880 BELL SYSTEM TECHNICAL JOURNAL 



new slope. In order to achieve this new slope a semi-infinite slope of mag- 

 nitude ko, commencing at frequency /i, must be added. This process is 

 continued up the frequency scale until the entire straight line approxima- 

 tion is constructed. 



The total phase d{f) at a particular frequency/ is then given by the sum : 

 of the phase at frequency/ associated with each of the semi-infinite constant 

 slopes of attenuation which together make up the straight line ap- 

 proxmation. 

 Thus: 



Oil) - ^1^0 - kA + kidi - kiSi + hG2 - hds + kids - kSi 



or for the general straight line approximation having slopes 



ki , ki J • • • k„ 



d{f) - h (do - ^l) + h (dl - ^2) + • • • ^n (^„-l - On) 



where : 



dn is the phase at frequency/ associated with the semi-infinite unit slope 

 of attenuation commencing at frequency /„ and extending to / = 00 

 and is read from Tables I or III for / < /« and Tables II or IV for 



/ >/", 

 and 



kn is the slope of the straight line approximation between /„_i and /„ \ 

 given by: 



7 ■'^n .^n— 1 



20 log f- 



Jn-l 



where: 



An is, the attenuation at frequency /„ on the straight line approximation. 



Note that in Fig. 4a the attenuation is constant from zero frequency to 

 the first critical frequency /o- In many problems, there is a constant slope 

 below frequency /i to frequency zero. In that event, the initial critical 

 frequency, /o, will be zero, and 60 will be 90°. (/o// = at all finite fre- 

 quencies.) When this occurs, ^1 must be determined by choosing a finite 

 frequency /o and taking the ratio of attenuation change between /o and /i 

 to 20 log of the ratio of /i to/o. Similarly, the attenuation is constant in 

 the illustration from the top critical frequency fi to infinity, whereas in 

 many problems the attenuation will have a constant slope extending from 

 the top critical frequency to infinity. In these cases, the top critical fre- 

 quency will be infinity and the final angle 0„ will, of course, be zero. Here 

 again the final slope k„ must be determined over a finite portion of this 

 infinite slope. 



