THEORETICAL FINDAMEXTALS OF PULSE TRANSMISSION 735 



Comparison of (2.13) with (2.09) shows that: 



R- + R+ = COS if P(t), 



(2.14) 

 Q.-Q+ = sin ^ P(t), 



tan ^= (Q_- Q+)/(7?_ + R+). (2.15) 



To find the coiTcspoiKliiifi; components when cor is changed to co/, 

 equation (2.13) may he written 



P(t) = COS[w// - ^Pr' - (co/ - 0:r)t + (.A/ - 4^r) " <p\ P{t) 



= cos(co// - ip/ - <p') P{t), (2.16) 



where <p' = (p'(t) is given bj^: 



<p' = f -\- (co/ — COr)^ — {\pr' " "Ar), 



(2.17) 



= ^ + CO,/ — lAi/ • 



Thus, when the reference freciuency is changed by Wy and its phase by 

 \f/y , the corresponding in-phase and cjuadrature components become: 



RJ + 7?+' = cos (^ + oiyt - yfyy) P{t), and 



(2.18) 

 QJ - QV = sin (<p + c^yt - ^,) P(0 



To summarize, when the in-phase and cjuadrature components have 

 been determined for any reference frequency co^ from (2.10) and (2.11), 

 and the envelope P together with the function (p from (2.12) and (2.14), 

 the in-phase and c^uadrature components for another reference frequency 

 co/ can readily be determined with the aid of (2.18). In the particular 

 case where the amplitude characteristic has even and the phase charac- 

 teristic odd symmetry Avith respect to the midband frequency, the 

 ciuadrature component disappears with respect to the midband fre- 

 quency, so that ^ = and (2.18) simplifies to 



R- +/?+' = cos (coyt - yPy) P(t), and 



(2.19) 



Q- - Q+' = sin (C^yt - XPy) P(t). 



The above relations (2.18) and (2.19) facihtate comparison of trans- 

 mission performance as the reference or carrier frequency is changed, for 

 example the comparison of double with vestigial sideband transmission, 

 as illustrated in section 14. 



