# 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". Dec 9, 2020 at 22:15

## 1 Answer

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

They are both constant over the whole frequency range

That's only true for the spectrum of the Dirac. For white noise, it's more complicated. The FFT of a chunk of white noise does NOT have constant spectrum, it's actually white noise as well (the real part and the imaginary part). It only becomes constant if you average every over a large enough number of chunks (or a long enough period of time) or if you apply some sort of spectral smoothing.

• i think, in the limit, the Dirac delta has infinite energy even though it has finite non-zero length. And white noise has infinite power. these are good reasons to consider them solely as theoretical or hypothetical signals. Dec 9, 2020 at 22:18
• Hilmar, I think there are some grammar error in your explanation, can you please check and fix them? Dec 10, 2020 at 21:26
• A more precise way of describing the spectrum of white noise is to say that the expected value of the spectrum is constant. The spectrum of any sample of a white noise process is, itself, white. Dec 11, 2020 at 3:06