## 2 System Concept and Dual One-Way Measurements
In the same manner, a similar expression can be formulated for the measurement of the Frequency of Arrival (FOA) of the signal [19]:  
\[(TOA_{j}-ToT_{i}) =\frac{\rho_{j,i}}{c}=\frac{\|\mathbf{r}_{j}-\mathbf{r}_{i}\|}{ c}+(\delta t_{j}-\delta t_{i})+\epsilon_{j,i} \tag{2}\]0  
where \(\alpha\) is the effective duration, which is defined as [19]:  
\[(TOA_{j}-ToT_{i}) =\frac{\rho_{j,i}}{c}=\frac{\|\mathbf{r}_{j}-\mathbf{r}_{i}\|}{ c}+(\delta t_{j}-\delta t_{i})+\epsilon_{j,i} \tag{2}\]1  
where \(s(t)\) is the received signal. In the case of pseudo-random code and Delay Look Loop (DLL), the TOA error model can also be refined by taking into account many of the receiver and signal parameters, by the following expression [18]:  
\[\sigma_{DLL}\approx \begin{cases}\sqrt{\frac{B_{n}}{2C/N_{0}}D\Big{[}1+\frac{2} {T_{int}C/N_{0}(2D)}\Big{]}},&D\geq\frac{\pi R_{c}}{B_{fc}}\\ \sqrt{\frac{B_{n}}{2C/N_{0}}\Big{(}\frac{1}{B_{fc}T_{c}}+\frac{B_{fc}T_{c}}{ \pi-1}\Big{(}D-\frac{1}{B_{fc}T_{c}}\Big{)}^{2}\Big{)}\Big{[}1+\frac{2}{T_{int }C/N_{0}(2D)}\Big{]}},&\frac{R_{c}}{B_{fc}}<D<\frac{\pi R_{c}}{B_{fc}}\\ \sqrt{\frac{B_{n}}{2C/N_{0}}\Big{(}\frac{1}{B_{fc}T_{c}}\Big{)}\Big{[}1+\frac{ 1}{T_{int}C/N_{0}}\Big{]}},&D\leq\frac{R_{c}}{B_{fc}}\end{cases}  \tag{11}\]  
where \(B_{n}\) is the code loop noise bandwith, \(B_{fc}\) is the double-sided front-end bandwidth, \(T_{int}\) is the predetection integration time, \(T_{c}\) is the chip period, \(R_{c}=1/T_{c}\) is the spreading code rate, \(C/N_{0}\) is the carrier to noise power ratio, and \(D\) is the early-to-late correlator spacing. Moreover, Ref. [20] demonstrates that using the PLL filter to estimate the Doppler frequency and the range rate gives the following error due to thermal noise (for the range rate):  
\[\sigma_{PLL}=\frac{\lambda_{c}}{2\pi T_{int}}\sqrt{\frac{B_{d}}{C/N_{0}} \Big{[}1+\frac{1}{2T_{int}C/N_{0}}\Big{]}} \tag{12}\]  
where \(\lambda_{c}\) is the wavelength of the signal, \(T_{int}\) is the coherent integration time, \(C/N_{0}\) is the carrier to noise ratio, and \(B_{d}\) is the Doppler bandwidth defined as:  
\[B_{d}=\frac{1}{2}\int_{-0.5/T_{int}}^{0.5/T_{int}}|H_{nf}(e^{2\pi fT_{int}})|^ {2}\,df \tag{13}\]where \(H_{inf}\) is the frequency noise impulse responses transformed into the Z-domain. The Doppler bandwidth \(B_{d}\) summarizes in a single parameter the ability of the tracking loop to produce smooth frequency estimates. Having these formulations, it is possible to find the range and range-rate measurement errors knowing the receiver and signal parameters. Referring to the literature, in [21], the range measurement error between GPS satellites is assumed to be 1 m by using ranging measurementswith a maximum ranging distance of 50,000 km; Ref. [22] proposes a method for relative navigation for microsatellites using inter-satellite measurements, and the range error is assumed to be at a maximum of 10 cm for a maximum ranging distance of 10 km. Another example can be found in [23], where a precise ranging and time synchronization system was implemented based on dual one-way ranging with a precision of the ranging measures of 1.038 m. Ref. [24] proposes a solution for relative OD using inter-satellite range measurements, and the ranging measurement error is assumed to be Gaussian with a standard deviation of 1 m.  
Refs. [18; 25; 26] explain in more detail the range measurement assumptions and the relative models, and it is shown that the measurement error always depends on the carrier-to-noise ratio, \(C/N_{0}\), remaining below 1 m for most cases. Concerning the range rate, in [27], the measurement accuracy is assumed to be 0.01 mm/s, and in [28], an autonomous navigation system for satellites deployed in Martian orbits was proposed, and the worst accuracy in range rate was assumed to be below 4 \(\upmu\)m/s. The inter-satellite link, in this case, was in the X-band, and the Carrier-to-Noise ratio was assumed to be equal to 30 db-Hz. In this work, we assume the use of the Ka-band  
(\(f_{c}=26\) GHz) for the inter-satellite link.