Abstract-In this paper, a novel Extended Hammerstein model is presented to accurately mimic the dynamic nonlinearity of the wideband Radio Frequency Power Amplifiers (RFPA). Starting with a Conventional Hammerstein model scheme, which fails to predict the behavior of the RFPA with short term memory effects, two areas of improvements were sought and found to allow for substantial improvement. First, a Polar Feed-Forward Neural Network (FFNN) was carefully chosen to construct the memoryless part of the model. Then, the error signal between the output and the input signal of the memoryless sub-model was filtered and then post-injected at the model output. This extra branch, when compared to the Conventional Hammerstein scheme, allowed for an extra mechanism to account for the memory effects due to dispersive biasing network that was present otherwise. The excellent estimation capability of the Polar FFNN together with the additional filtered error signal post-injection led to remarkable accuracy when modeling two different RFPAs both driven with 4 Carrier (4C) WCDMA signals. Despite its simple topology and identification procedure, the Extended Hammerstein model demonstrated is capable in accurately predicting the dynamic AM/AM and AM/PM characteristics and the output signal spectrum of the RFPA under test.
Index Terms-Power Amplifier Behavioral Modeling, Power Amplifier Nonlinearity, Wiener/Hammerstein model, Artificial Neural Network.
INTRODUCTION
P ower Amplifiers (PA) are key building blocks of emerging wireless radios systems. They dominate their power consumption and the sources of distortions especially when driven with modulated signals. Several approaches were devised to characterize the nonlinearity of a PA. Among those, dynamic amplitude (AM/AM) and phase (AM/PM) distortion characteristics as a function of the input power, Adjacent Channel Power Ratio (ACPR), and Error Vector Magnitude (EVM) were widely used to characterize the PA nonlinearity and its effects on the output signal when driven with realistic modulated signals in power, frequency or time domains [1]-[3], respectively. PAs' inherent nonlinear behavior generally yield output signals with unacceptable quality (high EVM) and unacceptable level of out-of-band emission (low ACPR) that usually fail to meet the established performance standards. Traditionally, brute force PAs are forced to operate deeply in back-off from their power capability to pass the mandatory spectrum mask and EVM. Despite its simplicity, this solution is increasingly discarded as it leads to cost and power inefficient radios. Alternatively, several linearization techniques (feedback [4], feed-forward [5] and predistortion [6]) were devised to tackle the PA nonlinearity and consequently improve the achievable PA's linearity vs. power efficiency trade-off.
Among these linearization techniques, Digital Pre-Distortion (DPD) technique consists of incorporating an extra nonlinear function before the PA, to preprocess the input signal to the PA, so that the overall cascaded systems behave linearly. The overall linearity of the cascaded system (DPD+PA) relies primarily on the ability of the DPD function to produce nonlinearities that are equal in magnitude and out of phase to those generated by the PA. Hence, good understanding and accurate modeling of PA distortions is a crucial step in the construction of the adequate DPD function. Different PA model schemes were suggested in the past to deal originally with its static nonlinear namely Saleh model [7], Memoryless Polynomial model [8] or Look-Up-Table (LUT) model [9]. However, as emerging radio systems are increasingly evolving to wider bandwidth signal, sophisticated behavioral models have been developed to account for the additional memory effects [6] at the cost of extra complexity. As an example, despite its comprehensiveness, Volterra model [10] has always been criticized for its prohibitive complexity and restricted applicability to mildly nonlinear PAs. Memory Polynomial model [11], [12] is currently the most popular derivation of Volterra model that excluded the cross terms to alleviate the complexity. Nevertheless, the identification of the Memory Polynomial model parameters using Least Square Error (LSE) like algorithm is still relatively computationally complex. Additionally, two-box models [13]-[17], generally known as Wiener or Hammerstein models, have been employing a cascade of nonlinear function and a linear filter to model dynamic nonlinear systems. In the case of the PA, the first box of the Hammerstein scheme captures its static nonlinear behavior, while the second one is intended to take into account for its memory effect. In the literature, polynomial functions, LUT and neuro-fuzzy inference system has been used to construct the static nonlinear function of the Hammerstein/Wiener model. However, authors in [15] showed the limitation of Conventional Hammerstein/Wiener scheme in mimicking the behavior of wideband PA. In addition they suggested an augmented version of these models where the linear block is replaced with weakly nonlinear one with multiple filters to solve the modeling inaccuracy. Parallel Hammerstein/Wiener [16] models have been also suggested to address the traditional schemes limited capability in accounting for the memory effects by stacking extra branches in parallel. Yet, their parameters identification is too tedious.
