This article addresses the stable fault detection observer design problem for linear time-invariant continuous-time systems in finite-frequency domain. The fault detection filter design is a synthesised optimal Luenberger observer that guarantees two requested performance indexes of fault sensitivity and stability. With the aid of generalised Kalman-Yakubovich-Popov lemma and increasing dimensions of slack variable matrix, the stability and H- performance analysis of the closed-loop system with a fault detection observer has been translated into a convex linear matrix inequality (LMI) optimisation problem to avoid the complexity of system associated with weight functions. An iterative LMI algorithm has been presented for the fault detection observer design. The effectiveness of proposed approaches is demonstrated by two numerical examples.