Adaptive Predistortion Architecture for Nonideal ... - Semantic Scholar
Adaptive Predistortion Architecture for Nonideal Radio Transmitter Mika Lasanen, Adrian Kotelba, Atso Hekkala, Pertti Järvensivu, and Aarne Mämmelä VTT Technical Research Centre of Finland, P.O. Box 1100, FI-90571 Oulu, Finland [email protected] Abstract— A nonideal radio transmitter both distorts the desired signal and generates spectral regrowth causing interference to adjacent channels. Predistortion techniques can be used to minimize these effects. The joint effect of different types of nonidealities is not straightforward to analyze and has not yet been studied in detail in the literature. In this paper, we demonstrate how to adaptively predistort a nonlinear high power amplifier and an I/Q modulator, and to compensate the frequency response of a radio frequency filter. The study considers the spatial order of adaptive predistorters and the temporal sequence of adaptation steps. In this way, both the system capacity and the power efficiency are improved. Keywords-DC offset, I/Q imbalance, nonlinear effects
I. INTRODUCTION A cost-efficient radio transmitter may consist of highly nonideal analog components. A high power amplifier is nonlinear. Without a careful system design, its efficient use may cause severe distortion effects. An I/Q modulator may introduce interference in the form of I/Q imbalance and DC offset. In addition, an RF filter has a nonideal frequency response. Transmitter linearization enables efficient use of power amplifiers with nonconstant envelope modulation methods [1]. A modern approach is to use digital predistortion. An adaptive nonlinear predistortion approach used here is called indirect learning architecture [2] or postdistorter training [3]. The former uses the least squares (LS) solution and the latter employs the recursive least squares (RLS) algorithm. Indirect learning architecture for linear systems may be seen equivalent to inverse control architecture [4]. Linear transmitter filters are compensated in addition to the nonlinear power amplifier in [5]. Predistortion of I/Q imbalance and DC offset with a nonlinear amplifier is considered in [6], [7]. I/Q modulator nonidealities alone are compensated with an adaptive leastmean square (LMS) algorithm in [8]. A popular approach is to use specialized training patterns for estimation [7]. Some methods require only a data signal [6] - [8]. Commutability of nonlinear blocks is analyzed in [9]. Extensive theory for adaptive linear systems is presented in [4], [10]. In this paper, we consider digital predistortion of a nonideal radio transmitter in more detail than before. Our aim is to minimize the level of various distortions in the transmitted signal. The system model includes the predistortion of a nonlinear amplifier, and compensation of the frequency response of a linear RF filter, I/Q imbalance, and DC offset.
978-1-4244-1645-5/08/$25.00 ©2008 IEEE
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Randomness to the system is introduced by phase noise and quantization noise. We use a complex baseband signal in adaptation while an envelope detector was used e.g. in [6] - [8]. We see that the presented architecture enables a novel and useful insight to transmitter design. The architecture consists of the spatial order of several adaptive predistorters and temporal sequence of adaptation steps. As an extension, we detail a novel linear frequency response predistortion approach applicable to work with nonlinear predistortion. In addition, we diminish the effect of phase noise to adaptation via a compensation method. We present simulation results to verify the discussion. The rest of the paper is organized as follows. In Section II, we introduce the system model. Then, we discuss predistortion architecture 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 transmitter system model is presented in Fig. 1. We use a dual carrier wideband code division multiple access (WCDMA) signal as a signal source following closely the Third Generation Partnership Project (3GPP) specifications [11]. The complex valued baseband signal is frequency translated upward in an I/Q modulator. A linear RF filter is used to shape the spectrum of the signal. Then, the signal is amplified and transmitted. To compensate the nonidealities we need a feedback (FB) signal from the output of the power amplifier. In the FB loop, we assume that the signal is first translated from RF to an intermediate frequency (IF) of 96 MHz. In this work, we assume that the bandpass IF filter following the RF/IF mixer is ideal and it is omitted in Fig. 1. The analog-to-digital converter (ADC) digitizes the analog signal. We use undersampling with a sampling frequency of Fs = 76.8 MHz, which is the same that we assume in the signal source. As a result, the IF frequency is mapped to a normalized
Figure 1. Simplified system model of the transmitter.
