welcome back to our discussion on the
basics of radar in this video we're
going to focus on radar power and noise
and to do that i want to introduce the
radar equation
the radar equation is a good tool in the
early stages of radar design because it
combines several of the main parameters
of a radar system in a way that gives
you a general understanding of how the
system will perform
i think this is going to be a nice
visual introduction to the equation so i
hope you stick around for it i'm brian
and welcome to a matlab tech talk
radar sends out an electromagnetic
signal with some amount of transmit
power
and then that signal has to travel
through the environment reflect off an
object and return back to the radar
the return signal is a much much lower
strength than the transmit signal since
most of the power spreads out and is
lost to the environment or lost through
inefficiencies in the radar hardware
itself
and determining range velocity and
direction depends on the radar's ability
to receive the weak reflected signal and
pull it out of the noise
and that becomes more difficult as the
noise increases relative to the strength
of the signal
for example a transmit signal might look
like this in the frequency domain where
a lot of power is centered around a
certain frequency band
when the radar receiver is listening for
the reflection
it's going to pick up noise from the
environment and the electronics but it's
also going to get that reflected signal
which is going to have a lower power
than what was transmitted and
potentially be shifted to a different
frequency due to doppler and as we
learned in the last video our ability to
determine the parameters of an object
depends on us being able to find the
location of this peak
however as the noise floor increases or
the signal decreases our ability to
determine the peak location is reduced
and if the signal-to-noise ratio is too
low we will lose the peak altogether and
this is why increasing signal and
decreasing noise is so important
and this is where the radar equation
comes in at least the version that i
want to talk about in this video there
are other variations of this equation
that take into account a larger set of
physical phenomenon but i think this
version will provide a good basic
introduction into what the radar
equation is
and why it's so powerful
but before we jump into it i want to
head over to matlab and really quickly
show you the radar designer app
i'm going to load in one of the default
designs this one is for an airport radar
system
now i've left a link below that explains
this whole app and how to use it so i'm
not going to get into that in this video
but all i want to point out here is this
graph that shows the signal to noise
versus range
you can see that snr for this particular
radar design is about a hundred for an
object that's just a few kilometers away
and drops to about 15 at the maximum
range that's specified in the
performance requirements
these two horizontal lines show the
detectability of an object it's
basically at what s and r does the radar
fail to detect the object in the
reflected signal
we can see that this radar design meets
the range and detectability requirements
since the snr line doesn't pass through
the yellow or red boxes
however if i lower the peak transmit
power by half
we can see that the design no longer
meets the requirements
lower transmit power means lower
received signal power hence lower snr
and you can tell from all of these input
parameters that there's a lot that goes
into this full calculation however we
can get a good approximation of this
line
using the relatively simple radar
equation and an estimate of noise
which is helpful at the early stages of
radar design when we don't know many of
these other parameters
so with all of that being said let's put
the radar designer app away for just a
bit and walk through the radar equation
so you have a better feel for what it's
doing and hopefully when we come back to
this app
everything will make a lot more sense
okay so we're going to start at the
radar transmit antenna
when the antenna is transmitting it's
sending out an electromagnetic signal
that radiates from the antenna in all
directions
now i'm only drawing a 2d circle here
but this is really a 3d sphere that's
growing like a balloon being blown up as
the signal propagates out from the
antenna
the signal is transmitted with some
power which could be as high as the peak
power of the transmitter pt
and if the antenna is isotropic like i'm
drawing here then this power is spread
out equally in all directions
or another way of putting it an
isotropic antenna produces the same
power density across the entire surface
of the sphere
power density is calculated by taking
the peak transmitted power and dividing
it by the surface area of a sphere which
is 4 pi times the radius squared and so
power density decreases as an inverse
square of the distance from the radar
now for most tracking and surveillance
radar systems it doesn't make sense to
send a lot of power out in all
directions with an isotropic antenna
