Realization of correlated chaotic signals

We show how to realize one-dimensional chaotic signals with the given first order autoregressive (AR(1)) autocorrelation function (ACF) of the form r[k]=r[0]a^k, where 0<a<1. We consider the class of piecewise linear and “piecewise onto” maps defined from the unit interval [0,1] onto itself. We prove that these maps are chaotic using their topological conjugacy with known chaotic maps. The autocovariance function of sequences generated by these maps can be calculated analytically as an ensemble average. The expression for the autocovariance function leads to a way of designing chaotic signals with the given ACF. ACF estimates (temporal averages) of typical signals are close to the theoretical values (ensemble averages)