# A Novel Signal Design and Performance Analysis in NavCom Based on LEO Constellation
## 4 Performance Evaluation
### Evaluation Criteria  
The performance evaluation of GNSS spread-spectrum modulation mainly focuses on an analysis of the performance of the proposed scheme on the accuracy potential indicator, compatibility ability indicator, anti-multipath error ability indicator, and anti-interference ability indicator. Gabor bandwidth (also called RMS bandwidth) is a core indicator that is commonly used to test the tracking accuracy of receivers. It can be defined as:  
\[\beta_{RMS}=\sqrt{\int_{-B_{r}/2}^{B_{r}/2}f^{2}G_{s}(f)df} \tag{10}\]  
Figure 7: CE–OFDM–PM (64) is compared with NSCC (19) in terms of normalized power spectral density. In the normalized PSD, the scarlet solid line is significantly larger than the blue, and the former supports the multiplexing of more users in the same bandwidth and suppresses the power generated by other signals at the same band. This is in line with typical green radio techniques and helps to improve the energy efficiency. where \(B_{r}\) is the bandwidth of the receiver RF front-end, \(G_{s}\) (_f_) is the normalized power density spectrum, and \(f\) is the carrier frequency. The measurement index of spreading code tracking error can be divided into coherent early minus-late (EML), coherent early minus-late power (EMLP), and Cramer-Rao lower bound (CRLB). The coherent EML spreads the code tracking error in additive white Gaussian noise (AWGN) and is based on the spread spectrum code tracking loop. As the product of the coherent time and the tracking loop bandwidth approaches zero and the coherent time difference between the early minus late in reference signal is very tiny, CRLB can be defined as:  
\[\sigma_{CRB}^{2}=\frac{B_{L}}{\left(2\pi\right)^{2}(C/N_{0})\int_{-B_{r}/2}^{ B_{r}/2}f^{2}G(f)df} \tag{11}\]  
In Equation (11), \(\sigma_{CRB}\) is the Cramer-Rao lower bound, \(B_{L}\) is the tracking loop bandwidth, \(\Delta f_{Gabor}\) is the Gabor bandwidth, \(B_{r}\) is the receiver RF front-end bandwidth, and \(C/N_{0}\) is the carrier-to-noise ratio. The term SSC is used to calculate the amount of noise created when an external signal introduces the receiver together with the input signal in ITU-R M.1831. It is used to measure the interaction and interference between different signals in-band. \[\chi_{I}^{s}=\int_{-B_{r}/2}^{B_{r}/2}G_{s}(f)G_{I}(f)df \tag{12}\]  
In Equation (11), \(B_{r}\) is the bandwidth of the receiver RF front-end, \(G_{s}\) (_f_) is the normalized PSD with the desired signal, and \(G_{I}\) (_f_) is the normalized PSD with the interference signal. The term CT-SSC is a function used to estimate the influence of interference signals on the tracking performance of the desired signal under a specific spreading code tracking loop discriminator. When the correlation distance \(d\) is relatively small, CT-SSC can be expressed as:  
\[\kappa =\frac{\int_{-B/2}^{B/2}G_{s}(f)G_{I}(f)(d\pi f)^{2}df}{\int_{-B/2}^{B/2} G_{s}(f)(d\pi f)^{2}df} \tag{13}\]  
where \(B\) is the transmission bandwidth, \(G_{s}\) (_f_) is the normalized power density spectrum of the desired signal, \(G_{I}\) (_f_) is the normalized PSD in the interference signal, and \(f\) is the carrier frequency. The mathematical model of the multipath error envelope (MEE) can be expressed as:  
\[\varepsilon_{\tau}\approx\frac{\pm\int_{-B/2}^{B/2}G(f)\text{sin}(\pi fd)\left[ \sum\limits_{i=1}^{m}a_{i}\text{sin}(2\pi f\tau_{i})\right]df}{2\pi\int_{-B/2}^ {B/2}f\cdot G(f)\text{sin}(\pi fd)\left[1\pm\sum\limits_{i=1}^{m}a_{i}\text{ cos}(2\pi f\tau_{i})\right]df} \tag{14}\]  
In Formula (14), \(B\) is the pre-filter bandwidth, \(G(f)\) is the normalized PSD of the signal, \(d\) is the correlation distance, \(\alpha_{i}\) and \(\tau_{i}\) are the amplitude and delay with the \(i\)-th multipath of the NLOS signal, and \(m\) is the total number of multipaths.