dyfR (r)dr (8)f R (r) =e −κ r (σ 2 e −r2 rs)2σ 2 I 0σ 2 , r ≥ 0 (9)and I 0 (rs/σ 2 )= ∑ ∞c=0 (r2c s 2c /(2σ 2 ) 2c (c!) 2 ). Changing theorder of integration and after some simplification, the analyticalexpression for the BER can be derived as 1[P e = 1 ∑ ∞1 − e −κ s 2c2(2σ 2 ) c c!c=0√2aσ 2σ 2 n + aσ 2( c∑ ((2l)! σ 2 ) l ( ) )](2l+0.5)n 1·(l!) 2 σ 2 l=0n + aσ 2 . (10)21 The 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 2Here, we derive the BER expression using the CF approach.The probability of error conditioned on instantaneous SNR γ =|Hk u|2 /σn, 2 which is denoted by P e (γ), is given byP e (γ) =Q ( √ aγ) = 1 ππ∫20( ) −aγexp2sin 2 dφ (11)φwhere 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 by1ψ γ (jω)=(1 − jωγ 01+κ) exp( jωκγ01+κ1 − jωγ 01+κ)(12)where γ 0 is the average SNR given by γ 0 =(s 2 +2σ 2 )/σ 2 n.Now, unconditioning over the random variable γ, we get theunconditional BER expression asP e = 1 π= 1 ππ∫20π∫20( −aψ γ2sin 2 φ)dφsin 2 φ(sin 2 φ + aγ 02(1+κ)⎧⎨) exp⎩−κaγ 02(1+κ)(sin 2 φ + aγ 02(1+κ)Let η Δ =(aγ 0 /2(1 + κ))/ sin 2 φ +(aγ 0 /2(1 + κ)); thenP e = 1 ππ∫20⎫⎬)⎭ dφ.(13)(1 − η)exp(−κη)dφ. (14)Using Taylor series expansion, the integrand can be written as(1 − η)exp(−κη) =(1 − κη + κ2 η 2− κ3 η 3 )···2! 3!− η(1 − κη + κ2 η 2− κ3 η 3 )···2! 3!( κ=1− η(κ +1)+η 2 22! + κ )1!( ) κ− η 3 33! + κ2···2!=1+∞∑( )κ(−η) c cc! + κ(c−1). (15)(c − 1)!c=1Substituting (15) in (14) and exchanging the integration andsummation, we getP e = 1 ∞2 + ∑( κ(−1) c cc=1c! + κc−1(c − 1)!) 1ππ∫20η c dφ. (16)

RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4395The inner integral evaluates as1ππ∫20η c dφ =∑c−1·l=0(2( c − 1Finally, we get⎡P e = 1 ⎢∞∑⎣1+2c=11) c−1+ 2(1+κ)12aγ 0l)( 2ll)( 1+κ2aγ 0) l, for c ≥ 1. (17)(−1) c( ) c−1+ 2(1+κ)12aγ 0( κc)c! + κc−1(c − 1)!⎤∑c−1( )( )( ) l c − 1 2l 1+κ ⎥·⎦. (18)l l 2aγ 0l=0Although 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 γ 0 =2σ 2 /σn,2and( √P e = 1 1 −2aσ 2σ 2 n + aσ 2 )= 1 2( √a21 −γ )01+ a 2 γ 0(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 CARRIERFREQUENCY OFFSETS AND TIMING OFFSETSIn this section, we consider the case of imperfect synchronization,where both the CFOs and TOs are nonzero. Let ɛ u ,u =1, 2,...,K denote the uth user’s residual CFO normalizedby the subcarrier spacing |ɛ u |≤0.5, ∀ u, and let μ u , 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 formY uk= H u k,kX u k + ∑ q∈Suq≠kv=1,v≠uH u k,qX u q +u∑H u,Ik,q Xu,I qq∈S u} {{ }self interference (SI)+K∑ ∑Hk,qX v q v + H v,Ik,q Xv,I q +Wk uq∈S v(20)} {{ }MUIwhere Xqu and Xqu,I are the symbols from the current andinterfering frames, respectively, of the uth user, and Wk u is theoutput noise of variance σn.IftheTOis−ve, 2 the interferingframe will be the previous frame; if the TO is +ve, the interferingframe will be the next frame. Coefficients H k,q ’s dependon the CFO and TO values. To write the expressions for thesecoefficients for N g = L − 1, we need to consider four differentcases of TOs, which are referred to as cases