## 2 Receiving and Evaluation Methodology  
We used static test equipment shown in Figure 1 to collect signals in Xi'an and Sanya, China, which will be introduced later. The extremely high sampling rate of 250 MHz and the 80 MHz bandwidth of the front-end filter meet the ultra-narrow correlation proposed by [PERSON] and [PERSON] [27]. Based on the SDR designed by [PERSON] [28], we designed a new SDR, called GNSS Receiver, uses the L1C signal transmitted on GPS III and QZSS to processing off-line and takes into account multiple differences in their signal structure defined in these documents [3; 11; 12]. Figure 2 is the flow chart of tracking algorithm in GNSS Receiver. Many of its receive-related parameters are flexible and adjustable, including correlator spacing, bandwidth and damping factor for Delay Lock Loop (DLL) and Phase Lock Loop (PLL). The local signal generated by GNSS Receiver for tracking is full bandwidth unfiltered. The tracking method of GNSS Receiver uses Bump-Jump proposed by [PERSON] and [PERSON] [29], and the five pairs of correlators independently demodulate the data component and the pilot component by a coherent integration of 1 ms. The squaring operation of the non-coherent integration destroys the zero-mean of the thermal noise and brings about the squaring loss, so it is not used. Continuously tracking the carrier frequency and code phase of the L1 CD demodulates the navigation massage bit stream, and GNSS Receiver demodulates the overlay code by tracking L1 CP. GNSS Receiver uses the overlay code to complete the frame synchronization of the navigation message [30; 31]. Pseudorange measurements of GNSS Receiver can be selected in TMBOC(6,1,4/33) ranging and BOC(1,1) ranging. Figure 3 is the flow chart of the ranging and positioning after tracking algorithm. For joint positioning, the ephemeris needs to be modified in the algorithm to ensure that the orbit is correctly calculated. Figure 1: The test equipment for collecting signals. Concerning the reception performance, the BOC(1,1) and TMBOC(6,1,4/33) modulations used in the L1C signal are better improved [32; 33]. Their auto-correlation function is shown in Figure 4. The auto-correlation function of TMBOC(6,1,4/33) is sharper than BOC(1,1) in several intervals, including 0-1/12 chips, 1/6-1/4 chips which is helpful to improve the reception of TMBOC (6,1,4/33). Therefore, the choice of correlator spacing is considered when using a software-defined receiver for ranging and positioning, which will be discussed later. Figure 3: Flow chart of the ranging and positioning after tracking algorithm. Figure 2: Flow chart of DLL (Bump-Jump) and PLL used by tracking algorithm in the GNSS Receiver. [PERSON] et al. [18] have demonstrated that setting different correlator spacing has different effects on tracking accuracy. A non-coherent early-late discriminator is used in GNSS Receiver. Assuming the use of a locally generated replica is identical to the unfiltered received signal, the theoretical code-tracking error variance caused by thermal noise [34; 35] is given by:  
\[ \begin{split}\sigma_{IDLL}=c\cdot\sqrt{\frac{B_{L}(1-0.5B_{L}T) \int_{B/2}^{B/2}G_{L}(f)\sin^{2}(\pi fdT_{c})df}{(2\pi)^{2}(C/N)\cdot\left[ \int_{-B/2}^{B/2}G_{L}(f)\sin^{2}(\pi fdT_{c})df\right]^{2}}}\\ \cdot\sqrt{1+\frac{1}{T_{P}}\frac{B_{L}(1-0.5B_{L}T_{P})\int_{-B/ 2}^{B/2}G_{L}(f)\cos^{2}(\pi fdT_{c})df}{(C/N)\cdot\left[\int_{-B/2}^{B/2}G_{L }(f)\cos^{2}(\pi fdT_{c})df\right]^{2}}}\end{split}  \tag{1}\]  
where the meaning of all symbols is shown in Table 1.  
The code-tracking error for BOC(1,1) and TMBOC(6,1,4/33) using different correlator spacing is shown in Figure 5. Theoretically, the loop tracking error of BOC(1,1) increases with the correlator spacing. In the sharper range of the TMBOC(6,1,4/33) autocorrelation function reflected in Figure 1, TMBOC(6,1,4/33) tracking is more accurate than BOC(1,1). Figure 4: Autocorrelation Function for TMBOC(6,1,4/33) and BOC(1,1). Figure 5: Code-tracking error with different correlator spacing. (**a**) Overall view; (**b**) Close-up view.