THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 733 



equation (2.07) can be written: 

 Pit) = cos [o^rt - rPr)[R-{t) + /?+(/)] 



+ sin U( - 'Ar)[Q-(0 - Q+m 



/?_ = -/ a(- u) cos [ut + ^'(— u)] du, 



TT Jo 



R+ = - a{u) COS [(// - ^(?/)] ^^<, 

 X Jo 



Q_ = - I (i(— It) sin [ui + ^(— w)] d«, and 



IT Jo 



Q^ = - / a(?0 sin [w/ — ^(w)] (^it. 



TT Jo 



(2.09) 



(2.10) 



(2.11) 



The envelope P(t) of the impulse transmission characteristic is given 

 by 



Pit) = [(i?_ + 7^)"' + iQ- - Q^)V". (2.12) 



Comparison of (2.09) with (2.07) shows that R- and R+ can be identi- 

 fied with the impulse characteristics of low-pass systems having the 

 same frequency characteristics as the bandpass system below and above 

 av . The impulse characteristics Q_ and Q+ which arise from asymmetry 

 in the transmission characteristic with respect to av are not present in 

 low-pass systems, since by definition the amplitude characteristic has 

 even symmetry and the phase characteristic odd sjnnmetry with respect 

 to zero frequency. 



The first and second components of (2.09) are referred to as the in- 

 phase and quadrature components of the impulse characteristic of 

 band-pass systems.^ The transmission-frequency characteristic may cor- 

 respondingly be regarded as made up of a component with even sym- 

 metry and another component with odd symmetry about Wr , as indicated 

 in Fig. G. These two components, together with the in-phase and quadra- 

 ture components, will depend on the choice of w, . How'ever, P(t) as given 

 by (2.09) and the envelope as given by (2.12), will remain the same, since 

 a single impulse characteristic is associated with a given transmission- 

 frequency characteristic. 



With the customary pulse transmission methods, the reference fre- 

 quency cor may be identified \\ith a modulating or carrier frequency, 

 which has a special significance when the envelope of a sequence of 

 received pulses is considered. Although for a single pulse the envelope 



