## 2 Modeling of Receiver Pulsed Environment Interference Effect Based on BER
### BER Formula under Pulse Signal Interference
Whether or not an error occurs mainly depends on the amplitude of the GWN at the decision time. From a statistical point of view, the level is mainly determined by the power spectral density of the GWN signal. The pulsed interference signal is periodic in the time domain, so it only interferes with part of symbols, and it is considered whether there will be a misjudgment under the condition that the symbols are interfered. So, the calculation step of the receiver's BER under pulsed interference is the conditional probability calculation: firstly consider the probability of symbol interfered, and secondly consider the probability of misjudgment generated by the symbol under the condition of interference. For different interference patterns, the derivation idea of BER formula under pulsed signal interference mentioned in this paper can still be used. Different parameters of the pulse signal and receiver will make the repetition period and the receiver symbol period have a different relative magnitude relationship. When the pulse width is smaller than the symbol period of the receiver, the pulse signal will interfere with at least one symbol and at most two symbols; when the pulse width is larger than the symbol period, the pulse signal will interfere with multiple symbols. After mathematical deduction, fortunately, in the case of long pulse and short pulse, the calculation results of interference probability are consistent:  
\[P_{1}=\frac{W+T_{s}}{T_{PRP}} \tag{11}\]  
where \(W\) is the pulse width, \(T_{s}\) is the symbol period, \(T_{PRP}\) is the repetition period. Under the condition that the symbol is determined to be interfered, the interference signal at this time can be equivalent to noise, and the probability of misjudgment can be deduced according to the traditional theory. Finally, the product of the two, \(P_{e}=P_{1}\cdot P_{2}\), is used as the BER formula of the receiver under pulse signal interference. For example, assuming that the receiver adopts Quadrature Amplitude Modulation (QAM) or Quadrature Phase Shift Keying (QPSK) modulation and coherent demodulation, when only the pulse signal interference is considered and the symbol is determined to be interfered, the probability of misjudgment can be derived as:  
\[P_{2}=\frac{1}{2}erfc\sqrt{\frac{1}{2}\cdot SI} \tag{12}\]  
\[erfc(x)=1-erf(x)=\frac{1}{\sqrt{\pi}}\int\limits_{x}^{\infty}e^{-\eta^{2}}d\eta \tag{13}\]  
Figure 6: Noise interference and pulse interference contrast diagram. where \(SIR\) is the signal-to-interference ratio. It is considered that during the operation of the receiver, the receiver will not only receive functional signals, but also receive interference signals in the external electromagnetic environment. The SIR is usually used to characterize the relationship between the function signal and the interference signal from the perspective of energy, which can be expressed as follows:  
\[SIR=P_{signal}(dB)-P_{interfrence}(dB) \tag{14}\]  
where \(P_{signal}\) is the received function signal power, and \(P_{interfrence}\) is the received interference signal power. SIR is related to the performance of the receiver itself. When SIR is higher than a certain value, according to Formula (14), that is, when the functional signal power is higher than a certain value of the interference signal power, the receiver can work normally. Otherwise, it cannot work normally, which means interference occurs. The interference effect of pulse signal amplitude on the receiver can be analyzed by SIR. Therefore, the BER of the receiver adopting QAM under pulsed interference \(P_{e,QAM}\) can be expressed as follows:  
Figure 4:0  
For the probability of it being interfered, it can be decomposed into the following two parts:  
Figure 4:1  
where \(\frac{W}{T_{PRP}}\) is the duty cycle, and \(\frac{T_{s}}{T_{PRP}}\) is the ratio of the receiver symbol period to the interference repetition period.