frequency of ¼ as in [12]. However, due to aliasing the undersampling limits the observable bandwidth in the FB loop to 38.4 MHz, which is a half of that available in the transmitter branch. We assume an ideal digital I/Q demodulator to form the complex baseband signal. The ideal demodulator is composed of a Hilbert transformer followed by a multiplication with a complex exponential function. For adaptation, signal blocks are stored in to a memory. We use a block length of 8000 samples. We remove the amplifier gain from the FB signal. The nonideality model of the study is depicted in Fig. 2. First, the I/Q modulator introduces DC offset, I/Q imbalance and phase noise. For the DC offset and I/Q imbalance effects we assume the model [6] and have y(t) = Mx(t) + Ma, where y(t) = [yI(t), yQ(t)]T and x(t) = [xI(t), xQ(t)]T are output and input signals of the model represented with the inphase (I) and quadrature (Q) components, and § α cos(φ / 2) β sin(φ / 2) · §a · ¸¸, a = ¨¨ 1 ¸¸, M = ¨¨ © α sin(φ / 2) β cos(φ / 2) ¹ © a2 ¹
(1)
Į and ȕ are the gains of I and Q components, φ is the phase splitter error between the components, and a1 and a2 are the DC offsets for both the components. In our model, the DC-tosignal power ratio is 0 dB before compensation. The parameter values for (1) are α = 1.005, β = 0.995 and φ = 1º. Using the image attenuation definition from [13], we can calculate for the matrix M and the given parameters that the ratio between the desired and the introduced image signal powers is about 40 dB. We call this relationship the signal-to-image ratio. We model phase noise as a sum of 1/f 2 noise [14] and a random filtered white Gaussian noise [15]. In the I/Q modulator, the average magnitude of the phase change at the sampling rate is about 0.0019º. Linear filters have a nonideal frequency response that can be shaped in certain limits via design parameters. For the RF power amplifier, we assume Saleh’s model [16]. The model introduces both AM/AM and AM/PM distortion where AM refers to amplitude modulation and PM refers to phase modulation. We use such a parameter and input backoff combination that gives adjacent channel powers from about -40 dBc to -50 dBc for the two most adjacent channels (see Fig. 8). In the FB loop, the phase noise is further affecting the signal in the RF/IF mixer. This time more severe phase noise can be tolerated if the adaptation process can cope with it. An average magnitude of the phase change at the sampling rate is about 0.0027º. Finally, quantization noise is modeled by a uniformly distributed white random process. The signal-toquantization noise ratio is 50 dB during data transmission. III.
I/Q phase noise
DC offset Signal source
I/Q imbalance
Linear RF filter
RF power amplifier FB phase noise
Input for adaptation
nq, quantization noise
Figure 2. Nonidealities.
predistorter with the new postdistorter. The DC offset predistorter is the only exception here. It must be updated by a residual new estimate because we take the reference signal for its adaptation before the predistorter. The adaptation is not performed in real time but offline using stored signals. In adaptation, the FB and reference signals are used. We emphasize temporal sequence of adaptation via shading of some of the blocks and the switch in Fig. 3. We use shaded blocks alone in acquisition or calibration phase at first. Acquisition is followed by a tracking phase. This phase uses all the blocks except the identification block. In acquisition, we use a downscaled signal to minimize the nonlinear distortion introduced by the power amplifier. A similar idea to identify separately the linear and nonlinear parts of an amplifier is detailed in [17]. In our model, downscaling corresponds to an increase in the experienced signal-toquantization noise ratio. We use 10 dB downscaling. Acquisition employs a wideband training signal. We use a bandlimited white Gaussian noise signal having a bandwidth of 36.6 MHz. Two signal blocks are sent during acquisition in series. The first one is used for identification of the coarse DC offset and the frequency response of the linear filter. Based on this, the DC offset is coarsely predistorted and the second signal block is transmitted for postdistortion training. In tracking, phase noise compensation is activated. Furthermore, we do not adapt the nonlinear amplifier postdistorter at the same time with other postdistorters. This is emphasized by the switch in Fig. 3. Tracking is started by amplifier postdistorter training. We use, e.g., three WCDMA signal blocks for this. Our goal is that other postdistorters would only see their nonidealities after the convergence of amplifier postdistortion. Then we train linear filter, I/Q imbalance and DC offset postdistorters at the same time and
x1
x2
COMPENSATION ARCHITECTURE
We present a compensation architecture for the discussed system model in Fig. 3. The upper blocks are predistorters each of which compensates its own nonideality. We adapt each predistorter in the corresponding postdistorter training block. After the desired postdistorter has processed the adaptation signal block, the corresponding predistorter is updated with the copy of the postdistorter. The update simply replaces the old
Φpn u1 u2
Figure 3. Compensation architecture.
