Nonlinear Identification Of Continuous-Time Radio Frequency Power .

A. Identification algorithmParameter identification is based on the definition ofa model. For power amplifier, we consider the previousmathematical model (Eqs. 1-7) and we define the parametervector: Tθ a0 · · · an 1 b0 · · · bm c1 · · · c2P 1(8)where[.]Tdenotes transposition operation.Assume that we have measured K values of input vector (t), Q (t)) with t k ·(Iin (t), Qin (t)) and output vector (IoutoutTe and 1/Te is the sampling rate. The identification problemis then to estimate the values of the parameters θ . Thus, wedefine the output prediction errors:½ IˆεIk Ioutoutk (θ̂ , Iin , Qin )k(9)εQk Q outk Q̂outk (θ̂ , Iin , Qin )where predicted outputs Iˆoutk and Q̂outk are obtained bynumerical simulations of the PA model and θ̂ is an estimationof true parameter vector θ . ³Jθ′ 2 Kk 1 ε TIk · σ Ik,θ ε TQk · σ Qk,θ³ Jθ′′θ 2 Kk 1 σ Ik,θ · σ TIk,θ σ Qk,θ · σ TQk,θλ is the monitoring parameter,σ Ik,θ Iˆout θan output sensitivity on I axis, Q̂out θand σ Qk,θ B. Sensitivity functionsThe sensitivity functions σ are important elements inthe identification procedure. The positive realness of thesefunctions ensures the stability and the convergence of anunbiased identification algorithm. In comparison, sensitivityfunctions are equivalent to the regressors in the linear case[12]. Thus, it is necessary to attach a great importance to thecalculation of these functions.For nonlinear grey-box identification, consider a generalcontinuous-time state-space model structure :ẋ(t) g (x(t), θ , u(t))y(t) f (x(t), θ , u(t))OscillatorModulationIin*Iout0 0 PAWe consider a SISO non-linear system only to simplifythe equations; there is no restrictive assumption on thedimensions of u and y.90 *QoutQinIn the proposed formulation, it is necessary to distinguishtwo kinds of sensitivity functions:PA Model AM/AMIdentificationAlgorithmAs a general rule, parameter estimation with OE techniqueis based on minimization of a quadratic multivariable criterion defined as :J ε I ε TQ ε Q(10)We obtain the optimal values of θ by Non Linear Programming techniques. Practically, we use Marquardt’s algorithm[28] for off-line estimation:θ̂ i 1 θ̂ i {[Jθ′′θ λ · I ] 1 .Jθ′′ }θ̂ θ y θ σx,θ x θσx,θ : vector of output sensitivity functions (I 1)used in the NLP algorithm: matrix of state sensitivity functions (N I)such as:σ x,θ1···σ x,θi···σ x,θI The sensitivity functions σx,θi are obtained, for each parameter θi , by partial differentiation of equation (12). ThusPA identification scheme(εI 2k εQ 2k ) ε TI k 1σ y,θ AM/PMK H(s) 00H(s)Fig. 2.(12)(13)where g and f are nonlinear functions. x(t) is the state vector(dim(x) N), u(t) and y(t) are input and output signals, andt denotes time. Finally θ is the vector of unknown parameters(dim(θ ) I).Demodulation90 an output sensitivity on Q axis.i(11)Jθ′ and Jθ′′θ are respectively gradient and hessian such as: ẋ g (x, θ , u) x g (x, θ , u) σ̇ x,θi θi x θi θi(14)So, σx,θi is the solution of the nonlinear differential system:σ̇ x,θi g (x, θ , u) g (x, θ , u)σ x,θi x θi(15)Finally, we obtain the output sensitivity functions used inNon Linear Programming algorithm by partial differentiationof equation (13), we get:¶µ y f (x, θ , u) T f (x, θ , u)σy,θi σ x,θi (16) θi x θi

