In this short note, the problem of locally reconstructing the state of a nonlinear system is studied. To avoid computational difficulties arising in the numerical differentiation of the output, the so-called delay coordinates are considered. The assumptions of analyticity and (local) observability of the system are shown to imply that a family of mappings, induced by the delay coordinates and parameterized by a time delay parameter, gives a local diffeomorphism for generic values of such delay parameter on a certain set. A worked-out example illustrates the result.