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using three signal blocks. The FB signal must go through the amplifier postdistorter before the adaptation due to the point the reference signal is taken from. We repeat the adaptation switching e.g. five times. Then we use e.g. ten signal blocks before each switch. We emphasize that other number of signal blocks may be used as this is one possible example. The nonlinear blocks are not in general commutative [9]. Therefore the spatial order of the blocks in the predistorter is the reverse of that in the nonideality model. In this way, each nonideality can be compensated separately. In a similar manner, each trained postdistorter should ideally see only the nonideality it tries to compensate. Hence, the optimal spatial order of postdistorters is identical with that of predistorters. In Fig. 3 we have one exception for this. The order of I/Q imbalance and residual DC offset postdistorters should ideally be different. However, we see that the residual DC offset must be minimized before I/Q imbalance postdistorter training. The reference signals for adaptation are taken after the corresponding predistorters. As discussed above, the DC offset postdistorter is again an exception for this. We detail each adaptation block more in detail below. A. DC Offset Estimation and Linear Filter Identification In our model, the DC offset is at a high level. Its effect must be minimized to avoid nonlinear distortion in the acquisition phase. While estimating the DC offset, we also identify the linear filter to define constraints for linear filter postdistortion training. We use an adaptive linear finite impulse response (FIR) system identification procedure extended with an averaging filter to estimate the DC offset which can be considered as a plant drift [4, p. 283]. The model to be identified is K −1
K −1
K −1
k =0
k =0
k =0
y (i ) = ¦ hk [x1 (i − k ) + d ] = ¦ hk x1 (i − k ) + d ¦ hk
(2)
where hk are the coefficients of the FIR filter system model, x1 is the input signal before activating any predistorters, i is the time index of the discrete signal, d is the DC offset, and K is the number of coefficients in the FIR model. We use K = 31. We estimate the coefficients hk via the LMS algorithm as depicted in Fig. 4. In addition, we use a conventional recursive averaging filter v(i+1) = v(i) + µe(i), where µ is the adaptation step size and e is an error signal. Using v as the converged v(i), we obtain a coarse DC offset predistorter
(
K −1
d c = −v ¦k =0 hk
)
−1
(3)
where the sum of the estimated coefficients hk from (2) approximates the amplitude gain of the filter at the zero frequency and thus amplification of the DC offset as seen at the x1
output of the filter. To coarsely compensate the DC offset, the estimate (3) is added to the input signal of the analog transmitter as shown in Fig. 3. Experienced phase noise forces us to use larger averaging filter and LMS adaptation step sizes than otherwise desirable. B. Linear Filter and Residual DC Offset Postdistortion We present the acquisition of the linear filter and the residual DC offset postdistorters in Fig. 5. The I/Q imbalance postdistorter is active only during tracking. Coarse DC offset is compensated before this stage. Since the bandwidth of the FB loop is limited to 38.4 MHz, a constrained adaptation is introduced to control the frequency response of the predistorter for frequencies |F| > 19.2 MHz. We use a modified Frost constrained LMS approach [18]
(
(4)
where w[i] is the column vector with N filter coefficients at time i, P = I – CH(CCH)-1C is the N×N projection matrix, µ is the step-size parameter, e[i] is the error signal, u2[i] (see Fig. 3) is the FB signal vector consisting of N samples, the asterisk * refers to the complex conjugate, I is the identity matrix, H refers to the Hermitian transpose, and G = CH(CCH)-1g. When the constrained adaptive filtering is applied to adaptive predistortion, we need to formulate the constraint matrix C and the gain vector g. As the constraint matrix, we get C = [cmn ]M ×N = [exp(− j 2πf m n)]M ×N
(5)
where n = 0, …, N - 1 and fm = Fm /Fs form a set of equally spaced normalized frequencies with 18.3 MHz < |Fm| < 35 MHz. Constraints are indexed with m = 0, …, M - 1. Then we define the gain vector ª exp(− j 2πf τ ) m g = [g m ]M ×1 = « K −1 − j 2πk sgn( f )f «¬ ¦ k =0 hk e m
e
º » »¼ M ×1
(6)
where fe = ±18.3MHz/Fs are the normalized corner frequencies of the acquisition training signal, and τ is the desired group delay of the system. We use positive fe when fm is positive and negative fe otherwise. In the adaptive postdistorter FIR, we use N = 63 and M = 30. We force the group delay τ to 32 sample periods. In this way, the predistorter also provides delay compensation between the baseband and FB signals for further adaptation. For the residual DC offset postdistorter, we use a similar I/Q imbalance compensator
FIR filter
FIR filter
u2
dr
x1
v LMS
)
w[i + 1] = P w[i ] + µe[i ]u *2 [i ] + G
1
e
-
-
LMS
1 LMS
u1
Figure 5. Training of linear filter, residual DC offset and I/Q imbalance postdistorters.
Figure 4. Adaptive identification structure with coarse DC offset estimator.
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recursive averaging filter as above in the system identification. In this case, the residual DC offset postdistorter result dr is introduced in predistortion without further manipulation. The DC offset predistorter uses the sum of (3) and dr. Due to phase noise, we use large adaptation step sizes during acquisition. During tracking we change to smaller values because phase noise compensation allows longer averaging periods. C. Phase Noise Compensation Before any adaptation from data transmission, phase noise is compensated. For this we use the phase estimate §
n+L / 2
· ∗ x 2 (i )u1 (i ) ¸ © i =n− L / 2 ¹
φ pn (n) = arg¨
¦
(7)
