This article addresses the fault detection observer design issue for linear time invariant (LTI) systems with additive or multiplicative uncertainties, which are also subject to unknown disturbances. The observer design is investigated under the H-infinity/H_ index framework using the generalised KYP lemma in the finite-frequency domain. Sufficient conditions for the existence of such a fault detection observer are given in terms of linear matrix inequalities (LMIs). The threshold design issue is discussed and a method for estimating the worst undetectable fault size is proposed. The effectiveness of the proposed algorithms is illustrated by numerical simulation examples.