Summary: The purpose of this paper is to show explicitly the spectral distribution function of some stationary stochastic processes as \[ X_t= F(X_{t-1}), \quad \text{for} \quad t \in \mathbb{Z}, \] where \(F\) is a deterministic two-dimensional invertible map. The invertible map \(F\) that will be considered in this paper is the natural extension of a map \(T\) on a class \({\mathcal F}_2\) (see Section 1 for definition) of one-dimensional piecewise linear expanding monotonic transformations. Any expanding map preserves an absolutely continuous invariant probability \(\mu\), also called Bowen-Ruelle-Sinai measure. In more precise terms, we are able to obtain explicitly a function \(\gamma\), in such way that the spectral density has the form \[ f_X (\lambda) ={1\over 2 \pi \text{Var}(X_t)} \bigl [\gamma (e^{i\lambda}) +\gamma (e^{-i \lambda}) -E(X^2_t) \bigr], \quad \text{for any} \quad \lambda\in (-\pi, \pi], \] when the random variable is position on the \(x\) axis, that is when the stationary process \(X_t\) given by \(\mu,T\), \(X_t= \varphi (T^t(x))\) and where \(\varphi (x)= x\). Any nonlinear expanding piecewise monotonic transformation \(g\in {\mathcal F}_1\) (see Section 1 for definition) can be approximated by a map \(T \in {\mathcal F}_2\). From the structural stability of the maps we consider here, it will follow that the spectral density function of the natural extension of any nonlinear expanding piecewise monotonic transformation \(g\in {\mathcal F}_1\), can be approximated by explicit expressions obtained for the spectral density function of the natural extension of maps \(T\) in \({\mathcal F}_2\). Results for the one-dimensional map \(T\) can be obtained from results for the two-dimensional map \(F\). It is known that the poles of the zeta function can be obtained from the poles of the spectral density function. This is a good reason for being interested in the spectral density function. An explicit formula for the spectral density can give precise information about resonances in the stationary process. We also show in the last section that the periodogram of expanding maps is a good estimator.