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This paper deals with the fault detection (FD) problem in finite frequency domain for linear time-invariant discrete-time systems with bounded disturbances. A fault detection observer is designed by employing two finite frequency performance indexes which are used to increase the fault sensitivity and attenuate the effects of disturbances. Faults are considered in the low frequency domain while disturbances are considered in certain finite frequency domain. With the aid of the Generalized Kalman-Yakubovich-Popov (GKYP) lemma, the design methods are presented in terms of solutions to a set of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed method.