Preamble Design For An Efficient Iq Imbalance

Preamble Design for an Efficient I/Q Imbalance Compensation in OFDM Direct-Conversion Receivers Marcus Windisch and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Technische Universit¨at Dresden, 01062 Dresden, Germany Email: [email protected]

Abstract— The growing number of different mobile communications standards calls for inexpensive and highly flexible receiver architectures supporting these standards. The direct-conversion receiver is a very attractive candidate for reaching 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. A digital estimation and compensation of the unwanted effects of the I/Q imbalance is possible. In this paper we consider an I/Q imbalance parameter estimation based on the preamble of OFDM systems. General constraints for the design of the preamble are derived, which enable a robust and computationally efficient I/Q imbalance compensation. Design examples are presented and evaluated.

I. I NTRODUCTION 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 Qbranch of the receiver [2]. A very promising approach for coping with these analog impairments is to compensate them digitally. Different concepts for the digital estimation and compensation of the I/Q imbalance in OFDM direct conversion receivers have been proposed in the literature. The most simple approach is to perform an off-line calibration based on the injection of analog test signals [4]. Alternatively, an on-line parameter estimation is possible by using a priori information about the transmitted signal, such as pilot symbols or the preamble [3], [7], [5]. However, in addition to the receiver I/Q imbalance, the transmitted reference symbols are corrupted by other effects, such as the wireless channel and various impairments at the transmitter and the receiver. Because all these effects have to be considered, an accurate parameter estimation based on reference symbols is computationally costly. The dependency on reference symbols can be dropped by considering the symmetric properties of the I/Q imbalThis 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

ance distortion. It has been shown in [8], that this symmetry enables a blind I/Q imbalance parameter estimation by evaluating the statistics of the received OFDM symbols. In practice, the statistics required for the parameter estimation have to be approximated by a sufficiently large number of observed OFDM symbols. Hence, the number of transmitted OFDM symbols required for the parameter estimation is usually increased compared to the reference-based approaches. This drawback is a consequence of the deliberate waiver of any presumed reference symbols. On the other hand, pilots and/or preambles are fundamental building blocks in many practical OFDM wireless LAN systems. The existence of reference symbols is essential for the channel estimation and the synchronization in time and frequency [6]. The accuracy of the blind I/Q imbalance parameter estimation in [8] can be significantly increased, if pilot symbols with ”good” properties are available [9]. Therefore, the goal of this paper is to combine the benefits of the flexible, low-complexity parameter estimation scheme in [8] with the potential of a more accurate estimation due to a proper design of the preamble. The outline of this paper is as follows: In section II an equivalent OFDM baseband model is presented, which models the effects of a noisy frequency-selective fading channel, I/Q imbalance and phase errors, such as carrier frequency offset. Based on the original I/Q imbalance parameter estimation scheme in [8], general constraints for the design of a ”good” preamble are derived in section III. 3 exemplary preambles based on the IEEE 802.11a WLAN system are designed and evaluated in section IV, followed by the conclusions in section V. II. S YSTEM M ODEL The purpose of any receiver architecture is to convert a radio frequency (RF) signal down to the base band (BB). The real RF signal at the antenna of the receiver can be written as r(t) = y(t)e+j2πfC t + y ∗ (t)e−j2πfC t ,

(1)

where y(t) denotes the complex base band equivalent of the frequency band of interest, fC denotes its carrier fre-

zI (t)

Antenna and RF Front-End

LO

x(t)

LPF zQ (t)

g

cos[2πfC t + c(t)]

LPF

Xm (n)

90◦ + φ

z(t)

y˜(t)

y(t)

Phase Errors (e.g. CFO) c(t), Cm (n)

Wireless Channel h(t), Hm (n)

Y˜m (n)

Ym (n)

I/Q Imbalance K1 , K2 Zm (n)

Fig. 2. Distortion of the BB signal due to both the wireless channel and direct conversion with phase errors and I/Q imbalance

