TABLES OF PHASE 899 



20 log 

 log 



is plotted in Fig. 10 for d = +1/4 and it is apparent than 20 

 for d = —1/4 is identical. This identity does not hold for 9, 



Zoi2 

 Zoi2 



z^ 



however. This is shown in Fig. 11 where 6 for d = +1/4 and 6 for d = 

 — 1/4 are plotted. 



The real characteristic of Fig. 10 was then approximated by a series of 

 straight lines determined by the critical points listed and the phase asso- 

 ciated with this straight line approximation summed. The phase so deter- 

 mined is plotted as individual points in Fig. 11. It is seen that this summa- 

 tion determined the phase of the function in question for d = +1/4 but 

 completely failed to do so for d = — 1/4. The function for d = — 1/4 is an 

 example of a non-minimum phase function for which the above technique 

 fails to determine the phase of the function from its attenuation 

 characteristic.'" 



There are certain instances where the above technique can be usefully 

 applied in connection with non-minimum phase systems in spite of the 

 failure of the method to predict the total phase. ^^ However, the necessity 

 of checking for non-minimum phase conditions and, if such exist, deter- 

 mining whether the above method of computing phase is at all applicable, is 

 illustrated by the non-minimum phase example above. 



1" This is the anticipated result since the function is identified as a non-minimum phase 

 function by the fact that it has two zeros falling in the right half p plane. 



" Bode, "Network Analysis and Feedback Amplifier Design," Chap. XIV, page 309. 



