Numerical modeling on the bit error rate of EHF communication in time-varying hypersonic plasma sheath

Once an object is moving hypersonically in near space, the neutral air in front of the object will be intensively compressed. The compression of the air results in aerodynamic thermal heating. The temperature of the gas surrounding the object rises significantly. The temperature could be up to thousands of kelvins. The particles of the neutral gas may be ionized due to the high temperature, which consequently form a weakly ionized gas layer called the plasma sheath. The plasma sheath envelopes the whole object. The upload and download communication signals could be shielded by the plasma sheath. As a result, the communication blackout occurs.^1,2 Due to the blackout, not only the communication signals but also the navigation signals cannot reach the hypersonic vehicle.³ The blackout problem has been considered as a serious threat to manned re-entry vehicles and modern hypersonic cruise flights since the 1950s.⁴ Unfortunately, it is still an unsolved problem.

In the recent couple of decades, extremely high frequency (EHF) communication has been considered as a potential solution to the blackout problem. The EHF band is from 30 to 300 GHz. The electron density in the plasma sheath could be up to 10²⁰ m⁻³, whose corresponding cutoff frequency is 89 GHz. The electromagnetic waves (EM) at the frequency higher than 89 GHz are able to penetrate the plasma sheath. In other words, it makes sense to use a communication system operating in the EHF band to mitigate the blackout effectively.

The transmission characteristics of the EHF signal in a plasma sheath have been studied theoretically and experimentally.^5,6 It has been found that the EHF signals can penetrate the plasma sheath.^7–10 On the other hand, the collisions between electrons and neutral particles play very significant roles in EHF signal attenuation in plasma sheaths.^11,12 The EHF signal attenuation in plasma sheaths increases with electron density and electron collision frequency.¹³ The spatial distribution of plasma also impacts the EHF signal attenuation in plasma sheaths.¹⁴ The EHF signal attenuation could vary with the variation of flight conditions since not only electron densities but also electron collision frequencies are impacted by the flight conditions.^15,16 In addition, the signal attenuation varies with the temporal variation of the plasma sheath.¹⁷ There are several approaches to reduce the attenuation of the EHF signal in plasma sheath, including adjusting angle of attack (AOA), using higher carrier frequency, and introducing external magnetic field to modulate the dispersion relationship of EHF waves in plasma sheath.^18–21 

It should be appreciated that the blackout is a problem of wireless communication, i.e., all the signals propagating in plasma sheaths are modulated. However, most of the electromagnetic waves involved in previous studies are sinusoidal waves only, which means those waves cannot be simply treated as “signals.” In modern wireless digital communications, the BER is a very important parameter in communication systems. The communication quality and data transmission rate are largely dependent on the BER.²² In addition, the plasma sheath is inhomogeneous in space,^23,24 and it always changes over time.^25,26 Wei et al. pointed out that the BER varies with the evolution of plasma sheath.²⁷ In the work of Wei, the plasma parameters measured in the radio attenuation measurement C-II (RAMC-II) re-entry experiment was employed to investigate the impact of the evolving plasma sheath on the BER. The period of plasma sheath evolution was about 23 s in that work. The frequencies taken were 1.575, 2.3, 4.0, 8.4, 32, and 100 GHz. The modulation modes investigated in that study were M-ary Phase-Shift Keying (MPSK) and Non-Coherent demodulation M-ary Frequency-Shift Keying (NC-MFSK), respectively. According to that study, the NC-MFSK (M ≥ 8) modulation was recommended for the communication system serving re-entry vehicles. Terahertz communication has better performance in dynamic scenario.

On the other hand, previous studies have shown that the time-variation is the intrinsic characteristic of hypersonic plasma sheath. The plasma sheath evolves even without any external perturbation. The time scale of the variation could be smaller than 10⁻⁵ s.²⁸ While the communication system is operating in the EHF band, the period of the carrier wave is much smaller than the time scale of plasma sheath evolution. Therefore, it is worth investigating how the BER varies with time in the period of milliseconds while external perturbation is absent. In addition, the plasma sheath is inhomogeneous. As a result, it could be expected that the location of the onboard antenna would make significant contribution to the condition of the EHF communication channel in the plasma sheath.

