# Practice Question on Computing the Fourier Transform of a Continuous-time Signal

Compute the Fourier transform of the signal

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

Guess $\chi(\omega)=2\pi\delta(\omega-2\pi k)\,$ such that $\mathfrak{F}^{-1}=e^{jk2\pi t}$

check:

$\mathfrak{F}(2\pi\delta(\omega-2\pi k))=\frac{1}{2\pi}\int_{-\infty}^\infty2\pi\delta(\omega-2\pi k)e^{j\omega t}d\omega=e^{jk2\pi t}$

Therefore,$\chi(\omega)=2\pi\delta(\omega-2\pi k)\,$

--Cmcmican 20:47, 21 February 2011 (UTC)

Instructor's comments: Take a look at your answer: it depends on k. However, the input does not depend on k. By the way, you can use the "mathcal" font to produce the curly X. Like this: ${\mathcal X}$. And if you use the inline class, it is aligned with the line like this: ${\mathcal X}$. -pm

I'll try this again, using the formula for Fourier transform of a periodic signal.

$\mathfrak{F}\Bigg(\frac{1}{2}e^{j2\pi t}+\frac{1}{2}e^{-j2\pi t}\Bigg)=\frac{2\pi}{2}\delta(\omega-2\pi)+\frac{2\pi}{2}\delta(\omega+\pi)$

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

--Cmcmican 17:38, 23 February 2011 (UTC) 