This paper considers the concept of input and state observability, that is, conditions under which both the unknown input and initial state of a known model can be determined from output measurements. We extend previous results on input and state observability for discrete-time systems by considering a more general framework. Particularly, we introduce a delay in the input estimation, that is, we consider estimation of delayed input instead of the current input. Furthermore, we derive results for cases in which partial inputs may be available. Finally, we present several illustrative examples, including a system in which partial inputs are available.