Fourier Transform of a Sum of Integrals

**cp05** · July 2nd 2012, 12:19 PM

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!

**BAdhi** · July 10th 2012, 08:53 AM

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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