IDENTIFICATION OF POWER AMPLIFIER USING MIMO - XLIM-SIC
Transcription
IDENTIFICATION OF POWER AMPLIFIER USING MIMO - XLIM-SIC
IDENTIFICATION OF POWER AMPLIFIER USING MIMO TIME-DOMAIN REPRESENTATION Smail Bachir ∗ Mourad Djamai ∗ Claude Duvanaud ∗ Régis Ouvrard ∗ ∗ Laboratoire d’Automatique et d’Informatique Industrielle 4 avenue de Varsovie, 16021 Angoulême cedex Email : [email protected] URL : http://laii.univ-poitiers.fr Abstract: In this paper, we present a new technique for modeling and characterization of Power Amplifier (PA) by parameter estimation using continuous-time representation. Firstly, we study a continuous model which takes into account nonlinearities and PA filtering. The filter structure includes a non linear polynomial representing amplitude and phase conversion in addition to a MIMO Laplace plant. Then, we propose a new approach for PA model characterization based on parameter estimation with several excitations. Using time-domain measurements, this method deduces recursively an optimal estimation with Non Linear Programming technique. The experimental results show good agreement and demonstrate the possibility of this technique to explain power amplifier dynamics. Keywords: Parameter estimation, Non Linear Programming, initialization problem, PA modeling, continuous filter, time-domain envelope. 1. INTRODUCTION Power Amplifiers (PA) are an important component in modern communication systems, providing the transmit signal levels needed to overcome the loss between the transmitter and receiver. However, they also introduce problems. The amplifier can consume a major fraction of the power used by the system and also distort the transmitted signal, introducing additional spectral components within the signal frequency band. Thus, the key challenge amplifier design for mobile communication systems is the design of amplifiers, which provide high output power, linearity and efficiency. For that, the modeling step is indispensable to a good comprehension of amplifier transmittance effects. The standard technique for PA modeling uses a nonlinear structure with fixed delay taps and complex coefficients (Clark et al., 1998)(Launay et al., 2002). A Volterra series (Kenington, 2000) and Wiener model (Ku and Kenney, 2003) are largely employed in quantifying Radio frequency (RF) effects. A number of algorithms exist performing the coefficient adaptations like the Least Mean Squares (LMS) technique which is the most commonly used algorithm (Young, 1981)(Ljung, 1999)(Johansson, 1994). However, the parameters computation for a nonlinear system is often difficult and time consuming for strongly nonlinear devices. The model considered in this paper is described on continuous-time domain. This structure is similar to PA discrete-time representation which is including nonlinear transfer function and continuous filter (Clark et al., 1998). The first branch is set to a memoryless amplitude (AM/AM) and phase (AM/PM) conversion. Conventionally, the power series model is used to considered these transfer functions. To describe a PA dynamics, an nth MIMO filter is inserted. This element operates on modulating input and represents a lowpass equivalent in envelope signals (Djamai et al., 2005). The parameters of this model have been identified by Output-Error technique (Richalet et al., 1971)(Trigeassou et al., 2003). In practice, identification technique should be adapted to the use objectives. In our case, PA characterization is realized by a parameters follow-up. Then, it is necessary to work in continuous time representation because achieved parameters have physical significance (gain conversion, cut-off frequency, ...). However, parameters initialization supposes a preliminary learning to know approximatively the parameter values. For this aim, a new methodology of PA identification based on different excitations has been adapted to parameters extraction in an effective way. A special experimental setup dedicated to radio frequency with baseband transmission has been performed and used in order to validate this technique. Experimental investigation exhibit the good agreement and confirm the possibility to PA characterization using continuous-time representation. 