136 BELL SYSTEM TECHNICAL JOURNAL 



of the volume displacement, and as such it is proportional to the transfer 

 displacement admittance of the system. W^en we are interested in the 

 charge rather than in the current, the admittance takes the form of a dis- 

 placement admittance, related to the ordinary admittance by a factor of the 

 frequency co. That Carson's original equations apply to such a system with 

 little if any change may be easily demonstrated. The term A{t) may be 

 used to denote any of these forms of indicial admittance or indicia! response. 

 The form of the applied voltage assumed is shown by Fig. 1. This form, 

 defined by Heaviside as the unit function, is a function of time equal to zero 

 before, and unity after the time t = 0. More properly, however, it may be 

 regarded as an increment in voltage closely analogous to Isaac Newton's 

 concept of infinitesimal elements of rectangular area, the summation of which 

 forms the basis of the integral calculus. The successive application of small 

 increments of voltage Hkewise forms the basis of the operational calculus, or 

 more particularly, the basis of the Carson extension theorem. 



TIME AXIS 



THE UNIT FUNCTION 



Fig. 1 



The Carson Extension Theorem 



Having obtained the indicial response, either experimentally or theoreti- 

 cally, we have the key to the more general problem where the applied voltage 

 e{t) may be of any form, such as that of speech waves. Let e(/). Fig. 2, be 

 any arbitrary voltage wave corresponding to speech^. Let a series of con- 

 secutive increments of voltage, differing in time by Ar be applied, of such 

 magnitude as to build up the form of the curve e{t). By analyzing each of 

 these components in terms of the indicial admittance A{i), and synthesizing 

 them again, the instantaneous sound pressure may be related to the voltage 

 producing it and the indicial admittance A{r) by the Carson extension 

 equation^: 



Pit) =jj A(T)e(t-T)dT 



I When the above integration is carried out, the term t disappears and is 

 replaced by /. The above sound pressure p(t) represents the sound pressure 

 generated by the receiver in a closed coupler due to an applied voltage e(t). 



