126 BELL SYSTEM TECHNICAL JOURNAL 



Squaring (4.5-1) gives the double series 



1 if? if? 



V N =^ - / J / J Cm Cn I 



4 —00 —00 



4-co +00 



4: A;=— 00 n=— o 



^ -TOO -roa 



Suppose we wish to consider the component of 1'a- of frequency /a- = ki^f. 

 It is seen to be 



4 +00 



Ak cos (coA,/ — i/'yt) = - X c^-«c„ cos {kat — <pk-,i — fn) (4.5-3) 



/ n= — 00 



The power spectrum W(f) of / at frequency //; is a times the coefficient of 

 A/" in the mean square value of (4.5-3) where the average is taken over the 

 ^'s. Thus 



2 +00 +00 



W(fk)Af = ^ Z) S Ck-nCnCk-mCm 

 4: —00 —00 



X ave. cos {kat — (pk-n — ^Pn) cos (kat — iPk-m — (r^n) 



where the summations extend over m and n. Let ii be fixed and consider 

 those values of m which give an average different from zero. We see that 

 m = n and m = k — n are two such values. The only other possibilities 

 are m = —n and m = — ^ + w, but these lead to terms containing (except 

 when 11 or k equal zero) three different angles, (pn , <Pk-n , and ipk+n which 

 average to zero. Using the fact that the average of cosine squared is one- 

 half and that for a given n there are two such terms, we get 



Wifk)Af = j E cLnCl 



n=—co 



+=o 



(4.4-5) 



= a'Af E '^(h -fn)w{fn)Af 



«=— 00 



where in the last step we have used 



fk-n = (k - n)Af = fk - fn 



and have implied, from c_„ = c„ , that 



is equal to w(/„). 



Thus, from (4.5-4), we get for the power spectrum of I 



/+00 

 w(x)w{f - .v) dx (4.5-5) 



■00 



