## 2 Signal and Channel Model
### Direct-FFT Method  
According to the previous analysis, in order to estimate the multipath parameters using (4), we need to separate CTF from the nominal PSD. A direct way of separating is to perform spectrum division between the CPSD and the nominal PSD of local replica. If we consider the system noise here, (4) can be rewritten as:  
\[G(f)=Ae^{i\varphi_{e}}G_{0}(f)\left(1+\sum_{i=1}^{P}\alpha_{i}e^{i\varphi_{i} }e^{-j2\pi f\tau_{i}}\right)+G_{noise}(f) \tag{6}\]where \(G_{noise}(f)\) is assumed to be the CPSD of the local replica and the additive Gaussian noise (antenna noise and thermal noise from the RF chain) with variance of \(\sigma^{2}\). Then the result of spectrum division is shown as:  
\[G(f)/G_{0}(f)=Ae^{i\varphi_{e}}CTF+G_{noise}(f)/G_{0}(f) \tag{7}\]  
Generally, there are various methods can be used to extract the multipath parameters from (7). Least square fitting method and its modified method recursive least square (RLS) algorithm [28; 29] can be seen as an approach like MEDLINE method in the frequency domain by curve fitting in the least squares sense. It is capable to obtain the multipath parameters \((\alpha_{i},\tau_{i})\), but unable to get the number of multipath signals. Thus, it is needed to find out ways to estimate or assign the number of multipath signals prior to applying this method. Cepstral analysis and Subspace Estimation Technique, such as the MUSIC method, can be used to search for the multipath number and delay, but they are both insensitive to estimate the amplitude of multipath signals and vulnerable in front of system noise. Besides, they require large-scale matrix transformations and/or singular value decomposition, which will also significantly increase the computational complexity.  
Another way to get multipath channel characteristics from CTF is directly translating (7) to the time-domain by taking IFFT of (7). If we assume \(A=1\) and \(\varphi_{e}=0\), the results after IFFT operation is given by:  
\[\begin{array}{l}F^{-1}[G(f)/G_{0}(f)]\\ =\delta(t)+\sum\limits_{i=1}^{p}\alpha_{i}e^{i\varphi_{i}}\delta(t-\tau_{i})+F ^{-1}(noise(f)/G_{0}(f))\end{array} \tag{8}\]  
where \(F^{-1}\) represents the IFFT operator. We can see from (8) that after IFFT there will be \(P+1\) impulses in the time domain. The first impulse is at the original point with amplitude of 1. It represents the transformation result of LOS. The other \(P\) impulses correspond to the transformation results of multipath signals. For the \(i\)-th path, its impulse appears at the epoch where \(t\) equals to \(\tau_{i}\) with amplitude of \(\alpha_{i}e^{i\varphi_{i}}\), so we can estimate the parameters of both LOS and reflected signals by searching for these impulses after IFFT. Since LOS is always in front of the multipath signals, it is easy to pick out the impulse of LOS by searching for the first impulse with very large amplitude. Then the time offset of this impulse can be obtained which is exactly the code delay of LOS. After that, we can also search for the other impulses and identify \(\tau_{i}\) and \(\alpha_{i}\) for each path.  
It should be noted that although the method of directly taking IFFT is able to estimate all parameters of multipath signals, it is quite sensitive to the noise. A main reason is that the actual GNSS signal is approximately 20 dB weaker than the thermal noise power in a 2 MHz bandwidth receiver, so the power spectra obtained directly from FFT without statistical averaging will be too noisy to be used for multipath parameter estimation. Besides, to estimate the multipath signals accurately, the system sampling rate must be quite high which results in large data volume. Then, direct taking FFT and IFFT of the entire data will be highly computationally expensive. All the above factors result in its unfeasibility and then prevent it from practical application.