2.1 Nonlinear Statics functions A nonlinear amplifier can be modeled by two parallel nonlinear elements. One is a nonlinear amplitude element which represent AM/AM conversion and another is a nonlinear phase element which represent AM/PM conversion. In our study, the identification is performed to characterize the PA amplitude and phase nonlinearities with polynomial power series described by scalars parameters {ck } and {dk } (Kenington, 2000)(Launay et al., 2002), such as: P X k V = ck Vin e k=1 Q X k ϕ = ϕ + dk Vin e in (1) k=1 where Ve and ϕe are respectively the amplitude and phase of the distorted signal. Vin and ϕin resp. the input envelope amplitude and phase of the Radio Frequency (RF) signal obtained by the trigonometric relations : q 2 + Q2 Vin = Iin in Qin ϕin = arctan Iin where Iin and Qin are the baseband inputs. 2. CONTINUOUS POWER AMPLIFIER MODEL 2.2 Continuous filter The nonlinear amplifier model used in this paper is an extension of the discrete time-model at continuous one (Clark et al., 1998). The major disadvantage of the discrete representation systems is that the parameters used have no physical significance, contrary to continuous representation where parameters keep their real aspect (Unbehauen and Rao, 1997). This is very important when advanced PA applications are considered such as linearization or real time control. AM/AM Ι Q in V V in Gain and phase conversion ϕ ϕ in e I I e in out e Baseband conversion Q H e 0 ( s ) 0 H As shows in fig. 1, the inputs to outputs relationship of this nth order filter may be represented with a differential equation: n−1 m X X dn dk dk I + a I = b Ie out k out k dtn dtk dtk k=0 k=0 (2) n−1 m X X dn dk dk ak k Qout = bk k Q e dtn Qout + dt dt k=0 k=0 Ie (t), Qe (t) and Iout (t), Qout (t) are respectively the filter input vector and output one where: ( s ) Q out AM/PM Fig. 1. High frequency power amplifier model As shown in fig. 1, the MIMO model structure is a nonlinear transfer function added to an nth order Laplace plant. A similar filtering H(s) operates on the two baseband inputs I/Q. The nonlinear functions defined an AM/AM and AM/PM characteristics and is placed before the filter to complete the PA model. Ie = Ve cos(ϕe ) Qe = Ve sin(ϕe ) The coefficients {ak } and {bk } are real scalars that define the model. Note that the filter structure is the same on the two axes I and Q, which gives a decoupled MIMO plant. Thus, the input-output relation can be expressed in Laplace-domain with the transfer-function H(s), as so: H(s) = Pm k k=0 bk · s P n−1 sn + k=0 ak sk (3) 2.3 Identification algorithm Oscillator Parameter estimation is the procedure that allows the determination of the mathematical representation of a real system from experimental data. Two classes of identification techniques can be used to estimate the parameters of continuous time systems: Equation Error and Output Error (Richalet et al., 1971)(Ljung, 1999)(Trigeassou et al., 2003) I Modulation Demodulation I 0° 0° PA 90° 90° Q Q PA Model • Equation Error techniques are based on the minimization of quadratic criterion by ordinary least-squares (Ljung, 1999). The advantage of these techniques is that they are simple and require few computations. However, there are severe drawbacks, especially for the identification of physical parameters, not acceptable in PA characterization, such as the bias caused by the output noise and the modeling errors. • Output Error (OE) techniques are based on iterative minimization of an output error quadratic criterion by a Non Linear Programming (NLP) algorithm (Young, 1981). These techniques require much more computation and do not converge to an unique optimum. But, OE methods present very attractive features, because the simulation of the output model is based only on the knowledge of the input, so the parameter estimation are unbiased (Trigeassou et al., 2003). Moreover, OE methods can be used to identify non linear systems. For these advantages, the OE methods are more appropriate for PA characterization. Parameter identification is based on the definition of a