Communication in the Presence of Noise — Probability 

 of Error for Two Encoding Schemes 



By S. O. RICE 



Recent work by C. E. Shannon and others has led to an expression for the 

 maximum rate at which information can be transmitted in the presence of ran- 

 dom noise. Here two encocHng schemes are described in which the ideal rate is 

 approached when the signal length is increased. Both schemes are based upon 

 drawing random numbers from a normal universe, an idea suggested bj' 

 Shannon's observation that in an efficient encoding system the typical signal 

 will resemble random noise. In choosing these schemes two requirements were 

 kept in mind: (1) the ideal rate must be approached, and (2) the problem of 

 computing the probability of error must be tractable. Although both schemes 

 meet both requirements, considerable work has been recjuired to put the expres- 

 sion for the probability of error into manageable form. 



1. Introduction 



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 \og2 (1 + Ws/Wj,) bits/sec. (1-1) 



In this expression F is the width of the frequency band used for signaling 

 (which we suppose to extend from to Z*" cps), PFs 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^/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 proj)- 

 erties 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 jjroduct FT large enough the ideal rate of signaling may be a{)i:)roached 

 in either case* and we are interested in the probabihty 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 /cero 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. I. K. E., Se])t. 1949, p. 1031) indicates 

 that the ideal rate of signaling may also be approached by quantized PPM under 

 suitable conditions. 



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