## 3 Multipath Parameter Estimate Algorithm
### Multipath Parameter Estimation  
Employing the CPSD obtained during stage I, we perform multipath parameter estimation in stage II. When we consider (10) and (4), and also assume \(A=1\) and \(\varphi_{e}=0\), the CPSD of the \(m\)-th segment can also be written as:  
\[ \begin{array}{ll}G_{r}^{m}(k)&=\frac{L}{M}\left(S^{m}(k)S_{nom}^{*}(k)+Noise ^{m}(k)S_{nom}^{*}(k)\right)\\ &=G^{m}(k)+G_{noise}^{m}(k)\end{array}  \tag{16}\]  
where:  
\[G^{m}(k)=G_{0,L}(k)\left(1+\sum\limits_{i=1}^{P}\alpha_{i}e^{i\varphi_{i}}e^{ -j2\pi\frac{k}{L}f_{i}\tau_{i}}\right) \tag{17}\]  
and \(G_{0,L}(k)\) is the PSD of an ideal BOC signal with \(L\)-point length. Finally, the averaged CPSD using total \(M\) segments is given by:  
\[ \begin{array}{ll}G_{r}(k)&=\frac{1}{M}\sum\limits_{i=0}^{M-1}G_{r}^{m}(k)\\ &=G_{0,L}(k)\left(1+\sum\limits_{i=1}^{P}\alpha_{i}e^{i\varphi_{i}}e^{-j2\pi \frac{k}{L}f_{i}\tau_{i}}\right)+\frac{1}{M}\sum\limits_{i=1}^{M}G_{noise}^{ m}(k)\end{array}  \tag{18}\]  
In order to remove the effect of the PRN code, averaged CPSD is divided by the nominal PSD. Then we obtain the CTF as well as a noise term, which is given by:  
\[\frac{G_{r}(k)}{G_{0,L}(k)}=\left(1+\sum\limits_{i=1}^{P}\alpha_{i}e^{i\varphi _{i}}e^{-j2\pi\frac{k}{L}f_{i}\tau_{i}}\right)+\frac{1}{M}\sum\limits_{i=1}^{M }\frac{G_{noise}^{m}(k)}{G_{0,L}(k)} \tag{19}\]  
Finally, we take the IFFT operation of the result and obtain the impulse response function in time domain as follows:  
\[F^{-1}\left[\frac{G_{r}(k)}{G_{0,L}(k)}\right]=\delta(t)+\sum\limits_{i=1}^{P }\alpha_{i}e^{i\varphi_{i}}\delta(t-\tau_{i})+F^{-1}\left[\frac{1}{M}\sum \limits_{i=1}^{M}\frac{G_{noise}^{m}(k)}{G_{0,L}(k)}\right] \tag{20}\]  
Consider the result in (20). The useful parts, the first two parts, are independent of \(M\), whereas the noise term is the averaging of \(M\) terms. Since \(G_{0,L}(k)\) and \(G_{noise}^{m}(k)\) are both computed for \(L\) points, the result \(G_{noise}^{m}(k)/G_{0,L}(k)\) is equivalent to \(noise(f)/G_{0}(f)\) in (8). Further, \(G_{noise}^{m}(k)\) is given by \(Noise^{m}(k)S_{nom}^{*}(k)\) and \(G_{0,L}(k)\) is defined as \(S_{nom}(k)S_{nom}^{*}(k)\). Then the \(m\)-th noise term \(G_{noise}^{m}(k)/G_{0,L}(k)\) can be simplified as \(Noise^{m}(k)/S_{nom}(k)\), where \(Noise^{m}(k)\) is varying in amplitude and phase for different \(m\), so we can see that the noise terms are isotropic, and thus the averaging operation \(\frac{1}{M}\sum\limits_{i=1}^{M}(\cdot)\) finally reduces the noise, making sure the noise power after averaging is smaller than \(G_{noise}^{m}(k)/G_{0,L}(k)\) and also the results in (8), which brings the performance gain compared with the direct-FFT method. In order to declare the presence of multipath, we need to set a threshold. Once the amplitude of a certain impulse excesses the threshold, a multipath-presence decision is made and a ray of multipath is found; otherwise, a multipath-absence decision is made. Generally, the threshold value is a function of the designated false alarm probability and the level of thermal noise. For a designated C/N\({}_{0}\), the final false alarm rate (\(P_{fa}\)) and detection rate (\(P_{d}\)) can also be seen as functions of the threshold value. In order to calculate \(P_{fa}\) and \(P_{d}\), the distributions of the amplitude of impulses for both multipath-absent and multipath-present circumstances have to be obtained. However, the distributions of the IFFT results are quite complicated. Besides, they also depend on the window size used in the method, the receiver bandwidth, sampling frequency and the specific multipath circumstance. It is impractical to derive the analytical expression of the probability density functions. One possible way to set the threshold is using the statistical method. The distribution of the IFFT result is first analyzed and computed under the circumstance of clear signal without multipath. The impulse of LOS is removed before calculation. Thus, we can obtain the distribution of the IFFT result that only depends on the system noise.