a–d of TOs, where0 < −μ u ≤ N g for case a, −μ u >N g for case b, 0 N g + μ u(23)(n b1 ,n b2 )=(l − μ u − N g ,N − 1), ∀l (24){(0,N − 1 − μu + l), for 0 ≤ l ≤ μ(n c1 ,n c2 )=u − 1(0,N − 1), for l ≥ μ(25)(n d1 ,n d2 )=(0,N − 1+l − μ u ) ∀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 H u,Ik,q’s for cases a and b can be written asH u,I)q=ej2π(μu+Ng Nk,qL−1∑l=N g +μ u +1h u l e −j2πlqN Γu,lqk (0,n α 1−1) (27)where n α1 in (27) is n a1 for case a and n b1 for case b. Forcase a, paths l ≤ N g + μ u do not contribute to previous frameinterference, which corresponds to Γ u,lqk (0,n α 1− 1) = 0 forthese paths. For case b, all paths contribute to previous frameinterference. Thus, Γ u,lqk ≠0for all paths. Thus, Hu,Iqkfor casesa and b can be written asH u,Ik,qj2π(μu+Ng )q= e NL−1∑l=0h u l e −j2πlqN Γu,lqk (0,n α 1− 1). (28)Likewise, the expressions for H u,Ik,qs for cases c and d can bewritten as∑μ u −1H u,Ik,q =e−j2π(Ng−μu)q Nl=0h u l e −j2πlqN Γu,lqk (n α 2+1,N−1) (29)where n α2 in (29) is n c2 for case c and n d2 for case d.For case c, paths l ≥ μ u do not contribute to next frameinterference, which corresponds to Γ u,lqk (n α 2+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 2011interference. Thus, Γ u,lqk ≠0for all paths. Thus, Hu,Iqkfor casesc and d can be written as∑L−1H u,Ik,q =e−j2π(Ng−μu)qNl=0h u l e −j2πlqN Γu,lqk (n α 2+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 bys u kqΔ= E [ L−1Hk,qu ] j2πμuq∑= e n E [h u l ] e −j2πlqN Γu,lqk (n α 1,n α2 ).l=0(31)The first path of each cluster is dominant nonrandom 2and contributes to the mean. Thus, E[h u l] ≠0 for l ∈{T0 u ,T1 u ,...,TN u c u −1 }, ands u kq =e j2πμuqNs u,IkqNc u −1∑i=0Similarly[ ]Δ= E H u,Ik,q⎧⎪⎨=⎪⎩j2π(μu+Ng )qe N·Γ u,T u i−j2π(Ng −μu)qe N√Ωu0i e jθu i e−j2πT u i qN Γ u,T u iqk(n α1 ,n α2 ). (32)N∑c u −1i=0i=0√Ωu0i e jθu i e −j2πT u i qNqk(0,n α1 − 1), for μ u < 0.N∑c u −1 √Ωu0i e jθu i e −j2πT i u qN·Γ u,T u iqk(n α2 +1,N − 1), for μ u > 0(33)2) The coefficients of any given user u, (i.e., Hk,q u s) arecorrelated, whereas the coefficients of any two differentusers (i.e., Hk,q u s and Hv k,qs, u ≠ v) are uncorrelated.Computation of the exact BER would involve M-fold integralin 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 Hk,q u s are correlated, we obtain an estimate of Hu k,qand H u,Ik,q , in terms of Hu k,k .2) Obtain expressions for the variances of SI/MUI and theSINR conditioned on |Hk,k u |.3) Obtain an expression for the BER conditioned on |Hk,k u |by assuming the estimation errors in Hk,q u s and Hu,Ik,q stobe Gaussian, and uncondition it to obtain unconditionalBER.Step 1 To obtain an estimate of Hk,q u in terms of Hu k,k ,weuse the fact that, if two nonzero mean complex Gaussianrandom variables X and Y having means m x and m y ,2 The 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.Y ukrespectively, are correlated, an estimate of one variable(e.g., Y ) can be obtained, in terms of the other variable,as [23]Ŷ = C X,Yσ 2 X(X + m y − C )X,YσX2 m X(34)with an