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Plasma photonic crystals (PPCs) have the potential to significantly expand the capabilities of current millimeter wave technologies by providing high speed (microsecond time scale) control of energy transmission characteristics in the GHz through low THz range. Furthermore, plasma-based devices can be used in higher power applications than their solid-state counterparts without experiencing significant changes in function or incurring damage. Plasmas with periodic variations in density can be created externally, or result naturally from instabilities or self-organization. Due to plasma's diffuse nature, PPCs cannot support rapid changes in density. Despite this fact, most theoretical work in PPCs is based on solid-state photonic crystal methods and assumes constant material properties with abrupt changes at material interfaces. In this work, a linear model is derived for a one-dimensional cold-plasma photonic crystal with an arbitrary density profile. The model is validated against a discontinuous Galerkin method numerical solution of the same device configuration. Bandgap maps are then created from derived group velocity data to elucidate the operating regime of a theoretical PPC device. The bandgap maps are compared for one-dimensional PPCs with both smooth and discontinuous density profiles. This study finds that bandgap behavior is strongly correlated with the density profile Fourier content and that density profile shapes can be engineered to produce specific transmission characteristics.