Paul Tagliamonte: Writing a simulator to check phased array beamforming
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While working on hz.tools, I started to move my beamforming
code from 2-D (meaning, beamforming to some specific angle on the X-Y plane for
waves on the X-Y plane) to 3-D. I ll have more to say about that once I get
around to publishing the code as soon as I m sure it s not completely wrong,
but in the meantime I decided to write a simple simulator to visually
check the beamformer against the textbooks. The results were pretty rad,
so I figured I d throw together a post since it s interesting all on its
own outside of beamforming as a general topic.
I figured I d write this in Rust, since I ve been using Rust as my primary
language over at zoo, and it s a good chance
to learn the language better.
hz.tools will be tagged
#hztools.
If you're on the Fediverse, I'd very much appreciate boosts on my toot!
This post has some large GIFs
It make take a little bit to load depending on your internet connection. Sorry about that, I'm not clever enough to do better without doing tons of complex engineering work. They may be choppy while they load or something. I tried to compress an ensmall them, so if they're loaded but fuzzy, click on them to load a slightly larger version.
This post won t cover the basics of how
phased arrays work
or the specifics of calculating the phase offsets for each antenna,
but I ll dig into how I wrote a simple simulator and how I wound up
checking my phase offsets to generate the renders below.
It make take a little bit to load depending on your internet connection. Sorry about that, I'm not clever enough to do better without doing tons of complex engineering work. They may be choppy while they load or something. I tried to compress an ensmall them, so if they're loaded but fuzzy, click on them to load a slightly larger version.
Assumptions
I didn t want to build a general purpose RF simulator, anything particularly
generic, or something that would solve for any more than the things right
in front of me. To do this as simply (and quickly all this code took about
a day to write, including the beamforming math) I had to reduce the
amount of work in front of me.
Given that I was concerend with visualizing what the antenna pattern would look
like in 3-D given some antenna geometry, operating frequency and configured
beam, I made the following assumptions:
All anetnnas are perfectly isotropic they receive a signal that is
exactly the same strength no matter what direction the signal originates
from.
There s a single point-source isotropic emitter in the far-field (I modeled
this as being 1 million meters away 1000 kilometers) of the antenna system.
There is no noise, multipath, loss or distortion in the signal as it travels
through space.
Antennas will never interfere with each other.
2-D Polar Plots
The last time I wrote something like this,
I generated 2-D GIFs which show a radiation pattern, not unlike the
polar plots you d see on a microphone.
These are handy because it lets you visualize what the directionality of
the antenna looks like, as well as in what direction emissions are
captured, and in what directions emissions are nulled out. You can see
these plots
on spec sheets for antennas in both 2-D and 3-D form.
Now, let s port the 2-D approach to 3-D and see how well it works out.
Writing the 3-D simulator
As an EM wave travels through free space, the place at which you sample
the wave controls that phase you observe at each time-step. This means,
assuming perfectly synchronized clocks, a transmitter and receiver exactly
one RF wavelength apart will observe a signal in-phase, but a transmitter
and receiver a half wavelength apart will observe a signal 180 degrees
out of phase.
This means that if we take the distance between our point-source and
antenna element, divide it by the wavelength, we can use the fractional
part of the resulting number to determine the phase observed. If we
multiply that number (in the range of 0 to just under 1) by
tau, we can generate a complex number by taking the
cos and sin of the multiplied phase (in the range of 0 to tau), assuming
the transmitter is emitting a carrier wave at a static amplitude and all
clocks are in perfect sync.
let observed_phases: Vec<Complex> = antennas
.iter()
.map( antenna
let distance = (antenna - tx).magnitude();
let distance = distance - (distance as i64 as f64);
((distance / wavelength) * TAU)
)
.map( phase Complex(phase.cos(), phase.sin()))
.collect();
At this point, given some synthetic transmission point and each antenna, we
know what the expected complex sample would be at each antenna. At this point,
we can adjust the phase of each antenna according to the beamforming
phase offset configuration, and add up every sample in order to determine
what the entire system would collectively produce a sample as.
