This paper addresses the fault detection observer design problem for linear parameter-varying systems. Two finite frequency performance indexes are introduced to measure the fault sensitivity and the disturbance robustness. First, the H- index fault sensitivity condition in finite frequency domain is obtained by generalised Kalman-Yakubovich-Popov lemma and new linearisation techniques. Then, with the aid of Kalman-Yakubovich-Popov lemma and projection lemma, the stability and robustness conditions are derived. It turns out that the non-convexity problem which is caused by dealing with the above three conditions can be translated into a bilinear matrix inequality optimisation problem by increasing the dimensions of slack variable matrix. An iterative linear matrix inequality algorithm is proposed to get the solution. The effectiveness of the filter is shown via three numerical examples.