66 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



Fifth Order Products 



When all possible combinations of n operating channels are activated 

 five at a time 



n\n - l)(n' - n + 4) 

 12 



fifth order products will be formed and they will lie between A — 

 2{n — 1) and A + 3(n — 1) where the operating band lies between 

 channels A and A + (n — 1). This means that the bandwidth of the 

 intermodulation products is very nearly five times that of the operating 

 channels. The products are sjonmetrically distributed with respect to 

 the midpoint of the operating band. It has been found empirically that 

 if all products known to fall on transmitting channels are excluded, 

 there can still be formed 



3^[6n4 _ 37^3 _^ 575^2 _ 2214n + 3922] 



potentially interfering products which fall within the operating band. 

 This is strictly true only for values of n that exceed 8 and are not mul- 

 tiples of 3. There will remain 



K2[^' - 14n4 + I79n3 - 1154n2 + 4428n - 7844] 



products that will fall either outside the band or on transmitting chan- 

 nels within the band. Since relatively few of these products fall on trans- 

 mitting channels within the band the above expression gives to a high 

 degree of approximation the number of products that will fall outside 

 the operating band. 



Numbers of Potentially Interfering 3rd Order and 5th Order Products 



The formulas given in the two preceding sections were developed by 

 considering individually the various types of 3rd order and 5th order 

 products. Table I shows the general formulas which apply to these 

 individual types of 3rd and 5th order products from which the summa- 

 tion formulas given in the preceding sections were obtained. The number 

 of potentially interfering 3rd and 5th order products is shown in Table 

 II for specific transmission bands containing 10 and 20 consecutive 

 channels. 



PROBABILITY OP INTERMODULATION INTERFERENCE WHEN OPERATING 

 CHANNELS ARE PICKED AT RANDOM 



In an uncoordinated radio communication system it may be assumed 

 that p operating channels are assigned on a random basis in a band con- 



