Abstract
Aimed at the antijamming needs of the space tracking, telemetry command (TT&C) receiver under a low signal-to-noise ratio, the anti-interference advantage of the dual-sequence frequency hopping (DSFH) communication system is applied. The channel amplitude fading influence of the receiver operating characteristics (ROC) of the TT&C receiver based on the DSFH is studied. Firstly, based on the typical channel model of the Rayleigh fading without direct path transmission, the conditional Fokker-Planck equation (FPE) is obtained by analyzing the statistical independence of the Rayleigh fading signal and SR output particle moments. Secondly, the probability density function (PDF) of the DSFH signal via channel Rayleigh fading enhanced by stochastic resonance (SR) is obtained by introducing the decision time. Thirdly, the detection probability, false alarm probability, ROC, and system bit error rate (BER) of the DSFH signals enhanced by SR under the Rayleigh fading conditions are obtained, under the minimum BER criterion. Finally, the conclusions are reached: one is that the DSFH signals via channel Rayleigh fading can still be detected by the SR system under low SNR, and the other one is that the SNR can reach the -13 dB by the reception of DSFH signal enhanced by SR, when the Rayleigh fading parameter is 0.2042.
1. Introduction
The space tracking, telemetry and command (TT&C) receiver plays a pivotal role of the connection between the orbiting spacecraft and the ground station which mainly completes the tasks of remote control, telemetry, ranging, and speed measurement. With the intensification of space electromagnetic interference conflicts, the antijamming performance of the TT&C receiver at a low signal-to-noise ratio (SNR) is becoming increasingly urgent [1, 2]. At the same time, under the low SNR, the small-scale fading in the process of space and sky signal transmission has a great impact on the quality of the received signal, thus affecting the antijamming performance of the TT&C receiver. The dual-sequence frequency hopping (DSFH) communication mode uses the idea of “channel is message” [3, 4]. Two groups of pseudorandom sequence-controlled frequency hopping (FH) carrier channels are selected, respectively, through the transmission of symbol 0 or 1. The selected channel is used as the communication channel, while the unselected channel is used as the dual channel. The receiver judges the transmission symbol by detecting the channel occupation [5, 6]. This “dominant modulation” communication mode can effectively combat the tracking interference and other interference patterns and has a certain degree of innate anti-interference performance [7, 8]. Elsayed and Yousif proposed free-space optical (FSO) communication links using the modified pulse-position modulation (MPPM) and spatial pulse-position modulation (SPPM) for reducing the interference and applied the FSO links to UAV to UAV [9–12], which achieved good results. Unlike the traditional communication mode-modulated information in the baseband, the symbol information of DSFH is carried by the presence of RF signals. Therefore, its spatial signal is a single-frequency sine signal, which can be used as the driving signal of the stochastic resonance (SR) system.
SR is a nonlinear physical phenomenon. When the signal, noise, and SR system characteristic are matched, noise plays a positive role in signal detection through the SR system, breaking the traditional view that noise is always harmful in signal detection. Benzi et al. [13] put forward the SR concept in their research on the changes of the Earth’s glacial period first, and then, the phenomenon was verified in the fields of physics, biology, and electronics [14–18]. So the SR is used to the signal detection and reception when the signal strength is lower than the decision threshold [19–26]. The output probability density function (PDF) and receiver operating characteristic (ROC) curve of SR driven by an ideal DSFH signal are studied [6, 27]. And the conclusion that SR can be applicable to DSFH signal detection at very low SNR is obtained. However, the signal will inevitably be distorted due to channel transmission, resulting in that the input signal enhanced by the SR system is not an ideal sine signal. But there is little research on the SR output characteristics driven by distorted signals at present.
Aiming at the reception of the DSFH signals via channel fading enhanced by SR, the conditional Fokker-Planck equation (FPE) is obtained by analyzing the statistical independence of the Rayleigh fading signal and particle moments. Then, the output PDF of the signal via the Rayleigh fading enhanced by SR is obtained. Finally, the detection probability, false alarm probability, ROC, and system bit error rate (BER) of the DSFH signals enhanced by SR under the Rayleigh fading conditions are obtained. And the conclusion that the DSFH signal via the Rayleigh fading can be enhanced by the SR system is reached.
2. The System of the Reception of the DSFH Signals via Channel Fading Enhanced by SR
2.1. Subsection
The superheterodyne reception is adopted in the DSFH communication mode, which can be described as shown in Figure 1.
At the radio front (RF), receiving the signal is mixed and band-pass filtered by the carrier signal controlled by two pseudorandom sequences. The waveform and frequency of the IF signal are the same obtained of the two branches, described as where is the frequency of the IF signal, is the FH period, is the step function, is the signal amplitude, and is the signal phase.