where we use an averaging interval of L = 513 and signals x2 and u1 are shown in Fig. 3. The approach weights large signal amplitudes more than small signal amplitudes. This gives more accurate results than considering phases alone because quantization noise affects more severely small amplitude signals. The idea is similar to maximum ratio combining. Phase noise compensation is not employed during calibration. D. Nonlinear Amplifier Postdistortion After the acquisition, we start to use the WCDMA signal and first train the nonlinear postdistorter using the RLS algorithm [3]. We use the approach slightly differently as we do not update the predistorter in real-time for each input sample. We use odd polynomial orders 1, 3, and 5. In the beginning, we use a small RLS forgetting factor for the first adaptation signal block to have faster convergence. Then we continue with a large value to introduce more averaging. We support the averaging by using the postdistorter and the recursion matrix obtained from the previous signal block. E. I/Q Imbalance Postdistortion The linear filter and DC offset postdistorters are adapted as during calibration with one exception. Before these operations, we put the FB signal through the nonlinear postdistorter (see Fig. 3). We now include also I/Q imbalance postdistorter training as in Fig. 5. The I/Q imbalance postdistorter Mc training is performed via four one-tap coefficients adapted with the LMS algorithm; i.e. one coefficient per each element of M. A background for a similar type of adaptive systems is presented in [10]. In addition, the I/Q imbalance postdistorter uses a different error signal than the linear filter and DC offset postdistorters. With this approach, we try to avoid stability problems that could originate from having multiple loops tracking each other. We assume that the dedicated error signal for the linear filter and DC offset postdistorters can be justified because M and Mc are approximately identity matrices and hence the input and output signals of the I/Q imbalance compensator are approximately equal. Finally, we justify using Mc directly in predistortion because ideally McM = I = MMc. IV. RESULTS Simulation results in Figs. 6 - 10 demonstrate that the nonidealities are considerably suppressed with the presented architecture. Fig. 6 shows magnitudes of linear frequency responses of the compensated RF filter and the predistorter
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obtained in acquisition. We can see a low variation both at the signal and higher frequencies. We show results for phase noise estimation after convergence of other adaptive algorithms in Fig. 7. We see that the approach can diminish the variation of several degrees to some tenths of degrees. In this way, we can use averaging in adaptation more efficiently. Fig. 8 shows power spectral densities for the signal source and the high power amplifier output signal without nonlinear predistortion. We depict the performance of nonlinear predistortion in Fig. 9. We can observe that adjacent channel powers are reduced from about -39 and -50 dBc given in Fig. 8 to less than about -70 and -80 dBc. Fig. 10 shows results for the DC offset and I/Q imbalance predistorters. Two first acquisition steps reduce the DC-to-signal power ratio considerably from 0 dB to about -70 dB. Tracking with data signal blocks gives further improvement. The signal-to-image ratio is calculated in the same way for the matrix MMc as for M in Section II. We can observe that the signal-to-image ratio improves relatively slowly compared to other predistorters. According to the results, this slowness of convergence is not critical for other predistorters and we finally obtain more than 35 dB improvement in the experienced signal-to-image ratio. V.
CONCLUSION
Digital predistortion techniques allow the use of nonideal analog transmitters. Several nonidealities may be faced in wideband transmitters. In this paper, we presented an adaptive compensation architecture to significantly reduce both linear and nonlinear distortions via predistortion. Further work could be done in stability analysis of the adaptation procedures as well as finding possible limits of the architecture via more challenging nonidealities. REFERENCES [1]
P. B. Kenington, “Linearized transmitters: An enabling technology for software defined radio,” IEEE Commun. Mag., vol. 40, pp. 156–162, Feb. 2002. [2] L. Ding et al., “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun., vol. 52, pp. 159–165, Jan. 2004. [3] 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. [4] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1985. [5] H. W. Kang, Y. S. Cho, and D. H. Youn, “On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter,” IEEE Trans. Commun., vol. 47, pp. 522–526, Apr. 1999. [6] D. S. Hilborn, S. P. Stapleton, and J. K. Cavers, “An adaptive direct conversion transmitter,” IEEE Trans. Veh. Technol., vol. 43, pp. 223– 233, May 1994. [7] J. K. Cavers, “New methods for adaptation of quadrature modulators and demodulators in amplifier linearization circuits,” IEEE Trans. Veh. Technol., vol. 46, pp. 707–716, Aug. 1997. [8] R. Marchesani, “Digital precompensation of imperfections in quadrature modulators,” IEEE Trans. Commun., vol. 48, pp. 552–556, Apr. 2000. [9] A. Mämmelä, “Commutation in linear and nonlinear systems,” Frequenz: Journal of RF-Engineering and Telecommunications, vol. 60, pp. 92-94, May/June 2006. [10] B. Widrow and E. Walach, Adaptive Inverse Control. Upper Saddle River, NJ: Prentice Hall, 1996. [11] Base Station (BS) Conformance Testing (FDD) (Release 5), 3GPP Technical Specification 25.141 version 5.8.0, 2003.
0 -10
ACP Low = -38.8 ACP Low 2 = -50.1 Low 2
-20 Power spectral density [dB]
[12] A. I. Hussein and W. B. Kuhn, “Bandpass SD modulator employing undersampling of RF signals for wireless communication,” IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process., vol. 47, pp. 614–620, June 2000. [13] M. Valkama, M. Renfors, and V. Koivunen, “Advanced methods for I/Q imbalance compensation in communication receivers,” IEEE Trans. Signal Process., vol. 49, pp. 2335-2344, Oct. 2001. [14] N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/ f α power law noise generation,” Proc. IEEE, vol. 83, pp. 802–827, May 1995. [15] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum Press, 1997. [16] A. A. M. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans Commun., vol. 29, pp. 1715–1720, Nov. 1981. [17] C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology, and Techniques. 2nd ed., New York: Kluwer Academic/Plenum Press, 2000. [18] L. Frost III, “An algorithm for linearly constrained adaptive array processing,” Proc. IEEE, vol. 60, pp. 926-935, Aug. 1972.
