White noise

White noise is a very special kind of stationary random process. It is a model very useful in communications, which describes, for example, thermal noise generated by electronic devices, or noise in many band-limited communication channels.

Definition: X(t) is white noise in strict sense IFF X(t₁) and X(t₂) are independent random variables for any t₁ and t₂.

Definition: X(t) is White noise in wide sense IFF the autocovariance function is zero:

C_{X X} (t_{1}, t_{2}) = R_{X X} (t_{1}, t_{2}) - η_{x} (t_{1}) η_{x} (t_{2}) = 0, \forall t_{1}, t_{2}

If a process is white noise in strict sense, it is also white noise in wide sense. Therefore, for white noise, the autocorrelation function is equal to the product of means, for any $t_{1} \neq t_{2}$ :

R_{X X} (t_{1}, t_{2}) = η_{X} (t_{1}, t_{2}) - η_{x} (t_{1}) η_{x} (t_{2}) = η_{X}^{2}, \forall t_{1} \neq t_{2}

The autocorrelation function of white noise can be expressed as follows:

$R_{X X} (τ) = σ_{X}^{2} δ (τ) + η_{X}^{2}$ (continuous time white noise)

$R_{X X} [m] = σ_{X}^{2} δ [m] + η_{X}^{2}$ (discrete time white noise)

The power spectral density is constant plus a delta function in the origin, due to the mean value of the process in the time domain.

Review of Basic Concepts of Probability

White noise