Hello everyone, I have a problem I'm trying to solve and I would appreciate any input that may be available. I am not so much looking for the exact solution, as I am looking for ideas on how to go about solving it. I can workout the details after, I just need to figure out exactly what it is I'm trying to do. The Equipment: I have a sensor moving through the ocean from surface to 200m depth in a saw-tooth pattern. As it moves, it records its depth every 4 seconds, it also records values for dissolved oxygen. The Problem: The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I look at the depth measurements, the corresponding oxygen measurements are not from that depth, they are from some depth before hand. I would like to correct for this, so I know exactly what depth my oxygen measurements correspond to. Someone suggested something about fourier transforms, or modelling it with e^-t/tau where tau is my 25seconds... but I am unsure of what they actually meant. If someone could provide either an outline of what it is im trying to do, or some keywords that I could google, it would be greatly appreciated! thanks for your time, - Charlie
signal processing: adjust for settling time of instrument.
Started by ●January 20, 2007
Reply by ●January 20, 20072007-01-20
Charlie wrote:> Hello everyone, > > I have a problem I'm trying to solve and I would appreciate any input > that may be available. I am not so much looking for the exact solution, > as I am looking for ideas on how to go about solving it. I can workout > the details after, I just need to figure out exactly what it is I'm > trying to do. > > The Equipment: > > I have a sensor moving through the ocean from surface to 200m depth in > a saw-tooth pattern. As it moves, it records its depth every 4 seconds, > it also records values for dissolved oxygen. > > The Problem: > > The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I > look at the depth measurements, the corresponding oxygen measurements > are not from that depth, they are from some depth before hand. > > I would like to correct for this, so I know exactly what depth my > oxygen measurements correspond to. > > Someone suggested something about fourier transforms, or modelling it > with e^-t/tau where tau is my 25seconds... but I am unsure of what they > actually meant. > > If someone could provide either an outline of what it is im trying to > do, or some keywords that I could google, it would be greatly > appreciated! > > thanks for your time, > > - Charlie >Charlie, what is the depth descent/ascent rate of the sensor (meters/sec)? By this I mean how long does it take for the sensor to move from surface to 200m depth and back up to the surface again?
Reply by ●January 20, 20072007-01-20
"Charlie" <charliebishop@gmail.com> wrote in message news:1169321507.939239.58370@q2g2000cwa.googlegroups.com...> Hello everyone, > > I have a problem I'm trying to solve and I would appreciate any input > that may be available. I am not so much looking for the exact solution, > as I am looking for ideas on how to go about solving it. I can workout > the details after, I just need to figure out exactly what it is I'm > trying to do. > > The Equipment: > > I have a sensor moving through the ocean from surface to 200m depth in > a saw-tooth pattern. As it moves, it records its depth every 4 seconds, > it also records values for dissolved oxygen. > > The Problem: > > The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I > look at the depth measurements, the corresponding oxygen measurements > are not from that depth, they are from some depth before hand. > > I would like to correct for this, so I know exactly what depth my > oxygen measurements correspond to. > > Someone suggested something about fourier transforms, or modelling it > with e^-t/tau where tau is my 25seconds... but I am unsure of what they > actually meant. > > If someone could provide either an outline of what it is im trying to > do, or some keywords that I could google, it would be greatly > appreciated! > > thanks for your time, > > - CharlieCharlie, It seems like you might approximate the correct measurement by using a system model something like this: The output of a simple lag to a step change input from zero initial state is: To = Ti * [1 - e^-(t/tau)] where tau is the system time constant, in your case ~25 seconds. So, in simple terms you might write this as: Ti = To / [1 - e^-(t/25)] ... again