In some engineering applications, such as chaotic encryption, chaotic maps have to exhibit required statistical and spectral properties close to those of random signals. However, most of the papers dealing with synchronization and observer synthesis consider maps exhibiting poor statistical and spectral properties. Moreover, most of the time these properties, however essential for the chaotic encryption, are simply neglected. Unlike these papers, in our work we present the analysis of a new ultra weakly coupled maps system introduced by Lozi. The model is a deterministic one, but exhibits spectral properties (spectrum, correlation and autocorrelation) close to those of random signals, and successfully passed all the statistical tests for closeness to random signals (NIST). Two different observers have been designed. The convergence rate has been discussed in the case of affine maps, and the conditions to decrease the convergence rate by a factor of 16 have been presented, based on the locally linear behaviour of the weakly coupled map.