Influence of stealth aircraft dynamic RCS peak on radar detection probability

Xiaoqiang LU, Jun HUANG, Yacong WU, Lei SONG

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

KEYWORDS Maneuver;Penetration;RCS;RCS fluctuation statistical model;RCS peak exposure time;RCS peak width;Stealth;Trajectory planning

Abstract For modern stealth aircraft,it is important to analyze the influence of Radar Cross Section(RCS)peak exposure on penetration for guiding stealth design and penetration trajectory planning,which needs to reflect the RCS statistical uncertainty and the RCS difference with the change of incident angle.Based on the RCS characteristics of typical stealth aircraft,this paper established a simplified RCS dynamic fluctuation statistical model with the parameters log mean and log standard deviation.According to the detection probability algorithm in radar signal processing field,this paper built the algorithm of radar detection probability based on the RCS dynamic fluctuation statistical model.The analysis of examples concluded that the key to successful penetration is to shorten the RCS peak exposure time, which can be reduced by decreasing the RCS peak width or increasing velocity.Based on the conclusion,this paper proposed the method of turning maneuvering to reduce RCS peak exposure time dramatically.

1.Introduction

In the stealth design of modern aircraft,radar echoes are often concentrated in several small non-head angular regions to ensure the stealth performance during frontal penetration.However,in the real penetration process,it is difficult to maintain the frontal penetration, which will inevitably make the peak angle area toward the radar.Besides, the developing metasurface technology could control the angular distribution of electromagnetic wave energy,1,2and might change the angular position of Radar Cross Section (RCS) peak.Therefore, it is necessary to analyze the influence of RCS peak exposure on penetration, in order to guide the stealth design and penetration trajectory planning.

To analyze the penetration, radar detection probability should be used as an index to evaluate trajectory.3As a result,the problem is transformed into the research on the influence of stealth aircraft dynamic RCS peak on radar detection probability.

Since the RCS of stealth aircraft changes drastically with the azimuth angle,4–7and the incidence of radar waves in dynamic flight is a random event, the process of the analysis needs to reflect the RCS statistical uncertainty and the variation of radar incident angle with time during the dynamic flight.

Firstly, to reflect the RCS statistical uncertainty, researchers usually divide the aircraft into multiple angular regions in the circumferential direction, and use fluctuation statistical model8–13with different parameters in each angular region to carry out the analysis.Chen14proposed a 30-degree-angularregion fluctuation statistical model, and developed a set of detection probability algorithm suitable for various specific aircraft targets.Li et al.15proposed target circumferential RCS scattering model.They used this model to calculate the probability of radar discovery.This kind of method is mainly used to describe the stealth performance of the aircraft itself,and to measure the penetration capability of the aircraft by calculating the radar discovery probability in a frontal penetration scenario(facing the radar,fast approaching).Although the model parameters are different in each angular domain,they do not change continuously with the azimuth angle.The focus is on the description of the aircraft’s own capabilities.That is, it pays attention to radar stealth performance rather than trajectory planning.However,considering the trajectory planning factors,the radar incident angle of the aircraft is constantly rapidly changing under a specific penetration threat and flight trajectory.Therefore, when conducting the simulation of the radar detection process in a given scenario,it is necessary to use a statistical model of RCS fluctuations which can reflect the difference in radar incident angle.

Secondly, to reflect the variation of radar incident angle with time during the dynamic flight,researchers establish algorithm and calculate the radar detection probability by different typical maneuver penetration methods.Yuan et al.16used the RCS deterministic model(the method in which the RCS is constant on a specific radar incident azimuth)to express the target characteristics of the aircraft, and analyzed the beneficial effects of the serpentine maneuver on the penetration.However,for the RCS deterministic model,the calculation interval of radar incident angle is in a dilemma.If the interval is not small enough,the drastic RCS changes caused by small differences in azimuth angles may cause large errors;if the interval is small enough, the calculation time will be unacceptably long.Zhang et al.17and others superimposed jitter information on the RCS deterministic model to characterize the statistical characteristics of the echo, but its radar detection probability is calculated based on Swerling I model, which is different from the typical echo statistical characteristics of stealth aircraft.The methods above reflect the time-varying radar incident angle, mainly to obtain the calculation results of typical scenarios, and select a better penetration trajectory for stealth aircraft.However,the real trajectory planning process is highly nonlinear, and the optimization efficiency can be accelerated by adding appropriate constraints.Therefore, it is necessary to explore the influence of the change of parameters on the penetration effect, analyze the causes of the influence, and use it to constrain the trajectory planning,which can also provide some guidance for the stealth design.

