Standard-Independent I/Q Imbalance Compensation in OFDM Direct-Conversion Receivers

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1 Standard-Independent I/Q Imbalance Compensation in OFDM Direct-Conversion Receivers Marcus Windisch, Gerhard Fettweis Dresden University of Technology, Vodafone Chair Mobile Communications Systems, D-0106 Dresden, Germany Abstract The growing number of different mobile communications standards calls for inexpensive and highly flexible receiver architectures supporting all standards. The direct conversion receiver is a very attractive candidate to reach this goal. However, unavoidable imbalances between the I- and the Q- branch of the I/Q demodulator lead to a significant performance degradation at the reception of OFDM signals. In this paper a novel compensation algorithm is presented to overcome this problem. The novel approach does not depend on any standard-specific signal components, such as pilots or a preamble. Instead, a blind I/Q imbalance parameter estimation is performed during the ordinary receive mode. Therefore, the algorithm is applicable to a wide range of present and future OFDM communications standards. Index Terms OFDM, Direct-conversion receiver, I/Q imbalance, Blind compensation, IEEE 80.11a I. INTRODUCTION Advanced receiver architectures based on I/Q signal processing are very attractive because the need for a bulky analog image rejection filter is avoided. However, one of the drawbacks is I/Q imbalance, resulting from imperfect matching of the analog components in the I- and the Q-branch of the receiver. The estimation and compensation of the I/Q imbalance in OFDM direct-conversion receivers has been addressed in the literature before. The most intuitive approach for an estimation of the I/Q imbalance parameters is to feed the I/Q mixer with dedicated calibration or training signals. In [1] this is This work was partly supported by the German Ministry of Education and Research (BMBF) within the project Wireless Gigabit with advanced multimedia support (WIGWAM) under grant 01BU370 done by feeding the RF part off-line with a receivergenerated calibration signals. The effort of additional hardware in the receiver can be avoided if components of the received signal itself, such as known pilots, are used for a calibration [], [3]. However, such techniques are limited to a certain class of communications standards with the presumed pilots. Furthermore, in a practical scenario the pilots are likely to be affected not only by the receiver I/Q imbalance, but also by other impairments, such as the transmission channel and various RF impairments at the transmitter and the receiver. Because all these effects have to be considered, an accurate pilot based I/Q imbalance compensation is likely to be very complex. The dependence on known pilots is avoided by applying blind signal processing techniques. The compensation of I/Q imbalance in the time domain based on Blind Signal Separation (BSS) has been proposed in [4]. In addition to the computational complexity of BSS, the drawback of BSS is one of its fundamental assumptions, that at most one of the source signals to be reconstructed is Gaussian distributed. However, both the I- and the Q- component of the OFDM time domain signals are asymptotically Gaussian distributed as the number of subcarriers and the modulation order gets larger. This property limits the application of BSS based techniques for OFDM applications. In order to cope with receiver I/Q imbalance in a wide range of present and future OFDM communications standards an advanced estimation and compensation technique is required. In this paper we will present a novel approach for a blind estimation of the unknown I/Q imbalance. It does not require any pilot signals, which makes it independent from the targeted communications standard. This paper is structured as follows: Section II

2 introduces a model for the I/Q imbalance, wich is used in this paper. A novel approach for the blind estimation and compensation of the I/Q imbalance is presented in sections III and IV. In section V we demonstrate the capabilities of this novel compensation technique exemplary for the IEEE 80.11a standard, followed by the conclusions in section VI. a) b) Ym(n) Y m(n) f LO K e +jπflot Z m (n) Z m (n) Y m (n) Y m (n) f LO K 1 e jπflot f II. I/Q IMBALANCE MODEL The purpose of any receiver architecture is to convert a frequency band of interest of an RF signal down to base band. The real RF signal just before down-conversion can be written as r(t) = y(t)e +jπf Ct + y (t)e jπf Ct, (1) where y(t) denotes the complex base band equivalent of the frequency band of interest, f C denotes its carrier frequency and ( ) denotes (complex) conjugation. The principle of direct-conversion is to use a local oscillator (LO) with the frequency f LO = f C and to apply complex (I/Q) mixing. For reference, we will first consider the case of no I/Q imbalance. In this case, the down-conversion yields the frequency band of interest y(t) = LP {r(t)x LO (t)}, () where