This letter derives a new exact and general closed-form expression involving a two-dimensional joint Gaussian Q-function for the symbol error rate (SER) of M-ary Phase Shift Keying (MPSK) under an additive white Gaussian noise (AWGN) channel. By using two rotations of coordinates the correlation coefficient between two Gaussian random vectors is provided, then with the derived correlation coefficient that characterizes the two-dimensional joint Gaussian Q-function, a new expression for the SER of MPSK is presented. The derived new SER expression offers a convenient method to evaluate the performances of MPSK for various cases of practical interest.

This paper introduces a new digital ATC (Automatic Train Control device) system. In the current ATC, the central ATC logic device calculates permissive speed of each blocking section and controls speed of all trains. On the other hand, in the new digital ATC, the central logic controller calculates each position to which a train can move safely, and sends the information on positions to all trains. On each train, the on-board equipment calculates an appropriate braking pattern with the information, and controls velocity of the train. That is, in the new system, the device on each train autonomously calculates permissive speed of that train. These special features realize ideal speed control of each train making full use of its performance for acceleration and deceleration, which in turns allows high-density train operations.