THEORETICAL FUNDAMENTALS OF PULSE TRANSMISSION 741 



4. IDEALIZED CHARACTERISTICS WITH GRADUAL CUTOFF 



The iderflizcd transmission characteristics discussed above are of 

 principal interest in that they indicate the physical Hmitations on pulse 

 transmission rates for a given bandwidth. Even if these impulse char- 

 acteristics could be realized without undue difficulties from the stand- 

 point of phase equalization, they would be impracticable in most applica- 

 tions. Their oscillatory nature would entail the use of discrete pulse 

 positions and precise synchronized sampling at fixed intervals, and 

 would preclude certain methods of pulse modulation and detection. 



The non-linearity in the phase characteristic as well as the oscillations 

 in the impulse characteristic can be reduced with a gradual rather than 

 a sharp cut-off, as illustrated in Fig. 10. It is assumed that an ideal 

 characteristic with a sharp cutoff is supplemented by an amplitude 

 characteristic (ii which has odd symmetry about the cutoff frequency 

 coi , i.e., (2i ( — w) = —Gil (u). 



If the latter component alone is considered, and a linear phase charac- 

 teristic assumed, it follows from (2.09) with coi — cor that the effect of 

 this component on the pulse transmission characteristic is given by 



Pi(t) = -Qisincoi^o, (4.01) 



where t{> = t — Td and 



Qi = - Gtiiii) sin uto du. (4.02) 



The function Pi(t) will be zero at the same points as the original pulse 

 transmission characteristic with a sharp cut-off at coi and under certain 

 conditions also at other points. It will modify the original impulse char- 

 acteristic by reducing the oscillatory tail, as illustrated in Fig. 10, but 

 the zero points remain unchanged. 



With the above modification, the resultant impulse characteristic ob- 

 tained by superposition of (3.01) and (4.02) becomes 



P(t) = - sin o)ifo ( — — 2 / (li(w) sin uto du. I , 



TT V^ii Jo / 



= - sin o}itoFii), 



(4.03) 



where 



Fit) = 



1 r"^ 



— — 2 / (li{u) sill 7/^0 du 

 to Jo 



(4.04) 



In the following the expression for F(t) is given for the case when the 



