Bit Error Rate Calculation for OFDM with Synchronization Errors in Time and Frequency Selective Fading Channels


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1 Bit Error Rate Cacuation for OFDM with Synchronization Errors in Time and Frequency Seective Fading Channes Marco Krondorf and Gerhard Fettweis Vodafone Chair Mobie Communications Systems, Technische Universität Dresden, D0106 Dresden, Germany Abstract In this paper we present an anaytica approach to evauate the bit error rate BER) of OFDM systems subject to carrier frequency offset CFO) and channe estimation error in Rayeigh fat fading as we as in time and frequency seective fading channes. Based on correct modeing of the correation between channe estimates and received signas with carrier frequency offset, the bit error rate can be numericay evauated by averaging bit error rates on different subcarriers using an anaytica expression of doube integras. The resuts iustrate that the anaysis can approximate the simuative performance very accuratey if the power deay profie of fading channes and carrier frequency offset are known. I. INTRODUCTION Orthogona Frequency Division Mutipexing OFDM) is a widey appied technique for wireess communications, which enabes simpe onetap equaization by cycic prefix insertation. Conversey, the sensitivity of OFDM systems to carrier frequency offset CFO) is higher than that of singecarrier systems. In present OFDM standards, such as IEEE80.11a/g or DVBT, preambe or piots) are used to estimate and compensate the carrier frequency offset CFO) and channe impuse response but after the CFO estimation and compensation, the residua carrier frequency offset sti destroys the orthogonaity of the received OFDM signas and the channe estimates, which worsen further the bit error rate of OFDM systems during the equaization process. In the iterature, the effects of carrier frequency offset on symbo error rate are mosty investigated under the assumption of perfect channe knowedge. The papers [6] and [3] consider the effects of carrier frequency offset ony without channe estimation and equaization) and give exact anaytica expressions in terms of SNRoss or bit error rates for the AWGN channe. The authors of [7] extend their anaysis to frequencyseective fading channes and derive the correspondent uncoded bit error rates for OFDM systems assuming aso perfect channe knowedge. In addition to the papers above, the authors of [1] tried to anayze the joint effect of carrier frequency offset and channe estimation error on uncoded bit error rate for OFDM systems, but the assumption in [1] channe estimates are uncorreated to channe estimation error or uncorreated to received signas) does not hod in rea OFDM systems, especiay when carrier frequency offset is arge. In this paper, after introducing the OFDM system mode Sec.II) and the probabiity density function anaysis [9] Sec.III), it wi be expained how to mode the correation between channe estimates and received signas for deriving the bit error rates of OFDM systems with carrier frequency offset in Rayeigh fat fading channes Sec.IV) and Rayeigh frequency and time seective fading channes Sec.V & Sec.VI) respectivey. It shoud be noted, that the terms bit error rate and bit error probabiity are used with equa meaning. This is due to the fact, that the bit error rate converges towards bit error probabiity with increasing observation time in a stationary environment. II. OFDM SYSTEM MODEL We consider an OFDM system with Npoint FFT. The data is MQAM moduated to different OFDM data subcarriers, then transformed to a time domain signa by IFFT operation and prepended by a cycic