The present study mainly focused on the BER of the EHF communication systems that serve hypersonic vehicles in an evolving plasma sheath. The evolution of the plasma sheath was solved with a numerical hypersonic hydrodynamic model. The total period of evolution is less than 1 ms. The time resolution of evolution is 10⁻⁵ s. Typical modulation modes, including 2ASK, 2PSK, 4QAM, and NC-2FSK, were concerned in the present study. The carrier frequencies involved in the present study are 94, 140, and 225 GHz. They are the so-called “atmospheric window” frequencies in the EHF band. The impact of antenna location on the BER will be investigated. The relationship between the signal power and the BER was also analyzed and discussed in the present study.

Once the EHF signal propagates in the hypersonic plasma sheath, the attenuation and phase shift of the carrier wave would be significantly affected by the electron density and electron collision frequency.⁴ As a result, the total signal attenuation and phase shift strongly depends on the electron density and electron collision frequency along the signal propagation path. Normally, it could be expected that the EHF signal is able to penetrate the plasma sheath while its carrier frequency is higher than the cutoff frequency of the plasma sheath. However, the EHF signal propagating in the plasma sheath would be weakened due to the electron–neutral particle collisions. If the signal is too weak to be detected by receiver, the blackout would still occur. In such a case, the time varying plasma sheath would results in time varying signal attenuation and phase shift and consequently leads to unstable channel conditions. In order to investigate the unstable channel conditions, the time varying hypersonic plasma sheath is modeled based on fluid theories first.

The time-varying plasma sheath was obtained by solving a hypersonic fluid mode.¹⁶ The hypersonic vehicle involved in the present study is a blunt coned object, whose shape is identical to the vehicle in the RAMC-II (radio attenuation measurement C-II) experiment and the length is 1.29 m.¹ The electron densities and the electron collision frequencies at 0.9, 1.2, and 1.5 ms are illustrated in upper panels (a)–(c) and lower panels (d)–(f), respectively. The electron collision frequency was obtained according to Bachynski formula²⁹ 

where ρ is the mass density of the ionized gas, T is the temperature, and ρ₀ = 1.288 23 kg/m³.

The angle of attack of the flight is assumed to be 0. In such a case, the plasma sheath could be considered to be axially symmetrical. Therefore, only the right half of the plasma sheath is illustrated in Fig. 1. According to Fig. 1, the spatial structure of the plasma sheath is inhomogeneous. In addition, both the electron density and the electron collision frequency of the plasma sheath evolve with time. Those yield time varying channel conditions for the modulated EHF signals. In the present study, six channels are selected, which are illustrated in Fig. 1(a). There are six dashed lines in Fig. 1(a). Each dashed line corresponds to a channel, which is labeled near the end of the dashed line. The other end of each dashed line is on the wall of the vehicle, which is the position of the onboard EHF antenna, labeled with digits 1 to 6. From antenna 1 to antenna 6, the axial distance to the nose of the vehicle is 0.45, 0.5, 0.75, 0.8, 1.0, and 1.05 m, respectively. Those antennas form three groups: antennas 1 and 2, antennas 3 and 4, and antennas 5 and 6, respectively. The three groups represent the antennas close to the nose, in the middle of the vehicle, and near the bottom of the vehicle, respectively.

In the present study, the lowest frequency of interest is 94 GHz, which is always higher than the maximum cutoff frequency along the signal propagation path. Therefore, the signals at the frequency of interest can always penetrate the plasma sheath. The received signal by on-board antenna can be written in the form of

where s_t is the transmitted signal, T_m is the multiplicative interference, and n₀ is the additive noise, which is supposed to be the systematic noise of the receiver in this study. The transmitted power, i.e., the power of transmitted signal, is considered to be constant. Other variables are the functions of time. n₀(t) is the Gaussian noise. T_m(t) is the concerned time varying channel condition.

