In this paper, a novel robust scheme is developed for detecting faults in nonlinear discrete time multi-input and multi-output systems in contrast with the available schemes that are developed in continuous-time. Both state and output faults are addressed by considering separate time profiles. The faults, which could be incipient or abrupt, are modeled using input and output signals of the system. By using nonlinear estimation techniques, the discrete-time system is monitored online. Once a fault is detected, its dynamics are characterized using an online approximator. A stable parameter update law is developed for the online approximator scheme in discrete-time. The robustness, sensitivity, and performance of the fault detection scheme are demonstrated mathematically. Finally, a Continuous Stir Tank Reactor (CSTR) is used as a simulation example to illustrate the performance of the fault detection scheme.