In this article, finite impulse response (FIR) control is addressed for H output feedback stabilisation of linear systems. The problem we deal with is the construction of an output feedback controller with a certain FIR structure such that the resulting closed-loop system is asymptotically stable and a prescribed H norm bound constraint is guaranteed. Some solvability conditions are suggested in this article. Sufficient conditions are derived to obtain a suboptimal solution of the H FIR control problem via convex optimisation. Also, an equivalent condition for the existence of H FIR control is presented in the set of linear matrix inequalities (LMIs) and a reciprocal matrices equality constraint. An effective computational algorithm involving LMIs is suggested to solve a concave minimisation problem characterising a local optimal solution of the H FIR control problem. Numerical examples demonstrate the validity of the proposed H FIR control and the numerical efficiency of the proposed algorithm for FIR control.