x LO (t) = e jπf LOt denotes the time function of the (perfectly balanced) complex LO and LP { } denotes low pass filtering. If the direct-conversion architecture is used for the reception of an OFDM signal, y(t) equals the received OFDM base band signal. This signal is further A/D-converted and its samples y(k) are used to compute the symbols on the subcarriers by applying the Discrete Fourier Transform (DFT) Y m (n) = 1 L DF T L DF T 1 l=0 y(l)e jπlm/ldf T, (3) were L DF T denotes the number of subcarriers. Y m (n) denotes the demodulated subcarrier symbol at the m th sumcarrier of the n th OFDM symbol. For a better understanding of the I/Q imbalance effects, the analyzed interval of subcarrier indices is set to m [ L DF T /; L DF T / 1], where m=0 denotes the DC subcarrier. Unfortunately, a perfect analog I/Q mixing is not achievable in practice. Unavoidable tolerances in the manufacturing process lead to deviations from the desired 90 phase shift and the desired equal gain in the I- and the Q-branch. These imperfections can be modelled by a complex LO with the time function x LO (t) = cos(πf LO t) jg sin(πf LO t + φ), Fig. 1. Frequency domain illustration of I/Q imbalance in OFDM direct-conversion receivers: a) Spectrum of the RF signal r(t), b) Spectrum of the base band signal z(t) where g denotes the amplitude imbalance and φ denotes the phase imbalance. Furthermore, we define the complex I/Q imbalance parameters K 1 = 1 + ge jφ, K = 1 ge+jφ, (4) in order to rewrite the time function of the complex LO with I/Q imbalance as x LO (t) = K 1 e jπflot + K e +jπflot. (5) This means, direct-conversion with I/Q imbalance can be interpreted as a superposition of a desired complex down-conversion (weighted by K 1 ) and an undesired complex up-conversion (weighted by K ). Consequently, the received base band signal with I/Q imbalance z(t) is a superposition of the desired frequency band y(t) and its image y (t): z(t) = LP {r(t) x LO (t)} = K 1 y(t) + K y (t). (6) This superposition translates to a mutual interference of symmetric OFDM symbols (see Fig. 1). With (3) and (6), the received OFDM symbols in the case of I/Q imbalance become: Z m (n) = K 1 Y m (n) + K Y m(n). (7) Equivalently, we can write for the mirror subcarrier Z m (n) = K 1 Y m (n) + K Y m(n). (8) Merging (7) and the conjugate of (8) leads to a convenient matrix description of the I/Q imbalance effects: [ ] [ ] [ Zm (n) Ym (n) K1 K Z m(n) = K Y m(n), K = K K 1 f ]. (9) Since the matrix K is always non-singular for realistic imbalance parameters, the desired OFDM

3 z(t) ADC DFT Z m (n), Z m (n) Collection of statistics ˆK 1 ˆK Ŷ m (n), Ŷ m(n) ˆK 1 Fig.. Structure of the proposed I/Q imbalance compensation algorithm symbols Y m (n) and Y m (n) can be perfectly reconstructed based on the interfered symbols Z m (n) and Z m (n) using the inverse K 1. It should be stressed, that the reconstructed symbols Y m (n) and Y m are not necessarily identical to the transmitted OFDM symbols. Instead, they might be corrupted by the channel or other RF impairments. The compensation of such distortions is beyond the scope of this paper. We focus on a cancellation of the I/Q imbalance effects, i.e. the goal is to provide OFDM symbols equivalent to those of a perfectly balanced direct-conversion. The key for the compensation is to gain knowledge about the unknown mixing matrix K. An analysis of the mixing equation (9) shows its structural similarity to the mixing equation in Low-IF receivers [5]. In fact, considering the effects of I/Q imbalance, the mutual interference between desired signal and image signal in Low-IF receivers corresponds to the mutual interference between symmetric subcarriers in OFDM direct-conversion receivers. It has been shown in [5], that a completely blind estimation of the unknown mixing matrix is possible for the Low-IF case. In this paper we will show, that the parameter estimation approach of [5] can be adapted to OFDM direct-conversion receivers. III. I/Q IMBALANCE COMPENSATION The proposed I/Q imbalance compensation approach consists of two steps: The blind determination of the product K 1 K, followed by the derivation of the inverse K 1. For the purpose of a clear presentation we will drop the sample time index (n) in this section. However, it should be kept in mind that the investigated OFDM symbols are generally time variant. In order to determine K 1 K, two statistics of the imbalanced symbols Z m and Z m will be evaluated, namely the cross-correlation E {Z m Z m } and the power of the sum E { Z m + Z m }, where E { } denotes expectation. The only assumption we introduce is, that E {Y m Y m } = 0 holds at the examined subcarrier index m, i.e. Y m and Y m are uncorrelated and have zero mean 1. This assumption is realistic at least for pairs of data-subcarriers, if a proper source and channel coding is applied. Analyzing the crosscorrelation term yields E {Z m Z m } = K 1 K E {Y m Ym} + K 1 K E { Y m Y m + K 1 E {Y m Y m } + K E { YmY m = K 1 K (P m + P m ), (10) where P m = E {Y m Ym} and P m = E { Y