082 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



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Fig. 9 — Saturated companding improvement for weakest signals as a function 

 of the degree of logarithmic compression {jx) . Given a value of yu, the corresponding 

 ordinate represents the reduction of quantizing error power (in db) for signals so 

 weak that their peaks satisfy the relation (e/F) <JC m~^- Thus weaker and weaker 

 signals are required to exploit the added improvement which follows from an in- 

 crease in IX. The auxiliary ordinate scale on the right exhibits the increase in the 

 number of digits per code group which may be shown to be equivalent to the linear 

 compression which gives rise to the saturated improvement of weak signals. 



improvement of weak signals (large C) is obtained at the price of im- 

 pairment of strong signals (small C). 



C. Companding Improvement for Cq = 



From the definition of D in (16) it follows that 



Mean Square Error Voltage Without Compandnig 





D 



Mean Square Error ^"oltage With Companding 

 Average Quantitizing Error Power Without Companding 



Average Quantizing Error Powder With Companding 



whence, the improvement (in dh) due to companding may be e\'aluated 

 from the expression 



10 logio {DJDf = 20 logio (A '£)) db (37) 



This quantity is plotted against C for variovis values of ^ in Fig. 8. 

 Substitution of D^_min for D in (37) jaelds the optimum ensemble limit- 

 ing curve. Just as in Fig. 6, each cui've is tangent to this limiting r\\v\Q 

 for a \alue of C corresponding to coincidence of /i and ii,. . 



