5-3] IMPULSE FUNCTIONS 243 



5-3 IMPULSE FUNCTIONS 



Impulse or deltajunctions (so called because they are often denoted by the 

 symbol 5) provide a most useful mathematical device in signal and noise 

 studies. These functions can be visualized as the limiting form of a function 

 whose integral is unity but which is 

 concentrated at a particular value of 

 its argument. Specific representa- 

 tions of impulse functions may take 

 a number of forms. One such form is 



shown in Fig. 5-2. In this figure a 



rectangular function of height A and ^ 



width 1 I A is shown centered at the pic. 5-2 Representation of an Impulse 

 point t\. As A becomes large, the Function, 

 function becomes very highly con- 

 centrated at the point A. For any finite value of A, though, the integral of 

 this function will be unity and independent of A. Thus, the limit of this 

 integral as ^^ ^ <» exists and is equal to the value of the integral. In 

 physical problems, it is conventional to suppose that these operations are 

 interchanged and that an impulse function denoted by hit — /i) whose 

 integral is unity is given by the limit of the function pictured in Fig. 5-2 as 

 A — ^ oo . This certainly seems reasonable in view of the fact that for any 

 finite A^ no matter how large, the integral is unity. Unfortunately, though, 

 integration over the singularity produced when A -^ ^ cannot be justified 

 in a mathematical sense, and these operations cannot correctly be inter- 

 changed. Thus, although we shall formally regard impulse functions con- 

 ventionally as being infinite in height with unit integrals, there is an implicit 

 understanding that the limiting operation must, in actuality, be carried out 

 after the finite function has been integrated. In this connection we note 

 that impulse functions acquire physical significance only after being 

 integrated and do not in themselves represent the end product of any 

 calculation. 



With these provisos, we proceed to a discussion of some of the properties 

 of impulse functions. Probably their most important characteristic is their 

 sampling property. The integral of the product of a continuous function 

 and an impulse is simply the value of the continuous function at the location 

 of the impulse. We can establish this relation with the aid of the represen- 

 tation pictured in Fig. 5-2: 



/" r (i+i/2^ 



/(/)6(/ - t,)dt = lim A J{t)dt = /(/:). (5-17) 



A^co Jt,-i/2A 



Additional properties can be established by finding the Fourier transform 

 of an impulse function : 



