# Fourier series

The Fourier series of a signal $$x(t)$$ defined in $$[0,1]$$ is given by $x(t) = \sum_{n\in \mathbf{Z}} X_n \exp(i 2\pi nt)$ with coefficients $X_n = \int_{-1}^1 x(t) \exp(-i2\pi nt) dt$ with convergence in $$L^2$$ if $$x(t)$$ has finite energy.

Check :

• the equivalence between regularity of the signal and decay of its Fourier coefficients,
• the influence of the amplitudes of the coefficients,
• the effect of the truncation of the series.
Real:
Truncation:
$$f_0$$ :