prefix, which is chosen to be onger than the maxima channe impuse response CIR) ength L. In this paper, it is assumed that the residua carrier frequency offset is a given deterministic vaue in practice, the residua carrier frequency offset is a Gaussianike distributed random variabe.) A oneofdmsymboong preambe is used for channe estimation. Furthermore, static channe characteristics during one OFDM symbo are assumed. The samped signa for the th subcarrier after the receiver FFT processing can be written as Y X H + N 1 k N, X k H k Ik ) + W 1) where X represents the transmitted compex QAM moduated symbo on subcarrier and W is a compex Gaussian noise sampe. The coefficient H k denotes the frequency domain channe transfer function on subcarrier k, which is the discrete Fourier transform of the CIR hτ) with maxima L taps H k hτ)e jπkτ/n ) τ0 the coefficient Ik ) represents the impact of the received signa at subcarrier k on the received signa at subcarrier due
2 to the residua carrier frequency offset [3] Ik ) e jπk )+ f)1 1/N) sinπk ) + f)) Nsinπk ) + f)/n) where f is the residua carrier frequency offset normaized to the subcarrier spacing. In addition, ater in this paper the summation N 1 wi be abbreviated as k N,. Equation 1) iustrates that f causes inter carrier interference ICI) as we as a phase shift and attenuation that is common for a subcarriers. Furthermore there is a time variant common phase shift for a subcarriers due to f as given in [7] that is not modeed here. This is due to the fact, that this time variant common phase term is considered to be robusty estimated and compensated by continuous piots that are inserted in the OFDM data symbos. Subsequenty, we consider the preambebased Least Square LS) frequency domain channe estimation to obtain the channe state information on subcarrier. Ĥ Y P, H + X P,kH k Ik ) + W 3) X P, X P, foowed by the frequency domain zeroforcing equaization before data detection Z Y Ĥ 4) where X P, and Y P, denote the transmitted and received preambe symbo on subcarrier. The Gaussian noise of preambe part W has the same variance as W of data part σw σ W ). The power of preambe signas and the average power of transmitted data signas on a subcarriers is normaized to one X P σx 1). The derived anaytica expressions for bit error and symbo error rates are compared with appropriate simuation resuts. In these numerica exampes, we consider a OFDM system with 64point FFT. The data is 4QAMmoduated to different subcarriers, then transformed to the time domain signa by IFFT operation and prepended by a 16tap ong cycic prefix. The data is randomy generated and one OFDM piot symbo was used for channe estimation. The used BPSK piot data in the frequency domain is given by X P, 1) for subcarrier index [ 3 : 1 : 31] In addition to Rayeigh fat fading channes, two types of frequency seective fading channes are considered as isted beow 1) Equa power deay profie PDP) στ 1, τ 0, 1,...,L 1 L ) Exponentia power deay profie σ τ 1 C e Dτ/L, τ 0, 1,...,L 1 where στ E{ h τ } and the factor C τ0 e Dτ/L is chosen to normaize the PDP as τ0 σ τ 1 or σh 1 on a subcarriers). The channe taps h τ are assumed to be compex Gaussian RV with uncorreated rea and imaginary parts. Additionay, the maxima channe impuse response is kept shorter/equa than the cycic prefix L 16) in the numerica exampes. Uness otherwise stated, 4QAM digita moduation is used in a the numerica exampes of this paper. III. PROBABILITY DENSITY FUNCTIONS ANALYSIS The author of [9] has suggested a correation mode regarding channe estimation for singecarrier systems and derived the correspondent symbo error rate and bit error rate