Fig. 1. Direct conversion receiver with phase errors and I/Q imbalance

where quency and (·)∗ denotes (complex) conjugation. The impact of the wireless communication channel can be modelled by the well know equivalent based band model: y(t) = x(t) ∗ h(t) + w(t),

(2)

where x(t) denotes the transmitted BB signal, ∗ denotes convolution, and h(t),w(t) denote the (complex valued) baseband equivalents of the channel impulse response and the channel noise, respectively. The fundamental principle of the so called directconversion receiver architecture is to perform the conversion from the RF down to the BB using complex (I/Q) signal processing [2]. In two parallel branches, the RF signal is multiplied by two orthogonal phases of a local oscillator (LO) signal. The frequency of the LO fLO is chosen equal to the carrier frequency of the desired RF signal. Ideally, the complex LO signal has the time function zLO (t) = e−j2πfC t , which corresponds to the desired down-conversion by fC . Unfortunately, in practice neither a perfect match with the desired frequency nor a perfect analog I/Q mixing is achievable. 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 (Fig. 1). These imperfections can be modelled by a complex LO signal with the time function z˜LO (t) = cos[2πfC t + c(t)] − jg sin[2πfC t + c(t) + ϕ], where g denotes the gain imbalance and ϕ denotes the phase imbalance between the I- and the Q-branch. The time variant function c(t) models deviations from the ideal phase, known as phase noise. The most dominant phase error in practice is the so called carrier frequency offset (CFO), which can be described by c(t) = c0 + 2πΔf t. In this case, c0 denotes the initial phase offset and Δf = fLO − fC denotes the carrier frequency offset. Based on g and ϕ, the complex valued I/Q imbalance parameters 1 + ge−jϕ , K1 = 2

1 − ge+jϕ K2 = 2

(3)

are defined, in order to rewrite the time function of the complex LO with I/Q imbalance and phase error as: z˜LO (t) = K1 e−j[2πfC t+c(t)] + K2 e+j[2πfC t+c(t)] . (4) By merging (1) and (4), the impact of an I/Q downconversion with a non-ideal complex LO can be described by the following equivalent BB model: ∗

z(t) = LP {r(t)˜ zLO (t)} = K1 y˜(t) + K2 y˜ (t),

(5)

y˜(t) = y(t)e−jc(t)

(6)

and LP {·} denotes low pass filtering. Summarizing, the impact of the channel, the phase error and the I/Q imbalance in the receiver can be modelled by a set of subsequent and linear operations, as indicated by equations (2),(6), and (5) (see also Figure 2). This general model can be applied to the transmission of OFDM signals. Exemplarily for the transmitted baseband signal, the frequency domain symbols Xm (n) can be computed from the discrete time samples x(l) of the continuous time waveform x(t) by applying the Discrete Fourier Transform (DFT) Xm (n) = DF T {x(l)} =

1

LDF T −1 

LDF T

−j L2πlm

x(l)e

DF T

,

l=0

were LDF T denotes the DFT length, which equals the total number of subcarriers. Xm (n) denotes the transmitted symbol at the mth subcarrier of the nth OFDM symbol. Because the DFT is a linear operation, the time domain BB model of subsequent impairments is transformed to a set of subsequent impairments in the frequency domain. For the lack of space, this property will only be briefly discussed, for a more detailed analysis the reader is referred to the literature. Assuming the duration of the channels impulse response does not exceed the length of the cyclic prefix, the convolution in the time domain (2) translates to a multiplication with the channel coefficients in the frequency domain [6]: Ym (n) = Xm (n)Hm (n) + Wm (n).

(7)

The multiplication with samples of the phase error (6) in the time domain translates to a cyclic convolution in the frequency domain, resulting in inter-carrier interference (ICI) [6]: Y˜m (n) = Ym (n) ∗ Cm (n). (8) Finally, the self-interference of the time domain BB signal (5) translates to a mutual interference between pairs of symmetric subcarriers [8]: ∗ Zm (n) = K1 Y˜m (n) + K2 Y˜−m (n).

