Symbo Error Rate Cacuation for Aamouti Space Time Coded OFDM in Direct Conversion Receivers Marco Krondorf Vodafone Chair Mobie Communications Systems, Technische Universität Dresden, D-6 Dresden, Germany krondorf@ifn.et.tu-dresden.de, http://www.ifn.et.tu-dresden.de/mns Gerhard Fettweis Vodafone Chair Mobie Communications Systems, Technische Universität Dresden, D-6 Dresden, Germany fettweis@ifn.et.tu-dresden.de, http://www.ifn.et.tu-dresden.de/mns Abstract In this paper we present an anaytica approach to evauate the M-QAM symbo error rate SER of Aamouti space time coded OFDM direct conversion receivers subject to carrier frequency offset CFO, channe estimation errors, outdated channe state information and fat receiver I/Q imbaance in Rayeigh time and frequency seective fading channes. Based on correct modeing of the correation between channe estimates and received signas with carrier frequency offset and receiver I/Q imbaance, the SER can be numericay evauated by averaging symbo 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, the receiver I/Q imbaance parameters and carrier frequency offset are known. I. INTRODUCTION Orthogona Frequency Division Mutipexing OFDM is a widey appied technique for wireess communications, which enabes simpe one-tap equaization by cycic prefix insertation. Conversey, the sensitivity of OFDM systems to carrier frequency offset CFO is higher than that of singe-carrier systems. In present OFDM standards, such as IEEE8.a/g or DVB-T, preambe or piots are used to estimate and compensate the carrier frequency offset CFO and channe impuse response. Unfortunatey, after CFO estimation and compensation, the residua carrier frequency offset sti destroys the orthogonaity of the received OFDM signas and corrupts channe estimates, which worsen further the symbo error rate of OFDM systems during the equaization process. In this paper we show how Aamouti space time coded STC OFDM signas are corrupted by various OFDM receiver impairments such as carrier frequency offset, channe estimation errors, fat receiver I/Q imbaance and outdated channe information. In the iterature, the effects of carrier frequency offset on Aamouti space time coded OFDM symbo/bit error rate are mosty investigated under the assumption of perfect channe knowedge such as in [] and []. The authors of [3] and [4] anayze the effects channe estimation errors on the performance of genera narrow-band Aamouti STC systems but do not extend their resuts toward impaired OFDM transceiver concepts. Additionay, receiver I/Q imbaance has been identified as one of the most serious concerns in the practica impementation of direct conversion receiver architectures. Since, transmitter I/Q imbaance is known to have much smaer effects on OFDM systems [7] than receiver I/Q imbaance, transmitter I/Q imbaance is not considered in this paper. The authors of [9] investigated the effect of receiver I/Q imbaance on SISO OFDM systems for frequency seective fading channes under the assumption of perfect channe knowedge and perfect receiver synchronization. Additionay, in order to cope with this impairment, the authors of [8] proposed a digita I/Q imbaance compensation method. To our best knowedge, there is currenty no work on the anaytica SER cacuation for Aamouti STC OFDM systems under the aggregate effect of a the mentioned impairments. Finay, we have to state that the knowedge about the quantitative reationship between receiver parameters such as I/Q imbaance, channe estimation error and residua CFO and performance metrics such as symbo error rate is essentia for the design and dimensioning of OFDM systems and receiver agorithms. In this paper, after introducing the genera impaired OFDM system mode Sec.II we give a detaied stochastic anaysis of Aamouti STC systems under