m Y m denote the power of the subcarriers. Furthermore, we can write Z m + Z m = (K 1 + K ) Y m + (K + K 1 ) Y m = Y m + Y m, (11) which holds, since definition (4) yields K 1 + K = K + K 1 = 1. (1) Now, the second expectation term can be written as E { Z m + Z m } = E { Y m + Y m } = E {Y m Ym} + E { Y m Y m + E {Y m Y m } + E { YmY m = P m + P m. (13) Merging (10) and (13) results in the backbone equation of the proposed imbalance parameter estimation scheme: E {Z m Z m } E { Z m + Z m } = K 1K (14) In order to derive the compensation matrix K 1, the product has to be split into its composing factors. Although K 1 and K cannot be separated directly, the parameters g and φ can be extracted. Definition (4) yields K 1 K = 1 ( 4 1 g ) j 1 g sin φ, which can be converted into: g = 1 4 Re {K 1 K } ( ) φ = arcsin g Im {K 1K }. (15) Re { } and Im { } denote the real part and the imaginary part, respectively. The values K 1 and K are now determinable using definition (4). Hence, at least one pair of uncorrelated subcarriers allows for a completely blind determination of the unknown I/Q 1 A residual cross-correlation E {Y m Y m } = 0 limits the achievable accuracy of the parameter estimation. A comprehensive theoretical analysis of how the parameter estimation in a Low-IF receiver is affected by a residual cross-correlation can be found in [6].

4 imbalance parameters. The structure of the resulting I/Q imbalance parameter estimation and compensation algorithm is depicted in Figure. IV. REALIZATION ASPECTS In a practical receiver the expectation terms of (14) have to be replaced by sample based approximations. This can be done by an averaging operation over multiple pairs of uncorrelated subcarriers. Furthermore, the I/Q imbalance parameters change very slowly with time. Hence, an averaging over time is also reasonable. The estimation can be formally written as ˆK 1 ˆK = m M m M n N Z m(n)z m (n) n N Z m(n) + Z m. (n) (16) M denotes the chosen subset of M (positive) subcarrier indices, N denotes the chosen subset of N sample time indices. Obviously, the accuracy of the estimation will be affected by the number of incorporated sample pairs M N. An increased subcarrier block size M raises the computational effort at each time instant n, whereas an increased temporal block size N raises the duration of the parameter estimation. Hence, the proposed parameter estimation allows for a flexible tradeoff between accuracy, computational effort and measurement time. The separation scheme of the product into the estimates ˆK 1 and ˆK is the same as sketched for the exact values, leading to the estimated I/Q imbalance compensation matrix [ ˆK 1 1 ˆK = 1 ˆK ] ˆK 1 ˆK ˆK. (17) ˆK 1 Using this blindly gained compensation matrix, a reconstruction of the unknown source signals is possible: [ ] [ ] [ ] Ŷm (n) Ŷ m(n) = ˆK 1 Zm (n) Z m(n) = ˆK 1 Ym (n) K Y m(n). (18) The computational effort of the proposed I/Q imbalance compensation technique is very low. In order to implement the parameter estimation (16), only 8M real additions and 6M real multiplications are required at the OFDM symbol rate. The more costly calculation of the compensation matrix ˆK 1 can be done at a much lower rate, depending on how fast the I/Q imbalance changes with time. SER 10 N=10 N=100 N= Fig. 3. Mean symbol error rate for an AWGN channel and I/Q imbalance parameter estimation based on all data-subcarriers (64-QAM, g=1.05, φ=5 ) V. SIMULATION RESULTS We will demonstrate the capabilities of the proposed I/Q imbalance compensation algorithm considering the IEEE 80.11a WLAN standard [7], which is a widely used OFDM-based wireless communications standard. The highest modulation order (64-QAM), which is also most sensitive to I/Q imbalances, is used in the simulations. For each simulation setting, we first perform a parameter estimation based on the proposed approach, followed by a measurement of the resulting symbol error rate (SER), if the estimated compensation matrix ˆK 1 is used for correction. The estimation is done based on all data-subcarriers, which results in M=4 for the IEEE 80.11a signal [7]. Because the parameter estimation is done based on the actual quasi-random realization of the incorporated symbols Z m (n), the accuracy of the estimation and consequently the resulting SER will vary. Therefore, instead of a single realization SER, it is more reasonable to plot the mean SER. The following results are calculated based on 1000 independent trails of the parameter estimation for each simulation setting. We start our analysis with the simple case of an AWGN channel. Figure 3 shows the performance degradation due to the I/Q imbalance for g=1.05, φ=5. The parameter estimation was done using all data-subcarriers (M =4) for a varying block size N. The simulations show a moderate sensitivity to I/Q imbalance for an AWGN channel (3 db SNRdegradation at SER= ). Even a small block size (N =100, equivalent to 0.4 milliseconds measurement time) is sufficient for the parameter estimation to push this SNR-degradation below 0.1 db.

