For the fault detection filter design of linear time-invariant systems, we propose a novel problem formulation, which not only makes a tradeoff between fault sensitivity and disturbance sensitivity, but also incorporates partial fault decoupling and disturbance decoupling. This formulation compensates the drawbacks of the available frameworks such as H-/H-infinity, H-2/H-infinity and H-infinity/H-infinity. One optimal fault detection filter design for this problem framework is derived. It is shown that the faults in a certain space have arbitrary sensitivities, while the faults in the complementary space have bounded and maximized sensitivities with the proposed filter. Both the decoupling and non-decoupling conditions are derived. Moreover, decoupling disturbances is discussed. Finally, an example is given to illustrate the results.