In the particular case of linear system described by thefollowing state space model :½ẋ A(θ ) x B(θ ) u(17)y CT (θ ) x D(θ ) uwe obtain :iihh σ̇ x,θ A(θ ) σ x,θ A(θ ) x B(θ ) u θ θiih ih i iTi σ CT (θ ) σ C(θ ) x D(θ ) uy,θix,θi θi θi(18)All discrete-time models are deduced from the continuousone by second order serie expansion of the transition matrix.C. Initialization problemsAn inherent problem of iterative search routines is thatonly convergence to a local minimum can be guaranteed.In order to converge to the global minimum, a good initialparameter estimate is important. Usually, for engineeringprocess, users have a good knowledge on physical parameters, necessary to initialize the iterative algorithm (Eq. 11).In our case, PA users have not sufficient information onparameter vector θ , especially on AM/AM and AM/PMparameters. It is then essential to define a global strategywhich makes it possible to obtain approximative values ofparameters. So we propose an optimal search method basedon Equation Error techniques to achieve initial values of nonlinear and filter parameters.1) Non linear parameters initialization:The first step consists in searching approximation of thecomplex parameters c2k 1 using the envelope magnitude andphase distorsions (Eqs. 1-3). Thus, the AM/AM and AM/PMcharacteristics are used to optimize a polynomial functionby Least Mean Square (LMS) algorithm [22]. A solution forthe coefficients is obtained by minimizing the mean-squared , Q ) and the modelederror between the measured (Ioutoutoutput (Iout , Qout ) under low frequency signal such as:θ̂ NL (φ H φ ) 1 φ H V outnon linear dynamics. The input-output curves are obtainedby measuring the output gain and phase as a function ofinput level.2) Filter parameters initialization:The second step is the determination of initial values forthe filter coefficients. They are obtained for an input signalwith low input level and large frequency bandwidth. Thesignal distorsion is then negligible, which makes it possibleto take into account only the linear filter effects. Thus, wedefine the filter parameter vector:θ f [ a0 a1 · · · an 1 b0 b1 · · · bm ]TParameter estimation is performed by iterative InstrumentalVariable based on Reinitilized Partial Moments RPM method(see also [12][14][17]). Used for continuous filter identification, this technique is included in the integral methods class.The main idea of this class is to avoid the input-output timederivatives calculation by performing integrations. In thisclass, the particularities of the RPM method1 is the use of atime-shifting window for the integration and to perform anoutput noise filtering.The main advantage of this estimation method to others isits relatively insensitivity to the initial conditions and roughsystem a priori knowledge.IV. PA SETUPThe measurement setup is shown in Fig. 3. The poweramplifier is a commercial ZHL-42 from M INI C IRCUITSmanufacturer. Input and output data are obtained fromYOKOGAWA D IGITAL O SCILLOSCOPE with a samplingperiod equal to 10 ns.DATA Acquisition(19)where :(.)H denotes transpose-conjugate transformationθ̂ NL [ c1 c3 · · · c2P 1 ]T is the vector of polynomialparameters,V out is the measured output,φ [ ϕ 1 ϕ 2 · · · ϕ K ] is the regression matrix,(20)QinQoutIinIoutPA0 0 90 90 Arbitrary fterϕ k [Vink Vink Vink 2 · · · Vink Vink 2P ]T is the regressionvector,and V ink is a kth sampled input.Noted that for these estimations, the regression vector ϕ k isnot correlated with the measured output V out .In practice, the PA characteristics is performed by a sinusoidal excitation applied on baseband inputs Iin and Qin atfixed low frequency and high input level. In these conditions,the PA filtering effects are assumed negligible according toLocal OscillatorFig. 3.PA setup1 C ONTSID M ATLAB TOOLBOX including the RPM estimation methodcan be downloaded from http://www.cran.uhp-nancy.fr/contsid/. The ivrpmfunction allows to obtain model estimation by iterative Instrumental Variable.

