Filter and inverse filter =/ timedelay?
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From: Demus on 26 Jul 2010 12:17 >> (By the same token, in my opinion the phase of a causal system can't really >> be said to be negative, at least not when the inputs and outputs are >> non-stationary signals...) > >Negative phase slope indicates delay, which is perfectly reasonable to >expect in a causal system. I think you're thinking of a positive phase >slope, indicating lead. You can have lead in a real system, but only at >the expense of a rising amplitude vs. frequency (think differentiation). > The amplitude 'distortion' is the price you pay for the 'prediction' >effect of differentiation. > Oh yeah, you're right... I had positive phases in mind. And yeah, of course that makes sense. Ok, but I'm not sure I understand this completely, even if dealing only with ideal linear systems (and stable, minphase) and uncorrupted signals. The answer is no, there shouldn't (ideally) be a pure time-delay in the above mentioned setup? >> Can someone please explain this situation to me, perhaps including how >> phase relates to causality? > >The slope of the phase vs. frequency characteristic of a linear system >transfer function indicates the effective delay of signals at that >particular frequency. It can be positive, but as I mentioned, if the >system is causal then the positive phase vs. frequency slope comes with >a cost in the amplitude vs. frequency relationship. So what does this mean in the time domain (still ideally - perfect linearity, no noise)? > >There's some relationship, but I can't remember the details, even if I >ever learned them. I know that if a system is minimum phase then its >phase response is the Hilbert transform of its magnitude, but I couldn't >even tell you if that's magnitude, magnitude squared, log magnitude, >etc. But if you really want to know, you should be able to at least >find a book reference with an internet search. Isn't this Bode's Integral Theorem? I brought this up a couple of months ago? I can verify (by the way) that the discrete time version is explicitly stated in the first edition of Oppenheims' Digital Signal Processing. Also, sorry, I posted my previous post before I saw your post. Thanks!
From: Demus on 26 Jul 2010 12:31 I think I understand the situation better now... I guess I tricked myself before. Thanks for the explanation!
From: Tim Wescott on 26 Jul 2010 12:38 On 07/26/2010 09:17 AM, Demus wrote: >>> (By the same token, in my opinion the phase of a causal system can't > really >>> be said to be negative, at least not when the inputs and outputs are >>> non-stationary signals...) >> >> Negative phase slope indicates delay, which is perfectly reasonable to >> expect in a causal system. I think you're thinking of a positive phase >> slope, indicating lead. You can have lead in a real system, but only at >> the expense of a rising amplitude vs. frequency (think differentiation). >> The amplitude 'distortion' is the price you pay for the 'prediction' >> effect of differentiation. >> > > Oh yeah, you're right... I had positive phases in mind. And yeah, of course > that makes sense. > > Ok, but I'm not sure I understand this completely, even if dealing only > with ideal linear systems (and stable, minphase) and uncorrupted signals. > The answer is no, there shouldn't (ideally) be a pure time-delay in the > above mentioned setup? Correct. In fact, for a linear, time-invariant, minimum-phase (no zeros on the imaginary axis), purely stable system (no poles on the imaginary axis) with a finite number of states, you can always find an inverse, and the cascade of the system with its inverse will always have a transfer function of one. This should contradict your intuition, before you even get to Simulink. >>> Can someone please explain this situation to me, perhaps including how >>> phase relates to causality? >> >> The slope of the phase vs. frequency characteristic of a linear system >> transfer function indicates the effective delay of signals at that >> particular frequency. It can be positive, but as I mentioned, if the >> system is causal then the positive phase vs. frequency slope comes with >> a cost in the amplitude vs. frequency relationship. > > So what does this mean in the time domain (still ideally - perfect > linearity, no noise)? That you can't make a predictor that's not both inaccurate and noisy. >> There's some relationship, but I can't remember the details, even if I >> ever learned them. I know that if a system is minimum phase then its >> phase response is the Hilbert transform of its magnitude, but I couldn't >> even tell you if that's magnitude, magnitude squared, log magnitude, >> etc. But if you really want to know, you should be able to at least >> find a book reference with an internet search. > > Isn't this Bode's Integral Theorem? No -- Bode's Integral Theorem is probably similar, but it has to do with the sensitivity of a control system to disturbances and what you can do with it. Although perhaps its a corollary -- or perhaps the one I'm familiar with is the corollary. > I brought this up a couple of months > ago? I can verify (by the way) that the discrete time version is explicitly > stated in the first edition of Oppenheims' Digital Signal Processing. Oppenheim & Schafer? Copyright 1975? I can't find a reference to it in the index or table of contents -- do you have a page number, or do I gave the wrong book? > Also, sorry, I posted my previous post before I saw your post. No problem -- it's just a race condition. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" was written for you. See details at http://www.wescottdesign.com/actfes/actfes.html
From: Vladimir Vassilevsky on 26 Jul 2010 12:45 H(z) x 1/H(z) === 1 This is scientific fact (except for the special case H(z) === 0). Neither H(z) nor 1/H(z) have to be casual. Neither H(z) nor 1/H(z) have to be stable. Neither H(z) nor 1/H(z) have to be minimum phase. Demus wrote: > Hello, > > This might be a silly question, but I can't figure out what implications > the phase of a transfer function has for causality. > > If I pass a signal to a filter and pass the output to the inverse filter... > my intuition of causality tells me then that the output of the inverse > filter can not be the same as what enters the first filter. Intuition may be good at gambling. > And by 'can not be the same' I mean to include it's place in time. It > should have the same shape but not occur at the same time, which would then > suggest that the result is a pure time delay. > > However, when I simulate this situation in simulink I get the opposite > result, the two signals have the same time coordinates as well, so I guess > my intuition is off. "Matlab does all thinking for us" (TM) > (By the same token, in my opinion the phase of a causal system can't really > be said to be negative, at least not when the inputs and outputs are > non-stationary signals...) > > Can someone please explain this situation to me, perhaps including how > phase relates to causality? There is no such relation. VLV
From: Demus on 26 Jul 2010 12:59
>No -- Bode's Integral Theorem is probably similar, but it has to do with >the sensitivity of a control system to disturbances and what you can do >with it. Although perhaps its a corollary -- or perhaps the one I'm >familiar with is the corollary. Yeah, the waterbed effect for "2-pole-excess" systems, right? They might have the same name sometimes but I know that one at least as Bode's Sensitivity Integral. >> I brought this up a couple of months >> ago? I can verify (by the way) that the discrete time version is explicitly >> stated in the first edition of Oppenheims' Digital Signal Processing. > >Oppenheim & Schafer? Copyright 1975? I can't find a reference to it in >the index or table of contents -- do you have a page number, or do I >gave the wrong book? I don't have the book, I borrowed it from the library last time (and then I only found it in the first edition, not the second edition). I might be way off here but I'm pretty sure it was either theorem 5.21 or 7.21. Can't remember. Again, thanks for your replies. You've always been very helpful. |