Understanding the Distribution of White Noise

White noise is a common term in the world of audio and signal processing. Scientifically, white noise is defined as a random signal that has an equal intensity at different frequencies. But what distribution does white noise have?

To understand the distribution of white noise, let's first look at the two main types of distributions: Continuous and discrete distributions. Continuous distributions can have an infinite number of possible values within a specified range, while discrete distributions have a finite number of values or resolvable states.

White noise, being a continuous signal, has a continuous distribution in both the time and frequency domains. From a statistical viewpoint, white noise is characterized by a uniform distribution in the frequency domain and a Gaussian distribution in the time domain.

Power spectral density of white noise in the Frequency Domain:
The power spectral density of white noise is constant over the entire range of frequencies it occupies. This means that the energy of the white noise signal is uniformly distributed across all frequencies, giving rise to the term 'white' noise (since white light contains all frequencies in the visible spectrum). In other words, there is an equal probability of any frequency within the specified range being present in the signal at any given time.

Gaussian distribution in the Time Domain:
In the time domain, white noise is often represented as a series of uncorrelated random variables. These random variables, or signal amplitudes in this context, follow a Gaussian (normal) distribution, which is a bell-shaped distribution centered around a mean amplitude value. In the case of white noise, this mean amplitude value is typically zero, resulting in a symmetrical Gaussian distribution around the zero-mean amplitude.

In summary, white noise has a uniform distribution in the frequency domain and a Gaussian distribution in the time domain. Its unique characteristics make it an essential tool for various applications, from audio engineering to scientific research, and even in the field of finance.