This paper presents a historical review of the many behavioral models actually used to model radio frequency power amplifiers and a new classification of these behavioral models. It also discusses the evolution of these models, from a single polynomial to multirate Volterra models, presenting equations and estimation methods. New trends in RF power amplifier behavioral modeling are suggested.

Modeling nonlinear systems has shown to be a challenge in different areas of science. Most natural phenomena and physical devices present a nonlinear behavior. In this sense, it is very useful to classify nonlinear systems, so the right model can be used for each system.

In [

asymmetric responses to symmetric input signal changes (ASYM),

generation of higher-order harmonics in response to a sinusoidal input (HARM),

input multiplicity, meaning that one steady-state response corresponds to more than one steady-state input (IM),

output multiplicity, meaning that one steady-state input corresponds to more than one steady-state output (OM),

generation of subharmonics in response to any periodic input (SHAM),

highly irregular responses to simple inputs like impulses, steps, or sinusoids (CHAOS),

input-dependent stability (IDS).

A nonlinear system is classified due to phenomena presence as follows:

mild: ASYM, HARM, and IM,

intermediate: IDS,

strong: OM, SHAM, and CHAOS.

In electrical engineering, signal amplifiers are very often used for different purposes. One of the main uses is for signal transmission, where a power amplifier (PA) is needed. A radio frequency (RF) power amplifier is a typical nonlinear system. Even when the transistor is operating in a quasilinear region, driven with small variance input signals, the output signal has nonlinear components, due to the physics of the transistor.

A PA behavioral model (BM) remains in the mildly nonlinear class. The known PAs to be modeled present these characteristics in normal operation conditions, when tested with sinusoid stimuli. None of the other phenomena (OM, SHAM, CHAOS, or IDS), which imply the need of intermediate or strong nonlinear dynamic models, were observed in amplifier measurements.

This paper will present a classification of BMs and discuss the evolution of BMs based on VS used in the modeling of RF PAs, from some of the simplest models to recent ones reported in the literature.

The classification of PA BMs used in this work is as in [

memoryless (ML): the output envelope reacting instantaneously to variations in the input envelope,

linear memory (LM): BMs that account for envelope memory effects attributable to the input and output matching networks’ frequency characteristics,

nonlinear memory (NLM): dynamic interaction of nonlinearities through a dynamic network.

Figure

A PA representation using a nonlinear feedback structure [

Although this classification was very complete by the time of this paper [

the pruned Volterra series (rVS1) [

the pruned Volterra series (rVS2) [

a parallel cascade model (PCM) composed of a static nonlinearity and a reduced Volterra model (PNLrVS) [

a parallel cascade subsampled reduced Volterra series with the first branch composed of a static nonlinearity and a rVS model and other branches being rVS models, all with the same memory depth (PssVS), as detailed in [

A timeline of publications related to the accuracy of VS models is presented in Figure

A timeline of Volterra series models based on accuracy.

A graphical representation of all these models from the initial classification of nonlinear systems to the modern VS models is presented in Figure

Representation of various classes of nonlinear models and behavioral models used for RF power amplifiers modeling.

This section presents the evolution of PA BMs and their equations from nonlinear memoryless models to reduced Volterra Series models.

The nonlinear part of an amplifier model represents the intermodulation distortion (IMD), or the static part, and is usually composed of polynomials or other nonlinear functions (e.g., tangent-sigmoids, look-up tables). These models do not account for dynamics of the system.

In this section, some of the memoryless nonlinear models will be covered.

A nonlinear system can be represented by a power series:

A simple form to estimate a power series is using linear regression methods, as polynomial coefficients are linear in parameters. The polynomial regression matrix

Then LS equation can be applied:

The Hessian CN can be improved if orthogonal polynomials are used. These polynomials are derived based on the input signal used in the system. Thus, the regressors are closer to the ideal situation for a Hessian (regressors mutually orthogonal).

For real valued input signals, Chebyshev (derived for single tones) and Hermite (derived for Gaussian distribution) polynomials are typically applied.

Although polynomial LS estimation is a reasonable possibility to calculate the IMD components, it generates also “out-of-band” harmonics, as shown as an example in Figure

Frequency-domain response of a nonlinear amplifier supplied with a two-tone test input signal.

