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Distant radio communication at high frequencies is difficult because the transmission medium, the earth ionosphere duct, is time variable, noisy, and shows dispersive or multipath transmission with consequent fading of the received signal. Because most of these factors are random and not under design control, one can treat the ionospheric communication problem only on a statistical basis in terms, for example, of such things as the probability that a transmitted pulse or bit of information will be received correctly. On this basis it is shown that, by a purely numerical experiment wherein random number tables are used to simulate fading and noise, it is possible to appraise various telegraph transmission systems without the costly process of building them and then testing their performance. An approximate, but simple, analysis is given of receiver signal detection, by which is meant generation of dc for operation of an output printing device. It is shown that the important parameter here is the average or expectation of the difference in dc between a received space and mark signal, divided by the square root of the variance of this difference. Incoherent square law detection is compared with coherent detection using the matched filter or, what is the same thing, correlation. The use of diversity transmission to overcome signal fading at the receiver is considered. The important statistical data regarding time-varying ionospheric transmission are obtained from the auto- and cross-correlation functions of the received signal envelopes for the various diversities. For the correlated fading of signal s in two transmission channels, a simple de sign formula is shown which predicts the improvement in the use of diversity. A brief description is also given of a simple acoustic ionosphere analog simulating time-varying multipath transmission.