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Gaussian Processes

Module by: Behnaam Aazhang

Summary: Describes signals that cannot be precisely characterized.

Gaussian Random Processes

Definition 1: Gaussian process
A process with mean μ X t μ X t and covariance function C X t 2 t 1 C X t 2 t 1 is said to be a Gaussian process if any X= X t 1 X t 2 … X t N T X X t 1 X t 2 … X t N formed by any sampling of the process is a Gaussian random vector, that is,
f X x=12πN2det Σ X 12ⅇ-12x-μXT Σ X -1x-μX f X x 1 2 N 2 Σ X 1 2 1 2 x μ X Σ X x μ X (1)
for all x∈ℝn x n where μX= μ X t 1 ⋮ μ X t N μ X μ X t 1 ⋮ μ X t N and Σ X = C X t 1 t 1 … C X t 1 t N ⋮⋱ C X t N t 1 … C X t N t N Σ X C X t 1 t 1 … C X t 1 t N ⋮ ⋱ C X t N t 1 … C X t N t N . The complete statistical properties of X t X t can be obtained from the second-order statistics.
    Properties
  1. If a Gaussian process is WSS, then it is strictly stationary.
  2. If two Gaussian processes are uncorrelated, then they are also statistically independent.
  3. Any linear processing of a Gaussian process results in a Gaussian process.
Example 1 
XX and YY are Gaussian and zero mean and independent. Z=X+Y Z X Y is also Gaussian.
φ X u=ⅇⅈuX¯=ⅇ-u22 σ X 2 φ X u u X u 2 2 σ X 2 (2)
for all u∈ℝ u
φ Z u=ⅇⅈuX+Y¯=ⅇ-u22 σ X 2 ⅇ-u22 σ Y 2 =ⅇ-u22 σ X 2 + σ Y 2 φ Z u u X Y u 2 2 σ X 2 u 2 2 σ Y 2 u 2 2 σ X 2 σ Y 2 (3)
therefore ZZ is also Gaussian.

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