Читать книгу Digital Communications 1 - Safwan El Assad - Страница 19

First order Markov sources play an important role in several domains, for example, in (Caragata et al. 2015), the authors used such a source to model the cryptanalysis of a digital watermarking algorithm.

It is characterized by:

[2.5]

with:

The probability = pl,k is called transition probability from state l to state k, and:

[2.6]

The probability that at time n the source is in the state k is:

[2.7]

By introducing matrix notations:

[2.8]

Taking into account the relationship [2.8], the relation [2.7] is also written as:

[2.9]

where Tt is the transposed matrix of transition probabilities.

Moreover, if the source is, stationary, then:

in other words, the probabilities of the occurrence of the symbols do not depend on n. It is the same for the transition probabilities pl,k, then the relation [2.9] is written:

[2.10]

where P₀ is the matrix of probabilities governing the generation of symbols by the source at the initial instant n = 0.