let beamformed_phases: Vec<Complex> = ...;
let magnitude = beamformed_phases
.iter()
.zip(observed_phases.iter())
.map( (beamformed, observed) observed * beamformed)
.reduce( acc, el acc + el)
.unwrap()
.abs();
Armed with this information, it s straight forward to generate some number
of (Azimuth, Elevation) points to sample, generate a transmission point
far away in that direction, resolve what the resulting Complex sample would be,
take its magnitude, and use that to create an (x, y, z) point at
(azimuth, elevation, magnitude). The color attached two that point is
based on its distance from (0, 0, 0). I opted to use the
Life Aquatic
table for this one.
After this process is complete, I have a
point cloud of
((x, y, z), (r, g, b)) points. I wrote a small program using
kiss3d to render point cloud using tons of
small spheres, and write out the frames to a set of PNGs, which get compiled
into a GIF.
Now for the fun part, let s take a look at some radiation patterns!
1x4 Phased Array
The first configuration is a phased array where all the elements are
in perfect alignment on the y and z axis, and separated by some
offset in the x axis. This configuration can sweep 180 degrees (not
the full 360), but can t be steared in elevation at all.
Let s take a look at what this looks like for a well constructed
1x4 phased array:
And now let s take a look at the renders as we play with the configuration of
this array and make sure things look right. Our initial quarter-wavelength
spacing is very effective and has some outstanding performance characteristics.
Let s check to see that everything looks right as a first test.
Nice. Looks perfect. When pointing forward at (0, 0), we d expect to see a
torus, which we do. As we sweep between 0 and 360, astute observers will notice
the pattern is mirrored along the axis of the antennas, when the beam is facing
forward to 0 degrees, it ll also receive at 180 degrees just as strong. There s
a small sidelobe that forms when it s configured along the array, but
it also becomes the most directional, and the sidelobes remain fairly small.
Long compared to the wavelength (1 )
Let s try again, but rather than spacing each antenna of a wavelength
apart, let s see about spacing each antenna 1 of a wavelength apart instead.
The main lobe is a lot more narrow (not a bad thing!), but some significant
sidelobes have formed (not ideal). This can cause a lot of confusion when doing
things that require a lot of directional resolution unless they re compensated
for.
Going from ( to 5 )
The last model begs the question - what do things look like when you separate
the antennas from each other but without moving the beam? Let s simulate moving
our antennas but not adjusting the configured beam or operating frequency.
Very cool. As the spacing becomes longer in relation to the operating frequency,
we can see the sidelobes start to form out of the end of the antenna system.
2x2 Phased Array
The second configuration I want to try is a phased array where the elements
are in perfect alignment on the z axis, and separated by a fixed offset
in either the x or y axis by their neighbor, forming a square when
viewed along the x/y axis.
Let s take a look at what this looks like for a well constructed
2x2 phased array:
Let s do the same as above and take a look at the renders as we play with the
configuration of this array and see what things look like. This configuration
should suppress the sidelobes and give us good performance, and even give us
some amount of control in elevation while we re at it.
Sweet. Heck yeah. The array is quite directional in the configured direction,
and can even sweep a little bit in elevation, a definite improvement
from the 1x4 above.
Long compared to the wavelength (1 )
Let s do the same thing as the 1x4 and take a look at what happens when the
distance between elements is long compared to the frequency of operation
say, 1 of a wavelength apart? What happens to the sidelobes given this
spacing when the frequency of operation is much different than the physical
geometry?
Mesmerising. This is my favorate render. The sidelobes are very fun to
watch come in and out of existence. It looks absolutely other-worldly.
Going from ( to 5 )
Finally, for completeness' sake, what do things look like when you separate the
antennas from each other just as we did with the 1x4? Let s simulate moving our
antennas but not adjusting the configured beam or operating frequency.
Very very cool. The sidelobes wind up turning the very blobby cardioid into
an electromagnetic dog toy. I think we ve proven to ourselves that using
a phased array much outside its designed frequency of operation seems like
a real bad idea.