model. For power amplifier, we consider the previous mathematical model (Eqs. 1-3) and we define the parameter vector: θ = [ a0 · · · an−1 b0 · · · bm c1 · · · cP d1 · · · dQ ] T AM/AM 0 0 _ Q̂ ˆ I + + _ AM/PM Identification Algorithm Fig. 2. Continuous-time PA identification schemes As a general rule, parameter estimation with OE technique is based on minimization of a quadratic multivariable criterion defined as : J= K X (εI 2k + εQ 2k ) = εTI εI + εTQ εQ (5) k=1 We obtain the optimal values of θ by Non Linear Programming techniques. Practically, we use Marquardt’s algorithm (Marquardt, 1963)(Trigeassou et al., 2003) for off-line estimation: 00 θ̂ i+1 = θ̂i − {[Jθθ + λ · I]−1 .Jθ0 }θ̂=θ i (6) 00 with Jθ0 and Jθθ are respectively gradient and hessian such as: PK Jθ0 = −2 k=1 εTIk · σ Ik,θ + εTQk · σ Qk,θ PK 00 Jθθ ≈ 2 k=1 σ Ik,θ · σ TIk,θ + σ Qk,θ · σ TQk,θ λ : monitoring parameter T where [.] denotes transposition operation. σ Ik,θ = Assume that we have measured K values of input vector (Iin (t), Qin (t)) and output vector ∗ (Iout (t), Q∗out (t)) with t = k · Te and 1/Te is the sampling rate, the identification problem is then to estimate the values of the parameters θ. Thus, we define the output prediction errors: ∗ εIk = Iout − Iˆoutk (θ̂, Iin , Qin ) k ∗ εQk = Qoutk − Q̂outk (θ̂, Iin , Qin ) (4) where predicted outputs Iˆoutk and Q̂outk are obtained by numerical simulation of the PA model and θ̂ is an estimation of true parameter vector θ. σ Qk,θ = ∂ Iˆout ∂θ ∂ Q̂ out ∂θ : output sensitivity on I axis : output sensitivity on Q axis where the sensitivity functions σ are obtained, for each parameter, by numerical integration of their differential system (Bachir, 2002). All discrete-time models are deduced from the continuous one by second order serie expansion of the transition matrix (Bachir, 2002). 2.4 Initialization problems Usually, for engineering process, one has good knowledge on physical parameters, necessary to initializing the recursive algorithm (Eq. 6). In our case, PA users have not sufficient information on parameter vector θ, especially on AM/AM and AM/PM parameters. It is then essential to define a global strategy which makes it possible to obtain an approximative parameters values. So we propose an optimal search method based on Equation Error techniques to obtain initial values of PA parameters with two steps. 2.4.1. Initialization of AM/AM and AM/PM parameters The first step consists in searching the nonlinear parameters {ck } and {dk } using the envelope magnitude and phase (Eq. 1). Thus, the AM/AM and AM/PM characteristic is employed to optimize a polynomial function by Least Squares technique (Ljung, 1999). A solution for the coefficients is obtained by minimizing the mean-squared error ∗ between the measured (Iout , Q∗out ) and the modeled output (Iout , Qout ) such as: ( θ̂ c = (φTc φc )−1 φTc V ∗e θ̂ d = (φTd φd )−1 φTd (ϕ∗e − ϕin ) (7) the linear filter effects. Thus, we define the filter parameter vector: θf = [ a0 a1 · · · an−1 b0 b1 · · · bm ]T (8) Parameter estimation is performed by iterative Instrumental Variable based on Reinitilised Partial Moments (see also (Trigeassou, 1987)(Garnier et al., 2003)). 3. TESTS RESULTS In this section, we illustrate through experimentations, performance of the PA time-continuous characterization method based on parameter estimation. In our experimental study, the Cartesian structure (I and Q axes) has been used in order to characterize the power amplifier. This transmission structure overcomes the problems associated with the wide bandwidth of the signal phase by applying modulation in I and Q components. Since the I and Q components are the natural outputs of a modern DSP, the Cartesian loop is widely used in mobile-radio systems. θ̂c = [ c1 c2 · · · cp ]T : AM/AM parameters T θ̂d = [ d1 d2 · · · dQ ] : AM/PM parameters ∗ e ϕ∗e V and respectively magnitude and phase of output signal h i φc = ϕ1 ϕ2 · · · ϕK : AM/AM regression c c c matrix T 2 P ϕk = Vink Vin · · · Vin : regression vector k k c i h φd = ϕ1 ϕ2 · · · ϕK : AM/PM regression d d d matrix h iT Q 2 ϕk = Vink Vin · · · V : regression vector in k k 3.1 Experimental setup