It can be seen that we need to combine the two perspectives above.Firstly, an RCS fluctuation statistical model should be established under the time-varying radar incident azimuth to analyze the change of radar echo during the dynamic flight,and the process of radar detecting aircraft should be described accurately.Secondly, a variety of penetration examples in given scenarios need to be calculated.Based on the results, it is necessary to analyze the factors that affect the radar detection probability in penetration, and find the design methods and maneuver measures that are beneficial to penetration.

In this paper,based on the RCS data of a typical stealth aircraft, we introduce a simplified RCS dynamic fluctuation statistical model.For this model, we define statistical model parameters to characterize the dynamic stealth performance of the aircraft, including log means, log standard deviation,and peak width.Referencing the detection probability algorithm in radar signal processing, we construct an algorithm of the radar detection probability based on the simplified RCS dynamic fluctuation statistical model.Using this method,we explore the influence of changes in aircraft dynamic stealth performance parameters on the radar detection probability in different penetration trajectories, and obtain the influence of aircraft RCS peak exposure and maneuver on the penetration effect.Meantime, we make suggestions for maneuver method that can be used for trajectory planning.

2.Method

2.1.Simplified RCS dynamic fluctuation statistical model and basic assumptions

2.1.1.Simplified RCS dynamic fluctuation statistical model

According to the analysis of the angular domain RCS fluctuation model,18,19we choose the lognormal model to build a simplified model of stealth aircraft dynamic target characteristics.

The RCS lognormal distribution model could be expressed as

where Λ represents the radar incident azimuth, and σRCSrepresents the RCS value.The parameter μ(Λ) represents the azimuth-varying log mean of the RCS, and the parameter s(Λ) represents the azimuth-varying log standard deviation of the RCS.

According to the RCS data of typical stealth aircraft,20in the non-RCS-peak angular region, the RCS fluctuates with a low level, corresponding to a small parameter μ; in the RCSpeak-center angular region, the RCS fluctuates with a high level, corresponding to a large parameter μ.At the edge of the RCS-peak angular region (the high value-low value junction),the RCS value fluctuates significantly,so the corresponding log standard deviation parameter s is relatively large.

In order to make the simplified RCS dynamic fluctuation statistical model be factual, we referenced the RCS data of a flying wing, and the schematic diagram is shown in Fig.1.

Based on the information above,a simplified RCS dynamic fluctuation statistical model of stealth aircraft is established.The peak value of the RCS model is assumed to be 10 m2.4 peaks symmetrically distribute in the circumferential direction of RCS, and the azimuth of the peak center is ±35° and±145°.The valley value of the RCS model is assumed to be 0.01 m2.

The parameters of the simplified RCS dynamic fluctuation statistical model are shown in Fig.2.

Fig.1 Typical RCS data of a flying wing.

Fig.2 Parameters of fluctuation model.

As shown in Fig.2, the peak value μpeakand valley value μvalleyof the azimuth-varying log mean are

According to the previous analysis,the azimuth-varying log standard deviation of the model is defined as:svalleyis 0.5 at the valley, sfootis 1.01 at the junction of peak and valley, speakis 0.01 at the peak, the largest log standard deviation is at the junction of peak and valley,and the smallest one is at the center of peak.

The definition of peak width is shown in Fig.2.The peak angular region of the model is 34.4°–35.6°, and the RCS Peak Width (RPW) is 1.2°.

2.1.2.Basic assumptions

To analyze the factors influencing penetration, this paper is based on the following basic assumptions:

(1) The simplified RCS dynamic fluctuation statistical model is used to reflect the RCS fluctuation of aircraft.

(2) According to the typical radar detection level, the radar model is defined as being able to detect 1 m2target at 240 km(i.e.20 dB signal-to-noise ratio).The pulse accumulation number is 2;the correlation coefficient is 1;the scanning period is 12 s; the false alarm probability pfis 1×10-6.

(3) The calculation of radar detection probability is based on the single-detection-found assumption, that is, the aircraft is found at any point in the trajectory, which means the failure of the penetration trajectory.

(4) The influence of roll angle on target characteristics is not considered when the aircraft maneuvers.

(5) The turning radius is restricted by 2G load factor when the aircraft maneuvers.

2.2.Algorithm of radar detection probability

The establishment process of the method is as follows:

(1) Based on the mathematical and physical model of radar detection,21,22the probability of radar’s detection of a specific attitude aircraft at a specific spatial position is obtained.