Filter identification algorithm needs large frequency bandwidth excitation signal to provide appropriate estimation.Indeed, modulated signals are required to excite both steadystate (low frequency) and process dynamics (medium tohigh frequency). This excitation is performed with a PseudoRandom Binary Sequence (P.R.B.S) baseband pulse as theinput modulation to the transmitter. All data processing arecarried using M ATLAB M ATH W ORKS then are downloadingto a BASEBAND WAVEFORM G ENERATOR . The quadraturemodulator AD8349 and demodulator AD8347 are inserted atthe input and output of the PA. They are standard commercialunits from Analog Devices.Modulation signals I and Q are delivered by a TTi 40 MHzArbitrary Waveform Generator connected to PC control. Thelocal oscillator frequency is 900 MHz obtained from DigitalModulation Signal Generator (A NRITSU MG 3660A).The identification procedure is performed in three steps :Initialization of nonlinear parameters, initialization of filterparameter and global identification of the PA’s model.Figure (4) allows a comparison between measured I andQ outputs waveforms and their estimations. As can be seen,even if the amplifier is driven near saturation, the LMSalgorithm converge to the optimum values with a maximumoutput estimation error less than 0.008 V.AM/AM and AM/PM characteristics are given in figure(5). Thus, we can clearly see that the non linear behavioralof the amplifier is successfully described by a traditionalthird polynomial series.Output amplitude (V)AM/AM characteristic0.2Measured 250.3Input amplitude (V)Output phase (rad)AM/PM characteristic3Measured data2EstimationA. Experimental results10Nonlinear parameters ck are extracted from the input/output transfer function. The AM/AM and AM/PM measured characteristics are obtained by sweeping the powerlevel of an input signal at a frequency located at the centerof the PA bandwidth. In our case, we used the 3th ordercomplex polynomial:¡ V NL c1 c3 · V in 2 c5 · V in 4 ·V in 1Input Amplitude (V) 2 30.050.10.15Input amplitude (V)0.2Fig. 5. Comparison between the measured and estimated AM/AM andAM/PM functionsThus, we define the estimated parameter complex vector:Fig. a0.3After running a LMS algorithm (Eq. 19), we obtained : ĉ1 1.222 0.115 jĉ 0.0918 0.0299 j 3ĉ5 0.017 10 2 0.062 10 2 jI axis0.10 0.1 0.20246810Time (µs)1214161820Fig. b0.3Measured dataEstimation0.2Magnitude (V)(21)Magnitude (V)0.2θ NL [ c1 c3 c5 ]TAmplitude (V)00.10 0.1Measured data0.2Estimation 0.20.10246810Time (µs)12141618200 0.1 0.200.050.10.150.20.25Time (ms)0.30.350.40.450.5Q axisAmplitude (V)Measured data0.2Estimation0.150.10.050 0.05 0.1 0.15 0.20Fig. 4.0.050.10.150.20.25Time (ms)0.30.350.40.450.5Comparison of time-domain measurement and estimationFig. 6. (a) Input signal. (b) Comparison of time-domain measurement andestimationThe initial values of the linear filter parameters areobtained by applying a Pseudo Random Binary Sequence(PRBS) signal with small amplitude level. The filter formis achieved using the RPM method for different plants. Aquadratic error comparison allows to obtain an appropriateorder. Then, the 3rd order filter are defined in the Laplacedomain as:H(s) b1 s b0s3 a2 s2 a1 s a0(22)

Thus, we define the estimated parameter vector:B. PA global identification :The model parameters obtained in the previous section willbe used to initialize the nonlinear identification algorithm.The unknown system in this case is the global PA modelcomposed by both: non-linear complex polynomial functions. 3rd order filter system.