Somebody told me that a differentiator is classified as an uncausal system because it has more zeros than poles. This would mean that it needs information from the future to effect the present which is nonsense. Whilst I agree you cannot implement a pure differentiator, you can implement an approximation over a limited frequency range and they have to be causal.
Causality
Started by ●October 8, 2013
Reply by ●October 8, 20132013-10-08
On Tue, 08 Oct 2013 10:52:15 -0700, gyansorova wrote:> Somebody told me that a differentiator is classified as an uncausal > system because it has more zeros than poles. This would mean that it > needs information from the future to effect the present which is > nonsense. Whilst I agree you cannot implement a pure differentiator, you > can implement an approximation over a limited frequency range and they > have to be causal.I'm assuming that you're talking about the Laplace domain, where a differentiator is H(s) = s. Somebody was wrong, or dancing on the edge of wrongness, or objecting to the wrong part of the wrongness, or something. Whether or not a pure differentiator is noncausal or not depends on how you classify "causal". If you insist that a system's response must be strictly after any input for the system to be causal, then yes, the differentiator is noncausal. If, on the other hand, you insist that a system's response must not be _before_ the input, then a pure differentiator is causal. If you really want to split hairs, raise one eyebrow in a superior sort of way and say "oh no, my good man, a differentiator is _meta_causal!" What a pure differentiator _is_, regardless of what excuse you find for calling it so, is impossible to build in real life. I suppose that causality in the strict sense may roll into this. But unless you're just playing pointless mathematical games that's kind of bullshit. It's bullshit because when you're talking real-world differentiators you can't just blithely specify a differentiator with any old time constant: the time constant of the differentiator is dependent on the technology that you're using to realize it. So in the _real_ world you never get a chance to bump into the causality argument, because in the _real_ world if there's not some band-limiting process that comes into play long, long before you have as much bandwidth as you want, there's enough noise at the input to the differentiator that you have to band limit it out of self defense. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●October 9, 20132013-10-09
On Wednesday, October 9, 2013 9:57:26 AM UTC+13, Tim Wescott wrote:> On Tue, 08 Oct 2013 10:52:15 -0700, gyansorova wrote: > > > > > Somebody told me that a differentiator is classified as an uncausal > > > system because it has more zeros than poles. This would mean that it > > > needs information from the future to effect the present which is > > > nonsense. Whilst I agree you cannot implement a pure differentiator, you > > > can implement an approximation over a limited frequency range and they > > > have to be causal. > > > > I'm assuming that you're talking about the Laplace domain, where a > > differentiator is H(s) = s. > > > > Somebody was wrong, or dancing on the edge of wrongness, or objecting to > > the wrong part of the wrongness, or something. > > > > Whether or not a pure differentiator is noncausal or not depends on how > > you classify "causal". If you insist that a system's response must be > > strictly after any input for the system to be causal, then yes, the > > differentiator is noncausal. If, on the other hand, you insist that a > > system's response must not be _before_ the input, then a pure > > differentiator is causal. > > > > If you really want to split hairs, raise one eyebrow in a superior sort > > of way and say "oh no, my good man, a differentiator is _meta_causal!" > > > > What a pure differentiator _is_, regardless of what excuse you find for > > calling it so, is impossible to build in real life. I suppose that > > causality in the strict sense may roll into this. > > > > But unless you're just playing pointless mathematical games that's kind > > of bullshit. It's bullshit because when you're talking real-world > > differentiators you can't just blithely specify a differentiator with any > > old time constant: the time constant of the differentiator is dependent > > on the technology that you're using to realize it. So in the _real_ > > world you never get a chance to bump into the causality argument, because > > in the _real_ world if there's not some band-limiting process that comes > > into play long, long before you have as much bandwidth as you want, > > there's enough noise at the input to the differentiator that you have to > > band limit it out of self defense. > > > > -- > > > > Tim Wescott > > Wescott Design Services > > http://www.wescottdesign.comYes could not agree more. It's like Physics gangs up and prevents you building a pure integrator anyway and you end up with a pole and a zero which is causal. It is mathematical masturbation.
