## 3 The Signal Existence Verification (SEV) Process for GPS Signal Detection
Figure 4(a) shows an example of a single trial decision where the blue and red curves represent the probability density functions (pdfs) of noise with signal absent and present, respectively. The pdf for noise with signal absent has a zero mean, and the pdf for noise with signal present has a non-zero mean. A satellite signal is declared present if the estimated SNR in the acquisition process is beyond the threshold. However, a false alarm of signal present might occur as the blue area of Figure 4(a). The signal detection process would have a probability of miss-detection (\(P_{md}\)) as shown in the red area of Figure 4(a). The false alarm probability (\(P_{fa}\)) is defined under the hypothesis \(H_{0}\) of signal absence or code delay/Doppler frequency shift mismatch as [1,2]:  
\[P_{fa,V_{t}^{\prime}}=P_{fa}\left(S(\tau_{\max},f_{D,\max})>V_{t}\left|H_{0}\right.\right) \tag{8}\]  
It is the probability that the decision variable \(S(\tau_{\max},f_{D,\max})\) passes a fixed threshold \(V_{t}\) under \(H_{0}\). On the contrary, the detection probability (\(P_{det}\)) is defined under the hypothesis \(H_{1}\) of signal presence and perfect delay and frequency alignment between the received signal and the local replica [1,2]:  
\[P_{\det,V_{t}^{\prime}}=P_{\det}\left(S(\tau_{\max},f_{D,\max})>V_{t}\left|H_{1 }\right.\right) \tag{9}\]  
In general, a required false alarm probability (\(P_{fa}\)) is chosen first, and the corresponding threshold and detection probability (\(P_{d}\)) can be estimated [2]. In addition, the GPS signal processing is sensitive to noise. For the low received power satellite signal case, the pdfs curves of noise with signal absent and present are close as shown in Figure 4(b). The probability of miss-detection would be increased while the required false alarm probability is strict as that of the signal with a typical strength. As a result, the satellite signal with low received power might be declared as signal absent while the acquired SNR is below the threshold. The number of acquired satellite therefore decreased and then might not be sufficient for positioning. This paper therefore designs a scheme, called the signal existence verification (SEV) process, based on the time-frequency approach to verify whether the satellite signal is absent. Note that the power of GPS signals is spread over a much wider bandwidth than that of the original information because GPS is a DSSS system [4]. As a result, the GPS signals present power spectral densities are completely hidden under the noise floor, as shown in the left plot of Figure 5(a). In other word, it is difficult to observe the useful information of satellite signals on the spectrogram if the GPS raw signal is mapped into the time-frequency plane directly, as shown in the right plot of Figure 5(a). Thanks to the processing gain of the spreading code, the power spectral density of original information could be despread beyond the noise floor. The proposed SEV method therefore takes the characteristic to perform the verification of the low received power satellite signal, as shown in Figure 5(b). The C/A code effect is removed from the received satellite signal by multiplying a local replica C/A code generated according to the acquired code delay information, \(\tau_{\max}\), in the acquisition process. From Equation (1), if the signal is present, the remaining signal with a length of 1 millisecond would consist of a carrier and undesired noise terms as:  
\[S_{L1}^{(k)}[n]=\cos[\omega_{I_{R}}n]+\epsilon[n] \tag{10}\]  
Note that the navigation data is the same value within 1 millisecond. The remaining signal is subsequently mapped into the time-frequency plane by a time-frequency tool. Ideally, the characteristic of the sinusoidal wave can be further detected around the Doppler frequency of \(f_{D,max}\) from the resulting spectrogram, as shown in the time-frequency plane of Figure 5(b). On the contrary, it would be difficult to observe the helpful information in the generated spectrogram if the satellite signal is absent.