Estimation of the gain factor for a discrete linear time-invariant system when the input and output series each have zero mean

The problem of gain factor estimation, for a discrete linear time-invariant system in which both the input and output series have zero means, is investigated. Two methods, based on the theorem of moments, are presented and tested in a simulation study on both error-free and contaminated output data. A third method, based on least-squares estimation for a simple discrete rectangular unit impulse response, is also considered and compared with the standard method of estimating the gain factor directly from the least-squares estimate of the non-parametric unit impulse response series, using both the data of the simulation study and also real catchment data from seven catchments.