of QAMmoduated signas transmitted in fat Rayeigh and Ricean channes. In this section, a short review of the contribution of [9] wi be given in order to further extend these resuts to OFDM systems for time and frequency seective fading channes with both carrier frequency offset and channe estimation error. The singecarrier transmission mode without carrier frequency offset for fat Rayeigh fading channes can be written as y hx + w 5) where y, h, x and w denote the compex baseband representation of the received signa, the channe coefficient, the transmitted data and the Gaussian noise with variance σw respectivey. In [9], the channe estimate ĥ is assumed to be biased and used for zero forcing equaization as foows z y with ĥ αh + ν 6) ĥ where α denotes the mutipicative bias in the channe estimates and ν is a zeromean compex Gaussian noise with variance σν. The channe coefficient h and Gaussian noise ν are assumed to be uncorreated. Since y and ĥ are correated, we can write y ryĥσ y ĥ + η 7) σĥ where the zeromean Gaussian random variabe η with variance σ η E{ η } σ y 1 r yĥ ) is uncorreated to ĥ and r y ĥ E{yĥ } σĥσ y with σ ĥ E{ ĥ } and σ y E{ y } Substituting equation 7) into equation 6), the equaized signa takes the form where b ryĥσy σĥ signa z and ǫ η ĥ z ryĥσ y + η ĥ b + ǫ 8) σĥ E{yĥ } σ ĥ represents the bias of equaized denotes the zero mean noise part of the equaized signa z. Interestingy, the term E{yĥ } xα λ)σ h contains the anaytica expression of the time seective channe properties where the channe gain is expressed by σ h E{ h }. In the time interva λ between channe estimation and data detection, the channe variation is expressed in terms
3 of correation coefficient λ) of i.i.d. channe reaizations given by λ) E{ht)h t + λ)} σ h 1,0) x I z) 0,0) x 1 It shoud be noted that the bias b is highy correated to the transmitted data symbo x and can be formuated by R z) bx) R{b} + ji{b} b r x) + jb i x) α x λ)σh ) α σh + σ ν Subsequenty, due to the fact that zeromean Gaussian random variabes η and ĥ are uncorreated, we have ǫ η ĥ η ejφη ĥ ejφ ĥ η ĥ ejφη φ ĥ ) ǫ e jφǫ where the phase φ η and φĥ are uniformy distributed. Hence, the phase φ ǫ is aso uniformy distributed, i.e. its pdf is f φǫ φ ǫ ) 1 π and the probabiity density functions of ǫ is given by [10] with f ǫ ǫ ) a x) σ η σ ĥ 9) 10) ǫ a ǫ + a ) 11) σ y 1 r yĥ ) σ ĥ x α σ 4 h 1 r h λ)) + σ νσ h α σ h + σ ν ) σ w + α σh + σ ν 1) Furthermore, it is we known from [5] that ǫ ǫ e jφǫ can be expressed in cartesian coordinates ǫ ǫ r + jǫ i with joint pdf f ǫr,ǫ i ǫ r, ǫ i ) 1 ǫ f ǫ ǫ )f φǫ ϕ) 13) ǫ ǫ r +ǫ i Substituting Eq.8), Eq.10) and Eq.11) into Eq. 13), the joint pdf of z z r + jz i with given transmitted symbo x in cartesian coordinates [9] can be derived by a x) f z x z x) π z bx) + a x)) 14) zzr+jz i Additionay, the antiderivative of Eq.14) w.r.t. z z r + jz i is given by ) z z i b i x))arctan r b rx) a F z x z x) x)+z i b ix)) π 15) a x) + z i b i x)) ) z z r b r x))arctan i b ix) a + x)+z r b rx)) π a x) + z r b r x)) B 1,1 x3 4 1,1) x 0,1) Fig QAM consteation diagram  decision region for one bit position of symbo x 1 The resut of 15) can be used to cacuate the bit error rate of a given MQAM consteation [9]. In a MQAM consteation there are M dm) different possibe bit positions with respect to the MQAM consteation. This fact is iustrated in Fig.1 using a 4QAM consteation. Taking a coser ook to the 4QAM exampe,we have four consteation points x , +1), x 1 1, +1), x , 1) and x 4 1 1, 1). For each consteation point x m, where m denotes the consteation point index, the parameters bx m ) b m,r + jb m,i and parameter a x m ) a m has to be