## 2 NN Models for HPA Pre-Distortion
### HPA Pre-Distortion Concept  
The model for a HPA can be characterized by the AM/AM and the AM/PM distortion, as it can be seen in Figure 1. The output \(y(t)\) of the non-linear amplifiers TWTA and SSPA corresponding to \(x(t)\) input, is expressed as  
\[y(t)=AM(|x(t)|)e^{i[PM(|x(t)|)+\phi(x(t))]} \tag{2}\]  
where \(AM(.)\) and \(PM(.)\) are, respectively, the AM/AM and AM/PM distortion functions. According to [PERSON]'s model [26], the AM/AM and AM/PM conversions can be written as  
\[AM(|x(t)|)=\frac{\alpha_{||}|x(t)|}{(1+\beta_{||}|x(t)|^{2})} \tag{3}\]  
\[PM(|x(t)|)=\frac{\alpha_{PM}|x(t)|^{2}}{(1+\beta_{PM}|x(t)|^{2})} \tag{4}\]  
where \(\alpha_{||}\) is the small signal gain, \(A_{sat}=\frac{1}{\sqrt{\beta_{||}}}\) is the input saturation voltage of TWTA or SSPA, and \(A_{max}=\frac{\alpha_{||}}{2\sqrt{\beta_{||}}}=\frac{\alpha_{||}A_{sat}}{2}\) is the maximum output amplitude.  
The modified Rapp model [27] is used instead for the SSPA case. Where, the AM/AM and AM/PM conversions is given as  
\[AM^{\prime}(|x(t)|)=g\frac{|x(t)|}{(1+(\frac{|x(t)|}{\bar{A}_{sat}})^{2s})^{ \frac{1}{2s}}} \tag{5}\]  
\[PM^{\prime}(|x(t)|)=\frac{\alpha|x(t)|^{c_{1}}}{(1+(\frac{|x(t)|}{\bar{B}})^{ c_{2}})} \tag{6}\]  
Figure 1: HPA’s operating principle.  
where \(g\) is the small gain signal, \(s\) is a smoothness factor parameter, \(A_{sat}\) is the saturation level with a similar meaning as in the TWTA, and parameters \(\alpha\), \(\beta\), \(c_{1}\) and \(c_{2}\) are adjusted to match the amplifier's characteristics.  
A common parameter in HPA is the Input Back Off (_IBO_), which is defined as  
\[IBO=10\log_{10}\frac{P_{sat}}{P_{avg}} \tag{7}\]  
where the \(P_{sat}\) represents the saturation input power and \(P_{avg}\) denotes the average input power. The _IBO_ accounts for how much power need to be reduced to obtain a low level of distortion output signal, and it is usually understood as a loss in link budget analysis. A summary of these operating principle is illustrated in Figure 1.  
The concept of pre-distortion is to compensate the AM/AM and AM/PM distortion with an inverse non-linearity. With the formulated AM/AM and AM/PM pre-distortion functions, the amplifier input can be re-written as  
\[y(t)=AM^{-1}(|x(t)|)e^{j[PM^{-1}(|x(t)|)+\phi(x(t))]} \tag{8}\]  
where \(AM^{-1}(|x(t)|)\) and \(PM^{-1}(x(t))\) are the AM/AM and the AM/PM pre-distorter functions, respectively. The pre-distorter output can be expressed as  
\[z(t)=AM^{-1}(|y(t)|)e^{j[PM^{-1}(|y(t)|)+\phi(y(t))]} \tag{9}\]  
It should be highlighted here that, although these pre-distortion schemes allow the use of lower _IBO_ than other techniques, we still need an _IBO_ to absorb the near-flat part of the curve due to the hard saturation in AM/AM characteristic (see Figure 2). In the case of the amplifiers in Figure 2, it should be around 4 dB of _IBO_ for the SSPA (worst case), around 2 dB for TWT2 and 1 dB for TWTA1, while other techniques would probably need an _IBO_ of around 8-10 dB. Of course, it will depend on the specific HPA characteristics and the hard saturation point. Since a minimum _IBO_ is needed, the dynamic range of the HPA is reduced. However, the higher the _IBO_ the lower the dynamic range. Thus, with pre-distortion techniques, since the _IBO_ can be lowered, so the the dynamic range can be larger than with other techniques.