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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?

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?