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Phase, Polarity, and Delay or Why a Loudspeaker Crossover Is Not a Time Machine

Phase delay and group delay functions are well known in lumped system theory but they are not well understood, particularly in the context of non-minimum phase lumped systems. This paper presents four paradoxes that arise from an overly-literal interpretation of the phase delay function. The paradoxes are very simply illustrated using the example of a first order loudspeaker crossover. A new system response model is presented to resolve these paradoxes. The model is extended to higher order systems and to non-minimum phase systems. Applications and implications for system analysis are discussed.

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