672 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



simulated closely by using closely spaced point couplings. In order to do 

 this intelligently we need a theory for multi-element point couplings. 



The most general symmetrical point coupling distribution for parallel 

 coupled lines is illustrated in Fig. 10. The letters Oo , ai , 02- • •«„ desig- 

 nate the strength of the couplings, and di , d^ , • • • L represents the 

 spacings between them. The transform for the total coupling distribu- 

 tion is 



Fr = f"\re'''''"' dx. (13) 



J-L/2 



Ft = do -\- 2ai cos 7i -f 2a2 cos 72 + 2a3 cos 73 + • • ■ 2a„ cos 6 in 

 which 



and 



e = .L('--l)or.L('- + '-), 

 \\i X2/ \Xi X2/' 



depending on whether forward wave discrimination or directivity is re- 

 quired. The discrimination function is then 



Discnmmation = — ^ =-^ . (14) 



b T 



Let us take as an example the familiar 1-3-3-1 binomial distribution 

 of amplitudes for equally spaced couplings. In the terminology of eciua- 

 tion (13), ao = 0, ai = 3, 02 =. 1, a^ = for k > 2,di = L/3, and di = L. 

 Then (14) yields 



Discrimination = 7- = — - — — . (15) 



6 cos e/d -f 2 cos cos^ d/S 



CENTER OF ARRAY 



J 



80 3 1 



|„ L H 



Fig. 10 — Schematic of point coupling distributions. 