instead we want to concentrate the power
in a specific direction using a directed
antenna like a dish or a phased array
antenna
so for the exact same peak power the
power density has decreased relative to
the isotropic antenna in most directions
but importantly has increased in the
direction of the beam
and this increase in power density
relative to an isotropic antenna is the
antenna gain which will label gt
note that the surface area of the sphere
is still growing as the signal range
increases and therefore the power
density is still dropping by the square
of the distance in our equation it just
starts from a value that is scaled
higher by the transmit gain
all right so this is the equation for
the power density
at some range
given the peak power of the transmitter
and in the direction of the maximum
antenna gain
now the signal is propagating through
the environment which might be free
space if you're lucky but for
earth-based radar systems it's usually
air and it might be obstructed with snow
or rain or who knows what and all of
that has the potential to attenuate some
of the power
the amount of power loss depends on the
transmit frequency and the type of
weather and obstructions and other
things but at this point in our design
we can account for all of those losses
by just adding a generic loss term to
our equation
a number greater than one means that
we're losing some proportion of our
power to the environment
all right eventually this signal reaches
an object and part of the signal is
reflected back towards the radar
the amount of power that is reflected
depends on the radar cross section or
rcs
rcs has the symbol sigma in our equation
and has units of meter squared since it
represents an area
so what exactly is the radar cross
section well let me provide a rather
informal explanation of how you can
think of what the radar cross section is
compared to the physical cross section
all right so when this transmitted
signal
intercepts an object the total power
that is intercepted is the physical area
of the object times the power density of
the signal at the object
so a larger object would intercept more
power
as an example let's say that this object
has a physical cross section of 10
square meters and the signal has a power
density of 10 watts per square meter
this object would then intercept 100
watts of power
now some materials might absorb a little
bit of that power and therefore the
reflected power is less than 100 watts
in our case but usually most of that
power is just re-radiated back out into
the environment
so let's assume that this object
re-radiates the whole 100 watts of power
and it does so isotropically so an equal
power density across the entire surface
of the sphere
i'm choosing an isotropic radiation
pattern here because it's tied to the
definition of rcs
rcs is what the area of the target would
be
if the total power that the object
intercepts was re-radiated isotropically
in this example with the object acting
like a perfect isotropic antenna the rcs
area is exactly the same as the physical
area both are 10 square meters
all right so let me copy this picture so
we remember what an isotropic pattern
looks like
and now let's say that the object is a
different shape and it re-radiates the
hundred watts in a non-isotropic way
much like how the directional antenna
concentrates the power in a specific
direction
an object might re-radiate the power
unevenly across the surface of a sphere
now both of these objects intercept 100
watts of power since they both have the
same physical cross-sectional area
however the object on the right
re-radiates the power unevenly
and if we look at the power density of
the reflected signal back in the
direction toward the radar we can see
that it's lower than what an isotropic
antenna would produce
in fact we would have to lower the area
of the isotropic object by about half in
order to match the same power density
in this way we would say that the rcs of
the right object is only about half of
what the physical area is
however we can see that rcs is direction
dependent this object looks small in the
direction back towards the transmitter
but if we view it from a different angle
it looks really large since it
re-radiates so brightly in other
directions
it would take an object about twice as
large to produce this brightness in the
reflected signal under the assumption
that the object was an isotropic
radiator
so rcs is an area that basically
represents how much signal is
re-radiated back towards the radar
receiver
and up in our equation here we've
already calculated the power density as
the signal reaches the object and when
we multiply that by the radar
cross-section we get the total reflected
power of the ideal isotropic emitter
but we're not done yet because now that
reflected power has to travel back to
the radar and the entire time that wave
front is spreading out and the power
density is dropping again by the square
of the distance
this means that the round-trip power
loss is a function of the distance to
the object raised to the power of 4.
