# Near-Nyquist-Limit Optical Communication and Ranging Method Based on Waveform Matched PPM
## 2 Deep-Space Optical Communication and Ranging Method Based on PPM
### Ranging Method  
The essence of ranging is the measurement of time-of-flight (TOF). The measurement variables of time involved in the ranging method are shown in Figure 3. The transmission time of the signal in free space can be obtained by recording the sending time and receiving time. Due to non-integer oversampling ratios, the frequency of TX Clk is not equal to that of RX. It is impossible to add or subtract the sending time recorded by TX Clk and the receiving time recorded by RX Clk because the time starting points of the two are difficult to align. To solve this problem, a self-looping link is designed for self-transmitting and self-receiving, which can realize the record of the sending time by RX Clk. In addition, the unmeasured inherent delay in the system can be eliminated with the help of the self-looping link. The detailed description is as follows. The ground terminal generates an uplink signal at time \(t_{\text{G0}}\) and sends the signal after the transmission delay \(\tau_{\text{G},\text{tx}}^{d}\). The ground terminal receives the signal through the cable delay \(\tau_{\text{G}}^{\text{A}}\) and the reception delay \(\tau_{\text{G},\text{rx}}^{d}\) of the self-looping link at time \(t_{\text{G1}}\). The receiving time \(t_{\text{G2}}\) of the downlink signal is recorded after a processing delay \(\tau_{\text{G},\text{rx}}^{d}\). The space terminal generates a downlink signal at time \(t_{\text{S0}}\), and sends the signal after the transmission delay \(\tau_{\text{S},\text{tx}}^{d}\). The space terminal receives the signal through the cable delay \(\tau_{\text{S}}^{\text{A}}\) and the reception delay \(\tau_{\text{S},\text{rx}}^{d}\) of the self-looping link at time \(t_{\text{S1}}\). The receiving time \(t_{\text{S2}}\) of the downlink signal is recorded after a processing delay \(\tau_{\text{S},\text{rx}}^{d}\). The transmission time of uplink and downlink signals in free space is \(\tau_{d}\). The relationships between the variables of time involved in Figure 3 are given by:  
\[t_{\text{G2}}-t_{\text{S0}} =\tau_{\text{S},\text{tx}}^{d}+\tau_{d}+\tau_{\text{G},\text{rx}} ^{d}, \tag{3}\] \[t_{\text{S2}}-t_{\text{G0}} =\tau_{\text{G},\text{tx}}^{d}+\tau_{d}+\tau_{\text{S},\text{rx}} ^{d}, \tag{4}\]  
Figure 3: Time relationships for ranging. \[t_{\text{G1}}=t_{\text{G0}}+\tau_{\text{G,tx}}^{d}+\tau_{\text{G,rx}}^{d}, \tag{5}\]  
\[t_{\text{S1}}=t_{\text{S0}}+\tau_{\text{S,tx}}^{d}+\tau_{\text{d}}+\tau_{\text{S, rx}}^{d}. \tag{6}\]  
Based on (3)-(6), the transmission time \(\tau_{d}\) can be expressed as:  
\[\tau_{d}=\frac{(t_{\text{G2}}-t_{\text{G1}})+(t_{\text{S2}}-t_{\text{S1}})}{2} +\frac{\tau_{\text{G}}^{\Delta}+\tau_{\text{S}}^{\Delta}}{2}. \tag{7}\]  
Thus, the distance between the space terminal and the ground terminal is given by:  
\[D=\text{c}\times[\frac{(t_{\text{G2}}-t_{\text{G1}})+(t_{\text{S2}}-t_{\text{ S1}})}{2}+\frac{\tau_{\text{G}}^{\Delta}+\tau_{\text{S}}^{\Delta}}{2}], \tag{8}\]  
where c is the propagation speed of light in vacuum. The basis of accurate ranging is to achieve accurate time measurement, i.e., the measurement accuracy of \(t_{\text{G1}}\), \(t_{\text{G2}}\), \(t_{\text{S1}}\) and \(t_{\text{S2}}\) determines the accuracy of ranging. Assume the time difference between the signal arrival time \(t_{\text{G2}}\) and the sampling time interval \(T_{s}\) is \(\Delta t\), as shown in Figure 4. \(T_{s}=1/F_{s}\), where \(F_{s}\) denotes sampling rate. The phase of a slot is quantized and \(\varepsilon\) indicates the left phase of the last slot of the previous frame. The target variable of time that needs to be measured is \(\Delta t\). The signal is sampled at a frequency that is relatively prime for the slot rate. \(\Delta t\) fluctuates within the sampling time period \(T_{s}\); hence, there are slight fluctuations in the synchronization process of the signal. Higher ranging accuracy can be obtained by estimating these fluctuations more accurately.