channe estimation error in Sec.III-A and extend our findings toward impaired Aamouti space time coded OFDM signas for deriving the symbo error rates w.r.t. carrier frequency offset, channe estimation errors and I/Q imbaance in Rayeigh frequency and time seective fading channes Sec.III-B. It shoud be noted, that the terms symbo error rate and symbo error probabiity are used with equa meaning. This is due to the fact, that the symbo error rate converges towards the symbo error probabiity with increasing observation time in an stationary environment. Finay, we end up with the verification of the anaytica resuts in Sec.III-C and Concusions in Sec.IV. II. OFDM SYSTEM MODEL We consider an OFDM system with N-point FFT. The data is M-QAM 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. The samped discrete compex baseband signa for the th
subcarrier after the receiver FFT processing can be written as Y X H + W where X represents the transmitted compex QAM moduated symbo on subcarrier and W is a compex Gaussian noise sampe. The coefficient H denotes the frequency domain channe transfer function on subcarrier, which is the discrete Fourier transform of the CIR hτ with maxima L taps H hτe jπτ/n τ In this paper, it is assumed that the residua carrier frequency offset after frequency synchronization is a given deterministic vaue. A one-ofdm-symbo-ong preambe is used for channe estimation. Furthermore, static channe characteristics during one OFDM symbo are assumed. The samped CFOimpaired discrete compex baseband signa for the th subcarrier after the receiver FFT processing can be written as Y X H I + N k N,k X k H k Ik + W 3 where the coefficient Ik represents the impact of the received signa at subcarrier k on the received signa at subcarrier due to the residua carrier frequency offset [5] Ik e jπk + f /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 wi be abbreviated as k N,k k. In Eq.3 we can see, that residua CFO causes a phase rotation of the received signa I and inter carrier interference ICI. Furthermore, there is a time variant common phase shift for a subcarriers due to f, as given in [6], 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. I/Q imbaance of direct conversion OFDM receivers directy transates to a mutua interference between each pair of subcarriers ocated symmetricay with respect to the DC carrier [8]. Hence, the received signa Y at subcarrier is interfered by the received signa Y at subcarrier, and vice versa. Therefore, the undesirabe eakage due to I/Q imbaance can be modeed by [8], [] Ỹ Y + K Y 4 where. represents compex conjugation and K denotes a compex-vaued weighting factor that is determined by the receiver phase and gain imbaance [8]. The image rejection capabiities of the receiver on subcarrier can be expressed in terms of the image rejection ratio - IRR given by IRR 5 K In this paper, we consider fat I/Q imbaance, which simpy means IRR IRR,. Subsequenty, we consider preambebased frequency domain Least Square LS channe estimation to obtain the channe state information on subcarrier with Ĥ ỸP, IH + k X P,kH k Ik + W +K m X P,m H mi m + + W 6 where and ỸP, denote the transmitted and received preambe symbo on subcarrier. The Gaussian noise of preambe part W has the same variance as W of the data part σw σ W. The power of preambe signas and the average power of transmitted data signas on the data carriers set to be equivaent X P σx. The derived anaytica expressions for symbo error rates are compared with appropriate simuation resuts in Sec.III-C. In these numerica exampes, we consider an IEEE 8.a-ike OFDM system with 64-point FFT. The randomy generated data is M-QAM-moduated to different subcarriers, then transformed to the time domain signa by IFFT operation and prepended by a 6-tap ong cycic prefix. One