White noise is a random signal having equal intensity at all frequencies. And refers to a — statistical model — for signals, as opposed to any specific signal. In discrete time, white noise is a signal whose samples are regarded as a sequence of — serially uncorrelated — random variables with zero mean and finite variance. Where — any — zero‑mean distribution of values (without DC component) is possible.
Different Distributions
The most typical flavor of white noise being one whose samples are pulled from a normal distribution. For example, the heights of adults in a population tend to follow a normal distribution, with most people clustered around the average and fewer at the extremes. Or scores on standardized tests (e.g., STA, GRE) are often scaled to fit a normal distribution to compare performance.
But one can also pull samples from a uniform distribution, like a fair die, where every outcome is equally likely. And even flipping a fair coin, with only two possible outcomes, can serve as distribution for independent — serially uncorrelated — samples.
The video below features three distinctly different white noises whose samples are pulled from a normal, uniform, and Bernoulli (coin toss) distribution respectively. In each instance the average (RMS) level has been normalized to -18 dBFS.
Different Crest Factors
Notice that these white noises functionally sound the same, but their waveforms look different. Because each distribution yields a different crest factor.
For normally distributed white noise, the crest factor is oftentimes stated as four (or 12 dB on a logarithmic scale). But it can be both lower and higher depending on the number of samples drawn. And subsequently, the noise signal's duration. Bearing in mind that, the — longer — one observes an unbounded normally distributed random process, the more likely one is to spot an outlier, that is, peak.
Whereas for a (bounded) uniform distribution, the crest factor works out to be \(\sqrt{3}\) ≈ 1,73 or 4,8 dB. And for the coin toss, each sample's magnitude is the same, there is no single peak, but their — sign — changes at random (50/50). Therefore, its crest factor is one or 0 dB.
Further Considerations
And whether these crest factors persist, once each signal is passed through an actual loudspeaker (system). Or any audio chain for that matter, where the signal invariably becomes subject to phase shift, is worthy of separate consideration.
Regardless, because of their — serially uncorrelated — samples, white noises are incredibly robust. And if you shuffle their sample order, their noise spectra (evidently) remain white. Which is not the case, for example, for pink noise. Whose spectrum turns white after shuffling its sample order. Indicating a correlation between a given pink noise sample and its immediate neighbors. Which (contrary to white noise), when scrambled, sacrifices pink noise's defining property, that is, its pink spectrum.
Audio Files & MS Excel Workbook
The audio files (ZIP archive) and MS Excel Workbook used for this article can be downloaded using the links shown below.