estimation error E Y =Y − Ŷ of variance σ2 Y −CX,Y 2 /σ2 X , where C X,Y is the covariance of X and Y , andσX 2 and σ2 Y are the variances of X and Y , respectively.Using this, we can write all H k,q s in (20) in terms of H k,k ,to get= Hk,kX u ku (Cuk,q+ ∑ q∈Suq≠k(σ u k )2 Hu k,k++ ∑ q∈S u(Cu,Ik,q(σ u k )2 Hu k,k++K∑ ∑v=1,v≠u()s u kq− Cu k,q(σk u su )2 kk(s u,Ikq − Cu,I k,q(σk u su )2 kk)+E u q)+E u,IqX u q)X u,Iqq∈S vH v k,qX v q + H v,Ik,q Xv,I q + W u k (35)where Ck,q u = E[(Hu k,k − su kk )(Hu k,q − su kq )∗ ], C u,Ik,q =E[(Hk,k u −su kk )(Hu,I k,q − su,I kq )∗ ], (σk u)2 =E[(Hk,k u −su kk )(Hk,k u −su kk )∗ ], and (.) ∗ denotes the conjugate operation.Step 2 Now, defining (σq u ) 2 = Δ E[(Hk,q u − su kq )(Hu k,q − su kq )∗ ],(σ E u q) 2 =(σ Δ q u ) 2 −|Ck,q u |2 /(σk u)2 , and (σq,I u )2 =ΔE[(H u,Ik,q − su,I kq )(Hu k,q − su,I kq )∗ ], the total variance ofall the terms that are interference to the uth user’s symbolon the kth subcarrier in (35), conditioned on Hk,k u , isobtained as⎛∣σI|H 2 k,k=|Hk,k| u 2 ⎜∑ ∣C u k,q∣ 2 ∣ ∣∣C u,I⎝(σ +∑ k,q ∣ 2 ⎞⎟uq∈Su k )4 q∈S u(σk u ⎠)4q≠k} {{ }+ ∑ q∈Suq≠k(σuq) 2−∣ ∣∣C uk,q∣ ∣∣2Δ=A(σ +∑ (σu,Ik u)2 qq∈S u∣ ∣∣C u,I∣) 2−k,q ∣ 2(σ u k )2} {{ }∣Δ=B 1∣∣+ ∑ q∈Su∣ su kq − Cu k,q(σk u su )2 kk2 + ∑ ∣∣∣∣∣ 2s u,Ikq∣− Cu,I k,qq∈S u(σk u su )2 kk∣q≠k} {{ }Δ=B 2K∑ ∑ ( )+ σv 2+ ( )q σv 2K∑ ∑ ∣ ∣ ∣ q,I ∣s vkq 2 ∣∣s +v,Ikq ∣ 2v=1, q∈S v=1,v q∈S vv≠uv≠u} {{ } } {{ }+Δ=B 3∣.Δ=B 4(36)

RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4397TABLE IEXPRESSIONS FOR A AND B 1 FOR DIFFERENT TIME OFFSET CASES a–dTABLE IIEXPRESSIONS FOR B 3 FOR DIFFERENT TIME OFFSET CASES a–dAssuming that, among K users in the system, K λ usersbelong to case λ, λ∈{a, b, c, d}, the expressions for termsA and B 1 in (36) for different TO cases are given inTable I, where we have (37)–(39), shown at the bottomof the page. Similarly, the expressions for the term B 3 in(36) for the different TO cases are given in Table II. Now,(σ u k ) 2 =Nc u −1∑i=0Nc ( ) u −1σu 2∑q =i=0N∑c⎧⎪ u −1( ) ⎨σu 2q,I =⎪ ⎩i=0N∑c u −1i=0N u p,i −1 ∑p=1N u p,i −1 ∑p=1N u p,i −1 ∑p=1Np,i u ∑−1p=1∣kk(n λ1 ,n λ2 )∣Ω u u+p)pi ∣ Γu,(T i∣qk(n λ1 ,n λ2 )∣Ω u u+p)pi ∣ Γu,(T iΩ u u+p)pi ∣ Γu,(T iΩ u u+p)pi ∣ Γu,(T i22∣2qk(0,n λ1 − 1)∣ for μ u < 0∣2qk(n λ2 +1,N − 1)∣ , for μ u > 0(37)(38)(39)

4398 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011defining B =B 1 +B 2 +B 3 +B 4 +σn, 2 the SINR at the kthsubcarrier of the uth user conditioned on Hk,k u , which wasdenoted by γ H u , is given byk,k∣ ∣ ∣∣H u ∣∣2γ H u = k,k k,k∣A ∣ 2 . (40)+ B∣H u k,kHere, E[Hk,k u ]=su kk, and the variance of the real andimaginary parts of Hk,k u , which was denoted by σ2 e,is1/2(σk u)2 .Letκ ′ = |s u kk |2 /2σe.2Step 3 Now, assuming the estimation errors Equ and Equ,I 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 1Here, we carry out the unconditioning in Step 3 usingthe pdf approach. The conditional BER, which wasdenoted by P e (γ H u ), can be written as P e(γk,k H ue −y2 /2 dy. Unconditioning over the