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Math Help - Fourier Transform Help

  1. #1
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    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.
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    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.
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  3. #3
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    Re: Fourier Transform Help

    Quote Originally Posted by Ackbeet View Post
    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:

    Fourier Transform Help-screen-shot-2011-08-05-10.03.48-pm.png

    So must use the following to solve my problem?

    Fourier Transform Help-inverse-fourier-transform.png
    Last edited by TheFirstOrder; August 5th 2011 at 04:40 AM.
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    Re: Fourier Transform Help

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

    Click image for larger version. 

Name:	Screen shot 2011-08-05 at 10.03.48 PM.png 
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    So must use the following to solve my problem?

    Click image for larger version. 

Name:	Inverse Fourier Transform.png 
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    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?
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  5. #5
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    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
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    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?
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  7. #7
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    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.
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    Re: Fourier Transform Help

    Right. So, can you write down the integral you must compute?
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  9. #9
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    Re: Fourier Transform Help

    Quote Originally Posted by Ackbeet View Post
    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:


    Fourier Transform Help-screen-shot-2011-08-05-11.55.44-pm.png

    But this doesn't seem right...
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    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?
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  11. #11
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    Re: Fourier Transform Help

    Quote Originally Posted by Ackbeet View Post
    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:

    Fourier Transform Help-screen-shot-2011-08-06-12.14.12-am.png

    I am also required to sketch it. Is there anything I need to look out for when sketching it?
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  12. #12
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    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?
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  13. #13
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    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:

    Fourier Transform Help-screen-shot-2011-08-06-12.26.02-am.png
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  14. #14
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    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.
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  15. #15
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    Re: Fourier Transform Help

    Oh yes, how silly of me! That 'a' shouldn't be there. Thanks for picking that up. Simplifying would get:

    Fourier Transform Help-screen-shot-2011-08-06-12.43.20-am.png

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