Principles of Radar

My name is Chris Allen. I'd like to keep this informal, so if you have some questions along the way or I'm starting to use a little jargon and you don't know what it means, interrupt and I'll take as long as I need to in order to explain it to your satisfaction.

I'm just giving you an introductory view. It's not a lot of depth. Hopefully, it will give you some background on this topic.

Basic Concepts

The basic concepts we'll be talking about are:
1. A little bit of electromagnetic signal transmission. Don't get scared, we're notgoing to go into a great deal of depth on that, but I want you to get an appreciation for the complexity of the signals we're dealing with.

2. We're going to talk about signal reception.How we're going to infer information about the target by comparing the signal we've received
with the signal we transmitted.The beauty of radar is we have full control over what we're transmitting and then we can receive the signal as that goes off of various things - off the targets.And then we can compare them. From that we can infer several things like: a) How big is the target? b) How far away is the target? c) How fast is it moving? d) Is it spinning? All kinds of things like that.

Why are we calling them "targets"? Radar was originally invented or developed by military folks. They are the ones that made the capital and intellectual investment in this originally, and everything they looked at was potentially a target, so the name "target" stuck even when we're looking at trees and shrubs and bushes and stuff like that.We still call them targets just because that's the history. Everything's a target to a radar.

Electromagnetic Signals

Physics gives us the fact that we have all these forces of nature. We have gravitational forces, nuclear forces, and we have electric fields and magnetic fields and so forth. Light is an example of electromagnetic fields and so forth. Magnets and all that kind of stuff is stuff that
makes current flow in circuits. That's all electromagnetism. That's what we're exploiting for radar. You can take the same principles involved,
most of the same principles to other things. If I apply them to acoustic signals, that would be sonar. In medical offices, the same principles are used for ultrasound. But we're just using electromagnetic waves.

When we're talking about electromagnetic waves, they're characterized by the frequency, that is, what's the period and how fast does it repeat itself?And that's correlated to the wavelength, based on the speed of light. I'll talk about that in a minute.It's polarized, unlike acoustic signals.

This is polarization because the field we're dealing with varies as a function of orientation. And everything happens at speed of light, which is pretty darn fast, but it's not infinite. People assume the speed of light is as fast as things go, that's probably true, but it's not infinite. It takes a finite amount of time for a signal to go from Point A to Point B, and that's kind of important.

We've got an electromagnetic wave propagating through space and it's going to interact with our target.We're going to assume our target is in free space and N1 is the index refraction of free space.If you refer to our target down here, it's got a different index of refraction depending on what it is - whether it's wood, or water, or ice or metal. Whatever it is, it has different electrical parameters and we have a clean boundary between the two.

Speed of Light

So what is the propagation at the speed of light? Typically we use the symbol "C": which is 3 times 10 to the 8th meter per second, if you want to round off to one or two significant figures. This is a big number. People have a hard time getting a grasp on how big is 3 times 10 to the eighth meter per second.

Basically you can go around the Earth 7 times in a second. Or, if you don't want to think about how big the earth is,you might want to think about how big a foot is. It takes 1 nanosecond or 10 to the minus 9 seconds for
a signal to propagate that far at the speed of light. And that's a useful reference to keep in the back of your mind. One foot - look your foot or my foot. It takes 1 nanosecond, or 10 to the minus 9, or a billionth of a second, for the signal to propagate that far.

Electromagnetic Signal Propogation

We have an incoming electric field propagating this direction. Here's the E vector. I told you this has a polarization, so let's say the E field is oriented this way, and propagates this way, but the E field is measured in this direction. It's impinging on the surface. We're going to get reflection. We're going to get some transmissioninto the material.

These angles are equivalent to each other. We're assuming this is a linear medium. And when we propagate across this boundary we get some refraction. It takes a whole messy equation to deal with the magnitude of this E field here relative to what was reflected and what was transmitted. I'm not going to go into that. But that's part of the physics behind the electromagnetics.

Reflection and Refraction

So to review this slide, we're dealing with reflection and refraction. In this bit of the slide here, we have an E field which is in the plane of the interaction with this boundary. In this field, the E field is perpendicular. This is supposed to be the tail on this arrow, sticking out of the screen here, so this is perpendicular. So here's the plane of the interaction.The E field is perpendicular to that.

Now in this case, the E field is parallel to that. So we have to be able to account for both polarizations. This is the polarization distance right here. So that's how we take into account the magnitude of how much gets reflected versus how much gets transmitted. What's the relative phase? We'll talk about phase here in a bit as well. that all changes based on polarization, so that's the polarization aspect. And that's as much as I'm going to talk about polarization today.

Attenuation

Attenuation is how much loss of signal power can be experienced as the signal propagates through this medium or through this material, or what have you.So, we're talking about basically a loss of signal energy.

