520 BELL SYSTEM TECHNICAL JOURNAL 



Appendix II 



A nalytical Discussion of Phase Characteristics 

 By means of the Fourier Integral any signal or wave whatever may be 

 regarded as the sum of an infinite number of steady state sinusoidal 

 frequency components which have existed and will exist for all time. 

 Their amplitudes are infinitesimal and they are separated by dif- 

 ferentially small amounts in their frequency spectrum. The finite 

 wave is the sum of the infinitesimal components and is determined by 

 their relative amplitudes and phases. The general Fourier Integral 

 for the wave /« may be written 



la = S y cos {(Jit + d) dco, (5) 



where co = 2irf, f being the frequency, y and 9 are functions of oo. 

 If the amplitude is altered by a constant factor at all frequencies there 

 is no amplitude distortion. For the purpose of discussing phase dis- 

 tortion we shall assume this factor unity. Let the angle be modified 

 by any network by an angle B which is a function of frequency. The 

 expression for the received wave will then be 



h= f y cos (co/ + - B) doi. (6) 



Let us assume a simple case where B is proportional to frequency, 

 i.e. 



B = aico. (7) 



Then 



lb = S y cos (co/ -\- Q — a\<jj)doi 



= f y cos (co{t - ai) + d)dco (8) 



— S y cos (co/' + Q)dw, 

 where 



t' = t- ax. (9) 



This then is identical with the original wave form but occurs at a 



time fli later given by 



B dB . , ^. 



- = J- = rt,. (10) 



CO aco 



Such a phase curve then gives no distortion but a delay.^^ 



The phase characteristic of a low pass filter or cable in the transmit- 

 ting range may be written 



B = aico + aoco"^ + aaco' • • •. _ (11) 



'" It can be readily seen that for any portion of the phase characteristic we may 

 have B = aiw ± Nw. N is an integer where A'^ is ev^en the results will still be the 

 same as for A^ = since cos {Nw + 6) = cos 6. If iV is odd the only difference is a 

 change in sign of /;,. 



