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The objective of this paper is to present a theory, constraints, and a design method for nonlinear-phase halfband and Mth-band filters. Based on a time-domain property, the constraints and properties in the frequency domain are derived. They are the generalization of the well-known conditions for linear-phase Mth-band filters. Having found all necessary conditions, we present the design method based on an eigenfilter that minimizes the mean-squared errors. The design method is also extended to the design of nonlinear-phase Mth-band filters with properties of R-regularity, or equiripple stopband attenuation, or impulse responses that have complex coefficients. Design examples of various Mth-band filters with different properties are presented, discussed, and compared with the linear-phase case