OFDM Link Performance Analysis under Various Receiver Impairments. Accepted for publication

We present a methodology for OFDM link capacity and bit error rate calculation that jointly captures the aggregate effects of various real life receiver imperfections such as: carrier frequency offset, channel estimation error, outdated channel state information due to time selective channel properties and flat receiver I/Q imbalance. Since such an analytical analysis is still missing in literature, we intend to provide a numerical tool for realistic OFDM performance evaluation that takes into account mobile channel characteristics as well as multiple receiver antenna branches. In our main contribution, we derived the probability density function (PDF) of the received frequency domain signal with respect to the mentioned impairments and use this PDF to numerically calculate both bit error rate and OFDM link capacity. Finally, we illustrate which of the mentioned impairments has the most severe impact on OFDM system performance.


I. INTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM) is a widely applied technique for wireless communications, which enables simple one-tap equalization by cyclic prefix insertion. Conversely, the sensitivity of OFDM systems to various receiver impairments is higher than that of singlecarrier systems. Furthermore, for OFDM system designers it is often desirable to have easy to use numerical tools to predict the system performance under various receiver impairments. Within this article the term performance means both link capacity and uncoded bit error rate (BER). Mostly, link level simulations are used to obtain reliable performance measures of a given system configuration. Unfortunately simulations are highly time consumptive especially when the parameter space of the system under investigation is large. Therefore, the intention of this article is to introduce a stochastic/analytical method to predict the performance metrics of a given OFDM system configuration. To get realistic performance results, our approach takes into account a variety of receiver characteristics and impairments as well as mobile channel properties such as: • residual carrier frequency offset (CFO) after synchronization • channel estimation errors • outdated channel state information due to time selective mobile channel properties • flat receiver I/Q imbalance in case of direct conversion receivers • frequency selective mobile channel characteristics • multiple receiver branches to realize diversity combining methods such as maximum ratio combining (MRC) In present OFDM standards, such as IEEE802.11a/g or DVB-T, preamble (or pilots) are used to estimate and to compensate the CFO and channel impulse response. Unfortunately, after CFO estimation and compensation, the residual carrier frequency offset still destroys the orthogonality of the received OFDM signals and corrupts channel estimates, which worsen further the performance of OFDM systems during the equalization process. In the literature, the effects of carrier frequency offset on bit error rate are mostly investigated under the assumption of perfect channel knowledge. The papers [6] and [5] consider the effects of carrier frequency offset only (without channel estimation and equalization imperfections) and give exact analytical expressions in terms 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 systems in case of CFO under the assumption of perfect channel knowledge. Additionally, the authors of [1] tried to analyze the joint effects of CFO and channel estimation error on uncoded bit error rate for OFDM systems, but the used Gaussian channel estimation error model does not hold in real OFDM systems, especially when carrier frequency offset is large (see Sec.V). Additionally, receiver I/Q imbalance has been identified as one of the most serious concerns in the practical implementation of direct conversion receiver architectures (see for example [12]). Direct conversion receiver designs are known to enable small and cheap OFDM terminals, highly suitable for consumer electronics. The authors of [11] investigated the effect of receiver I/Q imbalance on OFDM systems for frequency selective fading channels under the assumption of perfect channel knowledge and perfect receiver synchronization. Additionally, in order to cope with this impairment, the authors of [10] proposed a digital I/Q imbalance compensation method. To our best knowledge, there is currently no literature available that describes a calculation method for OFDM BER and link capacity under the aggregate effect of all the mentioned impairments. Therefore, our intention is to describe the quantitative relationship between OFDM parameters, receiver impairments and performance metrics such as bit error rate and link capacity. Furthermore, we intend to provide a useful system engineering tool for the design and dimensioning of OFDM system parameters, pilot symbols and receiver algorithms used for frequency synchronization, channel estimation and I/Q imbalance compensation. The structure of this article is as follows. After some general remarks on our proposed link capacity evaluation method in Sec.II, we introduce our OFDM system model in Sec.III followed by a general probability density function analysis in Sec.IV. In Sec.V, it will be explained how to model the correlation between channel estimates and received/impaired signals to derive uncoded bit error rates of OFDM systems with carrier frequency offset and I/Q imbalance in Rayleigh frequency and time selective fading channels. It should be noted, that the terms bit error rate and bit error probability are used with equal meaning. This is due to the fact, that the bit error rate converges toward bit error probability with increasing observation time in an stationary environment. Finally we introduce our link capacity calculation method in Sec.VI and conclude in Sec.VII.

