OFDM Link Performance Analysis under Various Receiver Impairments

Transcription

1 OFDM Link Perforance Analysis under Various Receiver Ipairents Marco Krondorf and Gerhard Fettweis Vodafone Chair Mobile Counications Systes, Technische Universität Dresden, D-006 Dresden, Gerany Abstract In this article, we present a ethodology for OFDM link capacity and bit error rate calculation that jointly captures the aggregate effects of various real life receiver iperfections such as: carrier frequency offset, channel estiation error, outdated channel state inforation due to tie selective channel properties and flat receiver I/Q ibalance. Since such an analytical analysis is still issing in literature, we intend to provide a nuerical tool for realistic OFDM perforance evaluation that takes into account obile channel characteristics as well as ultiple receiver antenna branches. In our ain contribution, we derived the probability density function PDF of the received frequency doain signal with respect to the entioned ipairents and use this PDF to nuerically calculate both bit error rate and OFDM link capacity. Finally, we illustrate which of the entioned ipairents has the ost severe ipact on OFDM syste perforance. I. INTRODUCTION Orthogonal Frequency Division Multiplexing OFDM is a widely applied technique for wireless counications, which enables siple one-tap equalization by cyclic prefix insertion. Conversely, the sensitivity of OFDM systes to various receiver ipairents is higher than that of singlecarrier systes. Furtherore, for OFDM syste designers it is often desirable to have easy to use nuerical tools to predict the syste perforance under various receiver ipairents. Within this article the ter perforance eans both link capacity and uncoded bit error rate BER. Mostly, link level siulations are used to obtain reliable perforance easures of a given syste configuration. Unfortunately siulations are highly tie consuptive especially when the paraeter space of the syste under investigation is large. Therefore, the intention of this article is to introduce a stochastic/analytical ethod to predict the perforance etrics of a given OFDM syste configuration. To get realistic perforance results, our approach takes into account a variety of receiver characteristics and ipairents as well as obile channel properties such as: residual carrier frequency offset CFO after synchronization channel estiation errors outdated channel state inforation due to tie selective obile channel properties flat receiver I/Q ibalance in case of direct conversion receivers frequency selective obile channel characteristics ultiple receiver branches to realize diversity cobining ethods such as axiu ratio cobining MRC In present OFDM standards, such as IEEE80.a/g or DVB-T, preable or pilots are used to estiate and to copensate the CFO and channel ipulse response. Unfortunately, after CFO estiation and copensation, the residual carrier frequency offset still destroys the orthogonality of the received OFDM signals and corrupts channel estiates, which worsen further the perforance of OFDM systes during the equalization process. In the literature, the effects of carrier frequency offset on bit error rate are ostly investigated under the assuption of perfect channel knowledge. The papers [6] and [5] consider the effects of carrier frequency offset only without channel estiation and equalization iperfections and give exact analytical expressions in ters of SNR-loss and OFDM bit error rate for the AWGN channel. The authors of [8] extend the work of [5] toward frequency-selective fading channels and derive the correspondent bit error rate for OFDM systes in case of CFO under the assuption of perfect channel knowledge. Additionally, the authors of [] tried to analyze the joint effects of CFO and channel estiation error on uncoded bit error rate for OFDM systes, but the used Gaussian channel estiation error odel does not hold in real OFDM systes, especially when carrier frequency offset is large see Sec.V. Additionally, receiver I/Q ibalance has been identified as one of the ost serious concerns in the practical ipleentation of direct conversion receiver architectures see for exaple []. Direct conversion receiver designs are known to enable sall and cheap OFDM terinals, highly suitable for consuer electronics. The authors of [] investigated the effect of receiver I/Q ibalance on OFDM systes for frequency selective fading channels under the assuption of perfect channel knowledge and perfect receiver synchronization. Additionally, in order to cope with this ipairent, the authors of [0] proposed a digital I/Q ibalance copensation ethod. To our best knowledge, there is currently no literature available that describes a calculation ethod for OFDM BER and link capacity under the aggregate effect of all the entioned ipairents. Therefore, our intention is to describe the quantitative relationship between OFDM paraeters, receiver ipairents and perforance etrics such as bit error rate and link capacity. Furtherore, we intend to provide a useful syste engineering tool for the design and diensioning of OFDM syste paraeters, pilot sybols and receiver algoriths used for frequency synchronization, channel estiation and I/Q ibalance copensation. The structure of this article is as follows. After soe general

