PHASE DISTORTION AND PHASE DISTORTION CORRECTION 201 



signal or the transmission is distortionless. Accordingly, we have 

 the following proposition.'' The necessary and sufficient condition for 

 the practically distortionless transmission of signals in communication 

 systems is that, over the essential range of frequencies contained in the 

 transmitted signal, the transfer impedance of the transmission circuit he 

 equalized both as regards amplitude and phase; that is, the amplitude 

 must be constant and the phase angle linear in the frequency, with a value, 

 when the frequency is zero, of ± mr, where n = 0, I, 2, • • - . 



For many years the variation of the phase angle with frequency 

 was ignored. Research in distortion correction was directed to 

 devising networks ^ so designed that \Z(io})\ would be a constant, R, 

 over the range of essential frequencies. Assuming that this condition 

 is fulfilled by the transducer but that 



B(o}) = cor + cr(co) ± nir, 



where o-(c«j) is non-linear in the frequency, we may write (4) as 



/(/) = ± ^ r F{o:) cos [o;(/ - r) + 0(co) - (t(co)]Jco. (7) 



1 r 



In formula (7) the amplitudes of the component frequencies of the 

 arrival curve are, within a constant, the same as those in the impressed 

 signal /(O- The wave form of the arrival curve, owing to the presence 

 of the phase o-(co), may, however, be widely different from that of the 

 impressed signal.^ 



2. Examples of Phase Distortion in Transmission Systems 



Let us consider the frequency-phase angle characteristic of the two 

 important transmission systems, the submarine telegraph cable and 

 the loaded line. 



The cable of characteristic impedance k — -yfiR -j- ioiL) /iwC and 

 propagation constant y = VCi? + icoL)iooC (with negligible leakage) 

 is assumed terminated in its characteristic impedance at :)c = / so 

 that reflection is suppressed. The transfer impedance Z(ico) is then 



Ziico) ^ ke"^ 



= \Ziico)\e'^^"\ (8) 



^ See reference 9. 



* See reference 10. 



^ In telephone transmission it is not at all certain that preservation of wave 

 form is essential. It is essential, however, that the components of different fre- 

 quencies build up at approximately the same time. It is further demonstrated in 

 the section on 'Loading Systems' below that (x{w) = is the necessary and sufficient 

 condition to fulfill the latter requirement. 



