Adaptive Time Misalignment Compensation in Envelope Tracking Amplifiers Atso Hekkala, Mika Lasanen, and Adrian Kotelba VTT Technical Research Centre of Finland, P.O. Box 1100, FI-90571 Oulu, Finland
[email protected]
Abstract—This paper considers the compensation of time misalignment between RF and envelope signals of envelope tracking (ET) amplifier. In particular, we propose methods to compensate the time misalignment and compare them to a previously presented method. Our simulation results indicate that the proposed methods give very accurate timing estimates. In the presence of optimal predistorter, we achieve with 90 percent probability adjacent channel power (ACP) level of -80 dB. With adaptive predistorters, we cannot see any degradation of ACP comparing to case of ideal timing. On the other hand, the proposed methods give roughly 10 dB better ACP than the previously presented method. Keywords-envelope tracking amplifier; predistortion; time misalignment compensation
I. INTRODUCTION In wireless communication, high power amplifier (HPA) is a critical component in the transmitters due to traditional tradeoff between its linearity and efficiency. The problem is more demanding, when the need of high data rates requires exploiting spectrally efficient linear modulation techniques, such as wideband code division multiple access (WCDMA) or orthogonal frequency division multiplexing (OFDM), where both the phase and the amplitude of the signal carry the information. An envelope tracking (ET) amplifier has been seen one of the promising power efficient structure [1]. However, the ET structure has some disadvantages, e.g. sensitivity to time misalignment between RF and envelope signals and nonlinearity due to dynamic supply voltage [1], [2]. A predistortion has been found as a suitable candidate for linearizing the HPA [3]. The time misalignment compensation in ET structure has been studied in [2]. The authors propose to use crosscovariance and phase comparison to estimate the time misalignment. It is assumed as a one-shot method and in addition, we find out that we cannot apply it because we have a different system model. In this paper, we analyze and further develop the time misalignment compensation methods presented originally in [2]. Furthermore, we demonstrate in detailed manner the performances of the time misalignment compensation methods for linear filters and ET amplifier with optimal as well as adaptive predistorters. This work is an extension of [4] where we studied the compensation of linear and nonlinear distortions of ET amplifier using a more general system model than in [2].
The rest of the paper is organized as follows. In Section II, we introduce the system model focusing on the most important blocks. Then, we discuss time misalignment compensation in Section III. In Section IV, we show some simulation results. Finally, Section V includes the conclusion of the paper. II.
SYSTEM DESCRIPTION
A. Simplified system model We present the simplified block diagram of the system model in Fig. 1. The system model is based on WCDMA base station transmitter. The actual transmitter consists of a signal source, linear filters, and high power amplifier. In addition to these, we introduce compensation blocks to compensate the time misalignments and distortions introduced by the analog parts of the transmitter. The adaptive algorithms calculate the parameters of the compensation blocks by comparing a HPA output signal z and reference signal r. Reference model and fixed delay block are delay blocks with delay IJ. To model unrecognized delays in envelope branch, we introduce an unknown delay block with delay range between zero and one. The tunable delay filters are allpass fractional-delay filters implemented as IIR gathering structures with Thiran approximation [5]. The order of the filter is 20 and approximation order is 3. A complex baseband model is utilized. We measure analog parts of the transmitter and model them into digital domain using sampling frequency of 76.8 MHz. Dual carrier WCDMA signal is used as an input signal u, i.e. any special training signal is not used. We follow closely the 3rd Generation Partnership Project (3GPP) specifications [6] when constructing the signal model. Crest factor reduction is applied to the generated signal. We assume that subband sampling is used in feedback branch, limiting the useful bandwidth of the forward branch to 38.4 MHz. In our system model, target gain for the predistorted amplifier is 13 dB. Time misalignments and adjacent channel powers are used as performance measures. In the next subsections, we introduce briefly linear filters, ET amplifier, and predistorters. For more information of the system model, please refer to [4]. B. Linear filters We characterize the linear filters using measurements of real analog RF filters. We utilize direct-design method in frequency domain [7] to model them as a non-minimum phase infinite-impulse response (IIR) filters.
This work has been partially performed in the framework of the project CHESS, which is partly funded by Tekes (decision number 40453/05).
978-1-4244-2204-3/08/$25.00 ©2008 IEEE
Tunable delay tENV
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Figure 1. Block diagram of the simplified system model with adaptive control structures.
