728 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



constant for w/coc > 1, in which case the attenuation characteristic will 

 be somewhat different for w/wc < 1 and the phase characteristic different 

 for co/coc > 1, as illustrated in Fig. 3. (The solution for the latter case is 

 given in Reference 2.) It will be noticed that there is a comparatively 

 minor difference betw^een the attenuation characteristics for w/ojc < 1 

 in the above cases, so that the attenuation characteristic for co/coc > 1 

 has a relatively minor effect, provided there is no discontinuitj^ near 

 w/coc = 1. The transmission loss characteristics shown in Fig. 3 represent 

 a close approximation to the type of characteristic employed in pulse 

 transmission systems, as will be sho\\Ti later. 



In the above examples low-pass characteristics were assumed. For 

 high-pass characteristics the algebraic sign of the phase is reversed with 

 respect to the amplitude characteristic as indicated in Fig. 4, which also 

 illustrates relationships for band-pass characteristics. The band-pass 

 characteristics are obtained by connecting low-pass and high-pass net- 

 works in tandem. The resultant attenuation and phase characteristics 

 are obtained by adding the low and high-pass attenuation and phase 

 characteristics, as illustrated in the figure. In the second case showTi in 

 the figure, the band-pass characteristic is assumed to have a linear phase 

 characteristic in the transmission band, in which case the attenuation 

 characteristic will not be symmetrical about the midband frequency, 

 unless the latter is high in relation to the bandwidth. The third case 

 illustrates the type of band-pass characteristic encountered in wire 

 systems with a low-frequency cutoff. There will then be phase distortion 

 at the low end of the band, since it is not feasible with a fairly sharp 

 low-frequency cutoff to obtain a linear phase characteristic in the trans- 

 mission band. 



If the amplitude or attenuation characteristic of a transmission system 

 is modified, it will be accompanied by a modification in the phase 

 characteristic. Of basic importance are cosine modifications in the 

 attenuation and amplitude characteristics. Let the modified amplitude 

 characteristic be of the form 



A(co) = ^o(co)e'"=°'"^ (1.09) 



where ^o(w) is the original amplitude characteristic. The modified at- 

 tenuation characteristic is then 



^nA(co) = ^nAoiu) + a cos cor. (1-10) 



In accordance with (1.05) the modified phase characteristic becomes, 



,0/ N 1 r* /n.4.o(co) , a f°° cos cot 



lA (co) = - / du + - / du, . . 



TV J-oo Oi — U TT J-M OJ — U U-ii/ 



— ^o(w) + a sin wr, 



