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BELL SYSTEM TECHNICAL JOURNAL 



pass band characteristic is of the ripple type with equal maxima and equal 

 minima. In the attenuation region the valleys of loss are of equal value. 



The general form for the insertion power ratio to obtain the desired char- 

 acteristic is: 





[1 + (e'"" - 1) cosh' Bi] 



In this equation R\ and R2 are the resistive terminations and ap is the 

 maximum ripple in the pass band as shown in Fig. 11. 



di represents a function of frequency so chosen that cosh di is an odd 

 or even rational function of frequency. Also di must be a pure imaginary 

 throughout the passing band and must be of the form («/ + nWi) in the 

 attenuation region. The term ai is real at all attenuation frequencies 

 becoming infinite at those required by the specification of minimum a„ 

 in Fig. 11. 



Darlington further showed that di closely conforms with the image 

 transfer constant of an image parameter filter if the effective pass band of 

 the insertion loss filter coincides with the theoretical pass band of the image 

 filter. Based on this conclusion a design method was formulated which 

 permits a reference filter derived from image parameters to be used as the 

 basis of the insertion Joss filter. There is, of course, no correspondence 

 between the elements of the reference image filter and the insertion filter. 

 This reference filter is not a requisite to the development of the insertion 

 theory but it does offer a convenient and well known transfer constant which 

 is the right functional form for use in the insertion power ratio stated above. 



Referring again to Fig. 11, the approximate minimum loss, aa , deter- 

 mines the number of peak sections required in the reference filter from the 

 relationship : 



aa = 20 log (e'"" - 1) - 10(2w + 1) log 9 - 18 



where "w" is the number of peaks required and ap is the band ripple func- 

 tion as before. The new term introduced here is "9" which is directly 

 tied up with the selectivity demanded of the filter, i.e., the amount of fre- 

 quency space avilable between the last useful frequency or ejfective cut-off 

 and the first frequency at which attenuation equal to aa is needed. The 

 relationships are as follows: 



1 Tl - a/F^""* 

 q = iz\ — — ^.-- I + 



16 Ll + 



where K' = Vl - K"^ 



and K 



_ h-h 

 /4-/1 