II. THE APPROACH
We choose link capacity, measured in [Bit / Channel Use], as an important performance metric for OFDM system designs. This information theoretic metric allows system designers to characterize the system behavior subject to real life receiver impairments independently from any kind of channel coding and iterative detection methods. As explained in Sec.VI and illustrated in Fig.1, the OFDM transceiver chain including channel and receiver properties can be characterized as effective channel between source and detector, often called the modulation channel. The modulation channel is characterized by its conditional PDF f Z|X (z|x) that describes the statistical relationship between the discrete input symbols x and the continuously distributed decision variable z. Using any given complex M-QAM constellation alphabet X, the link capacity can be expressed as mutual information between source and sink that only depends on the input statistic of X and f Z|X (z|x). Since our performance analysis framework intends to describe the mutual information (and hence the link capacity under a given input statistic), we propose the following work flow: 1) We show how to derive f Z|X (z|x) under receiver impairments, given channel properties and OFDM system parameters. 2) We use the derived f Z|X (z|x) for uncoded BER calculation to verify its correctness by comparing the BER prediction results with those obtained from simulation. 3) We calculate the mutual information, i.e. OFDM link capacity, using the verified statistic f Z|X (z|x).

III. OFDM SYSTEM MODEL
We consider an OFDM system with N-point FFT. The data is M-QAM modulated to different OFDM data subcarriers, then transformed to a time domain signal by IFFT operation and prepended by a cyclic prefix, which is chosen to be longer than the maximal channel impulse response (CIR) length L. The sampled discrete complex baseband signal for the lth subcarrier after the receiver FFT processing can be written as where X l represents the transmitted complex QAM modulated symbol on subcarrier l and W l represents complex Gaussian noise. The coefficient H l denotes the frequency domain channel transfer function on subcarrier l, which is the discrete Fourier transform (DFT) of the CIR h(τ ) with maximal L taps In this paper, it is assumed that the residual carrier frequency offset (after frequency synchronization) is a given deterministic value. Furthermore, static (non time selective) channel characteristics are assumed during one OFDM symbol. The CFO-impaired complex baseband signal subcarrier l can be written as The complex coefficients I(k − l) represent the impact 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] where ∆f is the residual carrier frequency offset normalized to the subcarrier spacing. In addition, later in this paper the summation N 2 −1 k=− N 2 ,k =l will be abbreviated as k =l . In Eq.(3) we can see, that residual CFO causes a phase rotation of the received signal (I(0)) and inter carrier interference (ICI). Furthermore, there is a time variant common phase shift for all subcarriers due to CFO as given in [8], that is not modeled here. This is due to the fact, that this time variant common phase term is considered to be robustly estimated and compensated by continuous pilots that are inserted among the OFDM data symbols. I/Q imbalance of direct conversion OFDM receivers directly translates to a mutual interference between each pair of subcarriers located symmetrically with respect to the DC carrier [10]. 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 imbalance can be modeled by [10], [12] where (.) * represents the complex conjugation and K l denotes a complex-valued weighting factor that is determined by the receiver phase and gain imbalance [10]. The image rejection capabilities of the receiver on subcarrier l can be expressed in terms of image rejection ratio -IRR given by In this paper, we consider flat I/Q imbalance which simply means IRR l = IRR , ∀ l. Subsequently, we consider preamblebased frequency domain least square (FDLS) channel estimation to obtain the channel state information (Ĥ l ) on subcarrier where X P,l andỸ P,l denote the transmitted and received preamble symbol on subcarrier l. The Gaussian noise of the preamble part W ′ l has the same variance as W l of the data part (σ 2 The channel estimate is used for frequency domain zero-forcing equalization before data detection where Z l is the decision variable that is feed into the detector/decoder stage. The power of preamble signals and the average power of transmitted data signals on all carriers is equivalent (|X P | 2 = σ 2 X ). In case of multiple (N Rx ) receiver branches, maximum ratio combining (MRC) is used at the receiver side. Therefore, the decision variable Z l on subcarrier l is given by where κ denotes the receiver branch index. We assume, that there is the same IRR and CFO on all branches, what is reasonable when considering one oscillator used for down-conversion in each branch. Furthermore, we assume uncorrelated channel coefficients among the branches 1 , i.e.