if initial conditions are zero. Since the initial conditions aren't zero, you might model the input as a series of tiny steps that are additive (like a staircase input). By superposition, the output is the sum of the results of all those little inputs: To1 = Ti0 To2 = Ti1 * [1 - e^-(T/25)] where T is the step size in seconds. T03 = Ti1 * [1 - e^-(2T/25)] + Ti2 * [1 - e^-(T/25)] . . or, in matrix form something like: [E][Ti]=[To] and solve for Ti So, you can build a system of linear equations in some number of output samples to compute the input samples. All of the [1 - e^-(nT/25)] terms are known. You may find that N equations in N unknowns is good or you may find that M>N equations in N unknowns is better solved using a least squares approximation. To be reasonably accurate, T is going to have to be rather small compared to tau and the number of samples is going to have to add up to around 3 tau or better I should think. So, let's say T is 2.5 and you go to 75 seconds, that's 30 samples and, thus 30 equations in 30 unknowns. But then I'm a numerical guy and there may be a way to do this analytically that's easier. But, beware: I know that depth may be a ramp but that's not going to also be true for temperature - so a ramp input is not a good assumption. You may well be stuck with a numerical solution as above. Mind you, the "system" is a lowpass filter. So, changes in temperature that aren't monotonic are going to result in some information being filtered out. This is a bit arm-waving but I hope you get the idea and can fill in the details and make it all correct... Fred
Reply by ●January 20, 20072007-01-20
"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:adKdnf9IRu4DEi_YnZ2dnUVZ_q6vnZ2d@centurytel.net...> > "Charlie" <charliebishop@gmail.com> wrote in message > news:1169321507.939239.58370@q2g2000cwa.googlegroups.com... >> Hello everyone, >> >> I have a problem I'm trying to solve and I would appreciate any input >> that may be available. I am not so much looking for the exact solution, >> as I am looking for ideas on how to go about solving it. I can workout >> the details after, I just need to figure out exactly what it is I'm >> trying to do. >> >> The Equipment: >> >> I have a sensor moving through the ocean from surface to 200m depth in >> a saw-tooth pattern. As it moves, it records its depth every 4 seconds, >> it also records values for dissolved oxygen. >> >> The Problem: >> >> The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I >> look at the depth measurements, the corresponding oxygen measurements >> are not from that depth, they are from some depth before hand. >> >> I would like to correct for this, so I know exactly what depth my >> oxygen measurements correspond to. >> >> Someone suggested something about fourier transforms, or modelling it >> with e^-t/tau where tau is my 25seconds... but I am unsure of what they >> actually meant. >> >> If someone could provide either an outline of what it is im trying to >> do, or some keywords that I could google, it would be greatly >> appreciated! >> >> thanks for your time, >> >> - Charlie > > Charlie, > > It seems like you might approximate the correct measurement by using a > system model something like this: > > The output of a simple lag to a step change input from zero initial state > is: > > To = Ti * [1 - e^-(t/tau)] > > where tau is the system time constant, in your case ~25 seconds. > > So, in simple terms you might write this as: > > Ti = To / [1 - e^-(t/25)] ... again if initial conditions are zero. > > Since the initial conditions aren't zero, you might model the input as a > series of tiny steps that are additive (like a staircase input). > By superposition, the output is the sum of the results of all those little > inputs: > To1 = Ti0 > To2 = Ti1 * [1 - e^-(T/25)] where T is the step size in seconds. > T03 = Ti1 * [1 - e^-(2T/25)] + Ti2 * [1 - e^-(T/25)] > .oh ... I think the terms should be in terms of (Ti2-Ti1),(Ti3-Ti2), etc. so the small steps add. To1 = Ti0> To2 = (Ti2-Ti1) * [1 - e^-(T/25)] where T is the step size in seconds. > T03 = (Ti2-Ti1) * [1 - e^-(2T/25)] + (Ti3 - Ti2) * [1 - e^-(T/25)]then you combine all the coefficients for each value of Ti ... Fred
Reply by ●January 20, 20072007-01-20
