Suppose I have a discrete time signal $x_t$ sampled with a frequency $f_s$. I know that if I take the discrete fourier transform of the signal and compute the power, I can easily (by visual inspection) determine which frequencies are the most important ones in the signal. I know that the signal is the superposition of several sine and cosine funtions, but I want to determine the amplitudes of each of these functions; i.e. the coefficients $a_n$ and $b_n$ in front of the sine and cosine functions, respectively.
Is there a way to estimate these fourier coefficients of a discrete time signal? How can I approximate them? Is there a way to estimate them using only the information from the power spectrum from discrete fourier transform?