Timeline for What characterizies 'causality' for a finite FFT?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 10, 2020 at 12:06 | comment | added | Dan Boschen | Yes by zero padding you are converting cyclical convolution to linear convolution; although what you said triggered my memory where someone else had good insight on this in terms of more efficient estimation by dealing with the cyclical nature properly. I will try to find it. | |
| May 10, 2020 at 6:06 | vote | accept | oliver | ||
| May 10, 2020 at 6:03 | comment | added | oliver | Yes, that is exactly the method I am currently using (although I didn't know that the Least-Squares equations were called 'Wiener-Hopf equations' in this case, thanks for the hint). The reason of my question was that I wasn't sure if this method could yield a causal response function at all when I am computing the correlation matrices in (cyclic) Fourier space (for reasons of efficiency). Although I understand that by zero-padding the original series, their correlations (convolutions) are exactly the same as if computed explicitely, the extra step over 'cyclic' is still kind of weird to me. | |
| May 9, 2020 at 12:07 | answer | added | hotpaw2 | timeline score: 1 | |
| May 8, 2020 at 20:33 | answer | added | Dan Boschen | timeline score: 1 | |
| May 8, 2020 at 20:09 | comment | added | Dan Boschen | Would classical least squared channel estimation approaches work for you such as that described in this post: dsp.stackexchange.com/questions/31318/… It utilizes the Wiener-Hopf equations and I would say the conditional requirement is that you can generate a known spectrally rich sounding pattern as the input. (Since it will resolve a solution for a frequency transfer function as long as there is an output vs input to compare. | |
| May 8, 2020 at 18:19 | comment | added | oliver | I want to derive a phenomenological transfer function from input measurements $x$ and output measurements $y$ in a linear prediction way (solution of Yule-Walker equations, well at least kind of...). I am worried that what I find describes a non-causal (and hence, unphysical) system. To rule that out, I need a sound understanding of what causality means for the solution space (which is discrete periodic if I compute the auto-/cross-correlations of $x$ and $y$ from FFT). I could also compute $G(\omega)=|y|^2(\omega)/|x|^2(\omega)$, but this would not be causal because it is centered around t=0. | |
| May 8, 2020 at 16:50 | comment | added | Dan Boschen | Yes I am referring to open loop systems - nothing to do specifically with control systems. A discrete physical system will have a response to its output given an impulse at its input; the minimum phase system is the one that has the quickest response time (decays the fastest) compared to any other system that has the same magnitude response in frequency. (And min phase means all the zeroes are inside the unit circle). But what is it you are looking to do ultimately that is related to the Kramers-Kronig relation? As if the system isn’t minimum phase you cannot derive the phase from the amp resp | |
| May 8, 2020 at 16:37 | comment | added | oliver | Note also that I have found something about the discrete form of Kramers-Kronig/Hilbert transform in this paper: researchgate.net/publication/…. But it doesn't seem to explain what the formula means in terms of causality for a discrete series either. | |
| May 8, 2020 at 16:18 | comment | added | oliver | Since I am still learning about control theory, I am not sure if I understand you correctly. Note that I don't want to design a control system, but I want to model a physical system (say, 'open-loop'). Is minimum phase the term that corresponds to causality for the continuum? Then, yes, that's what I am interested in. But while I exactly know what causality means, I don't know what minimum phase means for a (discretized) physical system... | |
| May 8, 2020 at 15:01 | comment | added | Dan Boschen | Interesting question and appears that the notion of causality is lost in the Discrete Fourier Transform specifically since both time and frequency become periodic and thus extend (unroll) to the equivalent of non-causal periodic functions only. However I believe the Kramers-Kronig relations would still apply (I need to confirm) in that the imaginary and real portions of a system's frequency response are related by the Hilbert Transform for minimum phase systems only -- is that relationship your interest in this? | |
| May 8, 2020 at 13:03 | history | edited | oliver | CC BY-SA 4.0 |
added 57 characters in body
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| May 8, 2020 at 12:45 | history | asked | oliver | CC BY-SA 4.0 |