OFDM piot symbo was used for channe estimation. The used BPSK piot data in the frequency domain is given by for subcarrier index [ 6 : : 6] A. Mobie Channe Characteristics Athough our anaysis is not imited to a specific type of frequency seective channe, frequency seective fading channes having an exponentia power deay profie PDP are considered in this paper with σ τ C e Dτ/L, τ,,..., L where στ E{ h τ } and the factor C τ e Dτ/L is chosen to normaize the PDP as τ σ τ or σh on a subcarriers. The channe taps h τ are assumed to be zero-mean compex Gaussian RV with uncorreated rea and imaginary parts. Additionay, the maxima channe impuse response is kept shorter/equa than the cycic prefix L 6. The cross-correation coefficients of the channe transfer functions between subcarrier k and in frequency seective fading channes are defined as r k, E{H kh } σh, k 7 where σh is equivaent for a subcarriers. Assuming mutua uncorreated channe taps of the CIR and appying Eq., one get r k, στe jπk τ/n 8 τ The cross correation properties of the channe coefficients can be written as foows H k r k, H + V k, 9
where V k is a compex zero-mean Gaussian with variance σ V k, σ H r k, and E{V k, H }. In current OFDM systems such as 8.a/n or 8.6, 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 incorporate the effect of outdated channe information into the anaytica SER derivation. Therefore we have to define the auto-correation properties of channe coefficients H. The auto-correation coefficient of subcarrier is defined as foows Appying Eq. we get E{H th t + λ} r H, λ E{H th t + λ} σ H { E hτ, th ν, t + λ e τ ν τ ν π N When assuming uncorreated channe taps it foows E{H th t + λ} } r h τ, λστ τ For sake of simpicity, it is assumed that a channe taps have the same auto-correation coefficient r h τ, λ r h λ, τ L. Together with the reation τ σ τ σh, we obtain r H, λ r h λ 3 For the numerica evauation of the anaytica SER resuts in Sec.III-C, the time seectivity of the compex Gaussian channe taps was modeed as foows with hτ, t + λ r h τ, λhτ, 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 στ r hτ, λ and E{hτ, tvτ,λ }. 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. B. System Mode Statistics Firsty, we can rewrite the channe estimates of subcarrier in Eq.6 with respect to the frequency seective fading characteristic given in Eq.9 to be k Ĥ IH + r k,x P,k Ik +η K I m r m, X P,m I m + I + ν 4 where η denotes the term e jφ. This is due to the fact that the compex Gaussian channe coefficient can be written as H H e jφ. Hence, we get H H e jφ η, where φ is an equay distributed RV in the interva [ π, π]. From Eq.4 we obtain Ĥ α H + ν 5 by defining the effective channe H IH with σ H I σh and effective mutipicative bias α as r k, X P,k Ik + ηk rm, X P,m I m + α + k m I where α is a stochastic quantity with given subcarrier index, a set of deterministic preambe symbos X P,k, a fixed predetermined frequency offset, a given IRR constant K and RV η e jφ. In order to simpify the anaysis, we negect the stochastic properties of α induced by I/Q imbaance and the parameter η e jφ and finay use the heuristic φ. This method is quite reasonabe under reaistic receiver characteristics with IRR 3 db. The AWGN part ν of the channe estimate can be written as ν W + K W + k X P,kV k, Ik +K m X P,m V m, I m + 6 For σ ν, which represents the AWGN variance of the channe estimates, we have σ ν X P,k XP,nIk I n k n r k,n r k, rn,σ H + K XP,kX P,n I k + In + k n rk,n rk,r n, σh +σ W + K 7 Appying the same method as above for Eq.3 and Eq.4, the same definition of the effective channe H can be used to obtain Y H k X + r k,x k Ik 8 +η K I m r m, X mi m + I + W H X + W 9