For the experimental investigation, a commercial 700-MHz/4,2-GHz MINI-CIRCUITS is used (ZHL-42 Model). Inputs and outputs data are obtained from YOKOGAWA Digital Oscilloscope with a sampling period equal to 10 ns. DATA Acquisition Qin Qout Iin Iout d Vink is a k th sampled input. Noted that for these estimations, the regression vector is not correlated with the output. In practice, the PA characteristics is performed by a sinusoid input at fixed low frequency and higher input level. In these conditions, the PA filtering characteristics are assume neglected according to non linear dynamics. The input-output curves are often obtained by measuring the output gain and phase as a function of input power. PA 0° 0° 90° 90° Arbitrary Waveform Generator φ Modulation De-modulation Phase shifter Local Oscillator Fig. 3. PA Experimental setup 2.4.2. Filter initialization The initial values of the filter coefficients are obtained for small input (linear case). Indeed, in this case, the signal distorsion is negligible, which makes it possible to take into account only Identification algorithm needs persistent excitation to provide appropriate estimation. Indeed, modulated signals are required to excite both steady-state (low frequency) and process dynamics (medium to high frequency). This excitation is performed with a Pseudo Random Binary Sequence (P.R.B.S) baseband pulse as the input modulation to the transmitter (fig. 5.a). AM/AM characteristic 0.25 Measured Data 0.2 Output amplitude (V) All modulation signals I and Q are delivered by a TTi 40 MHz Arbitrary Waveform Generator connected to PC control. The local oscillator frequency is 900 MHz obtained from 300-KHz/2,75GHz Digital Modulation Signal Generator (Anritsu MG 3660A). Estimation 0.15 0.1 0.05 0 0 3.2 Experimental results 0.05 0.1 0.15 0.2 Input amplitude (V) 3.2.1. AM/AM and AM/PM characteristics identification Polynomial parameters are extracted from the input/output transfer function. The measured characteristics is obtain by sweeping the power of an input signal in the center-band frequency equal to 900 MHz of the RF PA band width. In our case, we used the 3th and 2nd order polynomial respectively to amplitude and phase conversion (Djamai et al., 2005): 3 X k V = ck Vin e Fig. 4. AM/AM response and estimation After running a recursive algorithm, we obtained: â0 = 3, 38.1013 â = 3, 12.106 G = 1, 325 1 ⇒ fc = 0, 925 MHz b̂0 = 4, 48.1013 b̂1 = 1, 84.106 where the PA gain G and the resonant frequency fc are calculated using θ̂ values. Noted that the gain in the two steps are in the same proportion. 0.2 Fig. a Magnitude (V) 0.1 k=1 2 X k ϕ = ϕ + dk Vin in e 0 −0.1 k=1 where objective is to obtain the initialization values of ck and dk using Least Squares algorithm. Table 1 shows the estimation results. It is shown that the PA gain is equal to G = ĉ1 = 1, 52. −0.2 0 5 10 15 20 25 30 35 40 45 50 45 50 Fig. b Magnitude (V) Measured data Estimation 0.3 0.2 0.1 Table 1. Estimation results of AM/AM and AM/PM functions θ θ̂ c1 1, 52 c2 1.7.10−4 c3 −9, 61 d1 −1, 1.10−4 d2 3, 1.10−5 For illustration, it may be observed in fig. 4 a good agreement between a measured AM/AM characteristic and estimation. This confirms that a Taylor series is a realistic way to approximate a PA nonlinearities. 3.2.2. Filter identification In this step, we search an initialization of a following resonant 2nd order system which is a dynamical relationship between baseband inputs (Iin , Iin ) and outputs one (Iout , Qout ) in Laplace domain: Qout (s) b1 s + b 0 Iout (s) H(s) = = = 2 Iin (s) Qin (s) s + a1 s + a 0 Thus, we define the estimated parameter vector: θ f = [ a 0 a 1 b0 b1 ] T 0 −0.1 −0.2 −0.3 0 5 10 15 20 25 30 35 40 Time (µs) Fig. 5. (a) Input signal. (b) Comparison of timedomain measurement and estimation For small power, figure (5.b) proves that the PA dynamic is equivalent to a 2nd order resonant system. 