# Propagation Characteristics of Modulated EHF Signal in the Wake Region of Plasma Sheath
## 3 Numerical Simulations on the BER  
In this section, the time-varying BER characteristics of EHF signals in a plasma sheath with carrier frequencies of 94 GHz, 140 GHz, and 225 GHz and four modulation modes of 2 ASK, 2 PSK, 4 QAM (QPSK), and 2 FSK are studied by Monte Carlo simulation. Firstly, we randomly generate \(10^{5}\) digital signals and modulate them to the above three carrier frequencies using four different modulation modes. Then, the modulated signals are transmitted to the plasma sheath at the incident angle set in Section 2, and Gaussian white noise is added. Next, the modulated signals mixed with noise are demodulated. The plasma sheath causes signal attenuation and phase shift, resulting in bit errors in the received signal. Finally, the BER is statistically counted. Here, we take 2 PSK modulation simulation system as an example to introduce the simulation process used. Assume that only one symbol duration is considered in 2 PSK, and that the power of the signal transmitter is set to 1 mW. The carrier amplitude is set to \(\sqrt{0.002}\). The 2 PSK modulated signal can be expressed as Equation (5). \[s_{1}=\left\{\begin{array}{l}\sqrt{0.002}cos\omega_{c}t\ \mbox{send}\ \mbox{send}\ \mbox{\"1\"}\right. \tag{5}\]  
Make the signal \(s_{1}\) pass through the plasma sheath module, Gaussian white noise module, and band-pass filter module in turn, and then obtain the signal \(s_{2}\) to be demodulated, which can be expressed as Equation (6). \[s_{2}=\left\{\begin{array}{l}a_{1}\Big{(}\sqrt{0.002}Acos\omega_{c}tcos\varphi +n(t)\Big{)}\ \mbox{send}\ \mbox{\"1\"}\\ a_{1}\Big{(}-\sqrt{0.002}Acos\omega_{c}tcos\varphi+n(t)\Big{)}\ \mbox{send}\ \mbox{\"0\"}\right. \tag{6}\]  
where \(A\) and \(\varphi\) are the total transmission coefficient and phase shift of the signal propagating in the plasma sheath, respectively, \(a_{1}\) represents the preliminary processing result of the phase shift caused by the sheath, and \(n(t)\) is a narrow-band Gaussian process which can be orthogonally decomposed. \[a_{1}=\left\{ \begin{array}{l}1\ |\varphi|\leq 90^{\circ}\\ -1\ |\varphi|>90^{\circ}\end{array} \right. \tag{7}\]  
\[n(t)=n_{c}(t)cos\omega_{c}t-n_{s}(t)sin\omega_{c}t \tag{8}\]Finally, the signal \(s_{2}\) is coherently demodulated, and the output signal can be expressed as Equation (9). \[s_{3}=\left\{ \begin{array}{l}a_{1}\Big{(}0.001 Acos\varphi+\sqrt{0.0005}n_{c}(t) \Big{)}\ \textit{send \"1\"}\\ a_{1}\Big{(}-0.001 Acos\varphi+\sqrt{0.0005}n_{c}(t)\Big{)}\ \textit{send \"0\"}\end{array} \right. \tag{9}\]  
where \(n_{c}(t)\) is the in-phase component of the narrowband white Gaussian noise output from the bandpass filter. The BER of channels 3 and 6 are shown in Figure 3. The discontinuity in Figure 3 indicates that the BER is 0. Figure 3 shows that, under the same carrier frequency and noise level, the BER performance of channel 6 is better than that of channel 3, when 2 ASK, 2 PSK, and 4 QAM (QPSK) are adopted. When 2 FSK is adopted, the BER performance of channel 6 is better than that of channel 3 when the carrier center frequency is 94 GHz and 140 GHz, however the situation is just the opposite when the carrier center frequency is 225 GHz. When 2 ASK is adopted, the minimum BER magnitude is bigger than \(10^{-1}\) when the carrier frequency is 94 GHz, and between \(10^{-1}\) and \(10^{-2}\) when the carrier frequency is 140 GHz and 225 GHz. Under the same noise level, the BER performance improves with an increase in carrier frequency. When channel 6 and 225 GHz carrier frequency are selected, BER can reach optimum performance. When 2 PSK is adopted, in channel 3, the minimum BER magnitude is between \(10^{-3}\) and \(10^{-4}\) when the carrier frequency is 94 GHz and 225 GHz, and between\(10^{-4}\) and \(10^{-5}\) when the carrier frequency is 140 GHz. The BER of channel 6 can reach zero at all three carrier frequencies, but the BER remains zero for a longer time when the carrier frequencies are 140 GHz and 225 GHz.