# Radio Frequency Compatibility Evaluation of S Band Navigation Signals for Future BeiDou
## 2 Radio Frequency Compatibility Evaluation Methods
### S Band Signals  
In the S band, the IRNSS is simultaneously broadcasting BPSK(1) and BOCs(5,2) for open and authorized services respectively [9; 10]. The Galileo also exploits a series of S band candidates such as CBOC(6,1,1/11), BPSK(1), BPSK(4), and BPSK(8) for introducing alternate and complementary capabilities to some services in operation or under deployment [5; 6; 11]. The Globalstar employs the same S band frequency slot to transmit downlink signals by multi-beam antennas allowing frequency reutilization [5]. Each beam contains 13 frequency division multiple access (FDMA) channels with each channel taking up 1.23 MHz wide. Code division multiple access (CDMA) with a chipping rate of 1.2288 Mcps is implemented inside every FDMA channel. Before modulation of the carrier, the Globalstar signal is filtered by SRRC filter with roll-off factor \(\rho\) of 0.2, then Globalstar signal PSD of the each beam can de expressed by  
\[G_{Globalstar}(f)=\sum_{k=-6}^{6}G_{SRRC}^{k}(f) \tag{18}\]  
where PSD of SRRC at the \(k\)th FDMA channel is indicated as  
\[G_{SRRC}^{k}(f)=\left\{\begin{array}{cc}1,&\left|f-kB_{f}\right|\leq\frac{f_ {c}}{2}(1-\rho)\\ 0,&\left|f-kB_{f}\right|\leq\frac{f_{c}}{2}(1+\rho)\\ g(f),&\frac{f_{c}}{2}(1-\rho)\leq\left|f-kB\right|\leq\frac{f_{c}}{2}(1+\rho) \end{array}\right., \tag{19}\]  
with  
\[g(f)=\frac{1+\cos\Bigl{(}\frac{\pi}{\rho f_{c}}\Bigl{(}|f|-\frac{(1-\rho)f_{c }}{2}-kB_{f}\Bigr{)}\Bigr{)}}{2} \tag{20}\]  
where the \(f_{c}\) is a chip rate of 1.2288 Mcps, \(B_{f}\) is a FDMA bandwidth of 1.23 MHz and \(\rho\) is a roll-off factor of 0.2. The S band signal PSDs of IRNSS, Galileo candidate and Globalstar are shown in Figure 1a-c.  
To take full advantages of S band's superiorities and reserve resources for BDS, signal modulation schemes in S band have attracted extensive research attentions. The legacy modulations of BPSK(8) and BOCs(4,4) are suggested for BDS in S band [12]. Besides, a promising modulation called CPM with a chip rate of 8.184 Mcps, frequency pulse \(g(t)\) of RC, modulation order \(M=2\) and pulse length \(L=2\) denoted by CPM(8) and MSK-BOCs(4,4) are also recommended as candidate S band BDS signals [13; 14], because their inherent properties of constant envelope and phase continuity contribute to greatly reducing the nonlinear distortion due to the saturating characteristic of the high power amplifier and linear bandpass distortion due to nonideal bandpass characteristic in the satellite navigation applications. The PSD of CPM can be expressed as  
\[\begin{array}{ll}P(f)&=2\Bigl{\{}\int_{0}^{LT}\Re(\tau)\cos 2\pi f\tau d \tau+\frac{1-\psi(jh)\cos 2\pi fT}{1+\varphi^{2}(jh)-2\varphi(jh)\cos 2\pi fT} \cdot\int_{LT}^{(L+1)T}\Re(\tau)\cos 2\pi f\tau d\tau\\ &-\frac{\psi(jh)\sin 2\pi fT}{1+\varphi^{2}(jh)-2\psi(jh)\cos 2\pi fT}\cdot\int_{LT} ^{(L+1)T}\Re(\tau)\sin 2\pi f\tau d\tau\Bigr{\}}\end{array}, \tag{21}\]  
with  
\[\psi(jh)=\sin M\pi h/M\sin\pi h, \tag{22}\]where the \(\Re(\tau)\) refers to the autocorrelation function of CPM, i.e.,  
\[\Re(\tau)=\frac{1}{T}\int_{0}^{T}\prod_{k=1-L}^{\lfloor\tau/T\rfloor}\frac{1}{M} \frac{\sin 2\pi hM[q(t+\tau-kT)-q(t-kT)]}{\sin 2\pi h[q(t+\tau-kT)-q(t-kT)]}dt, \tag{23}\]  
where \(\lfloor\cdot\rfloor\) is the floor rounding operator. The PSD of MSK-BOC is written as  
\[G_{\text{MSK-BOCs}(n,m)}(f)=\left\{\begin{array}{l}\frac{2f_{\text{s}}^{2}f_ {\text{s}}\sin\left(\pi/f_{\text{c}}\right)}{\pi^{2}\left(f_{\text{s}}^{2}-f ^{2}\right)^{2}},2n/m\text{ is even}\\ \frac{2f_{\text{s}}^{2}f_{\text{c}}\cos\left(\pi/f_{\text{c}}\right)}{\pi^{2} \left(f_{\text{s}}^{2}-f^{2}\right)^{2}},2n/m\text{ is odd}\end{array}\right., \tag{24}\]  
where the \(f_{\text{s}}\) is subcarrier frequency of \(n\)\(\times\) 1.023 MHz and the \(f_{\text{c}}\) is chip rate of \(m\)\(\times\) 1.023 MHz. The Figure 1d shows all S band candidates for BDS with similar spectrum occupation in main lobe. In views of the severe spectrum overlap in S band, the compatibility is a particularly critical issue that must be considered for BDS signal design in S band.