Observability is an important characteristics of a system, which allows to address several problems when only the output function can be measured. Many authors have focused on the observer design problem starting from linear systems to nonlinear ones, adding then delays. Several differences already arise between continuous-time and discrete-time nonlinear delay-free systems (see, for example, Andrieu and Praly 2006; Besançon 1999; Bestle and Zeitz 1983; Califano et al. 2009; Krener and Isidori 1983; Xia and Gao 1989). When dealing with systems affected by delays additional features have to be taken into account. Some aspects are discussed in Anguelova and Wennberg (2010), Germani et al. (1996, 2002).