It can be caused by a couple of factors: Absorption of protons - dealing with these little tiny particles of electromagnetic energy. You can think about it in terms of protons, you can think about it in terms of waves. Here we're just going to think about protons as free packets of energy.They can be absorbed by the molecules and turned into heat. So, the molecules are vibrating more and more. They're heating up. You can't really measure it visually except in a microwave oven. So they can be diverted that way. That means that proton is lost, we can't recover it. It doesn't come back to the receiver, or it can be "scattered".

"Scattered" just means that it's not going to come back in our direction, it's going to go off in some other direction.And our receiver is not over there, it's over here and we're not going to receive those protons.

So those are attenuation factors which means that the signal we receive is going to be weaker than the signal we transmitted and that can cause some challenges.

Scattering

Scattering depends on, again, the electrical and magnetic properties of the target, or the scatterer, and the size of the scatterer relative to the wavelength.

I didn't really talk about wavelength so much yet. But wavelength is a function of the frequency, and frequency times wavelength equals speed of light. So for a really low frequency, say of mH,the wavelength is very, very large because the product has to equal its constant - the speed of light. So if one member gets smaller, the other member has to get much larger. So I say that wavelength's the inverse of frequency, if that helps you at all.

Anyway, the scattering depends on the size of the scatterer relative to the wavelength. So for a particle about this size (hands moderately far apart), it's going to respond very well to a wavelength which is about on this same order. Very, very short wavelengths or very, very long wavelengths are going to respond much differently based on relative size of the target in respect to the wavelength.

And a good example answers the question,"Why is the sky blue? The sky is blue because the air particles in the sky are on the same order of wavelength as blue light. The sunlight coming in is basically white light. It is covering the whole spectrum. But it's going to preferentially scatter the blue light as opposed to the red, orange and yellow lights. So that means the whole sky looks blue because all the blue light is scattered. Again this is because of the size of the air molecules.The red light and orange light and yellow light pretty much experience less scattering, so they come straight on through but the whole sky has a background of blue.

The same physics is going on. It scales to microwave wavelengths and stuff like that. But it's the same thing going on.

Radar Components

In every radar we have to have something that's going to control the timing and

We have to generate a waveform. Whether the waveform is just a simple little pulse or it's very, very complex, some little box inside our radar is responsible for making that waveform.

Then we have to take that waveform, do some processing perhaps it's just boost up the signal amplitude. We have to have a transmitter and some electronics to go along with that In order to make this useful, we have to couple this waveform, this signal we've generated, we have to couple it into free space, outer space or into the ground.

Wherever we want to couple it, we have to launch it. So we have to have an antenna. Then it's in God's hands. God takes the signal and interacts with the targets and then we get echoes back and now we can receive the echoed signals with the receive antenna. It could be the same antenna or it could be different antennas. We'll talk about that in a bit as well.

Then we have a very, very weak signal. I'll talk about why it's so weak here in a bit as well. We have a very weak signal coming back. We are going to amplify it, filter it, do all kinds of crazy processing to it.

And the receiver electronics -- At some point along the way we're going to want to digitize it, so we're going to have a data acquisition system. We're going to turn analog multi - ____ currents and so forth into zeros and ones through an analog to digital converter. And then we're going to do all kinds of digital signal processing to it, which happens here.

We will probably have some ancillary sensors. An example would be Global Positioning Satellite receiver so that we know where we were and what we were doing when we collected the signal.

And ultimately we have to store all this data that we're collecting in a data storage device or transmit it to somebody else who has a data storage device because we have to record what we did before we measure it.

Block Diagram

The timing and control is right down here. It's the first one in our list. It's generating all kinds of complex signals and ___ signals and whatnot. It's telling this digital waveform synthesizer when to begin generating this digital waveform.

And then, what we have coming out of there is called "chirp." Not all radars use chirp. We'll talk about chirp another day, but that's the waveform that they chose to use here.

We're doing some frequency conversion. We're up-converting it to the desired frequency. This frequency is a fairly low frequency. You can receive this signal with an FM radio. It's in that band. We're going to up-convert it to microwave frequencies - something you might use on the satellite.

And then we're going to run that through a transmitter which is going to clean it up and boost the power. It goes out to the antenna and interacts with the targets. Comes back in, goes to the receiver. The receiver does some filtering, frequency conversion. It's called video. Goes back to the baseband signal again. So we've gotten rid of all the microwave stuff, and then we're back to baseband signals. We digitize the signal at this point and then we begin the complex process of interpreting what the radar was telling us.

So that's what this image formation processor is doing because ultimately this radar forms images. I will show you some examples of the images here in a bit. And we're recording this on a high-density tape recorder type system. We also have a real-time display. You can see looking at the screen what the radar was actually producing.

