INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 663 



AlthousH the oxpandor continues to collaborate; witii the coinpi'essor 

 in inipio\iiig the (|uality of weak signals, it is now neithei' necessary nor 

 possible for it to perform the separate function of (luieting the circuit 

 in the absence of speech. Indeed, apart from instrumentational diffi- 

 culties which might arise, it is conceptually sound to transfer the PCM 

 expander to the transmitting terminal, with expansion taking place 

 subsequent to quantization but prior to encoding and transmission. 

 Another interesting peculiarity of the PCM expandor is the restriction 

 of its operation, b}^ quantization, to a finite number of discrete operating 

 points on the continuous characteristic. 



The use of companding to reduce the quantizing error which owes its 

 very existence to, and is therefore a function of, the signal, is thus sig- 

 nificantly different from the use of companding to reduce the effects of 

 an independent source of noise in the transmission medium. 



F. AppUcability of the Present Analysis 



Before we proceed to a detailed analj^sis, it is important to emphasize 

 certain restrictive conditions required for the meaningful application of 

 the results to be derived. 



1 . Signal Spectrum 



A signal with a sufficiently complex spectrum, such as speech, is re- 

 quired to justify consideration of the total quantizing error power with- 

 out regard to the detailed composition of the error spectrum. Although 

 it is known that quantization of simple signals (e.g., sinusoids) results in 

 discrete harmonics and modulation products deserving of indi\'idual 

 attention,*' ^ Bennett has shown that the error spectrum for complex 

 signals is sufficiently noise-like to justify analysis on a total power ba- 

 sis.2' 12 



2. Sampling Rate 



The consistent comparison of signal power with the total quantizing 

 error power, rather than with the fraction of the latter quantity appro- 

 priate to the signal band, might at first appear to impose serious limita- 

 tions on the present analysis. Furthermore, the role of sampling has not 

 been discussed explicitly. It is therefore important to note that the justi- 

 fication for this treatment, in the situation of actual interest, has also 

 been given by Bennett.' We need o\\\y add the standard hypothesis'"^ 

 that the sampling rate chosen for a practical sj-stem would equal the 