2 Coding & Sybol Mapping Discrete coplex input alphabet X OFDM Modulation Mobile Channel... OFDM Deodulation & Cobining continuous coplex detector input alphabet Z Detection & Decoding, SIMO Modulation Channel representing PHY ipairents, Modulation Characteristics and Mobile Channel Properties Fig.. The odulation channel concept used for capacity evaluation. rearks on our proposed link capacity evaluation ethod in Sec.II, we introduce our OFDM syste odel in Sec.III followed by a general probability density function analysis in Sec.IV. In Sec.V, it will be explained how to odel the correlation between channel estiates and received/ipaired signals to derive uncoded bit error rates of OFDM systes with carrier frequency offset and I/Q ibalance in Rayleigh frequency and tie selective fading channels. It should be noted, that the ters bit error rate and bit error probability are used with equal eaning. This is due to the fact, that the bit error rate converges toward bit error probability with increasing observation tie in an stationary environent. Finally we introduce our link capacity calculation ethod in Sec.VI and conclude in Sec.VII. II. THE APPROACH We choose link capacity, easured in [Bit / Channel Use], as an iportant perforance etric for OFDM syste designs. This inforation theoretic etric allows syste designers to characterize the syste behavior subject to real life receiver ipairents independently fro any kind of channel coding and iterative detection ethods. As explained in Sec.VI and illustrated in Fig., the OFDM transceiver chain including channel and receiver properties can be characterized as effective channel between source and detector, often called the odulation channel. The odulation channel is characterized by its conditional PDF f Z X z x that describes the statistical relationship between the discrete input sybols x and the continuously distributed decision variable z. Using any given coplex M-QAM constellation alphabet X, the link capacity can be expressed as utual inforation between source and sink that only depends on the input statistic of X and f Z X z x. Since our perforance analysis fraework intends to describe the utual inforation and hence the link capacity under a given input statistic, we propose the following work flow: We show how to derive f Z X z x under receiver ipairents, given channel properties and OFDM syste paraeters. We use the derived f Z X z x for uncoded BER calculation to verify its correctness by coparing the BER prediction results with those obtained fro siulation. 3 We calculate the utual inforation, i.e. OFDM link capacity, using the verified statistic f Z X z x. III. OFDM SYSTEM MODEL We consider an OFDM syste with N-point FFT. The data is M-QAM odulated to different OFDM data subcarriers, then transfored to a tie doain signal by IFFT operation and prepended by a cyclic prefix, which is chosen to be longer than the axial channel ipulse response CIR length L. The sapled discrete coplex baseband signal for the lth subcarrier after the receiver FFT processing can be written as Y l = X l H l + W l where X l represents the transitted coplex QAM odulated sybol on subcarrier l and W l represents coplex Gaussian noise. The coefficient H l denotes the frequency doain channel transfer function on subcarrier l, which is the discrete Fourier transfor DFT of the CIR hτ with axial L taps L H l = hτe jπlτ/n. τ=0 In this paper, it is assued that the residual carrier frequency offset after frequency synchronization is a given deterinistic value. Furtherore, static non tie selective channel characteristics are assued during one OFDM sybol. The CFO-ipaired coplex baseband signal subcarrier l can be written as Y l = X l H l I0 + N k= N,k l X k H k Ik l + W l. 3 The coplex coefficients Ik l represent the ipact of the received signal at subcarrier k on the received signal at subcarrier l due to the residual carrier frequency offset as defined in [5] Ik l = e jπk l+ f /N sinπk l + f Nsinπk l + f/n where f is the residual carrier frequency offset noralized to the subcarrier spacing. In addition, later in this paper the suation N will be abbreviated as k= N,k l k l. In Eq.3 we can see, that residual CFO causes a phase rotation of the received signal I0 and inter carrier interference ICI. Furtherore, there is a tie variant coon phase shift for all subcarriers due to CFO as given in [8], that is not odeled here. This is due to the fact, that this tie variant coon phase ter is considered to be robustly estiated and

