INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 675 



When applied to (15) and (18), with the upper limit in (15) replaced 

 by oc with negligible error, (25) implies that 



.4 = in /VP = 1/a/2 = 0.707 (26) 



Hence, (17) will be replaced, for nvmierical calculations, by the relation 



VSND = log (1 + m)[1 + (C/m)' + \/2C/nf (27) 



The corresponding substitution of (25) into (23) yields, for the case of 

 eo F^O, 



lE~\ = eo-\- (?/2)' exp {-V2C/B) (28) 



where we have introduced the "bias parameter," 



B = V/eo (29) 



When (28) is combined with (13), (21), and (24), we find, after some 

 algebraic manipulation, that 



VsNDe = log (1 + m) 



•[1 + (C/n)\l + m/5)' + (\/2C/m) exp (- V2C/5)]' 



where De' = (o-g/e-). It is to be noted that De has been defined in terms 

 of the ratio of as to e^ rather than E-, so that 



^ 2 _ Mean Square Error Voltage . ^ 



Mean Square Speech Voltage 



_ Average Error Power ('iM.\ 



Average Speech Power 



Examination of (30) reveals that it has the re(iuired property of re- 

 ducing to (27) for ^0 = 0, i.e., for 5 -^ oo. Furthermore (27) and (30) 

 indicate that De ^ D so that the addition of a dc component increases 

 the quantizing error power when companding is used. The existence of 



