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Math Help - Deriving fourier transform of e^jwot

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    Deriving fourier transform of e^jwot

    Hi there! I have searched extensively online and in text books in attempts to learn how to derive the fourier transform of Exp(jwot), that is fourier transformer of e to the power of 'j times omega-subscript nought times t', where 'j' is the imaginary number symbol (otherwise known as 'i'), and wo is 2*pi*fo.

    According to fourier transform 'tables', the fourier transform of e^jwot is 2.pi.delta(w-wo), where w is the angular frequency variable, and delta is the 'impulse' function). So the answer is saying that the fourier transform of e^jwot is an impulse in the frequency domain, and the impulse is at frequency of wo.

    I know that the equation to start off with for the fourier transform for e^jwot is integral (from -infinity to +infinity) of e^-j(w-wo)t. dt

    But from there on, the text books all typically then go straight to the 'answer' - namely 2.pi.delta(w-wo). But I'm very keen to try understand how they reach the answer with the delta function. I'm thinking that I'm missing some maths theory to get there, so thought I'd like to ask the experts for advice to point me in the right direction to understanding the derivation (without using the fourier transform look-up tables).

    Thanks in advance!
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    Re: Deriving fourier transform of e^jwot

    Hey SouthPark.

    Did you calculate the integral of the fourier transform of the function? (Hint: e^a * e^b = e^(a + b) and integral of e^[ax + b] = 1/a*e^[ax+b] which includes a and b being complex.)
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    Re: Deriving fourier transform of e^jwot

    Hi Chiro! I did attempt to go through to write down the result of my integral. The general equation for the fourier transform of a function f(t) is integral from -infinity to +infinity of f(t) e^[-jwt]dt.

    So if my function f(t) is e^[jwot], then the integral equation is the integral from -inf to +inf of e^{[jwot].[-jwt]}dt
    Which then leaves me with the integral from -inf to +inf of e^[-j(w-wo).t]dt

    Then when I integrate, I think that the result is: [1/{-j(w-wo)}] x e^[-j(w-wo).t] , in which then I assumed that I would need to plug in the upper limit (+infinity) for t and lower limit (-infinity) for t in order to get a result. But I can't see a way to reach a result with the delta (ie impulse) function. I'm guessing that I'm missing some maths theory somewhere along the line.

    Is there any source out there on the net or from a text book that shows how we get 2.pi.delta(w-wo) from the integral (from -inf to +inf) of e^[-j(w-wo).t]dt ?
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    Re: Deriving fourier transform of e^jwot

    I just refreshed myself on the Fourier transform and I'm not even sure if you can transform this function:

    The reason is that Integral [-infinity,+infinity] |f(x)|dx is not finite for this particular function.

    Here is the source for the information:

    Fourier Transform -- from Wolfram MathWorld
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    Re: Deriving fourier transform of e^jwot

    Hi again Chiro. I'm thinking that the function that I'm trying to get the fourier transformer derivation for -- is related to the fourier transform of cos(wo.t), which is on the second row of the table from Fourier Transform -- from Wolfram MathWorld

    cos(wo.t) is equal to (1/2){e^[jwot] + e^[-jwot]}. So for cos(wo.t), we have two of these exponential terms - one being e^[jwot] and the other being e^[j.(-wo).t]. So the fourier transform of cos(wo.t) results in two of these delta (impulse) functions in the frequency domain, one impulse at frequency wo and the other impulse at frequency -wo.

    Text books and wikipedia indicates that the fourier transform of e^[jwot] is 2.pi.delta(w-wo), such as in :
    Fourier transform - Wikipedia, the free encyclopedia (reference number '303' in the table, linked to "Fourier transform non-unitary, angular frequency").

    I'm very interested about finding out the steps needed to link up the integral of e^[jwot] with the delta function (without just referring to the look-up table for the fourier transform pairs). I was thinking that the derivation would be readily available in text books, online sources etc, but looks like it is not the case. Thanks Chiro!
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    Re: Deriving fourier transform of e^jwot

    Are you trying to find the transform of the delta function? How does the delta function play into your problem (or relate to it)?
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    Re: Deriving fourier transform of e^jwot

    Hi Chiro again. I'm not trying to find the transform of the delta function.

    I'm trying to find out how the fourier transform of e^[jwot] becomes 2.pi.delta[w-wo].

    The fourier transform equation of a function f(t) is integral from -inf to +inf of f(t).e^[-jwt]dt
    So if I set my function f(t) as being e^[jwot], then the fourier transform of e^[jwot] is then written as:
    integral from -inf to +inf of {e^[jwot].e^[jwt]}dt
    which reduces to :
    integral from -inf to +inf of e^[-j(w-wo).t]dt

    And the last equation above is given in text books. And the text books immediately give the answer to the above integral as being 2.pi.delta(w-wo). And they don't show how they arrive at that result.

    I haven't yet found any sources (teaching guides, text books etc) that teach us how we actually arrive at the impulse function 2.pi.delta(w-wo) as being the answer to the integral.

    So basically trying to find out the mathematical steps needed to reach the answer of 2.pi.delta(w-wo). Thanks Chiro.
    Last edited by SouthPark; January 29th 2013 at 10:43 PM.
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    Re: Deriving fourier transform of e^jwot

    I took a look at Wikipedia and it came up with the derivation of the Delta Function which will help you:

    Dirac delta function - Wikipedia, the free encyclopedia
    Thanks from SouthPark
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    Re: Deriving fourier transform of e^jwot

    Hey thanks a lot Chiro! I believe that it is just what I'm looking for. Something solid that ties all these things together. Thanks a lot for your help and going out of your way to help me out Chiro. Greatly appreciated!

    Looks like it took quite a bit of work from the mathematics (from the past) to get the relationship. According to that link you gave, at Dirac delta function - Wikipedia, the free encyclopedia

    They have the relation delta(x-alpha) = {1/(2.pi)} times integral (from -inf to +inf) of e^[j.p(x-alpha)].dp
    So if I set x as being 'w' (ie omega), and alpha as being 'wo' (ie omega subscript nought), and p being 't', then that results in the equation that I was interested in, which is integral (from -inf to +inf) of e^[-j(w-wo)t].dt is equal to 2.pi.delta(w-wo).

    But one thing about the wiki site is - there is an equation that follows the text - quoting "In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has"......

    I notice that the equation from the wiki link that immediately follows the above quoted text (from the wiki link) is : delta(epsilon2 - epsilon1). I could be wrong, but I think it should correctly be delta(epsilon1 - epsilon2). That is, the subscripts '1' and '2' with the brackets of the 'delta' function on that page should be interchanged. Might be a typo on the page?
    Last edited by SouthPark; January 30th 2013 at 03:28 AM.
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