3

I had this question in my mind for a long time but I was not sure if it makes sense to anyone.I would appreciate your valuable thoughts on this question.

  1. How to identify points of discontinuity of a function $f :\mathbb{R} \to \mathbb{R}$ given samples of the function (obtained by using Nyquist sampling or any other sampling technique with sampling frequency being as high as desired.)

  2. Is there any other alternate way of identifying points of discontinuity without evaluating the limit ?

EDIT 1: functions with a discontinuity are not strictly band-limited. But if we still go ahead by neglecting frequencies higher than certain limit, meaning bandlimiting, we observe the Gibb's phenomenon.

Rajesh D
  • 4,247
  • There is a close relationship between the degree of differentiability of a function and how quickly its Fourier coefficients decay at infinity; see this previous question. However, since a sampled function throws away information about the Fourier coefficients at higher frequencies, this does not help for your question. –  Mar 22 '11 at 05:37
  • @Rahul Narain : the second part of the question is not really about sampled function but the actual function. – Rajesh D Mar 22 '11 at 06:06

1 Answers1

3

No sampling method that leaves the samples greater than a certain distance apart can identify points of discontinuity unless you are given more information about the function. The simplest example is the step function:

$H(x) = \begin{cases} 0 \text{ if } x \le 0 \\ 1 \text{ if } x \gt 0 \end {cases}$

Unless you have samples approaching $x=0$ from above arbitrarily closely, you won't be able to tell this from a ramp

$s(x) = \begin{cases} 0 \text{ if } x \le 0 \\ 1/\delta \text { if } 0 \lt x \le \delta \\ 1 \text{ if } x \gt \delta \end {cases}$

as long as $\delta$ is less than your lowest positive sample point. If you know your function has no frequencies higher than a certain limit, you can rule it out.

Ross Millikan
  • 374,822
  • thank you for the answer. Please comment on the part 2 of the question. – Rajesh D Mar 22 '11 at 05:05
  • All I can say on point 2, assuming you are talking about a sampled function, is that seeing the samples of the step function (or something like it) you would suspect a discontinuity. But absent knowledge of the underlying function, you can't prove it, even at some lower derivative level. There are $C^{\infty}$ functions that look just like the step as long as you don't sample in the interval $0 \lt x \le \delta$ – Ross Millikan Mar 22 '11 at 05:11
  • functions with a discontinuity are not strictly bandlimited. – Rajesh D Mar 22 '11 at 05:13
  • I am not cosidering the sampled function but the actual function in the second part of the question. – Rajesh D Mar 22 '11 at 05:14
  • You are right. The ramp I defined will also have arbitrarily high frequency components unless you round off the corners. So if you know your function is bandlimited, you know it is continuous-one result of the Nyquist theorem. – Ross Millikan Mar 22 '11 at 05:17