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Designs for recursive and nonrecursive wide-band differentiators are presented. The coefficients for the recursive differentiators were optimally chosen to minimize a square-error criterion based on the magnitude of the frequency response. The coefficients for the nonrecursive differentiators were chosen using a frequency sampling technique. One or more of the coefficients were optimally selected to minimize the peak absolute error between the obtained frequency response and the response of an ideal differentiator. The frequency response characteristics of the recursive differentiators had small magnitude errors but significant phase errors. The nonrecursive differentiators required on the order of 16 to 32 terms for the magnitude error of the frequency response to be as small as the magnitude errors for the recursive differentiators; however, there were no phase errors for the nonrecursive case. The delay of the recursive differentiators was small compared to the delay of the nonrecursive differentiators.