mathematics//probability//stochastic process//white noise

White noise is a stochastic process whose samples are independent of one another, so that its autocorrelation is zero at every lag except zero and its power is spread evenly over all frequencies, and it is the model of the fresh part of a sensor's error, the part that averaging beats down. The name comes from light: white light carries every colour in equal measure, and white noise every frequency.

White noise is a stochastic process whose samples are independent of one another, so that its autocorrelation is zero at every lag except zero and its power is spread evenly over all frequencies, and it is the model of the fresh part of a sensor's error, the part that averaging beats down. The name comes from light: white light carries every colour in equal measure, and white noise every frequency.

Independence is what makes averaging work. Each sample brings an error unrelated to the last, so in an average of NNN of them the errors partly cancel and the standard deviation falls by N\sqrt NN​. An accelerometer whose individual samples scatter by 0.02 m/s20.02\ \mathrm{m/s^2}0.02 m/s2 gives averages of 100 samples that scatter by 0.002 m/s20.002\ \mathrm{m/s^2}0.002 m/s2. The square root is a law of diminishing returns: halving the noise costs four times as many samples, dividing it by ten costs a hundred times as many.

White is a statement about memory, whatever the size of the noise.

A small error with memory misleads a filter that assumes white noise far more than a large error without it, because the filter counts every reading as fresh evidence and grows confident on what is really one opinion repeated (colored noise).

White and Gaussian are separate properties. Gaussian describes the shape of one sample's distribution (normal distribution); white says the samples are independent in time. A sensor's noise can be either, both or neither, and a Kalman filter assumes both.

Datasheets give white noise as a density, in μg/Hz\mu g/\sqrt{\mathrm{Hz}}μg/Hz​ for an accelerometer or (∘/s)/Hz(^\circ/\mathrm s)/\sqrt{\mathrm{Hz}}(∘/s)/Hz​ for a gyroscope, because the scatter of one sample depends on how wide a band the sensor lets through: σ=density⋅bandwidth\sigma=\text{density}\cdot\sqrt{\text{bandwidth}}σ=density⋅bandwidth​. Raising the output rate and the internal filter's bandwidth makes each sample noisier while the information per second stays the same. (Continuous white noise with truly flat power at all frequencies would have infinite power; real white noise is flat up to the sensor's own filter.)

Integrated, white noise stops being white: its running sum is a random walk, and a gyroscope's angle random walk coefficient is its white noise density seen through that integral.

Whiteness is also what a model's leftovers should look like. The innovations of a healthy filter and the residuals of a good time-series model are white; any memory left in them means the model missed a piece of the dynamics (innovation).