750 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



band of the spectra. If the phase characteristic of the system is Hnear 

 over this band, the received pulses will have the same shape as the 

 impulse characteristics. It should be recognized, however, that there 

 may be appreciable phase distortion within the transmission band or 

 pulse spectrum, if there are amplitude discontinuities beyond the band 

 resulting from a sharp cut-off by filters. Nevertheless, the type of am- 

 plitude characteristic or frequency spectrum considered above has de- 

 cisive advantages from the standpoint of transmission distortion of the 

 pulses, as shown later, since appreciable phase distortion Avill ordinarily 

 be confined to the edges of the band where the frequency components 

 of the pulse spectrum have low amplitudes. 



Another type of amplitude characteristic resembling that shown in 

 Fig. 13 and frequently considered in connection with pulse transmission 

 is a Gaussian characteristic: 



yl(co) = e-'"'\ (5.06) 



The corresponding impulse characteristic is 



P(0 = ^-A_ e-'-'''\ (5.07) 



2(7rcr) 



1/2 



If it is assumed that the amplitude is reduced to 1 per cent of the peak 

 value after an interval ta = tt/wi , corresponding to the first zero point 

 of an ideal impulse characteristic, it is necessary that U /4o- = 4.6, or 

 (X = .54/col^ The corresponding amplitude and impulse characteristics 

 are 



A(co) = e-o-^^("/"i)', (5.08) 



and 



P{t) = -^ e-oA^ito<^0\ (5.09) 



In Fig. 18 a comparison is made of the two frec^uency characteristics 

 (5.01) and (5.08) considered above, and of the corresponding impulse 

 characteristics (5.02) and (5.09 V The comparison shows that for the 

 same pulse transmission rate and with negligible intersymbol inter- 

 ference, a somewhat wider band must be provided with a Gaussian 

 amplitude characteristic. This is a disadvantage, particularly when the 

 band is restricted within prescribed limits by considerations of inter- 

 ference in adjacent transmission bands, as radio pulse systems. 



