A simple numerical procedure is described for the construction of minimum phase selective band-pass digital filters satisfying simultaneous pass-band amplitude and phase constraints for any specified stop-band loss. It is shown that, if the pass-band loss of the filter is expressed as a linear combination of cosine functions, then the filter's group delay can be linearly expressed in terms of these functions together with any stop-band loss constraints. The simplex method of linear programming can then be invoked to minimize the pass-band delay deviation from constancy for any specified peak pass-band loss. Tlie_resulting transfer functions are shown to be less sensitive with reference to coefficients trancations. Illustrative examples are also given.