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I. INTRODUCTION A cost-efficient radio transmitter may consist of highly nonideal analog components. A high power amplifier is nonlinear. Without a careful system design, its efficient use may cause severe distortion effects. An I/Q modulator may introduce interference in the form of I/Q imbalance and DC offset. In addition, an RF filter has a nonideal frequency response. Transmitter linearization enables efficient use of power amplifiers with nonconstant envelope modulation methods [1]. A modern approach is to use digital predistortion. An adaptive nonlinear predistortion approach used here is called indirect learning architecture [2] or postdistorter training [3]. The former uses the least squares (LS) solution and the latter employs the recursive least squares (RLS) algorithm. Indirect learning architecture for linear systems may be seen equivalent to inverse control architecture [4]. Linear transmitter filters are compensated in addition to the nonlinear power amplifier in [5]. Predistortion of I/Q imbalance and DC offset with a nonlinear amplifier is considered in [6], [7]. I/Q modulator nonidealities alone are compensated with an adaptive leastmean square (LMS) algorithm in [8]. A popular approach is to use specialized training patterns for estimation [7]. Some methods require only a data signal [6] - [8]. Commutability of nonlinear blocks is analyzed in [9]. Extensive theory for adaptive linear systems is presented in [4], [10]. In this paper, we consider digital predistortion of a nonideal radio transmitter in more detail than before. Our aim is to minimize the level of various distortions in the transmitted signal. The system model includes the predistortion of a nonlinear amplifier, and compensation of the frequency response of a linear RF filter, I/Q imbalance, and DC offset.
978-1-4244-1645-5/08/$25.00 ©2008 IEEE
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Randomness to the system is introduced by phase noise and quantization noise. We use a complex baseband signal in adaptation while an envelope detector was used e.g. in [6] - [8]. We see that the presented architecture enables a novel and useful insight to transmitter design. The architecture consists of the spatial order of several adaptive predistorters and temporal sequence of adaptation steps. As an extension, we detail a novel linear frequency response predistortion approach applicable to work with nonlinear predistortion. In addition, we diminish the effect of phase noise to adaptation via a compensation method. We present simulation results to verify the discussion. The rest of the paper is organized as follows. In Section II, we introduce the system model. Then, we discuss predistortion architecture 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 transmitter system model is presented in Fig. 1. We use a dual carrier wideband code division multiple access (WCDMA) signal as a signal source following closely the Third Generation Partnership Project (3GPP) specifications [11]. The complex valued baseband signal is frequency translated upward in an I/Q modulator. A linear RF filter is used to shape the spectrum of the signal. Then, the signal is amplified and transmitted. To compensate the nonidealities we need a feedback (FB) signal from the output of the power amplifier. In the FB loop, we assume that the signal is first translated from RF to an intermediate frequency (IF) of 96 MHz. In this work, we assume that the bandpass IF filter following the RF/IF mixer is ideal and it is omitted in Fig. 1. The analog-to-digital converter (ADC) digitizes the analog signal. We use undersampling with a sampling frequency of Fs = 76.8 MHz, which is the same that we assume in the signal source. As a result, the IF frequency is mapped to a normalized
Figure 1. Simplified system model of the transmitter.
frequency of ¼ as in [12]. However, due to aliasing the undersampling limits the observable bandwidth in the FB loop to 38.4 MHz, which is a half of that available in the transmitter branch. We assume an ideal digital I/Q demodulator to form the complex baseband signal. The ideal demodulator is composed of a Hilbert transformer followed by a multiplication with a complex exponential function. For adaptation, signal blocks are stored in to a memory. We use a block length of 8000 samples. We remove the amplifier gain from the FB signal. The nonideality model of the study is depicted in Fig. 2. First, the I/Q modulator introduces DC offset, I/Q imbalance and phase noise. For the DC offset and I/Q imbalance effects we assume the model [6] and have y(t) = Mx(t) + Ma, where y(t) = [yI(t), yQ(t)]T and x(t) = [xI(t), xQ(t)]T are output and input signals of the model represented with the inphase (I) and quadrature (Q) components, and § α cos(φ / 2) β sin(φ / 2) · §a · ¸¸, a = ¨¨ 1 ¸¸, M = ¨¨ © α sin(φ / 2) β cos(φ / 2) ¹ © a2 ¹
(1)
Į and ȕ are the gains of I and Q components, φ is the phase splitter error between the components, and a1 and a2 are the DC offsets for both the components. In our model, the DC-tosignal power ratio is 0 dB before compensation. The parameter values for (1) are α = 1.005, β = 0.995 and φ = 1º. Using the image attenuation definition from [13], we can calculate for the matrix M and the given parameters that the ratio between the desired and the introduced image signal powers is about 40 dB. We call this relationship the signal-to-image ratio. We model phase noise as a sum of 1/f 2 noise [14] and a random filtered white Gaussian noise [15]. In the I/Q modulator, the average magnitude of the phase change at the sampling rate is about 0.0019º. Linear filters have a nonideal frequency response that can be shaped in certain limits via design parameters. For the RF power amplifier, we assume Saleh’s model [16]. The model introduces both AM/AM and AM/PM distortion where AM refers to amplitude modulation and PM refers to phase modulation. We use such a parameter and input backoff combination that gives adjacent channel powers from about -40 dBc to -50 dBc for the two most adjacent channels (see Fig. 8). In the FB loop, the phase noise is further affecting the signal in the RF/IF mixer. This time more severe phase noise can be tolerated if the adaptation process can cope with it. An average magnitude of the phase change at the sampling rate is about 0.0027º. Finally, quantization noise is modeled by a uniformly distributed white random process. The signal-toquantization noise ratio is 50 dB during data transmission. III.