Given Eq.8 and Eq.9, the effective symbo X can be defined, which is no onger a deterministic vaue but a stochastic quantity due to i.i.d. data symbos on the subcarriers k. Given a certain transmit symbo X and assuming randomy transmitted data signas X k with k, we can decompose the effective symbo X into X X + J which shows the stochastic nature of X due to the random interference part J resuting from ICI and I/Q imbaance. Appying the centra imit theorem, we assume that J is a compex zero-mean Gaussian random variabe J p + jq. The mutua uncorreated rea and imaginary parts p and q have the same variance for a consteation points k Ik r k, + K m Im + r m, σ J I where σ H I σ H and σ Ĥ α I σ H + σ ν. The term W of Eq.9 represents the effective noise of the received signa that consists of the AWGN parts W, W and ICI parts, respectivey. If we substitute Eq.3 and Eq.9 into Eq.4, we get W W + K W + k X k V k, Ik +K Xm V m, I m + m For an exact expression of σ W we take Eq., σv k, σh r k, together with the assumptions of mutuay uncorreated data symbos and obtain σw + K + σh Ik r k, σ W k + K σh Im + r m, m III. PERFORMANCE OF ALAMOUTI STC OFDM TRANSMISSION A. Stochastic Modeing In order to derive anaytica SER expressions for impaired Aamouti STC OFDM systems it is usefu to study the stochastic receiver characteristics of a simpe singe carrier Aamouti STC system impaired by channe estimation error. Therefore we have to reca the Aamouti STC signa mode using the fat channe assumption. Let y and y denote the received compex symbo at time instants and. Furthermore, there are two uncorreated fat zero-mean compex Gaussian channes having reaizations h and h. The corresponding channe estimates are given by ĥ and ĥ with ĥ p α p h p +ν p, p,, where α p denotes a mutipicative some of the basic derivations of this work can aso be found in [4] where the authors investigated non-ofdm Aamouti space time coded systems in static fat channe environments and BPSK digita moduation. However, in this work the basic derivations of [4] are extended toward impaired OFDM systems and compex mobie channe conditions. deterministic bias coming from the channe estimation scheme and ν p represents a zero-mean compex Gaussian of variance σν. It is reasonabe to write α α α since it is assumed that there is a stationary impairment situation during the channe estimation process and that the same channe estimation scheme and piot symbos are used for channe estimation for both ĥ and ĥ. Finay, the decision variabes z and z are given by [ ] [ ] [ ] z ĥ ĥ y z ĥ + ĥ ĥ y The variabes z and z are then fed into the decoder and detector stage where receiver imperfections and AWGN introduce bit/symbo errors. Assuming equa stochastic properties of z and z, it is reasonabe ony to investigate decision variabe z for sake of simpicity. Therefore, z reads z ĥ ĥ + ĥ y + ĥy 3 The vaues of y and y are compex Gaussian random variabes when considering fixed transmit symbos and compex Gaussian channe coefficients as we as AWGN. y h x + h x + w 4 y h x + h x + w 5 In the next step, we use the cross correation properties of the channe estimates and y and y and write ĥ y r y ĥ σ y ĥ + r y ĥ σ y σĥ y r y ĥ σ y ĥ + r y ĥ σ y σĥ σĥ ĥ + v 6 σĥ ĥ + v 7 where v and v represent zero-mean compex Gaussian RV with variance σ v σ y r y ĥ r y ĥ and σ v σ y r y ĥ r y ĥ. For the correation coefficients we find that r y ĥ E{y ĥ} σ y σĥ x α r h λ, σh σy σĥ, r y ĥ x α r h λ, σh σy σĥ8 where λ q,p represents the time duration between estimation of h p and reception of y q. Appying the same techniques as in Eq.8 we can summarize that r y ĥ x α r h λ, σh σy σĥ r y ĥ x α r h λ, σh σy σĥ 9 For sake of simpicity, we assume that λ q,p λ, and hence r h λ q,p r h λ, which is reasonabe under moderate time seective channe properties that we typicay find in mobie communications. It is easy to verify that and σ y σ y σ y σ xσ h + σ w σ ĥ σ ĥ σ ĥ α σ h + σ ν
