## 4 Improvement Analysis of Downlink Signal Power Utilization Efficiency  
Taking the X-band for example, the spacecraft downlink signal can be expressed as Formula (9) [27; 32]. For the S-band signal, there is only one group of DOR beacons, and the DOR\({}_{2}\) items should be removed from the following formula. \[s(t)=\sqrt{2P_{\rm T}}\cos\left( \begin{array}{c}2\pi f_{\rm c}t+m_{\rm TM}x_ {\rm TM}(t)+m_{\rm RG}x_{\rm RG}(t)+\\ m_{\rm DORI}\sin(2\pi f_{\rm DORI}t)+m_{\rm DOR2}\sin(2\pi f_{\rm DOR2}t) \end{array} \right) \tag{9}\]  
As shown in Formula (9), \(P_{\rm T}\) is the total power of the downlink signal, \(f_{\rm c}\), \(f_{\rm DORI}\) and \(f_{\rm DOR2}\) are the frequencies of the carrier and two groups of DOR beacon signals, respectively. \(m_{\rm TM}\), \(m_{\rm RG}\), \(m_{\rm DORI}\), and \(m_{\rm DOR2}\) are the modulation degrees of the telemetry signal, ranging signal, and two sets of DOR beacon signals, respectively. Notably, a higher modulation degree corresponds to an increased downlink power. \(x_{\rm TM}(t)\) and \(x_{\rm RG}(t)\) denote the telemetry signal and ranging signal, the detailed information of which can be referred to in reference [30]. For the pseudo-code ranging system, the sinusoidal signal is usually one of the subcarrier types, and then the downlink power of the pseudo-code ranging signal and the DOR beacon signal can be expressed as follows:  
\[\left\{ \begin{array}{l}P_{\rm RG}=P_{\rm T}\cos^{2}(m_{\rm TM})f_{1}^{2}(m_{ \rm RG})f_{0}^{2}(m_{\rm DORI})f_{0}^{2}(m_{\rm DOR2})\\ P_{\rm DORI}=P_{\rm T}\cos^{2}(m_{\rm TM})f_{0}^{2}(m_{\rm RG})f_{1}^{2}(m_{ \rm DORI})f_{0}^{2}(m_{\rm DOR2})\\ P_{\rm DOR2}=P_{\rm T}\cos^{2}(m_{\rm TM})f_{0}^{2}(m_{\rm RG})f_{0}^{2}(m_{ \rm DORI})f_{1}^{2}(m_{\rm DOR2})\end{array} \right. \tag{10}\]  
Suppose the modulation degrees of the telemetry signal, ranging signal, and the two sets of DOR beacon signals are 0.8 rad, 0.6 rad, and 0.5 rad (for the two sets of DOR beacon signals, the modulation degrees are usually same), the corresponding ratio of the power of the pseudo-code ranging signal and the two sets of the DOR beacon signals to the total downlink power is about \(-\)15.09 dB and \(-\)16.80 dB, respectively. The modulation of a downlink carrier signal by an increased number of signal types results in a more dispersed downlink signal power. If one of the modulated signals is detuned, the power of the remaining signals will increase even if the modulation degrees remain unchanged. Therefore, if the \(\pm\)DOR\({}_{1}\) beacon signal is not modulated to the downlink signal, the S/X band downlink signal can be expressed by Formula (11):  
\[\left\{ \begin{array}{l}s_{\rm S}(t)=\sqrt{2P_{\rm T}}\cos(2\pi f_{\rm c}t+m_{ \rm TM}x_{\rm TM}(t)+m_{\rm RG}x_{\rm RG}(t))\\ s_{\rm X}(t)=\sqrt{2P_{\rm T}}\cos\left(\begin{array}{c}2\pi f_{\rm c}t+m_{ \rm TM}x_{\rm TM}(t)+m_{\rm RG}x_{\rm RG}(t)+\\ m_{\rm DOR2}\sin(2\pi f_{\rm DOR2}t)\end{array}\right)\end{array} \right. \tag{11}\]  
The power of the pseudo-code ranging signal and the DOR beacon signal in the X-band downlink signal can be expressed as Formula (12) when the \(\pm\)DOR\({}_{1}\) beacon signal is not modulated:  
\[\left\{ \begin{array}{l}P_{\rm X,RG}=P_{\rm T}\cos^{2}(m_{\rm TM})f_{1}^{2}( m_{\rm RG})f_{0}^{2}(m_{\rm DOR2})\\ P_{\rm X,DOR2}=P_{\rm T}\cos^{2}(m_{\rm TM})f_{0}^{2}(m_{\rm RG})f_{1}^{2}(m_{ \rm DOR2})\end{array} \right. \tag{12}\]  
Under the same condition of modulation as Equation (9), the ratio of the power of the pseudo-code ranging signal and the DOR\({}_{2}\) beacon signal to the total power of the downlink signal are \(-14.54\) dB and \(-16.25\) dB, which are about \(13\%\) higher than that of the DOR\({}_{1}\) beacon signal being modulated to the downlink carrier.