With transition matrix p(y|x), such that

the diagonal elements, are equal to 1 minus epsilon,

and off diagonal elements are equal to epsilon.

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Definition 7.1 is the formal definition for Discrete Channel 1.

Let X and Y be discrete alphabets, and p(y|x)

be a transition matrix from X to Y.

A discrete channel p(y|x) is a single input, single output system,

with input random variable X taking values in X and output random variable

taking values in Y, such that the probability that X equals x,

and Y equals y, is equal to the probability that X

equals x times p(y|x), for all possible (x,y) pairs.

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We now introduce the next definition for a

discrete channel, which we call discrete channel 2.

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For such a channel, the input random variable X takes values in discrete

alphabet X, and output random variable Y, takes values in discrete alphabet Y.

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There is a noise variable Z, that takes values in discrete alphabet Z.

We assume that the noise variable Z,

is independent of the input random variable X.

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Alpha is a function that maps the input alphabet X,

and the noise alphabet Z, to the output alphabet Y.

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The channel is specified by the pair alpha Z.

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And the input output relation, is given by Y, the output random variable,

is equal to alpha of X, the input variable, and Z, the noise variable.

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Definition 7.2 is the formal definition of Discrete Channel 2.

Let X, Y, and Z be discrete alphabets.

Let alpha maps X times Z to

Y, and Z be a random variable, taking values in Z,

called the noise variable. A discrete channel alpha Z,

is a single input single output system,

with input alphabet X and output alphabet Y.