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Laplace Transform vs. Transfer Function

  1. Sep 6, 2010 #1

    I am confused between the difference between the Laplace transform and the Transfer function. I used to think that the Transfer Function was the Laplace transform of the Differential equation representation of a system, but in my readings it seems like that is incorrect - because according to my recent reading the transfer function is the laplace transform of the impulse response.

    eg: say a system is defined such as:
    [tex]\dot{y} + y = x[/tex]

    the transfer function of this is:
    [tex]H = \frac{1}{s+1}[/tex]
    and this is obtained from a simple replacement of the derivatives with s.

    so if I carry out inverse laplace transform of H, will it give me the impulse response of the system?

    Also, on a side note, I am unclear on teh difference between the laplace and fourier transforms - what is the difference between them?

  2. jcsd
  3. Sep 6, 2010 #2
    A transfer function is the output over the input. By taking the inverse laplace transform of the transfer function, you're going back into the time domain (or x-domain, in your case). An impulse response means your input is a unit step function, u(t), or one for t>0. So you're essentially multiplying the transfer function by one. So, in that case, once you've taken the inverse laplace transform, I think you'd have your impulse response.

    For fourier, that is more of switching into the frequency domain (magnitude and phase plots, filter stuffs, etc). It is important when you start getting digital with the Z domain, mainly signal processing.

    ....i think

    edit: I guess it doesn't have to be time domain. A transfer function states the output over the input. A Laplace transform is switching from one domain (I usually think of time) to the s-domain. Fourier Transform is (to me) usually time domain to frequency domain. And the inverse just go back the other way.
    Last edited: Sep 6, 2010
  4. Sep 6, 2010 #3
    Are you sure thats the case? the impulse response is the response when the input is the delta function.

    The transfer function is easily found from a linear differential equation (changing derivatives to s's).

    But, if you find the impulse response of a function and then take the laplace transform of that, do you get the same transfer function that you get using the method in the paragraph above?
  5. Sep 7, 2010 #4


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    I think that in general Laplace is an analysis of the transient response of a system, whereas Fourier is an analysis of the steady-state response of a system. Note that for Laplace, you can use just about any function and do the transform, but for Fourier you must have a periodic (repeating pattern) waveform. There are a number of situations where s <--> jw shows important details too.
  6. Sep 7, 2010 #5
    "An impulse response means your input is a unit step function, u(t), or one for t>0."

    An impulse response represents the transient response of the system. It is the system response for a an "instantaneous" input decaying with time. The step response is more like the steady state response of the system.

    Both Laplace and Fourier transforms are used for mapping time domain functions into the frequency domain (as s = o + jw). Fourier is a special case of the Laplace transform where o = 0 leaving s = jw. Laplace is better used for problems with ICs whereas Fourier is better used with boundary conditions (in fact it needs boundary conditions).

    The transfer function, as stated above, is the relationship between the input and output of the system. For your example above, an easy way to find it (as you've mentioned) is to take the Laplace transform of both sides of the equation and rearrange it so it's output/input. Note: It needs to be an LTI system.
    The inverse laplace of a transfer function will simply give you the time domain response of the system. In the case of your example above, the response is exp(-t)
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