In this paper, the improvement of the Conventional Hammerstein model ability in mimicking the behavior of wideband PAs with memory effects have been carried out from two different perspectives. In a first step, a Polar Feed Forward Artificial Neural Network (FFNN) has been carefully chosen for accurate construction of the static nonlinear part. The error signal between the input and the output signals of the memoryless module was then computed and used to feed a second FIR filter. The post-injection of the filtered error signal at the output of the Conventional Hammerstein allowed for substantial improvement of the modeling accuracy. Indeed, the extra error signal with the corresponding filter was found to complement the Conventional Hammerstein model by adding adequate mechanism to account for the memory effects in the PA behavior attributed to the dispersive bias network.
The remainder of this paper will be organized as follow. The Extended Hammerstein model will be first introduced in the second section. Then the construction of its two sub-modules will be explained in the third and fourth sections. In the last section, the performance of the Extended Hammerstein model in predicting the dynamic nonlinear behavior will be carefully evaluated and compared to previous modeling approaches one.
Extended Hammerstein Behavioral Model using Artificial Neural Networks
Conventional Hammerstein model is only able to partially predict the behavior of a PA when driven with a wideband signal. The first filter (FIR1) in Fig. 1 is only allowing Conventional Hammerstein to account for the memory effects that are attributed to the dispersive frequency response of a PA around the carrier frequency. In this paper, envelope post-injection method is proposed to extend the modeling capability of Conventional Hammerstein model by including an extra mechanism that will account for the memory effects attributed to the dispersive biasing network. For that, an error signal is first calculated by subtracting the input signal from the memoryless block output signal as depicted in Fig 1. It is worth mentioning that since the objective here is to capture the dynamic nonlinearity of the Device Under Test (DUT), in other words the deviation of the output signal from the ideal linear one, the measured output signal was first normalized by dividing it with the small signal gain of the DUT. Hence, the model of Fig. 1 and its memoryless sub-block have a small signal gain equal to 1. One can easily deduce the actual output signal by multiplying with the small signal gain.
(1)
Fig. 1. Extended Hammerstein PA model diagram.
The error signal represents the intermodulation distortion product generated by the static nonlinearity of the PA. The error signal is then fed to a second filter (FIR2) in Fig. 1. The frequency variation of the FIR2 in Fig. 1 will be used to introduce the frequency-dependency in the intermodulation distortion products that would be caused by the memory effects attributed to the base band frequency variation of the biasing network [18]. Hence, after post-injecting the filtered error signal, the output signal of the Extended Hammerstein model will include both dominant memory effects mechanisms and will consequently allow for better modeling accuracy than Conventional Hammerstein one, as will be demonstrated latter in the paper.
The memoryless block of Conventional Hammerstein model has been generally implemented using Polynomial functions, LUT or neuro-fuzzy inference systems. This paper combines the Extended Hammerstein model previously mentioned advantages with the efficiency of the Artificial Neural Network (ANN) by modeling the static nonlinear block with a FFNN. Indeed, ANNs are universal estimator [19] with excellent capability to learn the behavior of any nonlinear system based strictly on its input and output signals.
Neural network memoryless sub-model
This section will give the details of the construction of the FFNN sub-module from choice of its structure to its training. For that Polar and Rectangular signal representations were initially considered and their implication on the FFNN performance was carefully studied. The two FFNN structures, Polar FFNN (PNN) and Rectangular FFNN (RNN) were set as depicted in Fig. 2 and Fig .3, respectively. and designate the amplitude envelop of the input and output signals, respectively. denotes the phase difference between the output and input signals. and represent the in-phase and quadrature components of the input and output signals, respectively. In the Polar case, two separate and uncoupled real valued (i.e real weights and biases) FFNNs are used to model the output amplitude and phase distortions, as shown in Fig. 2. For the Rectangular representations, one real valued FFNN is used to model the in-phase I and quadrature Q components, as shown in Fig. 3. Hyperbolic tangent sigmoid functions defined in (2) are used in both FFNN structures.
(2)
Fig. 2. Polar Feed Forward Artificial Neural Networks.
Fig. 3. Rectangular Feed Forward Artificial Neural Networks.
FFNN training and validation results
Back Propagation Learning Algorithm (BPLA), available in MATLAB and Mathwork ANN toolbox, is used for training the two FFNN structures. This algorithm optimizes the network parameters in order to minimize the cost function E; defined in (3), over a training epoch.
(3)
where En is the instantaneous error, and represent the output measurement data (desired data), and represent the RNN output, and are the measured output signal's amplitude and phase distortion (desired data), and are those of the PNN output. N denotes the length of the training sequence.
The measured input and output signals of a 400Watt Doherty PA (2x Freescale MRF7S21170H) driven with a 4C-WDCMA with a Peak to Average Power Ratio (PAPR) of 7.412dB, synthesized in Agilent Design System (ADS) are used to train the FFNN structures previously presented, using the BPLA. In this paper the carrier frequency of the wideband test signal (bandwidth=20MHz) was set to 2.14GHz. Training and validation of the ANN were achieved using different segments of the signal to ensure the generality of the sub-models. 10k samples were used for the training phase and 20k different samples were used later for the validation phase. The same training data will be used later for the construction of the two remaining linear filters. In this paper, one hidden layer which contains of 15 neurons is used in both structures.