5 SER 10 N=10 N=100 N=1000 N= Fig. 4. Mean symbol error rate for an ETSI-A channel and I/Q imbalance parameter estimation based on all data-subcarriers (64-QAM, g=1.05, φ=5 ) SER 10 g=1.05, φ=5 g=1.0, φ= g=1.1, φ= Fig. 5. Mean symbol error rate for an ETSI-A channel and I/Q imbalance parameter estimation based on all data-subcarriers (64-QAM, N=1000) The impact of I/Q imbalance is substantially more critical for a frequency selective channel. In our simulations we use the ETSI-A channel model [8]. Figure 4 shows, that even for a high SNR the SER never falls below a specific limit, which is predetermined by the parameters of the I/Q imbalance (here: 5.3 % SER for g=1.05, φ=5 ). Such a lower limit still exists if the proposed compensation is applied. However, depending on the requirements of the chosen communications standard, the error floor can be arbitrarily decreased by a proper choice of the block size N. Furthermore, the actual I/Q imbalance parameters of a practical receiver are hardly predictable. Instead, they will vary with the manufacturing process. As a consequence, the limiting SER error floor due to I/Q imbalance will be arbitrary (Fig. 5). In contrast, for a fixed choice of M and N, the mean SER with compensation is independent from the actual values of the I/Q imbalance parameters. The error floor due to I/Q imbalance is kept at an predetermined arbitrary low level. Therefore, by using the proposed digital compensation scheme, the demands to the analog part of the receiver can be drastically reduced. VI. CONCLUSION A novel approach for a blind estimation and compensation of I/Q imbalances in OFDM directconversion receivers has been proposed. Because the parameter estimation does not depend on any structures of the received signals (such as pilots), this technique is applicable to a wide range of present and future OFDM based communications standards. The accuracy of the I/Q imbalance parameter estimation and consequently the performance of the compensation is arbitrarily scalable, allowing for a flexible tradeoff between performance, computational effort and measurement time. In order to fulfill the requirements of a chosen communications standard, the effects of the I/Q imbalance can be compensated digitally, while scaling down the demands for the analog part of the receiver. REFERENCES [1] Sébastien Simoens, Marc de Courville, François Bourzeix, and Paul de Champs, New I/Q imbalance modeling and compensation in OFDM systems with frequency offset, in Proc. IEEE PIMRC 00, Sept. 00, vol., pp [] Andreas Schuchert, Ralph Hasholzner, and Patrick Antoine, A novel IQ imbalance compensation scheme for the reception of OFDM signals, IEEE Trans. Consumer Electron., vol. 47, no. 3, pp , Aug [3] Jan Tubbax, Boris Côme, Liesbet Van der Perre, Luc Deneire, Stéphane Donnay, and Marc Engels, Compensation of IQ imbalance in OFDM systems, in Proc. IEEE Intl. Conf. on Communications (ICC 003), May 003, vol. 5, pp [4] Piotr Rykaczewski, Volker Blaschke, and Friedrich K. Jondral, I/Q Imbalance Compensation for Software Defined Radio OFDM Based Direct Conversion Receivers, in Proc. 7th Intl. OFDM-Workshop, Hamburg, Germany, 003. [5] Marcus Windisch and Gerhard Fettweis, Blind I/Q Imbalance Parameter Estimation and Compensation in Low- IF Receivers, in Proc. 1st Intl. Symposium on Control, Communications and Signal Processing (ISCCSP 004), Hammamet, Tunisia, 1-4 Mar [6] Marcus Windisch and Gerhard Fettweis, Performance Analysis for Blind I/Q Imbalance Compensation in Low- IF Receivers, in Proc. 1st Intl. Symposium on Control, Communications and Signal Processing (ISCCSP 004), Hammamet, Tunisia, 1-4 Mar [7] IEEE, Part11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications, IEEE Std 80.11a-1999, [8] ETSI EP BRAN, Channel models for HIPERLAN/ in different indoor scenarios, Mar