MATHEMATICAL THEORY OF COMMUNICATION 627 



shown, in fact, that if a device is linear as well as invariant Fourier analysis 

 is then the appropriate mathematical tool for dealing with the problem. 



An ensemble of functions is the appropriate mathematical representation 

 of the messages produced by a continuous source (for example speech), of 

 the signals produced by a transmitter, and of the perturbing noise. Com- 

 munication theory is properly concerned, as has been emphasized by Wiener, 

 not with operations on particular functions, but with operations on en- 

 sembles of functions. A communication system is designed not for a par- 

 ticular speech function and still less for a sine wave, but for the ensemble of 

 speech functions. 



19. Band Limited En'sembles of Functions 



If a function of time f{i) is limited to the band from to PF cycles per 

 second it is completely determined by giving its ordinates at a series of dis- 

 crete points spaced — — seconds apart in the manner indicated by the follow- 

 ing result. 



Theorem 13: Let f{t) contain no frequencies over W. 

 Then 



-00 w{2Wt — n) 



where 



In this expansion /(/) is represented as a sum of orthogonal functions. 

 The coefficients Xn of the various terms can be considered as coordinates in 

 an infinite dimensional ''function space." In this space each function cor- 

 responds to precisely one point and each point to one function. 



A function can be considered to be substantially limited to a time T if all 

 the ordinates Xn outside this interval of time are zero. In this case all but 

 2TW of the coordinates will be zero. Thus functions limited to a band W 

 and duration T correspond to points in a space of 2TW dimensions. 



A subset of the functions of band IF and duration T corresponds to a re- 

 gion in this space. For example, the functions whose total energy is less 



on time series. This work, although chiefly concerned with the linear prediction and 

 filtering problem, is an important collateral reference in connection with the present paper. 

 We may also refer here to Wiener's forthcoming book "Cybernetics" dealing with the 

 general problems of communication and control. 



^ For a proof of this theorem and further discussion see the author's paper "Communi- 

 cation in the Presence of Noise" to be published in the Proceedings of the Institute of Radio 

 Engineers. 



