544 BELL SYSTEM TECHNICAL JOURNAL 



impedance equal to the average of that for the irregular line. This 

 rise varies slowly with frequency. The standard deviation of the at- 

 tenuation will also include not only the efifect of the reflections which 

 we have been considering but in addition one caused by the fact that 

 the attenuations of the successive elementary pieces are not alike, and 

 hence their sum, aside from any reflections, will also show a distribu- 

 tion. This additional contribution will vary only very slowly with 

 frequency. The standard deviation will be vAAi^ + AA2^ where A 

 represents the losses in the total line, the subscript 1 indicates the con- 

 tribution due to the reflections, and the subscript 2 that due to the 

 distribution of the individual attenuations. 



The same observation may be made about the attenuation that was 

 made about the terminal impedance, as regards measurements made 

 at one frequency on an ensemble of lines and measurements over a 

 range of frequencies on one line; except that the contribution to the 

 deviation caused by the distribution of individual attenuations varies 

 so slowly with frequency that on each individual line it will look like a 

 displacement from the average attenuation, over the whole frequency 

 range. For the purposes of the present paper only the contributions 

 from the reflections will be computed. 



When this information on irregularities is being used by a designer of 

 equalizers he is interested in two characteristics: first, how far each 

 attenuation curve for a number of lines will be displaced as a whole 

 from the average; and second, how "wiggly" each individual curve is 

 likely to be. While the observations above give the general amplitude 

 of the latter they do not tell how closely together in frequency the 

 individual "wiggles" are likely to come. To express this, the term 

 "sinuosity " has been defined as the standard deviation of the difi^erence 

 in attenuation (for the ensemble of lines) at two frequencies separated 

 by a given interval A/. By the previous observations this can be 

 extended to the attenuation difl^erences for successive frequencies 

 separated by the interval A/, for a range of frequencies in a single line. 



When the transmission line is used for certain types of communica- 

 tion, notably for telephotography or television, it is important to 

 equalize it accurately for envelope delay as well as attenuation. The 

 envelope delay is defined as 



T = d^/do) (1) 



where /3 is the phase shift through the line and co is 2x times the fre- 

 quency. For an ensemble of lines, the envelope delay at a given 

 frequency will also form an ensemble, the standard deviation of which 

 will be VaT^. By the observations which have already been made 



