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This paper introduces a new class of linear-phase FIR Nyquist ( th-band) filters composed of cascaded FIR subfilters with different periodicities in the frequency domain. Each one of the subfilters is itself a Nyquist filter or an th-band filter. - The composite filters have zero intersymbol interference and they provide a Chebyshev stopband behavior, thereby band-limiting the pulses optimally. A computationally efficient Remez-type procedure is presented for simultaneously optimizing the subfilters. This algorithm is equally applicable to the design of conventional single-stage FIR Nyquist ( th-band) filters. Examples demonstrate that the proposed multistage filters provide significant advantages over equivalent IIR filters and single-stage FIR filters in terms of reduced multiplication rate and reduced number of multipliers. In addition, it is shown that, in the case of single-stage implementations, the proposed algorithm gives FIR Nyquist filters having higher selectivities than those which have been designed using other methods.