A. Mobile Channel Characteristics
To obtain precise performance analysis results in case of subcarrier crosstalk induced by CFO and I/Q imbalance, it is desirable to use exact expressions of the subcarrier channel cross-correlation properties what is shown in more detail in Sec.V. The cross-correlation properties between frequency domain channel coefficients are mainly determined by the power delay profile of the channel impulse response (CIR) and the CIR tap cross-correlation properties. Furthermore, the discrete nature of the sampled CIR is modeled as tapped delay line having L channel taps. Although our analysis is not limited to a specific type of frequency selective channel, in our numerical examples we consider mobile channels having an exponential power delay profile (PDP) The channel taps h(τ ) are assumed to be complex zero-mean Gaussian RV with uncorrelated real and imaginary parts. Hence, after DFT according to Eq.(2), the channel coefficients are zero-mean complex Gaussian random variables as well. Additionally, the CIR length L is assumed 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 where σ 2 H is equivalent for all subcarriers. Assuming mutual uncorrelated channel taps of the CIR and applying Eq. (2), one get The cross correlation property of the complex Gaussian channel coefficients can be formulated to be where V k is a complex zero-mean Gaussian with variance In current OFDM systems such as 802.11a/n or 802.16, there is a typical OFDM block structure. A OFDM block consists of a set of preamble symbols used for acquisition, synchronization and channel estimation, followed by a set of serially concatenated OFDM data symbols. User mobility gives rise to a considerable variation of the mobile channel during one OFDM block (fast fading) what causes outdated channel information in certain OFDM symbols if there is no appropriate channel tracking. To be precise, during the time period λ between channel estimation and OFDM symbol reception the channel changes in a way that the estimated channel information used for equalization does not fit the actual channel anymore. If there is no channel tracking at the receiver side, our aim is to incorporate the effect of outdated channel information into the performance analysis framework. 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.(2) we get When assuming uncorrelated channel taps it follows For sake of simplicity, it is assumed that all channel taps have the same auto-correlation coefficient, For the numerical BER and link capacity evaluations done in Sec.V-B and Sec.VI-B, the time selectivity of the complex Gaussian channel taps was modeled as follows For sake of simplicity, it is assumed that the channel is stationary during one OFDM symbol but changes from symbol to symbol in the above defined manner. In our analysis, we intentionally avoid any assumptions on concrete fast fading models in order to obtain fundamental results. Anyway, one of the commonly used statistical descriptions of fast channel variations is the Jakes' model [7], where the channel auto-correlation coefficient r h (τ ) is given by and f D,max denotes the maximum Doppler frequency that is determined by the mobile velocity and carrier frequency of the system. It should be noted, that r h (τ ) is real due to uncorrelated i.i.d. real and imaginary parts of the CIR taps.