3 copensated by continuous pilots that are inserted aong the OFDM data sybols. I/Q ibalance of direct conversion OFDM receivers directly translates to a utual interference between each pair of subcarriers located syetrically with respect to the DC carrier [0]. Hence, the received signal Y l at subcarrier l is interfered by the received signal Y l at subcarrier l, and vice versa. Therefore, the undesirable leakage due to I/Q ibalance can be odeled by [0], [] Ỹ l = Y l + K l Y l where. represents the coplex conjugation and K l denotes a coplex-valued weighting factor that is deterined by the receiver phase and gain ibalance [0]. The iage rejection capabilities of the receiver on subcarrier l can be expressed in ters of iage rejection ratio - IRR given by IRR l =. 5 In this paper, we consider flat I/Q ibalance which siply eans IRR l = IRR, l. Subsequently, we consider preablebased frequency doain least square FDLS channel estiation to obtain the channel state inforation Ĥl on subcarrier l: Ĥ l = ỸP,l X P,l = I0H l + K l k l X P,kH k Ik l + W l X P,l +K l X P, H I + l + W l X P,l 6 where X P,l and ỸP,l denote the transitted and received preable sybol on subcarrier l. The Gaussian noise of the preable part W l has the sae variance as W l of the data part σw = σ l W l. The channel estiate is used for frequency doain zero-forcing equalization before data detection Z l = Ỹl Ĥ l 7 where Z l is the decision variable that is feed into the detector/decoder stage. The power of preable signals and the average power of transitted data signals on all carriers is equivalent X P = σx. In case of ultiple N Rx receiver branches, axiu ratio cobining MRC is used at the receiver side. Therefore, the decision variable Z l on subcarrier l is given by NRx κ= Z l = Yl,κĤ l,κ NRx κ= Ĥl,κ 8 where κ denotes the receiver branch index. We assue, that there is the sae IRR and CFO on all branches, what is reasonable when considering one oscillator used for down-conversion in each branch. Furtherore, we assue uncorrelated channel coefficients aong the branches, i.e. E{H l,κ H l,κ } = 0, if κ κ, l. For sake of readability, we only include the antenna branch index κ if necessary. A. Mobile Channel Characteristics To obtain precise perforance analysis results in case of subcarrier crosstalk induced by CFO and I/Q ibalance, it is desirable to use exact expressions of the subcarrier channel cross-correlation properties what is shown in ore detail in Sec.V. The cross-correlation properties between frequency doain channel coefficients are ainly deterined by the power delay profile of the channel ipulse response CIR and the CIR tap cross-correlation properties. Furtherore, the discrete nature of the sapled CIR is odeled as tapped delay line having L channel taps. Although our analysis is not liited to a specific type of frequency selective channel, in our nuerical exaples we consider obile channels having an exponential power delay profile PDP σ τ = C e Dτ/L, τ = 0,,...,L where στ = E{ hτ } and the factor C = L τ=0 e Dτ/L is chosen to noralize the PDP as L τ=0 σ τ =, what leads to σh = E{ H l } =, l. The channel taps hτ are assued to be coplex zero-ean Gaussian RV with uncorrelated real and iaginary parts. Hence, after DFT according to Eq., the channel coefficients are zero-ean coplex Gaussian rando variables as well. Additionally, the CIR length L is assued to be shorter/equal than the cyclic prefix. The cross-correlation coefficient of the channel transfer function on subcarriers k and l in case of frequency selective fading is defined as r k,l = E{H kh l } σ H, k l, 9 where σh is equivalent for all subcarriers. Assuing utual uncorrelated channel taps of the CIR and applying Eq., one get E{H k H l } στ e jπk lτ/n. 0 = L τ=0 The cross correlation property of the coplex Gaussian channel coefficients can be forulated to be H k = r k,l H l + V k,l where V k is a coplex zero-ean Gaussian with variance σ V k,l = σ H r k,l and E{V k,l H l } = 0. In current OFDM systes such as 80.a/n or 80.6, there is a typical OFDM block structure. A OFDM block consists of a set of preable sybols used for acquisition, synchronization and channel estiation, followed by a set of serially concatenated OFDM data sybols. User obility gives rise to a considerable variation of the obile channel during one OFDM block fast fading what causes outdated channel inforation in certain OFDM sybols if there is no appropriate channel tracking. To be precise, during the tie period λ between channel estiation and OFDM sybol reception the channel changes in a way that the estiated channel inforation used for equalization does not fit the actual channel anyore. If there is no channel tracking at the receiver side, our ai is to incorporate the effect of outdated channel inforation into the perforance analysis fraework.

4 Therefore, we have to define the auto-correlation properties of channel coefficients H l. The auto-correlation coefficient of subcarrier l is defined as follows Applying Eq. we get r H l, λ = E{H lthl t + λ}. E{H l th l t + λ} = E σ H { L L hτ, th ν, t + λ τ=0 ν=0 e πlτ ν N When assuing uncorrelated channel taps it follows E{H l th l t + λ} = }. 