On the Capacity of a Discrete Channel. I Mathematical expression of capacity of a channel which is disturbed by noise in its every one symbol and expressible in one state diagram

This report describes a general and concrete method of calculating the capacity in a noisy channel having one state which Dr. Shannon did not solve analytically. In order to express H x ( y ) concisely, the author introduces H x ( y ) characteristic equations, the solution of which characterizes the channel capacity, and the capacity problem is solved, considering whether the calculated probabilities of symbols are real or not. Then an analogy between capacity and rank is discussed and the capacity when H x ( y ) characteristic equations have no solution is also discussed. Numerical examples and a geometrical representation of capacity are described.