# Design and Performance Analysis of Navigation Signal Based on OFDM
## 3 ACF and PSD Derivation Verification of OFDM Navigation Signal Modulation System
### OFDM Navigation Signal PSD  
The PSD of the OFDM navigation signal can be obtained by a Fourier transform of the ACF:  
\[\begin{array}{ll}S(f)&=\mathcal{F}\big{\{}R_{p}(\tau)\big{\}}\\ &=\mathcal{F}\big{\{}R_{u}(\tau)+R_{ug}(\tau)+R_{gu}(\tau)+R_{g}(\tau)\big{\}},\\ &=S_{u}(f)+S_{g}(f)+S_{ug}(f)+S_{gu}(f)\end{array} \tag{14}\]  
where \(S_{u}(f)\) is the power spectral density of the OFDM symbol, \(S_{g}(f)\) is the power spectral density of the optical density of the OFDM and the spectral density of the OFDM. The spectral density of the mutual correlation function of the protection interval and OFDM. That is, the PSD of the OFDM navigation signal is composed of the PSD of the OFDM symbol, the PSD of the protection interval, and the PSD of the cross-correlation function of the OFDM symbol and the protection interval.  
Calculate the Fourier transforms of A, B, C, and D, respectively:  
\[\mathcal{F}\{\mathsf{g}_{1}(\tau)\}=T_{u}\operatorname{sinc}(fT_{u})e^{-j \pi T_{u}f}, \tag{15}\]  
\[\mathcal{F}\{\mathsf{g}_{2}(\tau)\}=T_{g}\operatorname{sinc}\bigl{(}fT_{g} \bigr{)}e^{-j\pi T_{g}f} \tag{16}\]  
\[\mathcal{F}\{\Gamma_{K}(\tau)\}=\sum\limits_{k=1}^{K}\delta(f-k\Delta f) \tag{17}\]  
\[\mathcal{F}\{\Psi_{K}(\tau)\}=\sum\limits_{k=1}^{K}\delta(f-k\Delta f)e^{j2\pi (T_{g}-T_{s})k\Delta f} \tag{18}\]  
\[\mathcal{F}\{\Phi_{K}(\tau)\}=\sum\limits_{k=1}^{K}\delta(f-k\Delta f)e^{j2\pi (T_{s}-T_{g})k\Delta f} \tag{19}\]Taking the PSD \(S_{u}(f)\) of the OFDM symbol as an example, the corresponding Fourier transform is as follows:  
\[\begin{array}{ll}S_{u}(f)&=\mathcal{F}\{R_{u}(\tau)\}\\ &=\mathcal{F}\Big{\{}\frac{\Gamma_{K}(\tau)}{K^{T}T_{u}}S_{1}(\tau)*\overline{g _{1}(-\tau)}\Big{\}}\\ &=\frac{1}{K^{T}T_{s}}\mathcal{F}\{\Gamma_{K}(\tau)\}*\mathcal{F}\Big{\{}g_{1} (\tau)*\overline{g_{1}(-\tau)}\Big{\}}\\ &=\frac{1}{K^{T}T_{s}}\Gamma_{K}(f)*G_{1}^{2}(f)\\ &=\frac{1}{K^{T}T_{s}}\bigg{[}\sum\limits_{k=1}^{K}\delta(f-k\Delta f)\bigg{]}* \Big{[}T_{u}\operatorname{sinc}(fT_{u})e^{-j\pi T_{u}f}\Big{]}^{2}\\ &=\frac{1}{K^{T}T_{s}}\sum\limits_{k=1}^{K}\Big{[}T_{u}\operatorname{sinc}((f- k\Delta f)T_{u})e^{-j\pi T_{u}(f-k\Delta f)}\Big{]}^{2}\end{array}, \tag{20}\]  
where \(\delta(f)\) is the shock response function and \(g_{1}(t)\) is the frequency domain function of \(G_{1}(f)\), \(\operatorname{sinc}(fT_{u})=\operatorname{sinc}(fT_{u})/fT_{u}\).  
Similarly, using the same derivation method, the PSD \(S_{g}(f)\),\(S_{gu}(f)\), and \(S_{ug}(f)\) can be obtained in turn as follows:  
\[S_{g}(f)=\frac{1}{K^{2}T_{s}}\sum\limits_{k=1}^{K}\Big{[}T_{g}\operatorname{ sinc}((f-k\Delta f)T_{g})e^{-j\pi T_{g}(f-k\Delta f)}\Big{]}^{2}, \tag{21}\]  
\[S_{gu}(f)=\frac{1}{K^{2}T_{s}}\Bigg{[}\sum\limits_{k=1}^{K}T_{g}T_{u} \operatorname{sinc}\big{[}(f-k\Delta f)T_{g}\big{]}\operatorname{sinc}[(-f+k \Delta f)T_{u}]e^{j\pi T_{s}(f-k\Delta f)}\Bigg{]}, \tag{22}\]  
\[S_{ug}(f)=\frac{1}{K^{2}T_{s}}\Bigg{[}T_{g}T_{u}\sum\limits_{k=1}^{K} \operatorname{sinc}(-(f-k\Delta f)T_{g})\operatorname{sinc}((f-k\Delta f)T_{ u})e^{-j\pi T_{s}(f-k\Delta f)}\Bigg{]}, \tag{23}\]  
In the substitution of Equations (21)-(23) into Equation (14), the PSD of the OFDM navigation signal is obtained.