What is the difference between the spectra of white Gaussian noise and of a Dirac impulse?

Perhaps it is a basic question, but

• can anyone explain the difference between the spectra (or power spectral density) of white Gaussian noise and a Dirac delta?

They are both constant over the whole frequency range, however there are is a clearly striking difference between these signals.

• Why do they have the same spectra?
• also, white noise (or any other color) is a "power signal" and the Dirac impulse (or more specifically a "nascent delta") is an "energy signal". – robert bristow-johnson Dec 9 '20 at 22:15

They have very different phase spectra. For a dirac the phase of the spectrum is zero for white noise, it's uniformly distributed over $$[0, 2 \pi]$$