73G 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



3. IDEALIZED CHARACTERISTICS WITH SHARP CUTOFF 



In pulse transmission theory, particularly in dealing with transmission 

 capacity of idealized transmission systems, an ideal low-pass transmis- 

 sion frequency characteristic is ordinarily assumed, with constant ampli- 

 tude and delay in the transmission band together with an abrupt cutoff 

 at the top frequency and zero amplitude beyond, as shown in Fig. 7. 

 As is evident from Fig. 1, this type of characteristic is an abstraction 

 which cannot be physically realized since it will have phase distortion 

 and infinite transmission delay. It can, however, be approached with 

 sufficiently elaborate phase equalization. 



For the above type of characteristic, A(oo) — 1 between oj = and 

 coi , while i/'(co) = coTd , where Td is the transmission delay. With these 

 values in (2.07) : 



Pit) = 



d(x)i sin coi^o 



TT COi^o 



(3.01) 



where ^o = ^ — t^ is the time referred to the peak amplitude of the 

 received pulse. 



The resultant pulse transmission characteristic is shown in Fig. 7, 

 with the factor Scoi/x omitted. The peak amplitude is attained after an 

 infinite time, since the above type of characteristic can be realized only 

 with Td — > 00 . The impulse characteristic is zero when coiio = ±nT, or 

 to = ±Ti , ± 2ri , • • • ± UTi where 



1 



(3.02) 

 Impulses can thus be transmitted at the latter intervals Avithout 



'' " 2f; 



(a) 



AMPLITUDE AND PHASE 

 CHARACTERISTICS 



(b) 

 IMPULSE CHARACTERISTIC 



-to"* — 1 — •■to 



K-PHASE 



AMPLITUDE 



SINT^toA 



FREQUENCY, CV , , , , - , ; , - , , i i 



Fig. 7 — Idealized low-pass characteristic with sharp complete cut-off. 



