Abstract : We investigate the spectral asymptotic properties of the stationary dynamical system $\xi_t=\varphi(T^t(X_0))$. This process is given by the iterations of a piecewise expanding map $T$ of the interval $[0,1]$, invariant for an ergodic probability $\mu$. The initial state $X_0$ is distributed over $[0,1]$ according to $\mu$ and $\varphi$ is a function taking values in $\R$. We establish a strong law of large numbers and a central limit theorem for the integrated periodogram as well as for Fourier transforms associated with $(\xi_t)$. Several examples of expanding maps $T$ are also provided.