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An algorithm for designing a Chebyshev optimal FIR filter that approximates an arbitrary complex-valued frequency response is presented. This algorithm computes the optimal filter by solving the dual to the filter design problem. It is guaranteed to converge theoretically and requires O(N2) computations per iteration for a filter of length N. For the first time, properties of the optimal filter are derived, and the case where the desired filter has arbitrary constant group delay is studied in detail