The propagation constant in the inhomogeneous plasma sheath could be expressed in the form of² 

where α_p(l) is attenuation coefficient, β_p(l) is phase coefficient, l is the distance to the onboard antenna, and $i=−1$⁠,

with

where ω_p is the angular plasma frequency, ω is the angular frequency for carrier wave, e is the electron charge, N_e is the electron density, m_e is the mass of electron, ν_e is the collision frequency between electrons and neutral particles, which is defined in Eq. (1). The multiplicative interference could be expressed as

where E_t and E_r are the electric components of transmitted and received signals, respectively. L is the total length of signal transmission path in the plasma sheath. The total transmission coefficient and phase shift for the carrier wave could be obtained from the real part and the imaginary part of T_m, respectively. The power transmission rate (T_p) and attenuation rate (Att) for the communication signal could be defined as

Figure 2 illustrates the total transmission coefficient. Panels (a)–(c) show signal attenuation for carrier frequencies of 94, 140, and 225 GHz, respectively. The six channels in the plasma sheath shown in Fig. 1(a) were investigated.

Although the variable illustrated in Fig. 2 is only the real part of the multiplicative interference, it is obvious that the multiplicative interference is impacted by the signal transmission path, i.e., the location of onboard antenna. For all the signal transmission paths, the multiplicative interference varies with time. In addition, the received signal strength increases with the carrier frequency. Another important feature is that the received signal weakens with decreasing axial distance from the antenna to the nose. According to Fig. 2, the total transmission coefficients for channels 1 and 2 are much lower than that of other channels. It could be expected that the signal transmitted via channels close to the nose of the vehicle would suffer severe signal attenuation. Installing the antenna near the nose would more likely yield communication blackout. Thus, channels close to the nose will no longer be concerned. On the other hand, the total transmission coefficients for channels 3 and 4 are very close, while those of channels 5 and 6 are close as well. Therefore, in the latter study, channels 3 and 6 will be selected to investigate the signal propagation properties and the BER. Channels 3 and 6 represent the transmission paths near the middle of wall and close to the bottom of the vehicle, respectively.

The phase shifts of carrier waves used to make significant contribution to BER for a communication system.¹⁹ Figure 3 illustrates the temporal variation of phase shifts for channel 3 and channel 6 at 94 GHz (a), 140 GHz (b), and 225 GHz (c). The vertical axes in Fig. 3 are from −π to π, which cover a whole phase cycle. According to Fig. 3, the phase shifts always vary with time. In other words, the plasma sheath is a time varying communication environment. Once the communication signal is propagating in the plasma sheath, both the received signal strength and the phase of the carrier wave keep varying. Such a character could make significant impact on the communication system. It will be investigated in more detail in Sec. III.

Obviously, the BER of wireless communication would be affected by not only temporal variation but also spatial inhomogeneity of plasma sheath. Such an impact will be investigated numerically in the present section. The transmission coefficient near the top of the vehicle is too low to maintain the communication. Therefore, the communication performance of different antenna locations is studied by selecting one location in the middle and one in the rear of the aircraft. The selected antenna locations are 3 and 6 according to Sec. II.

In this section, the influence of antenna location and signal-to-noise ratio (E_b/N₀) on the time-varying BER characteristics is evaluated by Monte Carlo simulation. A numerical model of wireless EHF signal transmission system is introduced. The signal will be modulated in different ways. The modulation involved in this section are 2ASK, 2PSK, 4QAM, and NC-2FSK. Coherent demodulations will be investigated.

Figure 4 shows a simple schematic for the working flow of the simulated EHF signal transmission system involved in the present study. The system is formed by signal source, modulation module, physical channel (the plasma sheath), Additive White Gaussian Noise (AWGN) module, demodulation module, and the sink module. Generally, there is always noise in a physical wireless signal transmission system. However, there is no real physical noise in a numerical simulation system. In such a case, the artificial noise, which is generated by AWGN module, is introduced into present simulations.

The workflow starts from the signal source module. The signal source generates random binary signals. The binary signals are modulated to the carrier waves. The signals are then transmitted to the plasma sheath module, which is the physical channel for wireless communication. Signal attenuations and phase shifts occur in the plasma sheath. After that, the white Gaussian noise is added by the AWGN module. The signals mixed with noise are transmitted into the onboard antenna and are received by the receiver. The received signals are demodulated and finally reach the sink module. The statistics for the BER estimation are carried out at the sink module.

