# Improving Structural Power Content Analysis Robustness for Satellite Navigation Applications +
## 3 Jammer Models  
Five different jamming signals were considered: continuous wave (CW), swept CW or chirp, additive white Gaussian noise (AWGN), amplitude modulation (AM), and frequency modulation (FM). The jammer signals were modeled according to the definitions presented in [5]. For the purpose of illustrating how jammer signals affect the SPCA algorithm, we considered the cases of a CW and AM jammer in detail. Let the CW and AM jammer signals be represented as \(s_{cw}(t)\) and \(s_{am}(t)\), as follows:  
\[s_{cw}(t)=A\times e^{i(2\pi f_{c}t+\phi_{c})}, \tag{7}\]  
\[s_{am}(t)=A\times[1+m\text{cos}(2\pi f_{m}t+\phi_{m})]e^{i(2\pi f_{c}t+\phi_{ c})}, \tag{8}\]  
where the carrier frequency and phase are represented as \(f_{c}\) and \(\phi_{c}\), respectively, and \(A\) is the jammer signal amplitude. In the case of the AM jammer, \(m\) is the modulation index and \(f_{m}\) and \(\phi_{m}\) are the message signal frequency and phase, respectively. The power in the CW jammer signal is concentrated entirely within the carrier frequency \(f_{c}\). When passed through the DAM operation, the output of a CW jammer signal has the following form:  
\[y_{cw}[n]=\left(A\times e^{i(2\pi f_{c}nT_{s}+\phi_{c})}\right)\left(A\times e ^{-i(2\pi f_{c}(n-k)T_{s}+\phi_{c})}\right)=A^{2}e^{i2\pi f_{c}kT_{s}}, \tag{9}\]  
which is constant in discrete time. The DAM operation, as expected, removed the carrier and shifted all the signal power to Direct Current (DC). The power in the AM signal is distributed across three frequencies, the carrier \(f_{c}\) and the lower and upper sidebands \(f_{c}\pm f_{m}\), and the relative power sharing between these channels is determined by the modulation index. A simple intermodulation analysis shows that the result of passing such an AM signal through the DAM operation results in a signal for which all the power is distributed across five frequencies: \(\pm 2f_{m},\ \pm f_{m},\ \text{and}\ \text{DC}\). The result for either jammer signal is an excess of power in the DC channel, resulting in an increase in the test statistic for the SPCA method and potentially a false positive detection. Furthermore, for an AM signal where the message signal frequency is a multiple of the GNSS PRN repeat rate, additional excess power appears at additional frequencies passed by the signal filter, resulting in a further increase in the test statistic. To illustrate these results, authentic GNSS signals were recorded using a software-defined radio (SDR) and, subsequently, different jammer signals were added digitally to the recorded \"clean\" signal. The result was then provided as the input to a software implementation of the SPCA algorithm and the impact of each jammer signal was analyzed. The desired probability of false alarms was set to 0.01 and the corresponding test statistic threshold was calculated, as described above. The results are shown in Figure 1.  
Figure 1a shows the power spectral density (PSD) of the DAM output, where jammer signal model parameters were selected such that the spectral distribution for each signal would be more easily discernable. The AM and FM message frequency were \(1\) kHz and 4 kHz, respectively, and for both the modulation index was set to 0.7. The chirp swept a bandwidth of 500 kHz with a repeat rate of 5.5 kHz. All the signals used a modulation frequency of 1 MHz above the nominal L1 carrier. In agreement with the analysis above, the DAM operation on the CW caused the signal power to concentrate at DC, and for the AM jammer, power appeared in the expected intermodulation products. The FM jammer and chirp jammer also produced spectral lines at the harmonics of the message frequency and sweeping rate, respectively. As expected, the AWGN jammer did not produce a spectral structure in the DAM output. These results indicate that, for a given jammer model from the set CW, chirp, AM, and FM, the proper selection of the parameters can produce signal structures that could cause the SPCA algorithm to falsely detect a spoofing signal.