On the Capacity of OFDM Systems with Receiver I/Q Imbalance

Transcription

1 On the Capacity of OFDM Systems with Receiver I/Q Imbalance Stefan Krone and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universität Dresden, 16 Dresden, Germany {stefan.krone, Abstract OFDM systems have gained outstanding popularity for high data rate wireless communications. In practice, however, the performance of OFDM systems is often limited due to hardware impairments that result from stringent cost requirements. Receiver I/Q imbalance is known to be one of the most serious hardware impairments degrading the performance of OFDM systems. Previous work has shown that receiver I/Q imbalance may severely increase the symbol error rate depending on the characteristics of the communications channel. In this paper we investigate the maximum data rate, i.e. the capacity of OFDM systems that are impaired by receiver I/Q imbalance. We derive the system capacity analytically and consider the upper bound for a fixed channel realization as well as for Rayleigh fading channels. I. INTRODUCTION Orthogonal frequency division multiplexing OFDM is a widely deployed modulation technique for modern wireless communications systems. It has been adopted by a number of well-known standards such as IEEE 8.11a [1] and DVB-T []. The key advantage of OFDM is that it enables high data rate transmission over frequency selective wireless communications channels at low complexity. The maximum data rate that a communications system can provide for reliable transmission, at least in theory, is defined by its capacity. Considering the pure transmission channel, the system capacity equals the channel capacity. In his famous work [3], C. E. Shannon derived the capacity of an additive white Gaussian noise AWGN channel. In the same paper, he proposed to compute the capacity of a frequency selective channel by slicing it into a large number of frequency flat channels. Interestingly, this approach corresponds exactly to the concept of OFDM. As soon as channel fading has to be taken into account, one has to distinguish between ergodic capacity and outage capacity [4]. Whenever the capacity of a communications system is considered, it is oftentimes based on the assumption of ideal transmitter and receiver hardware. However, this assumption does not hold for practical implementations. Typically, the employed hardware has to be produced at low cost and thus at the expense of accuracy. The resulting hardware impairments are then likely to degrade the original system capacity. For OFDM systems, the direct conversion receiver architecture has turned out to be very attractive. Since it does not require any bulky analog image rejection filter, it enables a complete monolithic integration [5]. But fabrication tolerances usually lead to a mismatching of the analog components in the I- and Q-branch of the receiver, which is referred to as receiver I/Q imbalance. Unfortunately, OFDM systems appear to be quite sensitive to this kind of impairment. A lot of research on I/Q imbalance in OFDM systems has already been done. An appropriate I/Q imbalance model as well as related compensation schemes can be found in [6], for example. Recently, the authors of [7] and [8] investigated the symbol error rate SER of OFDM systems that are impaired by receiver I/Q imbalance. It has been shown that receiver I/Q imbalance may cause a severe SER degradation depending on the characteristics of the communications channel. However, when channel coding is employed, which is typically the case in practice, the achievable data rate, i.e. the system capacity will be a much more valuable performance measure. To our best knowledge, there has still been no work on how receiver I/Q imbalance affects the capacity of OFDM systems. Hence, this paper aims at closing this gap The paper is organized as follows: In section II we briefly introduce the model of an OFDM system with receiver I/Q imbalance. We analyze this system model from a statistical point of view in section III and derive the system capacity in section IV. We finally wrap up with conclusions in section V. Throughout the paper we use the following notation: Random variables will be denoted by boldface letters and their respective realizations by normal letters. The probability density function PDF of a certain random variable X in terms of its realizations X will be denoted by px. We will omit any subscript at px for the benefit of readability. II. GENERIC SYSTEM MODEL We consider an OFDM system with N subcarriers, where the cyclic prefix is chosen to be longer than the maximum length L of the channel impulse response. Furthermore, we assume the transmitter and receiver to be perfectly synchronized in frequency and time. Let S n denote the complex valued transmit symbol at subcarrier n. Then, the transmission of each S n over a frequency selective channel can be modeled in baseband by [4] Y n = H n S n + W n, 1 where H n and W n represent the associated frequency domain channel coefficient and AWGN sample, respectively. The case of an AGWN channel corresponds to H n = 1 n.

