Channel Matrix aka conditional probability# table is a statistical model of a #Discrete Memoryless Channel (DMC) which is defined as the following formulae:
$$ P = \left[ \begin{array}{cccc} p(y_0|x_0) & p(y_1|x_0) & \cdots & p(y_{k-1}|x_0)\\ p(y_0|x_1) & p(y_1|x_1) & \cdots & p(y_{k-1}|x_1)\\ \vdots & \vdots & & \vdots\\ p(y_0|x_{j-1}) & p(y_1|x_{j-1}) & \cdots & p(y_{k-1}|x_{j-1}) \end{array} \right] $$
Where:
- Input alphabet \(X = \{ x_0, x_1, \ldots, x_{j-1} \}\)
- Output alphabet \(Y = \{ y_0, y_1, \ldots, y_{k-1} \}\)
- \(j\) is the size of \(X\)
- \(k\) is the size of \(Y\)
- \(\forall j \text{ and } k, 0 \le p(y_k|x_j) \le 1\)
Each row corresponds to a fixed channel input (\(X\)) and each column corresponds to a fixed channel output (\(Y\)). The sum for all conditional probabilities in \(k\) is 1 for all \(j\), that is all conditional probabilities in the same row, defined as:
$$ \forall j, \sum^{K-1}_{k=0} p(y_k|x_j) = 1 $$
With given a priori probabilities, i.e., \(p(x_j)\), we can get Join Probability Distribution#.