cacuated separatey for evauating the bit error rate using Eq.15) and Eq.16). The probabiity of an erroneous bit with respect to the mth QAM symbo x m can be cacuated by using the primitive of conditiona PDF Eq.15) and an appropriate decision region B m,ν for the bit position used for doube integration see Fig.1). Hence, the bit error probabiity P b x m ) takes the form P b x m ) 1 dm) dm) ν1 [[ Fz xm z x m ) ]] B m,ν 16) Finay, the bit error probabiity can be obtained by averaging over a possibe consteation points, when assuming equa probabe MQAM symbos as foows P b 1 M P b x m ) 17) M m1 IV. OFDM BIT ERROR RATE ANALYSIS FOR RAYLEIGH FLAT FADING CHANNELS In this section, the derivation of the bit error rates of OFDM systems with carrier frequency offset and channe estimation error in Rayeigh fat fading channes H H k H, k, [1,..., N]) wi be given. Additionay, time seective channe characteristics are not considered in this section. A. Mathematica Derivation Firsty, we can rewrite the channe estimates of subcarrier in Eq.3) to be Ĥ H 1 + P,kIk ) + W 18) X P, X P,
4 and from Eq.18), an Eq.6)ike expression can be given Ĥ α H + W X P, α H + ν 19) by defining effective channe H and effective bias α H H, α 1 + P,kIk ) X P, where α is a deterministic quantity with given subcarrier index, a set of preambe symbos X P,k and a fixed frequency offset. For σ ν, which represents the AWGN variance of the channe estimates, we have σ ν σw for a subcarriers, if X P, 1 for a subcarrier indexes. Appying the same method above for Eq.1), the same definition of effective channe H can be used to obtain an Eq.5)ike expression as foows Y H X + kik ) +W }{{} X H X + W 0) Given Eq.0), the effective symbo X can be defined, that is no onger a deterministic vaue but a stochastic quantity due to i.i.d. data symbos on subcarriers k.giving a certain transmit symbo X and assuming randomy transmitted data signas X k with k, we can decompose the effective symbo X as foows X X + X kik ) X + ICI which shows the stochastic nature of X due to the random ICI part. Appying the centra imit theorem athough, stricty speaking, X has discrete distribution), we assume that the intercarrier interference term is a compex zeromean Gaussian random variabe ICI p+jq. The mutua uncorreated rea and imaginary parts p and q have the same variance for a consteation points σ ICI 1 Secondy, according to Eq.9) and Eq.1), we cacuate the parameters b X ) b,r + jb,i and a X ) for MQAM effective data symbos X on subcarrier in Rayeigh fat fading channes. ) b X ) X α σ H α σ H + σ ν 1) a X ) σ ν X σ H σ Ĥ ) + σ W σ Ĥ ) where σ H σ H and σ Ĥ α σ H + σ W. As an exampe, for one 4QAM consteation point with the index m 1, X 1, 1 1, 1) 1 1+j), on subcarrier we need to cacuate b X 1, ) b 1,,r + jb 1,,i and parameter a X 1, ) separatey for each effective symbo reaization X 1, X 1, + p + jq j) + p + jq in Eq. 15) and Eq. 16). Subsequenty, the bit error rate on subcarrier for the mth consteation point can be expressed using Eq.15) and Eq.16) by the foowing doube integra P b X m, ) P b X m, + p + jq) π σ ICI e p +q σ ICI dp dq 3) Finay, to obtain the genera bit error rate we have to average Eq.3) over a subcarriers with index and MQAM consteation points with index m as foows P b 1 MN B. Numerica Resuts N/ 1 M N/ m1 P b X m, ) 4) Fig. presents the cacuated and simuated bit error rates vs. SNR with given carrier frequency offset f for fat Rayeigh fading channes. The resuts iustrate that the anaysis can approximate the simuative performance very accuratey. BER f 1% cac f 1% Sim f 10% cac f 10% Sim f 15% cac f 15% Sim f 18% cac f 18% Sim f 0% cac f 0% Sim Fig.. Comparison of