is a weighting function: 



E Q (*)= f W x (t) E i (t-r)tfT 



(1) 



The output E Q (t) is observed to be a weighted mean of the 

 past inputs. The system is physically realizable if it 

 responds only to inputs which have already occurred, and 

 the system is stable if every bounded input produces a 

 bounded output. The weighting function may be thought of 

 as the unit-impulse-response for the system. The method of 

 description utilizing the concept of the weighting function 

 is based on an analysis in the time domain. 



The concept of the frequency-response may be introduced 

 by the following procedure. Let the input E. (t) be given 

 as follows : 



E { (t)= J BexpC/2rt/i) (2) 



where,/' = v -1, / is in cycles per unit tine, and B is a 

 complex factor involving amplitude and phase. This complex 

 function represents the sinusoidal input. The output is 

 given by substituting this input in equation (l): 



E Q (t)= / B W{r) exp\J2nf(t-Tj)dT 

 Jo 



CO _ 



E (t)=Bexpj2xft I V{t) exp(v'2jt/T)dT 



E Q (t)= B S ,_ (/) exvj2xft 



where the complex quantity B (/) is called the frequency 

 response, and is defined by 



-Sj. (/> f V x (t) exp(-j2K/ T )dT 



\ 



(3) 



CO 



