GENERAL CIRCUIT CONSIDERATIONS 



221 



Here A{p) is a complex function of />, such that A{—p) is the complex con- 

 jugate of A{p) (this assures that F{t) is real). 



With F{t) expressed as in (5.2), we can rewrite (5.1) 



A{p) 



Tin 



e'^""-"" dp 



(5.3) 



Now, suppose, as indicated in Fig. 5.2, we apply the r-f pulse /(/) to the 

 input of a transmission system of length L with a phase constant ^ which 



Fig. 5.1 — A radio-frequenc}' pulse varying with time as/(/). The envelope varies with 

 time as F(t). The pulse might be produced by modulating a radio-frequency source 

 with F(t). 



PHASE CONSTANT /3{a)) 



F(t) 

 f(t)" 



G(t) 



'g(t) 



Fig. 5.2 — When the pulse of Fig. 5.1 is applied to a transmission system of length L 

 and phase constant /3(a)) (a function of co), the output pulse g{t) has an envelope G{1). 



is a function of frequency. Let us assume that the system is lossless. The 

 output g(t) will then be 



g(t) = ['\i(p)e''''''-'''-''' dp (5.4) 



J-Po 



We have assumed that pn is much smaller than co. Let us assume that over 

 the range co — po to o) -\- po , l3 can be adequately represented by 



/3 = ft + |^? 

 oco 



In this case we obtain 



g{t) = e^'"'-''>'' r A{p) 



*'-Po 



The envelope at the output is 



G{t) = r A{p) e^"''-''^'"-''' 



jp(t-(dfildw)L) 1. 



dp 



(5.5) 



(5.6) 



(5.7) 



