# Fourier Transform of a Sum of Integrals

• Jul 2nd 2012, 12:19 PM
cp05
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!
• Jul 10th 2012, 08:53 AM
Re: Fourier Transform of a Sum of Signals
if the signals are$\displaystyle f_1(x),f_2(x),...,f_n(x)$and their fourier transforms are $\displaystyle F_1(\omega),F_2(\omega),...,F_n(\omega)$ accordingly,

take,

$\displaystyle f(x)=f_1(x)+f_2(x)+...+f_n(x)$

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

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

by expanding,

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

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

I hope this will help