(2) In combination with the real flight situation, the radar incident angle is calculated during the aircraft motion simulation process.During this process,the RCS fluctuation characteristics in the angle domain of the aircraft are converted into the fluctuation characteristics in the time domain, which can be substituted into the detection.The probability of the aircraft being detected by a radar’s complete scan can be obtained with this model.

(3) The detection probabilities of multiple scans are superimposed to obtain the probability of detection of the aircraft during the entire process of flying over a specific trajectory.

2.2.1.Algorithm of radar detection probability in single position of aircraft

The most widely used in radar signal detection is the threshold detection based on the Neyman–Pearson criterion.Once a radar receives echoes, the receiving and detecting system first performs matched filtering on a single pulse signal in the intermediate frequency strip, and then N pulses are weighted accumulating.The cumulative output is compared with a specific threshold voltage.If the output envelope exceeds the threshold, the target is considered to exist, and vice versa.In this case, the detection probability of the target can be shown by

where pdis the probability of detection,r is the integrator output normalized to the average per pulse noise power, pr(r ) is the probability density function of r, and rbis the detection threshold, which can be obtained from the false alarm probability pf.The false alarm probability refers to the probability that the noise exceeds the threshold level and is mistaken for the signal.Lowering the threshold will increase the false alarm probability,while increasing the threshold will lose some weak signals.The Neyman–Pearson criterion, which maximizes the probability of detection under the constraint of false alarm probability, can help operators or electronic systems to select the appropriate threshold level rb, that is,

In order to obtain the probability density function pr(r ),according to Kanter’s correlation function model,21do the Laplace transform to pr(r )

In summary, the radar detection probability can be calculated with the RCS fluctuation.

2.2.2.Algorithm of single-scan radar detection probability in time domain

In the real penetration, during the process of radar scanning,the distance and azimuth of the aircraft relative to the radar will change greatly.Therefore, according to the specific flight trajectory and radar scanning time, the angle-domain RCS fluctuation characteristics should be transformed into the time-domain RCS fluctuation characteristics, that is, putting the change of azimuth with time into the probability density function in angle domain, and obtaining the probability density function in time domain.

where μ(Λ (t ))and s(Λ (t ))represent the parameters μ and s of fluctuation statistical model corresponding to a specific azimuth Λ(t ).

In the real calculation, because the time of scanning to the aircraft is uncertain in a single scan, the probability density function of a single scan is obtained by integrating the probability density function in time domain.

where tsand teare the start and end time of a single scan respectively.

Significantly, due to the numerical algorithm, in order to prevent the RCS fluctuation characteristics from changing too much in each infinitesimal segment and affecting the calculation results, it is necessary to adjust the start and end time interval of each infinitesimal segment in the integration, in order to make the azimuth change within the segment less than 0.1° and ensure the accuracy of numerical calculation.

Substituting time-domain RCS fluctuation characteristics into algorithm of radar detection probability in single position of aircraft (presented in Section 2.2.1), we have

Since there are multiple scans in the whole trajectory, the detection probability of the whole trajectory (pd_route) can be obtained by accumulating the detection probability of the multi-segment single scans in the trajectory,

3.Penetration evaluation under different influencing factors

3.1.Influence of RCS peaks and valleys on the probability of radar detection

In this section,the top-view schematic diagram of the penetration scene is represented by the Cartesian coordinate system XoY, in kilometer.

For the mission,

(1) Defining a typical mission, the aircraft needs to penetrate from (0, -100) to (0,100).

(2) Defining the Straight Trajectory:a straight-line segment from (0, -100) to (0,100).

For the threat environment:

(1) Defining the Radar Distance(R): the distance between radar position to (0,0).

(2) Defining the Valley Situation: the radar is located at(R,0), which means that the peak value is not exposed to the radar during the whole trajectory.

(3) Defining the Peak Situation:the radar is located at(Rsin(35°), Rcos(35°)),that is, when the aircraft flies over the point (0,0), its maximum peak angle (35°) is exposed to the radar.

Fig.3 shows the schematic diagram of two situations when the R equals 500 km.

In this part,the RPW of the aircraft is 1.6,and the velocity along the trajectory is 800 km/h.

By calculating the radar detection probability in the two threat environments, we can know the impact of RCS peak exposure on penetration.

As shown in Fig.4, we assume that the Radar Detection Range (RDR) is radar distance (R) when the radar detection probability is 90%.

In the Valley Situation and the Peak Situation,the RDR is about 213 km and 586 km respectively,and the detection probability when R equals 300 km is about 4.497% and 99.87%respectively.Thus when R is between 300 km and 600 km,the influence of peak exposure on radar detection is decisive.It can be seen that the peak exposure will sharply increase the radar detection probability and the RDR.For the 200-km-long trajectory, the peak exposure of this trajectory leads to the increase of the RDR by nearly 400 km.

