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Fourier Transform question

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tresca

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the question is
find the fourier transform of

g(t) = exp(-t)sin(2*pi*fc*t)u(t) where u(t) is the unit step function

I have the solution which is 2*pi*fc/ (1+4*pi^2 (f-fc)^2)

however, I'm getting stuck at a certain point.

I get as far as....

(1/2j)*( exp[j*2pi(fc-f)t-t] - exp[-j*2pi(fc+f)t-t] )

However when they integrate, they somehow make the fc+f into a fc-f, which I dont understand how they do it.

See attached file for question and solution. I dont understand the 4 line. Any thoughts ?

Thanks !
 

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who wrote that solution sheet? they did not do any type of error checking at all.

first off, step 1 is showing the wrong equation for the fourier transform. you don't multiply by sin()dx, you multiply by exp()dx. It is corrected in step 2, but it is very confusing.

step 3 is not correct. if you go through the math you just cannot get what is shown in step 3. how can you get a single exponential term when there is a difference between two exponential functions? the actual term after the integral sign should be:
{exp[j*2*pi*(fc-f)t-t]-exp[-j*2*pi*(fc+f)-t]}dt

from that you get what is in step 4 - but the denominator of the second term in step 4 is WRONG - it is not (fc-f) it is (fc+f)!!! how can you get a different term in the denominator from what is in the exponential???
 
I kinda knew that the solution was wrong pointing out some of the errors that you picked up as well...

however, most solutions ive come across are correct, though the math in between question and final solution may not be...so basically..i was doing it right all this time...

thanks..
 
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