Several robust fault detection problems for linear discrete time-varying systems have been given in this paper. It is shown that the optimal robust detection filters for those well-recognized robust fault detection problems, such as ℋ-/ℋ_∞, ℋ₂/ℋ_∞ and ℋ_∞/ℋ_∞ problems, have the same observer structure, whose gain can be obtained by solving a standard difference Riccati equation. Furthermore, the solution is also extended to the case when the initial state is nonzero. Finally, an example is used to illustrate our results.