Discrete Stochastic Processes and Optimal Filtering
Concerned with the founding principles of optimal filters, this text presents several reminders about both random vectors and Gaussian vectors, and allows readers to tackle digital filtering.
TABLE OF CONTENTS Discrete Stochastic Processes and Optimal Filtering
Chapter 1: Random Vectors
1.1 Definitions and general properties
If we remember that 
the set of real n-tuples can be fitted to two laws: 
making it a vector space of dimension n.
The basis implicitly considered on 
will be the canonical base ? 1 = (1,0, ,0), , ? n = (0, ,0,1) and 
expressed in this base will be denoted:
Definition of a real random vector
Beginning with a basic definition, without concerning ourselves at the moment with its rigor: we can say simply that a real vector 
linked to a physical or biological phenomenon is random if the value taken by this vector is unknown and the phenomenon is not completed.
For typographical reasons, the vector will instead be written X T = ( X 1, , X n) or even X = ( X 1, , X n) when there is no risk of confusion.
In other words, given a random vector X and 
we do not know if the assertion (also called the event) ( X ? B) is true or false:
However, we do usually know the "chance" that X ? B; this is denoted P( X ? B) and is called the probability of the event ( X ? B).
After completion of the phenomenon, the result (also called the realization) will be denoted
when there is no risk of confusion.
An...
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