3 L τ, λστ. τ=0 For sake of siplicity, it is assued that all channel taps have the sae auto-correlation coefficient, i.e. τ, λ = λ, 0 τ L. Substituting the relation L τ=0 σ τ = and Eq.3 into Eq., we obtain σ H r H l, λ = λ. 5 For the nuerical BER and link capacity evaluations done in Sec.V-B and Sec.VI-B, the tie selectivity of the coplex Gaussian channel taps was odeled as follows with hτ, t + λ = τ, λhτ, t + v τ,λ, E{ hτ, t } = E{ hτ, t + λ } = σ τ where v τ,λ is a coplex Gaussian RV with variance σv τ,λ = στ τ, λ and E{hτ, tvτ,λ } = 0. For sake of siplicity, it is assued that the channel is stationary during one OFDM sybol but changes fro sybol to sybol in the above defined anner. In our analysis, we intentionally avoid any assuptions on concrete fast fading odels in order to obtain fundaental results. Anyway, one of the coonly used statistical descriptions of fast channel variations is the Jakes odel [7], where the channel auto-correlation coefficient τ is given by τ = J 0 πf D,ax τ, and f D,ax denotes the axiu Doppler frequency that is deterined by the obile velocity and carrier frequency of the syste. It should be noted, that τ is real due to uncorrelated i.i.d. real and iaginary parts of the CIR taps. IV. PROBABILITY DENSITY FUNCTIONS ANALYSIS The author of [9] suggested a correlation odel regarding channel estiation for single-carrier systes and derived the correspondent sybol error rate and bit error rate of QAModulated signals transitted in flat Rayleigh and Ricean channels. In this section, a short review of the contribution of [9] will be given in order to further extend these results to OFDM systes for tie and frequency selective fading channels with CFO, I/Q ibalance and channel estiation error. The single-carrier transission odel without carrier frequency offset for flat Rayleigh fading channels can be written as y = hx + w 6 where y, h, x and w denote the coplex baseband representation of the received signal, the channel coefficient, the transitted data sybol and the additive Gaussian noise with variance σw respectively. In [9], the channel estiate ĥ is assued to be biased and used for zero forcing equalization as follows z = y with ĥ = αh + ν 7 ĥ where α denotes the deterinistic ultiplicative bias of the channel estiates and ν represents zero-ean coplex Gaussian noise with variance σν. The channel coefficient h and Gaussian noise ν are assued to be uncorrelated. Hence, the case of perfect channel knowledge can be easily odeled by α = and σν = 0. In [9], the joint PDF of the decision variable z = z r + jz i in case of transit sybol x is derived in cartesian coordinates and can be written as a x f Z X z x = π z bx + a x. 8 z=zr+jz i The PDF ainly depends on the coplex paraeter bx, given by [9] and [] bx = R{b} + ji{b} = b r x + jb i x α = x λσh α σh + σ ν 9 and the real paraeter ax that can be written according [9] and [] as a x = x α σ h λ + σ νσ h α σ h + σ ν σ w + α σh +. 0 σ ν Additionally, the closed for integral of Eq.8 with z = z r + jz i is given by [9] to be z z i b i xarctan r b rx a F Z X z x = x+z i b ix π a x + z i b i x z z r b r xarctan i b ix a x+z r b rx + π. a x + z r b r x In case of N Rx receiver branches, axiu ratio cobining MRC is used for decision variable coputation what can be forulated as NRx κ= z = yκĥ κ NRx κ= ĥκ where κ represents the antenna branch index and the κ-th channel estiate can be written according to the case as ĥ κ = α k h κ + ν κ. Since it is quite reasonable to assue that the sae channel estiation schee is used in each receive antenna branch, we

5 ,0 I z 0,0 when assuing equal probable M-QAM sybols as follows x x P b = M P b x. 5 M = B, x3, x 0, R z Fig.. The QPSK constellation diagra, showing the decision region for one bit position of sybol x. have α κ = α, κ. The authors of [9] also derived the PDF of z in case of transit sybol x and N Rx receiver branches that is given by N Rx a x N Rx f Z X,NRx z x, N Rx =. 3 π z bx + a NRx+ x It easy to observe that the PDF Eq.3 for the MRC case takes the for of Eq.8 in case of N Rx =. Additionally, the closed for integral F Z X,NRx z x, N Rx of f Z X,NRx z x, N Rx can be found in [9] that also takes the for Eq. in case of N Rx =. To enhance readability and to siplify our notation, we oit the receiver branch nuber N Rx in the conditional PDF and its closed for integral, i.e. in the following we write f Z X z x instead of f Z X,NRx z x, N Rx. Finally, the result of can be used to calculate the bit error rate of a given M-QAM constellation. In a M-QAM constellation there are Mlog M different possible bit positions with respect to the M-QAM constellation. The probability of an erroneous bit with respect to the -th QAM transit sybol x can be calculated by using the closed for integral Eq. and an appropriate decision region B,ν for the ν- th bit position see Fig. that takes into account the bit apping of the QAM constellation. In the paper, we always use Gray apping in our nuerical results, but it is worth entioning that the described ethod can be used for arbitrary bit appings as well. As already stated, we propose to use bit error rate prediction to verify the correctness of the derived probability density function f Z X z x that is later used to deterine the OFDM link capacity of a given transceiver configuration. Therefore, the bit error probability P b x takes the for P b x = log M log M ν= [[ FZ X z x ]] B,ν where [[ F Z X z x ]] B,ν denotes the -diensional evaluation of the closed for integral F Z X z x subject to the decision region B,ν. Finally, the bit error probability can be obtained by averaging over all possible constellation points, V. OFDM BIT ERROR RATE ANALYSIS In this section, the derivation of the bit error rate of OFDM systes with carrier frequency offset, I/Q ibalance and channel estiation error in Rayleigh frequency and tie selective fading channels will be given. The central idea of our BER derivation is to ap the OFDM syste odel of Sec.III to the statistics given in Sec.IV. To be precise, we have to ap the OFDM syste odel to the paraeters α, a Eq.0 and b Eq.9 as explained below. A. Matheatical Derivation Firstly, we can rewrite the channel estiates of subcarrier l in Eq.6 with respect to the frequency selective fading characteristic given in Eq. to be k l Ĥ l = I0H l + r k,lx P,k Ik l +e K l I0X P,l r,l X P, I + l I0X P,l + ν l 6 where e denotes the ter e jφ l. This coes due to the fact that the coplex Gaussian Channel coefficient can be written as H l = H l e jφ l. Hence, we have H l H l = e jφ l = e, where φ l is an equally distributed RV in the interval [ π : π]. Fro Eq.6 we obtain an Eq.7-like expression as follows: Ĥ l = α l Hl + ν l, 7 by defining effective channel Hl = I0H l and effective bias α l as r k,l X P,k Ik l + ek l r,l X P, I + l α l = + k l I0X P,l where α l is a stochastic quantity with given subcarrier index l, a set of deterinistic preable sybols X P,k, a fixed predeterined frequency offset, a given IRR constant K l and RV e = e jφ l. It should be noted, that the stochastic part of α l is negligible in case of oderate I/Q ibalance IRR 30 db and oderate CFO. Hence, we have that ek l r,lx P,I + l 0 and α l can be well odeled to be a deterinistic quantity. This is due to the fact that the pilot sybols X P,k as well as the CFO are given deterinistic values and the channel cross correlation coefficients r k,l can be calculated using Eq.9 and Eq.0. The noise part ν l of the channel estiate can be written as ν l = W l + K lw l + k l X P,kV k,l Ik l X P,l +K l X P, V,l I + l X P,l. 8

6 For σ ν l, which represents the additive Gaussian noise variance of the channel estiates, we obtain σ ν l = X P,k XP,nIk li n l k l n l r k,n r k,l rn,lσ H + K l XP,kX P,n I k + lin + l k n rk,n rk,lr n,l σh +σ W + K l. 9 Applying the sae ethod as above for Eq.3 and Eq., the sae definition of effective channel Hl can be used to get an Eq.6-like expression as follows Y l = H k l l X l + r k,lx k Ik l 30 +e K l I0 r,l X I + l I0 + W l = H X l + W l. 3 Given Eq.30 and Eq.3, the effective sybol Xl can be defined, that is no longer a deterinistic value but a stochastic quantity due to i.i.d. data sybols on subcarriers k l: r k,l X k Ik l + ek l r,l X I + l k l X l = X l + I0 }{{} stochastic part of the effective transit sybol 3 Assuing a certain transit sybol X l and assuing randoly transitted data sybols X k with k l, we can decopose the effective sybol Xl as follows: X l = X l + J l, which shows the stochastic nature of Xl due to the rando interference part J l due to ICI and I/Q ibalance. Applying the central liit theore, we assue that the interference J l ter is a coplex zero-ean Gaussian rando variable J l = p+jq. The utual uncorrelated real and iaginary parts p and q have the sae variance for all constellation points k l σ J l = Ik l r k,l + K l I + l r,l I0. 33 According to Eq.9 and Eq.0, we calculate the paraeters b l = b l,r +jb l,i and a l for M-QAM effective data sybols X l on subcarrier l in frequency and tie selective fading channels: b l X l = a l X l = X l α l λσ H α l σ H + σ ν, 3 X l α l σ Hl r h λ + σ ν l σ Hl σ Ĥ l + σ Wl σ Ĥ l 35 where σ Hl = I0 σ H and σ Ĥ l = α l I0 σ H + σ ν l. Fro Eq.3 and Eq.35 one can observe, that the paraeter σ Wl has to be calculated exactly to obtain reliable results. The ter W l represents the effective noise of the received signal that consists of AWGN parts W l, W l and ICI parts respectively. If we substitute Eq.3 and Eq. into Eq., we get W l = W l + K l W l + k l X k V k,l Ik l +K l X V,l I + l. 36 For an exact expression of σ Wl, we take Eq.36, σv k,l = σh r k,l together with the assuptions of utually uncorrelated data sybols and obtain = σw + K l + σh Ik l r k,l σ Wl k l + K l σh I + l r,l. 37 l As an exaple, for one QPSK constellation point with index = on subcarrier l, X,l =, = +j, we need to recalculate b l X,l and paraeter a l X,l separately for each effective sybol realization X,l = X,l + p + jq = + j + p + jq to use the closed for integral and Eq. for BER calculation. Subsequently, the bit error rate on subcarrier l for the -th constellation point can be expressed using Eq. by the following double integral involving the Gaussian PDFs of p and q: P b X,l = P b X,l + p + jq π σ J l e p+q σ J l dp dq. 38 Finally, to obtain the general bit error rate we have to average Eq.38 over all N C data subcarriers with index l and M-QAM constellation points with index as follows: P b = MN C N C/ M l= N C/ = P b X,l. 39 B. Bit Error Rate Perforance - Nuerical Results In this section, the derived analytical expressions for bit error rate are copared with appropriate siulation results for both single input single output OFDM transission as well as SIMO single input ultiple output OFDM using MRC and two receiver antenna branches. Furtherore, we consider an IEEE 80.a-like OFDM syste [3] with 6-point FFT. The data is 6-QAM odulated to the data subcarriers, then transfored to the tie doain by IFFT operation and finally prepended by a 6-tap long cyclic prefix. The data is randoly generated and one OFDM pilot sybol was used for channel estiation. The used BPSK pilot data in the frequency doain is given by X P,l = l for subcarrier index l = [ 6 : : 6], l 0.

7 The data and pilot sybols are odulated on 5 data carriers. The DC carrier as well as the carriers at the spectral edges are not odulated and are often called virtual carriers. For siulation and nuerical BER analysis we use an 8 taps exponential PDP frequency selective Rayleigh fading channel with D = 7 see Sec.III. Furtherore, we choose statistical independent channel realizations for the two antenna branches in case of SIMO OFDM transission. The double integral of Eq.38 is evaluated nuerically using Matlab build-in integration functions having a nuerical tolerance of 0 8 and upper/lower integration bounds of ±0. Fig.3 illustrates the calculated and siulated 6-QAM BER vs. SNR σx /σ W with given carrier frequency offset f in % subcarrier spacing and IRR = 30dB under non tie variant obile channel conditions. Fig. illustrates the calculated and siulated 6-QAM BER vs. SNR σx /σ W with given carrier frequency offset f in % subcarrier spacing and IRR = 30dB under non tie variant obile channel conditions. In Fig.5, we use a fixed f of 3% to investigate 6-QAM BER vs. SNR for tie variant obile channel properties, characterized by the channel tap auto correlation coefficients λ. BER IRR = 30 db, Siulation IRR = 0 db, Siulation IRR = 30 db, Calculation IRR = 0 db, Calculation = Fig.. The coparison of siulated and calculated uncoded BER vs. SNR for 6-QAM OFDM with residual CFO of 3% under non-tie selective channel conditions under IRR = 30 db / 0 db BER 0 BER Siulation f = %, Calculation f = 5%, Calculation f = 7%, Calculation = Fig. 3. The coparison of siulated and calculated uncoded BER vs. SNR for 6-QAM OFDM under residual CFO in non-tie selective channel environent and IRR = 30dB. The results illustrate that our analysis can approxiate the siulative perforance very accurately, if the channel power delay profile, the iage rejection ratio of the direct conversion receiver and carrier frequency offset are known. VI. CAPACITY ANALYSIS OF IMPAIRED OFDM LINKS To perfor OFDM link capacity analysis, it sees andatory to review the ain principles and basic equations of how to calculate average utual inforation between source and sink of a odulation channel. An excellent overview of this topic can be found in [7] that is suarized in the following. In an OFDM syste we have a nuber of parallel channels, i.e. data subcarriers. Hence we propose to 0 3 Siulation λ = 0.99, Calculation λ = 0.995, Calculation λ = 0.998, Calculation = Fig. 5. The coparison of siulated and calculated uncoded BER vs. SNR for 6-QAM OFDM with residual CFO of 3% and IRR = 30 db under tie selective channel conditions. calculate the utual inforation for each of the parallel data carriers independently and to finally average the link capacity aong the data carriers. Let s consider real input and output alphabets X and Z. Both alphabets can be characterized in ters of inforation content carried by the eleents of each alphabet what leads to the concept of inforation entropy HX and HZ. The entropy of the discrete alphabet X having eleents X with appropriate probability PX is given by HX = PX log PX. 0 Conversely, Z is assued to be a real continuously distributed RV having realizations z. As a result, Z can be characterized by its differential entropy as HZ = f Z zlog f Z z dz Z

8 where f Z z denotes the PDF of Z. Finally, the utual inforation IX;Z of X and Z can be forulated as [7] IX; Z = PX f Z X z X Z f Z X z X log n f dz. Z Xz X n PX n It can be seen fro Eq. that IX;Z requires knowledge of a-priory probabilities PX and conditional PDFs f Z X z X only. Mostly we have that PX = /M in case of M-ary constellations. Since the above defined utual inforation calculation schee assues one-diensional output variables and z is a two-diensional coplex RV of real part z r and iaginary part z i, we have to solve a double integral to obtain the corresponding utual inforation as follows IX; Z = PX f Z X z r + jz i X 3 Z i log Z r f Z X z r + jz i X n f Z Xz r + jz i X n PX n A. Mutual Inforation under Carrier Crosstalk dz r dz i. Recalling the two-diensional conditional PDF f Z X z r + jz i X on subcarrier l as given in Sec. IV, we have that a l f Z X z r + jz i X = X π z r + jz i b l X + a l X where a l X and b lx contain the entire OFDM link ipairent inforation channel estiation error, I/Q ibalance, CFO, outdated channel inforation and channel power delay profile. According section IV, the coplex valued transit sybol is stochastic by nature due to CFO and I/Q ibalance carrier crosstalk and can be expressed as X +J = X +p+ jq, where represents the constellation point index while p and q represent the effects of I/Q ibalance and residual CFO. Both, p and q can be odeled as i.i.d. zero-ean Gaussian RV as done in Sec. IV. Additionally, both paraeters a l X and b l X are subcarrier-dependent. As