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A technique is presented whereby a class of recursive digital filters can be designed to approximate simultaneously given magnitude and linear phase characteristics. The underlying approach is to linearize the inherently nonlinear approximation problem, and thereby use linear programming to carry out the approximation. The linear phase is specified in terms of a desired constant group delay. Therefore, the constraints of the linear program become a function of desired group delay. Thus the design algorithm consists of carrying out a univariant search within a range of group delay values to obtain a minimum error of approximation. Several examples of design are presented to illustrate the usefulness of the method.