The simulation systems were realized in Simulink. The simulation system for signals of 2ASK modulation is illustrated as an example in Fig. 5. In this example, the coherent demodulation is employed. Once the signals reach the receiver, they would be demodulated one by one. The binary signals would be obtained after the demodulation process. In an idealized wireless signal transmission system, the digits of binary signals obtained at the sink are always expected to be identical to those of the transmitted signals. However, the bit error always exists in a realistic wireless signal transmission system. In the system shown in Fig. 5, the demodulated signals are compared with the output signals monitored at the signal source module. The BER can be obtained statistically in the sink. In the simulated signal transmission experiments, the total number for the transmission bits was 10⁵. The simulation systems for 2PSK, 4QAM, and NC-2FSK modulations were also designed and utilized in the present study.

Figure 6 illustrates the temporal variations of the BER exploiting 2ASK modulation and coherent demodulation. The impact of different levels of noise was investigated. The on-board antennas was supposed to be installed at antenna locations 3 and 6 corresponding to channel 3 and channel 6, as shown in Fig. 1(a). In this case, the signals received by the two antennas penetrate the plasma sheath via different paths [see the white dashed lines in Fig. 1(a)]. These two antenna locations will also be applied in the latter simulations for the other three modulation systems. According to Fig. 6, the BER always varies with time. The BER also changes with the location of onboard antenna. These characteristics are attributed to the spatiotemporal variation of the plasma sheath. For all frequencies of interest, the BER increases with the noise level. The BER is very large in channels 3 and 6 when the carrier frequency is 94 GHz. The BER varies significantly when the carrier frequencies are 140 or 225 GHz. At the same noise level, the BER is relatively small. In addition, the BER measured in channel 3 is the minimum when the carrier frequency is 225 GHz. It is worth being noticed that the decision threshold may be different for different antenna locations, carrier frequencies, and demodulation methods. In the present study, the criterion for choosing a decision threshold is to achieve a minimum BER. For the 2ASK modulation, the optimal decision threshold is half of the amplitude of the received signal.

The temporal variations of BER under 2PSK modulation are shown in Fig. 7. The coherent demodulation was employed in the 2PSK modulation. In order to achieve the same level of BER at 94 GHz, the required Signal-to-Noise Ratio (SNR) of 2ASK modulation is much greater than that of 2PSK modulation. Additionally, the BER at 94 GHz is more stable than that at 140 and 225 GHz. The minimum BER is obtained in channel 6 while the carrier frequency is 140 GHz. Basically, the BER for 140 GHz is smaller than that of 94 and 225 GHz. In channel 3, the measured variation range of BER for the carrier frequency of 225 GHz is greater than that of 94 GHz, which indicates a less stable BER. Yet, the value of BER for 225 GHz is generally smaller than that of 94 GHz. In contrast, in channel 6, the BER at 225 GHz is generally higher but more stable than that of 94 GHz.

The temporal variations of BER for 4QAM modulation are illustrated in Fig. 8. The disconnected positions in Figs. 8(b) and 8(e) indicate that the measured BER in the numerical experiment is 0. With the 4QAM modulation and coherent demodulation, the fluctuation amplitude of BER for 94 GHz is greater than that of 225 GHz, but more stable than that of 140 GHz. For the carrier frequency of 140 GHz, the BER reaches its minimum. Yet, the stability of BER for 140 GHz is worse than that of 94 and 225 GHz, i.e., the greatest fluctuation of BER was observed at 140 GHz. For the carrier frequency of 225 GHz, the BER achieved the best stability, yet its value is greater than that of 94 and 140 GHz.

The temporal variations of BER for NC-2FSK modulation are illustrated in Fig. 9. According to Fig. 9, the BER performance at 140 GHz is better than that of 94 and 225 GHz. It can be seen from Figs. 6–9 that the BER changes with time and the location of the vehicle antenna. By comparing Figs. 6–9, the BER performance of NC-2FSK seems more stable than that of the other three modulation modes. The BER performance of NC-2FSK modulation is much poorer than that of 2PSK modulation mode while the SNRs are at the similar level, regardless of the location of onboard antenna. Compared with the 2ASK modulation, in order to achieve the same level of BER, the required SNR of NC-2FSK modulation is lower than that of 2ASK. Hence, the BER performance of NC-2FSK modulation is generally better than that of 2ASK modulation.