2 It has been shown in [6] that the I/Q imbalance of a direct conversion receiver in OFDM systems translates into a mutual interference between subcarriers that are located symmetrically to the DC carrier. More precisely, the receive symbol Y n of subcarrier n will be interfered by the complex of subcarrier n, and vice versa. Following the notation in [7], the impact of receiver I/Q imbalance can thus be modeled by conjugated receive symbol Y n R n = K 1,n Y n + K,n Y n, where R n is the actually received symbol at subcarrier n. The complex valued weighting factors K 1,n and K,n are defined through the receiver gain and phase imbalance, which may in general be frequency selective. Most frequently, the complex conjugated receive symbol Y n of the interfering subcarrier n is referred to as image symbol. The I/Q imbalance is hence usually quantified in terms of a subcarrier dependent imageleakage-ratio ILR n given by ILR n = K,n. 3 K 1,n Contemporary semiconductor technologies lead to an ILR n of 3 db to 4 db, whereas the ideal case of no receiver I/Q imbalance corresponds to K 1,n = 1 and K,n =, i.e. ILR n = db n. III. STATISTICAL SYSTEM ANALYSIS In order to derive the system capacity it is necessary to consider the deterministic system model from a statistical point of view. Therefore, we model the transmit symbols as well as the AWGN samples of the individual subcarriers by means of complex valued random variables [9] denoted by S n and W n, respectively. Moreover, we assume each channel coefficient H n to be a realization of the complex valued random variable H n. We restrict ourselves to channels that are slowly varying with time, i.e. each time the channel is used for data transmission, the channel coefficients H n will be fixed. The I/Q imbalance parameters K 1,n and K,n are assumed to remain constant over time. Using 1 and we can express the statistical system model for a fixed channel realization as R n = K 1,n H n S n + K,n H n S n + K 1,n W n 4 + K,n W n, where R n is the complex valued random variable belonging to the actually received symbols at subcarrier n. Based on this statistical model we will now focus on the system parameters that are required for the derivation of the system capacity. A. Interference and Noise Characteristics As can be seen from 4, each received symbol R n results from the superposition of the transmit symbol S n, the interfering transmit symbol S n and the AGWN samples W n and W n. Consequently, there is a corruption due to interference and noise. The average amount of this corruption can be stated in terms of a signal-to-interference-noise-ratio SINR n, which is in our case given by SINR n = 5 E { K 1,n H n S n } E { K,n H n S n + K 1,n W n + K,n W n } for each subcarrier n, where E { } denotes expectation [9]. Note that we still consider a fixed channel realization here. The random variables S n, S n, W n and W n can be assumed to be independent of each other. We ignore n = since the DC carrier is typically not used for data transmission. In addition, it is reasonable to assume the S n and W n of the individual subcarriers to be identically distributed with zero means and variances σs =E{ S n } and σw =E{ W n }, respectively. With these assumptions the SINR n in 5 yields SNR n SINR n = 1 + ILR n SNR n + 1, 6 where the ILR n has been defined in 3 and the SNR n is the signal-to-noise-ratio without receiver I/Q imbalance given by SNR n = H n σ S σ W for each subcarrier n. As illustrated by the numerical results depicted in Fig. 1, the SINR n strongly depends on the ILR n but also on the ratio SNR n /SNR n, which is determined by the channel coefficients H n and H n. Compared to a frequency flat channel, where SNR n /SNR n = db n,the SINR n will be further decreased for a given ILR n,ifthe SNR n is less than the SNR n of the interfering subcarrier. On the other hand, it will be increased in the converse case. SINR n in db ILR n = 5 db ILRn = 3 db ILR n = 1 db SNR n /SNR. n = 3dB SNR data n /SNR n = db data3 SNR n /SNR n = 3 db SNRn in db Fig. 1. Impact of receiver I/Q imbalance on the signal-to-interference-noiseratio SINR n for different ILR n and ratios SNR n/snr n Notwithstanding that, there is always a performance degradation due to receiver I/Q imbalance, which results from 7 SINR n SNR n. 8 It remains to analyze the SINR n for the two special cases of a very high and vanishing noise variance σ W, respectively.

