How Can I Prove This Fourier Transform Pair for a Rectangular Function?

[thedean515]

Homework Statement

I'd like to prove a F/T pair and to confim if they are correct.

s(t) = A Sin[w0 t] * rect[t/T - T/2] ... (1)

it's Fourier transform is

S(f) = exp(-j w T)*T/2*A* {Sinc[(w+w0)T/2/Pi] + Sinc[(w-w0)T/2/Pi]} ...(2)

where rect is rectangular function

Homework Equations

I can prove rect[t/T] -> T Sinc[Pi f T]

The Attempt at a Solution

I tried to use mathematica, but it didn't give me as good results as (2)

Somebody can prove it?

 

[christianjb]

You know what the FT of rect and sin(wt) is. Use convolution theorem to get the FT of rect*sin.

 

[thedean515]

Hi, thanks chistianjb. I was going to using convolution, but seems too much maths involved. Because rectangular has only value within a range, this will simplfy the integration lots.

I worked out the range of t is between (T+T^2)/2 and (T^2-T)/2, am I right?

sb can try to integrate[Sin[w0 t], {t, (T+T^2)/2, (T^2-T)/2}]?