Hi.
I'm trying to figure out if the motion in this video is a physical embodiment of time-domain aliasing. (Watch the video for at least 60 seconds.)
It's an excellent demo of phase! Each string is slightly longer as it gets closer to the camera, so the frequency of oscillation of each ball is slightly different. It reminds me of dispersion in a plasma - low frequencies travel slower and after a half wave length a pulse looks like it has vanished into noise, but at a full wave length it comes back! This is the same thing in time. Very cool video, thanks for pointing it out.
I would say it is frequency domain aliasing linked to the fact that the number of balls is N= 15=3x5. The strings (or mass of the balls) have been adapted so that the frequencies of oscillations are in linear order for the k=1..15 (fortran) or in C or Python that I will prefer here, k=0..N-1 , N= 15 balls. F=k, phi= k* t in 2pi full turn unit.
At time t=7.5 (=15.0/2) the balls are at a phase phi = k x t = k x 7.5 -> phi= 0 for k=0, phi= 0.5 for k=1, 0.0 for k=2,.. well 0 for k even and 0.5 for k odd. So you get two straight lines.
At time t = 3.0 (=15/5): k=0 phi=0; k=1, phi=1/3.0; k=2, phi =2/3.0; k=3, phi = 3 modulo 1= 0; k=4, phi=4/3.0 = 1/3.0 ....you get 3 lines of five balls.
Similarly at time t=5.0 you get 5 lines of 3 balls.
And same (second half of the cycle) at t=15-3 =12, 3 lines of five balls
And at t=15-5=10 you get 3 lines of 5 balls.
And of course at + or - 1 of these 'echoes' you see a sinus or cosinus of one period or so in the 'frequency' (space) domain.
Note: I have been doing Magnetic Resonance Imaging for half my life, where a gradient of magnetic field creates a linear dependence of resonance frequency along on dimension in the space domain. This experiment is an illustration of that technique in case the object is composed of a regular lattice. AND I THINK THIS VIDEO SHOULD BE SAVED FOR TEACHING PURPOSE, as QUITE remarkable. It would also useful for music learners for N= 2x1, 3x2, 4x3, (3x5)..Stairway for heaven is also a good choice (for my age).
Hello precesseur, it looks like you have posted two very similar replies by mistake. Do you mind deleting one of them? Thanks!
I try to. But .. Maybe I am missing something: hitting "delete" on one of them (the last one say) does not avail to anything. Is it that I am doing that from France (although I have a very fast connection here). This is typical behavior of a non minimum phase system. The best guess/action in this case is to .." wait and see".. (so see you tomorrow).
Hi Rick,
It certainly is aliasing but above in the guy who plays the strings
Rick-
In that case would there be high frequencies we can't see in between balls ? Could that be tested by hanging some very light weight but stiff strings in between each ball ? For example a balsa wood stick, and see if it's swinging faster than the balls on either side ?
-Jeff
I'm baffled. I found an example of "time aliasing" on this site, but it was in relation to FIR filtering, which puts the time domain samples in the frequency domain and then back into the time domain (if I'm not mistaken). When that is undersampled in the frequency domain, it results in an undersampled time domain response.
But I challenge you to show me where there's the equivalent of a convolution in the video. Sure, the string length and Newtonian dynamics is kind of a filter if all the balls are the same weight, size, and never impinge on one another, but at no time does that system leave the time domain, right? That whole system should be relatively simple to emulate in a simulation. And the simulation can result in something similar to aliasing in this context, namely quantitization error. But once information (resolution) is gone, it's gone for good. It can't be undone. There may be ways to reconstruct the original from a truncated signal by knowing the resolution, but there's no way to prove that it will work in every case.
One of the most interesting things I have noticed in my previous work in ILS signal generation and monitoring, is that when one takes a long view of a large ensemble of ILS data measurements, the pattern of the time-to-time jitter resembles the ILS waveform itself when the measurement time is plotted on the time axis. (Background: an ILS signal is just a 90 and 150 Hz AM signal with each modulated by 20% at "center line". This results in a composite signal that has a 30 Hz rate). At first, that seemed strange, but then when one sees that this is a grossly time-shifted version of the radiated signal, it makes sense that this is the result of the ever so slight frequency difference (beat) between the radiated signal and the demodulated signal's sampling frequency whose timing accuracy are based on their 50 ppm crystal clocks' accuracy. I have seen this "image" appear when difference filtering the measurements from their ideal values.
The beat period is the difference between the two signals. When they are very close (and good 50 ppm crystals are much closer than that when first used to allow for aging and temperature drift), it may take thousands of measurements at about 250 msec each to see that pattern. And that's not time aliasing (or is it?).
What I see is different sine waves beating against each other so every {N1,N2,N3,....Nnumballs} oscillations all the balls come back to 0 deg phase offset. It strikes me as a nice demo of how residue numbers sort of work. A new form of modulation?
But the real genius is how the demonstrator got the string lengths so accurately set that after 3m 50s they still all come back to 0 deg phase offset. Now if someone could get him or her to show how it was done, that would be an experimental treat to see.
Lovelly demo.
I'm guessing that - after the initial calculations and string clamping to length - that this setup was tuned more like a harp than anything else. ...with much trial and error. I would think that it would be rather difficult to set the lengths of every string with that level of precision any other way. Swing. Watch. Adjust those whose frequency is off with a clamp or tuning key. Repeat. A strobe might help.
Yes to wonderful demo and effect.
Hi Rick,
Very cool video. Watched it twice.
Where is there relevant sampling being done to cause aliasing?
Dirk
To me, who tends to see things through DFT colored glasses, this looks like a representation of the real part of DFT basis vectors. Insofar as pendulum swings represent simple harmonic motion.
Think of the release as the start of a DFT frame. Now, when the balls all align again is the end of the frame. In that time they have all had a whole number of cycles, all different from each other. Since they start at an extreme, they are like the real part, the cosine.
There's a well known formula for the period of a pendulum, so calculating the string length shouldn't be hard. Probably needs some "real world" calibration.
That someone would think of that and work it out is very impressive. Thumbs up, I'll be sharing this with my pool league buddies.
Ced
I saw a live demo of such a huge wave pendulum at a light festival. Something like this:
Hi friedman. Thanks for sharing that video with us!
