skyeriding
900
I believe Nyquist-Shannon is not applicable here, as that context implies "measurement" being sampling and reproducing frequency data
Apr 10, 2022,04:42 AM
The context of "measurement" in Nyquist-Shannon theory is applicable if you're using a sampling signal, and want to perfectly recreate the sampled signal. If you have an audio track going up to 22kHz (ignoring any higher freqs), measuring the amplitude of the measured signal (sampling) at every 1/44100s (44.1khz sampling) can perfectly recreate the information of the audio track up to 22kHz (i.e. the frequency domain info are all captured perfectly up to 22kHz). If I have an oscillating pendulum and use a high-speed camera to capture its amplitude 100 times a second, I can perfectly simulate its motion of its frequency components up to 50Hz in the frequency domain.
The "measurement" in this context is simpler - it is simply measuring one instance of time duration, and not amplitude of a measured "signal/input". We are just taking one time sample/observation, whenever we stop the chronograph. The watch is designed to beat at 5Hz, or 10 beats a second (due to the back and forth swing). The number of teeth of the intermediate wheels of the geartrain does not matter - they simply serve the role of gear reduction purposes, with the only exception being the teeth "resolution" of the engagement of the lateral clutch of the chronograph (which adds extra uncertainty to the accuracy of measurement, but can be put aside for now for the remaining argument). Assuming the watch has a perfect beatrate of +-0s/d, it serves as a perfectly usable device to measure time up to the nearest 0.1 seconds. If I had a ruler with hashmarks going down to 1mm, I can be confident my resolution of measurement (not counting user error) is accurate up to the next 1mm - a 2.06cm measurement will round up to 2.1cm, while a 2.03cm measurement will round down to 2.0cm. It is similar here - the watch ticks away at 5Hz and provides hashmarks for 0.1second intervals - when you stop the chronograph, measurement rounds up/down to the nearest 0.1second.
Forcing the context of "measurement" with Nyquist-Shannon using a watch is saying that, I have a 5Hz chronograph and I want to sample the amplitude measurement of an oscillating pendulum, recreating its motion by determining its frequency components in the frequency domain. If at every 0.1second (10Hz sampling using a 36000bph watch) I measure the pendulum's amplitude (which, the watch is the wrong tool/context for the job - the chronograph measures only one instance of time, it doesn't measure the pendulum's amplitude continuously - for that I'd be using a camera instead), I can perfectly recreate/simulate that pendulum's motion/oscillation on a computer up to a frequency range of 5Hz (because I'm sampling amplitudes at 10Hz, and Nyquist-Shannon dictates I can recreate the frequency information up to 5Hz). I lose any information of higher oscillation frequency components above 5Hz in my recreated simulation. A more technically correct advertisement analogy would be "This is my high-resolution DSLR camera that captures 10 frames a second - it can sample oscillating pendulum amplitudes and thus you can recreate its waveform information up to 5Hz".
Thus, the claim that the Patek 5470P, the Zeniths, or any chronograph with a 5hz balance wheel can measure up to 1/10th of a second to me is perfectly reasonable, since the watches on paper are designed to accurately measure 36000 beats per hour. Of course, there's many sources of errors that can be introduced - the human reaction time and trigger of the stop pusher, the engagement of teeth of a lateral clutch chronograph, an imperfect beatrate of the watch, etc., but this doesn't take away the fact I can use the above chronographs to measure up to 1/10th of a second, or a ruler with 1mm gradations to measure up to the nearest 1mm. All these additional sources of errors start to make chronographs with even faster beatrates and higher resolutions (1/100th, 1/1000th s) a purely horological/theoretical exercise, since even the human error alone vastly outweighs the accuracy resolution of those devices.
That said, I do find the print of "1/10 SECOND" on the dial completely unnecessary - the dial doesn't have to constantly remind owners what it can do...
Regards,
skyeriding