The consensus-based distributed fusion estimation problem with communication bandwidth constraints is investigated. An optimization problem which sets the consensus filter gains and the sum of the traces of all sensors' finite horizon estimation error covariance matrices to be the decision variables and the cost function, respectively, is established. For given consensus weights, sufficient conditions for the existence of the consensus filter gains which makes the dynamics of the estimation errors without noise asymptotically stable are given. Then, a set of sub-optimal consensus filter gains are computed by minimizing an upper bound of the cost function. Finally, simulation example is given to illustrate the effectiveness of the proposed approach.