Shulou(Shulou.com)11/24 Report--

Imagine you are flying a plane on a secret mission. How can you try your best to avoid being detected by radar by enemies on the ground? This is an issue discussed by mathematicians, industry experts and members of the Defense Science and Technology Laboratory (DSTL) at a recent meeting. The working group gave a solution to this thorny practical problem with sophisticated mathematics, and the general idea was inspired by some basic geometry. This is exactly what this article wants to share.

The basic rules allow us to consider the following slightly simplified situation. Imagine a radar system containing a transmitter (transmitter), in which the transmitter can send short radar pulses propagating in all directions. This pulse is called an acoustic pulse. The radar system also includes a number of receiving stations (receivers) that can detect acoustic pulses. Each receiving station knows how far away it is from the transmitting station and the exact time when the transmitting station emits sound pulses.

The transmitter sends a signal that can be detected by the receiver. Now imagine that the plane you are flying is within the detection range of the transmitter and receiver. When radar pulses hit an aircraft, they scatter in many different directions. If a receiving station is on the propagation path of the reflected sound pulse, it will detect the sound pulse. Compared with the acoustic pulse transmitted directly to the receiving station, the acoustic pulse reflected to the receiving station travels a longer distance, and the latter naturally arrives at the receiving station later, so the receiving station can tell which one is the reflected signal. The total distance of the reflected sound pulse is the distance between the transmitting station and the aircraft plus the distance between the receiving station and the aircraft.

An acoustic pulse reflected by an aircraft. The total distance traveled by the acoustic pulse is equal to the distance between the transmitting station and the aircraft plus the distance between the receiving station and the aircraft. The plane you are flying is in the sky and the radar system is on the ground, so we are faced with a problem in three-dimensional space. In order to simplify the problem, we assume that this problem is only considered in two-dimensional space. This means that the plane you are flying is also flying on the ground! Of course, this will not be the case, but this simplification will not affect the problem we want to solve. We can extend this situation back to 3D, which will be implemented at the end of the next section.

How do they locate? you now imagine that a receiving station detects an acoustic pulse reflected by an object. Sound pulses travel at the speed of light c, that is, 300,000 kilometers per second. The reason for this is that the radio waves emitted by radar are essentially the same electromagnetic radiation as visible light, so they travel at the speed of light.

Since the receiving station knows the exact time of the signal sent by the transmitting station, it only needs to see when the signal was received to calculate how long the sound pulse has spread.

The propagation distance d of the sound pulse is obtained by multiplying the propagation time t by the propagation velocity c, that is,

As explained earlier, this distance d is equal to the sum of the distance between the transmitting station and the object and the distance between the receiving station and the object.

This means that the position of the object (such as the plane you are flying) may need to meet the condition that the sum of the distances to the transmitting and receiving stations is d. Long before radar was invented, the ancient Greeks knew that the figure made up of these possible locations (in two dimensions) was an ellipse.

Let's introduce the mathematical definition of ellipse. Given two points T and R in the two-dimensional plane, and then given a distance d, the distance d should be greater than the distance between T and R. Then all the points whose sum of distances from T and R are d form an ellipse. The two points T and R are called the focus of the ellipse.

The possible positions of the aircraft make up an ellipse focusing on the transmitting and receiving pads. in our example, points T and R are equivalent to the positions of the receiving and transmitting stations. So the receiving station knows that the object reflecting the sound pulse is somewhere on an ellipse, and the focus of the ellipse is the position of the receiving station and the transmitting station. If the coordinate system is established in the area of interest, the receiving station can write an equation describing the ellipse. Then it can know the exact position of each point on the ellipse, that is, every position where the object may be.

Now imagine that another receiving station has also detected the reflected waves of the sound pulses of the signal. As in the case of the first receiving station, it knows that the object is on an ellipse focusing on its location and the location of the transmitter. We now know that there are two different ellipses, and the object must be on both ellipses, which means that the object must be in the position where the two ellipses intersect. In general, two ellipses can intersect at four points (you can imagine that two ellipses cross like a cross). However, in our actual example, two ellipses share a focus, and they intersect at most two points. You can confirm this by software drawing or mathematical deduction.

In this way, the possible position of the object is locked at no more than two points: the intersection of two ellipses.