3 For σw approaching infinity we obtain from 6 and 8 that SINR σ n = SNR n. 9 W 1 + ILR n In this case, the SINR n depends linearly on the SNR n but becomes independent of the SNR n.forσw approaching zero we get SINR σ n W = SNR n 1. 1 SNR n ILR n Hence, the SINR n is upper-bounded for a given ILR n depending on the ratio SNR n /SNR n. In case of a frequency flat channel the upper bound is just the inverse of the ILR n. B. Channel Characteristics So far, we have assumed the channel coefficients to be fixed and concentrated on the statistics of the transmit symbols and AWGN samples only. This already enables us to derive the system capacity for fixed channel realizations. However, it is expedient to have a look at the channel statistics, too, in order to derive the average system capacity of all possible channel realizations. Given the taps h l of the time domain channel impulse response in baseband, we get the frequency domain channel coefficients H n by means of the discrete Fourier transform, which writes H n = h l e jπnl/n. 11 l= We treat each tap of the channel impulse response as a realization of the complex valued random variable h l with zero mean and variance σh l =E{ h l }. Assuming the h l of the individual taps to be mutually independent, which is a common assumption for wireless communications channels, the variance of the frequency domain channel coefficients yields =E{ H n } = σh l 1 for any subcarrier n. Thus, the variance of the frequency domain channel coefficients simply derives from the power delay profile [4] of the channel. In addition, the cross correlation of two channel coefficients yields E{H n H k} = σh l e jπn kl/n. 13 l= From the statistical system model in 4 it can already be anticipated that the average system capacity will depend on the cross correlation of channel coefficients of subcarriers located symmetrically to the DC carrier. Hence, it is convenient to define the cross correlation coefficient ρ n which is given by E{H n H ρ n = n} E{ Hn } E{ H n } = E{H n H n} 14 for each subcarrier n. We will especially focus on the two cases of fully correlated and uncorrelated channel coefficients, i.e. on ρ n = 1 and ρ n =, corresponding to frequency flat and strongly frequency selective channel fading, respectively. l= IV. SYSTEM CAPACITY The derivation of the system capacity builds on the mutual information between transmitted and actually received symbols [1]. Considering a single subcarrier n, the mutual information IR n, S n writes IR n, S n = 15 pr n S n ps n log pr n S n pr n ds n dr n, where ps n and pr n denote the PDFs of the random variables S n and R n, pr n S n denotes the conditional PDF of R n given a certain realization of S n and pr n = pr n S n ps n ds n. 16 For each subcarrier n, the system capacity C n is defined as the maximum mutual information IR n, S n with respect to all transmit symbol distributions, i.e. C n = max IR n, S n. 17 ps n Considering the whole set of subcarriers that are available for data transmission, the overall system capacity derives as the sum of the individual C n. Hence, it will be sufficient if we just concentrate on the system capacity C n of a single subcarrier in the following. In general, there will be two approaches to compute C n.on the one hand, we may assume a certain modulation scheme, such as QAM [1], and obtain C n by maximizing IR n, S n over all possible transmit symbol distributions of the modulation alphabet. This requires to determine the conditional PDF pr n S n but might not yield the absolute system capacity, since only a predetermined modulation scheme is considered. On the other hand, we can base the computation of C n on the results of [3], where it has already been shown that the capacity of an AWGN channel is achieved by using Gaussian distributed transmit symbols. OFDM transforms a frequency selective channel into a set of frequency flat channels, which in terms of capacity boils down to an AWGN channel for each subcarrier n [4]. Hence, in the ideal case of no hardware impairments, the system capacity C n is given by the channel capacity derived in [3], which yields for a fixed channel realization C n = log 1 + SNR n 18 in bits per channel use. Note that we consider complex valued transmit symbols here. To compute the system capacity with receiver I/Q imbalance, we proceed with the latter approach presuming complex Gaussian distributed transmit symbols with zero mean and variance σs. Thus, we aim at deriving the maximum data rate that can be achieved at least in theory. We will first again consider a fixed channel realization and extend the derivation for Rayleigh fading channels afterwards.

