In this article, a new event-triggered H-i/H-infinity optimization fault detection (FD) scheme is proposed for linear discrete time systems. A linear discrete time system with varying sampling periods is first presented to describe the dynamics of an event-triggered system. Based on this, an observer-based fault detection filter (FDF) is constructed as an event-triggered residual generator so that the generated residual signal is completely decoupled from the event-triggered transmission error. The design of event-triggered FDF is formulated as a multiobjective optimization problem denoted by H-i/H-infinity and an optimal solution can be obtained by recursive calculation of Riccati equations. In addition, an event-triggered residual evaluation strategy is also presented. Comparing with the existing event-triggered FD schemes, the major contribution of this article is to decouple the residual signals from the event-triggered transmission errors completely and, consequently, the FD performance can be improved significantly. Moreover, an optimal tradeoff between the robustness of residual to unknown input and the sensitivity of residual to fault is achieved in the sense of H-i/H-infinity optimization, while the design of FDF and event generator can be carried out independently. To demonstrate the effectiveness of the proposed method, a vehicle lateral dynamic system is used as a simulation example.