Reply by ●October 9, 20132013-10-09
On Wednesday, October 9, 2013 6:44:24 PM UTC+13, gyans...@gmail.com wrote:> On Wednesday, October 9, 2013 9:57:26 AM UTC+13, Tim Wescott wrote: > > > On Tue, 08 Oct 2013 10:52:15 -0700, gyansorova wrote: > > > > > > > > > > > > > Somebody told me that a differentiator is classified as an uncausal > > > > > > > system because it has more zeros than poles. This would mean that it > > > > > > > needs information from the future to effect the present which is > > > > > > > nonsense. Whilst I agree you cannot implement a pure differentiator, you > > > > > > > can implement an approximation over a limited frequency range and they > > > > > > > have to be causal. > > > > > > > > > > > > I'm assuming that you're talking about the Laplace domain, where a > > > > > > differentiator is H(s) = s. > > > > > > > > > > > > Somebody was wrong, or dancing on the edge of wrongness, or objecting to > > > > > > the wrong part of the wrongness, or something. > > > > > > > > > > > > Whether or not a pure differentiator is noncausal or not depends on how > > > > > > you classify "causal". If you insist that a system's response must be > > > > > > strictly after any input for the system to be causal, then yes, the > > > > > > differentiator is noncausal. If, on the other hand, you insist that a > > > > > > system's response must not be _before_ the input, then a pure > > > > > > differentiator is causal. > > > > > > > > > > > > If you really want to split hairs, raise one eyebrow in a superior sort > > > > > > of way and say "oh no, my good man, a differentiator is _meta_causal!" > > > > > > > > > > > > What a pure differentiator _is_, regardless of what excuse you find for > > > > > > calling it so, is impossible to build in real life. I suppose that > > > > > > causality in the strict sense may roll into this. > > > > > > > > > > > > But unless you're just playing pointless mathematical games that's kind > > > > > > of bullshit. It's bullshit because when you're talking real-world > > > > > > differentiators you can't just blithely specify a differentiator with any > > > > > > old time constant: the time constant of the differentiator is dependent > > > > > > on the technology that you're using to realize it. So in the _real_ > > > > > > world you never get a chance to bump into the causality argument, because > > > > > > in the _real_ world if there's not some band-limiting process that comes > > > > > > into play long, long before you have as much bandwidth as you want, > > > > > > there's enough noise at the input to the differentiator that you have to > > > > > > band limit it out of self defense. > > > > > > > > > > > > -- > > > > > > > > > > > > Tim Wescott > > > > > > Wescott Design Services > > > > > > http://www.wescottdesign.com > > > > Yes could not agree more. It's like Physics gangs up and prevents you building a pure integrator anyway and you end up with a pole and a zero which is causal. > > It is mathematical masturbation.I meant differentiator of course!
Reply by ●October 9, 20132013-10-09
On 10/9/13 1:43 AM, gyansorova@gmail.com wrote:>> It's like Physics gangs up and prevents you building a pure [differentiator] anyway and you end up with a pole and a zero which is causal. >> >> It is mathematical masturbation.unless you are modeling the resistance in wires, i don't think Physics prevents one from building a pure differentiator nor a pure integrator. ideally, with the lumped element model, there are straight-forward op-amp circuits with a capacitor that have s or 1/s as the transfer function. but, practically, it's not perfectly doable. i think analog canonical circuits use integrators (s^-1) instead of differentiators (s) because the practical circuits are less noisy. i dunno, i think that's it. the cool thing, in comparing analog to digital (or, more precisely, "discrete-time") is that both have elements that add (or subtract) signals and both have elements that scale signals by various constant coefficients. and analog canonical forms have elements we call "integrators" and are s^-1 while the digital counterparts have elements that are "unit delays" and are z^-1. so the techniques to deal with either H(s) or H(z) regarding cascading and partial fractions and poles and zeros and canonical (or non-canonical) forms such were the same. and, for both H(s) and H(z), the three constituent element classes (for H(s) it's adders, scalers, and integrators, for H(z) it's adders, scalers, and unit delays) all have the exponential function as an eigenfunction. from that (and Euler's formula) all of Fourier analysis and Linear System Theory are derived. i think that's kinda cool; exponential function goes into an LTI system means an exponential function comes out (with the same "alpha" as the exp going in). doesn't matter that it's an H(s) or H(z). -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●October 10, 20132013-10-10
robert bristow-johnson <rbj@audioimagination.com> wrote: (previously snipped person wrote)>>> It's like Physics gangs up and prevents you building a >>> pure [differentiator] anyway and you end up with a pole >>> and a zero which is causal.>>> It is mathematical masturbation.> unless you are modeling the resistance in wires, i don't think Physics > prevents one from building a pure differentiator nor a pure integrator.You can get rid of resistance (superconductors) but not inductance. There is an example in "Feynman Lectures on Physics", I will guess volume 2, discussing RLC circuits. He starts with an LC circuit and asks how to increase the resonant freguency. (It has a parallel plate capacitor and a small wirewound inductor.) First you decrease the capacitance by spreading the plates. You can do that only as far as the inductors leads will allow. Now you decrease the inductance. First, reduce the turns until it is just a straight wire. Then, to decrease it even more, put more and more wires in parallel until the two plates are completely surrounded by straight wires. The result is a cylindrical can, which microwave electronics calls a resonator. Also, consider an idea coax cable (R=0, G=0). It will have an inductance per unit length and capacitance per unit length. Make it with a nice straight center conductor, and nice hollow tube outer conductor to minimize inductance. Now compute the capacitance per unit length and inductance per unit lenght and multiply them together. You can only decrease the product so far before you hit a physical limit.> ideally, with the lumped element model, there are straight-forward > op-amp circuits with a capacitor that have s or 1/s as the transfer > function.The minimum delay through a capacitor is the time it takes light (or any electromagnetic wave) to get from one plate to the other.> but, practically, it's not perfectly doable.> i think analog canonical circuits use integrators (s^-1) instead of > differentiators (s) because the practical circuits are less noisy. i > dunno, i think that's it.> the cool thing, in comparing analog to digital (or, more precisely, > "discrete-time") is that both have elements that add (or subtract) > signals and both have elements that scale signals by various constant > coefficients. and analog canonical forms have elements we call > "integrators" and are s^-1 while the digital counterparts have elements > that are "unit delays" and are z^-1.-- glen
Reply by ●October 10, 20132013-10-10
On Thursday, October 10, 2013 8:27:16 AM UTC+13, robert bristow-johnson wrote:> On 10/9/13 1:43 AM, gyansorova@gmail.com wrote: > > >> It's like Physics gangs up and prevents you building a pure [differentiator] anyway and you end up with a pole and a zero which is causal. > > >> > > >> It is mathematical masturbation. > > > > unless you are modeling the resistance in wires, i don't think Physics > > prevents one from building a pure differentiator nor a pure integrator. > > > > ideally, with the lumped element model, there are straight-forward > > op-amp circuits with a capacitor that have s or 1/s as the transfer >an op-amp will oscillate if you try and implement a pure differentiator.
