776 THE BELL SYSTEM TECHNICAL JOURNAL, ^L\Y 1954 



For the particular case of sharp cut-offs at coo and coi 



Ao(a}) = 1 < w < coi , 



A(aj) — Ao(cji) =—1 0<a)<coo, and 



Po(0) = coi/tt ti = 7r/wi , 



and 



/ \l/2 / \l/2 



C7 = !L(^«) =(^) . (9.06) 



It A\'ill be noticed that the same result is obtained from (8.07) with the 

 amplitude deviation a = [A(w) — ylo(co)] = — 1 between and wo . 



In actual systems, the low-frequency cut-off will be gradual between 

 oj = and coo , rather than abrupt as assumed above. With a linear 

 variation in the amplitude characteristic between and coo , A (co) — 

 Ao(co) = (-1 + w/a;o)Ao(0) and U = (coo/Scoi)'''. 



If a sufficient number of pulses of one polarity is transmitted in suc- 

 cession at intervals n = l/2/i the received pulses will as noted before in 

 the limit be reduced to zero amplitude by the low-frequency cut-off. 

 The maximum pulse distortion resulting from pulse overlaps when a 

 train of pulses as transmitted is thus equal and opposite to the amplitude 

 Po(0) of the received pulses in the absence of a low-frequency cut-off, so 

 that peak intersymbol interference U = — 1 . If rms intersymbol inter- 

 ference is held at one-quarter the peak value, i.e., U = 0.25, the prob- 

 ability of encountering the maximum tolerable inters^nnbol interference 

 and resultant errors in reception is low enough to be disregarded. On this 

 basis the ratio coo/wi would in accordance with (9.06) have to be less than 

 0.0625. Actually a substantiallj^ smaller ratio would be required because 

 of intersymbol interference from other imperfections in the transmission 

 characteristic and noise. Furthermore, a low-frequency cut-off will 

 be accompanied by phase distortion at the low end of the transmission 

 band, disregarded in the above evaluation. The requirements imposed 

 on the low-frequency cut-off will thus be rather severe for a pulse 

 system as assumed above in which random sequences of pulses are 

 transmitted at intervals ri = l/2/i . Two pulse amplitudes were as- 

 sumed above, and with a greater number of amplitudes the require- 

 ments would be more severe. 



From Fig. 33 it is evident that the effect of a low-frequency cut-off 

 on a received pulse train can be reduced by transmitting pulses at longer 

 intervals than n = l/2/i considered above. For example, with a two-fold 



