GPS Signal Availability Augmentation Utilizing the Navigation Signal Retransmission Via the GEO Comsat

Abstract

In some special circumstance such as local conflict area or thick forests, the signal availability to the GPS receiver may greatly be decreased, and this will accordingly degrade the performance of the receiver. To improve this problem, we present a method to augment the GPS signal availability utilizing the navigation signal retransmission via the geosynchronous communications satellite (GEO comsat). The general implementation process of the proposed method is that at the transmitting side, we first upconvert the local-generated power-controlled GPS signal in the selected comsat frequency band, and then transmit it to the GEO comsat for its retransmission. While at the receiving side, we first downconvert the received GPS signal which may be overlapped with the strong communication signal to baseband. And then we make a cancellation on the strong communication signal with blind adaptive frequency-shift filtering to decrease its effect on the retransmitted GPS signal. Finally with the separated GPS signal, we can easily get the despreading and the demodulation results for the GPS receiver, and thus the goal of augmenting the signal availability for the GPS receiver is achieved. The final numerical results validate the effectiveness of the proposed method.

Appendix

1.1 Derivation of Eq. (11)

For the satellite communication receiver, suppose its received signal is similar as Eq. (6), then with the carrier $2\cos \left[ {2\pi \left( {f_0 +{f}_d^{\prime }}\right) t+\phi _c}\right]$ generated by the receiver, its demodulation result in time interval $[0,T_b^{(c)}]$ can be written as

$$\begin{aligned} Y&= \int _{0}^{T_b^{(c)}} {y(t)\left\{ {2\cos \left[ {2\pi \left( {f_0 +{f}_d^{\prime }}\right) t+\phi _c}\right] }\right\} dt}\nonumber \\&= A_c b_c (0)T_b^{(c)} +A_g b_g (0)c_1 T_b^{(c)} \hbox {sinc}\left( 2\pi f_b T_b^{(c)}\right) \nonumber \\&+\int _{0}^{T_b^{(c)}} {n(t)\left\{ {2\cos \left[ {2\pi \left( {f_0 +{f}'_d } \right) t+\phi _c}\right] }\right\} dt} \end{aligned}$$

(19)

where $\hbox {sinc}(x)$ is sinc function, ${f}_d^{\prime }$ is the Doppler frequency of the communication receiver. In Eq. (19), we have used the conditions $T_b^{(c)} \le T_p \ll T_b^{(g)}$. Since $f_b T_b^{(c)} \ll 1$, i.e., $\hbox {sinc}\left( {2\pi f_b T_b^{(c)}}\right) \approx 1$ in the time interval $[0,T_b^{(c)}]$, then Eq. (19) can be simplified as

$$\begin{aligned} Y\approx A_c b_c (0)T_b^{(c)} +A_g b_g (0)c_1 T_b^{(c)} +\int _{0}^{T_b^{(c)}} {n(t)\left\{ {2\cos \left[ {2\pi \left( {f_0 +{f}_d^{\prime }}\right) t+\phi _c}\right] }\right\} dt}. \end{aligned}$$

(20)

Let $L_n \buildrel \Delta \over = \int _{0}^{T_b^{(c)}} {n(t)\left\{ {2\cos \left[ {2\pi \left( {f_0 +{f}_d^{\prime }}\right) t+\phi _c}\right] }\right\} dt}$, we can easily get that $L_n$ follows the Gaussian distribution with mean 0 and variance $N_0 T_b^{(c)}$, i.e., $L_n \sim \mathcal{N}(0,N_0 T_b^{(c)})$, then the demodulation BER of the communication signal can be written as

$$\begin{aligned} P_e^{(c)}&= \Pr \left\{ {\left. {Y<0}\right| b_c (0) = 1}\right\} \Pr \left\{ {b_c (0) = 1}\right\} +\Pr \left\{ {\left. {Y>0}\right| b_c (0) = -1}\right\} \Pr \left\{ {b_c (0) = -1}\right\} \nonumber \\&\approx \frac{1}{2} \left[ {\begin{array}{l} \Pr \left\{ {A_c T_b^{(c)} +A_g b_g (0)c_1 T_b^{(c)} +L_n <0}\right\} + \\ \Pr \left\{ {-A_c T_b^{(c)} +A_g b_g (0)c_1 T_b^{(c)} +L_n >0}\right\} \\ \end{array}} \right] \end{aligned}$$