Fred Marshall wrote:> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message > news:adKdnf9IRu4DEi_YnZ2dnUVZ_q6vnZ2d@centurytel.net... >> "Charlie" <charliebishop@gmail.com> wrote in message >> news:1169321507.939239.58370@q2g2000cwa.googlegroups.com... >>> Hello everyone, >>> >>> I have a problem I'm trying to solve and I would appreciate any input >>> that may be available. I am not so much looking for the exact solution, >>> as I am looking for ideas on how to go about solving it. I can workout >>> the details after, I just need to figure out exactly what it is I'm >>> trying to do. >>> >>> The Equipment: >>> >>> I have a sensor moving through the ocean from surface to 200m depth in >>> a saw-tooth pattern. As it moves, it records its depth every 4 seconds, >>> it also records values for dissolved oxygen. >>> >>> The Problem: >>> >>> The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I >>> look at the depth measurements, the corresponding oxygen measurements >>> are not from that depth, they are from some depth before hand. >>> >>> I would like to correct for this, so I know exactly what depth my >>> oxygen measurements correspond to. >>> >>> Someone suggested something about fourier transforms, or modelling it >>> with e^-t/tau where tau is my 25seconds... but I am unsure of what they >>> actually meant. >>> >>> If someone could provide either an outline of what it is im trying to >>> do, or some keywords that I could google, it would be greatly >>> appreciated! >>> >>> thanks for your time, >>> >>> - Charlie >> Charlie, >> >> It seems like you might approximate the correct measurement by using a >> system model something like this: >> >> The output of a simple lag to a step change input from zero initial state >> is: >> >> To = Ti * [1 - e^-(t/tau)] >> >> where tau is the system time constant, in your case ~25 seconds. >> >> So, in simple terms you might write this as: >> >> Ti = To / [1 - e^-(t/25)] ... again if initial conditions are zero. >> >> Since the initial conditions aren't zero, you might model the input as a >> series of tiny steps that are additive (like a staircase input). >> By superposition, the output is the sum of the results of all those little >> inputs: >> To1 = Ti0 >> To2 = Ti1 * [1 - e^-(T/25)] where T is the step size in seconds. >> T03 = Ti1 * [1 - e^-(2T/25)] + Ti2 * [1 - e^-(T/25)] >> . > > oh ... I think the terms should be in terms of (Ti2-Ti1),(Ti3-Ti2), etc. so > the small steps add. > > > To1 = Ti0 >> To2 = (Ti2-Ti1) * [1 - e^-(T/25)] where T is the step size in seconds. >> T03 = (Ti2-Ti1) * [1 - e^-(2T/25)] + (Ti3 - Ti2) * [1 - e^-(T/25)] > > then you combine all the coefficients for each value of Ti ...Fred, it's a slow DO sensor, not temperature. That doesn't affect your argument, though. The first thing I would try to do is get a faster sensor. Minimizing a problem is always good, even when the need to compensate remains. I see this as a deconvolution problem. We know how the slow sensor will respond to a step and to an impulse. We want to apply a deconvolution filter that will approximate an impulse at the output when its input is the result of the exponentially decaying sensor output. There may be useful material at http://www.dspguide.com , page 300 in Chapter 17. Once we know the filter, we know it's delay and can adjust the timestamp -- hence the depth -- accordingly. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●January 20, 20072007-01-20
Charlie wrote:> Hello everyone, > > I have a problem I'm trying to solve and I would appreciate any input > that may be available. I am not so much looking for the exact solution, > as I am looking for ideas on how to go about solving it. I can workout > the details after, I just need to figure out exactly what it is I'm > trying to do. > > The Equipment: > > I have a sensor moving through the ocean from surface to 200m depth in > a saw-tooth pattern. As it moves, it records its depth every 4 seconds, > it also records values for dissolved oxygen. > > The Problem: > > The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I > look at the depth measurements, the corresponding oxygen measurements > are not from that depth, they are from some depth before hand. > > I would like to correct for this, so I know exactly what depth my > oxygen measurements correspond to. > > Someone suggested something about fourier transforms, or modelling it > with e^-t/tau where tau is my 25seconds... but I am unsure of what they > actually meant. > > If someone could provide either an outline of what it is im trying to > do, or some keywords that