# Is a system characterized by it's transfer function?

## Main Question or Discussion Point

This isnt really an electrical engineering question, bit I have other EE-related questions about it.

First off, If you are given a system's transfer function, is it unique? Or can other different functions have the same transfer function.

Second, does the Transfer function have anything to do with the Fourier transfer of its imput or output?

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MATLABdude
First off, a transfer function is the Laplace transform of the linear time-invariant system. The Fourier transform is a specific case of a Laplace transform (the one in which only imaginary frequencies are considered), consequently, you can use the Transfer function in Fourier space (making the substitution s=j*w)

As for the first question, given that the transfer function is just the Laplace transform of the system, your question is equivalent to asking whether or not a Laplace transform is unique for a given input function (there is a proof underneath the Uniqueness section):
http://math.fullerton.edu/mathews/c2003/LaplaceTransformMod.html [Broken]

Hope that helps!

EDIT: The link I gave above regarding uniqueness of the Laplace Transform refers you to the book for the proof of uniqueness; if you're willing to take their word for it (and http://mathworld.wolfram.com/LaplaceTransform.html" [Broken]) that's okay, but if you're looking for the mathematical proof, take a look in a Ordinary Differential Equation book (e.g. Nagle and Staff, Fundamentals of Differential Equations)

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First off, a transfer function is the Laplace transform of the linear time-invariant system. The Fourier transform is a specific case of a Laplace transform (the one in which only imaginary frequencies are considered), consequently, you can use the Transfer function in Fourier space (making the substitution s=j*w)

As for the first question, given that the transfer function is just the Laplace transform of the system, your question is equivalent to asking whether or not a Laplace transform is unique for a given input function (there is a proof underneath the Uniqueness section):
http://math.fullerton.edu/mathews/c2003/LaplaceTransformMod.html [Broken]

Hope that helps!

EDIT: The link I gave above regarding uniqueness of the Laplace Transform refers you to the book for the proof of uniqueness; if you're willing to take their word for it (and http://mathworld.wolfram.com/LaplaceTransform.html" [Broken]) that's okay, but if you're looking for the mathematical proof, take a look in a Ordinary Differential Equation book (e.g. Nagle and Staff, Fundamentals of Differential Equations)
Thanks. Im not really looking for a proof, Ill take your word for it :) I do have Nagel/Staff's Diffeq book lying around, Ill look it up in that if I ever need to.

So the Transfer Function is the Laplace Transform of the system. So to find the Transfer function, you'd characterize the function in the time domain then Laplace Transform it?

And Taking the fourier transfer of the function would be the same as taking the Laplace function, then substituting...what? (you mentioned subbing s=iw?)

The Fourier transform shows the frequency content, and takes a function from the time domain to the frequency domain. Similarly, taking the laplace transform maps a function from the time domain to what domain? that is, what physical quantity does the Laplace transform reveal?

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MATLABdude
Thanks. Im not really looking for a proof, Ill take your word for it :) I do have Nagel/Staff's Diffeq book lying around, Ill look it up in that if I ever need to.

So the Transfer Function is the Laplace Transform of the system. So to find the Transfer function, you'd characterize the function in the time domain then Laplace Transform it?
You could do that. Most often, you have some differential equation, and then you take the Laplace transform of that (why? differentiation is a 1/s factor, and integration is an s multiplier--reduces your differential equation into an algebraic one). But coming up with the differential equation (with, say, an RLC circuit) is, in fact, characterizing the system in time.

And Taking the fourier transfer of the function would be the same as taking the Laplace function, then substituting...what? (you mentioned subbing s=iw?)

The Fourier transform shows the frequency content, and takes a function from the time domain to the frequency domain. Similarly, taking the laplace transform maps a function from the time domain to what domain? that is, what physical quantity does the Laplace transform reveal?
Yes, but as I mention, a Fourier transform is just a special case of the Laplace transform. In a Laplace transform, you use complex frequencies $$s=\sigma+j*\omega$$, while in the Fourier transform, $$\sigma$$ is 0 (purely imaginary--i.e. a sine, or cosine, or some such, the basis set of the Fourier transform!)