3.2.Influence of RCS peak width on radar detection probability

The typical characteristic of aircraft RCS peak is its peak width, that is, the RPW.To study the influence of the RPW on radar detection probability, the dynamic stealth performance of aircraft can be evaluated by using the RPW.

As shown in Fig.5,the trajectory is a straight-line segment from (0, -100) to (0,100).The velocity along the track is 800 km/h.The threat environment is Peak Situation.

In the calculation, the RCS Peak Width (RPW) of the aircraft is taken as 0.4°, 0.8°, 1.2° and 1.6°, and the radar detection probability of targets with different RPW on the same trajectory is calculated respectively.

In Fig.6,it can be seen that when R is greater than 250 km,the influence of RPW on detection probability begins to show.When R is greater than 300 km,the effect of RPW on detection probability is pronounced.In the penetration process of a specific situation (e.g.R = 500 km), 10 radar scans when the aircraft is nearby the location(0,0)are drawn out for analysis.Through the detection probability of the 10 radar scans, how RPW affects the detection probability can be analyzed.

Fig.3 Schematic diagram of two situations.

Fig.4 Detection probability comparison between Peak Situation and Valley Situation.

Fig.5 Schematic diagram of penetration scene.

Fig.6 Probability of detection with different RPW.

As shown in Fig.7, the curve of ordered-scan detection probability shows a peak shape.When the RPW is 0.4°, there are 2 scans which can detect the aircraft probabilistically, and the total radar detection probability is 46.76%;when the RPW is 1.6°, there are 6 scans which can detect the aircraft probabilistically,and the total radar detection probability is 96.42%.

Fig.7 Probability of detection by ordered scan with different RPW (R = 500 km).

At this distance, the influence of peak exposure on radar detection is decisive (shown in Section 3.1).Combined with Fig.7,the number of scans which can detect the aircraft probabilistically is related to the RCS Peak Exposure Time(RPET).The RPW is positively correlated with radar detection probability, and its essence is that RPW changes the RPET.

Therefore, on the one hand, from the perspective of evaluation, when other parameters are the same, the smaller the RPW is, the better the dynamic stealth performance is; on the other hand, from the perspective of design, stealth design can be made with minimum RPW as the goal.However, in the trajectory planning of the specified aircraft, the RPW of the aircraft cannot be changed.What can be changed at this time is the RCS peak exposure time in the trajectory.

3.3.Influence of RCS peak exposure time on radar detection probability

The trajectory for calculation is also straight trajectory, same as Fig.5 in Section 3.2.The threat environment is Peak Situation.The RPW of aircraft target is 0.4°.The radar distance R equals 500 km.10 radar scans when the aircraft is nearby the location (0,0) are drawn out for analysis.Through the detection probability of 10 radar scans, how RPET affects the detection probability can be analyzed.

As shown in Fig.8, v represents the velocity of aircraft.With the increase of v, the RPET decreases, and the probability of radar detection also decreases from 96.58%at 200 km/h to 46.76% at 800 km/h.This shows that reducing the RPET can directly reduce the probability of radar detection.In fact,reducing RPW is a way to reduce RPET for stealth design.In the case of a certain peak height,the detection probability at a single point is also certain.The shorter the time is, the smaller the total detection probability is.Therefore, when aircraft has to expose its RCS peak to the radar, the target of trajectory planning is to reduce RPET.

3.4.Influence of turning maneuver on radar detection probability

3.4.1.Definition and principle of turning maneuver

In a real trajectory, the velocity of aircraft has an upper limit.Therefore, if we want to reduce the RPET and the probability of detection,we need to make the RCS peaks quickly‘‘sweep”the radar.

Fig.8 Probability of detection by ordered scan at different velocities (R = 500 km).

Therefore, we select a typical case (the RPW is 0.4°, the velocity is 800 km/h, Peak Situation) for maneuver planning:

When the radar incident angle is 34°,the aircraft turns right till facing the radar;when reaching the y=0 line,the aircraft makes a turning maneuver,so that the trajectory of the second half is the mirror image of the first half.

Fig.9 shows the schematic diagram of 2 trajectories when R is 300 km.

The maneuver changes the Straight Trajectory when radar incident angle is 1°less than 35°(34°).Thus,we define that the maneuver angle field is 35±1°,the Turning Angle(TA)is 2°,and the trajectory is ± 1° Avoidance Maneuver Trajectory.

