Timeline for Maximum Magnitude Deviation between DFT and DTFT
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Sep 19, 2022 at 5:54 | comment | added | robert bristow-johnson | It's the same thing, @KnutInge. I like the expressions I have better than how JOS puts it (with the $\alpha$, $\beta$, and $\gamma$). But that's just a personal preference. | |
| Sep 19, 2022 at 5:50 | comment | added | Knut Inge | I have no idea who was first, but this was the first place I was exposed to the idea: ccrma.stanford.edu/~jos/sasp/… | |
| Sep 19, 2022 at 3:29 | comment | added | robert bristow-johnson | I think I got it from some other source than JOS. But it's gone now. Sure glad I copied the image while it was there. @KnutInge it's just a simple, take three points (the maximum point must be the middle point), define the only possible quadratic parabola that hits those three points (like Lagrange interpolation) and the peak of that parabola is your interpolated peak. It's very simple and straight-forward. But maybe Julius said to do this first, but it seems to me a natural to go from discrete data to a continuous location for the peak. | |
| Sep 19, 2022 at 2:17 | comment | added | Knut Inge | The link seems to be broken. Is this the Julius Orion Smith approach, @robertbristow-johnson? | |
| Sep 29, 2018 at 21:47 | comment | added | robert bristow-johnson | it still depends on your assumptions. i'll have to work on another answer to spell this out. | |
| Sep 29, 2018 at 21:44 | comment | added | Jiro | I'm interested as defined in the original post in the relation of $m_c$ and $m_d$ for a general signal. The term peak is misleading, but it is defined properly above. | |
| Sep 29, 2018 at 21:28 | comment | added | robert bristow-johnson | what does the peak mean? "it depends on your assumptions." does the peak mean a sinusoid? if it is assumed that the peak is there because there is a sinusoid there, then it depends on the window function because, in the DTFT domain, the sinusoid is a Dirac spike convolved with the Fourier Transform of the window function. | |
| Sep 29, 2018 at 21:20 | comment | added | Jiro | I agree on your arguments for a single sinusoid. However, my question is for a general real valued signal. How bad can all this interference become in terms of the peak maximum? | |
| Sep 29, 2018 at 20:37 | comment | added | robert bristow-johnson | again, if your window is gaussian and if you look at the log-magnitude (such as dB) rather than just the magnitude (as we did above) and if there aren't any nearby frequency components to interfere, the quadratic interpolation is perfect in this log-magnitude domain. | |
| Sep 29, 2018 at 20:35 | comment | added | robert bristow-johnson | it depends on your assumptions. can we start with the assumption that you're trying to determine the frequency and amplitude of a single sinusoid? and, since the FFT is finite in size, your sinusoid will necessarily be windowed (no windowing is actually rectangular windowing). in the frequency-domain, windowing has tails and sidelobes that interfere with other spectral features. then, with the assumption that there are no other nearby sinusoids that interfere, based on the window function, there are results for how bad the quadratic interpolation is. | |
| Sep 29, 2018 at 13:35 | comment | added | Jiro | Thanks for the answer. I'm aware of the quadratic approximation, but what I'm interested is: what is the worst case error of this quadratic approximation? Are there special signals where this approximation fails considerably? | |
| Sep 29, 2018 at 3:50 | history | edited | robert bristow-johnson | CC BY-SA 4.0 | added 2124 characters in body |
| Sep 28, 2018 at 23:13 | history | answered | robert bristow-johnson | CC BY-SA 4.0 |