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Course by: Behnaam Aazhang. E-mail the author

# Gaussian Processes

Module by: Behnaam Aazhang. E-mail the author

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(12(xμX)T Σ X -1(xμX)) f X x 1 2 N 2 Σ X 1 2 1 2 x μ X Σ X x μ X
(1)
for all xRn 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=ejuX-=e(u22 σ X 2 ) φ X u u X u 2 2 σ X 2
(2)
for all uR u
φ Z u=eju(X+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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#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks