## Example of Computation of Fourier transform of a CT SIGNAL

A practice problem on CT Fourier transform

Let the signal x(t) be equal to:

$ x(t) = cos(2\pi t) \, $

The Fourier Transform of a signal in Continuous Time is defined by:

$ X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt \, $

Using this, we obtain:

$ X(\omega) = \int_{-\infty}^{\infty}cos(2\pi t)e^{-j\omega t}dt \, $

Knowing that cos(t) is equal to: $ \frac{e^{jt}+e^{-jt}}{2} $:

$ X(\omega) = \int_{-\infty}^{\infty}\frac{1}{2}(e^{j2\pi t}+e^{-j2\pi t})e^{-j\omega t}dt \, $

$ X(\omega) = \pi \delta(\omega - 2\pi) + \pi \delta(\omega - 2\pi) \, $

## Comments/questions

- Note: This could also be done knowing that the Fourier transform of $ e^{j\omega_0 t} = 2\pi \delta(\omega - \omega_0) \, $.