624 BELL SYSTEM TECHNICAL JOURNAL 



2. The set of all functions of time containing no frequencies over W cycles 

 per second. 



3. The set of all functions limited in band to W and in amplitude to A. 



4. The set of all English speech signals as functions of time. 



An ensemble of functions is a set of functions together with a probability 

 measure whereby we may determine the probability of a function in the 

 set having certain properties.-^ For example with the set, 



feit) = sin (/ + d), 



we may give a probability distribution for 6, P{d). The set then becomes 

 an ensemble. 



Some further examples of ensembles of functions are: 



1. A finite set of functions /;b(/) (k = 1, 2, ■ • • , n) with the probability of 

 fk being pk . 



2. A finite dimensional family of functions 



/(ai , a.2, ' ■ ' ,an ;t) 

 with a probability distribution for the parameters ai : 



p{ai , ' ■ ■ , an) 

 For example we could consider the ensemble defined by 



n 



fidi , ■ • - , an , di , • • • , On ', t) = J2 fln sin n{cot + dn) 



n = l 



with the amplitudes ai distributed normally and independently, and the 

 phrases di distributed uniformly (from to 27r) and independently. 



3. The ensemble 



r/ A V^°° sin 7r(21^/ - n) 

 n=-oo 7r{2Wt — n) 



with the di normal and independent all with the same standard deviation 

 \/N. This is a representation of "white" noise, band-limited to the band 

 from to W cycles per second and with average power N. 



^ In mathematical terminology the functions belong to a measure space whose total 

 measure is unity. 



2 This representation can be used as a definition of band limited white noise. It has 

 certain advantages in that it involves fewer limiting operations than do definitions that 

 have been used in the past. The name "white noise," already firmly intrenched in the 

 literature, is perhaps somewhat unfortunate. In optics white light means either any 

 continuous spectrum as contrasted with a point spectrum, or a spectrum which is flat with 

 wavelength (which is not the same as a spectrum flat with frequency) . 