a result Eq. has to be reforulated for subcarrier l as f Z X,P,Q z r + jz i X + p + jq = 5 a l X + p + jq π z r + jz i b l X + p + jq + a l X + p + jq. Hence, the calculation of p/q-independent conditional arginal PDFs can be done via nuerical double integration as f Z X z r + jz i X = f Z X,P,Q z r + jz i X + p + jq f Q q f P p dp dq. 6 According Sec. V, we have the Gaussian distribution for each p and q: f P p = f Q q = e p,q σ J πσ J where σj is given in Eq.33. In case of MRC ulti-antenna reception, we have to proceed in the sae anner. B. OFDM Link Capacity - Nuerical Exaples The quantitative relationship between receiver ipairents, OFDM syste paraeters and link capacity is an essential piece of inforation for the diensioning of I/Q ibalance copensation algoriths as well as frequency synchronization ethods. Moreover, the effects of tie selective obile channels on link capacity can be used to design scattered pilot structures for channel estiation and tracking as done in []. Generally, link capacity indicates the axiu data rate that can be achieved with strong channel coding under a given input constellation and a specified receiver architecture. The nuerical exaples of average utual inforation are chosen such, that we illustrate the effects of channel estiation error, outdated channel state inforation CSI, residual CFO and flat receiver I/Q ibalance on the link capacity of and SIMO OFDM links. Therefore, we choose the sae IEEE 80.a-like OFDM syste paraeters as introduced in Sec.V-B, assue an 8 taps exponential PDP obile channel and the use of 6-QAM odulation on each data carrier. Again, statistical independent channel realizations for the N Rx antenna branches in case of SIMO OFDM transission are assued. The utual inforation easured in [Bit / Channel Use] is averaged aong the data carriers and plotted over SNR σ X /σ W. Mutual Inforation in Bit / Channel Use QAM upper bound, perfect CSI =, perfect CSI, FDLS =, FDLS Fig. 6. The utual inforation, averaged over all data carriers, coparison between perfect channel state inforation and real FDLS channel estiation for and SIMO OFDM, CFO = 0%, no I/Q ibalance, static Rayleigh fading channel. In Fig.6, we illustrate the effect of real life frequency doain least square FDLS channel estiation on the link capacity of and SIMO OFDM respectively, assuing no I/Q ibalance, a perfect frequency synchronization CFO = 0% and static non-tie selective channel properties. As reference, we plotted the case of perfect channel state inforation, that can easily be odelled by α l = and σ ν l = 0. In Fig.7, we show the aggregate effect of I/Q ibalance and FDLS channel estiation under static channel conditions and

9 Mutual Inforation in Bit / Channel Use 6 QAM upper bound = no I/Q ibalance IRR = 30 db IRR = 0 db Fig. 7. The utual inforation averaged over all data carriers under the aggregate effect of I/Q ibalance and FDLS channel estiation for OFDM, 6-QAM, CFO = 0%, static 8 taps exponential PDP Rayleigh fading channel. perfect frequency synchronization. It is easy to see that I/Q ibalance has only little effect on the averaged utual inforation perforance, what is especially the case at realistic iage rejection ratios above 30 db. Interestingly, a worst case IRR of 0 db heavily ipacts the perforance but causes only a sall perforance loss in case of receiver diversity cobining. Mutual Inforation in Bit / Channel Use 6 QAM upper bound = no CFO CFO = 3% CFO = 0% Fig. 8. The utual inforation averaged over all data carriers under CFO, FDLS channel estiation is assued, 6-QAM odulation on all subcarriers, no I/Q ibalance, tie variant 8 taps exponential PDP Rayleigh fading channel. Fig.8 depicts the effect of CFO on averaged link capacity under real FDLS channel estiation and no I/Q ibalance under static channel conditions. It can be shown that a oderate CFO of 3% causes only a negligable degradation of and SIMO OFDM link capacity. The worst case perforance in case of CFO = 0% is plotted to illustrate the lower sensitivity of the SIMO link copared to the link. Nevertheless, we have to state that in case of realistic frequency synchronization techniques it is highly iprobable to have a residual CFO larger than 3% at oderate SNR > 0dB. This fact is also entioned in [] where the authors derived the PDF of the residual CFO in case of real frequency synchronization under Rayleigh fading channels and given SNR. Mutual Inforation in Bit / Channel Use 6 QAM upper bound = λ = λ = λ = λ = Fig. 9. The utual inforation averaged over all data carriers under tie selective channel properties and FDLS channel estiation for OFDM, 6-QAM, CFO = 0%, IRR = 30 db, tie variant 8 taps exponential PDP Rayleigh fading channel. Fig.9 depicts the effect of outdated channel state inforation quantified by appropriate