# Approximating the fourier coefficients from a discrete time signal

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?

• "Is there a way to estimate them using only the information from the power spectrum from discrete fourier transform?" you'll lose your phase information so you would only be able to surmise $\sqrt{a_n^2+b_n^2}$ and not $a_n$ and $b_n$ individually. – robert bristow-johnson Apr 30 '15 at 18:53
• is the frequency of your periodic (or is it "quasi-periodic"? with is this? a musical note, perchance?) or its period known in advance? could the period be a non-integer number of samples long? – robert bristow-johnson Apr 30 '15 at 18:57
• @robertbristow-johnson: Thank you for confirming my suspicion that information would be lost. The signal that I'm analyzing converges to a periodic signal as $t\rightarrow\infty$, I'm analyzing the signal after a long time has passed and the signal is virtually periodic. However, the period is not known a priori. Yes, the signal can be non-integer samples long. – Paul Apr 30 '15 at 19:03
• then this becomes the same problem as "pitch detection" (to get the period that is not known a priori) followed by the interpolation necessary to make your period an integer number of (new, interpolated) samples. once you do that, you can use the DFT to get your Fourier coefficients. – robert bristow-johnson May 1 '15 at 18:21