Calculate Fourier series of $f(x)=x^2$ , $x \in \ [-\pi,\pi]$ and determine module and phase spectrum

$$f(x)=\frac{a_0}{2}+\sum_{n=1}^{+\infty} a_n \ \cos(nx) \ + \ b_n \ \sin(nx)$$

$$a_0=\frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \ dx=\frac{2}{3} \pi^2$$

$$a_n=\frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \ \cos(nx) \ dx=\frac{1}{\pi} \ \frac{2(\pi^2 n^2-2) \sin(n \pi)+4 \pi n \cos(n \pi)}{n^3}=\frac{4 (-1)^n}{n^2}$$

$$b_n=0 \qquad \forall n\ge 1$$

because $f$ is even

$$f(x)=\frac{\pi^2}{3}+4 \ \sum_{n=1}^{+\infty} \frac{(-1)^n}{n^2} \ \cos(nx)$$

How can I determine module and phase spectrum?

Should I calculate the Fourier series coefficients in different values of n, then calculate module and phase of the result?


  • 2
    $\begingroup$ The module is simply $A_n=\sqrt{a_n^2+b_n^2}$ and the phase is $\phi_n=-\arctan\dfrac{b_n}{a_n}$. You should be careful with the $\arctan$ and find the proper value. $\endgroup$ – Galc127 Jan 9 '17 at 9:40


The amplitude spectrum is $$\{|a_{|i|}|\},\quad i=-\infty\dots\infty.$$ The phase spectrum is $$\{(-1)^i\pi\},\quad i=-\infty\dots\infty.$$ Both of the spectra are discrete. If time sequence is measurement in seconds then step of the spectra is 1 Hz.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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