## 3 The Signal Existence Verification (SEV) Process for GPS Signal Detection  
This paper focuses on the GPS signal transmitted on L1 frequency band (1575.42 MHz). After down-conversion in the front-end and sampling by an A/D converter, the GPS L1 signal received from the \(k^{\text{th}}\) satellite, \(S_{ L1}^{(k)}[n]\), can be described as [23]:  
\[S_{ L1}^{(k)}[n]=\sqrt{P_{c}}C^{(k)}[n]D^{(k)}[n]\cos[\omega_{ \textit{\tiny{IF}}}n]+\textit{e}[n] \tag{6}\]  
where \(P_{c}\) is the power of signal associated with C/A code. \(C^{(k)}\) is the 1023-chip coarse/acquisition (C/A) code sequence with a chip rate of 1.023 MHz assigned to satellite number \(k\). The C/A code is also called the pseudorandom noise (PRN) code, and each PRN number could represent an identification of a specific GPS satellite [1; 2; 3]. \(D^{(k)}\) is the navigation data sequence of a binary value with a data rate of 50 Hz assigned to satellite number \(k\). \(\omega_{IF}\) is the intermediate frequency (IF) after front-end down-conversion, and \(n\) denotes the \(n^{\text{th}}\) sample of signal in discrete time representation. \(\textit{e}[n]\) is the undesired signal, which might consist of interference or Gaussian noise. The signal detection process of a GPS receiver is implemented in the signal acquisition stage, which estimates the PRNs, code delay chips, and Doppler frequency of satellites in the received GPS signal. It is required to search the possible Doppler frequency shift and code delay chip in the range of \(\pm\)5 kHz and 0-1023 chips, respectively [1; 2; 3; 24]. In general, the conventional acquisition process implemented in a software defined receiver (SDR) utilizes the parallel code phase search acquisition method [24], as shown in Figure 3. The correlation process between the GPS raw IF signal and the locally generated replica is done based on a method of performing circular correlation through Fourier transform [3; 25]. The incoming GPS signal is first multiplied by a locally generated carrier signal. Multiplications with the carrier signal and a 90\({}^{\circ}\) phase shifted version of the carrier signal generate the inphase (I) and quadrature (Q) signals, respectively. Figure 2: An example of the non-sinusoidal part effect in Fourier spectrum (adopted from [22]). The I and Q signals are combined to form a complex input signal, \(x[n]\), to the FFT function, and \(x[n]\) can be expressed as  
\[x[n]=I[n]+jQ[n] \tag{7}\]  
The generated PRN code is transformed into the frequency domain by FFT and the result is complex conjugated. The Fourier transform of the input GPS signal is subsequently multiplied with the Fourier transform of the PRN code. The multiplication result is transformed into the time domain by an IFFT. As a result, the absolute square value of the output of the IFFT represents the correlation result between the input GPS signal and the local replica of a specific PRN. The correlation result is a two-dimensional search pattern, \(S(\tau,f_{{}_{D}})\), called the ambiguity function, which is a function of the code delay chip \(\tau\) and Doppler frequency bin \(f_{{}_{D}}\)[1, 2]. Each cell of the search pattern represents the correlation value at the corresponding code delay chip and Doppler frequency bin. If a correlation peak \(S(\tau_{\max},f_{{}_{D,\max}})\) can be found on the search pattern, the index \((\tau_{\max},f_{{}_{D,\max}})\) of this peak denotes the code delay sample and the Doppler frequency for the specific PRN (_i.e.,_ satellite). With the acquired information, the receiver tracks and subsequently decodes the navigation bits, and the user position can be estimated accordingly [1, 2, 3, 23]. In general, a SNR of a satellite signal can be estimated from the maximum correlation peak [1, 2, 3, 24]. In theory, the  
Figure 3: The GPS signal acquisition process using the parallel code phase search acquisition method. signal detection in the acquisition process is a statistical process, because each cell either contains noise with the signal absent or noise with the signal present [2].