Rician1/ √ 2π ∫ √ ∞ aγH uk,kpdf of R = |Hk,k uk,k )=|, we get an unconditional BER expression asP e = √ 1 ∫∞2π0c=0∫ ∞√ aγH uk,ke −y22 dyfR (r)dr. (41)Following the derivation steps similar to those given in theAppendix, (41) can be simplified as∑ ∞(s u c∑( ) lP e =1− e −κ′ kk )2c 1 b(2σe) 2 c c! l! 2σe2 I l (42)whereI l = √ 1∞(∫y 2l2π (a −Ay 2 ) l e−y20l=0)12 + B2σe 2(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 thereforeit is enough to evaluate this integral from 0 to ⌊ √ a/A⌋.B. Approach 2Here, we uncondition the BER using the CF approach. UsingCraig’s formula, the conditional BER, which was denoted byP e (γ H u ), can be written as P e(γk,k H u )=1/π ∫ π/2k,k 0exp(−aγ H u /k,k2sin 2 φ)dφ. Unconditioning over the r.v, Z = γ H u , we getk,kP e = 1 ππ∫20( )] −azE Z[exp2sin 2 dφ. (44)φUsing Taylor series expansion for the exponential in (44)P e = 1 ππ∫20= 1 2 + 1 π( ) azE Z[1 −2sin 2 + 1 ( ) 2 azφ 2! 2sin 2 φπ∫20− 1 3![ ∞∑l=1( ) 3 az2sin 2 + ···]dφφ](−a) l E Z (z l )2 l l!sin 2l dφ. (45)φDefine g = Δ |Hk,k u |2 . Then, z = g/Ag + B.For0 ≤ g

RAGHUNATH et al.: BER ANALYSIS OF UPLINK OFDMA IN THE PRESENCE OF CFOs AND TOs 4399Combining (45), and (48)–(50), we getP e = 1 2 + e−κ′ππ∫20[ ∞∑l=1(−a) l2 l l!sin 2l φ∞∑( ) l + k − 1(−1) k kk=0· (2σ ( )e2 ) (k+l) Ak ∑ ∞B k+l⎛· ⎝1 − e−B2Aσe2(k+l+c)∑n=0c=0(κ ′c (k + l + c)!(c!) 2) nB2Aσe2n!⎞ ]⎠ 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 = B 3 + B 4 + σ 2 n. The BER in this case is given by(10) or (18), with σ 2 n replaced by B 3 + B 4 + σ 2 n, which is anexact closed-form BER expression. For M-QAM (M > 4),the conditional BER is of the formFig. 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, N p,0 =10, ζ 0 =0.25,andK 0 =0, 5, 10 dB.Analysis versus simulation.)P e(γ H u =k,k√∑M−1(√ )c j Q d j γ H u . (52)k,kj=1For example, for 16-QAM, the expression for conditional BERis [24])(√ ) ⎛√⎞P e(γ H u = 3 γHk,k4 Q uk,k+ 1 5 2 Q 9γ H uk,k⎝ ⎠5− 1 ) (√5γ4 Q H u . (53)k,kThe unconditional BER can be obtained by averaging out eachterm using either the pdf or the CF approach.V. R ESULTS AND DISCUSSIONSWe numerically evaluated the analytical BER performanceand compared them with the simulated performance. In computing(51), the number of moments summed in the infinitesum is 100 (i.e., l =1to 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, ɛ u = μ u =0for all users) uplink OFDMAwith QPSK, N =64, K =4, M =16in single-cluster (N c =1) multipath Rician fading channel with N p,0 = L =10, andexponential delay profile factor ζ 0 =0.25. Rice factors of 0, 5,and 10 dB are considered. Similar performance plots for a twoclustermodel (with parameters N c =2, L =20, β =0.083,N p,0 =10, N p1 =5, and ζ 0 = ζ 1 =0.25) areshowninFig.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. 