Instantaneous Companding of 

 Quantized Signals 



By BERNARD SMITH 



(Manuscript received October 8, 1956) 



Instantaneous companding may be used to improve the quantized approx- 

 imation of a signal by producing effectively nonuniform quantization. A 

 revision, extension, and reinterpretation of the analysis of Panter and Dite 

 permits the calculation of the quantizing error power as a function of the 

 degree of companding, the number of quantizing steps, the signal volume, 

 the size of the ^'equivalent dc component" in the signal input to the com- 

 pressor, and the statistical distribution of amplitudes in the signal. It ap- 

 pears, from Bennett's spectral analysis, that the total quantizing error power 

 so calculated may properly be studied without attention to the detailed com- 

 position of the error spectrum, provided the signal is complex (such as speech 

 or noise) and is sampled at the minimum information-theoretic rate. 



These calculations lead to the formulation of an effective process for choos- 

 ing the proper combination of the number of digits per code group and com- 

 panding characteristic for quantized speech communication systems. An 

 illustrative application is made to the planning of a hypothetical PCM sys- 

 tem, employing a common channel compandor on a time division multiplex 

 basis. This reveals that the calculated companding improvement, for the 

 weakest signals to be encountered in such a system, is equivalent to the addi- 

 tion of about 4 to 6 digits per code group, i.e., to an increase in the number 

 of uniform quantizing steps by a factor between 2^ = 16 and 2^ = 64- 



Comparison with the results of related theoretical and experimental studies 

 is also provided. 



TABLE OF CONTENTS 



(Passages marked with an asterisk contain mathematical details which 

 may be omitted in a first reading without loss of continuity.) 



Page 

 I. Introduction 655 



A. Fundamental Properties of Pulse Modulation 655 



1 . Unquantized Signals 655 



2. Quantized Signals (PCM) 655 



B. Quantizing Impairment in PCM Systems 656 



653 



