## 3 Transmitter Hardware Impairments or 'RF Features' Overview
### PA Non-Linearity Models  
The power amplifier is an important element in the wireless communications system, and its non-linearity behaviour varies from device to device. It is expected that PA non-linearities can also be used as differentiating features between GNSS satellite transmitters and spoafers or jammers, due to the fact the GNSS PAs are high-cost high power amplifiers (HPA), such as solid state power amplifiers (SSPA) or a travel-wave tube amplifier (TWTA) [39], while non-genuine GNSS transmitters typically have low-power amplifiers (LPA) [40]. The highest PA power efficiency is achieved at the saturation point, where heavy non-linearity occurs in all PA models [48].  
There are typically two classes of models for PA non-linearities [49]: the memoryless non-linear models and the non-linear models with linear memory. The memoryless non-linear model of a system with input \(x(t)\) and output \(y(t)\) (assuming a continuous-time model) is given by the \(L_{th}\) order polynomial:  
\[y(t)=\sum_{l=1}^{L}\alpha_{l}x^{l}(t) \tag{4}\]  
where \(\alpha_{l},l=1,\ldots,L\) is the \(l_{th}\) coefficient of a PA non-linearity of order \(L\). When the wideband signals pass through the power amplifier, the bandwidth of signals is not negligible compared with the inherent bandwidth of the amplifier, and therefore a frequency-dependent behaviour occurs. This behaviour is called a memory effect. Regarding the non-linear model with linear memory, the two most encountered models are the Wiener model and the Hammerstein model, as described in [49]. We illustrated these two models in Figure 4. The corresponding mathematical expressions (this time in the discrete-time domain) are, respectively:  
Wiener model:  
\[y_{Wiener}(s)=\sum_{n=0}^{N}c_{n}\left[\sum_{q=0}^{Q-1}h(q)x(s-q)\right]^{n}\] (5a) Hammerstein model: \[y_{Hammerstein}(s)=\sum_{q=0}^{Q-1}h(q)\left[\sum_{n=0}^{N}c_{n}x^{n}(s-q)\right] \tag{5b}\]  
where \(s\) is the sample index (assuming the \(x(t)\) signal was sampled at a sampling rate \(1/T_{s}\), namely at \(t=s/T_{s}\) time instants), \(h(q)\) denotes the \(q\)-th coefficient of a finite impulse response (FIR) filter, and \(c_{n}\) denotes the \(n_{th}\) order coefficient in the polynomial memory model.  
Due to the difficulties of estimating the coefficients for FIR filters in both the Wiener and Hammerstein model, the memory polynomial [50] has become a popular model for the behaviour of power amplifiers. The expression of MP is given by [50],  
\[y_{MP}(n)=\sum_{k=0}^{K-1}\sum_{m=0}^{M}a_{km}x(n-m)|x(n-m)|^{k} \tag{6}\]  
where \(a_{km}\) are the model parameters. In our work, we used the memory polynomial to model PA in navigation payload with user-defined model parameters \(a_{km}\) which were considered different for each transmitter (i.e., satellite and spoofer transmitters).  
In order to maximize the power efficiency and the lifespan of the satellite payload, the GNSS signals are usually designed to exhibit a (quasi) constant complex envelope. For instance, this is achieved by including an inter-modulation product among the signal components. For this reason, it is reasonable to expect that the PA non-linearities will not significantly distort the genuine GNSS signals. This might not hold for many spoofing signals, which may simplify the signal generation by only emulating some of the signal components and/or omit the inter-modulation product. However, it shall be noted that a spoofer usually needs to generate low-power levels, hence it is easier to ensure linearity with LPA. The fact that the spoofer needs to transmit at a lower power than the GNSS transmitters is due to the fact that spoofers are usually within the range of a few tens of meters to a few km away from the GNSS receivers, while GNSS satellites are at more than 20,000 km away from the receivers.