Fourier Transform !

Discussion in 'Math' started by F(t), Jul 18, 2012.

  1. F(t)

    Thread Starter New Member

    Jul 7, 2012
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    How to prove Fourier Transform of a Fourier Transform of a function is the Function Inverse?

    Thanks !
     
  2. WBahn

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    If I'm understanding what you are saying, I don't think it is.

    Give an example or two and perhaps I can figure out either what you are meaning to say or how I should be interpreting what you did say.
     
  3. t_n_k

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    Suppose one has an input signal as a simple sinusoid function.

    Take the Fourier transform of this input signal and then pass the Fourier transformed output data as a 'pseudo time' sequence input to a second Fourier transform process.

    The result from the second Fourier transform has the same sinusoidal form as the original signal.

    Other input signals such as square waves and other non sinusoidal periodic waves work as well.

    I've not attempted a proof - I've simply used a Labview simulation method to verify the result.
     
  4. steveb

    Senior Member

    Jul 3, 2008
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    I have a sneaking suspicion that you mean a "Time Reversed" function (assuming you have signals as a function of time), and not an "Inverse Function". Can you confirm?

    If you do confirm, then the proof is simple. Since this looks like a homework problem, I won't answer, but I'll give a hint. What is the difference between the "Fourier transform of a Fourier transform" and the "Inverse Fourier transform of a Fourier transform"? How would each of these operations affect a function that is symmetrical in time (or, if you don't have time based signals, symmetrical in x, or whatever variable you are interested in)?
     
  5. MrChips

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    The Fourier Transform exhibits reciprocity behaviour, i.e. the inverse transform of the transform gives the original signal.
     
  6. steveb

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    True, but that's not what he is asking. It's also clear that what he is asking is not worded correctly.
     
  7. F(t)

    Thread Starter New Member

    Jul 7, 2012
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    Thank You all. I have just started with the Signals and Fourier and I am sorry if I am not stating things correctly.
    Here it is again to prove, F[F(s)]= inverse Fourier Transform or the transform of F(signal) reversed.
     
  8. WBahn

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    Are you sure you aren't being asked to show that:

    Given:

    <br />
X(s) = F\{x(t)\}<br />

    Prove that:

    <br />
x(t) = F^{-1}\{X(s)\}<br />

    This is not as trivial as just saying that the inverse transform, by definition, must recover the original time-domain equation from the Fourier transform of that equation. If this is what was asked, then I suspect you are expected to take the mathematical definitions for each operation and show that they effectively cancel out.

    But F[F(s)] is not, in general, going to implement the inverse transform. If it did, then there would not be a different mathematical definition for the inverse transform. The difference may only be a minus sign in the exponential, but it is there for a reason.
     
  9. t_n_k

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    I think the task is to show that if F(u) is the Fourier transform of f(x) then

    F(F(u))=f(-x)

    or

    F(F(f(x)))=f(-x)

    Where F(F(u)) means "the Fourier transform of F(u)"
     
    Last edited: Jul 19, 2012
  10. steveb

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    This is exactly what I said above, but the OP chose to ignore my queries. He did not confirm nor deny that this is the question and if he did not understand, he didn't think it worth asking what I meant.

    I guess when your name is F(t), any discussion about F(-t) is off limits.
     
  11. t_n_k

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    Point taken - thanks. Also that's very clever / funny about the "off limits" comment.:)
     
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  12. F(t)

    Thread Starter New Member

    Jul 7, 2012
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    I believe t_n_k is able to correctly interpret what I am asking. But Steve says it should be a time reversed signal. Again what if its an inverse signal ?
    Steve, I am sorry I didn't reply to you ,plainly because I didn't understand it then. Anyway using your wits didn't help much.
     
  13. MrChips

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    Keep it clean guys, this is a G rated forum.
     
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  14. steveb

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    Yes, it did help me a lot! It made me feel better about being ignored, and it got you to tell me that you didn't understand what I said. :)

    When you don't understand, it's best to just ask for clarification. I did not expect that my answer would be complete and fully understandable, but I meant it as a way to start a conversation. So, please allow me to try and explain.

    By "time reversed" signal, I simply mean what t_n_k was saying in equation form. That is, f(t) gets transformed to f(-t). This means that the time direction is reversed, and more specifically, the function is spun 180 degrees around the vertical axis on a graph.

    What is an inverse signal? This term is general and can have multiple interpretations. (I think this is what caused the confusion) Strictly, the inverse signal would be the result of applying an inverse operation to a signal. However, there are many possible inverse operations to apply to a signal.

    For example, the reciprocal is an inverse operation. If you have f(x) and you take the reciprocal, you get 1/f(x). Now, take the reciprocal of this and you get f(x) again.

    Also, a sign change is an inverse operation. Take f(x) and make it negative to get -f(x). Now, make this negative and you get f(x) back again.

    Even the time reversal operation is an inverse operation. Take the time reversal of your name F(t), and you get your alter ego F(-t). Now, do another time reversal operation and you regain your sanity and become F(t) again.

    Basically, one can define many inverse operations. A Fourier transform is not an inverse operation. That is why we need to define an Inverse Fourier Transform to get us back to the original signal.

