Peak of Analytical Fourier Transform

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  • Thread starter Luke Tan
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  • #1
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Summary:

Finding the peak frequency in an analytical fourier transform

Main Question or Discussion Point

In a numerical fourier transform, we find the frequency that maximizes the value of the fourier transform.

However, let us consider an analytical fourier transform, of ##\sin\Omega t##. It's fourier transform is given by
$$-i\pi\delta(\Omega-\omega)+i\pi\delta(\omega+\Omega)$$
Normally, to find the value of ##\omega## that maximizes this function, we would differentiate with respect to ##\omega## and set to 0. However, in this case, the derivative of a dirac delta function cannot be evaluated. Hence, we are unable to find the peak frequency to be at ##\Omega## unlike what we would in a numerical, discrete-time fourier transform.

Is there any way around this, to find the peak frequency of the fourier transform of a function?
 

Answers and Replies

  • #2
DrClaude
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Te Dirac delta is 0 everywhere except for ##\delta(0) \neq 0##. It shouldn't be too har to find where the FT has a peak :smile:
 
  • #3
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Te Dirac delta is 0 everywhere except for ##\delta(0) \neq 0##. It shouldn't be too har to find where the FT has a peak :smile:
oh so this is a special case, and does not disqualify differentiating and setting to 0 as a general method of finding the peak frequency?
 
  • #4
DrClaude
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oh so this is a special case, and does not disqualify differentiating and setting to 0 as a general method of finding the peak frequency?
Yes, it's a special case because the Dirac delta is not a real function, but a distribution, so you can't apply the same methods as you would normally use.
 
  • #5
osilmag
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What frequency is the sine wave oscillating at? That is where your delta function will be and at ##2 \pi - \Omega##

You could always use Matlab or etc. to find maxes and mins of Fourier's.
 

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