And I talked about ancillary sensors as well. We have an IMU, "Inertial Measurement Unit." It's kind of like a gyroscope, so that it can measure displacements and rotations and that kind of stuff. It has a GPS receiver, Global Positioning Satellite receiver, so it knows, to a fair degree of accuracy, things like: Where on the earth are we? How fast are we moving? In which direction? We also have a gimbal, because the antenna has to be stabilized in roll, pitch and yaw. And that's what the gimbal does. It keeps the antenna stationary while the airplane is doing all kinds of _____ maneuvers, but that's its purpose in life.

So that pretty much goes through a block diagram. Not every radar you would see would have the same block diagram but they should have the same classes of elements.

Target Range

The simplest thing we can measure is range. That's where radar got its name, Radio Detection and Ranging (RADAR). So the first thing people figured out you could do with this crazy thing called a radar is to measure the range. So that's what we're talking about first.

So the transmitted signal (let's assume a very, very simple system)... So the transmitted signal is a gated sinusoid so here's our sinusoid. It's a cosine but phase shift makes it turn back to a sine. It's a pulse duration Tau _____ a tau of almost 4. The pulse duration, that's how long the actual pulse is in existence.

It has an amplitude A that's calling the whole signal, "s" and this is only valid for time t=0 to t=Tau. After that it's zero. So this is the waveform that we're sending out of the antenna.

"f" is the frequency, it might be envisioned in gigahertz, billions of cycles, or it could be megahertz or ... This is the radio frequency, how many oscillations per second. This is just an arbitrary phase which should be constant for that pulse and like I say, it's just a time-gated sinusoid. That's what goes out to the antenna.

That's what we transmit. The signal goes out through free space and propagates through space.

The propogation characteristics are: The signal propagates, bounces off a target, we get the echo back. The received signal is going to look like this but it has a different amplitude. B will typically be much, much smaller than A. It still has a sinusoidal relationship. It's a linear process so we're not changing frequencies. We're not changing the waveform significantly. We still have the same fundamental frequency. That did not change. We still preserve the phase, but we have another phase that we are adding, which is based on the round-trip travel time.

The signal took a finite amount of time when it left the antenna to go out, hit the target and come back and that's captured, in part, by this phase difference, but also look at the domain over which this expression is valid.

It's valid over T to T plus tau, so the pulse duration is still tau, (the difference between this point and this point is still tau), but there's a finite amount of time between when the transmit happened and the echo came back This is the echo function or the echo's expression.

Altimeter

One system that might be used - just a simple ranging system would be called an altimeter. Almost all the aircraft have them. They're flying along and they want to make sure they're not going to run into the ground, so they have a radar that looks straight down. It's measuring the height above the local terrain because to know the height above sea level, that's good (that's what a barometer can tell you among other things), but to know the height above the local terrain is kind of important so you don't run into the mountains and stuff like that.

So here's an example of a system. It's called Geosat. So the satellite is looking straight down so it's measuring the height of whatever is directly beneath it So it measures this thing, this "h" it's called, and that can be measured relative to the sea level which is the measure of the whole planet, formed this geoid, which is fairly complex, higher order function and you can reference your measurements and use that geoid so you know, "Are we above sea level, below sea level," and so forth?

Then it radios a signal down to a station on the earth and we can form maps. That's the basic idea of an altimeter, but at its heart is a radar that measures range.

If you fly one of these altimeters over the contiguous US, and it's got a very small footprint so you can discriminate what's happening (you can see kind of the valleys and this kind of stuff), you can make elevation maps of things.

So here's a map of the U.S. I believe it was measured with an altimeter. And it's kind of color coded, but it's not calibrated so you can't see how deep things are and so forth. But you can see large relief patterns. You can see the Rocky Mountains, you can see some large drainage patterns here, you can see the Grand Canyon. You can see river valleys and so forth. So this range over here. And Kansas would be over here. Anyway, you can see a lot of features. The sea is essentially flat, at least to this level of relief, it's fairly flat. I'll show you an example here where it's not really flat, but this is something an altimeter can do from radar on a satellite. You know the orbit of the satellite very well. The differences in elevation are due to the relief of the terrain beneath it.

Antenna

This is just a radar map like the Weather Channel and everybody else puts out, and we can measure scatterers. The scatterers in this case are precipitation - raindrops, hailstones, and sometimes snow - depending on the time of year.

As a function of position, we can measure range. The radar beams up the center of the circle, so it's around here and we can discriminate from this point to this point to this point based on the fact that we can measure range very, very accurately.