I/Q phase noise
DC offset Signal source
I/Q imbalance
Linear RF filter
RF power amplifier FB phase noise
Input for adaptation
nq, quantization noise
Figure 2. Nonidealities.
predistorter with the new postdistorter. The DC offset predistorter is the only exception here. It must be updated by a residual new estimate because we take the reference signal for its adaptation before the predistorter. The adaptation is not performed in real time but offline using stored signals. In adaptation, the FB and reference signals are used. We emphasize temporal sequence of adaptation via shading of some of the blocks and the switch in Fig. 3. We use shaded blocks alone in acquisition or calibration phase at first. Acquisition is followed by a tracking phase. This phase uses all the blocks except the identification block. In acquisition, we use a downscaled signal to minimize the nonlinear distortion introduced by the power amplifier. A similar idea to identify separately the linear and nonlinear parts of an amplifier is detailed in [17]. In our model, downscaling corresponds to an increase in the experienced signal-toquantization noise ratio. We use 10 dB downscaling. Acquisition employs a wideband training signal. We use a bandlimited white Gaussian noise signal having a bandwidth of 36.6 MHz. Two signal blocks are sent during acquisition in series. The first one is used for identification of the coarse DC offset and the frequency response of the linear filter. Based on this, the DC offset is coarsely predistorted and the second signal block is transmitted for postdistortion training. In tracking, phase noise compensation is activated. Furthermore, we do not adapt the nonlinear amplifier postdistorter at the same time with other postdistorters. This is emphasized by the switch in Fig. 3. Tracking is started by amplifier postdistorter training. We use, e.g., three WCDMA signal blocks for this. Our goal is that other postdistorters would only see their nonidealities after the convergence of amplifier postdistortion. Then we train linear filter, I/Q imbalance and DC offset postdistorters at the same time and
x1
x2
COMPENSATION ARCHITECTURE
We present a compensation architecture for the discussed system model in Fig. 3. The upper blocks are predistorters each of which compensates its own nonideality. We adapt each predistorter in the corresponding postdistorter training block. After the desired postdistorter has processed the adaptation signal block, the corresponding predistorter is updated with the copy of the postdistorter. The update simply replaces the old
Φpn u1 u2
Figure 3. Compensation architecture.
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using three signal blocks. The FB signal must go through the amplifier postdistorter before the adaptation due to the point the reference signal is taken from. We repeat the adaptation switching e.g. five times. Then we use e.g. ten signal blocks before each switch. We emphasize that other number of signal blocks may be used as this is one possible example. The nonlinear blocks are not in general commutative [9]. Therefore the spatial order of the blocks in the predistorter is the reverse of that in the nonideality model. In this way, each nonideality can be compensated separately. In a similar manner, each trained postdistorter should ideally see only the nonideality it tries to compensate. Hence, the optimal spatial order of postdistorters is identical with that of predistorters. In Fig. 3 we have one exception for this. The order of I/Q imbalance and residual DC offset postdistorters should ideally be different. However, we see that the residual DC offset must be minimized before I/Q imbalance postdistorter training. The reference signals for adaptation are taken after the corresponding predistorters. As discussed above, the DC offset postdistorter is again an exception for this. We detail each adaptation block more in detail below. A. DC Offset Estimation and Linear Filter Identification In our model, the DC offset is at a high level. Its effect must be minimized to avoid nonlinear distortion in the acquisition phase. While estimating the DC offset, we also identify the linear filter to define constraints for linear filter postdistortion training. We use an adaptive linear finite impulse response (FIR) system identification procedure extended with an averaging filter to estimate the DC offset which can be considered as a plant drift [4, p. 283]. The model to be identified is K −1
K −1
K −1
k =0
k =0
k =0
y (i ) = ¦ hk [x1 (i − k ) + d ] = ¦ hk x1 (i − k ) + d ¦ hk
(2)
where hk are the coefficients of the FIR filter system model, x1 is the input signal before activating any predistorters, i is the time index of the discrete signal, d is the DC offset, and K is the number of coefficients in the FIR model. We use K = 31. We estimate the coefficients hk via the LMS algorithm as depicted in Fig. 4. In addition, we use a conventional recursive averaging filter v(i+1) = v(i) + µe(i), where µ is the adaptation step size and e is an error signal. Using v as the converged v(i), we obtain a coarse DC offset predistorter
(
K −1
d c = −v ¦k =0 hk
)
−1
(3)
where the sum of the estimated coefficients hk from (2) approximates the amplitude gain of the filter at the zero frequency and thus amplification of the DC offset as seen at the x1
output of the filter. To coarsely compensate the DC offset, the estimate (3) is added to the input signal of the analog transmitter as shown in Fig. 3. Experienced phase noise forces us to use larger averaging filter and LMS adaptation step sizes than otherwise desirable. B. Linear Filter and Residual DC Offset Postdistortion We present the acquisition of the linear filter and the residual DC offset postdistorters in Fig. 5. The I/Q imbalance postdistorter is active only during tracking. Coarse DC offset is compensated before this stage. Since the bandwidth of the FB loop is limited to 38.4 MHz, a constrained adaptation is introduced to control the frequency response of the predistorter for frequencies |F| > 19.2 MHz. We use a modified Frost constrained LMS approach [18]
(
(4)