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, N p,0 =10, N p,1 =5, ζ 0 = ζ 1 =0.25, andK 0 = K 1 =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 [ɛ 1 ,ɛ 2 ,ɛ 3 ,ɛ 4 ]=[0.1, 0.2, −0.15, −0.3]and [μ 1 ,μ 2 ,μ 3 ,μ 4 ]=[−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 B 1 to B 4 in (36) and the noise variance σ 2 n, i.e.,B = B 1 + B 2 + B 3 + B 4 + σ 2 n, and A in the denominator isalso a function of interference variances. Thus, even if σ n =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 2011Fig. 4. BER performance of uplink OFDMA with nonzero CFOs andTOs for the single-cluster channel for different values of Rice factorsand QPSK. N =64, K =4, M =16, L =10, N p,0 =10, ζ 0 =0.25, K 0 =0, 5, 10 dB, [ɛ 1 ,ɛ 2 ,ɛ 3 ,ɛ 4 ]=[0.1, 0.2, −0.15, −0.3], and[μ 1 ,μ 2 ,μ 3 ,μ 4 ]=[−1, −5, 1, 5]. Analysis versus simulation.Fig. 6. BER performance of uplink OFDMA with nonzero CFOs andTOs for the single-cluster channel for different values of Rice factorsand 16-QAM. N =64, K =4, M =16, L =10, N p,0 =10,ζ 0 =0.25, K 0 =0, 5, 10 dB, [ɛ 1 ,ɛ 2 ,ɛ 3 ,ɛ 4 ]=[0.1, 0.2, −0.15, −0.3], and[μ 1 ,μ 2 ,μ 3 ,μ 4 ]=[−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, N p,0 =10, N p,1 =5, ζ 0 = ζ 1 =0.25, K 0 = K 1 =0, 5, 10 dB, [ɛ 1 ,ɛ 2 ,ɛ 3 ,ɛ 4 ]=[0.1, 0.2, −0.15, −0.3],and [μ 1 ,μ 2 ,μ 3 ,μ 4 ]=[−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 interferencecancellation/equalization algorithms to reduce these errorfloors is an interesting topic. In particular, message-passing algorithmson graphical models are quite promising and attractivein such systems.VI. CONCLUSIONWe have analyzed the BER performance of uplink OFDMAon Rician fading channels in the presence of nonzero CFOsand TOs. We have considered a multicluster multipath channelFig. 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, N p,0 =10, N p,1 =5, ζ 0 =ζ 1 =0.25, K 0 =K 1 =0, 5, 10 dB, [ɛ 1 ,ɛ 2 ,ɛ 3 ,ɛ 4 ]=[0.1, 0.2, −0.15, −0.3], and[μ 1 ,μ 2 ,μ 3 ,μ 4 ]=[−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 characterizationof 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 4401APPENDIXDERIVATION OF (10)Substituting the expression for I 0 (.) in (9), and (9) in (8) andchanging the order of integration, we can writeP e = √ e−κ ∑ ∞s 2c2πc=0(2σ 2 ) 2c (c!) 2∫ ∞·0√σ 2 n y2aSubstituting r 2 /2σ 2 = t, (54) becomesP e = e−κ√2π ∞ ∑c=0∫0s 2c ∫∞(2σ 2 ) c (c!) 20r 2c+1σ 2 e −r22σ 2 dre −y22 dy. (54)σ 2 n y22aσ 2∫0t c e −t dte −y22 dy. (55)With U = σ 2 ny 2 /2aσ 2 , the inner integral in (55) evaluates as∫ U0t c e −t dt = c! − e −USubstituting (56) in (55), we getP e = √ e−κ ∑ ∞s 2c2π (2σ 2 ) c (c!)c=0∫ ∞0e −y22 dy− √ e−κ ∑ ∞s 2c2π (2σ 2 ) c (c!)c=0c=0c∑l=0l=0c∑l=01l!c!l! U l . (56)∫ ∞0U l e −y22 dy. (57)Noting that 1/ √ 2π ∫ ∞ /20dy =1/2 and κ = s 2 /2σ 2 ,wee−y2have ∑ ∞c=0 (s2c /(2σ 2 ) c (c!)) = e κ . With this, the first term in(57) evaluates to 1/2, and henceP e = 1 2 − ∑ ∞s 2c c∑( )1 σ2 lne−κ(2σ 2 ) c (c!) l! 2aσ 2 I l (58)where I l =1/ √ 2π ∫ ∞0 y2l e −y2 /2 (1 + σ 2 n/aσ 2 )dy. With b Δ =1/2(1 + σ 2 n/aσ 2 ), I l can be written asI l = 0.5 √2π∫∞−∞y 2l e −by2 dy. (59)Using the notation F (.) to denote the Fourier transform, wehave∫ ∞−∞y 2l e −by2 dy = 1 b l d 2l F (e −t2 )df 2l ∣ ∣∣∣∣f=0. (60)Evaluating (59) and (60) evaluates to( (2l+1.5) ( ) 1 2aσ2 (l+0.5)(2l)!I l =2)aσ 2 + σn2 . (61)l!Substituting (61) in (58), we get (10).REFERENCES[1] R. Fantacci, D. Marabissi, and S. Papini, “Multiuser interference cancellationreceivers for OFDMA uplink communications with carrier frequencyoffset,” in Proc. IEEE GLOBECOM, Dec. 2004, pp. 2808–2812.