Please forgive me if this has already been asked. Let us assume an example with $x(t) = \sum_{i=1}^N A_i \sin(2\pi f_i t) $ under a given sampling frequency $f_s$, frequencies $\omega_i$ and Amplitudes $A_i$. My ideas was to calculate the DFT transform $\tilde{x}(\omega)$, find the peaks of $|\tilde{x}(\omega)|$ and infer the original amplitudes from them.
However, there are several obstacles.
- There is leakage, and the Maximum of the peak is a bad indicator for the Amplitude of the signal; the normalization factor seems to be dependent on several factors.
- The Peaks overlap.
- I need a window function to get rid of the side lobes, which again changes my normalization.
Do you have experience in doing this? What is the best practice? Any useful literature?