Why symmetry matters in the discrete Fourier transform : structural fidelity, phase invariance, and complementarity

Although the discrete Fourier transform (DFT) is universally regarded as the digital realization of the continuous Fourier transform (FT), the structural fidelity of this correspondence has remained largely unexamined. Here we show that the ordinary DFT, through its asymmetric indexing and implicit choice of temporal origin, systematically violates fundamental symmetry relations, parity complementarity, and intrinsic phase structure inherent to the continuous transform. These violations are not matters of convention or numerical implementation, but constitute genuine structural artifacts introduced by discretization. By reformulating the DFT within a strictly centered and symmetric framework, we demonstrate that these artifacts can be eliminated, restoring the natural correspondence between even-odd signal decomposition, sampling parity, and phase behavior. The resulting symmetric DFT yields phase spectra invariant under temporal shifts and preserves the symmetry properties essential for physical interpretability. Our findings call into question the long-assumed equivalence between the continuous FT and its standard discrete implementation, and suggest that symmetry-preserving discretization is a necessary condition for faithful spectral representation across scientific and engineering applications. These results indicate that symmetry preservation is not a representational choice but a necessary structural condition for any discrete transform intended to faithfully represent continuous physical spectra.