# Fourier Transform in the Form of Dirac-Delta Function

## Homework Statement

Given $x(t)=8cos(70\pi t)+4sin(132\pi t)+8cos(24\pi t)$, find the Fourier transform $X(f)$ in the form of $\delta$ function.

## Homework Equations

$X(f)=\int ^{\infty}_{-\infty}x(t)e^{-j\omega _0t}dt$
$cos(\omega t)=\frac{e^{j\omega t}+e^{-j\omega t}}{2}$
$sin(\omega t)=\frac{e^{j\omega t}-e^{-j\omega t}}{2j}$
$\int ^{\infty}_{-\infty}cos(\omega _0t)e^{-j\omega t}dt=\frac{\pi}{2}(\delta (\omega +\omega _0)+\delta (\omega -\omega _0))$
$\int ^{\infty}_{-\infty}sin(\omega _0t)e^{-j\omega t}dt=\frac{\pi}{j2}(\delta (\omega +\omega _0)-\delta (\omega -\omega _0))$

## The Attempt at a Solution

$X(f)=\frac{8\pi}{2}(\delta (\omega +70\pi)+\delta (\omega -70\pi))+\frac{4\pi}{j2}(\delta (\omega +132\pi)-\delta (\omega -132\pi))+\frac{8\pi}{2}(\delta (\omega +24\pi)+\delta (\omega -24\pi))$

Simplifying: $X(f)=4\pi (\delta (\omega +70\pi)+\delta (\omega -70\pi))+\frac{2\pi}{j} (\delta (\omega +132\pi)-\delta (\omega -132\pi))+4\pi (\delta (\omega +24\pi)+\delta (\omega -24\pi))$