## 4 Navigation Signal Processing
### Signal Acquisition  
In the signal acquisition stage, only coarse estimations of the delay and Doppler frequency need to be provided. The received continuous-time baseband signal is denoted as \(r(t)\), and the discrete-time samples \(r[q]\) obtained after the analog-to-digital converter (ADC) serve as the input to the acquisition module.  
The block diagram of the DDBDM signal acquisition using the parallel code-phase search (PCS) method is shown in Figure 5. This structure is similar to that of a traditional navigation receiver, with the difference being the generation of the local replica, namely the part enclosed in the dashed box. The local replica in TF domain is as follows:  
\[\mathbf{X}_{\text{tf,loc}}=\begin{bmatrix}\mathbf{F}_{M_{\text{s}}}\mathbf{X} _{\text{loc}}\mathbf{F}_{N}^{H}\\ 0_{(M_{\text{s}}-M_{\text{m}})\times N}\end{bmatrix} \tag{19}\]  
where \(\mathbf{X}_{\text{loc}}\) is the local PRN code in the DD domain defined in (5), \(M_{\text{s}}\in\mathbb{Z}_{+}\) and satisfies \(M_{\text{s}}>M_{\text{m}}\), and the zero padding in (19) is used to oversample the PRN code [46]. Therefore, the sampling rate \(f_{\text{s}}=M_{\text{s}}\Delta f\) is an integer multiple of the subcarrier spacing. Since the sampling rate no longer needs to be greater than the entire signal bandwidth \(M\Delta f\), the signal processing overhead for navigation is reduced.  
Figure 5: The block diagram of signal acquisition.  
Then, IFFT and P/S operations are employed to generate the oversampled local replica:  
\[\mathbf{s}_{\text{loc}}=\text{vec}\left(\mathbf{F}_{M_{\text{s}}}^{H}\mathbf{X}_ {\text{tf,loc}}\right) \tag{20}\]  
The discrete-time local replica in the above equation can be pre-stored in the receiver, without the necessity for real-time generation.  
The orthogonality of the basis functions and the periodicity of the PRN code enable receivers to estimate the delay and Doppler frequency in the time domain. The correlation function between the cyclic shift and the conjugate version of the received signal and the local replica is as follows:  
\[R_{\text{rs}}[n]=\sum_{q=0}^{M_{\text{s}}N-1}r[q]\cdot s_{\text{loc}}^{*}[q+n] _{M_{\text{s}}N} \tag{21}\]  
where the number of signal samples in one period is \(M_{\text{s}}\times N\). The correlation values at the delay of each sample in the above equation can be quickly calculated using the FFT/IFFT-based PCS method:  
\[\mathbf{R}_{\text{rs}}=\mathbf{F}_{M_{\text{s}}N}^{H}\big{(}\mathbf{F}_{M_{ \text{s}}N}\mathbf{r}\circ\overline{\mathbf{F}_{M_{\text{s}}N}\mathbf{s}_{ \text{loc}}}\big{)} \tag{22}\]  
where \(\circ\) denotes the Hadamard product operator.  
The carrier numerically controlled oscillator (NCO) is used to adjust the Doppler frequency of the received signal. After the cross-correlation operations for all possible frequency bins, a correlation value matrix is generated. The maximum value obtained from the incoherent combination of the real and imaginary parts of the elements in the matrix is then compared with the detection threshold to determine whether the signal is present [50].  
If the signal is detected, the maximum likelihood (ML) estimate of the delay can be calculated by finding the peak value of the correlation corresponding to the current frequency bin:  
\[\hat{\tau}=\frac{1}{f_{\text{s}}}\cdot\text{argmax}_{n}\Big{\{}|R_{\text{rs}} [n]|^{2}\Big{\}} \tag{23}\]  
Finally, since the accuracy of the frequency bin determined during the signal detection process typically does not meet the precision for initializing the tracking loop, a fine Doppler frequency \(\hat{\
u}\) can be estimated using methods such as spectral analysis [51].