Performance Evaluation of a Memory-Polynomial Model for Microwave Power Amplifiers

This paper is focused on a model for the power amplifier considering the memory effects in short terms, this work was developed for periodic signals through Matlab software and was implemented in Simulink. This model for the power amplifier implements a Memory-Polynomial model. Memory-polynomials prove to be both accurate and easy to implement and was compared with Ghorbani and Saleh quasimemoryless models to demonstrate its precision.


Introduction
An amplifier is a device designed to increase signal power levels.There are mainly two types of amplifiers in Radio Frequency (RF) front end circuits; these are power amplifiers (PA), and low noise amplifiers (LNA).Power amplifiers are mainly present in the transmitters, and are designed to raise the power level of the signal before passing it to the antenna.This power boost is crucial to achieve the desired signal to noise ratio at the receiver, and without which received signals would not be detectable [1].

Nonlinearity is an inherent property of High
Power Amplifier (HPA), in wideband applications, HPAs exhibit memory effect as well, which means the current output of an amplifier is stimulated by not only the current input but also previous input.Volterra series are a precise behavioral model to describe moderately nonlinear HPAs [2].However, high computational complexity continues to make methods of this kind rather impractical in some real applications because the number of parameters to be estimated increases exponentially with the degree of nonlinearity and with the memory length of the system.

Effects
The Volterra series can be used to describe any nonlinear stable system with fading memory.Memory effects due to the existence of components which store energy, such as inductors and capacitors, impedance of inductors and capacitors is relevant to frequency.PA memory effect is reflected as a non-linear distortion associated with the signal bandwidth and power [8].However, its main disadvantages are the dramatic increase in the number of parameters with respect to nonlinear order and memory length, which causes drastic increase of complexity in the identification of parameters.
As the input signal bandwidth becomes wider, such as in WCDMA (Wideband Code Division Multiple Access), the time span of the power amplifier memory becomes comparable to the time variations of the input signal envelope.A Volterra series is a combination of linear convolution and a nonlinear power series so that it can be used to describe the input/output relationship of a general nonlinear, causal, and time-invariant system with fading memory [6].The Volterra series in discrete-domain can be represented as equation (1).
where (*) denote the complex conjugation and x(n) and y(n) represents the input and output of the model.It can be observed that the number of coefficients of the Volterra series increases exponentially as the memory length and the nonlinear order increase making it unpractical for modeling power amplifiers in real time applications [4].y(n) is the output complex base-band signal.a k,q are complex valued parameters.

The Memory-Polynomial
Q is the memory depth.
K is the order of the polynomial.

Implementing of the Memory-Polynomial Model
As it was expressed in equation ( 2), a memory-polinomial model can be rewritten as follows: where F q (n) can be expressed as: The equation ( 3) can be defined as block diagram as shown in the Figure 2    Based on the block diagram showed in the Figure 3 using the parameters gotten of a 2k-1,q is possible to create the same structure using the same sinewave and AM Modulation as can see briefly in the Figure 4b, was inserted a sinewave modulated in amplitude and sampled during 0.1 secs, in the Figure 5 is showed that after 0.1 seconds the amplifier is stable and is amplifying x(n).
In comparison with the Volterra series to calculate the parameters a 2k-1,q and the output y(n), there is one more internal cycle to generate the output and the parameters so was required more data processing.The

The Memory-Polinomial model
The Memory-Polynomial Model consists of several delay taps and nonlinear static functions.This model is a truncation of the general Volterra series, which consists of only the diagonal terms in the Volterra kernels.
Thus, the number of parameters is significantly reduced comparing to general Volterra series [1].The Figures 8 and 9 were made considering the intermodulation effect.

The Saleh model
The

The Ghorbani model
The     sinewave modulated in Amplitude.

Comparing Models
An RF Satellite Link (Fig. 15) was simulated.
In Figures (16- Model as Special Case of the Volterra series Performance Evaluation of a Memory-Polynomial Model for Microwave Power Amplifiers 15 The memory-polynomial model, [3] consists of several delay taps and nonlinear static functions.This model is a truncation of the general Volterra series, which consists of only the diagonal terms in the Volterra kernels.Thus, the number of parameters is significantly reduced compared to general Volterra series, the memory-polynomial model is considered as a subset of the volterra series.The model is shown in Figure 1.

Figure 6 Fig. 5 .
Figure 6 shows an overview of the implementation and the Figure 7 the amplified output y(n) of the Memory-Polinomial Model.
Figures 8 and 9 the Memory-Polynomial Model has an amplification slope between 0V and 1V , this is a good condition if the amplification is going to be used for applications that require just a little energy, i.e. telephones or micro and nano-devices.The Memory-Polynomial Model has the advantage that the AM/PM conversion isn't changing.
Ghorbani model uses eight parameters to fit the model to measurement data, this model is quasi-memoryless, and its AM-AM Performance Evaluation of a Memory-Polynomial Model for Microwave Power Amplifiers 19 and AM-PM conversions functions are described by the following equations: x 1 , x 2 , x 3 , x 4 , y 1 , y 2 , y 3 , y 4 ] are the models parameters, which are calculated from measurement data by means of curve fitting [10].The graphic representation of the equations 7 and 8 are showed by the Figures 12 and 13, respectively.

Fig. 14 .
Fig. 14.(a) Overview of Saleh and Ghorbani models,(b) Amplification made by the Saleh, Ghorbani and Memory-polynomial model to a 19) are plotted input, AM signal, Memory-polinomial model output, and demodulated output signals are plotted, respectively.This case is for the memorypolynomial model without memory depth.
his was at the Head of the team "Radiofrequency Devices, Circuits and Systems" of the LPM/INL, dealing with noises or parasitic disturbances in mixed complex 2D and 3D RF circuits and systems., in Cuernavaca, Mexico, in 2003, and the PhD degree from the Institut National des Sciences Appliquées de Lyon (INSA-Lyon), Villeurbanne France, in 2007.In first semester 2008, he was a Research Director at Advanced Technology Research S.A. de C.V. (ATR) in Guadalajara, Mexico, where he led a team of researchers working on networking, and telecommunication architectures.Currently, he is a Professor at the Centro de Investigación y Desarrollo de Tecnologia Digital (CITEDI) of Intituto Politecnico Nacional (IPN), in Tijuana, Mexico.He is the Research Coordinator in Telecommunications Department at CITEDI-IPN.His research interests include digital and analog circuits design, device physic modeling, Si/SiGe:C heterojunction bipolar transistor, VCO design, oscillator phase noise, high frequency circuits, DSP and FPGA design, circuit and system cosimulation, and electromagnetic compatibility.