In other words, when the radar system has two receiving stations, all possible positions of the aircraft can be limited to two points of intersection formed by the intersection of two ellipses. By analogy, if a third receiving station detects a reflected wave, it can finally determine the position of the object. The third receiving station introduces a third ellipse that is different from the first two, while the transmitting station is still at a focal point of the ellipse. The located object must be on all three ellipses-and there will be only one such point on the plane, at least as long as the third reflected pulse has a different propagation distance from the first two. If each receiving station sends its own inferred elliptic equation to a control center, then the control center can calculate the location of this point. In fact, the computer can calculate this location in an instant.

When there are three receiving stations, the position of the aircraft is determined: it must be at the intersection of three ellipses, and there is only one such point. We just showed that in two dimensions, as long as three enemy receivers receive sound pulses reflected from your aircraft, they can determine your location. If the enemy has enough reasonably arranged receiving stations, then any sound pulses reflected by objects flying over the nearby area will be detected by enough stations, which means you will be detected anyway.

Before we move on to the following section on how to deceive the enemy, let's briefly mention how to extend the above plot to 3D. In the three-dimensional case, a receiving station that receives the reflected signal can infer that the object is on an ellipsoid-a three-dimensional version of the ellipse, which can be obtained by rotating the ellipse around the long axis connecting its two focal points. At this time, the mathematical operation is more complex, but generally speaking, four intersecting ellipsoids, corresponding to four receiving stations, can accurately locate the position of the aircraft.

How do you deceive the enemy one way to hide yourself from the enemy is to attach a layer of material that significantly absorbs radar sound pulses and reflects only a small part of the surface of your aircraft, so that the reflection is not enough for the receiver to detect. But even if you don't have such special materials, you can deceive the enemy.

Imagine that you can detect and record radar sound pulses from the transmitter. You can rebroadcast the signal after a certain delay. In this way, your enemies will locate two of you and be in very different positions. This avoids being caught to some extent: if the enemy decides to follow you, they need to allocate resources between you and your imaginary image.

The delayed sound pulse misled the receiver into thinking that there was a second object. Because the delayed signal takes longer to reach the receiving station, the receiving station deduces that it has traveled a longer distance. This means that the image needs to be represented by a dotted line on another ellipse that contains the initial ellipse. When all three receiving stations are deceived by this delay signal, your enemy will think that the object is at the intersection of three "false" ellipses, which is different from the position of the original object. During the discussion at the meeting, mathematicians showed that you could go further. What we overlooked just now is that you are actually moving when you fly the plane. In order to confirm the trajectory of a moving object, your enemy may send repeated sound pulses. This will enable them to predict where you are going and try to intercept you.

Mathematicians say (assuming you move at a constant speed in a certain direction) you can adjust your delay in transmitting the signal so that the receiver thinks the second object is moving in a very different trajectory from your plane. By sending several sets of delay signals, you can even make them think that there are multiple objects, each moving in a different direction, which will seriously confuse your enemies and reduce your chances of being captured.

Considerations beyond basic knowledge the mathematics used by mathematicians at the meeting is more sophisticated than what we mentioned. Crucially, mathematicians take into account the uncertainties that always arise in reality. For example, because the receiving station cannot measure the flight time of the reflected signal 100% accurately, they cannot locate an object 100% accurately. In addition, the scattering and absorption behavior of electromagnetic radiation from complex objects such as aircraft will also bring errors, which will introduce uncertainty.

Mathematicians assume that in order to deal with uncertainty, the enemy will statistically analyze the received signals to determine the most likely position or trajectory of the detected object. This analysis is based on Bayesian theory. The theory allows you to speculate about the probability of something (such as where the plane is currently) based on new evidence (such as reflected signals) and information you already know (such as where you think the plane was just now).

In the context of modern communications, every country needs to operate effectively in many fields, such as space, sky, land, sea and network. In our example of radar, this operation uses the electromagnetic spectrum. The problem is that our "electromagnetic environment" is becoming more and more crowded and competitive due to the growing civilian and military demand for the electromagnetic spectrum (for example, the demand for 5G). The above-mentioned meeting is one of the activities aimed at seeking to apply new mathematics to the challenges posed by this problem.

Interestingly, despite the complexity of the problems we consider, we can still get inspiration for solving problems from basic mathematics that we have known for more than 2000 years.

Author: Marianne Freiberger

The cloud opens and the leaves fall.

Revision: there are interests in the future

Original link: Hiding in plane sight

This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop), author: Freiberger

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