Peak of Analytical Fourier Transform

[Luke Tan]

TL;DR Summary

Finding the peak frequency in an analytical fourier transform

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?

 

[DrClaude]

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

 

[Luke Tan]

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

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?

 

[DrClaude]

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.

 

[osilmag]

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.