Noise with temporal correlations, in contrast to white noise which has no memory.Colored noise is a type of random signal whose properties change over time in a way that is not completely unpredictable. Unlike white noise, which has no memory — each instant is independent of the last — colored noise carries correlations between moments separated by time. This means that if the signal is high now, it tends to stay high (or low) for a while, giving it a smooth, persistent character. The term 'colored' borrows from optics: white light contains all frequencies equally, while colored light emphasizes certain ranges. Similarly, colored noise has a frequency spectrum that is not flat, favoring some fluctuations over others.
The concept emerged from studies of electronic and acoustic noise in the mid-20th century, with the '1/f' label popularized by researchers like Benoit Mandelbrot in the 1960s, though the exact origin of the color analogy is diffuse.
The key is the autocorrelation function, which measures how much a value at one time resembles a value at a later time. For white noise, this function is zero for any nonzero lag — no memory. For colored noise, it decays slowly, often following a power law. A common family is '1/f noise' (pink noise), where the power spectral density falls as 1/frequency, meaning low-frequency (slow) changes are stronger than high-frequency (fast) ones. Other colors, like red noise (Brownian motion), have even steeper spectra, while blue noise has the opposite trend, emphasizing rapid fluctuations.
Colored noise models are used in climate science to simulate natural variability, in audio engineering to generate realistic ambient sounds, and in finance to describe market fluctuations.
Pink noise, with its 1/f spectrum, is found in everything from heartbeats to river flows, and its ubiquity has led some to call it 'the sound of nature.'