cacuated and simuated 4QAM BER vs. SNR under different f in Rayeigh fat fading channe V. OFDM BIT ERROR RATE ANALYSIS FOR RAYLEIGH FREQUENCY SELECTIVE FADING CHANNELS In this section the bit error rates of OFDM systems with carrier frequency offset and channe estimation error in frequency seective fading channes are derived. As in the section before, time seective channe characteristics are not considered. A. Mathematica Derivation Firsty, the crosscorreation coefficient of channe transfer functions between subcarrier k and in frequency seective fading channes are defined r k, E{H kh } σ H 1 k 5)
5 where the channe gains σh are the same for a subcarriers. Assuming mutua uncorreated channe taps of the CIR and appying Eq. ), one get { } r k, E hτ)h γ)e jπkτ/n e jπγ/n τ0 γ0 E { hτ) } e jπk )τ/n 6) τ0 where the vaues E{ hτ) } στ denote the tap power of the fading channes with different power deay profie. The cross correation properties of the channe coefficients is written as foows H k r k, H + V k where V k a compex zeromean Gaussian with variance σv k σh 1 r k, ) and E{V k H }0. Substituting the crosscorreation properties of the channe coefficients into Eq.3), the channe estimate can be written as Ĥ H 1 + P,kIk )r k, + ˇν X P, Ȟˇα + ˇν 7) where the effective channe can be defined as Ȟ H In addition, the effective noise term of the channe estimates with variance σ ˇν is given by ˇν 1 X P,k Ik )V k + W 8) X P, The factor ˇα represents the fact that the channe estimates are biased and is expressed by ˇα 1 + P,kIk )r k, X P, Secondy, substituting the crosscorreation properties of the channe coefficients into Eq.1), one obtain Y H X + kik )r k, + X k Ik )V k + W Ȟ ˇX + ˇW 9) where the effective noise term ˇW with variance σ ˇW can be written as ˇW X k Ik )V k + W 30) The stochastic effective transmit symbo ˇX is given by ˇX X + X kik )r k, 31) Appying the centra imit theorem, the ICI part of ˇX can be expressed as compex Gaussian random variabe as foows ICI X kik )r k, p + jq 3) The uncorreated rea part p and imaginary part q of the intercarrier interference have the same variance for a MQAM consteation points: ˇσ ICI Ik ) r k, According to Eq.9) and Eq.1), we cacuate the parameters b ˇX ) b,r + jb,i and a ˇX ) for MQAM effective data symbos ˇX on subcarrier in frequency seective fading channes. ) b ˇX ) ˇX ˇα σ Ȟ ˇα σ + 33) Ȟ σˇν a ˇX ) σˇν ˇX σ Ȟ σ Ĥ ) + σˇw σ Ĥ 34) where σ Ȟ σ H and σ Ĥ ˇα σ H + σˇν. From Eq.33) and Eq.34) one can observe, that the parameters σ ˇW and σ ˇν have to be cacuated exacty to obtain reiabe resuts. For an exact expression of σ ˇW one take Eq.30), σ V k σ H 1 r k, ) together with the assumptions of mutuay uncorreated data symbos and obtain σ ˇW σw + σh Ik ) Ik ) r k, 35) For variance of the effective AWGN term of the channe estimate according Eq.8) we get σ ˇν σ W + m X P,k X P,m Ik )I m ) r k,m r k, r m, σ H ) 36) Simiar to Sec.IV, given a certain effective symbo ˇXm, X m, + ICI X m, + p + jq together with an appropriate decision region of the reated MQAM data symbo X m, on subcarrier, where m is the consteation point index, the bit error probabiity is obtained by averaging over a subcarriers and MQAM consteation points with index m as foows ˇP b 1 MN N/ 1 B. Numerica Exampes M N/ m1 P b X m, + p + jq) πˇσ ICI e p +q ˇσ ICI dp dq 37) Fig. 3 present the cacuated and simuated bit error rates vs. SNR with given carrier frequency offset f for frequency seective fading channes with 8 tap equa power deay profie. The resuts iustrate that the anaysis can approximate the simuative performance very accuratey, if the power deay profie and the carrier frequency offset are known.
