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Introduction to Radar Systems – Lecture 2 – Radar Equation; Part 1 — Transcript

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0:00hello again this is lecture two of the
0:06introduction to radar systems course and
0:08in this lecture will be discussing the
0:12radar equation as I mentioned in the
0:16first lecture at the beginning of each
0:19that following lecture will bring up and
0:23show you the radar system block diagram
0:26and we will give you the context of that
0:30individual lecture within that radar
0:34system block diagram rather than
0:38focusing on one individual piece like
0:40the transmitters and receivers of the
0:42antenna propagation this lecture we'll
0:46discuss the radar range equation which
0:49connects all of the different pieces of
0:52the radar with the target and the
0:54distance from the radar to the target
0:57that radar equation as I just said
1:00connects the target properties which are
1:03the targets reflectivity or radar
1:06cross-section as we call it the radar
1:10characteristics such as transmitter
1:12power and antenna aperture and the
1:16distance between the target and the
1:19radar ie the range to the target and
1:22also the properties of the medium the
1:25antenna in atmospheric attenuation and
1:28that sort of thing okay first we're
1:34going to go over in introduction to the
1:36radar equation then we'll look at the
1:41surveillance form of that radar range
1:44equation we'll go over the different
1:46losses that can so-called I call it the
1:49humanity of the radar the inefficiencies
1:52and the different components and
1:55subsystems of the radar that contribute
1:57to the losses in the radar equation will
2:02look in detail at an example of
2:06the how the performance of an individual
2:08radar example is calculated and then
2:13we'll summarize okay now let's start off
2:17with deriving the radar range equation
2:21what we're going to do is to go over
2:25from basic physical principles how the
2:28radar equation evolves we're going to
2:33just need algebra in in our reasonable
2:36good physical intuition now let's start
2:39where you'd think off with the radar
2:42transmitting a pulse in the simplest
2:45possible way transmitting a spherically
2:48symmetric uniform pulse of energy with a
2:53given power in that pulse
2:55and it's written uniformly radiating out
2:58spherically that peak power of that
3:03pulse of energy we denote as P sub T the
3:07peak power of the transmitter and at any
3:10distance R away from that transmitter
3:15the density of power is given by the
3:20that peak power divided by the area of
3:24the sphere because that density of
3:27energy is going to diminish as the
3:30sphere gets larger and larger it's the
3:32power per unit area okay well if you say
3:38you had a small area on the sphere the
3:41power density honestly at a given point
3:45would be the overall power divided by
3:48all the area in the sphere so at a given
3:52arbitrary distance from the radar the
3:55power density is that P sub T divided by
4:004 PI R squared now in practical radars
4:05we don't transmit power out in all
4:08directions
4:08this one we use an antenna to shape the
4:13beam and send that energy preferentially
4:16in one direction
4:18and as we mentioned it went over in the
4:20first lecture that directivity that we
4:24give the the beam it is characterized by
4:28a quantity we call the gain and it's the
4:32power the group that you have in excess
4:34of the power that you'd have if you were
4:41transmitting in an isotropic spherical
4:43way so let's just slowly reiterate it
4:46the games the intake radiation intensity
4:49of the antenna in a given direction over
4:53that that you'd get from a uniformly
4:55radiating isotropic storms and that gain
4:59that can be written out as for 4pi times
5:03the area of the antenna divided by the
5:07wavelength squared I haven't given you a
5:09derivation or a physically intuitive
5:13understanding of that we'll talk about
5:15that later in the antenna section but
5:20that gain is that greater amount of
5:23energy you'll have over the spherical of
5:27radiating energy so if we want to write
5:30down the modify this above expression
5:32for the power density of an isotropic
5:34antenna the power density from a direct
5:37of the antenna is just the first
5:39expression multiplied by the the gain of
5:42the answer of the transmitter
5:45transmitting antenna excuse me
5:49okay now that's that wave going out
5:52towards the target is going to emit out
5:55until it gets to the target and that a
5:58power density will impinge on the target
6:02and the radar cross-section which is a
6:07like electromagnetically the size of the
6:09target
6:10it's the electromagnetic area that the
6:13target sees it's a measure of the energy
6:15that is radiated back towards the radar
6:19that's intercepted and scattered and