[2] D. Huang and K. B. Letaief, “An interference cancellation scheme forcarrier frequency offsets correction in OFDMA systems,” IEEE Trans.Commun., vol. 53, no. 7, pp. 1155–1165, Jul. 2005.[3] J.-H. Lee and S.-C. Kim, “Detection of interleaved OFDMA uplinksignals in the presence of residual frequency offset using the SAGEalgorithm,” IEEE Trans. Veh. Technol., vol. 56, no. 3, pp. 1455–1460,May 2007.[4] S. Manohar, D. 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4402 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011K. Raghunath received the B.Tech. degree in electronicsand communication engineering from SriVenkateswara University, Tirupati, India, the M.E.degree in communication systems engineering fromP. S. G. College of Technology, Coimbatore, India,and the Ph.D. degree in electrical communicationengineering 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 MentorR&D. His current research interests include the implementation of long-termevolutionsystems.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 Instituteof Science, Bangalore, India.His current research interests are low-complexityequalization in large-dimension multiple-input–multiple-output intersignal-interference channelsand underwater acoustic communications.A. Chockalingam (S’92–M’93–SM’98) was born inTamil Nadu, India. He received the B.E. (Honors) degreein electronics and communication engineeringfrom the P. S. G. College of Technology, Coimbatore,India, in 1984, the M.Tech. degree (with specializationin satellite communications) from the IndianInstitute of Technology, Kharagpur, India, in 1985and the Ph.D. degree in electrical communication engineeringfrom the Indian Institute of Science (IISc),Bangalore, India, in 1993.During 1986 to 1993, he was with the TransmissionR&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 WIRELESSCOMMUNICATIONS. He served as a Guest Editor for the IEEE JOURNAL ONSELECTED AREAS IN COMMUNICATIONS (Special Issue on Multiuser Detectionfor Advanced Communication Systems and Networks). Currently, he servesas a Guest Editor for the IEEE JOURNAL OF SELECTED TOPICS IN SIGNALPROCESSING (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 Technology,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 Laboratory,Hyderabad, India, working on missile andelectronic warfare projects. From November 1994 toJanuary 1996, he was a Faculty Member with theDepartment 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 MathematicalSociety; 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 ONWIRELESS COMMUNICATIONS and an Editor for the IEEE TRANSACTIONSON 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 EngineeringSciences.