608 BELL SYSTEM TECHNICAL JOURNAL 



It is now a straightforward, though somewhat tedious process, to 

 count the total number of possible products of each type falling in 

 individual channels. The arrangement of terms above is such that 

 I > m > n; this forms a convenient way of insuring that no product 

 is counted twice. We shall illustrate by taking a simple case — the 

 second order sum product, or (A + 5) -type. Let kp represent the 

 carrier frequency of the channel in which we wish to determine the 

 number of possible {A + B)-type products. This means that in (2) 

 we wish to find the number of terms in the third summation in which 

 m = n = k. The resulting sum becomes: 



^ {m^ k — m — m — I). (4) 



TO=ni+l 



That is, there are as many terms as there are integer values of n 

 satisfying the simultaneous inequalities, 



ni + 1 — m ^ Hi 



< m ^ ^ — Ml 



(5) 



The number of terms is zero li k > 2n2 — I, because the lower limit 

 of the second inequality exceeds the upper limit of the ftrst. If 

 wi + M2 — ife — 2^2 — 1, the upper limit of the first inequality and the 

 lower limit of the second inequality are governing, and the number of 

 terms is «2 — I{k/2), where I(x) is a symbolic representation for 

 the largest integer ^ x. If 2ni ^ k ^ iii + n^, the second inequality 



is governing and the number of terms is I ( — -z — ) — wi. If ^ :^ 2«i, 



the number of terms is zero. 



In a similar manner the more complicated sums representing the 

 number of third order products can be evaluated. It is to be noted 

 that contributions to a particular type can come from more than one 

 of the sums listed. For example, the {A + B - C)-type is made up 

 of the summations from the l + m — n,l — m + n, and I — m — n 

 terms. In fact all these are oi {A -\- B — C)-type except those in which 

 I — yyi — n \s negative. The latter, since only positive values of fre- 

 quency are significant, are oi {A — B — C)-type. An {A — B — C)- 

 type product differs from {A -^ B — C)-type not only in S.T.M.F., but 

 also in manner of addition of contributions from a multi-repeater line 

 as discussed in Section 8. 



As an alternative to an actual count of the products falling in a 

 channel, it is possible to approximate the sum by an integration 