6:21goes back to the radar and we call that
6:24sometimes the RCS the for the initials
6:28radar cross-section and it's usually
6:30denoted in equations with the Greek
6:33symbol Sigma a small Sigma Sigma and
6:36it's units are in meters squared or area
6:39remember I called it an effective area
6:41now if the power of the reflected signal
6:45at the target then would be the power
6:48density which we just had times that
6:52area power density times the area will
6:55be the power reflected at the target now
6:58that energy will be reflected back and
7:00again will undergo a diminishment of one
7:05over R squared as and times four pi as
7:10that wave expands out so that the power
7:13density received at the radar is given
7:18by this expression just the power of the
7:21reflected signal at the target divided
7:24by another factor of the area of the
7:27sphere back to the target four PI R
7:30squared and notice that the power
7:32density of the reflected signal falls
7:35off as one over R squared
7:39now back at the target the received
7:43power is just the power density at the
7:47radar which we calculated in the
7:49previous viewgraph times the area of the
7:53receiving antenna so I'm again
7:55multiplying the received power density
7:59times the effective area of the antenna
8:01this is ASA B and this gives us the
8:05power of the reflected signal at the
8:10radar very important factor so that's
8:13the power that's received back at the
8:15radar from the echo of the targets okay
8:22now competing with that power of the
8:25echo is background noise remember we we
8:30showed you a graph of the noise and B
8:34that the receiver would hear if there
8:36was no target no echo no transmitter no
8:39nothing just turn on the radio detector
8:42radar receiver turn up the volume and
8:44they'll be some ambient noise so what
8:47causes that ambient noise that we want
8:50to see that very small a few micro watts
8:53of power in now there are a number of
8:56different physical effects that cause it
8:59some of it is galactic noise that's
9:01noise that comes from other galaxies
9:03that's in the microwave reach frequency
9:05range noise from the Sun in the same
9:09that would be in the same frequency
9:11range that your listening and your radar
9:13noise that's generated in the atmosphere
9:16whitening which will generate some
9:18energy in that spectrum and a little
9:23cartoon here for lightning also that can
9:26be man-made interference interference
9:29from other nearby elec electromagnetic
9:32sources like radars radio stations
9:35things like that
9:36or it could be
9:39deliberate deliberate transmissions to
9:44raise the floor noise and then the
9:46receiver of the radar so that the radar
9:48would be ineffective we call those
9:50jammers okay and then noise that could
9:54come from lots of different sources
9:56reflect off the ground and go into the
9:59side lobes the the places in the antenna
10:02that don't have a huge amount of
10:03reflectivity but they all add in
10:05together and then of course this going
10:08to be noise that comes from the portions
10:11of the receiver and the waveguide until
10:14it gets back into the into the depths of
10:16the receiver okay now we characterize
10:20the noise power as Boltzmann's constant
10:23times the temperature and we have a
10:27bandwidth factor in here intuitively
10:30when you have like an atom it's moving
10:35back and forth when you heat it up it it
10:39uh it it gains energy and that amount of
10:45energy that it gets by being heated up
10:48is Boltzmann's constant times the
10:51temperature that you heat heat it up to
10:54okay and that's the amount of energy now
10:57the power would be the energy per unit
10:59time okay so what we're going to do is
11:03we're going to characterize all these
11:05different noise sources by an effective
11:07temperature that we're going to multiply
11:10by Boltzmann constant by but that's an
11:12energy and we have to divide that by a
11:14time to get the power and the time over
11:18which we're looking is just the pulse
11:21width of the radar that not the size of
11:23the pulse and one over that is a good it
11:26is the bandwidth that the frequency
11:28range over which we're operating and
11:30that the receiver is listening to and
11:33that's that B sub N and that's measured
11:35in Hertz so the effective power of the
11:38noise is Boltzmann's constant it's shown
11:42over here it's a universal constant and
11:45its measured in energy per degree Kelvin
11:49now so there's many different sources
11:56that the noise can come from and what we
12:00do is we represent them by a single
12:02noise source at the output of the
12:03antenna terminal now what are we left
12:07with
12:07in terms of the equations we've
12:10developed you've got the signal power
12:13right up here that's up and then we've
12:17got the noise power and the ratio of
