This article researches the problem of finite frequency (FF) H-infinity filtering for linear discrete-time state-delayed systems. The disturbance is assumed to reside in low/middle/high frequency ranges. To reduce the conservatism of the results, delay-partitioning idea is used to derive a new FF bounded real lemma (BRL). By applying the generalized Kalman-Yakubovich-Popov lemma, two equivalent approaches to the proof of the proposed FF BRL are given, respectively, starting from transfer function and Lyapunov-Krasovskii functional. A new FF H-infinity filter design method is proposed in terms of solving a set of linear matrix inequalities. Finally, a numerical example clearly demonstrates the merits and effectiveness of the proposed method.