Under different R, the radar detection probabilities of Straight Trajectory and Maneuver Trajectory are calculated respectively, and the results are shown in Fig.10.

When the R is more than 250 km,the effect of maneuvering turn begins to appear.When the R is more than 300 km, the effect of maneuvering turn is obvious.When the R is 300 km, the radar detection probability of Maneuver Trajectory is only 24.84%, which is far less than 73.84% of Straight Trajectory.The reason is that when the aircraft maneuvers,the RCS peak value quickly passes the radar, which is equivalent to greatly increasing the velocity of the peak direction relative to the radar, reducing the RPET, thus reducing the radar detection probability.

Fig.9 Schematic diagram of Straight Trajectory and Maneuver Trajectory.

Fig.10 Probability of detection in different trajectory.

In order to show the essence of turning maneuver, Fig.11 and Fig.12 show the change of radar incident azimuth with time during turning maneuver.

As shown in Fig.11, it can be seen that both curves cross the peak centerline (azimuth = 35°), but when crossing the peak centerline,the slope of curve in Maneuver Trajectory Situation is much greater than that in Straight Trajectory Situation.

The specific RPET can be seen from Fig.12.For the Straight Trajectory Situation, the RCS peak exposure is from 441.74 s to 458.18 s,and the RPET is 16.44 s.For the Maneuvering Trajectory Situation, the peak RCS exposure is from 457.21 s to 457.30 s, and the RPET is 0.09 s, which is two orders of magnitude lower than that of Straight Trajectory Situation.Therefore,the turning maneuver can greatly reduce the RPET, thus reducing radar detection probability.

3.4.2.Influence of velocity on turning maneuver effect

Under different R,the radar detection probabilities of different trajectory and velocity are calculated respectively, and the results are shown in Fig.13.

As shown in Fig.13,turning maneuver has a good effect on both high-velocity and low-velocity targets.For the same mission,when R is greater than about 300 km,the radar detection probability of low-velocity Maneuvering Trajectory is less than that of high-velocity Straight Trajectory, which indicates that the effect of turning maneuver is more obvious on the middle- and long-distance trajectory.It is consistent with the conclusion in Section 3.4.1.

Fig.11 Change of azimuth with time during turning maneuver(R = 500 km, overall).

Fig.12 Change of azimuth with time during turning maneuver(R = 500 km, partially magnified).

Fig.13 Probability of detection with different trajectory and velocity.

3.4.3.Influence of RPW and TA on turning maneuver effect

Under different R,the radar detection probabilities of different trajectory and RPW are calculated respectively,and the results are shown in Fig.14.

Fig.14 Probability of detection with different trajectory and RPW.

Fig.15 Probability of detection on different maneuver trajectory.

Fig.16 Probability of detection with different RPW and TA.

As shown in Fig.14, for a fixed maneuver mode (±1°Turning Maneuver Trajectory), the change of the RPW of the aircraft has little effect on the radar detection probability after maneuver.Therefore, we can infer that the maneuver planning only needs to make the maneuver angle domain cover the dangerous angle domain.

In order to verify this conjecture,it is necessary to calculate the radar detection probabilities of different RPW and TA.The schematic diagram of different maneuver is shown in Fig.15, and the results are shown in Fig.16.

As shown in Fig.16, the turning maneuver could have a very good effect, as long as the TA is greater than the RPW.Compared to Fig.14, when the TA is less than the RPW,the turning maneuver works, but could not reach the effect above.

4.Conclusions

In this paper, a simplified RCS dynamic fluctuation statistical model is constructed for stealth aircraft,and a radar detection probability algorithm based on the model is proposed.The dynamic stealth performance of the model is characterized by the azimuth-varying log mean, the azimuth-varying log standard deviation, and the peak width.By calculating different examples, the influence on the radar detection probability with different parameters is analyzed, and the following conclusions are obtained:

(1) The key to successful penetration is to keep the RCS peak unexposed or the exposure time short.The RCS Peak Exposure Time (RPET) is positively correlated with the radar detection probability.By decreasing the RCS Peak Width (RPW) or increasing velocity, the RPET can be reduced, so as to reduce the radar detection probability to a certain extent.

(2) The RPET can be greatly reduced by turning maneuver defined in this paper,so as to reduce the radar detection probability.Compared with increasing velocity, the results show that the turning maneuver has more obvious effects on the middle- and long-distance trajectory.Meanwhile, the turning maneuver could have a very good effect as long as the TA is greater than the RPW.Therefore, from the aspect of stealth aircraft design, there is no need to excessively pursue too small RPW and too large velocity.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.