12:19those two is the signal-to-noise ratio
12:22so we just take this set of quantities
12:25divided by that and we have this
12:27equation right here okay now the
12:32signal-to-noise ratio we call it s / N
12:36or SNR is the standard measure of a
12:39radars ability to detect a given target
12:43at a given range from a radar and the
12:47way we would state that is we'd say the
12:49signal-to-noise ratio of a certain radar
12:52is 13 DB but always we'd say it's on a 1
12:57square meter target as an example at a
13:00range of a thousand kilometers and that
13:03statement is a statement of the
13:05detecting detectability characteristics
13:08of of a radar notice if I take this
13:11equation and I plug in a certain
13:16cross-section and I plug in a certain
13:19range then all the other parameters are
13:22the properties of the radar itself
13:24innately okay so this this says that
13:31is is a statement of the death of the
13:35ability of a certain radar to detect a
13:39one square metre target at a thousand
13:41kilometers now I told you about the
13:47system noise temperature being the sum
13:49of a lot of different characteristics
13:51and this is how one calculates that
13:53total system noise temperature it's
13:56divided up into three components apart
13:59from the antenna apart from the that's
14:02the contributions from the components
14:05and between the antenna and the receiver
14:07and a part in the receiver itself the
14:12contribution from the antenna includes
14:15the apparent sky temperature and you can
14:17get that from a standard graph in a
14:19radar text and it depends on the angle
14:22you're looking in the sky and the
14:24frequency of the of the radar that sort
14:26of thing and also it includes heating
14:30ohmic losses so-called ohmic losses
14:33within the antenna itself then there's
14:36the contribution for the the microwave
14:39components to so-called radio frequency
14:42of microwave components between the
14:44antenna and the receiver and they're all
14:47lumped into one effective temperature
14:49and then there's a component for that
14:55that characterizes the actual noise
14:58that's an eighth in the receiver and
15:01there's a turn called the noise factor
15:05of the receiver that that is related to
15:10the to the temperature that would that
15:12receive or I'm not going to put that
15:14equation down for simplicity in this
15:16course later courses will go into that
15:18detail but it's effectively the
15:20temperature of the receiver and then
15:22also the loss of those input microwave
15:27components within the receiver so when
15:29you put that all together you'll come
15:31out with a certain temperature in
15:33degrees Kelvin that you plug in the
15:35radar
15:36raishin okay now we want to go to the
15:42surveillance form of the radar equation
15:44and why so much the surveillance radar
15:47equation because what we've just done is
15:49we have derived just previously and here
15:53it is the radar equation when the
15:57location of a target is known and the
16:00antenna is pointing towards the target
16:02if you think in that whole set of logic
16:05I went through to develop the radar
16:07equation I had the antenna pointing
16:09directly at the target so you might say
16:11well gee what if I know the antenna
16:13the target is up in the sky but I don't
16:16know where and my beam is relatively
16:19narrow I've got a look here listen look
16:22there listen look at another angle and
16:24listen look at a whole bunch of angular
16:27positions that might be in a rectangular
16:31solid angle or angular area or a
16:34horizontal set of beams we'd call a
16:39horizon fence and go back and forth
16:43looking for targets that form of the
16:47radar equation is called the
16:49surveillance form and it you have to
16:51manipulate this algebraically this to
16:55put one in the form of the other okay
16:58now when we do that we come out with
17:04this equation on the right and you can
17:07see what we want to do is we want to say
17:09for a given radar one of my parameters
17:13of saying how well the radar will work
17:15is I have to say how big a volume do I
17:19have to an Euler volume do I have to
17:21search and that's characterized by a
17:23solid angle which is the angular space I
17:27have to keep searching to find the
17:29target and it's also characterized by a
17:32term which is the time it takes to do
17:35that okay
17:37and and in this form of the equation we
17:41convert the peak power to the average
17:43power through the duty cycle and the
17:45time between pulses so that we talked
17:47about earlier and but this is the form
17:50of the search equation okay and this is
17:54the form of the track equation so when
17:56you could imagine when you build a radar
17:59you're going to have two different
18:01functions first I want to search for
18:03targets so you develop a radar that