Future Work
Now that I have a system to test things out, I m a bit more confident that
my beamforming code is close to right! I d love to push that code over
the line and blog about it, since it s a really interesting topic on its own.
Once I m sure the code involved isn t full of lies, I ll put it up on the
hztools org, and post about it here and on
mastodon.
Writing the 3-D simulator
As an EM wave travels through free space, the place at which you sample
the wave controls that phase you observe at each time-step. This means,
assuming perfectly synchronized clocks, a transmitter and receiver exactly
one RF wavelength apart will observe a signal in-phase, but a transmitter
and receiver a half wavelength apart will observe a signal 180 degrees
out of phase.
This means that if we take the distance between our point-source and
antenna element, divide it by the wavelength, we can use the fractional
part of the resulting number to determine the phase observed. If we
multiply that number (in the range of 0 to just under 1) by
tau, we can generate a complex number by taking the
cos and sin of the multiplied phase (in the range of 0 to tau), assuming
the transmitter is emitting a carrier wave at a static amplitude and all
clocks are in perfect sync.
let observed_phases: Vec<Complex> = antennas
.iter()
.map( antenna
let distance = (antenna - tx).magnitude();
let distance = distance - (distance as i64 as f64);
((distance / wavelength) * TAU)
)
.map( phase Complex(phase.cos(), phase.sin()))
.collect();
At this point, given some synthetic transmission point and each antenna, we
know what the expected complex sample would be at each antenna. At this point,
we can adjust the phase of each antenna according to the beamforming
phase offset configuration, and add up every sample in order to determine
what the entire system would collectively produce a sample as.
let beamformed_phases: Vec<Complex> = ...;
let magnitude = beamformed_phases
.iter()
.zip(observed_phases.iter())
.map( (beamformed, observed) observed * beamformed)
.reduce( acc, el acc + el)
.unwrap()
.abs();
Armed with this information, it s straight forward to generate some number
of (Azimuth, Elevation) points to sample, generate a transmission point
far away in that direction, resolve what the resulting Complex sample would be,
take its magnitude, and use that to create an (x, y, z) point at
(azimuth, elevation, magnitude). The color attached two that point is
based on its distance from (0, 0, 0). I opted to use the
Life Aquatic
table for this one.
After this process is complete, I have a
point cloud of
((x, y, z), (r, g, b)) points. I wrote a small program using
kiss3d to render point cloud using tons of
small spheres, and write out the frames to a set of PNGs, which get compiled
into a GIF.
Now for the fun part, let s take a look at some radiation patterns!
1x4 Phased Array
The first configuration is a phased array where all the elements are
in perfect alignment on the y and z axis, and separated by some
offset in the x axis. This configuration can sweep 180 degrees (not
the full 360), but can t be steared in elevation at all.
Let s take a look at what this looks like for a well constructed
1x4 phased array:
And now let s take a look at the renders as we play with the configuration of
this array and make sure things look right. Our initial quarter-wavelength
spacing is very effective and has some outstanding performance characteristics.
Let s check to see that everything looks right as a first test.
Nice. Looks perfect. When pointing forward at (0, 0), we d expect to see a
torus, which we do. As we sweep between 0 and 360, astute observers will notice
the pattern is mirrored along the axis of the antennas, when the beam is facing
forward to 0 degrees, it ll also receive at 180 degrees just as strong. There s
a small sidelobe that forms when it s configured along the array, but
it also becomes the most directional, and the sidelobes remain fairly small.
Long compared to the wavelength (1 )
Let s try again, but rather than spacing each antenna of a wavelength
apart, let s see about spacing each antenna 1 of a wavelength apart instead.
The main lobe is a lot more narrow (not a bad thing!), but some significant
sidelobes have formed (not ideal). This can cause a lot of confusion when doing
things that require a lot of directional resolution unless they re compensated
for.
Going from ( to 5 )
The last model begs the question - what do things look like when you separate
the antennas from each other but without moving the beam? Let s simulate moving
our antennas but not adjusting the configured beam or operating frequency.