After training, which take around 30 seconds in each case, the validation of the two networks yields very similar Normalized Mean Square Error (NMSE) of about -28dB. However, after examining closely Fig. 4 and Fig. 5, which show the AM/AM and AM/PM distortions obtained using the output signals predicted by the RNN and the PNN, one can conclude that the two structures behave differently.
The PNN leads to expected and smoothed AM/AM and AM/PM nonlinear characteristics as shown in Fig. 5. However, the RNN behavior is less accurate because it leads to different possible AM/AM and AM/PM distortions for a given input power level Pin, as shown in Fig. 4. Knowing that the RNN uses the current input signal's sample to predict the memoryless part of the PA behavior, the dispersion in its AM/AM and AM/PM characteristics is unexpected. Indeed, the dispersion in the measured characteristics is assumed to be fully attributed to the memory effects that yield to the dependency of the PA behavior on the current and past input samples. Further study carried out on the RNN model's output signal revealed an extra PM/AM and PM/PM distortions. These extra sources of distortion are non physical and only introduced by the RNN structure in its search for the best fitting of the training sequence that exhibit memory effects. As a proof of the last statement and as one can clearly observe in Fig. 6, where AM/AM and AM/PM characteristics of the RNN were drawn for different values of the input signal phase, different AM/AM and AM/PM curves are obtained for a given phase value. Therefore one can deduce that the RNN structure introduced a phase dependent source of distortion (PM/AM and PM/PM) as a result of the memory effects present in the training sequence.
One can conclude that the PNN structure is more suitable in accurately predicting the memoryless part of the PA behavior. Contrary to the RNN, the PNN structure prevented it from introducing unwanted and nonphysical mechanisms (e.g. PM/AM or PM/PM distortions in the RNN) that would affect the overall modeling accuracy.
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(b)
Fig. 4. AM/AM (a) and AM/PM (b) characteristics obtained using the measurement and RNN model data.
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(b)
Fig. 5. AM/AM (a) and AM/PM (b) characteristics obtained using the measurement and PNN model data.
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(b)
Fig. 6. AM/AM (a) and AM/PM (b) characteristics for different phases.
Linear Filter
Thus far only the static nonlinear block of the Extended Hammerstein model is constructed using a polar FFNN trained using the modulated input and output signal of the DUT. Determining the PNN module of Fig. 1 will allow for the calculation of its output signal which will be used later as an intermediate signal. will be also used to compute the error signal of Fig. 1. and together with the measured output signal will be used to identify the coefficients of the two FIR filters that are proposed to capture the memory effects due to the frequency-dependent biasing circuits and harmonic loading of the power transistors.
The relation between these three signals can be expressed as follow:
(4)
where and denotes the number of taps ( and ) in the first and second filter.
The identification of two FIR filters set of parameters ( and ) is performed using the LSE algorithm. The same range of data used in the FFNN training step is used here. For that, the number of taps of the two filters ( and ) were kept increasing until stable modeling accuracy is reached. Based on Fig. 7, the length of the two FIR filter can be set to be equal to 10. In the remainder of this paper, the validation of the extended Hammerstein model was carried out using 10 taps filters.
Fig. 7 NMSE vs. FIR length for the ANN based Extended Hammerstein behavioral model.
TABLE I
1st Power amplifier Modeling Comparison
Model
NMSE 1stPA at max compression
NMSE 1stPA at 2dB back-off
Memoryless
-28.1088
-28.5972
Con. Ham.
-32.9687
-35.0941
Aug. Ham.
-33.9871
-35.9412
Ext. Ham.
-34.1043
-36.2403
Extended Hammerstein Model Validation
The capacity of the extended Hammerstein model in predicting the responses of the RFPAs will be used to validate its accuracy in the power, time, and frequency domains. It is worth mentioning that model validation approaches suggested in [20] was used in this paper as it puts the emphasis on the capability of the modeling scheme in predicting the memory effects. For that a memoryless digital predistortion function was applied to the 4C WCDMA signal and its output is used to drive the RFPAs under test. Hence, the focus will be in assessing the capability of the Extended Hammerstein model in predicting the residual out of band emission dominated by the memory effects.
Furthermore, for extensive validation of Extended Hammerstein model was performed in different scenarios to measure its accuracy in predicting the dynamic nonlinearity of different PA topologies with different nonlinear characteristics. For that, the first amplifier was operated in two different operation power conditions namely maximum compression point and 2dB back off from that point. In addition, a different 130Watt-LDMOS Class AB amplifier was also used in the experiments. Besides, the 130Watt Class AB PA was also biased in two different biasing conditions i) Class AB close to A (Idq=0.9A) and ii) deeper class AB (Idq=0.55A).
