MA THEM A TIC A L A XA LYSIS OF RA XDOM NOISE 127 



with the understanding that /is not zero and 



ui-x) = liix). (4.5-6) 



The result which is obtained by using (2.8-6), involving the cosines and 

 only positive values of m, is 



W{f) = a' [ ivixyaif - .v) dx + 2a~ f w{x)w(f + x) dx (4.5-7) 



This contains only positive values of frequency. (4.5-5) and (4.5-7) are 

 equivalent and may readily be transformed into each other. 



The first integral in (4.5-7) arises from second order modulation products 

 of the sum type and the second integral from products of the difference 

 type. This may be seen by writing the current as 



00 00 



I = aV = a Zli Z2t ^mCn COS (cOto/ — ifrr) COS (w„ / — (^„) 



00 00 



= ^3 XI IZ f,„C„{C0S [(C0„ — O),,)/ — iPm + <p,^ (4.5-8) 



^ m.—\ n = l 



-f- COS [(a),„ + O),,)^ + (^m + <^,J} 



The power in the range /a- ,/a- + A/ is the power due to modulation products 

 of the difference type, coa+^ — wf , plus the power due to the modulation 

 products of the sum type, biu-i -\- o^( ■ In the first type / runs from 1 to ^o 

 and in the second type i runs from 1 to ^ — 1. 



Consider the difference tv-pe first, and for the moment take both k and / 

 to be fixed. The two sets m = k -\- C, n = /and m = f,}i = k-\--C are the 

 only values of m and n in (4.5-8) leading to ook+H — o:c . The two corre- 

 sponding terms in (4.5-8) are equal because cos ( — .v) is equal to cos x. The 

 average power contributed by these two terms is 



( ^ Ck+( cf) X {Average of (2 cos [{u^k+C — o}i)t — ^pk+t -\- <p(]f} 



\2 / (4.5-9) 



The power contributed to//, , //, -|- A/ by the difference modulation products 

 is obtained bv summing ( from 1 to oc : 



2 00 



a 



I 1=1 /=1 



-^2a'Af [ w{f, + f)w(f) df 

 Jo 



This leads to the second term in (4.5-7). 