(21)

where $\Pr \left\{ X\right\}$ denotes the probability of $X$. Since $c_1$ is the first chip of the spreading code, for the given retransmitted GPS augmentation signal, it is a fixed value and we can merge it to the random variable $b_g (0)$. Let ${b}_g^{\prime } (0)\buildrel \Delta \over = b_g (0)c_1$, we have ${b}'_g (0) = \pm 1$ with equal probability, then Eq. (21) can be further written as

$$\begin{aligned} P_e^{(c)}&\approx \frac{1}{2} \left[ {\begin{array}{l} \Pr \left\{ {A_c T_b^{(c)} +A_g T_b^{(c)} +L_n <0}\right\} \Pr \left\{ {{b}_g^{\prime } (0) = 1}\right\} + \\ \Pr \left\{ {A_c T_b^{(c)} -A_g T_b^{(c)} +L_n <0}\right\} \Pr \left\{ {{b}_g^{\prime } (0) = -1}\right\} + \\ \Pr \left\{ {-A_c T_b^{(c)} +A_g T_b^{(c)} +L_n >0}\right\} \Pr \left\{ {{b}_g^{\prime } (0) = 1}\right\} + \\ \Pr \left\{ {-A_c T_b^{(c)} -A_g T_b^{(c)} +L_n >0}\right\} \Pr \left\{ {{b}_g^{\prime } (0) = -1}\right\} \\ \end{array}}\right] \nonumber \\&= \frac{1}{4} \left[ {\begin{array}{l} \underbrace{\Pr \left\{ {A_c T_b^{(c)} +A_g T_b^{(c)} +L_n <0}\right\} }_{P1}+\underbrace{\Pr \left\{ {A_c T_b^{(c)} -A_g T_b^{(c)} +L_n <0} \right\} }_{P2}+ \\ \underbrace{\Pr \left\{ {-A_c T_b^{(c)} +A_g T_b^{(c)} +L_n >0}\right\} }_{P3}+\underbrace{\Pr \left\{ {-A_c T_b^{(c)} -A_g T_b^{(c)} +L_n >0}\right\} }_{P4} \\ \end{array}} \right] .\nonumber \\ \end{aligned}$$

(22)

For $P1$, since $L_n \sim \mathcal{N}(0,N_0 T_b^{(c)})$, we have

$$\begin{aligned} P1&= \Pr \left\{ {A_c T_b^{(c)} +A_g T_b^{(c)} +L_n <0}\right\} \nonumber \\&= \Pr \left\{ {L_n <(-A_c T_b^{(c)} -A_g T_b^{(c)})}\right\} = Q\left( {\sqrt{\frac{2E_b^{(c)} \left( {1+\sqrt{\textit{ISR}_c}}\right) ^{2}}{N_0}}}\right) . \end{aligned}$$

(23)

Similarly, we can get the results of other terms in Eq. (22) as

$$\begin{aligned} P4 = P1, P2 = P3 = Q\left( {\sqrt{\frac{2E_b^{(c)} \left( {1-\sqrt{\textit{ISR}_c}}\right) ^{2}}{N_0}}}\right) . \end{aligned}$$

(24)

Substitute Eqs. (23) and (24) into Eq. (22), then we can get the final result of $P_e^{(c)}$ as given in Eq. (11). Note that in the derivations of $P2$ and $P3$, we have used the condition $A_c>A_g$.