let's continue our discussion about
frequency modulated continuous wave
radar
in the last video we talked about how we
can use fmcw radar to determine the
distance and velocity of multiple
objects at once and the key to being
able to do that was with clever
frequency modulation schemes
in this video let's talk about how fmcw
radar is also able to determine the
direction to an object both horizontally
in azimuth and vertically in elevation
and to do that we need to move past
modulation schemes and turn our
attention to the antenna or more
precisely to the antenna array
and by the way i've left some links to
some fantastic resources that describe
the mathematics and equations of
measuring angles with continuous wave
radar and i hope you take some time to
check them out later because in this
video i want to forego a lot of that
math and focus mainly on concepts and
building a little intuition around the
problem i think it's really interesting
and so i hope you stick around for it
i'm brian and welcome to a matlab tech
talk
let's start by briefly discussing why
it's important for the radar to know the
direction to the detected object and as
an example let's take the forward-facing
radar that is used for an adaptive
cruise controller
a radar detection would indicate that
there's an object somewhere in front of
the vehicle and we would know the
distance and the relative speed to that
object
however the adaptive cruise controller
wouldn't work too well if it couldn't
determine if that detection came from an
object in the same lane as itself or the
lane next to it or worse still what if
it couldn't determine the difference
between a car parked on the side of the
road and a car stopped in its lane
so angle determination along the azimuth
is important but elevation angle
determination is equally important a
vehicle would like to know if a
stationary object is from a stopped car
or from an overpass
and beyond just detecting a single
object in azimuth and elevation the
radar system should be able to resolve
two or more objects in the same scene
and be able to distinguish one from the
other
we want a radar system that can resolve
a moving vehicle while it's driving next
to parked cars or when it's driving
under an overpass
and what i think is really cool is that
calculating direction and resolving two
different objects like this can be
accomplished with continuous wave radar
and to understand how it works let's
walk through a thought experiment
assume that we have a radar with an
isotropic antenna so the frequency
modulated waves that it sends out are
emitted equally in every direction
and we can pick up that reflection from
an object from any direction so we have
an isotropic receive antenna as well
now let's put a single object in the
scene at some distance in radial
velocity and we know from the last video
how to determine those two parameters
however the range measurement only means
that the object is somewhere on this
circle and the velocity measurement
means that we only know its speed and
direction relative to the radar
so knowing just range and velocity is of
limited use for tracking and
understanding where the object actually
is
now to reduce the region where the
object could be we need to change the
antenna
instead of transmitting with an
isotropic antenna we can use an antenna
that forms a more narrow beam
if the radar doesn't sense a detection
then we know that there isn't an object
in the direction of the beam
and we could scan through the
environment and note the direction the
beam is facing when a detection is made
in this way we can know the direction to
the object to within the width of the
beam
and scanning through the environment
like this could be done by physically
moving the antenna with motors that
gimbal in azimuth and elevation
or it can be done with a fixed antenna
where you electronically steer the beam
by adjusting the gain and phase of the
signals to each of the elements in the
array and we're going to talk about
phased arrays and beam forming in a
future video so we're going to come back
to this topic then
but what i want to talk about in this
video is that there is a way to
determine the angle to an object more
accurately than the width of the
transmit beam itself
in this way we can have a wide beam
that's fixed and still know where an
object is within that beam and the
solution is to have multiple receive
antennas
for example imagine we have two received
antennas that are separated by a
distance d
and this distance by the way is one of
the design parameters of the radar but
it's very small it's typically half of
the wavelength of the transmit signal
and the wavelength at 79 gigahertz is
about 3.8 millimeters so these two
receive antennas are only about 1.9
millimeters apart
so i don't want you to get the
impression that adding a second antenna
is somehow making the radar huge it's
still a compact form factor which is
important for these small vehicle
applications
all right so the transmit antenna sends
out a frequency modulated signal mostly
in the direction of the beam