Very cool. As the spacing becomes longer in relation to the operating frequency,
we can see the sidelobes start to form out of the end of the antenna system.
2x2 Phased Array
The second configuration I want to try is a phased array where the elements
are in perfect alignment on the z axis, and separated by a fixed offset
in either the x or y axis by their neighbor, forming a square when
viewed along the x/y axis.
Let s take a look at what this looks like for a well constructed
2x2 phased array:
Let s do the same as above and take a look at the renders as we play with the
configuration of this array and see what things look like. This configuration
should suppress the sidelobes and give us good performance, and even give us
some amount of control in elevation while we re at it.
Sweet. Heck yeah. The array is quite directional in the configured direction,
and can even sweep a little bit in elevation, a definite improvement
from the 1x4 above.
Long compared to the wavelength (1 )
Let s do the same thing as the 1x4 and take a look at what happens when the
distance between elements is long compared to the frequency of operation
say, 1 of a wavelength apart? What happens to the sidelobes given this
spacing when the frequency of operation is much different than the physical
geometry?
Mesmerising. This is my favorate render. The sidelobes are very fun to
watch come in and out of existence. It looks absolutely other-worldly.
Going from ( to 5 )
Finally, for completeness' sake, what do things look like when you separate the
antennas from each other just as we did with the 1x4? Let s simulate moving our
antennas but not adjusting the configured beam or operating frequency.
Very very cool. The sidelobes wind up turning the very blobby cardioid into
an electromagnetic dog toy. I think we ve proven to ourselves that using
a phased array much outside its designed frequency of operation seems like
a real bad idea.
Future Work
Now that I have a system to test things out, I m a bit more confident that
my beamforming code is close to right! I d love to push that code over
the line and blog about it, since it s a really interesting topic on its own.
Once I m sure the code involved isn t full of lies, I ll put it up on the
hztools org, and post about it here and on
mastodon.
let observed_phases: Vec<Complex> = antennas
.iter()
.map( antenna
let distance = (antenna - tx).magnitude();
let distance = distance - (distance as i64 as f64);
((distance / wavelength) * TAU)
)
.map( phase Complex(phase.cos(), phase.sin()))
.collect();
let beamformed_phases: Vec<Complex> = ...;
let magnitude = beamformed_phases
.iter()
.zip(observed_phases.iter())
.map( (beamformed, observed) observed * beamformed)
.reduce( acc, el acc + el)
.unwrap()
.abs();
y and z axis, and separated by some
offset in the x axis. This configuration can sweep 180 degrees (not
the full 360), but can t be steared in elevation at all.
Let s take a look at what this looks like for a well constructed
1x4 phased array:
And now let s take a look at the renders as we play with the configuration of
this array and make sure things look right. Our initial quarter-wavelength
spacing is very effective and has some outstanding performance characteristics.
Let s check to see that everything looks right as a first test.
Nice. Looks perfect. When pointing forward at (0, 0), we d expect to see a
torus, which we do. As we sweep between 0 and 360, astute observers will notice
the pattern is mirrored along the axis of the antennas, when the beam is facing
forward to 0 degrees, it ll also receive at 180 degrees just as strong. There s
a small sidelobe that forms when it s configured along the array, but
it also becomes the most directional, and the sidelobes remain fairly small.
Long compared to the wavelength (1 )
Let s try again, but rather than spacing each antenna of a wavelength
apart, let s see about spacing each antenna 1 of a wavelength apart instead.
The main lobe is a lot more narrow (not a bad thing!), but some significant
sidelobes have formed (not ideal). This can cause a lot of confusion when doing
things that require a lot of directional resolution unless they re compensated
for.
Going from ( to 5 )
The last model begs the question - what do things look like when you separate
the antennas from each other but without moving the beam? Let s simulate moving
our antennas but not adjusting the configured beam or operating frequency.
Very cool. As the spacing becomes longer in relation to the operating frequency,
we can see the sidelobes start to form out of the end of the antenna system.
2x2 Phased Array
The second configuration I want to try is a phased array where the elements
are in perfect alignment on the z axis, and separated by a fixed offset
in either the x or y axis by their neighbor, forming a square when
viewed along the x/y axis.
