744 BELL SYSTEM TECHNICAL JOURNAL 



Practical values for these elements may be chosen as C5 = 20 mm/ and Rl = 

 150 ohms. 

 Next, the transfer specifications for this illustration may be summarized as 



(a) Degree of equalization — k = 0.25 



(b) Useful band— 2.5 to 8.0 mc 



■ (c) Useful band distortion— < ±0.10 db 

 (d) Resistance efficiency — 65% 



The computation of the desired transfer characteristic Ke^^'^''^ begins with 

 the consideration of the degree of equalization. In order to equalize one- 

 quarter of the power loss between coaxial repeaters, the transfer character- 

 istic over the useful band must vary as Xe"''" where a' represents the com- 

 plete line loss between repeaters. If it is assumed that a is 4 nepers (34.7 dhY'^ 

 at 8.0 mc {x = 1) and varies as a' = j{x) = 4\/^, the transfer character- 

 istic over the range, < .t < 1, according to eq. (10), becomes 



j^ 2fc/(i) _ -2a5(l-v/i) _ -2(l-N/i) 



where a = kj{x) = \/x and ao = ^/(l) = 1- 



The specification of a useful band from 2.5 to 8.0 mc (or x = 0.3 to 

 .V = 1.0) in this example is chosen to illustrate the practical limitation on 

 the precision of equalization at low frequencies. The dashed curve of Fig. 23 

 indicates a low-frequency response which seems realistic for this illustration. 



The computation of the desired transfer characteristic is completed when 

 the out-band portion of the characteristic is chosen to satisfy the specified 

 resistance efficiency. The assumption of a linear cut-off characteristic is 

 suitable as an initial requirement. Hence, the transfer characteristic may be 

 summarized as shown in Fig. 23. The solid curve of this figure represents the 

 transfer characteristic which would be required for equalization over the 

 range, < x < 1, while the dashed curve indicates the modification in this 

 curve resulting from the choice of a conservative low-frequency response 

 and the specification of a useful band of 0.3 < x < 1. 



The solution of the approximation problem consists of three main oper- 

 ations. First, is the determination of the amplitude function a((p) from the 

 transfer characteristic specified in Fig. 23. Second, is the determination of 

 the Fourier cosine coefficients, ao ■ ■ ■ On , of the approximating function 

 /i(v?) and the calculation of the coefficients, Aq ■ •■ An, oif{x-). Third, is 

 the choice of the coefficient e- of the squared Tchebycheff polynomial. 



The amplitude function a((p) is calculated from the specified transfer 

 characteristic by using the relations expressed by eq. (22) . According to 

 eq. (11) of Section 3, the specification for B{x-) over the useful band, 



2^ This discrimination is correct for 4 or 5 miles of coaxial cable. The attenuation on a 

 coaxial line varies as the square root of frequency. 



