548 BELL SYSTEM TECHNICAL JOURNAL 



that in this form the equation is independent of e, <^, and n. It is in 

 this case that Didlaukis and Kaden found that the result is independent 

 of whether the reflection points are sharp and equally spaced or not. 



The sinuosity has been computed in Appendix III. When finally 

 simplified and measured in nepers, it amounts to 



V(AA.-ArO^ = ^^A/. (9) 



Expressed in terms of the impedance irregularities this amounts to 



v(nr^^ = [ ?]' J^-^' ^'"^ 



where T is, as mentioned before, the envelope delay of the whole line, 

 in seconds. 



In computing the above it is only the components of the echoes which 

 are in phase (or tt radians out of phase) with the main transmission 

 which affect the results. If the echo components at right angles to 

 the main transmission are considered, they will give phase shifts in 

 the resultant signal wave. Further, an echo component whose ratio 

 to the main transmission is x will, when tt radians out of phase with it, 

 give a loss of x nepers; and when at right angles to it, a phase shift of x 

 radians. Now the distribution of echo components in phase (or tt 

 radians out of phase) with the main transmission is substantially the 

 same as that of components at right angles to it. Consequently the 

 sinuosity is also numerically equal to the standard deviation of the 

 difference in phase shifts at two frequencies separated by the given 

 interval A/. Therefore if the interval is called Aco/2x and the resulting 

 numerical value of the sinuosity is divided by Aw it will give the 

 standard deviation of the envelope delay. This is 



(11) 



J 2Va 



The quantity which has been used in considering the suitability of a 

 line from a delay standpoint for transmitting pictorial signals is its 

 envelope delay distortion, or maximum departure in delay each way 

 from a fixed average in the frequency band studied. If we make the 

 usual assumption that the maximum departure ordinarily met (strictly 

 speaking, except in about 3 cases out of 1000) is three times the 

 standard deviation, then the delay distortion contributed by the ir- 

 regularities is ± 3 times the expression given in equation (11). 



