# Fourier Transform Help

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• Aug 4th 2011, 11:29 PM
TheFirstOrder
Fourier Transform Help
I just need some clarification. I have been asked how to find the inverse rectangular pulse signal x(t) but I am not sure how to go about finding it. I guess what I have trouble understanding is the "inverse rectangular pulse signal". I know how to solve for just for the "rectangular pulse signal" through the Fourier Transform, but how does it change for the inverse? Many thanks.(Wait)
• Aug 5th 2011, 01:08 AM
Ackbeet
Re: Fourier Transform Help
There are at least three definitions of the Fourier Transform. Which one are you using? I would interpret "inverse rectangular pulse signal" as "find the inverse Fourier Transform of the rectangular pulse signal". The definition of the inverse Fourier Transform depends on your definition of the Fourier Transform.
• Aug 5th 2011, 04:08 AM
TheFirstOrder
Re: Fourier Transform Help
Quote:

Originally Posted by Ackbeet
There are at least three definitions of the Fourier Transform. Which one are you using? I would interpret "inverse rectangular pulse signal" as "find the inverse Fourier Transform of the rectangular pulse signal". The definition of the inverse Fourier Transform depends on your definition of the Fourier Transform.

I have been given several definitions of the Fourier Transform, but the one I think I am suppose to use is:

Attachment 21970

So must use the following to solve my problem?

Attachment 21971
• Aug 5th 2011, 05:33 AM
Ackbeet
Re: Fourier Transform Help
Quote:

Originally Posted by TheFirstOrder
I have been given several definitions of the Fourier Transform, but the one I think I am suppose to use is:

Attachment 21970

So must use the following to solve my problem?

Attachment 21971

That is a fine Fourier Transform/Inverse Fourier Transform pair to use, and is one of the standard definitions. So what is the "rectangular pulse signal"? And I mean, what exactly is it? Can you write down a formula for it?
• Aug 5th 2011, 05:36 AM
TheFirstOrder
Re: Fourier Transform Help
Oh yes, sorry. The formula for the inverse rectangular pulse signal x(t) is defined by:

x(t)={ 0 |t|<a
={ 1 |t|>=a
• Aug 5th 2011, 05:40 AM
Ackbeet
Re: Fourier Transform Help
Ok, great. Next question: are you allowed to use tables to compute your inverse Fourier Transform, or are you supposed to compute from the definition?
• Aug 5th 2011, 05:43 AM
TheFirstOrder
Re: Fourier Transform Help
I don't think I am suppose to use the tables, so it going to have to be from the definition. :)
• Aug 5th 2011, 05:46 AM
Ackbeet
Re: Fourier Transform Help
Right. So, can you write down the integral you must compute?
• Aug 5th 2011, 05:54 AM
TheFirstOrder
Re: Fourier Transform Help
Quote:

Originally Posted by Ackbeet
Right. So, can you write down the integral you must compute?

Okay, this is the bit where I am a little unsure of what I am doing, but here it goes:

Attachment 21976

But this doesn't seem right...
• Aug 5th 2011, 06:04 AM
Ackbeet
Re: Fourier Transform Help
I don't see any problem with it. You're using the definition of the Inverse Fourier Transform. The rectangular function eliminates the tails of your integration region, because it's zero there. So, crank through. What do you get?
• Aug 5th 2011, 06:14 AM
TheFirstOrder
Re: Fourier Transform Help
Quote:

Originally Posted by Ackbeet
I don't see any problem with it. You're using the definition of the Inverse Fourier Transform. The rectangular function eliminates the tails of your integration region, because it's zero there. So, crank through. What do you get?

Skipping a couple steps of working out, I got:

Attachment 21978

I am also required to sketch it. Is there anything I need to look out for when sketching it?
• Aug 5th 2011, 06:21 AM
Ackbeet
Re: Fourier Transform Help
Hmm. That's not what I get. Your final answer should definitely have an a in it. Can you show your steps?
• Aug 5th 2011, 06:30 AM
TheFirstOrder
Re: Fourier Transform Help
Sorry, I realised that I got my omegas and 'a's mixed up, as well as leaving out an 'i.t' in the previous working out, but it should be fixed now:

Attachment 21979
• Aug 5th 2011, 06:37 AM
Ackbeet
Re: Fourier Transform Help
I agree with everything except the last line. There shouldn't be an 'a' multiplying the sine.

So, here's the next question: now that you've got this expression, can you write it in a more compact format? Hint: examine the sync function.
• Aug 5th 2011, 06:45 AM
TheFirstOrder
Re: Fourier Transform Help
Oh yes, how silly of me! That 'a' shouldn't be there. Thanks for picking that up. Simplifying would get:

Attachment 21980

whereby sinc is the sync function. Is this right? And then how do I sketch it?
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