Fourier Transform help (bit )

[benjamince]

Fourier Transform help! (bit urgent)

Hi there,

I'm having a recurring problem with my Fourier transforms that I have tried really hard to figure out but I feel like I'm missing something important. It keeps popping up in my communications and signal processing papers.

I keep getting FTs like:

∫e^(j2π(n/2T - f)t).dt

Does this reduce to anything? I've tried for a few hours to understand it and I'm pretty much stuck. If I had the fundamental frequency then it would make more sense to me, but I don't.

The question involves finding the FT of:

e^(jπn(t/T)


Thanks in advance for any help!

Ben

 

[chiro]

Hi there,

I'm having a recurring problem with my Fourier transforms that I have tried really hard to figure out but I feel like I'm missing something important. It keeps popping up in my communications and signal processing papers.

I keep getting FTs like:

∫e^(j2π(n/2T - f)t).dt

Does this reduce to anything? I've tried for a few hours to understand it and I'm pretty much stuck. If I had the fundamental frequency then it would make more sense to me, but I don't.

The question involves finding the FT of:

e^(jπn(t/T)


Thanks in advance for any help!

Ben

Hey benjamince and welcome to the forums.

Assuming all your other parameters are constant or not related to t, You should be able to use the fact that integrating e^(at)dt gives 1/a x e^(at) for the anti-derivative.

Is the above true or is your integral in terms of something more complicated where the a, above is in terms of a function of t?

 

[benjamince]

Thanks for the quick reply chiro!

Yeah, I did try that, but because I'm integrating over limits from -∞ to +∞ (the signal is a pulse train) then I get an undefined result:

1/(n/2T - f) * sin(2π*(n/2T - f)*∞)
- that's after converting from exponentials into sine form.

Should I be using these limits? I guess that the frequency content of a periodic signal is the same for each period right?

 

[JJacquelin]

I'm integrating over limits from -∞ to +∞ (the signal is a pulse train)

The function corresponding to a pulse train is not the same function that a sinusoidal continuous funtion from -infinity to +infinity.The writing of your integral is for the second, not for the first one.

 

[benjamince]

The function corresponding to a pulse train is not the same function that a sinusoidal continuous funtion from -infinity to +infinity.The writing of your integral is for the second, not for the first one.

Thanks for the reply.

Sorry I'm misunderstanding a bit. You can represent the pulse train as an infinite summation of sinusoids (hence the n in the equation), but I moved the summation sign outside the integral due to linearity properties of the FT - the pulse is actually a sinc function itself. (I think)

The problem I'm trying to solve doesn't say what the pulse train is doing, but it's likely being used for sampling, in which case I thought it theoretically did go from -infinity to +infinity.

The original question is:

Find the Fourier transform of:

(A/T) Ʃ sinc(πn(tau/T)) * e^(jπn(tau/T)) * e^(jπn(t/T))

T is the period, and tau/T is the mark-to-space ratio.

My answer is:

A Ʃ sinc(πn(tau/T)) * e^(jπn(tau/T)) * sinc(π(n/2T - f)T)

The answer isn't given..

Thanks again for the replies!