THEORETICAL FITNDAMKXTAT.S OK rrLSK TRANSMISSION t'2.) 



linear component cur,/ , which represents a constant transmission delay 

 Td for all frequencies, as in the case of an ideal dela.y netwoi'k. Ladder 

 tjq^e structures and transmission lines have phase characteristics which 

 can be represented by the a])ove two components. The third component 

 can be represented by a lattice structure ^vith constant amplitude chai'- 

 acteristic but varying phase. Such a network component may he i)resen1 

 in a transmission system or may be inserted intentionally for i)hase 

 equalization, i.e. to supplement the first component above so as to secure 

 a linear phase characteristic without altering the amplitude characteristic 

 of the sj'^stem. 



The following discussion is concerned with the relationship of the first 

 component to the amplitude characteristic of the .system, or conversely. 



The natural logarithm of the transmission-frequency characteristic 

 given by (1.01) is 



lnT(ico) = fnA(o)) - ti/'(co). (1.03) 



The component t7iA(co) is referred to as the attenuation characteristic, 

 and when expressed in decibels equals 8.69 hiA(oj). 



The followdng relations exist between the attenuation and phase char- 

 acteristics of minimum phase shift systems or system components: '" 



fnA{.) = -l f ±^ du = ^ I 2}^^ a,^ (1.04) 



and 



TT J-00 u — u IT Jo u^ — co- 



in the evaluation of these integrals, the principal \'alues are to be used, 

 i.e., re.sults of the form fn( — ii) are to be taken as In \ —u\ rather than 

 ^/i I w I + tV. 



As an example consider an attenuation characteristic as shown in Fig. 

 1, with A(o}) = Ao between co = and coc and .li between co = coc and 

 X. Equation (1.05) then becomes 



,0f N 2co 



TT 



Jo U" 



= - fn{Ao/A{)(n 



T 



Jo n- — CO- Ju 



COc -|- CO 



(LOG) 



In Fig. 1 is shoAvn the phase characteristic for Ao/A^ = 100, correspond- 

 ing to a 40 db cutoff at co = co^ . 



