# Volterra Series Models

After understanding the origins of memory effects, it is important to take them into consideration when analyzing the effect of power amplifiers on signal linearity. These effects are dependents on different factors related to the signal and power amplifier characteristics. While electrical memory effects are a function of the signal bandwidth, thermal memory effects are a function of the amount of power dissipation in the power amplifier and the cooling circuit used to dissipate the heat from this dissipated power.

**Figure 2.11 **Generic block diagram for behavioral model performance assessment

On one side, if a signal with relatively narrow bandwidth is used such that the responses of the amplifier and its matching networks around the envelope frequency are considered constant over the bandwidth of the signal, the output signal of the power amplifier is considered to have non-significant electrical memory effects. The electrical memory effect can then be neglected without affecting the analysis of the power amplifier behavior. In practice, if the signal has a bandwidth lower than 10 MHz, a careful power amplifier design guarantee nearly constant frequency response around the envelope frequency, the fundamental, and the harmonics. In this case, the electrical memory effect can be neglected.

On the other side, if the power amplifier’s junction temperature variation is small enough so that its effect on the gain of the power amplifier is insignificant, the thermal memory effect can be neglected. In practice, a temperature variation of few degrees will not introduce significant changes in the power amplifier gain. Therefore, to have negligible thermal memory effect, the variation in the power amplifier power dissipation defined in Equation 2.27 should satisfy: *AP _{disspated}* should be in the same order of magnitude or smaller than ^. This condition can be satisfied automatically for ideal class A power amplifiers where

*AP*is zero or for power amplifiers with low power dissipation, for example, efficient switching mode power amplifiers.

_{disspated}If a power amplifier has negligible electrical and thermal memory effects, it can be modeled using a nonlinear static model that does not have any memory effect. Often, Taylor series are used for such modeling and the output of the power amplifier, *x _{out}(n), *can then be related to the input

*x*by [3] and [22]:

_{in}(n)

However, if the bandwidth increases, the electrical memory effects can no longer be neglected. Therefore, memory effects should be taken into consideration when modeling power amplifiers. The Volterra series [23-25], can be used to accurately characterize a dynamic nonlinear system including linearity and the different types of memory effect. In Volterra series models, the output signal is related to the input signal as follows:

where *h _{p}(i_{1},* ... ,

*i*are the parameters (kernels) of the Volterra model,

_{p})*K*is the nonlinearity order of the model, and

*M*is the memory depth. Each kernel of the Volterra series models a given nonlinearity order and its corresponding memory effect. A к-th order kernel includes all possible combinations of a product of

*к*time shifts of the input signal. Therefore, it includes all possible forms of memory effect and is considered the most complete model to take into account linearity and any type of memory effect. However, it results in a large number of coefficients that increases exponentially with the degrees of the nonlinearity and memory depth of the system. The increase in the number of coefficients increases the computational complexity of the model. Therefore, in practice, the Volterra series model is limited to modeling systems with low nonlinearity and memory orders. To overcome the computational complexity of the Volterra series, different reductions of the Volterra series have been proposed [32, 33]. These complexity reduced models will be described in the following chapters.