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A procedure is described which can be used to design digital differentiators satisfying prescribed specifications. The procedure is based on the Fourier-series method for the design of nonrecursive digital filters, and uses the Kaiser window function for the minimisation of the amplitude of Gibbs oscillations. The approach is used to design a number of differentiators assuming various prescribed bandwidths and various prescribed inband errors. It is also used to design a wideband high-precision differentiator. The proposed method is compared briefly with that of McClellan and Parks, based on the theory of weighted Chebyshev approximation.