In case of symbo power E{ x } E{ x } σ x we get σ v σ v σ v σ x σ h + σ w α r h λσ4 h σ ĥ 3 Substituting a the mentioned reationships into Eq.3 we finay obtain [ z ĥ ĥ x + ĥ + ĥ ĥx α r h λσh + v σ ] ĥ + ĥ ĥ x ĥ x αr h λσh + v 3 σ ĥ In the genera impaired OFDM system mode anaysis of Sec.II-B, it has been shown, that x and x are no onger deterministic symbos, but stochastic quantities due to CFO and I/Q imbaance. Therefore, we perform the foowing decomposition: x x m, + p + jq where x m, denotes the deterministic transmit symbo and m represents the M-QAM consteation point index. It is assumed, that the stochastic part of x is characterized by the i.i.d. Gaussian random variabes p and q with E{ p +jq } σj. The same hods for x. Hence we obtain [ z ĥ + ĥ + ĥ + ĥ x m, r h λσ h σ ĥ α ĥ + α ĥ r h λσh +x m, σ α αĥ ĥ ĥ [ ĥ v + ĥv + r hλσh σ ĥ p + jq α ĥ + α ĥ ] + p + jq α αĥ ĥ 3 In the ast step, we observe from Eq.3 that the decision variabe is a compex Gaussian random variabe if ĥ, ĥ, x and x are considered to be deterministic vaues. Hence, z can be characterized by its conditiona compex mean m z and variance σ z. We derive both quantities using Eq.3 under the assumption that α α what is reasonabe under moderate impairments and with ĥ r and ĥ r as σ z r,r σ v + σ J r h λ σ4 h r + r r + r α r + αr 33 σ 4 ĥ m z x m,,r,r r hλσ h σ ĥ α r + αr x m, r + r ] 34 In order to derive the conditiona symbo error probabiity P e x m r, r of the m-th consteation point for given r and r we have to evauate the two-dimensiona integra over the appropriate decision region R m with z z r + jz i as foows P e x m r, r R m πσz r e z m z xm,r,r σ z r,r,r dz r dz i 35 For exampe, et s consider a QPSK consteation and assume that there is an equa conditiona symbo error probabiity P e r, r for a consteation points. Hence, it is enough ony to evauate the conditiona symbo error probabiity for the first QPSK symbo x + j. We simpify the notation m z xm,,r,r m z m r,z + jm i,z observe with Eq.35 that P e r, r P e r, r Q πσz m r,z σ z, σ z r,r σ z and e zr+jz i mr,z jm i,z σ z Q m i,z σ z dz r dz i 36 Since ĥ αh + ν is a compex Gaussian RV, r and r are exponentiay distributed RVs with PDF p r r p r r σ e r σ ĥ 37 ĥ Hence, the resuting symbo error probabiity for the QPSK scenario can be formuated as P e 4σ 4 ĥ B. OFDM Aamouti System Mode P e r, r e r+r σ ĥ dr dr 38 In this section we map the impaired OFDM system mode to the eementary stochastic performance anaysis of section III-A. At time instants and, the Aamouti STC OFDM receive signa on subcarrier is given by Y, X, H, + X, H, + W, 39 Y, X, H, + X, H, + W, 4 where X,p, H,p and W represent the effective stochastic symbo, the effective channe and the effective noise, respectivey. For an appropriate performance anaysis that captures a OFDM-reevant effects given in the introductory section, it is necessary to map the parameters of the ast section such as σz, m z onto Eq.39 and Eq.4 what eads to the foowing expressions m Z Xm,r,r r hλ I σ H σ Ĥ α X r + α r m r + r σz r,r σ V + σ J rh λ σ4 H r + r r + r α σ 4 r + α r Ĥ 4 4
where the parameters I, α, σj, σ H and σ are given in Ĥ Sec.II-B, whie the parameter σv can be cacuated simiar to Eq.3 as IRR 3 db Sim IRR 3 db cac IRR 4 db Sim IRR 4 db cac σ V σ X I σ H + σ W α r h λ I 4 σ 4 H σ Ĥ 43 The Equations 4 and 4 are especiay vaid under the assumption of mutuay uncorreated mobie channes between the transmit antenna and the two receive antennas, a having simiar power deay profies. Finay, the 4-QAM SER cacuation of the previous section III-A, can be formuated in case of impaired OFDM transmission as P e N C C. Numerica Resuts 4σ 4 Ĥ Q m i,z x,r,r σ Z r,r Q m r,z x,r,r σ Z r,r r +r σ e Ĥ dr dr 44 In our numerica exampes, we consider an Aamouti STC OFDM system as described in