Actually, given some of your questions, I may be getting ahead of myself... If you haven't taken your EE continuous-time Laplace / Fourier transforms class (often goes by the name Continuous Time Signals, or Fourier Transforms, or some such), or an introductory controls class, then that's where you'll be learning this more in detail. The Laplace transform does map to a frequency transform--just a complex one (a 'real' frequency means that a signal has a non-periodic portion).

Unfortunately, it's been a number of years since I've really any of this in great detail, so I may have to refer you back to the textbook, or one on Linear Systems (B. P. Lathi, but hopefully something a little better and less pricey). In the meantime, I'll point you to the Wiki regarding the relationship between the Laplace and Fourier transforms:
http://en.wikipedia.org/wiki/Laplace_transform#Fourier_transform

But coming up with the differential equation (with, say, an RLC circuit) is, in fact, characterizing the system in time.
Thanks, and yes thats what I was reffering to :).
Yes, but as I mention, a Fourier transform is just a special case of the Laplace transform. In a Laplace transform, you use complex frequencies $$s=\sigma+j*\omega$$, while in the Fourier transform, $$\sigma$$ is 0 (purely imaginary--i.e. a sine, or cosine, or some such, the basis set of the Fourier transform!)

Actually, given some of your questions, I may be getting ahead of myself... If you haven't taken your EE continuous-time Laplace / Fourier transforms class (often goes by the name Continuous Time Signals, or Fourier Transforms, or some such), or an introductory controls class, then that's where you'll be learning this more in detail. The Laplace transform does map to a frequency transform--just a complex one (a 'real' frequency means that a signal has a non-periodic portion).
I have taken a signals and systems class that covered DT and CT Fourier transforms, but I didn't invest much effort to remember the material since Its not my major. I think I am going to take the upper division version of this class though, since it seems that this sort of thing is very applicable to what I plan to study (control systems).

The problem I have now has to do with a automotive suspension (a model, not a control system), in which the suspension is the system. Im given the transfer function which maps input frequencies (vibrations in the road) to the vibrations the car actually feels.

So if my understanding is correct, if I were to take the inverse Leplace transform of this function, would it give me the differential equations governing the suspension's vertical motion based on the road, correct?

EDIT: I had more questions, but Ill just wait and see if they get answered next year in my controls class, as you suggested.

Thanks MATLABdude.

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MATLABdude
Thanks, and yes thats what I was reffering to :).

I have taken a signals and systems class that covered DT and CT Fourier transforms, but I didn't invest much effort to remember the material since Its not my major. I think I am going to take the upper division version of this class though, since it seems that this sort of thing is very applicable to what I plan to study (control systems).

The problem I have now has to do with a automotive suspension (a model, not a control system), in which the suspension is the system. Im given the transfer function which maps input frequencies (vibrations in the road) to the vibrations the car actually feels.

So if my understanding is correct, if I were to take the inverse Leplace transform of this function, would it give me the differential equations governing the suspension's vertical motion based on the road, correct?

EDIT: I had more questions, but Ill just wait and see if they get answered next year in my controls class, as you suggested.

Thanks MATLABdude.
I find that the control system guys (be it chemical, electrical, or mechanical) are the ones that end up using transfer functions and what not the most often (so learn the theory well). The inverse LAPLACE transform does indeed give you the suspension's differential equation (and yes, given that the system is a suspension, it's probably the vertical motion as you say).

To figure out the output given a particular road, you'd have to convolve the suspension transfer function with the road's vertical displacement function in the time domain. Or just multiply them in the frequency domain (which is why it's so useful). And inverse laplace transform the product, of course.

However, the other great thing about knowing the transfer function of the system is that you can find the frequency response of the system and plot a Bode plot, thus telling you how your system responds given inputs of various frequencies. For instance, if your transmission has a pole at, say, 50 Hz, when your car is going over a series of speed bumps at 50 rad/s, you're going to be in for a pretty rough ride:
http://en.wikibooks.org/wiki/Control_Systems/Bode_Plots

Pick up a controls textbook (or find out the one you're using in your intro class) it should cover this material in far greater depth than on an internet forum.