# Impact Evaluation of DME Beacons on BeiDou B2a Signal Reception Performance
## 3 Interference Evaluation Model
### B2a Signal Reception Performance Degradation  
Detection probability is a key parameter of signal acquisition and determines the acquisition performance [27]. If \(H_{0}\) represents noise only and \(H_{1}\) represents a composite signal consisting of both interference and noise, the power of the input signal, which is defined as the detection metric, can be expressed as  
\[z=\sum_{j=1}^{M}(I_{j}^{2}+Q_{j}^{2}). \tag{14}\]  
The intermediate frequency (IF) signal received by the receiver is mixed with the in-phase and quadrature carriers, respectively, to obtain the I-channel and Q-channel signals. \(I_{i}\) and \(Q_{i}\) are the coherent integration results for the \(I\) and \(Q\) paths, respectively, while \(M\) is the number of incoherent integrations. The detection threshold was \(V_{t}\), and the false alarm probability and detection probability of the signal can be expressed by  
\[P_{f}=\int\limits_{V_{t}}^{\infty}\frac{1}{2^{M}\Gamma(M)}z^{M-1}\exp(-\frac{ z}{2})dz, \tag{15}\]  
\[P_{d}=\int_{V_{t}}^{\infty}\frac{1}{2}(\frac{z}{\lambda})^{\frac{M-1}{2}}\exp (-\frac{z+\lambda}{2})I_{M-1}(\sqrt{z\lambda})dz, \tag{16}\]  
where \(\Gamma(M)=(M-1)!\) is the gamma function; \(\lambda=2M(S/N_{0})=2M(C/N_{0})T_{coh1}\) is the noncentral parameter of the noncentral chi-squared distribution; \(T_{coh1}\) represents the pre-detection integration time; and \(I_{M-1}=(\sqrt{z\lambda})\) is the first class of \(M-1\) order Bessel functions. In the practical acquisition process, the false alarm probability \(P_{f}\) for signal detection should be minimized; typically, it is set to \(P_{f}=1\times 10^{-5}\). And the detection probability of the signal under different detection integration times can be further calculated if \(P_{f}\) and \(C/N_{0}\) are known. The acquired navigation signal iswas passed into the carrier tracking loop, where its performance governed both tracking precision and dynamic response. The most commonly used carrier tracking ring was a phase locked loop (PLL), and its tracking quality was quantified by the carrier tracking measurement error [28]. The jitter sources of the PLL are generally transient for generic airborne civil aviation BDS receivers, and carrier tracking errors are typically attributed to thermal noise. When interference exists, its contribution to carrier tracking error must be additionally accounted for. The final carrier tracking error caused by thermal noise and interference can be determined by\[\delta_{PLL}=\frac{360}{2\pi}\sqrt{\frac{B_{P}}{C/N_{0}}(1+\frac{1}{2T_{\text{coh2 }}C/N_{0}})}, \tag{17}\]  
where \(B_{P}\) is the carrier loop noise bandwidth. In order to conservatively evaluate the performance degradation of the receiver under the DME interference scenario, \(B_{P}\) was taken to be 20 Hz for the simulation [29], and \(T_{coh2}\) was taken to represent the pre-detection integration time for carrier tracking. The data demodulation of the signal is important for the performance, accuracy, and reliability of the whole navigation system. The demodulation bit error rate (BER) quantifies the probability of bit errors (Pe) during the demodulation of navigation signal data, and it is a key metric for assessing signal demodulation performance [5]. When the B2a signal is affected by interference from DME pulses, the equivalent noise power is assumed to be \(\delta_{eq}^{2}\). The equivalent noise can be approximated as a Gaussian noise with mean 0 and a variance of \(\delta_{eq}^{2}\). Assuming the receiver's judgment threshold is 0, the Pe can be expressed as  
\[P_{\epsilon}=Q(\frac{\sqrt{C}}{\delta_{eq}})=Q(\sqrt{2T_{coh3}C/N_{0}}), \tag{18}\]  
where \(C\) represents the signal power, and \(T_{coh3}\) represents the integration duration of the carrier tracking loop.