2.2. The Analysis of the SR Output Driven by Random Amplitude Received Signal
The amplitude of the DSFH-received signal is not constant due to the multipath propagation via channel fading, which obeys a certain distribution. At the extremely low SNR, the distribution can be described by Rayleigh, which the amplitude signal is
Then, the SR system driven by the random signal can be represented by Langevin equation (LE) as where is the random amplitude, obeying Rayleigh, and is the white Gaussian noise.
When is constant, the relationship between LE and FPE can be obtained by the transition moments of particles and Kramers-Moyal Expansion [22]. So, , which is the probability density function (PDF) of the particle, obeys
When the signal amplitude is random, is dependent of the PDF of the SR output [19]. So the PDF of the SR system is
Further, in order to obtain the particles PDF of the SR system driven by the random signal of , the can be assumed as constant, and then, the conditional PDF as is obtained. So the is carried out at the statistical average according to formula (5).
3. The ROC of SR Driven by DSFH Signal via Channel Fading
3.1. Test Statistics
The test statistics of two hypotheses that the SR system is driven by the DSFH signal and noise or not can be described as where the definite integral is in the form of transcendental function, which can be solved by numerical solution.
Due to the dependence of the SR and the driven signal, the statistical average of formula (8) is carried out. So the average likelihood function of under hypothesis of is
where is the output signal of the SR system driven by noise singly and is the output signal of the SR system driven by the DSFH signal and noise.
The decision time [6], which represents the wave crest and trough of sine signals, is introduced. So the PDF of two hypothesis is
So the likelihood function of the two hypotheses is
The judgment function is where is the likelihood ratio decision threshold.
3.2. The Detection Structure under Random Amplitude
The detection structure of the output of the SR system driven by the received IF signal of DSFH after channel fading with the likelihood ratio decision is shown in Figure 2.
3.3. The Performance of Signal Detection
The detection probability with the decision threshold is
The false alarm probability is
The BER is
And the decision threshold with that the communication system generally adopts the minimum error probability criterion meets
It can be carried out with the necessary conditions for the extreme value of the function
So the numerical solution of can be obtained with formulas (12)–(14) and (16).
The BER of the SR system driven by DSFH signal via channel fading can be obtained by the numerical solution of and formula (14).
4. Simulation Results and Discussions
In this section, the theory and simulation are analyzed to demonstrate the detection improvement by the Simulink model of the DSFH signal enhanced by the SR system via channel fading. The simulation parameters are as follows: the frequency of the IF signal is 1 kHz; the sample frequency of the IF signal is 200 kHz; the frequency hopping section is 30-88 kHz; the sample frequency of the RF signal is 2000 kHz.
4.1. The Time-Frequency Characteristic of the DSFH Signals Enhanced by SR via Channel Fading
The IF signal is obtained and processed by superheterodyne demodulated and band-pass filtered of DSFH RF signal which goes through channel fading. Its waveform of time domain and frequency domain is described in Figures 3(a) and 3(b), and its waveform of time domain and frequency domain enhanced by the SR system is described in Figures 3(c) and 3(d). When the input SNR is -14 dB, its waveform of time domain and frequency domain is disordered and irregular, where there is not any characteristics of IF signal components. However, its waveform of time domain appears periodic characteristics in Figure 3(c) processed by the SR system that it is observed that obvious signal components appear at 1 kHz through the frequency domain in Figure 3(d). From a numerical perspective, the output SNR is -9.4316 dB, raising 4.5684 dB. That is to say, the SNR has improved by 4.5684 dB via the SR system. This is because that enhanced by the SR system, the white Gaussian noise with flat distribution will converge to the low-frequency region, making the energy in the low-frequency region larger. Together with the low-frequency IF signal via the Rayleigh fading, the particles will be driven to transition between the bistable potential wells. The bistable potential wells represent symbols 0 and 1, from a certain perspective. The time-domain signal has a certain periodic characteristic, which is more obvious in the frequency-domain observation. It changes the spectral structure of the noisy signal, which is shown as an increase in SNR on the macrolevel.