I could google, it would be greatly > appreciated! > > thanks for your time, > > - Charlie >Oh man, you're setting yourself up for a hard problem. If I were cruel I'd suggest you get a copy of Harry Van Trees' "Detection, Estimation and Modulation Theory". That was the textbook for the single hardest and most useful class in communications systems that I've ever taken -- and it's given me some lasting insights into control systems as well. You could, in theory, get the exact O2 concentration out of the sensor immediately simply by taking the output of the sensor times it's derivative times 25 seconds. That's the theory. In practice, you'll be taking the inevitable noise at the output of the sensor and magnifying it to absurd proportions. If you have a lot of confidence in the dynamics of your O2 sensor and the characteristics of it's output noise you can use a steady-state Kalman filter (AKA a Wiener filter). Your problem looks like it can stand some delay, so you would want an after-the-fact ("non-causal") state estimator. This will give you the World's Best Estimate for a linear, perfectly known system that's excited by Gaussian noise, as long as "best" means "least mean square error". But your system isn't excited by a Gaussian process (triangle != Gaussian), and I'll bet you don't know the O2 sensor dynamics perfectly either (they may even change with varying O2 concentrations). So you probably want an H-infinity filter, or you at least want to know enough to verify that a Kalman filter assuming ideal conditions is good enough. So search on "Kalman filter" and "Wiener filter". Also, seriously consider getting yourself a copy of Dan Simon's "Optimal State Estimation". It'll walk you through the whole process of constructing Kalman, Wiener, H-infinity and even more bizarre filters. It won't make your brain melt the way the Van Trees book would, and it has more than a section or two on the specific solution you need. As far as I can tell it should be just the book for you. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●January 20, 20072007-01-20
Tim Wescott <tim@seemywebsite.com> writes:> [...]> If I were cruel I'd suggest you get a copy of Harry Van Trees' > "Detection, Estimation and Modulation Theory". That was the > textbook for the single hardest and most useful class in > communications systems that I've ever taken -- and it's given me > some lasting insights into control systems as well.It just so happens that I'm taking this class at NCSU this semester. Read Dr. Van Trees biography from the back cover [vantrees]: Harry L. Van Trees, ScD, was Professor of Electrical Engineering at Massachusetts Institute of Technology. He served as Chief Scientist of the U.S. Air Force, Chief Scientist of the Defense Communications Agency, and Principle Deputy Assistant Secretary of Defense for C3I. He was President of M/A-Com-Linkabit Government Systems and a cofounder of CommQuest Technologies. He is currently Distinguished Professor of Electrical Engineering and Director of the C3I Center of Excellence at George Mason University, Fairfax, Virginia. I think I'll pay attention to what he has to say. --RY @book{vantrees title = "Detection, Estimation, and Modulation Theory: Part I", author = "Harry~L.~Van~Trees", publisher = "John Wiley and Sons", year = "2001"} -- % Randy Yates % "The dreamer, the unwoken fool - %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Eldorado Overture', *Eldorado*, ELO http://home.earthlink.net/~yatescr
Reply by ●January 21, 20072007-01-21
Tim Wescott wrote:> Charlie wrote: > >> Hello everyone, >> >> I have a problem I'm trying to solve and I would appreciate any input >> that may be available. I am not so much looking for the exact solution, >> as I am looking for ideas on how to go about solving it. I can workout >> the details after, I just need to figure out exactly what it is I'm >> trying to do. >> >> The Equipment: >> >> I have a sensor moving through the ocean from surface to 200m depth in >> a saw-tooth pattern. As it moves, it records its depth every 4 seconds, >> it also records values for dissolved oxygen. >> >> The Problem: >> >> The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I >> look at the depth measurements, the corresponding oxygen measurements >> are not from that depth, they are from some depth before hand. >> >> I would like to correct for this, so I know exactly what depth my >> oxygen measurements