IV. PROBABILITY DENSITY FUNCTIONS ANALYSIS
The author of [9] suggested a correlation model regarding channel estimation for single-carrier systems and derived the correspondent symbol error rate and bit error rate of QAMmodulated signals transmitted 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 systems for time and frequency selective fading channels with CFO, I/Q imbalance and channel estimation error. The single-carrier transmission model without carrier frequency offset for flat Rayleigh fading channels can be written as where y, h, x and w denote the complex baseband representation of the received signal, the channel coefficient, the transmitted data symbol and the additive Gaussian noise with variance σ 2 w respectively. In [9], the channel estimateĥ is assumed to be biased and used for zero forcing equalization as follows where α denotes the deterministic multiplicative bias of the channel estimates and ν represents zero-mean complex Gaussian noise with variance σ 2 ν . The channel coefficient h and Gaussian noise ν are assumed to be uncorrelated. Hence, the case of perfect channel knowledge can be easily modeled by α = 1 and σ 2 ν = 0. In [9], the joint PDF of the decision variable z = z r + jz i in case of transmit symbol x is derived in cartesian coordinates and can be written as The PDF mainly depends on the complex parameter b(x), given by [9] and [4] b and the real parameter a(x) that can be written according [9] and [4] as Additionally, the closed form integral of Eq.(18) with z = z r + jz i is given by [9] to be In case of N Rx receiver branches, maximum ratio combining (MRC) is used for decision variable computation what can be formulated as where κ represents the antenna branch index and the κ-th channel estimate can be written according to the SISO case aŝ Since it is quite reasonable to assume that the same channel estimation scheme is used in each receive antenna branch, we have α κ = α , ∀ κ. The authors of [9] also derived the PDF of z in case of transmit symbol x and N Rx receiver branches that is given by It easy to observe that the PDF Eq.(23) for the MRC case takes the SISO form of Eq.(18) in case of N Rx = 1.
Additionally, the closed form 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 SISO form Eq.(21) in case of N Rx = 1. To enhance readability and to simplify our notation, we omit the receiver branch number N Rx in the conditional PDF and its closed form 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 (21) can be used to calculate the bit error rate of a given M-QAM constellation. In a M-QAM constellation there are M log 2 (M ) different possible bit positions with respect to the M-QAM constellation. The probability of an erroneous bit with respect to the m-th QAM transmit symbol x m can be calculated by using the closed form integral Eq.(21) and an appropriate decision region B m,ν for the νth bit position (see Fig.2) that takes into account the bit mapping of the QAM constellation. In the paper, we always use Gray mapping in our numerical results, but it is worth mentioning that the described method can be used for arbitrary bit mappings 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 determine the OFDM link capacity of a given transceiver configuration. Therefore, the bit error probability P b (x m ) takes the form where F Z|X (z|x m ) Bm,ν denotes the 2-dimensional evaluation of the closed form integral F Z|X (z|x m ) subject to the decision region B m,ν . Finally, the bit error probability can be obtained by averaging over all possible constellation points, when assuming equal probable M-QAM symbols as follows

V. OFDM BIT ERROR RATE ANALYSIS
In this section, the derivation of the bit error rate of OFDM systems with carrier frequency offset, I/Q imbalance and channel estimation error in Rayleigh frequency and time selective fading channels will be given. The central idea of our BER derivation is to map the OFDM system model of Sec.III to the statistics given in Sec.IV. To be precise, we have to map the OFDM system model to the parameters α, a 2 (Eq.(20)) and b 2 (Eq.(19)) as explained below.

A. Mathematical Derivation
Firstly, we can rewrite the channel estimates of subcarrier l in Eq.(6) with respect to the frequency selective fading characteristic given in Eq. (11) to bê H l = I(0)H l 1 + k =l r k,l X P,k I(k − l) I(0)X P,l +e K l m r * m,l X * P,m I * (m + l) I(0)X P,l +ν l (26) where e denotes the term e −j2φ l . This comes due to the fact that the complex Gaussian Channel coefficient can be written as H l = |H l |e jφ l . Hence, we have whereα l is a stochastic quantity with given subcarrier index l, a set of deterministic preamble symbols X P,k , a fixed predetermined frequency offset, a given IRR constant K l and RV e = e j2φ l . It should be noted, that the stochastic part of α l is negligible in case of moderate I/Q imbalance (IRR ≥ 30 dB) and moderate CFO. Hence, we have that eK l m r * m,l X * P,m I * (m + l) ≈ 0 and α l can be well modeled to be a deterministic quantity. This is due to the fact that the pilot symbols X P,k as well as the CFO are given deterministic values and the channel cross correlation coefficients r k,l can be calculated using Eq.(9) and Eq.(10). The noise partν l of the channel estimate can be written as For σ 2 ν l , which represents the additive Gaussian noise variance of the channel estimates, we obtain σ 2 ν l = k =l n =l X P,k X * P,n I(k − l)I * (n − l) ×(r k,n − r k,l r * n,l σ 2 H ) +|K l | 2 k n X * P,k X P,n I * (k + l)I(n + l) Applying the same method as above for Eq.(3) and Eq.(4), the same definition of effective channelH l can be used to get an Eq.(16)-like expression as follows Given Eq.(30) and Eq.(31), the effective symbolX l can be defined, that is no longer a deterministic value but a stochastic quantity due to i.i.d. data symbols on subcarriers k = l: stochastic part of the effective transmit symbol (32) Assuming a certain transmit symbol X l and assuming randomly transmitted data symbols X k with k = l, we can decompose the effective symbolX l as follows: which shows the stochastic nature ofX l due to the random interference part J l due to ICI and I/Q imbalance. Applying the central limit theorem, we assume that the interference J l term is a complex zero-mean Gaussian random variable J l = p+jq. The mutual uncorrelated real and imaginary parts p and q have the same variance for all constellation points (33) According to Eq.(19) and Eq.(20), we calculate the parameters b l = b l,r + jb l,i and a 2 l for M-QAM effective data symbolsX l on subcarrier l in frequency and time selective fading channels: where σ 2 H l = |I(0)| 2 σ 2 H and σ 2 H l = |α l | 2 |I(0)| 2 σ 2 H + σ 2 ν l . From Eq.(34) and Eq.(35) one can observe, that the parameter σ 2W l has to be calculated exactly to obtain reliable results. The termW 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.(11) into Eq.(4), we getW For an exact expression of σ 2W l , we take Eq.(36), σ 2 V k,l = σ 2 H (1 − |r k,l | 2 ) together with the assumptions of mutually uncorrelated data symbols and obtain As an example, for one QPSK constellation point with index m = 1 on subcarrier l, X 1,l = 1 √ 2 (1, 1) = 1 √ 2 (1+j), we need to recalculate b l (X 1,l ) and parameter a 2 l (X 1,l ) separately for each effective symbol realizatioñ to use the closed form integral and Eq. (24) for BER calculation. Subsequently, the bit error rate on subcarrier l for the m-th constellation point can be expressed using Eq.(24) by the following double integral involving the Gaussian PDFs of p and q: 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 m as follows:

