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.