correspond to. >> >> Someone suggested something about fourier transforms, or modelling it >> with e^-t/tau where tau is my 25seconds... but I am unsure of what they >> actually meant. >> >> If someone could provide either an outline of what it is im trying to >> do, or some keywords that I could google, it would be greatly >> appreciated! >> >> thanks for your time, >> >> - Charlie >> > Oh man, you're setting yourself up for a hard problem. If I were cruel > I'd suggest you get a copy of Harry Van Trees' "Detection, Estimation > and Modulation Theory". That was the textbook for the single hardest > and most useful class in communications systems that I've ever taken -- > and it's given me some lasting insights into control systems as well. > > You could, in theory, get the exact O2 concentration out of the sensor > immediately simply by taking the output of the sensor times it's > derivative times 25 seconds. > > That's the theory. > > In practice, you'll be taking the inevitable noise at the output of the > sensor and magnifying it to absurd proportions. > > If you have a lot of confidence in the dynamics of your O2 sensor and > the characteristics of it's output noise you can use a steady-state > Kalman filter (AKA a Wiener filter). Your problem looks like it can > stand some delay, so you would want an after-the-fact ("non-causal") > state estimator. This will give you the World's Best Estimate for a > linear, perfectly known system that's excited by Gaussian noise, as long > as "best" means "least mean square error". > > But your system isn't excited by a Gaussian process (triangle != > Gaussian), and I'll bet you don't know the O2 sensor dynamics perfectly > either (they may even change with varying O2 concentrations). So you > probably want an H-infinity filter, or you at least want to know enough > to verify that a Kalman filter assuming ideal conditions is good enough. > > So search on "Kalman filter" and "Wiener filter". Also, seriously > consider getting yourself a copy of Dan Simon's "Optimal State > Estimation". It'll walk you through the whole process of constructing > Kalman, Wiener, H-infinity and even more bizarre filters. It won't make > your brain melt the way the Van Trees book would, and it has more than a > section or two on the specific solution you need. As far as I can tell > it should be just the book for you.We know that depth is triangular. If we also knew that DO were triangular, the problem would mostly go away. Assuming that the DO sensor has only one dominant time constant that is pretty independent of other influences (like pressure or salinity) that might affect it, It should be possible to determine the step response quite closely by applying a step after steady state and observing the output until the new steady state. The impulse response is simply the derivative of the step response (no matter how many time constants there may be), and an IIR filter can be designed to simulate it empirically if need be. Then we need to find a deconvolution filter that restores an impulse to the IIR to a semblance of an impulse at the output of the deconvolver. We then deconvolve the sensor output and backdate the data by the time from the impulse or step to the IIR or sensor, to the emergence of the impulse or step from the deconvolving filter. Easier said than done, I know, but I'm sure that's the way. To repeat: this works *provided* that the response of the DO sensor is independent of depth or other conditions that may vary with depth. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●January 21, 20072007-01-21
"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:l6GdnSsI9rXdCy_YnZ2dnUVZ_uuqnZ2d@centurytel.net...> > "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message > news:adKdnf9IRu4DEi_YnZ2dnUVZ_q6vnZ2d@centurytel.net... >> >> "Charlie" <charliebishop@gmail.com> wrote in message >> news:1169321507.939239.58370@q2g2000cwa.googlegroups.com... >>> Hello everyone, >>> >>> I have a problem I'm trying to solve and I would appreciate any input >>> that may be available. I am not so much looking for the exact solution, >>> as I am looking for ideas on how to go about solving it. I can workout >>> the details after, I just need to figure out exactly what it is I'm >>> trying to do. >>> >>> The Equipment: >>> >>> I have a sensor moving through the ocean from surface to 200m depth in >>> a saw-tooth pattern. As it moves, it records