4 A. System Capacity for a Fixed Channel Realization Allowing for complex Gaussian distributed transmit symbols, the interference at each subcarrier n caused by receiver I/Q imbalance will be complex Gaussian distributed as well. We can thus interpret the interference as part of the AWGN, which means that the original SNR n becomes the SINR n given in 6. Accordingly, we obtain the system capacity with receiver I/Q imbalance for a fixed channel realization from 18 by replacing the SNR n with the SINR n, which yields SNR n C n = log ILR n SNR n + 1 Note that C n simply results from a mapping of the SINR n with a logarithmic mapping function that is strictly monotonic increasing. Hence, the numerical results depicted in Fig. are very similar in curve shape to those in Fig. 1. However, from Fig. we can now observe how receiver I/Q imbalance affects the system capacity for a fixed channel realization. C n in bits per channel use ILR n = 5 db ILR n = 3 db ILR n = 1 db SNR n /SNR. n = 3dB data SNR n /SNR n = db data3 SNRn /SNR n = 3 db SNRn in db Fig.. Impact of receiver I/Q imbalance on the system capacity C n for different ILR n and fixed channel realizations, i.e. ratios SNR n/snr n Like the SINR n, the system capacity C n shows a strong dependence on the ILR n as well as on the ratio SNR n /SNR n. For a given ILR n, the system capacity decreases, if the SNR n becomes less than the SNR n of the interfering subcarrier and increases in the converse case. The lower and upper asymptotes of C n, that are apparent from Fig., can be derived analytically by recalling the two cases of a very high and vanishing noise variance σw discussed in section III. For σw approaching infinity we find C σ n = log W 1 + SNR n, 1 + ILR n which in fact converges to the lower asymptote of zero bits per channel use. For σw approaching zero we get C σ n = log W 1 + SNR n 1 1 SNR n ILR n as the upper asymptote. Consequently, receiver I/Q imbalance limits the system capacity for a fixed channel realization to an upper bound that depends on the ILR n but also on the ratio SNR n /SNR n. B. Ergodic System Capacity for Rayleigh Fading Channels In case of channel fading we will have to consider not only a fixed but all possible channel realizations. For this purpose, we can derive an average system capacity, where we assume that we are able to realize the instantaneous system capacity each time the channel is used for transmission. This average system capacity will then be called ergodic system capacity [4]. To ease the derivation, we rewrite the channel coefficients H n in terms of their absolute values Θ n, and phase values φ n π, π with H n =Θ n e jφn, where we treat Θ n and φ n as realizations of the real-valued random variables Θ n and φ n, respectively. Using 7 and, the system capacity for a fixed channel realization in 19 can be rewritten as Θ C n = log 1 n σ S + /σ W 1 + ILR n Θ n σs /σ W The ergodic system capacity C n is then for each subcarrier n given by the following double integral C n = C n pθ n, Θ n dθ n dθ n, 4 where pθ n, Θ n denotes the joint PDF of the random variables Θ n and Θ n. In order to compute C n, we need to determine pθ n, Θ n from the joint PDF ph n,h n of the complex-valued random variables H n and H n. For Rayleigh fading channels, which are of special interest for wireless communications systems, ph n,h n is given by the multivariate Gaussian distribution [1] ph n,h n = 1 π σ 4 H 1 ρ n exp H n + H n Re{H n H n ρ n} 1 ρ n, 5 where σ H and ρ n are the variance and cross correlation coefficient of the channel coefficients defined in 1 and 13, respectively, and Re{ } denotes the real part. By cartesian-topolar-transformation [9] we obtain from 5 the joint PDF pθ n, Θ n,φ n,φ n given in 6. The desired pθ n, Θ n finally derives as the marginal PDF of pθ n, Θ n,φ n,φ n, given by pθ n, Θ n = π π π π pθ n, Θ n,φ n,φ n dφ n dφ n. 7 In general, there will be no closed form solution for 7, which necessitates numerical integration. However, for the two special cases of fully correlated and uncorrelated channel coefficients, i.e. for ρ n = 1 and ρ n =, closed form solutions can be found. For ρ n = 1weget pθ n, Θ n ρn=1 = Θ n σ H exp Θ n σ H δθ n Θ n, 8