Sec.II. Fig. presents the cacuated and simuated 4-QAM SER vs. SNR E S /N with given carrier frequency offset f in % subcarrier spacing for uncorreated static non time variant 8 taps exponentia PDP frequency seective channes with D 7 see Sec.II. The resuts iustrate that our anaysis can approximate the simuative performance very accuratey, if the power deay profie of fading channes, the image rejection ratio of the direct conversion receiver and carrier frequency offset are taken to be predefined input parameters for ink eve anaysis. In Fig. we use a fixed f of 5% to investigate 4-QAM SER for time variant exponentia 8 taps PDP channes characterized by the channe tap auto correation coefficients r h λ. SER 3 4 5 6 f % sim f % cac f 5% sim f 5% cac f % sim f % cac f 3% sim f 3% cac IRR 3 db IRR 4 db 5 5 3 35 4 E S /N in db Fig.. Comparison of simuated and anaytica uncoded SER vs. SNR for 4-QAM Aamouti space-time coded OFDM under residua CFO in static exponentia PDP 8 taps channe environment IV. CONCLUSIONS In this paper we have shown how to anayticay evauate the symbo error probabiity/rate of Aamouti STC OFDM SER 3 4 5 r h λ.993 r h λ.997 r h λ 5 5 5 3 35 4 E S /N in db Fig.. Comparison of simuated and anaytica uncoded SER vs. SNR for 4-QAM Aamouti space-time coded OFDM with residua CFO of 5% under time seective exponentia PDP 8 taps channe conditions systems subject to carrier frequency offset, I/Q imbaance, outdated channe information and channe estimation error in frequency and time seective mobie fading channes. We were abe to show that the symbo error probabiity can be cacuated exacty when using a priori knowedge of mobie channe power deay profie and image rejection ratio of the direct conversion receiver. Therefore, our anaysis provides a usefu too for system engineers to cross-reference the performance of OFDM synchronization/acquisition agorithms with appropriate symbo error rates. REFERENCES [] C. R. N. Athaudage and K. Sathananthan. Probabiity of Error of Space-Time Coded OFDM Systems With Frequency Offset in Frequency- Seective Rayeigh Fading Channes. In Proc. IEEE Internationa Conference on Communications ICC 5, Seou, Korea, May 5. [] L. Brötje, S. Vogeer, and K.-D. Kammeyer. On Carrier Frequency Offsets in Aamouti-coded OFDM systems simiar to IEEE 8.a. In Proc. 8th Internationa OFDM Workshop InOWo3, Hamburg, Germany, August 3. [3] Z. Diao, D. Shen, and V. Li. Performance anaysis of space-time codes with channe information errors. In Proc. VTC 4, Los Angees, USA, September 4. [4] D. Gu and C. Leung. Performance anaysis of transmit diversity scheme with imperfect channe estimation. In Proc. IEEE Eectronic Letters, voume 39, pages 4 43, February 3. [5] K.Sathananthan and C. Teambura. Probabiity of error cacuation of OFDM with frequency offset. In Proc. IEEE Transactions on Communications, voume 49, pages 884 888, November. [6] L Rugini, P. Banei, and S. Cacopardi. Probabiity of Error of OFDM Systems with Carrier Frequency Offset in Frequency-Seective Fading Channes. In Proc. IEEE ICC 4, June 4. [7] Tim Schenk. RF Impairments in Mutipe Antenna OFDM, Chapter 4.3. T.C.W. Schenk, Eindhoven, 6. [8] M. Windisch and G. Fettweis. Standard-Independend I/Q Imbaance Compensation in OFDM Direct-Conversion Receivers. In Proc. 9th Int. OFDM Workshop InOWo, pages 57 6, Dresden, Germany, September 4. [9] M. Windisch and G. Fettweis. Error Probabiity Anaysis of Muti-Carrier Systems Impaired by Receiver I/Q Imbaance. In Proc. Internationa Symposium on Wireess Persona Mutimedia Communications WPMC 6, San Diego, USA, September 6. [] M. Windisch and G. Fettweis. Performance Degradation due to I/Q Imbaance in Muti-Carrier Direct Conversion Receivers: A Theoretica Anaysis. In Proc. IEEE Internationa Conference on Communications ICC 6, Istanbu, Turkey, June 6.