## 2 Fundamentals of the New Algorithms
### Analytical Expression of the CAF  
The CAF [15] is used in radar, sonar and other similar systems to estimate the time delay and the Doppler shift of an incoming signal. An accurate estimation of these signal parameters generally requires the evaluation of several CAF samples, at the cost of an increased computational complexity. In this paper we propose a family of methods that exploits the knowledge of an approximate expression of the analytical formula of the CAF, given in Reference [15], to reach a trade-off between accuracy and complexity. Following the approach presented in Reference [15], the CAF associated to the generic \(i\)-th satellite code, locally generated for each trial value of code delay \(\overline{\tau}\) and Doppler frequency \(\overline{f}_{d}\), can be written as  
\[S_{i}(\overline{\tau},\overline{f}_{d})=\sum_{m=1}^{N_{v}}R_{m,i}(\overline{ \tau},\overline{f}_{d}) \tag{11}\]  
where \(R_{m,i}(\overline{\tau},\overline{f}_{d})\) is the contribution to the CAF of the \(m\)-th signal \(y_{m}(nT_{s})\). Its analytical expression is [15]  
\[R_{m,i}(\overline{\tau},\overline{f}_{d}) =\frac{1}{T_{d}}\int\limits_{0}^{T_{d}}y_{m}(t)\cdot\{C_{i}(t- \overline{\tau})e^{j2\pi(f_{IF}+\overline{f}_{d})t}\}dt\] \[=\frac{A_{m}}{2T_{d}}e^{-j\varphi_{m}}\{F\{P_{T_{d}}(t)\}\ast F\{ b_{m,i}(t)\}\}_{f=\Delta f}\] \[=B_{m}\sum_{k=-\infty}^{\infty}a_{k(m,i)}\mathrm{Sinc}\left( \left(\Delta f-\frac{k}{T_{p}}\right)\pi T_{d}\right)e^{-j(\Delta f-\frac{k}{ T_{p}})\pi T_{d}} \tag{12}\]  
where \(B_{m}=(A_{m}/2T_{d})e^{-j\varphi_{m}}\), the subscript \(i\) denotes the \(i\)-th satellite code generated by the local generator, \(T_{d}\) is the integration time, \(F\{\}\) denotes Fourier transform, the symbol \(\ast\) denotes convolution operation, \(T_{p}\) is the code period, \(\overline{\tau}\) is the code delay estimate introduced in the local code, \(\mathrm{Sinc}(x)=\sin(x)/x\), \(\overline{f}_{d}\) is the Doppler frequency estimate introduced in the local carrier, \(\Delta f\) is the Doppler frequency estimate error, which can be expressed as  
\[\Delta f=f_{d,m}-\overline{f}_{d} \tag{13}\]  
\(b_{m,i}(t)\) is the product of two spreading codes, that is  
\[b_{m,i}(t)=C_{m}(t-\tau_{m})C_{i}(t-\overline{\tau}) \tag{14}\]  
and \(P_{T_{d}}\) is a window function defined as  
\[P_{T_{d}}=\left\{ \begin{array}{ll}1,&0<t<T_{d}\\ 0,&\mathrm{otherwise}\end{array} \right. \tag{15}\]  
Since \(b_{m,i}(t)\) is a periodic signal with a period equal to the code period, its Fourier transform leads to a line spectrum with coefficients given by  
\[a_{k(m,i)}=\frac{1}{T_{p}}\int_{T_{p}}b_{m,i}(t)e^{-jk(2\pi/T_{p})}dt \tag{16}\]  
and the convolution with the line spectrum leads to the summation in Equation (12). In Figure 2 the distribution of \(a_{k(m,i)}\) is shown in the case \(\tau_{m}-\overline{\tau}=0.2\)\(T_{ch}\), where \(T_{ch}\) is the chip duration. Thanks to the property of Pseudo Random Noise (PRN) code, as expected, \(a_{0(m=i)}\) predominates over the other \(a_{k(m,i)}\). To better understand the nature of the summation in Equation (12), let us refer to Figure 3 where for simplicity, we represent the quantity  
\[\sum_{k=-\infty}^{\infty}a_{k(i,i)}\left|\mathrm{Sinc}\left(\left(\Delta f- \frac{k}{T_{p}}\right)\pi T_{d}\right)\right| \tag{17}\]to demonstrate the relationship among different \(\mathrm{Sinc}\) functions. As Figure 3 shows, if \(a_{0}\) is the peak component in Equation (12) (the subscript \(i,i\) is omitted to simplify the notations) only the components strictly adjacent to \(a_{0}\) (_i.e_., \(a_{-2}\)\(a_{-1}\)\(a_{1}\)\(a_{2}\)...) affect the shape of the main lobe of Equation (12), while the contribution of faraway components can be ignored.