C. Envelope tracking amplifier From measurements of a real amplifier, we obtain amplitude-to-amplitude (AM/AM) and amplitude-to-phase (AM/PM) characteristics for a number of drain supply voltages Vdd. We use nonlinear functions to model AM/AM and AM/PM characteristics for the ET amplifier [4]. In the functions, the parameter values depend on Vdd. The AM/AM and AM/PM characteristics for different Vdd values are shown in Figs. 2 and 3. Measurement data are shown as well. Using the model [4], we obtain nominal curves for AM/AM and AM/PM characteristics with dynamically varying Vdd, see dashed lines in Figs. 2 and 3. The nominal curves are achieved when the signals in RF and envelope branches are time aligned. To keep the amplifier operational, we have to set a minimum possible drain supply voltage to a constant, i.e. Vmin = 3 V. D. Predistortion A linear predistorter is introduced to compensate linear filters. It is implemented using 63 taps finite impulse response (FIR) filter. An optimal solution is obtained by direct-design method in frequency domain [7]. For adaptive solution, the constrained LMS-algorithm is used to obtain the filter coefficient of the predistorter. An adaptive inverse control 15
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Figure 2. ET amplifier gain as the function of input signal power level [4].
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Figure 3. Phase shift introduced by the ET amplifier as the function of input signal power level [4].
technique [8] is utilized for predistorter training. Due to delays of the linear predistorter and linear filters, a fixed delay block is introduced in the envelope branch to time align signals in RF and envelope branches. In order to keep the response of the linear predistorter constant for frequencies higher than 19.2 MHz, constraints are introduced in those frequencies. Optimal solution for nonlinear predistorter is obtained by look-up-table (LUT) based predistorter. For adaptive compensation of the ET amplifier model, a polynomial based predistorter structure is utilized. The polynomial predistorter output v can be given by v ( n) =
K
¦ a u ( n) u ( n) k
k −1
(1)
k =1
where u is the predistorter input, K is the predistorter order, and ak is the polynomial coefficient. In this work, the adaptive identification of the nonlinear predistorter is done using indirect learning architecture [9], which is the form of nonlinear adaptive inverse control. To obtain nonlinear predistorter coefficients ak in (1), we utilize recursive least squares (RLS) algorithm developed in [9]. For more discussions on the linear and nonlinear predistorters used in this work, see [4] and [10]. III. TIME MISALIGNMENT COMPENSATION To compensate the time misalignment, we need to know the both of the RF and envelope delays (TRF and TENV) or the difference of them. We assume that we can use the phases and amplitudes separately for the compensation. We note that the envelope signal branch does not ideally affect the phase of the output signal, i.e. the change of the output signal phase is mainly affected by the input signal phase. This can be explained as follows. The change of the output signal due to change of envelope signal is small because the envelope signal and its delayed version is highly correlated. In particular, the correlation coefficient is greater than 0.9 for delays in range of ± 1 sample. Refer also Fig. 3, that the phase will be close the nominal curve all the time if the linear predistorter is converged.
TL = TRF + T FB .
(2)
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-1 Phase value (rad)
The amplitude is affected by both the envelope and RF delays. An exception is a case in which the envelope path is kept constant, i.e. when the amplitudes are small; see Fig. 2. Now before going to more detailed discussion, we conclude that we may be able to use the signal phases and small signal amplitudes to estimate RF + feedback delays (TRF + TFB). In addition, the amplitude of the signal may be utilized to get an estimate for a combined effect of all delays, i.e. TENV, TRF, and TFB. We notice by measurements that we are able to see phase difference between ET amplifier input and output signals. Assuming the phase difference is coming only due to delays in RF and feedback branches, we get delay estimate TL for visible delay of phases method [2]
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Figure 4. Principle of the visible delay of phases calculation.
We use the similar right triangle geometry to calculate TL. After some manipulations we find the following equation to calculate the visible delay of phases TL =
φi (l ) − φo (l ) φi (l ) − φi (l − 1)
(3)
where l is timing index going through all vector elements, and
φi and φo are phases of the ideal and FB signal vectors. Fig. 4
clarifies the calculation of TL. In order to exploit the similar right triangles we need to select the indices l in (3) with the following criteria l
=
{i; Δφ
i
Δφ o < 1.5, 0.9 < Δφo / Δφi < 1.1, u > 0.125}
TC = (TRF + TENV ) / 2 + TFB .
N
°
i =1
¯
1
N
½°°
(5)
j =1
¿¯
1
Ruz (t m + 1) − Ruz (t m − 1) 1 − tm 2 Ruz (t m + 1) − 2 Ruz (t m ) + Ruz (t m − 1)
N
½°
j =1
¿
(7)
where tm is the time lag which the cross-covariance function Ruz(t) achieves its maximum. With the delays TL and TC, it is possible to approximate delay difference (8)
(4)
We exploit cross-covariance of amplitudes to get an estimate of TC. A cross-covariance of amplitudes function between input and output is [2] 1 N
TC =
TENV − TRF = 2(TC − TL ) .
