742 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



band-edge is modified by a supplementary characteristic of the form 



Giiiu) = 2(1 ~ sill Tni/2(j)x) u < to^ , 



= 



U > coj 



(4.05) 



This form of Qi represents a close approximation to actual modifications 

 of band-edges by a gradual cutoff and also results in rather simple ex- 

 pressions for the modified impulse characteristic 

 With (4.05) in (4.04), 



1 f"-" . 



F{t) = - — (1 — sin Tni/2wx) sin uto du, 



to Jo 



1 — cos COx^O I cos COxto 



cos COx^O 



= Wi cos ooxto 





IT + 'IWxk TT — 20ixU. 



1 1 



(4.06) 



1 



_OixU T — 2oOxto TT + 2cOxio J ' 



cos CCxU 



^0 1 — {2u)xto/iry- 

 The impulse characteristic obtained from (4.03) is 



boix sin coi^o cos oixk 



Pit) 



T uito 1 — (2cOxio/7r)2 



(4.07) 



For the particular case shown in Fig. 11 the value of Ux is taken to be 

 wi/2. 



For a symmetrical bandpass characteristic, as shown in Fig. 12, 



P(0 = 2 cos {o:Jo - ypo) Pit). 



(4.08) 



Pit) is obtained by replacing coi by cos in (4.07), and \po is the phase 

 intercept at zero frequency as in connection wdth (3.03). This gives 



Pit) = 



5(j)s sin oisto 



cos COxto 



coJn 1 



(2cOxV7r)2- 



(4.09) 



For the particular case shown in Fig. 12, the value of Wx is taken to be 

 w,/2. 



The in-phase and quadrature components with respect to anj^ fre- 

 quency are obtained from (2.19) with xpy = (jiTd and are shown in Fig. 

 12 for the particular case in which the reference frequency is displaced 

 from the midband frequency by co^ = cos . 



