This paper reformulates the suboptimal (or level-gamma) H-infinity filtering problem into a Linear Matrix Inequality (LMI) problem by applying a bounded real lemma to the closed-loop transfer function. This formulation not only provides the condition of solvability but also constructs the suboptimal H-infinity filter. This formulation furthermore allows one to solve the ’epsilon-optimal’ H-infinity filtering problem via convex optimization techniques, where the ’epsilon-optimality’ implies that the H-infinity-norm of the resulting closed-loop transfer function is greater than or equal to the infimum, say gamma(inf) of the H-infinity-norms of all possible closed-loop transfer functions, but is less than or equal to epsilon + gamma(inf) for an extremely small positive number epsilon, which will depend on the accuracy of the algorithms and the precision of a computer.