# GNSS Signal Distortion Estimation: A Comparative Analysis of L5 Signal from GPS II and GPS III
## 2 Models and Methods  
### Navigation Payload Model  
As shown in Figure 1, the navigation signal generating unit completes the modulation of the spreading code, subcarrier, and data message to obtain an ideal infinite bandwidth constant envelope baseband signal. Next, the harmonics and mirror signal interference are filtered out for digital-to-analog conversion. Afterward, the out-of-band spurious and harmonic interference are filtered and then upconverted to the broadcast frequency. Through HPA amplification, it enters the multiplexer for signal multiplexing. Finally, it is broadcast by the antenna.  
As shown in Figure 1, the navigation payload transmission channel is mainly composed of digital and analog components, and each component has different effects on the signal quality [9; 10; 11; 12; 13; 14; 15]. The baseband signal generation and digital filtering usually introduce small linear distortion. D/A conversion and up-conversion are usually modeled as linear distortions because of the same source as baseband signal frequency synthesis [15]. Pre-band-limit filtering destroys the constant envelope characteristics of the signal, resulting in severe nonlinear distortion after HPA [15]. After output filtering and multiplexing, this becomes even worse. The navigation signal is an intuitive reflection of the characteristics of the transmission channel. Comparing the influence of different units on the signal, as shown in Figure 2, this work models the navigation payload signal distortion in equivalently two parts: digital distortion and analog distortion [10]. In this work, the influence before HPA is uniformly classified as digital distortion, whereby it is considered that digital devices only bring a time delay and do not affect signal waveform distortion. Hence, the digital distortion is represented using the classic TMA model, and the analog distortion acts on the digital distortion totally. The analog distortion is equivalently modeled as an N-order FIR filter due to the limited-band pre-filter of HPA. The amplitude and group delay are used to characterize the filter characteristics. The specific mathematical derivation is provided below.  
The ideal multiplexed signal is  
\[s(t)=\sum_{n=1}^{N}A_{n}\mathrm{e}^{i\theta_{n}}s_{n}(t)+IM(t), \tag{1}\]  
where \(s_{n}(t)\) is the ideal single-branch binary signal, \(A_{n}\) is the amplitude, \(\theta_{n}\) is the phase, and \(IM(t)\) is the intermodulation term. It is defined that the digital distortions between each branch are independent and band-unlimited, and the multiplexed signal can be rewritten as  
\[x(t)=\sum_{n=1}^{N}A_{n}\mathrm{e}^{i\theta_{n}}s^{\bigtriangleup}_{n}(t)+IM(t), \tag{2}\]  
Figure 1: Navigation payload transmission channel model.  
Analog distortion is introduced on the basis of digital distortion. In this work, the analog distortion is modeled as an N-order linear FIR filter; hence, the signal is the convolution of the input signal and the analog impulse response.  
\[y(t)=x(t)\otimes h(t), \tag{3}\]  
where \(h(t)\) is the impulse response of the filter, and \(\otimes\) represents the convolution calculation.  
Assuming that all branches can be completely separated, after analog distortion, the single branch signal is  
\[\widetilde{s}_{n}(t)=s^{\triangle}{}_{n}(t)\otimes h(t). \tag{4}\]  
Correspondingly, the correlation with the local ideal signal is  
\[\begin{array}{l}\hat{R}_{n}(\tau)=\frac{1}{T_{p}}\int_{T_{p}}\widetilde{s}_{ n}(t)s_{n}^{*}(t-\tau)dt\\ =\widetilde{s}_{n}(\tau)\otimes s_{n}^{*}(\tau)\otimes h(\tau)=R_{n}(\tau) \otimes h(\tau),\end{array} \tag{5}\]  
where \(T_{p}\) indicates the period (or code epoch duration) of the signal components, \(R_{n}(\tau)\) represents the autocorrelation function of the \(n\)th branch, and \(*\) is a conjugate matrix.