Reply by ●October 10, 20132013-10-10
On 10/9/13 9:31 PM, gyansorova@gmail.com wrote:> On Thursday, October 10, 2013 8:27:16 AM UTC+13, robert bristow-johnson wrote: >> On 10/9/13 1:43 AM, gyansorova@gmail.com wrote: >> >>>> It's like Physics gangs up and prevents you building a pure [differentiator] anyway and you end up with a pole and a zero which is causal. >>>> >>>> It is mathematical masturbation. >> >> unless you are modeling the resistance in wires, i don't think Physics >> prevents one from building a pure differentiator nor a pure integrator. >> >> ideally, with the lumped element model, there are straight-forward >> op-amp circuits with a capacitor that have s or 1/s as the transfer function. >> > an op-amp will oscillate if you try and implement a pure differentiator.i'm a little dubious about that, because in the feedback path, it's the same resistor as you would have for a typical inverting amplifier. (ASCII art, mono font required) .------- Rf -------. | | | |\ | | | \ | Vin ---- R1 ----'---|- \ | | \ | | >---------'------ Vout | / .------|+ / | | / | |/ | GND so, how does that oscillate when R1 is replaced by a C ? same Rf in the loop. gy, i haven't breadboarded anything for the past 3 decades, but i would think that every EE student has implemented a simple differentiator in the form of a non-inverting op-amp circuit in the lab. i certainly did on some day in the 1970s. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●October 10, 20132013-10-10
On Wednesday, October 9, 2013 6:52:15 AM UTC+13, gyans...@gmail.com wrote:> Somebody told me that a differentiator is classified as an uncausal system because it has more zeros than poles. This would mean that it needs information from the future to effect the present which is nonsense. Whilst I agree you cannot implement a pure differentiator, you can implement an approximation over a limited frequency range and they have to be causal.If you do an open-loop bode-plot of the op-amp plus the feedback term you get no phase-margin. What happens when you try and implement it is that it will differentiate but an oscillation is superimposed on top of it. You ened a series resistance to fix this (series to the capacitor). It's not mentioned in all text books mind you but most talk of a practical differentiator. If you have your op-amp, in open-loop it behaves as a massive gain plus -20dB/decade rolloff (1st order system ignoring higher order poles). ie G=A/(1+sTa) where Ta is related to the cross-over frequency of the op-amp. For a typical resistor feedback config as you have initially the feedback transfer function beta is just a constant. You then plot a Bode-Plot and you find 90 degrees phase margin. Now change the input resistor to a capacitor and the beta term becomes beta=1/(1+sT) where T=1/2pi*CR where R is the feedback resistor. (in the case of an integrator this beta term is 1+sT ). Put the G and beta together and get G*Beta = A/(1+sTa) * 1/(1+sT) What now happens is that the Bode plot is -20dB/decade followed by another -20dB/decade and the phase approaches -180 degrees. The frequency where the plot intersepts 0dB is the oscillation frequency. If you put a series resistor Rs with the capacitor you get a new feedback term beta = (1+sCRs)/(1+sC(Rs+R)) (R is feedback resistor) and this bit has a -20dB/decade roll-pff followed by a zero which makes it go flat. The net Bode Plot now goes -20dB/decade and then -40dB/decade and then back to -20dB/decade passing through 0dB and hence stable again. This is like a simple lag compensator.
Reply by ●October 10, 20132013-10-10
On Friday, October 11, 2013 6:09:40 AM UTC+13, robert bristow-johnson wrote:> On 10/9/13 9:31 PM, gyansorova@gmail.com wrote: > > > On Thursday, October 10, 2013 8:27:16 AM UTC+13, robert bristow-johnson wrote: > > >> On 10/9/13 1:43 AM, gyansorova@gmail.com wrote: > > >> > > >>>> It's like Physics gangs up and prevents you building a pure [differentiator] anyway and you end up with a pole and a zero which is causal. > > >>>> > > >>>> It is mathematical masturbation. > > >> > > >> unless you are modeling the resistance in wires, i don't think Physics > > >> prevents one from building a pure differentiator nor a pure integrator. > > >> > > >> ideally, with the lumped element model, there are straight-forward > > >> op-amp circuits with a capacitor that have s or 1/s as the transfer function. > > >> > > > an op-amp will oscillate if you try and implement a pure differentiator. > > > > > > i'm a little dubious about that, because in the feedback path, it's the > > same resistor as you would have for a typical inverting amplifier. > > > > > > (ASCII art, mono font required) > > > > > > .------- Rf -------. > > | | > > | |\ | > > | | \ | > > Vin ---- R1 ----'---|- \ | > > | \ | > > | >---------'------ Vout > > | / > > .------|+ / > > | | / > > | |/ > > | > > GND > > > > > > > > so, how does that oscillate when R1 is replaced by a C ? same Rf in > > the loop. > > > > > > gy, i haven't breadboarded anything for the past 3 decades, but i would > > think that every EE student has implemented a simple differentiator in > > the form of a non-inverting op-amp circuit in the lab. i certainly did > > on some day in the 1970s. > > > > -- > > > > r b-j rbj@audioimagination.com > > > > "Imagination is more important than knowledge."Actually I found a better explanation and simulation http://stabilityissues.net/2012/12/26/differentiator-stability/ http://stabilityissues.net/2012/12/29/differentiator-stability-2/
