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I was reading "Discrete and continuous Fourier transforms: analysis, applications and fast algorithms" written by E.Chu and, at some point, I found something that I could not completly understand.

If $x(t)$ is not continuous, then it is possible that 2 different functions share the same transform $X(f)$. Therefore, the function $x(t)$ cannot be uniquely determined by inverse trasforming $X(f)$ unless it is continuous or it satisfies 2 more conditions:

1- $x(t)$ has only a finite number of maxima and minima on any finite interval

2- $x(t)$ has on any finite interval at most a finite number of discontinuities, each of which is a jump discontinuity.

In the latter case, the inverse Fourier transform of $X(f)$ produces $\hat{x}(t)$, which agrees with $x(t)$ at every $t$ at which $x(t)$ is continuous, and $\hat{x}(t)$ equals the average of the left-hand limit $x(t^-_{\alpha})$ and the right-hand limit $x(t^+_{\alpha})$ at every $t_{\alpha}$ at which a jump discontinuity occurs.

So, here my doubts.

First of all, I am not sure I got the meaning of the statements related to the 2 conditions listed. This is probably due to the fact that I am familiar only with signals coming from structural tests (mainly accelerations), therefore I think it could be really helpful if someone of you can provide me some example I can focus on to understand the concept (for example, how can be possible to have infinite maxima and minima in a signal? And what are the discontinuities it is talking about?).

Second, measured signals (for example, the aformentioned acceleration measurements coming from a test) are not continuous (since they are sampled at a given sampling frequency), so if I want to apply the IFFT algorithm to retrieve the time signal does it mean I have always to check if the 2 other conditions are satisfied? And how can I check them? Visual inspection? Or do I have to look for some specific features the signal must have?

Are there classes of signals which always satisfy the 2 conditions?

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  • $\begingroup$ It is possible to theoretically formulate pathological functions which can't exist as signals in the real world, and have too great a bandwidth to be sampled anyway. But some of them have an FT anyway. $\endgroup$ – hotpaw2 Jan 4 '15 at 0:19
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The conditions you mention only apply to the continuous-time Fourier transform, and they only have theoretical relevance. All signals occurring in engineering applications will satisfy these conditions. But if you want an example of a function with an infinite number of maxima and minima on a finite interval, here it is:

$$x(t)=\sin(1/t)$$

This function has an infinite number of minima and maxima on the interval $(0,a]$ for any real-valued $a>0$.

And as for your measured signals, they are all discrete-time signals of finite length, and the above conditions do not apply. So you don't need to worry about them when using the DFT/FFT, because the DFT is defined by a finite sum, unlike the continuous-time Fourier transform, which is defined by an improper integral.

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