Fourier Transform H(t).cos(w0t)

[tsumi]

Homework Statement

Fourier Transform f(t)=H(t).cos(ω0t) ,using the transform of H(t)

H(t)=Heaviside function (also known as signal function if I ain't wrong)

Homework Equations

(1) FT[f(t)] = ∫ f(t).e^-(iωt) dt

(2) FT[H(t)] = pi.δ(ω) + 1/iω

(3) δ(ω) = Delta Dirac Function

(4) FT[cos(ω0t)] = pi (δ(ω-ω0) + δ(ω+ω0))

The Attempt at a Solution

I used equation (1) to get it, integrating by parts. The first part of parts integration yealds 0, then the new integral has the derivative of H(t), which is not bad since we can use the derivative propertie of the transform, but integrating cos(ω0t).e^-(iωt) yealds something quite far from the expected result. Am I on the right way? Any suggestions?

Thank you for any help

 

[vela]

Please show the details of your work.

 

[tsumi]

I got it! You have to make cos(ω0t) = 1/2 (e^(iω0t) + e^-(iω0t)) (euler formula)

Then use the linearity propertie of the transform and then the modulation propertie.

Thanks for such a quick response anyway =)