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A direct synthesis method is presented for obtaining a discrete compensator to produce a finite-settling-time transient in response to a prescribed input with no intersample transient terms after the finite settling time. The method is applicable to a single constant-frequency sampler in a feedback system with any type of hold for any polynomial input. The method uses a fictitious sampler at the output with sample time to scan the intersample times and to obtain criteria for deadbeat transients. The criteria are derived from root locus considerations but are applied to Laplace transforms containing both functions of and functions of so that the only operations are analytic ones.