channel auto-correlation coefficients λ, FDLS channel estiation and I/Q ibalance under 8 taps exponential PDP Rayleigh fading channel conditions and perfect frequency synchronization. Again, the perforance loss in case of diversity cobining is saller than the loss that we have in case of conventional receiver designs. Moreover, we have to state that even in case of very sall deviations of λ fro the ideal static case λ =, the effect of outdated channel state inforation causes uch larger perforance losses than realistic CFO and I/Q ibalance. Finally, we want to highlight the fact that in case of oderate receiver ipairents the perforance loss ainly coes due to channel estiation errors. This iportant observation is illustrated in Fig.0 where we plotted averaged utual inforation vs. SNR under CFO = 3% and IRR = 30 db assuing static channel properties. As reference we use a plot without any I/Q ibalance, CFO or channel estiation error. Interestingly the ipairent plots in case of perfect CSI are alost equivalent to the reference curves but we observe a severe perforance degradation in case of real FDLS channel estiation. VII. CONCLUSIONS In this paper, we show how to analytically evaluate the uncoded bit error rate as well as link capacity of OFDM systes subject to carrier frequency offset, channel estiation error, outdated channel state inforation and flat receiver I/Q ibalance in Rayleigh frequency and tie selective obile fading channels. The probability density function of the frequency doain received signal subject to the entioned

10 Mutual Inforation in Bit / Channel Use 6 QAM upper bound = CFO = 3%, IRR = 30 db and real FDLS CFO = 3%, IRR = 30 db and perfect CSI no ipairents, perfect CSI Fig. 0. The utual inforation averaged over all data carriers, coparing the effect of receiver ipairents in case of perfect CSI and real FDLS channel estiation, 6-QAM odulation on all subcarriers, static 8 taps exponential PDP Rayleigh fading channel. ipairents is derived. Furtherore, this PDF is verified by eans of bit error rate calculation. We show that our approach can be used to exactly evaluate uncoded bit error rates when a-priori knowledge of the obile channel power delay profile, the iage rejection ratio and receiver CFO is used. Furtherore, we show how to use the derived PDF to calculate OFDM link capacity under the aggregate effects of receiver ipairents and obile channel characteristics. Finally, we highlight the fact that channel uncertainty induced by channel estiation errors as well as outdated channel state inforation have uch severer ipact on OFDM capacity than CFO or I/Q ibalance. REFERENCES [] H. Cheon and D. Hong. Effect of channel estiation error in OFDMbased WLAN. In Proc. IEEE Counication Letters, volue 6, pages 90 9, May 00. [] I. Cosovic and G. Auer. Capacity Achieving Pilot Design for MIMO- OFDM over Tie-Varying Frequency-Selective Channels. In Proc. IEEE Int. Conference on Counications 007 ICC 007, Glasgow, June 007. [3] IEEE. Part: Wireless LAN Mediu Access Control MAC and Physical Layer PHY specifications. In IEEE Std 80.a-999, 999. [] M. Krondorf and G. Fettweis. Bit Error Rate Calculation for OFDM with Synchronization Errors in Tie and Frequency Selective Fading Channels. In Proc. European Wireless Conference, April 007. [5] K.Sathananthan and C. Tellabura. Probability of error calculation of OFDM with frequency offset. In Proc. IEEE Transactions on Counications, volue 9, pages , Noveber 00. [6] T. Pollet, M. van Bladel, and M.Moeneclaey. BER sensitivity of OFDM systesto carrier frequency offset and Wiener phase noise. In Proc. IEEE Transactions on Counications, volue, pages 9 93, 995. [7] John G. Proakis. Digital counications, th Ed. McGraw-Hill Inc., 00. [8] L. Rugini and P. Banelli. BER of OFDM systes ipaired by carrier frequency offset in ultipath fading channels. In IEEE Transactions on Wireless Counications, volue, pages 79 88, Septeber 005. [9] S. K. Wilson and J. M. Cioffi. Probability Density Functions for Analyzing Multi-Aplitude Constellations in Rayleigh and Ricean Channels. In IEEE Transactions on Counications, volue VOL. 7, NO. 3, March 999. [0] M. Windisch and G. Fettweis. Standard-Independend I/Q Ibalance Copensation in OFDM Direct-Conversion Receivers. In Proc. 9th Intl. OFDM Workshop InOWo, pages 57 6, Dresden, Gerany, Septeber 00. [] M. Windisch and G. Fettweis. Error Probability Analysis of Multi- Carrier Systes Ipaired by Receiver I/Q Ibalance. In Proc. International Syposiu on Wireless Personal Multiedia Counications WPMC 06, San Diego, USA, Septeber 006. [] M. Windisch and G. Fettweis. Perforance Degradation due to I/Q Ibalance in Multi-Carrier Direct Conversion Receivers: A Theoretical Analysis. In Proc. IEEE International Conference on Counications ICC 06, Istanbul, Turkey, June 006.