Table I, Fig. 8, Fig. 9 and Fig. 10 summarize the performance of the Extended Hammerstein in mimicking the first RFPA under test as compared to other alternative modeling approaches. On one hand Fig. 8 depicts the measured and predicted AM/AM and AM/PM characteristics of the 1st DUT. The very similar dispersion in the measured and predicted characteristics gives a first order pledge in the capacity of the Extended Hammerstein to mimic the PA behavior.
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Fig. 8. AM/AM (a) and AM/PM (b) characteristic comparison between the PNN behavioral and Measurement behavioral.
The Extended Hammerstein model prediction capacity was also confirmed in Table I which summarizes the NMSE of the different modeling alternatives considered in this paper. In the other hand, the spectra of the predicted signals obtained using the (b) memoryless model, (c) Conventional Hammerstein model, (d) Augmented Hammerstein model and (e) Extended Hammerstein model were compared to the measured one as shown in Fig. 9. One can clearly observe the superior modeling performance of the FFNN based Extended Hammerstein model. To support this statement, the error spectra of the different modeling approaches were traced in Fig. 10. Based on this figure, the Conventional Hammerstein was only able to reduce the in band error; however, its performance was equal to the memoryless one in the out of band frequency range. This supports our initial statement in the paper where the Conventional Hammerstein was supposed to predict the memory effects due to the non constant frequency response of the PA around the carrier. The traces (d) and (e) in Fig. 9 are confirming the superiority of the Augmented and Extended Hammerstein models which improved the out of band spectra error. Further examination of the Fig. 10 revealed extra modeling accuracy of the Extended Hammerstein as compared to the Augmented Hammerstein model in the frequency range from -0.6 to 0.6. The same modeling accuracy was obtained at 2dB back off as one can observe in Fig. 11. Hence one can conclude the capacity of the Extended Hammerstein to predict the 1st DUT behavior from mild compression to strong nonlinearity regions.
Fig. 9. Normalized PSD of the 1st PA output signals predicted by b) memoryless model, c) conventional Hammerstein, d) Augmented Hammerstein, e) Extended Hammerstein and the a) measured one- maximum operation power case.
Fig. 10. Normalized PSD of the error signal of the a) memoryless model, b) conventional Hammerstein, c) Extended Hammerstein, d) Augmented Hammerstein-maximum operation power when predicting the response of the 1st PA at maximum compression
Fig. 11. Normalized PSD of the error signal of the a) memoryless model, b) conventional Hammerstein, c) Extended Hammerstein, d) Augmented Hammerstein- when predicting the response of the 1st PA at 2dB back-off from maximum operation power
The results of the assessment of the model accuracy in predicting the response of the 2nd PA are summarized in the Fig. 12 and Fig. 13. As one can clearly observe from these two figures, the Extended Hammerstein maintained its modeling accuracy, which is superior to the Augmented Hammerstein and the Conventional Hammerstein, when predicting the response of the 2nd PA operating at its maximum compression being in light class AB or deep class AB conditions.
Fig. 12. Normalized PSD of the error signal of the a) memoryless model, b) conventional Hammerstein, c) Augmented Hammerstein, d) Extended Hammerstein when predicting the response of the 2nd PA (light class AB) operating at maximum operation power
Fig. 13. Normalized PSD of the error signal of the a) memoryless model, b) conventional Hammerstein, c) Augmented Hammerstein, d) Extended Hammerstein when predicting the response of the 2nd PA (deep class AB) operating at maximum operation power
TABLE II
2nd Power amplifier Modeling Comparison
Model
NMSE 2ndPA (High class AB)
NMSE 2ndPA (deep class AB)
Memoryless
-30.4201
-34.5056
Con. Ham.
-34.7894
-37.9620
Aug. Ham.
-35.5554
-38.4444
Ext. Ham.
-35.9469
-38.7013
Conclusion
In this paper an ANN based Extended Hammerstein model has been proposed to predict the dynamic nonlinear of wideband PAs. The excellent and universal estimation capability of FFNN when coupled with the extra post-injection of the filtered error signal allowed for excellent accuracy in predicting the PA behavior with memory effects. Polar signal representation was preferred over the Rectangular representation in the construction of the FFNN structure. Indeed, the RNN was found to be not adequate for the modeling of the memoryless nonlinearity as it introduces non physical mechanisms (PM/AM and PM/PM distortions) as results of the memory effects in the training sequence.
Although the second module of the novel Extended Hammerstein model was kept linear, the added filtered error signal was found to improve substantially the accuracy of the Conventional Hammerstein by offering an extra mechanism to account for the memory effects due to the memory effects attributed to dispersive biasing network.
