To sidestep the math, here's a graphing exercise that seems to work for my students: Draw a sine wave. Pick off N equally spaced samples per cycle, then try to reconstruct by eye the original sine from the samples. If N exceeds 2, you can do it. If N = 2, you can't (consider if the samples all happen to be at the zero crossings). If N is less than two, you can reconstruct a sine, but it will be at a lower frequency than the original waveform -- we call that aliasing, since the lower-frequency reconstruction is falsely passing itself off as the original (we exploit this in sampling scopes).
The reconstruction problem is really nothing more than an interpolation problem: "What is the ideal way to interpolate between the samples?" That's where the math comes in. Once you connect the math to filter design, you find that the ideal interpolator is a filter with constant transmission up to half the sampling frequency, and then infinite attenuation above it. The finite rolloff characteristics of practically realizable filters forces the use of N significantly in excess of 2.
-- Tom
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Prof. Thomas H. Lee
Allen Ctr., Rm. 205
350 Jane Stanford Way
Stanford University
Stanford, CA 94305-4070
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On 11/22/2021 08:52, Jeff Dutky wrote:
(not that I really understand all the math)
a very simplistic analysis says that you would not be able to resolve a sine wave from discrete samples unless you had at least four (very well placed) samples per cycle. I would expect (again, not based on any real mathematical understanding) that you would really need more like ten samples per cycle at your maximum frequency.
I have heard (but, again, don't understand the math) that you can use some kind of interpolation (sine x over x?) to get better looking waveforms than you really have sampled data for, but I also understand that this is a relatively recent innovation on DSOs (i.e. within the past 20 years).
-- Jeff Dutky
