Go4 THE BELL SYSTEM TECHXIfAL JOX'RXAL, MAY 1957 



C. Physical Implications of Nonuniform Quantization 657 



1. Quantizing Error as a Function of Step Size 657 



2. Properties of the Mean Square Excited Step Size 658 



D. Nonuniform Quantization Through Uniform Quantization of a Com- 

 pressed Signal 659 



E. The Mechanism of Companding Improvement in Various Communica- 

 tion Systems 661 



1. Syllabic Companding of Continuous Signals 661 



2. Instantaneous Companding of Unquantized Pulse Signals 662 



3. Instantaneous Companding of Quantized Signals 662 



F. Applicability of the Present Analj'sis 663 



1. Signal Spectrum 663 



2. Sampling Kate 663 



3. Number of Quantizing Steps 664 



4. Subjective Effects Beyond the Scope of the Present Analysis. . . . 664 



II. Evaluation of the Mean Square Quantization Error {<t) 665 



A.* Generalization of the Analysis of Panter and Dite 665 



B. Operational Significance of o- 667 



III. Choice of Compression Characteristic 667 



A. Restriction to Logarithmic Compression 667 



B. Comparison with Other Compandors 671 



IV.* The Calculation of Quantizing Error 672 



A. Logarithmic Companding in tlie Alssence of "DC Bias" 672 



B. Logarithmic Companding in the Presence of "DC Bias" 673 



C. Application to Speech as Represented bj' a Negative Exponential 

 Distribution of Amplitudes 674 



D. Uniform Quantization : fx = 676 



V. Discussion of General Results 676 



A. Quantitative Description of Conventional Operation (eo = 0) 677 



1 . Number of Quantizing Steps (N) 677 



2. Compandor Overload Voltage (F) 677 



3. Relative Signal Power (C) 677 



4. Average Absolute Signal Amplitude (| e |) 678 



5. Degree of Compression (ju) 679 



B. Optimum Compandor Ensemble 679 



C. Companding Improvement for eo = 682 



D. Companding Improvement for eo 9^ 684 



VI. Application of Results to a Hypothetical PCM S3^stem 686 



A. Speech Vohmies 686 



B. Choice of Compression Characteristic 687 



1 . Ideal Behavior for Speech 687 



2. Practical Limitations on Companding Improvement 688 



(a) Mismatch Between Zero Levels of Signal and Compandor. . . 688 



(b) Background Noise Level 690 



C. Choice of the Number of Digits per Code Group 690 



1. Ideal Behavior for Speech 690 



2. Practical Limitations 693 



(a) Mismatch of Zero Levels of Signal and Compandor 693 



(b) Background Noise Level 695 



D. Possibility of Using Automatic Volume Regulation 697 



E. Comparison with Previous Experimental Results 697 



VII. Conclusions 698 



Acknowledgments 698 



Appendix — The Minimization of Quantizing Error Power 698 



References 708 



