Timeline for Given a signal that is not bandlimited, how do you properly take the FFT?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2022 at 14:56 | comment | added | MBaz | @OverLordGoldDragon You are reading things in my post that I didn't say. Also: the sampling theorem (what the original question is about) is all about reconstruction from the samples. And: most signals of interest tend to zero both as $|t| \rightarrow \infty$ and $|f| \rightarrow \infty$. | |
| May 11, 2022 at 14:02 | comment | added | OverLordGoldDragon | That's not a problem with sampling or the signal, that's a limitation of just one of possible postprocessing tools. Sinc interpolation isn't the only continuous reconstruction method, nor is it necessarily the best, nor do we need a continuous reconstruction in the first place. "Fourier bandlimited" requires infinite sampling rate from big bang to heat death, which is a useless criterion and makes the use of Fourier transform in this way useless, like saying $x^2 = -1$ can't be solved. It also makes your point on energy rolloff irrelevant, as finite signals never roll off in CFT. | |
| May 11, 2022 at 13:05 | comment | added | MBaz | @user2 I recommend picking up Richard Lyon's book on DSP, it's excellent and approachable. To your questions: it's not that the distinction is arbitrary; rather, it depends on the requirements of your design, regarding how much aliasing is tolerable, what sampling rates are available, etcetera. Finally, indeed a low-pass filter is used to match the bandwidth of the analog, continuous-time signal to the sampling rate of your analog-to-digital converter. Filtering must happen before sampling; otherwise aliasing will occur and can't be fixed. Search for "anti alias filtering". | |
| May 11, 2022 at 3:49 | comment | added | user62718 | To add on to my previous comment, I guess the distinction between the two cases is somewhat arbitrary, and in the case #2, you probably apply a low-pass filter or something like that. Is that correct? But why do you sample afterward? I thought the signal was already sampled in the time domain, you take the FT, and then apply the filter to the output. Could you sample first and then apply the filter? Sorry for being dense, but I don't have a signal processing background. | |
| May 11, 2022 at 3:12 | comment | added | user62718 | @MBaz +1 Thank you very much. Could you please elaborate on how to determine the proper sampling in either of the two cases you mentioned, as well as how to determine what the filter should be? Signal processing is quite new to me. | |
| May 10, 2022 at 17:23 | comment | added | MBaz | The problem with the 1Hz sine is when you try to re-generate the continuous-time signal from its samples: the interpolation will fail at the start and end of the signal. Regarding distortion, the fact that this distortion is acceptable in many cases is what makes DSP useful in the first place. In a well-designed system aliasing is an irrelevant technicality. It was not clear from the OP's question whether the aliasing in their system is irrelevant or significant. | |
| May 10, 2022 at 16:48 | comment | added | OverLordGoldDragon | I don't understand why such aliasing isn't treated as an irrelevant technicality. A signal can be perfectly 1Hz over a finite duration, and if we sample it over that duration, then we have a perfect 1Hz sine, without distortions that are implied by aliasing. Just because "finite = infinite bw for continuous FT" doesn't mean "all real world signals are distorted", as such descriptions imply. Isn't it better to reserve the word for when we have actual uncertainty for peak frequency over a finite duration? | |
| May 10, 2022 at 14:17 | history | answered | MBaz | CC BY-SA 4.0 |