# Positioning Using IRIDIUM Satellite Signals of Opportunity in Weak Signal Environment
## 3 The Estimation of Iridium Satellite Signal Doppler-Shift in Weak Signal Environment
### The Principle of Estimating Iridium Satellite Signal Doppler-Shift in Weak Signal Environment
 \tag{3}\]  
We can see that \(r_{d}(t)\) have four parts, the base band signal is \(\frac{3}{2}p_{d}^{2}\), the \(4f_{0}\) IF signal is \(-\frac{1}{2}p_{d}^{2}\cos(8\pi f_{0}t+4\phi_{d})\), the \(2f_{0}\) IF signal is \(2p_{d}^{2}d_{i}(t)d_{q}(t)\sin(4\pi f_{0}t+2\phi_{d})\), and the noise signal. The \(4f_{0}\) IF signal and \(2f_{0}\) IF signal can be obtained, respectively by filtering. However, the \(d_{i}(t)d_{q}(t)\) in \(2f_{0}\) IF signal can cause the phase flip, which means the coherent integration in signal acquisition will be affected. This situation will not happen for \(4f_{0}\) IF signal, which means we can use this signal to estimate Doppler-shift. After filtering, we get:  
\[m_{d}(t)=-\frac{1}{2}P_{d}^{2}\cos(8\pi f_{0}t+4\varphi_{d})+N_{d}(t) \tag{4}\]  
where \(m_{d}(t)\) denotes the \(4f_{0}\) IF signal and \(N_{d}(t)\) is the noise. The expression of \(s_{uni}(t)\) is:  
\[s_{uni}(t)=\sqrt{2P_{uni}}d_{uni}(t)cos(2\pi f_{0}t+\varphi_{uni})+n_{uni}(t) \tag{5}\]  
where \(P_{uni}\) is the signal power, \(f_{0}\) is the carrier frequency, \(\varphi_{uni}\) is the initial phase. \(d_{uni}(t)\) is the binary information code (789 hex). \(n_{uni}(t)\) is the noise signal. We get the quadratic square signal \(r_{uni}(t)\) by squaring the \(s_{uni}(t)\) twice, and the \(r_{uni}(t)\) can be written as:  
\[\begin{array}{lll}r_{uni}(t)&=\frac{3}{2}P_{uni}^{2}+\frac{1}{2}P_{uni}^{2} \cos(8\pi f_{0}t+4\varphi_{uni})+2P_{uni}^{2}\cdot\cos(4\pi f_{0}t+\\ &&2\varphi_{uni})+(2\sqrt{2P_{uni}}d_{uni}(t)n_{uni}(t)\cos(2\pi f_{0}t+\varphi_ {uni})+n_{uni}^{2}(t))\cdot\\ &&[2P_{uni}\cos(4\pi f_{0}t+2\varphi_{uni})+P_{uni}]+(  
signal and \(2f_{0}\) IF signal are both without binary information code sequences, which means they will not cause phase flip in signal acquisition. If we want use both information signal and unique signal to estimate Doppler-shift, their center frequency needs to be the same. As a result, \(4f_{0}\) IF signal is used here and the \(2f_{0}\) IF signal is removed by filtering. Then, we get:  
\[m_{uni}(t)=\frac{1}{2}P_{uni}{}^{2}\cos(8\pi f_{0}t+4\varphi_{uni})+N_{uni}(t) \tag{7}\]  
where \(m_{uni}(t)\) denotes the \(4f_{0}\) IF signal, and \(N_{uni}(t)\) is the noise. Let \(m_{tone}(t)\) denotes the \(4f_{0}\) IF signal of the quadratic square tone signal. Then, the quadratic square Iridium burst signal \(M_{tri}(t)\) can be written as:  
\[M_{tri}(t)=\left\{ \begin{array}{l}m_{tone}(t);t_{0}\leq t\leq T_{tone}\\ m_{uni}(t);T_{tone}<t\leq T_{uni}\\ m_{d}(t);\ T_{uni}<t\leq T_{d}\end{array} \right. \tag{8}\]  
Let \(u(t)\) denotes the local signal, the expression is:  
\[u(t)=\exp(-j2\pi\cdot 4 ft) \tag{9}\]  
The signal, after multiplication with the local carrier, assumes the form:  
\[M_{tri}(t)\cdot u(t) = P^{2}\cdot\cos(8\pi f_{0}t+4\varphi_{uni})\cdot\exp(-j2\pi\cdot 4 ft)+ \tag{10}\] \[N(t)\cdot\exp(-j2\pi\cdot 4 ft)\]  
In (10), \(P\) is the signal power and \(N(t)\) is the noise. The high-frequency signal and noise of I and Q arms are removed by integration of the low pass filter. The expression of the result of the coherent integration can be written as [15]:  
\[r_{p}(n)=\frac{a\cdot\sin(\frac{1}{2}(2\pi f_{e}\cdot T_{coh}))}{\frac{1}{2}( 2\pi f_{e}\cdot T_{coh})}\exp^{j[2\pi f_{e}(t_{1}+\frac{T_{coh}}{2})+\theta_{ e}]} \tag{11}\]  
In (11), \(a^{2}\) is the mean power of the real input signals, \(t_{1}\) denotes the starting time of the coherent integration, \(f_{e}\) denotes the difference between real signal frequency and local signal frequency, and \(\theta_{e}\) is the phase difference. The length of the time of the coherent integration is \(T_{coh}=T_{ton}+T_{uni}+T_{d}\).