some rays intercept the object and
bounce off and then some of those rays
are reflected back towards the receive
antennas
you could imagine that if the object is
far away like on the order of a few
centimeters or more and you form the
triangle to the two antennas that are a
few millimeters apart then the angle
from the object to the two antennas will
be really small
it's so small in fact that we can often
assume it's zero and that the two rays
are parallel
and they're definitely not parallel as
i've drawn them here so let's move the
object further away which puts it off
the screen but now we have these nice
nearly parallel lines
so basically the reflected signal is
arriving at both antennas from the same
direction
all right so what does this do for us
well let's think about the reflected
signal which consists of two orthogonal
sinusoids that are 90 degrees out of
phase the so-called iq signals that we
talked about in the last video
and since both rays are from the same
source they move in phase with each
other at the same distance from the
object so at this point here which is
perpendicular to the direction of the
rays the signals are in phase
but since this object is off to one side
one of the received signals needs to
travel just a little bit longer to get
to the farther antenna and therefore
each antenna measures slightly different
iq values
in fact we can show with some trig that
the signal has to travel an extra d sine
theta farther
again where d is half the wavelength of
the signal and theta is the angle
between the object and the direction the
radar is facing that's the angle we're
trying to measure
so if we sample the iq signals from each
antenna at the same time we can
determine what the phase difference is
between the two which allows us to
determine how much farther one signal
traveled versus the other
and therefore the angle to the object
how cool is that i mean we're just using
the difference in values between the two
antennas to determine angle
and what's even better is that the
accuracy of this calculation comes down
to the signal to noise ratio and not the
beam width of the transmit antenna
and to illustrate that let's start with
an example that has no noise
imagine the first antenna at time t
measures these iq values
and the second antenna measures these iq
values
again there's no noise or interference
whatsoever in this system these are
perfect values and so we could line up
an ideal signal with the measured values
from the first antenna
and then starting from that position we
could add phase until the iq signal from
the second antennas line up however far
we shifted the signal is the phase
difference between the two antennas and
assuming no noise in this system then we
can determine this phase shift really
accurately
of course there is noise and so accuracy
is degraded and it's easy to see this by
assuming that there's some noise that
slightly raises the value of the
quadrature signal of the second antenna
you can see that in order to line up
these values it requires us to add
slightly more phase and therefore the
angle to the object would come out as a
higher value than it actually is
of course we don't actually find the
phase shift by manually shifting signals
around like this this is just a way to
visualize how it's possible to calculate
angles from two separate antennas
instead of this we use a fourier
transform to calculate the angles and
we're going to come back to that in just
a bit so hold that thought
because before we move on i want to talk
a little bit about angular resolution
that is the ability for the radar to
resolve two or more objects in the scene
if we have a low resolution radar then
two close objects would look like a
single large object which would not be
ideal if we want to distinguish between
cars that are next to each other in
different lanes
and we saw that accuracy is a function
of noise but resolution is a function of
the radar's aperture or the size of the
antenna array by adding more receive
antennas to the array we can increase
the angular resolution of the radar
and we can keep adding receive antennas
each set a half wavelength apart from
each other
now the signal has to travel an extra d
sine theta farther to each one and
therefore the delay to each antenna is
more as we move across the array
so how does this improve resolution
well i think showing is going to be a
lot easier with a simple matlab app that
i made so let's hop over to matlab and
check it out
all right in this example i'm using an
array of eight receive antennas and this
plot shows the iq signals from each of
them where the first line is the first
antenna the second and the third and so
on
and this red line indicates the iq
values that each antenna is seeing at
the moment that's what these red dots
are
and this plot up in the corner is just
plotting those iq values over distance
remember each antenna is a distance of
half a wavelength away so the whole
array is three and a half wavelengths
long
now currently the angle to the object is
zero degrees and so they're all
