# 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
If the maximum output power value exceeds the acquisition threshold, the frequency of the local carrier corresponding to the maximum power is the carrier frequency estimation value of the current signal. The Doppler-shift fine estimation can be easily calculated once the center frequency of the quadratic square Iridium signal is searched successfully. When the quadratic square processing module in Figure 3 is removed, the principle of the traditional Iridium SOPs Doppler-shift estimation in open sky comes out. In this case, only tone signal in an Iridium burst is used since the unique word signal and the information signal will lead to acquisition failure due to the phase flip. The traditional method of Doppler-shift estimation cannot be used in weak signal environment. It can cause the bursts missing due to the interference caused by signal blockage. The situation that the starting time of the tone signal cannot be precisely locked may appear when the burst is detected. If it happens, the tone signal may contain a unique word signal (BPSK) and information signal (QPSK) which can affect the correlation results of MLE. As a result, the traditional method fails to estimate Doppler-shift. The new method (QSA-IDE) can overcome the above problems since it squares the Iridium burst twice and the tone signal, unique word signal, and information signal instead of tone are used simultaneously to measure Doppler-shift. On the other hand, the signal power can still be accumulated in weak signal environment by increasing the coherent integration time of MLE since the whole Iridium burst is used. As a result, the QSA-IDE algorithm can measure Doppler-shift in weak signal environment. Figure 3: Basic Iridium SOPs positioning principle in weak signal environment. The Mathematical Model of Estimating Iridium Satellite Signal Doppler-Shift in Weak Signal Environment  
Let tone signal, unique word signal, and information signal in one burst be denoted by \(s_{tone}\), \(s_{uni}\), and \(s_{d}\). The corresponding time periods are \(T_{tone}\), \(T_{uni}\), \(T_{d}\), respectively \(t_{0}\) is the starting time of the burst. Then, the expression of the burst signal can be written as:  
\[S_{Intr}(t)=\left\{ \begin{array}{l}s_{tone}(t);t_{0}\leq t\leq T_{tone}\\ s_{uni}(t);T_{tone}<t\leq T_{uni}\\ s_{d}(t);\ T_{uni}<t\leq T_{d}\end{array} \right. \tag{1}\]  
The signals \(s_{tone}\), \(s_{uni}\), and \(s_{d}\) are needed to be squared twice, respectively before Doppler-shift estimation when we use the QSA-IDE algorithm. This process is easy for tone signal than for the unique word signal and information signal since the tone is without modulation. The expression of \(s_{d}(t)\) is:  
\[ \begin{array}{lll}s_{d}(t)&=&\sqrt{P_{d}}d_{i}(t)cos(2\pi f_{0}t+\varphi_{d}) +\sqrt{P_{d}}d_{q}(t)\cdot\\ &&sin(2\pi f_{0}t+\varphi_{d})+n_{d}(t)\end{array}  \tag{2}\]  
where \(P_{d}\) is the signal power, \(f_{0}\) is the carrier frequency, \(\varphi_{d}\) is the initial phase. \(d_{i}\) and \(d_{q}\) are the binary information code sequences of I and Q arms. \(n_{d}(t)\) is the noise signal. We get the quadratic square signal \(r_{d}(t)\) by squaring the \(s_{d}(t)\) twice, and the \(r_{d}(t)\) can be written as:  
\[ \begin{array}{lll}r_{d}(t)&=&\frac{3}{2}p_{d}^{2}-\frac{1}{2}p_{d}^{2}\cos (8\pi f_{0}t+4\phi_{d})+2p_{d}^{2}d_{i}(t)d_{q}(t)\sin(4\pi f_{0}t+\\ &&2\phi_{d})+2p_{d}\Big{\{}\sqrt{p_{d}}[d_{i}(t)\cos(2\pi f_{0}t+\phi_{d})+d_ {q}(t)\sin(2\pi f_{0}t+\]  
\[\phi_{d})n_{d}(t)+n_{d}^{2}(t)\Big{\}}+2p_{d}d_{i}(t)d_{q}(t)\sin(4\pi f_{0}t +2\phi_{d})\Big{\{}\sqrt{p_{d}}[d_{i}(t)\cdot\\ &&\cos(2\pi f_{0}t+\phi_{d})+d_{q}(t)\sin(2\pi f_{0}t+\phi_{d})]n_{d}(t)+n_{d} ^{2}(t)\Big{\}}+\\ &&\Big{\{}\sqrt{p_{d}}[d_{i}(t)\cos(2\pi f_{0}t+\phi_{d})+d_{q}(t)\sin(2\pi f _{0}t+\phi_{d})]n_{d}(t)+\\ &&n_{d}^{2}(t)\Big{\}}^{2}\end{array}