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In this paper are calculated the informational capacities and the optimum statistics of communication systems consisting of an amplitude- and/or phase-modulated transmitter; a medium, which may include the rf and IF portions of the receiver, and in which white thermal noise of the same bandwidth as the transmitted signal is added to the latter; and a receiver responding to the amplitude and/or phase of this received signal. Restriction of the transmitted signal or receiver response to amplitude modulation alone or to phase modulation alone reduces the informational capacity of the system below the full channel capacity found by Shannon, C = W log (1+r2), with W the bandwidth in cps and r the rms signal-to-noise ratio. It is found that at very high signal-to-noise ratios the capabilities of AM alone and of PM alone each account for half this full informational capacity of the channel. When the signal-to-noise ratio is much less than unity, however, PM has 79% of the full channel capacity, while AM can transmit no more than about 17% of the full channel capacity. If a receiver is used which responds to both amplitude and phase, on the other hand, the capacity at low signal-to-noise ratios equals the full channel capacity whatever type of transmitter is used. An AM signal received on a PM receiver is found to be able at low signal-to-noise ratios to convey as much information as a PM signal.