A power spectrum retains no phase information, so multiple signals share it

Evidence

One can go back and forth between the autocorrelation and the power spectrum, but one cannot get back unambiguously to the original signal because of the loss of phase information: for example, the amplitudes an of the original signal are squared, so their sign is lost. The original signal cannot be unambiguously constructed, but its power spectrum is unaffected by the sign of the amplitudes an or the phases ϕ of the sinusoids. — Eric J. Heller, Why You Hear What You Hear- An…, 381

Categorization

• DEFINE: Explains the core reason autocorrelation/power-spectrum processing loses information: squared amplitudes and dropped phases.
• XREF: This ‘phase information loss’ is exactly why averaged spectra can deceive the ear, relates to how phase matters in auditory perception vs. amplitude.