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A general analysis based upon information theory and the mathematical theory of probability is used to investigate the fundamental principles involved in the transmission of signals through a background of random noise. Three general theorems governing the probability relations between signal and noise are proved, and one is applied to investigate the effect of pulse length and repetition rate on radar range. The concept of "generalized selectivity" is introduced, and it is shown how and why extra bandwidth can be used for noise reduction. It is pointed out that most noise-improvement systems are based upon coherent repetition of the message information either in time or in the frequency spectrum. It is also pointed out why more powerful noise-improvement systems should be possible than have so far been made. The general mechanism of noise-improvement thresholds is discussed, and it is shown how they depend upon the establishment of a coherence standard. The reason for and the limitation of the apparent law that the maximum operating range of a communications system, for a given average power, is independent of the type of modulation used is then explained. General ways in which improvements in range of radar and communication systems may be made are also discussed. The possibility of using extra bandwidth to reduce distortion is pointed out. Finally, some possible relations of this work to biology and psychology are described.