These are uninteresting for predistortion purposes, the main objective of behavioral modeling. To solve this problem, the first-zone equivalent (or baseband) polynomial is necessary. It can be derived writing the input signal as [

So, a binomial based expression for

Only the terms where

Using the binomial property and the relation observed in (

Finally the first-zone filtered input signal can be found as

The component

If no bias is present in the input/output signals, the regression matrix for the estimation of the coefficients of the baseband polynomial can be written as

The complex Bessel approximation of a memoryless RF power amplifier is obtained by the periodic extension of the instantaneous voltage transfer characteristics by a complex Fourier series expansion. This derivation was extracted from [

Look-up tables (LUTs) are the most common type of nonlinear static models in real-world implementations [

The two-box modeling techniques are a possibility to represent the linear memory of an amplifier. They are also known as modular approaches [

The most frequently used configuration for the linear block of this model is a FIR filter. The nonlinear block is commonly represented by a polynomial [

Wiener (a) and Hammerstein (b) models.

If the linear dynamic block is represented by a FIR filter, the output of this block for the Wiener model is

For the Hammerstein model, the FIR filter output is

If the static nonlinearity block is represented by a power series, the output of this block can be formulated for the Wiener Model and for the Hammerstein Model as follows:

The overall model output is then the combination of these equations for each model:

Equations (

More complex models are necessary to estimate the nonlinear memory, like parallel models or Volterra series. Examples of these models are parallel cascade models.

Any system that can be represented by a truncated VS (

This technique is the association in branches of various models (Wiener, Hammerstein, Wiener-Hammerstein, etc.). The overall model structure becomes more complicated with each iteration, as each branch is composed of a single model. The value of the cost function decreases or stays constant with each additional branch [

Example of a parallel Wiener model.

This method combines the following favorable properties:

computationally efficient even for high-order models with large memory-bandwidth products,

allowing the direct extraction of the Volterra kernels,

offering the convenience to use different methods for the identification of the linear and nonlinear blocks [

As a drawback, this method is very sensitive to noise if too many paths are used [

Although the best estimation methods to identify the parallel Wiener model’s coefficients would be the nonlinear ones, Korenberg proposed initially linear methods with acceptable results, as described in his paper [

Volterra series accounts for a mildly nonlinear class of nonlinear systems and has the property of dynamic interaction of nonlinearities, so it is well suited for the description of PAs.

The finite, discrete VS model is given by [

The main disadvantage of a VS based BM is the number of parameters necessary to estimate and consequently to represent the model. A VS model using

By using the symmetry condition, the complexity of the Volterra kernels as a function of the order of nonlinearity is given by the binomial [

In order to obtain the best model performance, it is necessary to adapt the BMs under study to the modern PA input/output industry standard signals, once these models are designed for linearization purposes. The excitation signals are complex valued, and as a practical issue only first-zone filtered (baseband) equivalent BMs are frequently used, due to the difficulties to implement bandpass models in hardware.

A closed form for determining the number of independent terms for baseband VS using complex signals is the binomial [

As an example, the numbers of parameters of a complex valued baseband VS using

Several techniques are employed to estimate VS. If the system is memoryless, VS are reduced to a Taylor series and can be estimated as described in Section

As shown above, VS presents a very high number of coefficients. This can lead to ill-conditioned Hessian matrices, as shown in [

There are other models that take into account some physical knowledge of the device and include important interactions of the input signal, as in [

Joining pruning techniques of VS, multirate techniques, and also parallelization of models (cascade), a new model was developed. This model presented different models composing the cascade. The first branch is a nonlinear static block, and the next branches are reduced Volterra series, each one estimated at a different subsampling rate (PssVS). This model and its estimation method are fully explained in [

A PA behavioral model representation using multirate and reduced Volterra series in a cascade configuration.

Based on Figure

The second output signal,

This equation is also described as in (

A comparison of this model with other models here cited using simulated and measured data obtained from a LDMOS RF PA is firstly presented in [

These results show a trend in LDMOS RF PA behavioral models that can not be override. Multirate models allow simpler hardware to be used in linearization devices that are the end products of BM. They also present a higher accuracy than any other model reported so far for LDMOS RF PAs. Further efforts in this research direction can reveal even more accurate models that use fewer coefficients with simpler hardware.

This paper presented a classification of nonlinear systems and also a modern classification of behavioral models of RF power amplifiers. Then an evolution of behavioral models was also presented, including all equations that characterize these models. This review showed several models, from basic baseband power series to the recent parallel multirate pruned Volterra series models, commenting also on their accuracy. Remarks about future trends in power amplifiers behavioral models were also made.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the PPEE-UFJF,