θ f [ a0 a1 a2 b0 b1 ]TThe RPM algorithm gives the parameters values: â0 2.01 · 1023 â1 6.11 · 1023â2 9.60 · 107 b̂ 1.51 · 1023 0b̂1 1.79 · 1015For small power, figure (6) shows that the PA dynamic canbe modeled as a 3rd order resonant system.Magnitude (V)0.10.080.060.040.020 0.02 0.04The measurements are performed by an input signal obtained from the adding of some P.R.B.Sequences at differentlevels. The aim is to drive the amplifier in its overall levelrange (linear and non linear area). Figure (9.a) shows theinput signal applied to perform global PA identification. After8 iterations, we obtain the following parameters: ĉ1 1.181 8.452 · 10 3 jĉ 0.042 0.023 jθ̂ NL 3ĉ5 0.201 · 10 2 0.316 · 10 2 j â0 2.01 · 1023 â1 6.11 · 1015â2 9.65 · 107θ̂ f b̂0 1.51 · 1023 b̂1 1.79 · 1015 0.06Fig. a 0.080.2246810Time (µs)1214161820Magnitude (V)0.15 0.100.10.050 0.05 0.1Identification residuals 0.15 0.2As observed in figure (7), the identification residuals(estimation error) are negligible and dont exceed 0, 04 V.The dynamic behavioral of the PA system can be describedby a MIMO filter. The filter characteristic is representedin figure (8) by the gain and phase curves. The resonnantfrequency of the filter is around 9.8 MHz.0246810Time (µs)12141618201820Fig. bEstimationMeasured data0.20.15Magnitude (V)Fig. 7.0.10.050 0.05 0.1 0.15 0.20246810Time (µs)121416Bode DiagramFig. 9. (a) Input signal. (b) Comparison of time-domain measurement andestimation0Magnitude (dB) 20Model simulation with the achieved parameters exhibit goodapproximation of measured data (fig. 9.b). 40 60 80C. Model validation 100360315Phase (deg)27022518013590450610Fig. 8.7108Frequency (rad/sec)10910Frequency responses of the PA gain and the phase1010In this section, we validate the PA model by comparingpredicted and measured outputs for different modulationschemes. As a test signal, we use a QPSK digitally modulated signal shaped with a raised cosine filter with a Rollofffactor of α 0.25.Figures (10.a) and (10.b) compare the simulated modeloutput (dotted line) with the measured output for an excitation signal different of the one previously used foridentification (solid line). It can be seen that the simulatedoutput follows the measured one.

Fig. aMagnitude (V)Measured data0.2Estimation0.10 0.1 0.2024681012141618201820Time (µs)Fig. bMagnitude (V)0.2EstimationMeasured data0.10 0.1 0.20246810121416Time (µs)Fig. 10. Comparison of I/Q time-domain measurement and estimation forQPSK modulation0Measured dataEstimation 10Magnitude (dB) 20 30 40 50 60 70 80 30 20Fig. 11. 100Frequency (MHz)102030Measured and estimated output spectrumTo validate the proposed model, figure (11) compares themeasured and simulated output power spectral densities atspecific frequencies.V. C ONCLUSIONA model based on continuous-time representation is described which offers a simple way to modeling PA dynamics.This model is able of accounting the magnitude and phaseamplifier nonlinearities such as the saturation effects.Test results illustrate the efficiency of this technique foruse in off-line identification. The continuous approach wasfound to be accurate in predicting the dynamical responseof the power amplifier. Estimation results show that thedescribed amplifier acts like a resonant system coupled witha polynomial series.The proposed technique is based on continuous timedomain model. The model achieved can be used to developa continuous baseband method for the compensation ofnonlinearity of the RF front-end in a wireless transmitter.R EFERENCES[1] S. C. Cripps, Advanced Techniques in RF power Amplifier Design,Artech House, Boston 2002.[2] P. B. Kennington, High-linearity RF Amplifier Design, Artech House,Boston 2000.[3] C. J. Clark, G. Chrisikos, M. S. Muha, A. A. Moulthrop and C. P.Silva, Time-domain envelope measurement technique with applicationto wideband power amplifier modeling, in IEEE Trans. MicrowaveTheory Tech. VOL. 46, NO. 12, pp. 2531-2540, Dec. 1998.[4] M. Isakson, D. Wisell and D. Ronnow, A Comparative Analysisof behavioral Models for RF Power Amplifiers, in IEEE Trans.Microwave Theory Tech., VOL. 54, No. 1, pp. 348-359, Junuary 2006.[5] L . 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continuous-time representation. II. PA MODEL DESCRIPTION The nonlinear amplifier model used in this paper is an extension of the discrete time-model at continuous represen-tation [3][13][24]. The major disadvantage of the discrete representation is that the used parameters have no physical significance, contrary to continuous one where parameters