Communication in the Presence of Noice - Probability of Error for Two Encoding Schemes

In recent work concerning the theory of communication it has been shown that the maximum or ideal rate of signaling which may be achieved in the presence of noise is (1, 2, 3, 4, 5) Ri = F log2 (1 + Ws/WN) bits/sec. (1-1) In this expression F is the width of the frequency band used for signaling (which we suppose to extend from 0 to F cps), Ws is the average signaling power and WN the average power of the noise. The noise is assumed to be random and to have a constant power spectrum of W N / F watts per cps over the frequency band (0, F). This ideal rate is achieved only by the most efficient encoding schemes in which, as Shannon (1,2) states, the typical signal has many of the properties of random noise. Here we shall study two different encoding schemes, both of them referring to a bandwidth F and a time interval T. By making the product FT large enough the ideal rate of signaling may be approached in either case* and we are interested in the probability of error for rates of signaling a little below the rate (1-1). The work given here is closely associated with Section 7 of Shannon's second paper (2). In the first encoding scheme the signal corresponding to a given message lasts exactly T seconds, but (because the signal is zero outside this assigned interval of duration) the power spectrum of the signal is not exactly zero for frequencies exceeding F. In the second encoding scheme, the signal * A recent analysis by M. J. E. Golay (Proc. /. R. E., Sept. 1949, p. 1031) indicates t h a t the ideal rate of signaling m a y also be approached by quantized P P M under suitable conditions.