B. Bit Error Rate Performance -Numerical Results
In this section, the derived analytical expressions for bit error rate are compared with appropriate simulation results for both SISO (single input single output) OFDM transmission as well as SIMO (single input multiple output) OFDM using MRC and two receiver antenna branches. Furthermore, we consider an IEEE 802.11a-like OFDM system [3] with 64-point FFT. The data is 16-QAM modulated to the data subcarriers, then transformed to the time domain by IFFT operation and finally prepended by a 16-tap long cyclic prefix. The data is randomly generated and one OFDM pilot symbol was used for channel estimation. The used BPSK pilot data in the frequency domain is given by The data and pilot symbols are modulated on 52 data carriers. The DC carrier as well as the carriers at the spectral edges are not modulated and are often called 'virtual carriers'. For simulation and numerical BER analysis we use an 8 taps exponential PDP frequency selective Rayleigh fading channel with D = 7 (see Sec.III). Furthermore, we choose statistical independent channel realizations for the two antenna branches in case of SIMO OFDM transmission. The double integral of Eq.(38) is evaluated numerically using Matlab build-in integration functions having a numerical tolerance of 10 −8 and upper/lower integration bounds of ±10. Fig.3 illustrates the calculated and simulated 16-QAM BER vs. SNR (σ 2 X /σ 2 W ) with given carrier frequency offset ∆f (in % subcarrier spacing) and IRR = 30dB under non time variant mobile channel conditions. Fig.4 illustrates the calculated and simulated 16-QAM BER vs. SNR (σ 2 X /σ 2 W ) with given carrier frequency offset ∆f (in % subcarrier spacing) and IRR = 30dB under non time variant mobile channel conditions. In Fig.5, we use a fixed ∆f of 3% to investigate 16-QAM BER vs. SNR for time variant mobile channel properties, characterized by the channel tap auto correlation coefficients r h (λ). The results illustrate that our analysis can approximate the simulative performance very accurately, if the channel power delay profile, the image rejection ratio of the direct conversion receiver and carrier frequency offset are known.