where w[i] is the column vector with N filter coefficients at time i, P = I – CH(CCH)-1C is the N×N projection matrix, µ is the step-size parameter, e[i] is the error signal, u2[i] (see Fig. 3) is the FB signal vector consisting of N samples, the asterisk * refers to the complex conjugate, I is the identity matrix, H refers to the Hermitian transpose, and G = CH(CCH)-1g. When the constrained adaptive filtering is applied to adaptive predistortion, we need to formulate the constraint matrix C and the gain vector g. As the constraint matrix, we get C = [cmn ]M ×N = [exp(− j 2πf m n)]M ×N
(5)
where n = 0, …, N - 1 and fm = Fm /Fs form a set of equally spaced normalized frequencies with 18.3 MHz < |Fm| < 35 MHz. Constraints are indexed with m = 0, …, M - 1. Then we define the gain vector ª exp(− j 2πf τ ) m g = [g m ]M ×1 = « K −1 − j 2πk sgn( f )f «¬ ¦ k =0 hk e m
e
º » »¼ M ×1
(6)
where fe = ±18.3MHz/Fs are the normalized corner frequencies of the acquisition training signal, and τ is the desired group delay of the system. We use positive fe when fm is positive and negative fe otherwise. In the adaptive postdistorter FIR, we use N = 63 and M = 30. We force the group delay τ to 32 sample periods. In this way, the predistorter also provides delay compensation between the baseband and FB signals for further adaptation. For the residual DC offset postdistorter, we use a similar I/Q imbalance compensator
FIR filter
FIR filter
u2
dr
x1
v LMS
)
w[i + 1] = P w[i ] + µe[i ]u *2 [i ] + G
1
e
-
-
LMS
1 LMS
u1
Figure 5. Training of linear filter, residual DC offset and I/Q imbalance postdistorters.
Figure 4. Adaptive identification structure with coarse DC offset estimator.
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recursive averaging filter as above in the system identification. In this case, the residual DC offset postdistorter result dr is introduced in predistortion without further manipulation. The DC offset predistorter uses the sum of (3) and dr. Due to phase noise, we use large adaptation step sizes during acquisition. During tracking we change to smaller values because phase noise compensation allows longer averaging periods. C. Phase Noise Compensation Before any adaptation from data transmission, phase noise is compensated. For this we use the phase estimate §
n+L / 2
· ∗ x 2 (i )u1 (i ) ¸ © i =n− L / 2 ¹
φ pn (n) = arg¨
¦
(7)
where we use an averaging interval of L = 513 and signals x2 and u1 are shown in Fig. 3. The approach weights large signal amplitudes more than small signal amplitudes. This gives more accurate results than considering phases alone because quantization noise affects more severely small amplitude signals. The idea is similar to maximum ratio combining. Phase noise compensation is not employed during calibration. D. Nonlinear Amplifier Postdistortion After the acquisition, we start to use the WCDMA signal and first train the nonlinear postdistorter using the RLS algorithm [3]. We use the approach slightly differently as we do not update the predistorter in real-time for each input sample. We use odd polynomial orders 1, 3, and 5. In the beginning, we use a small RLS forgetting factor for the first adaptation signal block to have faster convergence. Then we continue with a large value to introduce more averaging. We support the averaging by using the postdistorter and the recursion matrix obtained from the previous signal block. E. I/Q Imbalance Postdistortion The linear filter and DC offset postdistorters are adapted as during calibration with one exception. Before these operations, we put the FB signal through the nonlinear postdistorter (see Fig. 3). We now include also I/Q imbalance postdistorter training as in Fig. 5. The I/Q imbalance postdistorter Mc training is performed via four one-tap coefficients adapted with the LMS algorithm; i.e. one coefficient per each element of M. A background for a similar type of adaptive systems is presented in [10]. In addition, the I/Q imbalance postdistorter uses a different error signal than the linear filter and DC offset postdistorters. With this approach, we try to avoid stability problems that could originate from having multiple loops tracking each other. We assume that the dedicated error signal for the linear filter and DC offset postdistorters can be justified because M and Mc are approximately identity matrices and hence the input and output signals of the I/Q imbalance compensator are approximately equal. Finally, we justify using Mc directly in predistortion because ideally McM = I = MMc. IV. RESULTS Simulation results in Figs. 6 - 10 demonstrate that the nonidealities are considerably suppressed with the presented architecture. Fig. 6 shows magnitudes of linear frequency responses of the compensated RF filter and the predistorter
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obtained in acquisition. We can see a low variation both at the signal and higher frequencies. We show results for phase noise estimation after convergence of other adaptive algorithms in Fig. 7. We see that the approach can diminish the variation of several degrees to some tenths of degrees. In this way, we can use averaging in adaptation more efficiently. Fig. 8 shows power spectral densities for the signal source and the high power amplifier output signal without nonlinear predistortion. We depict the performance of nonlinear predistortion in Fig. 9. We can observe that adjacent channel powers are reduced from about -39 and -50 dBc given in Fig. 8 to less than about -70 and -80 dBc. Fig. 10 shows results for the DC offset and I/Q imbalance predistorters. Two first acquisition steps reduce the DC-to-signal power ratio considerably from 0 dB to about -70 dB. Tracking with data signal blocks gives further improvement. The signal-to-image ratio is calculated in the same way for the matrix MMc as for M in Section II. We can observe that the signal-to-image ratio improves relatively slowly compared to other predistorters. According to the results, this slowness of convergence is not critical for other predistorters and we finally obtain more than 35 dB improvement in the experienced signal-to-image ratio. V.
