312 BELL SYSTEM TECHNICAL JOURNAL 



That portion of p arising frcm the components having frequencies between 

 /and/ + df will be denoted by w(f)df, and consequently 



= I Hf)df (2.1-1) 



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Since w(f) is the spectrum of the average power we shall call it the "power 

 spectrum" of I{t). It has the dimensions of energy and on this account is 

 frequently called the "energy-frequency spectrum" oiJ{t). A mathematical 

 formulation of this disctission leads to a clear cut definition of w(f). 



Let $(/) be a function of /, which is zero outside the interval < / < Tand 

 is equal to /(/) inside the interval. Its spectrum S(f) is given by 



5(/) = f I{t)e~'''^' dt (2.1-2) 



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The spectrum of the power, w(/), is defined as 



wif) = Limit Mfll (2.1-3) 



r-»oo i 



where we consider only values of / > and assume that this limit exists. 

 This is substantially the definition of w{f) given by J. R. Carson and is 

 useful when I{t) has no periodic terms and no d.c. component. In the 

 latter case (2.1-3) must either be supplemented by additional definitions or 

 else a somewhat different method of approach used. These questions will 

 be discussed in section 2.2. 



The correlation function ;/'(t) of /(/) is defined by the limit 



^{r) = Limit i f /(/)/(/ + r) dt (2.1-4) 



which is assumed to exist, ypir) is closely related to the correlation coeffi- 

 cients used in statistical theory to measure the correlation of two random 

 variables. In the present case the value of /(/) at time / is one variable and 

 its value at a different time ^ -f t is the other variable. 



The spectrum of the power w(/) and the correlation function \}/{t) are 

 related by the equations 



^{f) = 4 f ;^(t) cos lirfrdT (2.1-5) 



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^(t) = [ w{f) C05 Itt/t df (2.1-6) 



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15 "The Statistical Enerfry-Frequency Spectrum of Random Disturbances," B.S.T.J., 

 Vol. 10, pp. 374-381 (1931). 