> 0.2, φ o − φi < 1, Δφi < 1.5, ...
where φi and φo are the phases of the ideal and FB signals in radians, Δφi and Δφo model the derivates of the phases of the ideal and FB signals, and u is ideal signal with maximum amplitude normalized to one. The conditions try to select the phases, which are quite similar in terms of the actual values and derivatives. In addition, it is essential to keep the absolute value of derivates within a certain range when using the similar right triangle geometry. Moreover, the small amplitudes are restricted because the noise and the uncertainty of the predistorter have a large impact to the phase when the amplitudes are small. Finally, averaging of TL in (3) is performed over the indices selected according to (4). We notice that using the signal amplitudes we can obtain a combined estimate of all delays TC [2]
Ruz (t ) =
To estimate subsample accuracy for the maximum correlation point, we utilize quadratic approximation, which is defined as [11]
¦ ®° u(i ) − N ¦ u( j ) ¾°®° z(i − t ) − N ¦ z( j ) ¾° . (6)
Equation (8) is basically same as (36) in [2]. This approach could ideally be used in one-shot way. However, we assume that accurate estimation in one-shot may not be possible if preliminary errors are large. In addition, both TC and TL require their own estimations meaning that both estimation errors cumulate to the end result. Another approach could be to first adapt RF + FB delay near to zero and then start to track the envelope delay. In this case, an adaptive linear predistorter could be employed to obtain RF + FB delay synchronization. Originally, the linear predistorter has been developed to compensate linear filters. In addition to that, it can handle all the delays in RF and FB branches by adjusting the group delay of the branches to a certain value, i.e. the group delay of the reference model. Together with the adaptive linear predistorter and e.g. crosscovariance of amplitudes method, we are able to compensate time mismatches between RF and envelope signals. IV.
RESULTS
A. Simulations with optimal nonlinear predistorter In the simulations with the optimal nonlinear predistorter, we ignore the linear predistorter and linear filters to simplify the simulation model. We use about 8000 signal samples as an input for each estimation round. The number of simulation cases or events is 200 per plot.
In Fig. 5, we show the histograms of the timing estimate errors using visible delay of phases (VDP) and crosscovariance of amplitudes (CCA) methods. RF and envelope delays are random variables with uniform distribution between zero and one. FB delay is set to zero. Fig. 5(a) shows the RF + FB delay estimation errors of the VDP method. Almost all of the estimations give better than 0.015 samples accuracy. In Fig. 5(b), we present timing estimate errors of CCA method. The accuracy of the CCA method is a bit worse than the VDP method. Please note that the CCA method gives a combined delay estimates, just according to (5). It is also worth to note that the CCA method works well also without predistorter, whereas the VDP method does not. The VDP method suffers the phase distortion of the uncompensated amplifier. In addition, the phase noise can degrade, or even collapse, the performance of the VDP method. Accurate oscillators and/or a phase noise compensation is clearly needed in the real implementation. Considering Figs. 5(c) – 5(f), we can also see the performance of joint delay estimation of CCA and VDP methods. In the joint estimation, we can compensate the time mismatch between RF and envelope branches. We update delay compensation blocks in RF and envelope branches after each iteration according to (8). In addition to the delay difference compensation, using (2), we force the RF + FB delay to zero. Within three iterations, we achieve very accurate timing estimates. However, with more iteration even better performance is attained. We show also cumulative density functions (CDFs) for adjacent channel powers (ACPs) using the joint delay estimation in Fig. 6. After three iterations, we achieve with 90 percent probability less than 10 dB loss of ACP comparing to the input signal ACP, i.e. -88 dB. Also within two iterations, we get quite good ACPs: almost all the time below – 70 dB. Please note that using optimal predistortion with optimal time misalignment compensation, we achieve also -88 dB level of ACP. In the Figs. ACPs are mean values of lower and higher adjacent channel powers in decibels relative to the carrier power. For comparison purposes, we model the time misalignment estimator presented in [2]. That estimator is assumed to be a one-shot method. Therefore, we can compare that estimator to the proposed method after one iteration. In Fig. 6, we can see that the proposed method gives roughly 10 dB better ACPs. The complexities of the both estimators are roughly at the same level. We change FB delay to 0.66 samples and do the same simulations than previously. Comparing Figs. 5 and 7, we can say that the proposed methods work well also with the additional FB delay: the timing estimate accuracies are at the same level as in Fig. 5 or only a little bit worse. This is the expected result from (8). B. Simulations with adaptive predistorters In the simulations with adaptive predistorters, we first utilize calibration or acquisition of linear filters. In order to keep the ET amplifier as linear as possible in the acquisition, the downscaled signal level, i.e. the drain supply voltage remains constant, is used. After the acquisition, in tracking phase, we train the delay estimation, nonlinear and linear predistorters one after another several times. Physical and
temporal orders of the predistortion are discussed more by the authors in [10]. In [4], we have shown that using the system model with adaptive predistorters and ideal time misalignment compensation the best ACP we can achieve is close to -60 dB. We notice that the adaptive linear predistorter takes care of delays in the RF and FB branches and compensates them. Therefore, we can utilize the CCA method to estimate delay in envelope branch. Fig. 8 shows time misalignment and ACP using the CCA method with adaptive predistorters. We see that ACP stays about -60 dB level after five iterations, even though it takes more iterations for envelope delay estimator to converge. The time misalignment is calculated assuming ideal timing of 31 samples. Initial values for RF and envelope delays are 3.45 and 0.9 samples. FB delay is zero or 0.66 samples. We assume that the adaptive linear predistorter causes the little bias of the time misalignment in Fig. 8. Therefore, we introduce an additional delay estimator, i.e. the VDP method, for the RF branch. The role of the VDP method is to estimate the residual RF + FB delay, i.e. the delay after the compensation process of adaptive linear predistorter. In Fig. 9, we can see improved performances of delay estimations and ACPs. V.
