COUPLED AVAVE THEORY AND WAVEOUIDE APPLTOATIONS 



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m5 28 



%■ AT FIRST NULL WITH UNIFORM ARRAY 



Fig. 11 — For all spurious mode responses down ordinate db, abscissa is the 

 ratio of the required coupling length to the length required for constant amplitude 

 coupling to produce a first null at the same value of (1/Xi — IA2). (After C. L. 

 Dolph, Reference 4). 



which is the relation given by Mumford. The approach is perfectly gen- 

 eral, and henceforth the coupling distribution only will be given with 

 the understanding that the corresponding discrimination function can 

 be obtained from (13) and (14). 



For the case of tapered amplitudes and an even number of equally 

 spaced couplings, (13) can be simpUfied to 



2ai cos 



2n - 1 



+ 2a2 cos 



3d 



2n - 1 



+ • • • 2a„ cos 6. (1(3) 



This case is of interest because a solution has been worked out for the 

 analogous antenna problem to bring the spurious responses (the peaks 

 of the side lobes in the antenna case, or the peaks of the undesired mode 

 responses in the wave selector case) to the same lex'el relative to the de- 

 sired response. This makes the total length of the coupling array a minimum 

 for a given required degree of discrimination. The solution includes specifi- 

 cation of the Tchebysheff distribution of coupling strengths Oi , 02 , a^- • • 

 ttn that are rec^uired to achieve various levels of spurious response, and 

 the resulting increase in total array length required to place the first null 



