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This paper considers the design of FIR digital filters with the objective of obtaining a given complex-valued frequency response within error tolerance regions which are specified as a function of frequency. For a general class of phase-magnitude error tolerance specifications, it is shown that this filter design problem is equivalent to the requirements for minimax synthesis of a particular two-dimensional linear-phase FIR filter. As a result, design techniques developed for 2-D linear-phase FIR filters can be applied to solve the 1-D problem with complex specifications. A number of 2-D filter synthesis algorithms are reviewed and related to methods previously pro - posed for the design of 1-D FIR filters to joint phase and magnitude specifications. The relative computational efficiencies of several suboptimal 2-D design techniques are compared by example with optimal synthesis via linear programming.