5 pθ n, Θ n,φ n,φ n = 6 Θ n Θ n π 4 1 ρ n exp Θ n +Θ n Θ n Θ n Re{ρ n } cosφ n φ n +Im{ρ n } sinφ n φ n 1 ρ n where δ denotes the Dirac delta function [11]. In addition, for ρ =we find pθ n, Θ n = 4 Θ n Θ n ρn= 4 exp Θ n +Θ n. 9 Nevertheless, numerical integration is still required to compute the associated ergodic system capacity C n, see 4. Fig. 3 shows the obtained results, where the SNR denotes the average signal-to-noise-ratio without receiver I/Q imbalance given by SNR = σ H σ S σw 3 for each subcarrier n. It can be observed, that C n does not significantly depend on the correlation of the channel coefficients for practically relevant ILR n. Only for exceptionally high ILR n, uncorrelated channel coefficients will cause an alleviated capacity degradation. At first glance, this observation seems to be inconsistent with the results presented in [7], where uncorrelated channel coefficients were identified to significantly reduce the SER in the presence of receiver I/Q imbalance. However, this discrepancy can be traced back to the fact, that the authors of [7] focused on hard symbol detection with zero-forcing channel equalization, which we did not consider in our analysis to derive the system capacity in general. C n in bits per channel use ILR n = 5 db ILR n = 3 db ILR n = 1 db Uncorr. channel coefficients. ρ n = Corr. channel coefficients ρ n = SNR in db Fig. 3. Impact of receiver I/Q imbalance on the ergodic system capacity C n for different ILR n and Rayleigh fading channel scenarios, i.e. ρ n Like the system capacity for a fixed channel realization C n, the ergodic system capacity for Rayleigh fading channels C n shows lower and upper asymptotes that derive from the consideration of a very high and vanishing noise variance σ W,respectively. For σ W approaching infinity, C n converges to zero bits per channel use, similar to the lower asymptote of C n.the upper asymptote of C n, which is achieved for σ W approaching zero, depends on the ILR n and at least for a very high ILR n on the correlation of the channel coefficients, i.e. on ρ n.for the two special cases of ρ n = and ρ n = 1wefind C ρn=,σ n = log ILR n 31 W ILR n 1 and C ρn=1,σ n = log W ILR n by applying 1 with 8 and 9 to 4, respectively. According to Fig. 3, 31 delivers an entire upper bound of the ergodic system capacity for Rayleigh fading channels, that is determined by the receiver I/Q imbalance. V. CONCLUSIONS In this paper we derived the capacity of OFDM systems that are impaired by receiver I/Q imbalance. It can be stated that receiver I/Q imbalance limits the system capacity to an upper bound depending on the amount of interference raised at each subcarrier. We investigated the upper bound analytically and observed that in case of a Rayleigh fading channel the correlation of the channel coefficients has almost no impact on the ergodic system capacity. Our results will be relevant for OFDM system designers to verify which amount of receiver I/Q imbalance can be allowed to ensure reliable transmission at a given maximum data rate. REFERENCES [1] IEEE Std 8.11a-1999, Part11: Wireless LAN Medium Access Control MAC and Physical Layer PHY specifications, [] ETSI EN V1.5.1, Digital Video Broadcasting DVB; Framing structure, channel coding and modulation for digital terrestrial television, 4. [3] C. E. Shannon, Communication in the Presence of Noise, Proceedings of the I.R.E, pp. 1 1, January [4] A. Goldsmith, Wireless Communications. Cambridge University Press, 5. [5] B. Razavi, Design Considerations for Direct-Conversion Receivers, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, pp , June [6] M. Valkama and M. Renfors, Compensation of Frequency-Selective I/Q Imbalances in Wideband Receivers: Models and Algorithms, Proceedings of the IEEE Third Workshop on Signal Processing Advances in Wireless Communications, pp. 4 45, March 1. [7] M. Windisch and G. Fettweis, Error Probability Analysis of Multi- Carrier Systems Impaired by Receiver I/Q Imbalance, Proceedings of the International Symposium on Wireless Personal Multimedia Communications, pp , September 6. [8] M. Krondorf and G. Fettweis, Bit Error Rate Calculation for OFDM Direct Conversion Receivers with Synchronization and Channel Estimation Imperfections, Proceedings of the 16th IST Mobile and Wireless Communications Summit, July 7. [9] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes. McGraw-Hill, Inc.,. [1] J. G. Proakis, Digital Communications. McGraw-Hill, Inc., 1. [11] J. A. Gubner, Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press, 6.