INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS G67 



[(1.6) of Reference 2] which is equivalent to (6), when the average is ex- 

 pressed as an integral over a continuous probability distribution and 

 (Ae) is replaced by (de/dv)(Av), with (Ay) = const. This form of (6) is 

 the point of departure for the calculation in the Appendix. 



B. Operational Significance of a 



Manipulation of (2) may be shown to result in the expression 



J^(Tj = J2 ^fVi - \ cP{e) de, 



j 3 •' 



which is the difference between the mean square signal voltages following 

 and preceding quantization. Hence a is proportional to the difference 

 between the quantized and unciuantized signal powers. Since o- is intrin- 

 sically positive, the quantizing error power is added to the signal by quan- 

 tization and is, in principle, measurable as the difference between two 

 wattmeter readings. 



In addition to providing an operational interpretation of the quantiz- 

 ing error power, the rewritten expression for a reveals the equivalence of 

 0" to the "Sheppard correction" to the grouped second-moment in sta- 

 tistics,"^' where calculations are facilitated by grouping (i.e., uniform 

 quantization) of numerical data. The reader who is interested in a more 

 elaborate deduction of (7) from the Euler-Maclaurin summation formula, 

 as well as discussions of the validity of (7), may therefore consult the 

 statistical literature. 



III. CHOICE OF COMPRESSION CHARACTERISTIC 



A. Restriction to Logarithmic Compression 



We shall consider the properties of the logarithmic type of compression 

 characteristic,* defined by the equations f 



^ ^ F log 11 + WDI j^^ o^.SF • (8a) 



log (1 + /i) 



and 



^ ^ -VJog[l-Si^ f^^ -F^e^O (8b) 



log (1 + m) 



* The author first encountered this characteristic in the work of Panter and 

 Dite* and the references thereto cited by C. P. Villars in an unpublished memo- 

 randum. He has since learned that such characteristics had been considered by 

 VV. R. Bennett as early as 1944 (unpublished), as well as b}- Holzwarth"' in 1949. 



t Unless otherwise specified, natural logarithms will be used throughout. 



