Validity of differentiation property of Fourier transform

1. What is the main limitation of Fourier transform as a data analysis tool?
2. How do you overcome the limitation of a Fourier transform?
3. When can a Fourier series be differentiated?
4. What are the conditions for the existence of Fourier transform?

What is the main limitation of Fourier transform as a data analysis tool?

The major disadvantage of the Fourier transformation is the inherent compromise that exists between frequency and time resolution. The length of Fourier transformation used can be critical in ensuring that subtle changes in frequency over time, which are very important in bat echolocation calls, are seen.

How do you overcome the limitation of a Fourier transform?

The solution to this is time-frequency analysis, which is a field that deals with signal processing in both time and frequency domain.

When can a Fourier series be differentiated?

formally differentiating the Fourier series of f if and only if f(x) is almost every- where equal to a suitably chosen indefinite integral of f', and if in addition, fTf'(x)dx = 0. The necessity of these conditions hinges upon the theorem that two functions are equivalent if they have the same Fourier coefficients.

What are the conditions for the existence of Fourier transform?

Condition for Existence of Fourier Transform

The function x(t) has a finite number of maxima and minima in every finite interval of time. The function x(t) has a finite number of discontinuities in every finite interval of time. Also, each of these discontinuities must be finite.