Let s take a look at what this looks like for a well constructed
2x2 phased array:
Let s do the same as above and take a look at the renders as we play with the
configuration of this array and see what things look like. This configuration
should suppress the sidelobes and give us good performance, and even give us
some amount of control in elevation while we re at it.
Sweet. Heck yeah. The array is quite directional in the configured direction,
and can even sweep a little bit in elevation, a definite improvement
from the 1x4 above.
Long compared to the wavelength (1 )
Let s do the same thing as the 1x4 and take a look at what happens when the
distance between elements is long compared to the frequency of operation
say, 1 of a wavelength apart? What happens to the sidelobes given this
spacing when the frequency of operation is much different than the physical
geometry?
Mesmerising. This is my favorate render. The sidelobes are very fun to
watch come in and out of existence. It looks absolutely other-worldly.
Going from ( to 5 )
Finally, for completeness' sake, what do things look like when you separate the
antennas from each other just as we did with the 1x4? Let s simulate moving our
antennas but not adjusting the configured beam or operating frequency.
Very very cool. The sidelobes wind up turning the very blobby cardioid into
an electromagnetic dog toy. I think we ve proven to ourselves that using
a phased array much outside its designed frequency of operation seems like
a real bad idea.
Future Work
Now that I have a system to test things out, I m a bit more confident that
my beamforming code is close to right! I d love to push that code over
the line and blog about it, since it s a really interesting topic on its own.
Once I m sure the code involved isn t full of lies, I ll put it up on the
hztools org, and post about it here and on
mastodon.
Very cool. As the spacing becomes longer in relation to the operating frequency,
we can see the sidelobes start to form out of the end of the antenna system.
2x2 Phased Array
The second configuration I want to try is a phased array where the elements
are in perfect alignment on the z axis, and separated by a fixed offset
in either the x or y axis by their neighbor, forming a square when
viewed along the x/y axis.
Let s take a look at what this looks like for a well constructed
2x2 phased array:
Let s do the same as above and take a look at the renders as we play with the
configuration of this array and see what things look like. This configuration
should suppress the sidelobes and give us good performance, and even give us
some amount of control in elevation while we re at it.
Sweet. Heck yeah. The array is quite directional in the configured direction,
and can even sweep a little bit in elevation, a definite improvement
from the 1x4 above.
Long compared to the wavelength (1 )
Let s do the same thing as the 1x4 and take a look at what happens when the
distance between elements is long compared to the frequency of operation
say, 1 of a wavelength apart? What happens to the sidelobes given this
spacing when the frequency of operation is much different than the physical
geometry?
Mesmerising. This is my favorate render. The sidelobes are very fun to
watch come in and out of existence. It looks absolutely other-worldly.
Going from ( to 5 )
Finally, for completeness' sake, what do things look like when you separate the
antennas from each other just as we did with the 1x4? Let s simulate moving our
antennas but not adjusting the configured beam or operating frequency.
Very very cool. The sidelobes wind up turning the very blobby cardioid into
an electromagnetic dog toy. I think we ve proven to ourselves that using
a phased array much outside its designed frequency of operation seems like
a real bad idea.
Future Work
Now that I have a system to test things out, I m a bit more confident that
my beamforming code is close to right! I d love to push that code over
the line and blog about it, since it s a really interesting topic on its own.
Once I m sure the code involved isn t full of lies, I ll put it up on the
hztools org, and post about it here and on
mastodon.
Mesmerising. This is my favorate render. The sidelobes are very fun to
watch come in and out of existence. It looks absolutely other-worldly.
A new maintenance release 0.4.22 of 
Probably a trait of my family s origins as migrants from East Europe, probably
part of the collective trauma of jews throughout the world or probably
because that s just who I turned out to be, I hold in high regard the
preservation of memory of my family s photos, movies and such items. And it s a
trait shared by many people in my familiar group.
