📝 SummarXiv

Abstract

A conjecture is proposed concerning the recovery of a discrete magnitude spectrum through a nonlinear transformation involving the Newman's phase sequence. Given a discrete magnitude spectrum sampled from a continuous function, consider the process of applying a complex exponential with the Newman's phase sequence, computing the inverse discrete Fourier transform (IFFT), taking the absolute value of the result, and reversing the time-domain signal. The conjecture states that Newman's phase sequence, defined by a formula $\phi^{(N)}[k] = \frac{\pi (k-1)^2}{N}$, asymptotically recovers the original magnitude spectrum as the number of samples $N \to \infty$. Notably, the phase sequence is also independent of the input signal and is unique up to an overall constant phase shift. The broader implications of this conjecture remain to be fully understood, but the phenomenon raises fundamental questions about the role of phase in nonlinear spectral recovery.