CONCLUSION
Digital predistortion techniques allow the use of nonideal analog transmitters. Several nonidealities may be faced in wideband transmitters. In this paper, we presented an adaptive compensation architecture to significantly reduce both linear and nonlinear distortions via predistortion. Further work could be done in stability analysis of the adaptation procedures as well as finding possible limits of the architecture via more challenging nonidealities. REFERENCES [1]
P. B. Kenington, “Linearized transmitters: An enabling technology for software defined radio,” IEEE Commun. Mag., vol. 40, pp. 156–162, Feb. 2002. [2] L. Ding et al., “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun., vol. 52, pp. 159–165, Jan. 2004. [3] 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. [4] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1985. [5] H. W. Kang, Y. S. Cho, and D. H. Youn, “On compensating nonlinear distortions of an OFDM system using an efficient adaptive predistorter,” IEEE Trans. Commun., vol. 47, pp. 522–526, Apr. 1999. [6] D. S. Hilborn, S. P. Stapleton, and J. K. Cavers, “An adaptive direct conversion transmitter,” IEEE Trans. Veh. Technol., vol. 43, pp. 223– 233, May 1994. [7] J. K. Cavers, “New methods for adaptation of quadrature modulators and demodulators in amplifier linearization circuits,” IEEE Trans. Veh. Technol., vol. 46, pp. 707–716, Aug. 1997. [8] R. Marchesani, “Digital precompensation of imperfections in quadrature modulators,” IEEE Trans. Commun., vol. 48, pp. 552–556, Apr. 2000. [9] A. Mämmelä, “Commutation in linear and nonlinear systems,” Frequenz: Journal of RF-Engineering and Telecommunications, vol. 60, pp. 92-94, May/June 2006. [10] B. Widrow and E. Walach, Adaptive Inverse Control. Upper Saddle River, NJ: Prentice Hall, 1996. [11] Base Station (BS) Conformance Testing (FDD) (Release 5), 3GPP Technical Specification 25.141 version 5.8.0, 2003.
0 -10
ACP Low = -38.8 ACP Low 2 = -50.1 Low 2
-20 Power spectral density [dB]
[12] A. I. Hussein and W. B. Kuhn, “Bandpass SD modulator employing undersampling of RF signals for wireless communication,” IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process., vol. 47, pp. 614–620, June 2000. [13] M. Valkama, M. Renfors, and V. Koivunen, “Advanced methods for I/Q imbalance compensation in communication receivers,” IEEE Trans. Signal Process., vol. 49, pp. 2335-2344, Oct. 2001. [14] N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/ f α power law noise generation,” Proc. IEEE, vol. 83, pp. 802–827, May 1995. [15] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum Press, 1997. [16] A. A. M. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans Commun., vol. 29, pp. 1715–1720, Nov. 1981. [17] C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology, and Techniques. 2nd ed., New York: Kluwer Academic/Plenum Press, 2000. [18] L. Frost III, “An algorithm for linearly constrained adaptive array processing,” Proc. IEEE, vol. 60, pp. 926-935, Aug. 1972.
ACP High = -38.6 ACP High 2 = -49.6 High
Low
High 2
-30 Nonlinear distortion -40 -50 -60
No distortion
-70 -80 -90 -100
-1.5
-1
-0.5
0 0.5 Frequency [Hz]
1
1.5 7
x 10
Figure 8. Power spectral density for ideal signal and signal with nonlinear distortion.
0.8 ACP High
ACP Low
Compensated RF filter Predistorter
0.4
Adjacent channel power (ACP) [dBc]
Magnitude of the frequency response [dB]
0.6
0.2 0 -0.2 -0.4 -0.6
-30
-20
-10 0 10 Frequency (MHz)
20
30
-60
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-70
-70
-75
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50
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50
ACP Low 2
-0.8 -1 -40
-60
100
150
200
150
200
ACP High 2
-65
-65
-70
-70
-75
-75
-80
-80
40 -85
0
50
Figure 6. Magnitude responses of compensated RF filter and predistorter.
-85 100 150 200 0 50 Index of adaptation signal block
100
Figure 9. Adjacent channel power ratios for each adaptation block.
DC-to-signal power ratio [dB]
2 Phase noise
1
0.5
Estimation error
0
-0.5
-1 0
2000
4000
6000
-40 -60 -80 -100 -120
Signal-to-image ratio [dB]
Phase in degrees
1.5
-20
8000 10000 12000 14000 16000 18000 Sample index
Figure 7. Phase noise estimation demonstration.
0
20
40
60
80
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140
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60 80 100 120 140 Index of adaptation signal block
160
180
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Figure 10. DC-to-signal power ratios above and signal-to-image ratios for each adaptation block.
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