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White noise can be created by setting every sample in the time domain to a random number with a gaussian distribution. Every sample is random, so the frequency response is obviously random. Easy enough.

Next step: Set every sample to a random number between 0 and 1, but the numbers are equally distributed instead of using a gaussian distribution (please ignore the DC offsets, they don't really matter). Samples are random enough, so it still makes sense that it would sound like white noise.

Now, set every sample to either 0 or 1 with a 50/50 chance. This feels like it shouldn't sound like white noise anymore, but it still does.

Finally, set every sample to either 0 or 1 with a 90/10 chance. How does this still sound like white noise when it looks nothing like it?

EDIT
I did some experimenting (with sample rate 48000 Hz).
1/2 chance is basically white noise.
1/4 chance sounds almost like white noise, but if you know what you're looking for it's easy to tell the difference.
1/16 chance sounds like rain smattering against something (you can hear individual impulses), with white noise in the background.

Varying the amplitudes of the impulses did not seem to have any effect on the sound, other than volume.

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  • $\begingroup$ To clarify: Gaussian distribution is not required to be "white". White refers to the uniform distribution in frequency and that every sample is independent/uncorrelated to the other samples. Gaussian refers to the distribution in magnitude which may or may not be for a white noise process. If you take the requirement or need to be "Gaussian" out of this, I think it will clear up your question. $\endgroup$ – Dan Boschen Sep 29 '18 at 13:58
  • $\begingroup$ @DanBoschen The first sentence I wrote wasn't supposed to frame gaussian as a requirement. It's more that I know that random samples with a gaussian distribution is white noise for sure. Editing it now. $\endgroup$ – usernumber Sep 29 '18 at 14:45
  • $\begingroup$ Yes but I thought that distinction may help clear up your confusion in “not looking the same”... in that even a set with just two elements can be used to generate a white noise sequence. $\endgroup$ – Dan Boschen Sep 29 '18 at 14:47
  • $\begingroup$ (I have never seen the term used but I am guessing in that case it would be "AWBN"!) $\endgroup$ – Dan Boschen Sep 29 '18 at 14:57
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White noise is defined as having a flat spectrum over the entire frequency range. Using uncorellated random numbers is just one of many methods to generate this, but by far not the only or the best one. Also, whiteness implies no gaussian or whatever distribution, so your first sentence is wrong twice.

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  • $\begingroup$ "by far not the only or the best one" Do you have a "best one" in mind? $\endgroup$ – usernumber Sep 29 '18 at 11:02
  • $\begingroup$ Best would be using a natural white noise source such as subsampling amplified thermal noise. The problem is in actual application we often want to repeat the "pseudo-random" sequence, so rely on algorithmic approaches to generating them. $\endgroup$ – Dan Boschen Sep 29 '18 at 13:55
  • $\begingroup$ What I found is this definition is not sufficient, meaning white noise will have a flat spectrum over the entire frequency range, but the converse is not necessarily true; you can have processes with a flat spectrum that is clearly not white. $\endgroup$ – Dan Boschen Sep 29 '18 at 15:01
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Two answer your question, a single unit sample with value 1 has a perfect flat spectrum, and so has a sequence of uncorrelated unit samples. The distribution does not matter here unless the pauses between samples are so long the they are perceived in time domain, assuming we deal with audio.

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  • $\begingroup$ So basically, as long as there are some samples in random places that aren't zero (and they aren't too sparse), it's white noise? $\endgroup$ – usernumber Sep 29 '18 at 11:01
  • $\begingroup$ As long as every sample is completely independent/uncorrelated to all other samples, it is white noise. If any correlation exists, the process is no longer white (even if it has a flat spectrum). $\endgroup$ – Dan Boschen Sep 29 '18 at 15:33

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