I'm looking for a way to compute the Fourier coefficients for a periodic wave, knowing its spectral decomposition (i.e., amplitudes $A_n$ for each $F_n$ harmonics).

By "Fourier coefficients", I mean the $\alpha$ and $\beta$ in this formula:

$$f(t) = \sum_{n\ge0} \alpha_n\cos(n\omega t) + \sum_{n>0} \beta_n\sin(n\omega t)$$

I've seen quite a lot of explanations on how to compute spectral analysis knowing the coefficients, but I'm looking for the reverse process. I'm not really an engineer or a physicist though, so my math skills are a bit rusty.


1 Answer 1


We know that $$A_n ={ \sqrt {\alpha_n ^2+\beta_n ^2}}\,\text{ and }\,\theta_n=-\tan^{-1}\frac{\beta_n} {\alpha_n}$$ therefore $$\alpha_n=A_n\cos(\theta_n) \,\text{ and }\,\beta_n=-A_n\sin(\theta_n)$$ where $A_n$ is amplitude & $\theta_n$ is phase of the harmonic frequency.

  • $\begingroup$ Thank you for your answer. In your definition of $\theta_n$ you use $a_n$ and $b_n$ but didn't define them, what are those ? $\endgroup$
    – corbin-c
    Oct 30, 2019 at 16:39
  • 1
    $\begingroup$ He means $\alpha$ and $\beta$, not $a$ and $b$ in the tan. The problem is to find the phase. Spectral intensity gives the $A_n$. More inflormation is required beyond spectral intensity to get the phase. This is huge problem with more than a century of research devoted to it. $\endgroup$
    – mike stone
    Oct 30, 2019 at 19:38
  • $\begingroup$ @mikestone thanks I didn't even notice my mistake. $\endgroup$ Oct 30, 2019 at 19:41
  • $\begingroup$ @mikestone thank you for the clarification. In my work, phase is not really an issue, i'll just consider all harmonics are in phase. $\endgroup$
    – corbin-c
    Oct 31, 2019 at 12:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.