According to the numerical simulations in Sec. II, despite the amplitude of the transmission rate variations within a small range with time, the variation of phase shifts with time are significant. In addition, both of them are impacted by the location of onboard antenna. As a result, the BER varies significantly with time, carrier frequency, and the location of onboard antenna.

The BER differs while the modulation modes are different. Once the demodulation modes are coherent, in order to achieve identical level of BER, the SNR required by the 2PSK system is the smallest. The BER performance of NC-2FSK system seems more stable than the other three. Unfortunately, its BER is almost always higher than 10⁻². Hence, the NC-2FSK modulation may not be the appropriate modulation mode for an EHF communication system serving hypersonic vehicles. In this section, the NC-2FSK modulation mode will not be discussed further.

The optimal decision thresholds for 2PSK and 4QAM systems are 0, which yields better BER performance for the two systems. The optimal threshold level of 2ASK system is related to the signal amplitude. When the channel characteristics vary, the optimal decision threshold level changes correspondingly. It may lead to the increase of BER.

Another significant potential characteristic to be concerned about is the bandwidth for the data transfer rate. In the present study, the bandwidths for the three modulation systems are identical. However, a system utilizing the 4QAM modulation has higher potential to broaden its data transfer rate. The complexities of sending devices in 2ASK and 2PSK systems are almost identical. However, the transmission equipment for a 4QAM system is more complex.

In such a case, if the anti-noise performance of the system is the first concern, the 2PSK system is suggested (see Figs. 7 and 8). On the other hand, if the band utilization ratio is the first concern, the 4QAM system can be selected. From Fig. 6, it can be seen that the anti-noise performance of 2ASK system is too poor. According to Figs. 7 and 8, in order to achieve good BER performance, the suggested location of onboard antenna is closer to the bottom of the vehicle. In addition, to operate the communication system at 140 GHz could be helpful to achieve good BER performance, although its signal attenuation is more severe than that of 225 GHz.

It should be realized that even the best BER illustrated in Figs. 6–9 is still higher than 10⁻⁵, which could be too high to be applied in realistic scenarios. Therefore, it is necessary to find the appropriate way to improve the BER performance. Figure 10 shows a very simple but effective way to improve the BER performance for the 2PSK system. The background of channel 6, which is the electron density and the temperature, is illustrated in Fig. 10(a). The BER vs signal power is exhibited in Fig. 10(b). At t = 1 ms, the maximum electron density of channel 6 in the plasma sheath is about 3 × 10¹⁹ m⁻³, and the maximum temperature is about 7000 K. In Figs. 6–9, the transmitted signals were fixed to be 1 mW. According to Fig. 10(b), the BER decreases (see the vertical axis) with the increasing signal power. Once the carrier wave is 140 GHz, the BER is lower than 10⁻⁶ while the signal power is about 1.5 mW. For the carrier frequencies of 94 and 225 GHz, in order to achieve the BER of 10⁻⁶, the necessary transmitted power is about 2.2 and 4.1 mW, respectively.

In this study, both the spatial inhomogeneity and the temporal variation of hypersonic plasma sheath were studied. Their impact on the BER of the EHF communication system utilizing hypersonic vehicles was investigated numerically. The BERs for 2ASK, 2PSK, 4QAM, and NC-2FSK modulation modes were compared. The concerned carrier frequencies are 94, 140, and 225 GHz, which are “atmospheric window” frequencies.

According to the study, the signal attenuation and the phase shifts for the carrier waves always vary with time. Such a time variation results in the variation of BER for the EHF communication systems. In addition, the location of onboard antenna makes significant contribution to the BER. To install the onboard antenna on the wall beyond the nose of the vehicle is helpful to achieve better BER performance. According to the present study, the EHF communication system operating at 140 GHz with 2PSK modulation mode has the best anti-noise performance. In order to achieve better data transfer rate, the 4QAM mode could be recommended, although the 4QAM mode requires higher SNR to achieve good BER performance compared with the 2PSK mode. Moreover, for a 2PSK system, the BER decreases with the increasing power of the transmitted signal. Therefore, the BER performance for a 2PSK EHF communication system serving hypersonic vehicles could be efficiently improved by increasing the transmitted power.

This work was supported by the National Natural Science Foundation of China under Grant No. 61861031.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.