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Fourier Transforms


Date: 10/19/2001 at 12:24:28
From: Stuart Park
Subject: Fourier Transforms

I can't get my head around the sinc function. I know what the sinc 
function is, but I can't convert my expression (in terms of sin) to an 
expression (in terms of sinc).

i.e.

sinc(x) = sin ((pi*X)/(pi*X))

What should I do to the right-hand term to get it into the form on the 
left? Do I divide by through by pi?

Thank you for taking the time to read my question.


Date: 10/19/2001 at 12:52:50
From: Doctor Douglas
Subject: Re: Fourier Transforms

Hi Stuart, and thanks for writing.

I'm not exactly sure what you mean by "get it into the form on the 
left." I'll make a guess, though, and if it doesn't answer your 
question, please feel free to write back:

   sinc(x) = sin(pi*x)/(pi*x)     Notice that your parentheses above
                                  yield a right-hand side of sin(1),
               sin(pi*x)          which is surely not what you meant.
           =  -----------
                pi * x

This formula gives us sinc(x) in terms of x. If we know what x is, we
simply plug it in here.

Now let's solve for sin(x) here. But to avoid confusion, let's say
y = pi*x, so x = y/pi, and rewrite the above as

  sinc(y/pi) = sin(y) / y         Looks much easier now, doesn't it?
 
So we now can solve for sin(y) as

  sin(y) = y * sinc(y/pi).

Of course, we can substitute any variable we like now for y.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Functions
High School Physics/Chemistry

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