(9)

Each set of frequency domain coefficients is determined by the corresponding set of time domain samples: Hm (n) = DF T {h(l)} ,

(10)

Wm (n) = DF T {w(l)} ,

(11)

−jc(l)

Cm (n) = DF T {e

}.

(12)

III. P REAMBLE DESIGN A proper digital signal processing is required in order to reconstruct the transmitted symbols Xm (n) from the received (distorted) symbols Zm (n). This reconstruction can be done by applying the inverse operations of the subsequent error effects (as shown in Fig. 2) in reverse order. In this paper we focus on the first step of the correction: the compensation of the I/Q imbalance. The subsequent compensation of the CFO (or more generally: phase errors) and the channel equalization is beyond the scope of this paper, see for example [6]. Prior to the compensation, the parameters of each distortion have to be estimated. The use of predetermined OFDM symbols, such as a preamble, is a very common technique for allowing a proper parameter estimation. By comparing the symbols before and after the distortion, the parameters of the distortion can be estimated. However, any I/Q imbalance parameter estimation based on reference symbols suffers from the fact, that the known transmitted symbols are corrupted (with unknown parameters) prior to the I/Q imbalance, as one can see from Fig. 2. These problems are avoided by applying blind estimation techniques. The term blind indicates, that actual values of the transmitted symbols Xm (n) are unknown to the receiver. Instead, the estimation is done based on the received, corrupted symbols Zm (n) only. Therefore, the goal of in this paper is to derive general design constraints of the preamble, such that a robust I/Q imbalance parameter estimation is possible even in the presence of a noisy, frequency-selective fading channel and phase errors. The estimation is reference-based in the sense, that it is done based on a fixed, predetermined preamble. At the same time, the estimation is blind in the sense, that the reference symbols are unknown to the receiver. A. Blind I/Q imbalance parameter estimation Let us assume, the receiver has collected a set of corrupted symbols Zm (n), Z−m (n) originating from one or multiple pairs of symmetric subcarriers m and −m. Based on this collected data, a two-step parameter estimation is possible [8]. In a first step, the product   m∈M n∈N Zm (n)Z−m (n) ˆ ˆ   K1 K2 = . (13) ∗ 2 m∈M n∈N |Zm (n) + Z−m (n)| is estimated. M denotes the chosen subset of M (positive) subcarrier indices, N denotes the chosen subset of N sample time indices. In a second step, the estimated product ist split into the desired estimates of the I/Q imbalance parameters ˆ − j βˆ ˆ − j βˆ ˆ2 = 1 − α ˆ1 = 1 + α , K , (14) K 2 2 where ˆ 2 }, ˆ 1K βˆ = −2 Im{K  (15) ˆ 2 }. ˆ 1K α ˆ = 1 − βˆ2 − 4 Re{K Re {·} and Im {·} denote the real and the imaginary part, respectively.

B. Design constraints It has been shown in [9], that the accuracy of the parameter estimation, and therefore the performance of the I/Q imbalance compensation, is very closely related to the term   Y˜m (n)Y˜−m (n). (16) = m∈M n∈N

For  → 0, the estimation approaches the desired value: ˆ 1K ˆ 2 → K1 K2 . Originally, it has been suggested K to perform the parameter estimation based on uncorrelated subcarriers, such as data subcarriers [8]. With E{Y˜m (n)Y˜−m (n)} = 0 at the evaluated pairs of subcarriers,  approaches zero for large values of M and/or N . Equivalently, if the parameter estimation is done based on a preamble, the symbols Xm (n) of the preamble should be designed such that (16) is small for arbitrary distortions due to the channel and due to phase errors. In order to derive reasonable constraints for the preamble design, we consider a noise-free frequency-selective fading channel and no phase errors. In this case, (16) results in:   Xm (n)X−m (n)Hm (n)H−m (n). (17) = m∈M n∈N

Obviously,  = 0 can be ensured for arbitrary realizations of the channel coefficients Hm (n), if the following condition holds: Xm (n)X−m (n) = 0

∀ m ∈ M, n ∈ N.