    So, your question just needed more clarification for us to help you. I was trying to get you to clarify. In fact, I think the wording of the original question (in post 1) is really not far off, and from a certain point of view is correct. However, it is open to interpretation, as proved by the fact that two very knowledgable people (t_n_k and WBahn) did not understand it immediately out of the gate (it confused me too).

    So, I ask once more. Do you think the problem is asking you to prove that the "Fourier transform of a Fourier transform" (in totality) is an inverse operator itself, since it is an operator that returns the time reversed version of the original signal? That is, f(t) becomes f(-t)? And, as I just mentioned above, the time reversal operator is an inverse operator.

    If this is the question, then we have just solved half the problem for you because now you know in what way it is an inverse operator. So, the multiple number of possible inverse operators has been narrowed down to one, which should make it easier for you to see the proof. The last step in the proof would read.

    * Therefore, the Fourier transform of a Fourier transform is equivalent to the time reversal operator (in the sense that t is replace by negative t), and since the time reversal operator is an inverse operator, the "Fourier transform of a Fourier transform" is an inverse operator.
     
    Last edited: Jul 20, 2012
  15. steveb

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    To say it more succinctly, consider the following. Above is your quoted question. I would think a better wording, that would lead to less confusion, would be the following.

    "How do you prove that the Fourier Transform of a Fourier Transform is an inverse operator, in the sense that using the operator twice on a function returns that function back?"

    Or, the following.

    How do you prove that the Fourier Transform of a Fourier Transform is an inverse operator, such that two such operations equals the identity operator?"
     
  16. F(t)

    Thread Starter New Member

    Jul 7, 2012
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    Okay Steve , Thank You. It is what you are quoting that Fourier Transform of a Fourier Transform is an Inverse operator. One last confusion I still have. When it is said the Reverse function, Inverse Function and the Negative Function. To what extent they are same or does anything goes beyond that ?
     
  17. t_n_k

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    Attached is a Labview simulation result of the Fourier transform [actually using an FFT algorithm] applied to a sawtooth function. The second transform is the 'reverse' of the input function. More correctly, if the input was f(t) then the second transform looks like a function f(-t).
     
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  18. steveb

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    OK, I can give my opinion about this, but I'm an engineer and not a mathematician, so this may not be perfectly correct. The thing we have to stress here is that clear definitions are very important in mathematics.

    Personally, I find "reverse function" and "inverse function" too vague to use without a definition given directly. It is much safer to define these terms when they are being used.

    I discussed the inverse function above, and provided a working definition for this context. I also pointed out that to be clearer, you would want to say what kind of inverse operation is being done. The definition of "inverse" is clear in mathematics, but it is just that there is more than one type of "inverse", and there is more than one meaning of "inverse function". You can say that the inverse function is one that was operated on by an inverse operator, and application of that same inverse operator again will return the same starting function back. That's how we define it here.

    Mathematicians also talk about inverse functions in the sense that you can have one function that does undoes the effect of another function. So, if one function multiplies by two, the the inverse function divides by two. In equation form you might have f(x)=2x, and the inverse function would be g(x)=x/2. We see the same thing with transforms, hence you have a Fourier Transform and an Inverse Fourier Transform. These are operations that are the inverse of each other, and this is a little different than inverse operators that have an operation that equals the inverse operation. (This is very confusing, isn't it!). For example, the reciprocal f(x)=1/x has an inverse function g(x)=1/x, but here f(x)=g(x), so it's a special case.

    The reverse function may not be an accepted mathematical term ( I could be wrong about that), but in any context one is free to define terms. My best guess would be that a reasonable definition of reverse function is substituting -t in for t, as we spoke about. Even my use of "time reversed function" is probably too vague without a definition because although it is reasonably clear that the flow of time is backwards, it is not clear that there is a 180 degree rotation about the vertical axis on a graph, at t=0. For example, a physicist might discuss time reversal symmetry, but he might not care where you define t=0. A mathematician might care very much that t=0 is the point of symmetry. In engineering we also might care where t=0 is, if the system is not time invariant. So, clear definitions matter.

    Using this definition, the reverse function is also one example of an inverse function, by the definition I'm using here.

    A "negative function" is an accepted term and the only reasonable interpretation is multiplication of the function by a negative one, or a simple sign change. Again, the negative function is an example of an inverse function, by the definition I'm using here.

    To summarize the answer to your question, if you choose these particular definitions above, the reverse function and the negative function are specific examples of inverse functions.
     
    Last edited: Jul 21, 2012
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  19. F(t)

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    Jul 7, 2012
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    The third graph , the reverse transform is like the Reverse Function itself .
    1.How generalized this can be?
    2.Most of the cases the Fourier Transform is used for a frequency varying result? How often time domain is used in Fourier Transform?
     
  20. F(t)

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    Jul 7, 2012
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    I understand it now, the relation between the each of Inverse , Reverse and a Negative Signal and the possibility - a mathematician , physicist or an engineer may have different ways of approaching each of the function. It all depends upon the conditions and the context. Bur now can it be stated
    "Fourier Transform of Fourier Transform" is either an Inverse, Reverse or a Negative Signal?
     
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