1. 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.

2. 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.

3. Re: Fourier Transform Help

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:

So must use the following to solve my problem?

4. Re: Fourier Transform Help

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:

So must use the following to solve my problem?

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?

5. 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

6. 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?

7. 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.

8. Re: Fourier Transform Help

Right. So, can you write down the integral you must compute?

9. Re: Fourier Transform Help

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:

But this doesn't seem right...

10. 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?

11. Re: Fourier Transform Help

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:

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

12. 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?

13. 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:

14. 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.

15. Re: Fourier Transform Help

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

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

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