Volterra series – RF power amplifier model

Key focus: Volterra series is an useful tool for modeling nonlinear time invariant systems in general, and they are commonly used to model RF power amplifiers with memory effects

Introduction

Power amplifier are nonlinear devices responsible for compensating for the signal power loss between a transmitter and receiver. The input power levels to the power amplifiers can be increased to compensate for the power loss during transmission. At low input power regime, an ideal power amplifier provides constant gain to the output signal.

However, in the high input power regime, nonlinear effects of the power amplifier come to play, the power of the output signal shows diminished gain (compressed) and starts to saturate.

1-dB compression point is the most important measure of quantifying the beginning of non-linearity of the power amplifier (Figure 1). It is the point where the RF output power level decreases 1 dB from the constant value. Therefore, 1-dB compression point is a measure of the beginning of distortion, where a perfect sine wave fed at the input of the power amplifier do not maintain the perfect shape at the output of the power amplifier.

It is therefore critical to model and understand the power amplifier in order to correct for the non-linear distortion it creates and not break the requirements defined by the agreed-upon standards like 3GPP.

Memoryless models: Taylor series

Memoryless models place emphasis on the power amplifier with memoryless nonlinearity, which means that the current output is determined solely by the current input via a nonlinear method. A nonlinear memoryless system can be modeled using Taylor series. For example, a discrete time system with input x[n] and output y[n], the nonlinearity is incorporated using nonlinear terms with corresponding coefficients/weights (ɑ_i) for each term.

Memory models : Volterra series

The bandwidth of the signals, such as in 3GPP LTE and 3GPP 5G NR, gets wider. When the wideband signals are input to high power amplifiers (such as in base stations), the power amplifiers start to exhibit memory effects. As a consequence, the power amplifier’s current output is affected not only by the current input, but also by previous input values. Volterra series and its derivatives are commonly used to model power amplifiers with memory.

Volterra series is an useful tool for modeling nonlinear time invariant systems in general, and they are commonly used to model RF power amplifiers with memory effects. The Volterra series is a polynomial based mathematical model that captures the nonlinear terms and memory effects present in these devices.

For a discrete time system with input x[n] and output y[n], the Volterra series with nonlinearity order K and memory depth Q is given by

where,

h_k(i₁, …, i_k) are the n^th order Volterra Kernels. It can be understood that it models any system by using the system’s impulse responses. Because of the high complexity and high resource requirements, Volterra series cannot be used in practical applications.

For practical applications, Volterra series has to be truncated in such a way that it has the least impact on the models performance. This can be accomplished by omitting some of the terms in equation (2) for each application. Special versions of Volterra series allow for use in power amplifier behavioral modeling. Memory polynomial model is one of the derivative of Volterra series that is easier to implement. We will discuss this model with an example simulation model in the next post.