its depth every 4 seconds, >>> it also records values for dissolved oxygen. >>> >>> The Problem: >>> >>> The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I >>> look at the depth measurements, the corresponding oxygen measurements >>> are not from that depth, they are from some depth before hand. >>> >>> I would like to correct for this, so I know exactly what depth my >>> oxygen measurements correspond to. >>> >>> Someone suggested something about fourier transforms, or modelling it >>> with e^-t/tau where tau is my 25seconds... but I am unsure of what they >>> actually meant. >>> >>> If someone could provide either an outline of what it is im trying to >>> do, or some keywords that I could google, it would be greatly >>> appreciated! >>> >>> thanks for your time, >>> >>> - Charlie >> >> Charlie, >> >> It seems like you might approximate the correct measurement by using a >> system model something like this: >> >> The output of a simple lag to a step change input from zero initial state >> is: >> >> To = Ti * [1 - e^-(t/tau)] >> >> where tau is the system time constant, in your case ~25 seconds. >> >> So, in simple terms you might write this as: >> >> Ti = To / [1 - e^-(t/25)] ... again if initial conditions are zero. >> >> Since the initial conditions aren't zero, you might model the input as a >> series of tiny steps that are additive (like a staircase input). >> By superposition, the output is the sum of the results of all those >> little inputs: >> To1 = Ti0 >> To2 = Ti1 * [1 - e^-(T/25)] where T is the step size in seconds. >> T03 = Ti1 * [1 - e^-(2T/25)] + Ti2 * [1 - e^-(T/25)] >> . > > oh ... I think the terms should be in terms of (Ti2-Ti1),(Ti3-Ti2), etc. > so the small steps add. > > > To1 = Ti0 >> To2 = (Ti2-Ti1) * [1 - e^-(T/25)] where T is the step size in seconds. >> T03 = (Ti2-Ti1) * [1 - e^-(2T/25)] + (Ti3 - Ti2) * [1 - e^-(T/25)] > > then you combine all the coefficients for each value of Ti ... > > Fred >Oops again. I should have said DO instead of TEMP. Same thing applies tho. But, listen to the other folks too! Fred
Reply by ●January 22, 20072007-01-22
Charlie wrote:> Hello everyone, > > I have a problem I'm trying to solve and I would appreciate any input > that may be available. I am not so much looking for the exact solution, > as I am looking for ideas on how to go about solving it. I can workout > the details after, I just need to figure out exactly what it is I'm > trying to do. > > The Equipment: > > I have a sensor moving through the ocean from surface to 200m depth in > a saw-tooth pattern. As it moves, it records its depth every 4 seconds, > it also records values for dissolved oxygen. > > The Problem: > > The oxygen sensor has a "settling time (63%)" of ~25seconds. So when I > look at the depth measurements, the corresponding oxygen measurements > are not from that depth, they are from some depth before hand. > > I would like to correct for this, so I know exactly what depth my > oxygen measurements correspond to. > > Someone suggested something about fourier transforms, or modelling it > with e^-t/tau where tau is my 25seconds... but I am unsure of what they > actually meant. > > If someone could provide either an outline of what it is im trying to > do, or some keywords that I could google, it would be greatly > appreciated! > > thanks for your time,Seems doable. From the description of the manufacturer, it seems like the sensor behaves like a first order lowpass, ie. its step response is s(t) = 1 - exp(-t / tau), and tau = 25 seconds. If the sensor really is such a simple linear (in the operating range) and time-invariant system (this hypothesis might need verification via system identification), then you can just apply an inverse filter to retrieve the input (O2 concentration). Specifically: You have a series of measurements (k T, mo_k, d_k), where k = 0, 1, ... (the sampling indeces), T = 4 seconds (the sampling period), mo_k = measured O2 concentration and d_k = measured depth of the sensor. If the above hypothesis holds, ie. the O2-sensor transfer function can for example be approximated by H(z) = alpha / (1 - beta z^-1), beta = tau/(T+tau) and alpha = T/(T+tau), then you can retrieve the O2 concentration o_k at depth d_k by the simple inverse filter o_k = 1/alpha (mo_k - beta mo_{k-1}). The approximation of the transfer function improves if you reduce T with respect to tau (you measure faster). Regards, Andor
