Boundary observer design of a system of nχ-ODEs coupled to nx-hyperbolic PDEs with positive convective speeds is studied. An infinite dimensional observer is used to solve the considered state estimation problem. The interconnection of the observer and the system is written in estimation error coordinates and analyzed as an abstract dynamical system on a specific Hilbert space. The design of the observer is performed to achieve global exponential stability of the estimation error with respect to a suitable norm. Sufficient conditions in the form matrix inequalities are given for the design of the observer. The effectiveness of the approach is shown in a numerical example.