VI. CAPACITY ANALYSIS OF IMPAIRED OFDM LINKS
To perform OFDM link capacity analysis, it seems mandatory to review the main principles and basic equations of how to calculate average mutual information between source and sink of a modulation channel. An excellent overview of this topic can be found in [7] that is summarized in the following. In an OFDM system we have a number of parallel channels, i.e. data subcarriers. Hence we propose to calculate the mutual information for each of the parallel data carriers independently and to finally average the link capacity among the data carriers.
Let's consider real input and output alphabets X and Z. Both alphabets can be characterized in terms of information content carried by the elements of each alphabet what leads to the concept of information entropy H(X) and H(Z). The entropy of the discrete alphabet X having elements X m with appropriate probability P (X m ) is given by Conversely, Z is assumed to be a real continuously distributed RV having realizations z. As a result, Z can be characterized by its differential entropy as The quantitative relationship between receiver impairments, OFDM system parameters and link capacity is an essential piece of information for the dimensioning of I/Q imbalance compensation algorithms as well as frequency synchronization methods. Moreover, the effects of time selective mobile channels on link capacity can be used to design scattered pilot structures for channel estimation and tracking as done in [2]. Generally, link capacity indicates the maximum data rate that can be achieved with strong channel coding under a given input constellation and a specified receiver architecture.
The numerical examples of average mutual information are chosen such, that we illustrate the effects of channel estimation error, outdated channel state information (CSI), residual CFO and flat receiver I/Q imbalance on the link capacity of SISO and SIMO OFDM links. Therefore, we choose the same IEEE 802.11a-like OFDM system parameters as introduced in Sec.V-B, assume an 8 taps exponential PDP mobile channel and the use of 16-QAM modulation on each data carrier. Again, statistical independent channel realizations for the N Rx antenna branches in case of SIMO OFDM transmission are assumed. The mutual information (measured in [Bit / Channel Use]) is averaged among the data carriers and plotted over SNR (σ 2 X /σ 2 W ). In Fig.6, we illustrate the effect of real life frequency domain least square (FDLS) channel estimation on the link capacity of SISO and SIMO OFDM respectively, assuming no I/Q imbalance, a perfect frequency synchronization (CFO = 0%) and static (non-time selective) channel properties. As reference, we plotted the case of perfect channel state information, that can easily be modelled by α l = 1 and σ 2 ν l = 0. In Fig.7 Fig.8 depicts the effect of CFO on averaged link capacity under real FDLS channel estimation and no I/Q imbalance under static channel conditions. It can be shown that a moderate CFO of 3% causes only a negligable degradation of SISO and SIMO OFDM link capacity. The worst case performance in case of CFO = 10% is plotted to illustrate the lower sensitivity of the SIMO link compared to the SISO link. Nevertheless, we have to state that in case of realistic frequency synchronization techniques it is highly improbable to have a residual CFO larger than 3% at moderate SNR (> 10dB). This fact is also mentioned in [4] where the authors derived the PDF of the residual CFO in case of real frequency synchronization under Rayleigh fading channels and given SNR.   Fig.9 depicts the effect of outdated channel state information quantified by appropriate channel auto-correlation coefficients r h (λ), FDLS channel estimation and I/Q imbalance under 8 taps exponential PDP Rayleigh fading channel conditions and perfect frequency synchronization. Again, the performance loss in case of diversity combining is smaller than the loss that we have in case of conventional SISO receiver designs. Moreover, we have to state that even in case of very small deviations of r h (λ) from the ideal static case r h (λ) = 1, the effect of outdated channel state information causes much larger performance losses than realistic CFO and I/Q imbalance. Finally, we want to highlight the fact that in case of moderate receiver impairments the performance loss mainly comes due to channel estimation errors. This important observation is illustrated in Fig.10 where we plotted averaged mutual information vs. SNR under CFO = 3% and IRR = 30 dB assuming static channel properties. As reference we use a plot without any I/Q imbalance, CFO or channel estimation error. Interestingly the impairment plots in case of perfect CSI are almost equivalent to the reference curves but we observe a severe performance degradation in case of real FDLS channel estimation.

VII. CONCLUSIONS
In this paper, we show how to analytically evaluate the uncoded bit error rate as well as link capacity of OFDM systems subject to carrier frequency offset, channel estimation error, outdated channel state information and flat receiver I/Q imbalance in Rayleigh frequency and time selective mobile fading channels. The probability density function of the frequency domain received signal subject to the mentioned impairments is derived. Furthermore, this PDF is verified by means 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 mobile channel power delay profile, the image rejection ratio and receiver CFO is used. Furthermore, we show how to use the derived PDF to calculate OFDM link capacity under the aggregate effects of receiver impairments and mobile channel characteristics. Finally, we highlight the fact that channel uncertainty induced by channel estimation errors as well as outdated channel state information have much severer impact on OFDM capacity than CFO or I/Q imbalance.