INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 671 



For sufficiently small input pulses, (f/Ae) becomes i)roportional to e, 

 as a result of the linearity of the logaritlnnic function in (8) for small 

 arguments. In view of our professed preference for logarithmic behavior, 

 with (e/Ae) = const., it is important to emphasize that the transition to 

 linearity is not peculiar to (8), but is rather an example of the linearity 

 to be expected of any suitably behaved (i.e., continuous, single-valued, 

 with {dv/de)e=o > 1) odd compression function, v(e), capable of power 

 series expansion, in the vicinity of the origin. In (8) this transition to 

 linearity takes place where (e/V) is comparable to /x~ • The extension of 

 the region where (e/Ae) ~ const, to lower and lower pulse amplitudes 

 requires an increase in n, and a concomitant reduction of the (e/Ae) ratio 

 for strong pulses. 



Further evidence of the significance of the parameter ju may be deduced 

 by evaluating the ratio of the largest to the smallest step size from the 

 asj^mptotic expressions for (e/Ae). Thus we find 



(Ae)e=v 



n for jiz » 1 



(Ae)e=o 

 which is a special form of the more general relation 



(Ae)e=v (dv/de)e=o , -, 



(Ae)e=o (dv/de)e=v 



which follows from our standard approximation of 



(de/dv) ^ (Av/Ae) 

 with Av = const. 



B. Comparison with Other Compandors 



An upper bound for companding improvement, which permits the 

 ciuantitative evaluation of the penalty incurred (if any) through the 

 restriction to logarithmic companding, is established in the Appendix. 

 Comparison of the results to be derived from (8) with this upper bound 

 will reveal that nonlogarithmic characteristics, which provide somewhat 

 more companding improvement at certain ^'oluraes, are apt to prove too 

 specialized for the common application to a broad volume range envi- 

 sioned herein. The ^t-characteristics do not suffer from this deficiency 

 since the equitable treatment of large samples, which we have hitherto 

 associated with an ''intuitive naturalness conjecture," will be seen to 

 tend to equalize the treatment of all signal volumes. 



Finally, it will develop that (8), when applied to (0), has the added 

 merit of calculational simplicity. 