CONCLUSION
In this work, we studied the compensation of the time misalignment between RF and envelope signals in envelope tracking amplifier. We proposed two methods to estimate the delays in the RF and envelope branches. We studied the performances of the proposed methods in the presence of optimal as well as adaptive predistorters. The results suggest that the proposed methods give very accurate timing estimates. In the presence of optimal predistorter, the ACP loss comparing to input signal ACP is with 90 percent probability less than 10 dB. With adaptive predistorters, we cannot see any degradation of ACP comparing to the case of ideal timing. ACKNOWLEDGEMENT The authors would like to thank Prof. Aarne Mämmelä and Pertti Järvensivu for valuable discussions and comments during the work. REFERENCES [1]
[2]
[3]
[4]
[5]
B. Sahu, “Integrated, dynamically adaptive supplies for linear RF power amplifiers in portable applications,” Ph.D. dissertation, Dept. Elect. and Comp. Eng., Georgia Inst. Tech., Antlanta, GA, 2004. F. Wang, A. H. Yang, D. F. Kimball, L. E. Larson, and P. M. Asbeck, “Design of wide-bandwidth envelope-tracking power amplifiers for OFDM applications,” IEEE Trans. Commun., vol. 53, pp. 1244-1255, Apr. 2005. W. Woo, “Hybrid digital/RF envelope predistortion linearization for high power amplifiers in wireless communication systems,” Ph.D dissertation, Dept. Elect. and Comp. Eng., Georgia Inst. Tech., Antlanta, GA, 2005. A. Hekkala, A. Kotelba, and M. Lasanen, “Compensation of Linear and Nonlinear Distortions in Envelope Tracking Amplifiers,” IEEE Int. Symp. Personal, Indoor, and Mobile Comm. 2008, to be published. M. Makundi, V. Valimaki, and T. I. Laakso, “Closed-form design of tunable fractional-delay allpass filter structures,” in Proc. IEEE Int. Symp. Circuits Syst., vol. 4, pp. 434–437, Sydney, Australia, May 6-9, 2001.
[6]
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Figure 7. Timing estimate error histograms using CCA and VDP methods after one (a) and (b), two (c) and (d), and three iterations (e) and (f). FB delay is 0.66 samples.
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Figure 5. Timing estimate error histograms using CCA and VDP methods after one (a) and (b), two (c) and (d), and three iterations (e) and (f). FB delay is 0 samples.
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Base Station (BS) Conformance Testing (FDD) (Release 5), 3GPP Technical Specification 25.141 version 5.8.0, 2003. [7] J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications. Upper Saddle River, NJ: Prentice Hall, 1996. ch. 8. [8] B. Widrow and E. Walach, Adaptive Inverse Control. Upper Saddle River, NJ: Prentice Hall, 1996. ch. 6. [9] R. Marsalek, “Contribution to the power amplifier linearization using digital baseband adaptive predistortion,” Ph.D. dissertation, Inst. Gaspard Monge, Univ. de Marne la Vallee, France, 2003. [10] M. Lasanen, A. Kotelba, A. Hekkala, P. Järvensivu, and A. Mämmelä, “Adaptive predistortion architecture for nonideal radio transmitter,” in Proc. IEEE Vehic. Tech. Conf. spring, pp. 1256-1260, Singapore, May 11-14, 2008. [11] G. Jacovitti and G. Scarano, “Discrete time techniques for time delay estimation,” IEEE Trans. Signal Process., vol. 41, pp. 525-533, Feb. 1993.
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Figure 6. Empirical CDFs of ACP using CCA and VDP methods after one, two, and three iterations. Results using estimator from [2] is plotted for comparison. FB delay is 0 samples.
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Figure 9. Time misalignments (a) and ACPs (b) using joint estimation of CCA and VDP methods.