6 BER f 1% cac f 1% Sim f 5% cac f 5% Sim f 10% cac f 10% Sim f 15% cac f 15% Sim f 30% cac f 30% Sim For sake of simpicity, it is assumed that a channe taps have the same autocorreation coefficient τ, λ) λ) 0 τ L 1. Together with the reation τ0 σ τ σ H, we obtain r H, λ) λ) with N N 1 41) With respect to the frequency seective derivations of Sec.VA and Eq.9) and Eq.1) the parameters b and a have to be reformuated according the time seective channe properties as foows ) b ˇX ) ˇX ˇα λ)σ Ȟ ˇα σ Ȟ + σˇν 4) Fig. 3. Comparison of cacuated and simuated 4QAM BER vs. SNR under different f in frequency seective channes with 8 taps equay distributed power deay profie a ˇX ) ˇX ˇα σ 4 Ȟ 1 r h λ)) + σˇν σ Ȟ σ Ĥ ) VI. OFDM PERFORMANCE ANALYSIS IN TIME AND FREQUENCY SELECTIVE FADING CHANNELS In current OFDM systems such as 80.11a/n, there is a typica OFDM bock structure.a OFDM bock consists of a set of preambe symbos used for acquisition, synchronization and channe estimation, foowed by a set of seriay concatenated OFDM data symbos. User mobiity gives rise to a considerabe variation of the mobie channe during one OFDM bock what causes outdated channe information in certain OFDM symbos if there is no appropriate channe tracking. To be precise, during the time period λ between channe estimation and OFDM symbo reception the channe changes in a way that the estimated channe information used for equaization does not fit the actua channe anymore. If there is no channe tracking at the receiver side, our aim is it to extend the BER derivation of Sec. V according considerabe channe variations between channe estimation and data detection. A. Channe AutoCorreation Anaysis To derive the OFDM bit error rates for time and frequency seective fading channes it is necessary to define the autocorreation properties of certain channe coefficients H. The autocorreation coefficient of subcarrier is defined as foows r H, λ) E{H t)h t + λ)} 38) Appying Eq.) we get σ H { E{H t)h t + λ)} E hτ, t)h ν, t + λ) τ0 ν0 e πτ ν) N When assuming uncorreated channe taps it foows E{H t)h t + λ)} } E{hτ, t)h τ, t + λ)} τ0 39) τ, λ)στ 40) τ0 + σˇw σ Ĥ 43) where σ Ȟ σ H and σ Ĥ ˇα σ H + σˇν according Sec.VA. B. Numerica Resuts For the numerica evauation of the anaytica BER resuts see. Fig.4), the time seectivity of the compex Gaussian channe taps was modeed as foows with hτ, t + λ) λ)hτ, t) + v τ,λ E{ hτ, t) } E{ hτ, t + λ) } σ τ where v τ,λ is a compex Gaussian RV with variance σv τ,λ στ1 λ) ) and E{hτ, t)vτ,λ } 0. For sake of simpicity, it is assumed that the channe is stationary during one OFDM symbo but changes from symbo to symbo in the above defined manner. BER 10 λ) 1 Sim λ) 1 cac λ) Sim λ) cac λ) Sim λ) cac λ) Sim λ) cac Fig. 4. Comparison of cacuated and simuated uncoded 4QAM BER vs. SNR under f 3% for different channe autocorreation coefficients in exponentia PDP 8 Taps channe
7 VII. RESIDUAL CFO DISTRIBUTION There is a number of agorithms to estimate the carrier frequency offset in OFDM receivers proposed by severa authors such as M. Morei/U.Mengai [4] and T.M.Schmid/D.C.Cox [8]. The most common way of ML CFO estimation is to use periodic piot sequences, meaning that they resut from the repetition of a fixed bock of sampes in the time domain. For convenience, we have a bock of ength Λ and an observation interva of pλ, where p [, 3,...]