In this work, we have studied the problem of designing dynamic observers (HDO) for discrete-time linear systems (DTLS) with time-varying delay (TVD) and disturbance. By designing an augmented Lyapunov-Krasovskii functional (LKF) with double summation terms using the Generalized reciprocally convex matrix inequality (GRCMI), as well as the Jensen-based inequality (JBI) and the Wirtinger-based inequality (WBI) that derive new less conservative time-dependent conditions. The resulting algebraic conditions form a set of linear matrix inequalities (LMIs) which can be solved by the LMI or YALMIP toolboxes. Furthermore, the observer under consideration has more degrees of freedom to be estimated, and is known as a generalized observer, where the proportional and proportional-integral observers are particular cases. Finally, two examples are given to demonstrate the validity and effectiveness of the findings.