receiving the exact same signal and
that's why this plot is just two
straight lines
at the top over here i have a slider
that lets me change theta and therefore
delay each of the signals into the eight
antennas accordingly
and notice as i increase theta these iq
signals from the antennas trace out a
sine wave with increasing spatial
frequency
and it's this relationship between theta
and spatial frequency that we can
exploit
because now we don't have to fiddle
around with phase shifts and distances
we can calculate the angle to the object
based on this spatial frequency the
faster it is the larger the angle to the
object and this is where the fast
fourier transform comes in
taking the fft of this signal produces
this plot where the x-axis is the sine
of theta and to find the angle we just
need to find where the max peak is which
is around sine theta of 0.5 or an angle
of 30 degrees and that's exactly where
the object is
and since we're dealing with iq signals
which means a real and imaginary
component we can determine whether the
angle is positive or if it's negative
and this is analogous to what we did in
the last video where we were only able
to tell velocity direction when we had
two iq signals if we only had a real
stage on this radar then we would only
be able to tell the magnitude of the
angle and not the sign
all right so with all of this where does
resolution come in
well take note of how narrow this peak
is with eight antennas
now as i remove antennas from the array
the peak stays in the same spot
indicating that the angle hasn't changed
but it gets wider with fewer antennas
and this wider peak is analogous to a
wider receive antenna pattern
and with a wide receive pattern you
can't distinguish between two close
objects it'll look just like one at
least compared to the narrower peak
so we really want these narrow peaks in
the fft in order to get better
resolution
let me show you what this looks like in
this example i have two objects that are
separated by about 30 degrees or so the
above plot shows the angle fft of a
radar system that uses 16 receive
antennas and it has these narrower peaks
the lower plot only uses eight and has
wider peaks and you can see that in both
cases the two objects can be resolved
since there are two distinct peaks
but as i slowly move the right objects
closer to the other one we can see how
the array with eight antennas has a
lower resolution
right about
here the two peaks merge into a single
peak for the eight antenna case and this
happened right around a separation of 15
degrees or so
this means that with eight antennas the
angle between the two objects can't be
resolved as there's only a single peak
in the fft and so it looks just like a
single object
there are still however two distinct
peaks with the 16 antenna array due to
the higher resolution
in fact we can resolve objects down to
about 8 degrees or so
so increasing the number of antennas is
good because it improves our ability to
resolve objects that are closer to each
other
the downside though is that you need a
larger and more complex antenna array
but there is something that we can do
about this what we've talked about so
far is a radar system with a single
transmit antenna and multiple receive
antennas but there's a concept called
mimo radar or multi-input multi-output
with mimo we can have a system with
multiple transmit antennas and multiple
receive antennas hence the name
and there's a very clear benefit to
moving to a mimo radar and that's
because you can form a virtual array
with a virtual array you can get the
resolution benefit of having more
receive antennas but with fewer overall
antennas
instead of say having eight receive
antennas in one transmit
we could have two transmit antennas and
four receive and form a virtual array
that is equivalent
if the arrangement of these antennas is
done just right this system performs the
exact same with six antennas as a system
does with nine
and this savings increases with larger
arrays two transmit nate receive
antennas can perform with 10 the exact
same as one with 16 received antennas
and one transmit
so about half as many but you get the
benefit of those really narrow peaks in
the fft that improves resolution
and all of this is a really cool concept
and there's a ton that i'm leaving out
in these two videos but i hope you check
out the more complete resources below if
you want to find out exactly how all of
this stuff works plus i've left some
links to matlab examples where you can
see all of the stuff i just talked about
in action
but that's where i'm going to leave this
video for now
in the next video we're going to move
past measuring the parameters of an
object and start to talk about the power
and noise of the radar itself and
introduce the range radar equation so if
you don't want to miss that or any other
future tech talk video don't forget to
subscribe and you can check out my
channel control system lectures where
you can find more control theory topics
there as well
thanks for watching and i'll see you
next time