# Peak of Analytical Fourier Transform

• I
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?

Mentor
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

osilmag
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?

Mentor
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.

Gold Member
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.