# A Cumulant-Based Method for Acquiring GNSS Signals
## 3 Cumulant-Based Acquisition Method
### FFT-Cumulant Method for GNSS Signal Acquisition  
As previously noted, the Gaussian nature of GNSS signals makes the direct application of the cumulant method largely ineffective. To address this challenge, we first apply an FFT to the signal and then compute its third-order cumulant in the frequency domain. To begin, we must define our signal model for the Galileo E1 signal. Let's assume that the receiver measurement \(y(t)\) satisfies the following equation:  
\[y(t)=\sum_{k\in N}c^{k}(t-\tau^{k})c_{sc}(t)\cos(2\pi(f_{1}+f_{D}^{k})t+\theta^ {k})+w(t),N\in[1,32] \tag{9}\]  
where superscript \(k\) denotes the satellite PRN number, \(c^{k}(t-\tau^{k})\) is the Galileo Primary code with different delay, and \(\tau^{k}\); \(c_{sc}(t)\) is the square wave sub-carrier used on BOC modulation; \(d(t)\) is the navigation data; \(\cos(2\pi(f_{1}+f_{D}^{k})t+\theta^{k})\) is the carrier signal on L1 band with satellite-to-receiver Doppler shift \(f_{D}^{k}\) and carrier phase \(\theta^{k}\) with respect to the local reference oscillator; and \(w(t)\) is an unknown noise source. We shall now drop the superscript for simplicity, resulting in the following:  
\[y(t)=c(t-\tau)c_{sc}(t)\cos(2\pi(f_{1}+f_{D})t+\theta)+w(t) \tag{10}\]  
The receiver's RF front-end performs several critical functions on the signal received from the satellite, including mixing, frequency down-conversion, and low-pass filtering. These operations produce a baseband signal that is suitable for further processing, as shown below:  
\[y^{\prime}(t)=c(t-\tau)c_{sc}(t)\cos(2\pi(f_{1F}+f_{D})t+\theta)+w(t) \tag{11}\]  
The method proposed in this study can be implemented using two different approaches, as described below:  
Approach #1:  
1. Carrier Signal Multiplication: The received signal is first multiplied by a locally generated carrier signal. This process generates two components: the in-phase (I) signal, by multiplying with the local carrier, and the quadrature (Q) signal, by multiplying with a phase-shifted version of the carrier. These two signals are then combined to form the complex input signal \(s(t)=I(t)+jQ(t)\). 2. PRN Sequence Multiplication: the complex signal is then multiplied by locally generated PRN (Pseudo-Random Noise) sequences. 3. Frequency-Domain Transformation: after multiplication with the PRN sequence, a Discrete Fourier Transform (DFT) is performed on the signal to convert it into its frequency-domain representation. 4. Cumulant Estimation: The third-order cumulant is estimated from the frequency-domain signal. The absolute value of the resulting output represents the correlation in the frequency-cumulant domain. The above operation is illustrated in Figure 3a. In the second step, the signal is multiplied by a known PRN sequence with a code phase delay ranging from 0 to 4091. In Step 1, the Doppler shift and carrier phase are effectively removed, resulting in a signal that is ready for further processing:  
\[\widetilde{y}(t)=c(t-\tau)c_{sc}(t)+w(t) \tag{12}\]  
This yields a cleaned signal, free from Doppler effects and aligned with the locally generated PRN sequence, allowing for accurate subsequent analysis. Next, by multiplying the signal from Equation (12) by a local replica of the PRN code, which is modulated by the sub-carrier reciprocal term, we obtain  
\[\widetilde{y}(t)c(t-\hat{\tau})c_{sc}^{-1}(t)=c(t-\tau)c_{sc}(t)c(t-\hat{\tau} )c_{sc}^{-1}(t)+w(t)c(t-\hat{\tau}) \tag{13}\]Here, \({}^{\sim}\) indicates estimation. If \(\hat{\tau}\) is shifted to align with the real \(\tau\), then  
\[\widetilde{y}(t)=c^{2}(t-\tau)+\hat{w}(t) \tag{14}\]  
It is easy to see that the term \(c^{2}(t-\tau)\) is a constant, where all elements are equal to one. When we take the Fourier transform in Step 4, it is well-known that the Fourier transform of a constant yields an impulse in the frequency domain. As a result, we obtain a sequence of data with one large value while all other values are equal to zero.