# GNSS Accuracy Under White Gaussian Noise Jamming +
## 2 The Impact of a Jammer on DLL Noise  
Pseudo-range errors are a result of the tracking error of the delay lock loop (DLL), so this is our starting point. The tracking error for a non-coherent DLL discriminator is given by the general expression [5; 6; 7]  
\[\sigma_{\tau}^{2}=\frac{B_{L}\int_{-B_{\tau}/2}^{B_{\tau}/2}S_{s}(f)\sin^{2}( \pi fdT_{c})\mathrm{d}f}{(2\pi)^{2}\frac{C}{N_{0}}T_{c}^{2}\Big{(}\int_{-B_{ \tau}/2}^{B_{\tau}/2}fS_{s}(f)\sin(\pi fdT_{c})\mathrm{d}f\Big{)}^{2}}\Bigg{[} 1+\frac{\int_{-B_{\tau}/2}^{B_{\tau}/2}S_{s}(f)\cos^{2}(\pi fdT_{c})\mathrm{d }f}{T\frac{C}{N_{0}}\Big{(}\int_{-B_{\tau}/2}^{B_{\tau}/2}S_{s}(f)\cos(\pi fdT _{c})\mathrm{d}f\Big{)}^{2}}\Bigg{]}, \tag{1}\]  
where \(S_{s}(f)\) is the normalized power spectral density of the signal used, \(d\) is the correlator spacing, \(T_{c}\) is the duration of a code chip, \(T\) is the coherent integration time, \(B_{L}\) is the DLL tracking bandwidth, and \(B_{\tau}\) is the receiver's front-end bandwidth. In (1), the noise and interference are assumed to be white Gaussian noise. In [1], a simplified version of (1), specifically for BPSK signals, is given:  
\[\sigma_{\tau}^{2}=\frac{B_{L}dT_{c}^{2}}{2\frac{C}{N_{0}}}\left(1+\frac{2}{T \frac{C}{N_{0}}}\right). \tag{2}\]  
As we see in these equations, the accuracy is dependent on the \(C/N_{0}\), amongst other parameters. The \(C/N_{0}\), however, is the only parameter that can be influenced by an interference source, such as a jammer. So, if we can determine the impact of a jammer on the \(C/N_{0}\), we will be able to predict the accuracy of the GNSS receiver while under attack from a jammer. In the literature, the following expression for the impact of interference with a received power \(J\) on the effective \(C/N_{0}\) can be found [7]:  
\[(C/N_{0})_{eff}=\frac{P_{B_{r}}}{\frac{N_{0}}{C}+\frac{T_{c}}{Q}\int\limits_{C} ^{T}}, \tag{3}\]  
where \(P_{B_{r}}\) is the normalized received power of the desired signal within the bandwidth of the front end of the receiver, and \(Q\) is a jamming resistance quality factor. This jamming resistance quality factor can be calculated as follows:  
\[Q=\frac{T_{c}P_{B_{r}}}{\int_{-B_{r}/2}^{B_{r}/2}S_{i}(f)S_{s}(f)df}, \tag{4}\]  
where \(S_{i}(f)\) is the normalized power spectral density of the interference; for an AWGN jammer,  
\[S_{i}(f)= \begin{cases}\frac{1}{B_{i}}&\text{for}-\frac{B_{i}}{2}\leq f\leq \frac{B_{i}}{2}\\ 0&\text{otherwise},\end{cases} ,\]  
where \(B_{i}\) is the bandwidth of the jammer. Further, the received power, \(P_{B_{r}}\), of the signal is expressed as  
\[P_{B_{r}}=\int\limits_{-B_{r}/2}^{B_{r}/2}S_{s}(f)df.\]  
The normalized power spectral densities for BPSK and CBOC are expressed as  
\[S_{BPSK}(f)=T_{c}\text{sinc}^{2}(\pi fT_{c}), \tag{5}\]  
and  
\[S_{CBOC}(f)=T_{c}\text{sinc}^{2}(\pi fT_{c})\bigg{[}\frac{10}{11}\text{tan}^{ 2}\bigg{(}\frac{1}{2}\pi fT_{c}\bigg{)}+\frac{1}{11}\text{tan}^{2}\bigg{(} \frac{1}{12}\pi fT_{c}\bigg{)}\bigg{]}. \tag{6}\]  
Finally, the received power of the jammer for the receiver at a distance \(R\) can be calculated by using the well-known Friis law, expressed as  
\[J=J_{0}G_{t}G_{r}\bigg{(}\frac{\lambda}{4\pi R}\bigg{)}^{2},\]  
where \(J_{0}\) is the transmitted jammer power; \(G_{t}\) and \(G_{r}\) are the antenna gains for the jammer and receiver, respectively; and \(\lambda\) is the wavelength. Upon inspecting (3), it becomes clear that both the duration of a chip, \(T_{c}\), and the jamming resistance quality factor, \(Q\), influence the impact of a jammer. A smaller \(T_{c}\) (a larger chiprate) will reduce the impact of a jammer, so the P(Y) code with a chiprate of 10.23 MHz will be less affected by a jammer than a C/A code with a chiprate of 1.023 MHz. This is, of course, exactly as expected. Similarly, a larger \(Q\) will also reduce the impact of the jammer. In Figure 1a, the value of \(Q\) is given for different GNSS signal modulations and a white noise interference source as a function of the normalized (by the chiprate \(f_{s}\)) bandwidth of the interference.