And now we can discriminate from this point to this point to this point to this point which are all basically the same range, but we can discriminate one from the other based on the fact we have this high-gain antenna narrow beam width so it can discriminate things based on their spatial position. So we can form a map of targets, in this case, it's weather events, as a function of position. That's using the spatial extent of the target.

We're still looking at the spatial extent of a target. I thought this was kind of interesting so I threw it up here as well. I'll bring in a different mode and so forth. There is no big storm coming in but they've got this very long feature going on here and it turns out that this is a flock of birds coming in, -- it's migrating, And this is south Texas. This is the Gulf of Mexico. So the birds are migrating up. This was taken in March, so this must be spring migration. You're not going to see an individual bird because the energy reflected off the individual bird is so small that it's going to look like noise. This is a bunch of noise out here. But when you have thousands of these birds, then they represent a fairly large signal and you can, in fact, see where the birds are, and what their distribution is.

Radial Velocity

So we've exploited range and we've exploited spatial variability in our targets. We can also measure velocity, the radial velocity only.

That is, take the point where my antenna is, then take the point of the scatterer. Draw a line between those. Any change in that range, the radial velocity, to the target can be measured with the radar as well.

We're getting into a little bit of math here. I apologize for that, but bear with me. Remember when we talked about the received signal, we have this "received signal phase" which is a function of range. It's dependent on the range to the target, it depends on the wavelength and here finally I've defined the wavelength as the wavelength (lamda) is the speed of light divided by the frequency. And then put 2 pi in there if you want to deal with radians. But basically we can measure the phase of the received signal in terms of the range to the target. So if the target is moving, relative to the radar it's going to result in a change in range.

Then this phase is going to change and if you remember physics, a change in phase corresponds to a change in frequency, or a frequency shift. So that's what's going on here.

If we have a timed rate of change of this phase, that corresponds to what's known as a Doppler shift, I'm sure you've all heard about Doppler shift before, And, so this is the basic expression for the Doppler shift, and now we can express it in terms of the same thing, this is the Doppler shift measured in frequency is a function of the radial velocity of the target relative to the scatterer - the wavelength - so if the wavelength changes dealing with one radar versus another radar you get a different frequency -- and the relative angle between you and the target. So here's an expression for the received radar signal that we've included Doppler frequency in the expression.

Let's say we have this aircraft drawn here, but not to scale, because this is measured in kilometers and the aircraft isn't nearly that big. The aircraft is flying at 1500 meters altitude over the surface of something. It's operating at 10 gigahertz (GH), so we have a wavelength of 3 cm. It's flying at 10 meters per second north, upwards.

These lines, these hyperbolic shapes, these are lines of constant Doppler shift. If I look at a point here, or a point here, or a point here, they're all going to be experiencing the same shift in frequency, due to the fact that we're moving relative to the scatter. The scatter is on the surface of whatever they are flying over. Let's say it's Greenland.

The aircraft velocity is going this way. So points that are exactly beneath the aircraft, or where this plane slices right through the aircraft, experience zero Doppler shift, because there's zero radial velocity there. If we measure the angle - zero, theta is 90 degrees, theta is 180 degrees, When theta is 90 degrees, you go back to that earlier slide and the cosine of 90 degrees is 0, so this line is zero Doppler. It has exactly the same frequency coming back as what we transmitted because there's no Doppler shift experienced. This point up here sees the maximum Doppler shift. This point is the minimum Doppler shift. We move from a positive Doppler shift, to zero, to a negative Doppler shift. These are called "Isodops," a fancy word for lines of constant Doppler shift. And that's what's going on.

Reflectivity

Last example here on what can be measured.

Target reflectivity - We talked about how the electromagnetic signals interact with the various targets based on the electrical properties, the magnetic properties, surface characteristics, the geometry, all these different factors are going to affect reflectivity. That is, I throw a whole bunch of radar photons over at this target - how many of them come back? That's a measure of reflectivity. The ratio of how much I send out to how much comes back is "reflectivity". We're calling that "Backscatter."

The backscatter depends on geometry, surface roughness, all these different effects. We can map the distribution. We can look at the reflectivity as it's seen here - a patch of ground here relative to here, relative to here, by exploiting our ability to measure range and velocity and using the antenna as a discriminator so we can have this side which is that side so here's a geometry that people would use to image things.

This is an imaging radar geometry. so we have the antenna hanging on the aircraft. The aircraft ground track is right here. So, the part that we are imaging is offset from the ground track of the aircraft. The aircraft has a ___ velocity in this direction.

You see these different pulses, we're transmitting pulses, which have a short extent. Remember the speed of light is very, very large, so these pulses have to be very, very short if we can confine them to a very short extent in space. They engage with the ground. To get echoes back we synchronize our receiver so we're receiving echoes at the appropriate time. We can measure the Doppler shift.