On the symmetry of FIR filter with linear phase

This paper deals with signal processing theory related to finite impulse response (FIR) filters with linear phase. The aim is to show a new and complete proof of the equivalence between linear phase and symmetry or antisymmetry of the real coefficients of the filter. Despite numerous proofs are available in the literature, they are usually incomplete, even though the result is commonly used by the signal processing community. We hereby address a pending issue in digital signal processing: we first prove the uniqueness of the group delay for any decomposition amplitude-phase of the frequency response. Based on this first step, we then derive a complete proof of the equivalence: a FIR filter has (anti)symmetric coefficients if and only if the phase is linear. It must be emphasized that this brief paper deals with theoretical aspects of FIR filters.