234 BELL SYSTEM TECHNICAL JOURNAL 



18 db. For other values of <p the distortion is, of course, proportional 

 to the cube of the total change in attenuation. 



This estimate is in good agreement with the computations made on 

 actual networks. It also covers the range of greatest practical interest, 

 since in most communication systems changes in attenuation of as 

 much as 18 db produce undesirable variations in the levels of the 

 different channels with respect to one another or to interfering signals. 

 If necessary, however, it appears to be possible to go considerably 

 farther. This possibility arises from the fact that in actual practice (p 

 will be complex, which means that the structure acts as a variable 

 equalizer with respect to both phase and attenuation. Ordinarily, 

 however, only the variation of the attenuation characteristic is of 

 interest. Since the real component of ip^ depends upon both the real 

 and imaginary components of <p, it is thus possible, by choosing the 

 proper relation between these latter two quantities, to eliminate, 

 effectively, the third order term as well as the even order terms in the 

 general expansion. The first disturbing term is then of the fifth order, 

 and has a very small coefficient. If we assume that the desired relation 

 between the real and imaginary components of if can be obtained with 

 sufficient precision in a physical network, it appears that this process 

 allows us to confine the distortion to 0.1 db for total variations in at- 

 tenuation as great as 30 or 35 db. Since the distortion now depends 

 upon the fifth power of the total variation in attenuation it is, of course, 

 very small for more moderate variations. For example, under the same 

 assumptions it is only about 0.001 db for a total variation of 12 db. 



All of these relations, of course, have no utility unless structures 

 meeting the general conditions laid down by equation (3) can be found. 

 The simplest structure for the purpose appears to be the 11 of fixed re- 

 sistances shown in Fig. 3. A set of illustrative characteristics, drawn 

 on the assumption that the parameter a equals 2 , is shown at the bottom 

 of the figure. 



At first sight, this may appear to be a trivial illustration, since ^o 

 and <p are merely constants, and the structure thus has only the proper- 

 ties of an ordinary gain control. It is possible, however, to introduce 

 auxiliary networks by means of which ^o and (p can be made prescribed 

 functions of frequency. For example, 0o can be altered by adding an 

 ordinary equalizer in tandem with either terminating resistance. The 

 modification which allows us to vary ^ may be somewhat less obvious. 

 It consists of the introduction of a symmetrical four-terminal network 

 having the image impedance i?o, between the variable resistance and 

 the terminals to which it was previously connected, as shown by 

 Fig. 4. 