(18)

In other words, a perfect I/Q imbalance parameter estimation can be ensured, if at least one symbol Xm (n) and/or X−m (n) is zero for each m, n. Therefore, (18) is referred to as the Zero Subcarrier design constraint. While the introduction of zero subcarriers is extremely advantageous for the I/Q imbalance parameter estimation, zero subcarriers should be avoided from the channel estimation point of view. Therefore, alternative design constraints are required, which avoid the introduction of zero subcarriers. Most practical OFDM systems are dimensioned such that the channel coefficients Hm (n) change very slowly in time (index n) and frequency (index m). Using this property, the set M of subcarrier indices is split into P subsets of mutually exclusive subsets M1 . . . MP and the set N of time indices is split into Q subsets of mutually exclusive subsets N1 . . . NQ , such that the channel coefficients are approximately constant within each subset: + − Hm (n) ≈ Hp,q , H−m (n) ≈ Hp,q

∀ m ∈ Mp , n ∈ Nq . (19)

By using (17), it can be easily shown, that  ≈ 0 for arbi+ − trary channel coefficients Hp,q , Hp,q , if   Xm (n)X−m (n) = 0 ∀ Mp , Nq (20) m∈Mp n∈Nq

holds for each subset. In contrast to (18), (20) can be fulfilled without the introduction of zero subcarriers. Therefore, (20) is referred to as the Complementary Subsets design constraint.

−15 Preamble A: Block of Zeros −20

−1.4

−25 −30

−20

−10

0

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20

30

Preamble B: Interlaced Zeros Xm

1.4 −1.4 −30

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−10

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Xm

Preamble C: Complementary Subsets

Preamble A @ AWGN Preamble A @ ETSI−A Preamble A @ ETSI−B Preamble B @ AWGN Preamble B @ ETSI−A Preamble B @ ETSI−B Preamble C @ AWGN Preamble C @ ETSI−A Preamble C @ ETSI−B

−30 Mean GC [dB]

Xm

1.4

−35 −40 −45 −50 −55

1 −60

−1 −30

−20

−10

0 10 Subcarrier index m

20

30

−65 −10

0

10

20 30 Channel SNR [dB]

40

50

60

Fig. 3. 3 exemplary preambles, which were generated under different design constraints (see text)

Fig. 4. Performance of the parameter estimation under different channel conditions (g=1.05, φ=5◦ )

IV. D ESIGN EXAMPLES The general design constraints derived in the previous section shall be be illustrated by design examples. The exemplary preambles, which are generated and evaluated in this section, are based on the parameters of the IEEE 802.11a WLAN standard. Each preamble consists of only one OFDM symbol (N =1), which is composed of 64 subcarriers, including 52 used subcarriers and 12 zero subcarriers. The preamble design aims on a suitable definition of the symbols of the 52 used subcarriers. Our analysis will be based on the following 3 design examples (see Fig. 3): 1) Preamble A: is the most simple approach based on the zero subcarrier design constraint by setting all negative subcarriers to zero. Obviously, the drawback of such a design is, that no channel estimation can be done for the negative subcarriers. 2) Preamble B: follows the zero subcarrier design constraint as well. However, in contrast to preamble A, the zero subcarriers are equally distributed over both negative and positive subcarrier indices. The design is chosen such that the positive subcarrier is set to zero for even subcarrier indices, while the negative subcarrier is set to zero for odd subcarrier indices. Hence, the channel estimation can be performed on equally distributed subcarrier indices over the whole band of used subcarriers. 3) Preamble C: avoids the need for zero subcarriers by following the design constraint of complementary subsets. The set of M =26 subcarrier indices is split into P =13 subsets. The small subset size (2 pairs of subcarriers in each subset) ensures high robustness against strongly frequency-selective fading channels. At the same time, preamble C enables a channel estimation for each individual subcarrier. For simplicity, we used BPSK modulated symbols in all non-zero subcarriers. The introduction of additional zero subcarriers in the preambles A and B is compen√ sated by scaling the BPSK symbols by a factor of 2. This scaling ensures, that the resulting OFDM symbol power is the same for all 3 preambles. The actual values of the BPSK symbols are chosen such that the result-

