Fourier Series

Discussion in 'Math' started by Mazaag, Jan 9, 2007.

  1. Mazaag

    Thread Starter Senior Member

    Oct 23, 2004
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    0
    Hey Guys..

    I was wondering if someone can give me a hand finding the Fourier Series Coefficients of the function f(t) = Acos(wt) .

    I tried using the following definition

    Xn = 1/T integral ( f(t) e^-jwnt )

    I converted the cos(wt) to its exponential form, then multiplied and combined and integrated.. however it sorta kept going on and on and doesn't seem to end :S

    does anyone have a link of the explanation on how to solve this.. ?

    thanks guys
     
  2. Dave

    Retired Moderator

    Nov 17, 2003
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    143
    You need to perform your integral over the time period T:

    Say T = π

    For f(t) = ACos(wt)

    Your even coefficients are given by:

    a(n) = 1/π [f(t)Cos(nt)] dt For -π < t < π

    Your odd coefficients are given by:

    b(n) = 1/π [f(t)Sin(nt)] dt For -π < t < π

    Substitute f(t) into the above equation and perform one integral using a suitable method (Parts).

    Because you are performing the integral over the period t, you should be able to derive a(n) and b(n) in terms of n. As a pointer, the Fourier Series of ACos(wt) will have and infinite number of even coefficents, a(n), and b(n) = 0. This should be clear because ACos(wt) is an even function.

    Clearer?

    Dave
     
  3. Mazaag

    Thread Starter Senior Member

    Oct 23, 2004
    255
    0
    Okay..

    So i got the fourier series of this Cosine wave. Its basically a 2 diracs at w0 and -w0 (fundamental frequency).

    now my question is this: what does this tell me ? like... does the fourier series of a signal give me the coefficients Xn of which I can construct that signal ? or am I getting things mixed up... ? If so , then what does the fouier series and fourier transform represent... ?

    Thanks guys
     
  4. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
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    The Fourier series gives you the frequency representation of a time-domain periodic signal. It decomposes your signal into a series of weighted sinusoidal harmonics that consistutes a mapping of the original signal. When in the frequency domain we can perform signal manipulation, enhancement, filtering etc. You should also be aware of the Discrete Fourier Transform (and FFT) when looking into Fourier analysis in the field of electroninc engineering.

    Dave
     
  5. Mazaag

    Thread Starter Senior Member

    Oct 23, 2004
    255
    0
    okay so in the case of the cosine function, when I took the transform I got 2 dirac deltas. (Wo and -Wo) what does that mean ? like what do these impulses tell me about the cosine wave.. and what does the y axis of the frequency spectrum represent..
     
  6. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
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    You get 2 Dirac deltas which are the real components at frequency Wo and -Wo, and no imaginary components. This tells you that you have a cosine function (because you only have even components) at a single frequency of Wo, and amplitude 2-times the height of the impulse at Wo (to take account of the fact you have an impulse at Wo and -Wo).

    Dave
     
  7. alitex

    Active Member

    Mar 5, 2007
    122
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    this is pdf file about Fourier Series
     
  8. Dave

    Retired Moderator

    Nov 17, 2003
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    Its an interesting article, would you happen to have the answers to the problems?

    Dave
     
  9. DrNick

    Active Member

    Dec 13, 2006
    110
    2
    if you are finding the exponential form of the fourier series they would be

    C_1 = 1/2
    C_-1 = 1/2

    so your series would be (in exponential form)

    f(t) = A [1/2 exp(-jwt) + 1/2 exp(jwt)]

    or in cosine form

    f(t) = A cos(wt)

    or in sine form

    f(t) = A sin(wt -pi/2)

    in a sense that is kindof a redundent question. Acos(wt) is ALREADY a fourier series representation of f(t).
     
  10. alitex

    Active Member

    Mar 5, 2007
    122
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    yes dave if i can solve problem why i don't put solving?
     
  11. Dave

    Retired Moderator

    Nov 17, 2003
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    What I mean is, do you have the solution set for the problems in the pdf article you attached above?

    Dave
     
  12. alitex

    Active Member

    Mar 5, 2007
    122
    0
    what do u mean dave?
    is the my pdf file useless
    actually i have alot of informations about many scopes ;)
     
  13. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
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    No the pdf is useful. However towards the end there are some questions (also referred to as problems), do you have the answers to these questions? It doesn't matter if you haven't I was only wondering, if you have please share them so others can learn from the attachments.

    Dave
     
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