(a) Time-domain waveform of input signal
(b) Frequency-domain amplitude of input signal
(c) Time-domain waveform of output signal
(d) Frequency-domain amplitude of output signal
4.2. The PDF of the DSFH Signals Enhanced by SR via Channel Fading
The theoretical and simulated PDF at different positions of the DSFH IF signals enhanced by the SR system via channel fading or not are shown in Figure 4. The PDF of the DSFH IF signals enhanced by the SR system is black color, the PDF of the DSFH IF signals enhanced by the SR system via channel fading is blue, and the PDF of the noise enhanced by the SR system is red. The results show that the driving force of the DSFH IF signal will increase the probability of particles’ transition to both sides of the steady state, increasing the residence time of particles in the steady state, increasing the difference between two assumptions, which is more conducive to distinguish between the two hypotheses with the DSFH signal or not and improve the detection probability of DSFH signal. At the same time, compared with the effect of the DSFH IF signal without channel fading, the is less different from . This is because the Rayleigh fading will cause the strength of the DSFH-received signal to decrease, leading to the strength of driving force driving the SR system to decrease, so the difference between the Rayleigh fading and nondriving force will decrease; that is, the Rayleigh fading will weaken the output response of the DSFH signal enhanced by the SR system. It is worth noting that there is a certain gap between theory and simulation. This is because the simulated IF signal always exists, leading to the driving force in the simulation being the comprehensive driving effect of the DSFH signal within one cycle. The black solid line is the output without channel fading. The theoretical value is only related to the driving force state at the current decision time, that is, the time when DSFH has the maximum driving effect, which is quite different from the average driving effect of the simulation. The solid blue line is the average of the Rayleigh fading of the DSFH signal within one cycle, which has little difference with the average driving effect of simulation.
(a)
(b)
The ROC curves of the DSFH IF signal enhanced by the SR system via channel fading or not are shown in Figure 5. It reflects the relationship between or and the or SNR, with the black line without channel fading and the red line with channel fading. It can be seen that different ROC curves correspond to different SNR, but they all pass through two points (0,0) and (1,1), which are convex curves located at the upper left of the line of with the higher the SNR, the higher the curve position. The signal is better detected, the higher the , with the higher the SNR, when is constant. However, when SNR is constant, the and are decreased with an increase of . For the detection problem with known, the tangent point of the slope line of and the curve is the and with the current SNR and . At the same time, the detection performance of the red line representing the Rayleigh fading is poorer than the black one without the Rayleigh fading. This is because the Rayleigh fading can decrease the received signal strength and the amplitude, leading to the decrease of the difference between the PDF with the DSFH signal or not. This is also in line with common sense, as the Rayleigh fading affects receiver performance via the SR system. Therefore, the ROC curve can completely describe the likelihood ratio detection performance of the DSFH signal enhanced by the SR system.
The relationship between and SNR of the DSFH IF signal enhanced by the SR system via channel fading or not is shown in Figure 6. It can be seen that the BER of the DSFH signal enhanced by the SR system via the Rayleigh fading is higher than that without fading. This is because the signal strength of the DSFH signal will become smaller and divergent after fading, which will weaken the effect of SR vibration. When SNR is extremely low ( dB without channel fading and dB with channel fading), the SR system can not get any useful information of DSFH, which is equivalent to random decision, so all four curves are about 0.5. And the SR system gets some useful information of DSFH, when SNR is equal to -18 dB without channel fading where . However, the SR system gets some useful information of DSFH via channel fading when the SNR is equal to -13 dB where . At the same time, compared with the theory and simulation values of the black one without channel fading, the two curves of the Rayleigh fading have smaller descending speed with SNR. This is because the Rayleigh fading will cause the amplitude of the received signal to diverge, so that the received signal will no longer be a deterministic signal but will become a fluctuating signal, resulting in the same SNR increase and a small increase in BER. Through experiments, it is concluded that DSFH signals via the Rayleigh fading can still be detected and received by the SR system under low SNR. Meanwhile, the Rayleigh fading will reduce the reception performance of the DSFH signals via SR processing. The theoretical boundary of reception performance of the DSFH signals with the Rayleigh fading via SR processing is also obtained. When the fading parameter is 0.2042, it is suitable for the reception of the DSFH signals via the Rayleigh fading with dB. Meanwhile, the Rayleigh fading is mostly applicable to multipath propagation, and there is a direct path in the TT&C communication in most cases, which restricts the applicability of the method.
5. Conclusions
SR can make use of noise to improve the application range of the DSFH signal under extremely low SNR, thereby improving the anti-interference performance of the TT&C receiver using the DSFH system. By analyzing the statistical independence of the Rayleigh fading and moments of SR particles, the receiver performance of the DSFH signal enhanced by the SR system via the Rayleigh fading is obtained. Further, the theoretical boundary of reception performance of the DSFH signals with the Rayleigh fading via SR processing has been obtained. The conclusion that when the fading parameter is 0.2042, it is suitable for the reception of the DSFH signals via the Rayleigh fading with dB be obtained. It provides a theoretical support for the practical application of SR in the reception of DSFH signals and provides a reference for the antijamming performance of space TT&C receiver under extremely low SNR. The conclusions can further guide the next experiment. At the same time, by assuming the distortion caused by signal transmission and the statistical independence of SR resonance output moments, the output PDF of SR driven by distorted signals is obtained. The processing method provides a reference for analyzing the influence of distorted signals on SR. Further research focuses on the reception of the DSFH signal via the SR system with a Rice fading.
Data Availability
The data used to support the findings of this study are available from the corresponding or submitting author upon request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Funding
The research was funded by the Grant No. 2100050079.