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The performance of phase modulation communication systems is investigated for the asymptotic case of a high signal-to-noise ratio. A simple and excellent approximation to the probability of error is derived for both coherent and phase-comparison demodulation. In addition, a simple expression is derived for the asymptotic degradation of phase-comparison demodulation. When the number of bits per sample of the signal is large, a combination of digital amplitude and phase modulation is found to make more effcient use of transmitter power than phase modulation alone. The number of amplitude levels may be optimized for either minimum peak power or minimum average power, using the approximate expressions for probability of error. While an analytical solution can be found for the peak power case, a numerical solution with restriction to powers of two appears preferable, and is necessary for the average power case. Accordingly, tables showing the optimum number of levels with this restriction are presented.
C. R. Cahn (Thu,) studied this question.