18:06would have an the appropriate power and
18:09aperture and to be able to have
18:11sufficient signal-to-noise ratio to
18:13perform its search function but then
18:16after you've you've developed the radar
18:19to do that search function you want to
18:22make sure that a contract it can perform
18:23the track function so you want to have
18:27it perform also and you'd use this form
18:29of the radar equation so when you do the
18:32design process in a radar you're going
18:34to use both forms of the radar equation
18:39now let's look at the search equation
18:41for a few minutes and a couple of
18:43different view graphs and what we'll do
18:45is well we'll just draw some measure
18:49from it about physically intuitive
18:52things and trade-offs and just to give
18:54you a feel for what what these algebraic
18:57quantities mean now we can take this
19:00algebra up here which in some sense it
19:03isn't let's just poke them and move over
19:05to one side all those parameters which
19:09are design parameters of the radar the
19:12power the aperture of the radar the
19:15system noise temperature and the losses
19:19they all have to do with what the how
19:22the engineers would build that radar and
19:24design it then on the other side of the
19:27equation there's a few constants in one
19:29case I put Boltzmann constant here
19:31because that's natural these sort of
19:33would
19:34system noise temperature enough that 4pi
19:37is over here but all the other
19:39quantities over here are performance
19:42parameters how big a solid angle can
19:45recover when we do search how far out
19:48can we perform search and range and
19:52what's the quality of our measurements
19:54the signal-to-noise ratio how much time
19:57is required to do that search and what
20:00size targets can we see these are all
20:03performance characteristics okay
20:06so you see on the one hand you've got
20:10the stuff you want to know the
20:11requirements you want the radar to do
20:14and on the other hand you've got the
20:16engineering characteristics that the
20:18designer has to build and if you want a
20:21certain set of performance parameters
20:23you've got these things at your beck and
20:27call that you have to build the radar
20:29with enough power big another aperture
20:32etc
20:33excuse me etc now let's rewrite that
20:38equation from its original form of
20:41signal-to-noise ratio is equal to all
20:44those parameters - one way we put power
20:47on the left the average power and
20:51everything else on the right okay and
20:54let's look and let's see hey the power
20:57that's required to do the job it's
21:01independent of wavelength you don't see
21:04wavelength appearing anywhere in here
21:07interesting point but if we brought back
21:09the I'll go back quickly to view graphs
21:13we go to the track equation the
21:17wavelength comes into play
21:20so we have the power it's independent of
21:26wavelength and it's a very strong
21:29function of our we'll get into that but
21:33everything else it's it's linear and
21:35everything else you know if you want W
21:39signal-to-noise ratio you've got a W
21:42power if you want a half the area
21:47you got a W power that sort of thing now
21:50let's look and see how strong that part
21:53of the fourth character that really
21:55makes a big difference say we have a
21:57radar it can do its job at a thousand
22:00kilometers it can do search out to a
22:02thousand kilometers of range well how do
22:06we have to modify that radar to be able
22:09to do the same job at two thousand
22:14kilometers well this one solution is we
22:20can increase to increase the range by a
22:23factor of two or three dB
22:26I'm gonna use this as an example to get
22:29you a little more used to using DBS if
22:31you're not I can I have to increase the
22:34power by a factor of sixteen if I double
22:38the power from two to four that range
22:42number goes up by a factor of sixteen
22:45which is 12 DB so I'd have to increase
22:50my power well over a factor of ten not a
22:56hundred but you know somewhere in
22:57between incredible huge amount I have to
23:01increase the door I could increase the
23:03diameter of the antenna by a factor of
23:07four 6 DB or the area by 12 dB
23:16okay big you know well over a factor of
23:1910 or I could increase the time I scan
23:24by 12 DB or increase this or decrease
23:29the solid angle that I can see if I want
23:33to see our father I can't look at as
23:35many different angle cells by a factor
23:37of well over a factor of 10
23:40oh I can take pieces out of each one of
23:42those but the thing I want to point use
23:44this to point out is that the radar
23:46range equation is a very very strong
23:49fact function of the very strong
23:53function of the the power required in
23:57the other priam is a very strong
23:59function of our

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