ing continuous-time Peak-to-Average-Power-Ratio1 (CTPAPR) is less than 6 dB. For reference, the CT-PAPR of the regular IEEE 802.11a preamble is 4.13 dB. A. Performance evaluation In order to evaluate the performance of the I/Q imbalance parameter estimation, a reasonable quality measure is required. Following the analysis framework presented in [9], we define the normalized image power gain 2  K K ∗ ˆ   2 ˆ 1∗ − K1 K 2 GC =   , ˆ ∗ − K2 ∗ K ˆ2   K1 K 1

(21)

which measures the residual error after (potentially imperfect) I/Q imbalance compensation. GC is the inverse of what is sometimes referred to as image rejection ratio. B. Simulation results We start our investigations with the case of no phase error. Figure 4 shows the performance of the parameter estimation under different channel conditions. In the presence of an AWGN channel, all preambles show the same performance. Clearly, the mean estimation error (expressed by GC ) reaches arbitrary low values as the Signal-to-NoiseRatio (SNR) of the channel raises. For low SNR’s the mean estimation error is upper bounded. In this case, each subcarrier is dominated by the additive channel noise. Because the additive noise in each subcarrier can be treated as uncorrelated, this scenario corresponds to a parameter estimation based on uncorrelated subcarriers, as originally proposed in [8]. According to the theoretical analysis in [9], the resulting expected GC is -20.2 dB, which matches exactly with in simulation results shown in Fig. 4. 1 The CT-PAPR denotes the PAPR of the OFDM symbol in the analog domain. In our simulations the CT-PAPR has been approximated by an 8 times oversampling of the (discrete) time-domain representation of the evaluated preambles.

0

−20

10

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Symbol Error Rate

Mean GC [dB]

−40

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−2

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−3

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−4

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With I/Q imbalance (no compensation)

−1

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Preamble A Preamble B Preamble C −1

−0.5

0 Normalized CFO Δf / fsc

0.5

1

10

Preamble A @ cfo=0 Preamble A @ cfo=5% Preamble A @ cfo=10% Preamble B @ cfo=0 Preamble B @ cfo=5% Preamble B @ cfo=10% Preamble C @ cfo=0 Preamble C @ cfo=5% Preamble C @ cfo=10% 20

30

No I/Q imbalance

40

50

60

SNR [dB]

Fig. 5. Performance of the parameter estimation as a function of the carrier frequency offset (ETSI-A channel @ 80dB SNR, g=1.05, φ=5◦ )

Fig. 6. Error rates of 64-QAM symbols with and without digital compensation of the I/Q imbalance (ETSI-A channel, g=1.05, φ=5◦ )