so it's really sensitive to range
now in addition to losses due to
scattering through the atmosphere when
the signal is traveling over certain
terrain it can be amplified or
attenuated even further
this can occur when the signal bounces
off the surface of the terrain and
reflects back towards the radar
since the distance traveled between
these multiple paths is different the
phase shift between them can
constructively or destructively add
together
the amount of this amplification or
attenuation depends on the elevation
angle of the object as well as the shape
and type of terrain
but we can account for this by just
adding a propagation factor to the
numerator of our equation a value larger
than one produces a boost in the signal
and less than one attenuates it
all right so now after all of that
traveling and reflecting and traveling
we have an equation for the power
density back at the receiver
and to determine the total power that
the receiver collects we can multiply
the power density by the area of the
antenna
and this is why larger antennas can pick
up fainter signals they collect more
power given the same power density
but the receive process isn't perfect in
addition to the losses that we've
already accounted for there are other
system losses within the radar
components including inefficiencies
within the antenna itself
so what we end up actually receiving as
useful signal is the ideal received
power scaled down by all of these losses
and if we want to simplify this equation
a bit we can take all of the losses and
just wrap them up into a single term
l
we now have our equation for received
power
however in some cases like at the
beginning of a project we might not want
this equation as a function of the area
of the antenna and would rather see it
in terms of antenna gain and signal
frequency since those tend to be the
parameters that we trade early in the
design
and as we talked about antenna gain is
the ratio of its power density compared
to the power density of an isotropic
antenna and for the same total power
this becomes a ratio of areas
now if the beam width of the antenna is
small enough we can assume that the beam
area on the sphere is small enough that
it's essentially a flat plane and
therefore we can make the following
approximations
so we get gain as a function of antenna
area and signal wavelength
now we can replace the antenna area in
our equation with lambda squared over 4
pi times the received gain
and in the case where the transmitter
and receiver are using the same antenna
this equation simplifies to the final
radar equation that we started with
so now we can see that this whole
equation is just calculating how much
power is received given how much power
was transmitted the antenna gain the
transmit frequency the rcs of the object
and the propagation through the
environment and radar components
this is signal
but we also want to talk about noise
noise comes from many different sources
including some that are external to the
radar like atmospheric noise and solar
noise and some that are internal like
the noise from the electronics in the
radar components and all of this noise
can be represented as a single noise
source whose power is equal to the
boltzmann constant times the system
noise temperature and the noise
bandwidth of the signal
we can now divide the signal power
equation by the noise power equation to
get the signal to noise ratio
and with it we can start to get some
preliminary understanding of how all of
these parameters affect each other
all right so i know that this was a
whirlwind introduction to the radar
equation but if we go back to the radar
designer app in matlab hopefully we can
start to make sense of some of these
parameters by seeing their effect on snr
as we change them
we left off with our snr being too low
to meet the requirements and we know
that by increasing transmit power we can
increase signal
but we could also relax some
requirements
for example we could double the minimum
radar cross-sectional area from 10
meters squared to 20.
and we could trade whether this was a
good compromise or not
and unfortunately all of this is
assuming perfect weather and if it's
raining heavily then our system once
again won't meet requirements
and since airport radar need to operate
during the rain
we should investigate how to boost snr
or we could lower the detectability
threshold by changing the signal
processing algorithms to be able to find
the signal peak even with lower snr
but note that this radar is operating in
free space if we change the environment
from free space to a curved earth with
standard atmosphere
well now it looks like we have plenty of
snr to meet our requirements so that's
great
but if we go to the vertical coverage
map which shows the propagation factor
that was f in our equation it shows it
as a function of elevation angle we can
see that the object being at an
elevation of half a degree produces a
large propagation gain it just so
happens that the signal is amplified
from the surface reflections at this
elevation
however if the object rises in altitude
such that the elevation angle is 0.9
degrees
then the surface reflections attenuate
the signal and our radar system violates
the requirements by a lot
and for an airport radar system we could
use propagation factor to our benefit by
placing these high gain lobes at the
typical approach angles of incoming
aircraft
all right so it's all really interesting
to figure out how snr is affected by all
of these parameters and like i said an
approximation can be made in the early
design process with the radar equation
and hopefully you have a little better
understanding of what the radar equation
is and are a little more motivated to
continue learning on your own
i've left links to a bunch of great
resources that cover the mathematics of
all of this in much greater detail
plus you should check out the radar
designer app in matlab where you can
easily investigate all of these
parameters quickly and see the results
of your design
alright so that's where i'm going to
leave this video in the next video we'll
spend some more time looking at
different waveforms we covered
continuous wave in the first two videos
and so in the next video i want to talk
about pulsed radar
so if you don't want to miss that or any
other tech talk video don't forget to
subscribe to this channel and if you
want to check out my channel i cover
other control theory topics there as
well
thanks for watching and i'll see you
next time