Shortly after my grandmother died 24 years ago, my mother did a large, loving
work of digitalization and restoration of my grandparent s photos. Sadly, the
higher resolution copies of said photos is lost but she took the work of not
just scanning the photos, but assembling them in presentations, telling a story,
introducing my older relatives, many of them missing 40 or more years before my
birth.
But said presentations were built using Flash. Right, not my choice of tool, and
I told her back in the day but given I wasn t around to do the work in what
I d chosen (a standards-abiding format, naturally), and given my graphic design
skills are nonexistant Several years ago, when Adobe pulled the plug on the
Flash format, we realized they would no longer be accessible. I managed to get
the photos out of the preentations, but lost the narration, that is a great part
of the work.
Three days ago, however, I read a post on 
In 2022 I read a post on the fediverse by somebody who mentioned that
they had bought on a whim a cute tiny book years ago, and that it
had been a companion through hard times. Right now I can t find the
post, but it was pretty aaaaawwww.
At the same time, I had discovered Coptic binding, and I wanted to do
some exercise to let my hands learn it, but apparently there is a limit
to the number of notebooks and sketchbooks a person needs (I m not 100%
sure I actually believe this, but I ve heard it is a thing).
So I decided to start making minibooks with the intent to give them
away: I settled (mostly) on the A8 size, and used a combination of found
materials, leftovers from bigger projects and things I had in the Stash.
As for paper, I ve used a variety of the ones I have that are at the
very least good enough for non-problematic fountain pen inks.
Thanks to the small size, and the way coptic binding works, I ve been
able to play around with the covers, experimenting with different styles
beyond the classic bookbinding cloth / paper covered cardboard,
including adding lace, covering food box cardboard with gesso and
decorating it with acrylic paints, embossing designs by gluing together
two layers of cardboard, one of which has holes, making covers
completely out of cernit, etc. Some of these I will probably also use in
future full-scale projects, but it s nice to find out what works and
what doesn t on a small scale.
Now, after a year of sporadically making these I have to say that the
making went quite well: I enjoyed the making and the creativity in
making different covers. The giving away was a bit more problematic, as
I didn t really have a lot of chances to do so, so I believe I still
have most of them. In 2024 I ll try to look for more opportunities (and
if you live nearby and want one or a few feel free to ask!)
Like each month, have a look at the work funded by 
I think while developing Wayland-as-an-ecosystem we are now entrenched into narrow concepts of how a desktop should work. While discussing Wayland protocol additions, a lot of concepts clash, people from different desktops with different design philosophies debate the merits of those over and over again never reaching any conclusion (just as you will never get an answer out of humans whether sushi or pizza is the clearly superior food, or whether CSD or SSD is better). Some people want to use Wayland as a vehicle to force applications to submit to their desktop s design philosophies, others prefer the smallest and leanest protocol possible, other developers want the most elegant behavior possible. To be clear, I think those are all very valid approaches.
But this also creates problems: By switching to Wayland compositors, we are already forcing a lot of porting work onto toolkit developers and application developers. This is annoying, but just work that has to be done. It becomes frustrating though if Wayland provides toolkits with absolutely no way to reach their goal in any reasonable way. For Nate s Photoshop analogy: Of course Linux does not break Photoshop, it is Adobe s responsibility to port it. But what if Linux was missing a crucial syscall that Photoshop needed for proper functionality and Adobe couldn t port it without that? In that case it becomes much less clear on who is to blame for Photoshop not being available.
A lot of Wayland protocol work is focused on the environment and design, while applications and work to port them often is considered less. I think this happens because the overlap between application developers and developers of the desktop environments is not necessarily large, and the overlap with people willing to engage with Wayland upstream is even smaller. The combination of Windows developers porting apps to Linux and having involvement with toolkits or Wayland is pretty much nonexistent. So they have less of a voice.
I will also bring my two protocol MRs to their conclusion for sure, because as application developers we need clarity on what the platform (either all desktops or even just a few) supports and will or will not support in future. And the only way to get something good done is by contribution and friendly discussion.
I ve mentioned this in a
The Rcpp Core Team is once again thrilled to announce a new release
1.0.12 of the