## 2 Modeling of Receiver Pulsed Environment Interference Effect Based on BER
### BER Analysis under GWN Interference  
At the first stage, let us consider GWN as the only interference and the BER can be calculated as follows. Digital modulation is mainly divided into amplitude modulation, frequency modulation and phase modulation, which correspond to three modes: amplitude-shift keying (ASK), frequency-shift keying (FSK) and phase-shift keying (PSK). If binary form is adopted, it corresponds to binary amplitude-shift keying (2 ASK), binary frequency-shift keying (2 FSK) and binary phase-shift keying (2 PSK) respectively. The difference between the three modulation modes is mainly the different ways in which the digital signal controls the change of parameters. 2 ASK is the modulation mode in which the digital signal controls the change of carrier amplitude. 2 FSK is the modulation mode in which the digital signal controls the change of carrier frequency, and 2 PSK is the modulation mode in which the digital signal controls the change of carrier phase.  
As an example, let us assume the 2 ASK has been used as the modulation scheme and the filter at the front end of the receiver has ideal rectangular frequency selection characteristics, then its output signal can be expressed as:  
\[y(t)=\left\{\begin{array}{cc}a\cos\omega_{c}t+n(t)&\mbox{when sending ``1''}\\ n(t)&\mbox{when sending ``0''}\end{array}\right. \tag{1}\]  
where \(n(t)\) is the narrowband GWN of the output after the filter, and \(n(t)\) can be denoted as the sum of the in-phase part and quadrature part.  
\[n(t)=n_{c}(t)\cos\omega_{c}t-n_{s}(t)\sin\omega_{c}t \tag{2}\]  
After substituting it to (1), it is mixed with the coherent carrier of the same frequency and phase, and then the signal is filtered through a low-pass filter to remove frequency components outside the pass-band, and the filtered signal is used as the input signal of the sampling decision device. The mathematical expression of the input is:  
\[x(t)=\left\{\begin{array}{cc}a+n_{c}(t)&\mbox{when sending ``1''}\\ n_{c}(t)&\mbox{when sending ``0''}\end{array}\right. \tag{3}\]  
where \(a\) is the component of the useful signal, \(n_{c}(t)\!\sim\!\!N(0,\sigma_{n}^{2})\), so it is obvious that \(x(t)\) also obeys a Gaussian distribution. Thus, \(x(t)\) is sampled as:  
\[x=x(kT_{B})=\left\{\begin{array}{cc}a+n_{c}(kT_{B})&\mbox{when sending ``1''}\\ n_{c}(kT_{B})&\mbox{when sending ``0''}\end{array}\right. \tag{4}\]  
Thus the \(k_{th}\) sampled value also obeys Gaussian distribution, and the probability distribution function is:  
\[\begin{array}{c}f_{1}(x)=\frac{1}{\sqrt{2}\pi\omega_{n}}\exp\{-\frac{(x-a)^{ 2}}{2\sigma_{n}^{2}}\}\\ f_{0}(x)=\frac{1}{\sqrt{2}\pi\omega_{n}}\exp\{-\frac{3^{2}}{2\sigma_{n}^{2}}\} \end{array} \tag{5}\]The BER of the system is:  
\[P_{e}=P(1)P(0/1)+P(0)P(1/0)=P(1)\int\limits_{-\infty}^{b}f_{1}(x)dx+P(0)\int \limits_{b}^{\infty}f_{0}(x)dx \tag{6}\]  
where \(b\) is the detection threshold. If \(P(0)=P(1)=1/2\), the best value of \(b\) equals to \(b^{*}=a/2\), finally the BER is:  
\[P_{e}=\frac{1}{2}erfc(\frac{a}{2\sqrt{2}\sigma_{n}})=\frac{1}{2}erfc(\sqrt{ \frac{r}{4}}) \tag{7}\]  
where \(erfc(x)=1-erf(x)=\frac{2}{\sqrt{n}}\int\limits_{x}^{\infty}e^{-\eta^{2}}d\eta\).