752 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



6, PULSE ECHOES FROM PHASE DISTORTION 



For any transmission — frequency characteristic the corresponding 

 impulse characteristics can be determined from the Fourier integral 

 relation (2.01). This, however, may involve the evaluation of compli- 

 cated integrals, which in general would require numerical integration 

 and would be a rather elaborate procedure. A preferable method of 

 sufficient accuracy in most engineering applications is to employ the 

 theoretical solutions given previously for various ideal transmission 

 characteristics with a hnear phase shift as a point of departure or first 

 approximation. A satisfactory second approximation can in many 

 instances be secured by evaluating the transmission distortion resulting 

 from a sinusoidal deviation in the phase characteristic. Furthermore, 

 any type of deviation in the phase characteristic can in principle be 

 represented by a Fourier series in terms of harmonic sinusoidal devia- 

 tions. 



Aside from the circumstance that in many cases a sine deviation in 

 the phase characteristic affords a fairly satisfactory approximation to 

 actual phase distortion it has the advantage in theoretical formulation 

 that it permits determination of the resultant pulse distortion by the 

 method of "paired echoes." In the usual application of this method only 

 small phase deviations are considered resulting in a single pair of pulse 

 or signal echoes of small amplitude, and the method is then particularly 

 simple.^' ^ When delay distortion is appreciable, however, as is fre- 

 quently the case in wire circuits, it becomes necessary to consider a 

 large number of pulse or signal echoes of considerable amplitude. Since 

 the amplitudes of the pulse echoes may be obtained from available tables 

 of Bessel Functions, the determination of the echoes is, nevertheless, 

 simple in procedure and the determination of the shape of the distorted 

 pulses or other signals not too elaborate. 



A given amplitude characteristic within the transmission band may 

 be associated wdth various phase characteristics, depending on the shape 

 of the amplitude characteristic outside the transmission band and also 

 on whether or not a minimum phase shift system is involved. It is 

 therefore permissible to consider the effect of various departures from 

 a given phase characteristic independent of the amplitude characteristic 

 within the transmission band. 



With a sinusoidal departure from a given phase characteristic \{/q(o:) 

 as shown in Fig. 19, the modified phase function becomes 



;/'(to) = ^o(w) — h sin wr. (6.01) 



With 



