## 3 Doppler Frequency Evaluation in the Absence of Noise
### True Solution Based on the Absolute Value of CAF  
In Reference [16], which is our initial work on this topic, we can also find the basic idea about the solution based on the absolute value of the CAF. Under the assumption that there is no data transition in the integration interval, based on Equations (19) and (20) we can write the following equations [16]  
\[ \begin{array}{l}\overline{S}_{A}=\frac{A}{2}\left|R(\Delta\tau)\right|\left| \mathrm{Sinc}\left(\pi LT_{s}(f_{d}-\overline{f}_{A})\right)\right|\\ \overline{S}_{B}=\frac{A}{2}\left|R(\Delta\tau)\right|\left|\mathrm{Sinc}\left( \pi LT_{s}(f_{d}-\overline{f}_{B})\right)\right|\\ \overline{S}_{C}=\frac{A}{2}\left|R(\Delta\tau)\right|\left|\mathrm{Sinc}\left( \pi LT_{s}(f_{d}-\overline{f}_{C})\right)\right|\end{array}  \tag{21}\]  
where the unknowns are the amplitude \(A\), the value of the code correlation function \(R(\Delta\tau)\), and the true Doppler frequency \(f_{d}\). We are going to show that the value of \(f_{d}\) can be easily computed from the above system of equations [16]. Figure 5: The plot of the column (\(\overline{\tau}\approx\tau\)). First of all we observe that, when the points are located in the main lobe (like points A, B and C in Figure 5), we can write \(\left|\mathrm{Sinc}(\pi LT_{s}(f_{d}-\overline{f}_{A}))\right|=\mathrm{Sinc}( \pi LT_{s}(f_{d}-\overline{f}_{A}))\). Now, according to Equation (13) we write \(\overline{f}=f_{d}-\Delta f\), so  
\[\overline{f}\bar{S} = (f_{d}-\Delta f)\overline{S} \tag{22}\] \[= f_{d}\overline{S}-\frac{A\left|R(\Delta\tau)\right|\sin(\pi LT_{ s}\Delta f)}{2\pi LT_{s}}\]  
Based on Equations (21) and (22), we can write  
\[\overline{f}_{A}\overline{S}_{A}+\overline{f}_{B}\overline{S}_{B}+\overline{ f}_{C}\overline{S}_{C}=f_{d}\overline{S}_{A}+f_{d}\overline{S}_{B}+f_{d} \overline{S}_{C}-S_{A,B,C} \tag{23}\]  
where  
\[S_{A,B,C}=\frac{A\left|R(\Delta\tau)\right|}{2\pi LT_{s}}\left(\sin(\pi LT_{s} \Delta f_{A})+\sin(\pi LT_{s}\Delta f_{B})+\sin(\pi LT_{s}\Delta f_{C})\right) \tag{24}\]  
Considering \(f_{sp}=2/(3 LT_{S})\):  
\[\overline{f}_{B} = \overline{f}_{A}+\frac{2}{3 LT_{s}}\ \Rightarrow\ \Delta f_{B}=\Delta f_{A}+\frac{2}{3 LT_{s}}\] \[\overline{f}_{C} = \overline{f}_{A}-\frac{2}{3 LT_{s}}\ \Rightarrow\ \Delta f_{C}=\Delta f_{A}-\frac{2}{3 LT_{s}} \tag{25}\]  
from which  
\[S_{A,B,C}=\sin\left(\pi LT_{s}\Delta f_{A}\right)+\sin\left(\pi LT_{s}\Delta f _{A}+\frac{2\pi}{3}\right)+\sin\left(\pi LT_{s}\Delta f_{A}-\frac{2\pi}{3} \right)=0 \tag{26}\]  
By substituting Equation (26) for \(S_{A,B,C}\) in Equation (23) we obtain \(\overline{f}_{A}\overline{S}_{A}+\overline{f}_{B}\overline{S}_{B}+\overline{ f}_{C}\overline{S}_{C}=f_{d}(\overline{S}_{A}+\overline{S}_{B}+\overline{S}_{C})\), from which we can write the true Doppler frequency as  
\[f_{d}=\frac{\overline{f}_{A}\overline{S}_{A}+\overline{f}_{B}\overline{S}_{B} +\overline{f}_{C}\overline{S}_{C}}{\overline{S}_{A}+\overline{S}_{B}+\overline {S}_{C}} \tag{27}\]  
This expression gives the correct value of the Doppler frequency \(f_{d}\), independently from the code delay error \(\Delta\tau\), in the absence of noise and other nuisances. This equation represents a weighted average of the three measured points, and can be used as a first promising estimator of the Doppler frequency even when the CAF is affected by noise. Notice that Equation (27) is valid only when the Doppler frequency step is \(f_{sp}=2/(3 LT_{S})\). More in general, if we choose the Doppler frequency step as  
\[f_{sp}=\frac{2}{\overline{\pi}LT_{s}} \tag{28}\]  
where \(\overline{n}\) is a positive integer, we can choose the cells as follows:  
* When \(\overline{n}\) is odd, we take \((\overline{n}-1)/2\) cells at each side of the peak cell, as shown in Figure 6(a). * When \(\overline{n}\) is even, we take \(\overline{n}/2\) cells at one side and \([(\overline{n}/2)-1]\) cells on the other side of the peak cell, as Figure 6(b) shows.