Now you're cutting and pasting random quotes and pictures from Nyquist sampling theory, but I'm waiting for an Ebay snipe to trigger, so I'll play:
I actually have three degrees A Bachelor's and Master's in Electrical Engineering, and a second Master's in Computer Science. But let's get back to CNC interpolation...Originally Posted by Evan
This quote is correct, but taken somewhat out of context. Nyquist sampling theory applies to digitally sampling any analog signal. It's a rule applied to digital signal processing, including audio and graphics.Originally Posted by Evan
Unfortunately, cutting a curve with a CNC toolpath is not sampling an analog signal, and therefore has nothing to do with Nyquist sampling theory.
You don't understand how Fourier Analysis works, or applies here. A single component signal, like the pure sine wave I proposed earlier in the thread, and which you have copied in your post from some random Nyquist tutorial, is a single term function, and not an infinite series.Originally Posted by Evan
An infinite series occurs when you're sampling a complex signal with an infinite number of frequency components. Fourier analysis breaks each frequency component into a separate sine wave, such that the sum of all the sine waves == the original signal.
If you have a mathematical discontinuity, such as a square wave signal, it takes an infinite series of frequency components to represent. But the reason I picked the pure sine wave for your challenge, and the reason it's in the Nyquist diagram you copied, is because a pure sine wave has a single frequency component, and therefore no Fourier analysis is necessary:
Originally Posted by lazloThat's not what that Nyquist diagram is showing Evan. That's the classic undergraduate Nyquist sampling diagram that I described to you in this post:Originally Posted by Evan
Which is exactly the example in the section of the Nyquist diagram you snatched. That's not a coincidence by the way, this is the same exact example you use to teach Nyquist to undergraduates:Originally Posted by Lazlo
2 Hz Sine Wave with 2 Hz Sampling:
And the third picture is with ideal Nyquist sampling: a 2 Hz pure sine wave with 4 Hz (Nyquist) sampling frequency, and the since wave is captured perfectly:
2 Hz Sine Wave with 4 Hz Sampling:
This is also the exact sine wave in the challenge I proposed to you to show you why Nyquist Sampling theory has nothing to do with CNC toolpaths: Take that 2 Hz Sine wave, and make a 2.5D or 3D model of it. Nyquist says, and your diagram shows, 2 sample points per cycle is necessary and sufficient to describe the sine wave. Now cut that sine wave on a CNC toolpath.
Despite the fact that's it's a pure sine wave, with a single frequency term, it will still take an infinite number of interpolated cutting steps to cut a smooth version of it.
So if you really understand Nyquist Sampling Theory, it will take you 10 minutes to generate that sine wave in CAD and create a GCode file of it. It will be immediately apparently to you that it will take an infinite number of vectors to cut a smooth version of it, as Swarf&Sparks aptly pointed out.Originally Posted by lazlo