. If we choose p the ML CFO estimate takes the form [4] f N πλ arg { Λ 1 } y k)yk + Λ) ν0 44) Using the resuts of Couson [], the residua CFO f after estimation Eq.44)) and compensation can be identified to be a zeromean Gaussian RV of variance σ f under AWGN channe conditions. In [4] and [8] it was shown that the variance of f reaches the CramerRao ower bound CRLB) in case of AWGN channe conditions and can be expressed for N pλ and p as var[ f] σ f N 4π Λ 3 SNR 45) where SNR E S /N 0. Since the PDF of the residua CFO is Gaussian under AWGN conditions, one can verify that in case of fat Rayeigh fading channe conditions the PDF can be formuated as p f f) σ f 1 f σ f ) 46) The anaytica bit error rate expressions obtained in the ast sections mainy depend on a specific f. Using the PDF in Eq.46) it is possibe to formuate a resuting bit error rate P b that is independent from any specific f and can be written as foows P b P b f)p f f) d f 47) In Fig.5, we show BER vs. SNR under rea synchronization conditions in a exponentia 8 taps channe scenario. A one OFDM symbo ong preambe was used for frequency synchronization, having two identica haves, 3 sampes each. Equation 44) was used for CFO estimation and compensation. The resuting BER was cacuated numericay using Eq.47). Athough Eq.46) is vaid ony in fat Rayeigh fading environments, it is shown in Fig.5 that our anaysis is sti very accurate even under frequency seective channe conditions. BER QAM cac 4QAM Simuation 16QAM cac 16QAM Simuation λ) λ) Fig. 5. Comparison of cacuated and simuated BER vs. SNR for exponentia PDP 8 taps channe under rea frequency synchronization and channe estimation conditions channes. We show that the bit error probabiity can be cacuated exacty when using a priori knowedge of mobie channe power deay profie. Therefore, our anaysis provides a usefu and necessary too for system engineers to crossreference the performance of OFDM synchronization/acquisition agorithms with appropriate uncoded bit error rates. REFERENCES [1] H. Cheon and D. Hong. Effect of channe estimation error in OFDMbased WLAN. In Proc. IEEE Communication Letters, voume 6, pages , May 00. [] A.J. Couson. Maximum Likeihood Synchronization for OFDM Using a Piot Symbo: Anaysis. In Proc. IEEE Journa on Seected Areas in Communications, voume 19, pages , December 001. [3] K.Sathananthan and C. Teambura. Probabiity of error cacuation of OFDM with frequency offset. In Proc. IEEE Transactions on Communications, voume 49, pages , November 001. [4] M. Morei and U. Mengai. CarrierFrequency Estimation for Transmissions over Seective Channes. In Proc. IEEE Transactions on Communications, voume 48, pages , September 000. [5] A. Papouis. Probabiity, Random Variabes, and Stochastic Processes. New York: McGrawHi, [6] T. Poet, M. van Bade, and M.Moenecaey. BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise. In Proc. IEEE Transactions on Communications, voume 4, pages , [7] L Rugini, P. Banei, and S. Cacopardi. Probabiity of Error of OFDM Systems with Carrier Frequency Offset in FrequencySeective Fading Channes [8] T. Schmid and D.C. Cox. Robust Frequency and Timing Synchronization for OFDM. In Proc. IEEE Transactions on Communications, voume 45, pages , December [9] S. K. Wison and J. M. Cioffi. Probabiity Density Functions for Anayzing MutiAmpitude Consteations in Rayeigh and Ricean Channes. In IEEE Transactions on Communications, March [10] M. Windisch and G. Fettweis. Performance Degradation due to I/Q Imbaance in MutiCarrier Direct Conversion Receivers: A Theoretica Anaysis. In Proc. IEEE Internationa Conference on Communications ICC 06), Istanbu, Turkey, June 006. VIII. CONCLUSIONS In this paper we show how to anayticay evauate the bit error probabiity/rate of OFDM systems subject to carrier frequency offset and channe estimation error in fat Rayeigh fading as we as frequency and time seective mobie fading
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