3.2.3. Global PA Model identification The unknown system in this case is a global PA model composed by: • non-linear AM-AM and AM/PM functions, • second order system. The input signals is performed by adding some P.R.B.Sequence at different levels (fig. 6.a). The aim is to sweep all AM/AM characteristic in order to take into account the linear and nonlinear area. Thus, the PA is driven near small power and saturation to exploit maximum power efficiency. Figure (6.a) shows the input signal applied to perform global PA identification. After 20 iterations, we obtained: â0 = 3, 42.1013 ĉ = 1, 55 −4 ˆ â1 = 3, 15.106 1 −4 d1 = −1, 5.10 ĉ = 1, 9.10 13 2 −5 ˆ b̂0 = 4, 53.10 d2 = 3, 6.10 ĉ3 = −9, 48 b̂1 = 1, 86.106 Fig. a Magnitude (V) 0.2 0 −0.2 0 0.4 5 10 15 20 25 30 35 40 45 50 45 50 Fig. b Magnitude (V) Measured data Estimation 0.2 0 −0.2 −0.4 0 5 10 15 20 25 30 35 40 Time (µs) Fig. 6. (a) Input signal. (b) Comparison of timedomain measurement and estimation For a different experiment, model simulation with the obtained parameters exhibit good approximation of measured data on I axe (fig. 6.b). 4. CONCLUSION A model is described which offers a simple way to modeling PA dynamics, based on continuous-time representation. This model is able of accounting the magnitude and phase amplifier nonlinearities such as the saturation effects. Test results illustrate the efficiency of this technique for use in off-line identification. The continuous approach was found to be accurate in predicting the PA response in dynamical mode. Estimation results show that the described amplifier acts like a resonant system coupled with a polynomial series. The proposed technique considers the transfer function approximately constant, which allows to describe the amplifier as memoryless. Our next objective will be to characterize PA with memory effects contributions. So, it will be necessary to develop a transfer function able to explain these additional dynamics. REFERENCES Bachir, S. (2002). Contribution au diagnostic de la machine asynchrone par estimation paramétrique. Ph.d. thesis. Université de Poitiers, France. Clark, C. J., G. Chrisikos, M. S. Muha, A. A. Moulthrop and C. P. Silva (1998). Timedomain envelope measurement technique with application to wideband power amplifier modeling. IEEE Tran. Microwave Theory Tech. 46, 2531–2540. Djamai, M., S. Bachir and C. Duvanaud (2005). Caractérisation et identification des amplificateurs de puissance à partir d’une structure de modulation/démodulation iq à conversion directe. In: 14ème Journées Nationales Microondes. Nantes, France. Garnier, H., M. Gilson and E. Huselstein (2003). Developments for the matlab contsid toolbox. In: 13th IFAC Symposium on System Identification SYSID’2003. Rotterdam, The Netherlands. Johansson, R. (1994). Identification of continuoustime model. IEEE Transactions on Signal Processing 42((4)), 887–896. Kenington, P. B. (2000). High-linearity RF Amplifier Design. Artech House Edtitions. London. Ku, H. and J. S. Kenney (2003). Behavioral modeling of nonlinear rf power amplifiers considering memory effects. IEEE Tran. Microwave Theory Tech. 51, 2495–2503. Launay, F., Y. Wang, S. Toutain, D. Barataud, J. M. Nebus and R. Quere (2002). Nonlinear amplifier modeling taking into account hf memory frequency. MTT-S Int. Microwave Symposium Digest 02, 865–868. Ljung, L. (1999). System identification: Theory for the user. 2nd ed, Prentice Hall. USA. Marquardt, D. W. (1963). An algorithm for leastsquares estimation of non-linear parameters. Soc. Indust. Appl. Math. 11, 431–441. Richalet, J., A. Rault and R. Pouliquen (1971). Identification des processus par la mèthode du modèle. Gordon & Breach, Thérie des systèmes. Trigeassou, J. C., T. Poinot and S. Bachir (2003). Estimation paramétrique pour la connaissance et le diagnostic des machines électriques. In: Méthodes de commande des machines électriques (R. Husson, Ed.). pp. 215–251. Hermès Publications. Paris. Trigeassou, J.C. (1987). Contribution à l’extension de la méthode des moments en automatique. Application à l’identification des systèmes linéaires. Ph.d. thesis. Université de Poitiers, France. Unbehauen, H. and G.P. Rao (1997). Identification of continuous-time systems : a tutorial. In: 11th IFAC Symposium on System Identification. Fukuoka, Japan. pp. 1023–1049. Young, P. (1981). Parameter estimation for continuous-time models - a survey. Automatica 17((1)), 23–39.