Next, a frequency-selective fading channel is considered by using the ETSI channel models A and B [1]. Despite a small degradation compared to the AWGN channel, the general trend of a linearly decreasing estimation error with a raising SNR is maintained. The only exception is preamble C, which exhibits an error floor for high SNR’s due to the approximation in (19). The coherency bandwidth of channel ETSI-B is smaller than the coherency bandwidth of channel ETSI-A, resulting in a higher error floor. The performance of the parameter estimation is strongly affected by phase errors. Figure 5 shows the impact of a carrier frequency offset (CFO) for an ETSI-A channel. The absolute CFO Δf is normalized by the subcarrier spacing fSC . As for a frequency-selective channel, the zero-based preambles A and B show the best robustness against a non-zero CFO. The robustness can be increased by ”grouping” as many zeros as possible (here: preamble A). This property can be explained by interpreting the CFO as inter-carrier interference (ICI). For typical values of the CFO, most of the ICI is caused by neighboring subcarriers. Hence, a desirable zero subcarrier is less affected by the CFO (i.e. its value remains close to zero) if most of its neighboring subcarriers are zero subcarriers as well. Finally, the impact on the resulting symbol error rate (SER) is considered. The frame structure is based on the IEEE 802.11a WLAN standard. Each transmitted frame is corrupted by the channel, the CFO and the I/Q imbalance. Normalized CFO values of 0, 5%, and 10% (corresponding to 0, 16 kHz, and 31 kHz absolute CFO) were used. For each frame, the I/Q imbalance parameters are estimated based on only one OFDM symbol (preamble A, B, or C, respectively). Using this estimate, a time-domain I/Q imbalance compensation is applied to the subsequent payload OFDM symbols, followed by a CFO correction and a channel equalization. The CFO correction and the channel equalization are assumed to be perfect, such that the I/Q imbalance is the only error source which degrades the system performance. Figure 6 shows that the I/Q imbalance can be perfectly compensated based on each of the 3 preambles, if no CFO

is present. The performance is slightly degraded due to a non-zero CFO. However, the error rate is still significantly reduced compared to the reference case of no I/Q imbalance compensation. The accuracy of the parameter estimation (and consequently the performance of the compensation) can be further increased by averaging over multiple received preamble symbols in time. V. C ONCLUSIONS General constraints for the preamble design in OFDM systems have been derived, which allow a low-complexity I/Q imbalance parameter estimation. The evaluation of exemplary preambles, which were generated using these constraints, indicate sufficient robustness to both channel distortions and phase errors, such as carrier frequency offset. R EFERENCES [1] ETSI EP BRAN. Channel models for HIPERLAN/2 in different indoor scenarios, March 1998. [2] Behzad Razavi. Design Considerations for Direct-Conversion Receivers. IEEE Trans. Circuits Syst. II, 44(6):428–435, June 1997. [3] Andreas Schuchert, Ralph Hasholzner, and Patrick Antoine. A novel IQ imbalance compensation scheme for the reception of OFDM signals. IEEE Trans. Consumer Electron., 47(3):313–318, August 2001. [4] S´ebastien Simoens, Marc de Courville, Franc¸ois Bourzeix, and Paul de Champs. New I/Q imbalance modeling and compensation in OFDM systems with frequency offset. In Proc. IEEE PIMRC 2002, volume 2, pages 561–566, September 2002. [5] A. Tarighat and A. H. Sayed. On the baseband compensation of IQ imbalances in OFDM systems. In Proc. of the IEEE ICASSP’04, volume 4, pages 1021–1024, May 2004. [6] John Terry and Johu Heiskala. OFDM Wireless LANSs: A Theoretical and Practical Guide. Sams Publishing, 2002. [7] Jan Tubbax, Boris Cˆome, Liesbet Van der Perre, Luc Deneire, St´ephane Donnay, and Marc Engels. Compensation of IQ imbalance in OFDM systems. In Proc. IEEE Intl. Conf. on Communications (ICC 2003), volume 5, pages 3403–3407, May 2003. [8] Marcus Windisch and Gerhard Fettweis. Standard-Independent I/Q Imbalance Compensation in OFDM Direct-Conversion Receivers. In Proc. 9th Intl. OFDM Workshop (InOWo), pages 57–61, Dresden, Germany, 15-16 September 2004. [9] Marcus Windisch and Gerhard Fettweis. On the Performance of Standard-Independent I/Q Imbalance Compensation in OFDM Direct-Conversion Receivers. In Proc. 13th European Signal Processing Conference (EUSIPCO), Antalya, Turkey, 4-8 September 2005.