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Math Help - Fourier Transform of a Sum of Integrals

  1. #1
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    Fourier Transform of a Sum of Signals

    Hi guys,
    I was wondering, is the Fourier transform of a sum of signals the same as the sum of the fourier transforms of each signal? I want to say it is, but how would I go about trying to prove that?
    Thanks a bunch!
    Last edited by cp05; July 2nd 2012 at 01:54 PM.
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  2. #2
    Senior Member BAdhi's Avatar
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    Re: Fourier Transform of a Sum of Signals

    if the signals are  f_1(x),f_2(x),...,f_n(x)and their fourier transforms are F_1(\omega),F_2(\omega),...,F_n(\omega) accordingly,

    take,

    f(x)=f_1(x)+f_2(x)+...+f_n(x)

    F(\omega)=\int_{-\infty}^{\infty} f(x)e^{-i\omega x}dx

    F(\omega)=\int_{-\infty}^{\infty} \left[f_1(x)+f_2(x)+...+f_n(x)\right]e^{-i\omega x}dx

    by expanding,

    F(\omega)=\int_{-\infty}^{\infty}f_1(x)e^{-i\omega x}dx+\int_{-\infty}^{\infty}f_2(x)e^{-i\omega x}dx+...+\int_{-\infty}^{\infty}f_n(x)e^{-i\omega x}dx

    F(\omega)=F_1(\omega)+F_2(\omega)+...+F_n(\omega)

    I hope this will help
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