730 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



teniiation characteristic * 



A(o:) = Ao(co) [1 + a COS cor], (1.12) 



and the corresponding phase characteristic becomes 



(nAoiu)[l + a cos ut] 



TT J-o 



dii. 



= iPM + 2 tan"' 



u 



r sin cor 



1 + ^^ cos COT ' 



lAoCw) + 2[r sin cor + — sin 2 cor 



(1.13) 



3 



7' 



+ - sin 3 cor + . , 

 o 



where 



= - [1 T Vl - a^], (1.14) 



and the minus sign is to be used. 



Thus, a cosine modification in the ampHtude characteristic is accom- 

 panied by an infinite series of sine deviations in the phase characteristic. 

 For sufficiently small values of a, r = a/2 and (1.13) reduces to (1.11). 



2. FREQUENCY AND IMPULSE TRANSMISSION CHARACTERISTICS 



In dealing with pulse transmission, it is customary to consider three 

 basic types of time variations of currents and electromotive forces, a 

 cisoidal variation, a unit impulse and a unit step. The cisoidal variation, 

 e'"', is basic in the solution of network and transmission problems in 

 terms of complex impedances and admittances. The unit impulse is a 

 current or electromotive force of very high intensity and short duration, 

 such that the area under the impulse is unity. The unit step is a current 

 or electromotive force which is zero for if < and unity thereafter. 



The time responses of networks or transmission systems to these three 

 basic time functions are interrelated so that each may be obtained when 

 one of the others is known. Furthermore, the time responses for electro- 

 motive forces or currents of arbitrary wave shape may be obtained from 

 the response characteristic for any one of these basic time functions. 



The pulses applied in pulse systems can usually be approximated by 

 impulses. Furthermore, with impulses certain simple relationships can 

 be established which are either obscured or more complicated when a 



