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Tuesday, July 18, 2006

I can't drive 55!

When sampling a signal, the Nyquist limit determines the maximum frequency of the bandwidth of the sampled signal that can be uniquely restored. If you are outside the Nyquist limit, the signal gets aliased back into sampled signal, leading to heartache and woe.



In the included movie (above. If you can't get it, here's an alternate link), the clock face is sampled at a (virtual) rate of 55 minutes per sample. Since it takes the minute hand one hour to complete a revolution, to represent it we should be sampling the signal at 30 minute intervals at a minimum. Because we don't, the minute hand gets visually aliased and it looks like it is moving backwards at a much slower rate. The minute hand frequency is aliased such that it looks to be almost the same frequency as the hour hand!

The hour hand, however, takes 12 hours to complete a full revolution, so sampling it at 55 minutes per sample is well under the Nyquist limit (6 hours per sample) needed to uniquely represent it. By sampling at 55 minute intervals, the minute hand is aliased to a rotational frequency very near that of the hour hand, making them appear to rotate at the same rate in the video.

The toy cars actually do represent a related topic. Because they are not "blurred" (something stop-frame animation lacks), we can tell that the capture pulse is a delta function relative to their motion. If there were blurring, as we see with regular movie camera type filming, we'd know that the size (in time) of the capture window was on the order of apparent angular motion relative to the lense.

There's a lot of math I could go into at this point, but it's already covered reasonably well in the links.

Burton MacKenZie /

Credits: The video was produced with the help of The GIMP, VirtualDub, Audacity and TMPGEnc; fine products, all.

Addendum: There are at least two problems with this first draft short. Will be fixed in sequels/directors cuts or whenever I get the time. (i.e. never)

11 comments:

Derek said...

You've captured some anomalous duct-tape jitter...

Derek said...

Would be neat if a window was visible, then you could see the motion of the sun too.

burton mackenzie said...

Actually, I shot all the frames at low artificial lighting conditions (and did half second exposures to compensate) specifically so I wouldn't see the motion of the light from the sun (and also because I couldn't control the way the light was reflecting brightly off of the clock face and swamping the image).

I've had a computer capturing frames of some street construction for about seven weeks now. When I make that into a video you'll see lots of light motion. I ponder how long it's going to be, as so far I have four data DVDs full of the construction images, and they're not finished. It's definitely going to be full video frame rate! (30/29.97fps)

andrea said...

would you not be able to create blur (in the car) if it were moving when you took the stop action shot for each frame? Difficult in practice (having it moving AT the exact spot where the shot will be taken) but would that achieve the blur? Try it, would ya?

burton mackenzie said...

Thanks for the suggestion, Andrea. I'll get right on it, but I have to wait for my order of Round Tuits to arrive, first. I need them before I can start on that effect.

Actually, I'm not prepared to cover for the audience the mathematics involved in using a non-delta sample function. It's much easier this way, and demonstrates more.

Derek Bair said...

Very cool!

geyser545 said...

Hey can you post the name of this house tune?

burton mackenzie said...

white album

zxber said...

I am discombobulated.

How many times per hour do you take a picture to get that effect.

Thanks

burton mackenzie said...

@zxber: one shot every 55 minutes. The sampling period of 55 minutes causes the aliasing of one rotational frequency (the minute hand), but not of the other (the hour hand). The aliased frequency of the minute hand is close in magnitude to that of the hour hand, so they appear to rotate at approximately the same rate. (Just in opposite directions)

Poetry said...

I've been spending the last few hours pouring over various formulas in relation to Nyquist rates and Sampling Theorems, this little video was a really cool break!