Quantization Noise Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Glossary of Symbols
- Acronyms and Abbreviations
- Part I Background
- Part II Uniform Quantization
- 4 Statistical Analysis of the Quantizer Output
- 5 Statistical Analysis of the Quantization Noise
- 6 Crosscorrelations between Quantization Noise, Quantizer Input, and Quantizer Output
- 7 General Statistical Relations among the Quantization Noise, the Quantizer Input, and the Quantizer Output
- 8 Quantization of Two or More Variables: Statistical Analysis of the Quantizer Output
- 9 Quantization of Two or More Variables: Statistical Analysis of Quantization Noise
- 10 Quantization of Two or More Variables: General Statistical Relations among the Quantization Noises, and the Quantizer Inputs and Outputs
- 11 Calculation of Moments and Correlation Functions of Quantized Gaussian Variables
- Part III Floating–Point Quantization
- Part IV Quantization in Signal Processing, Feedback Control, and Computations
- Part V Applications of Quantization Noise Theory
- Part VI Quantization of System Parameters
- APPENDICES
- Bibliography
- Index
7 - General Statistical Relations among the Quantization Noise, the Quantizer Input, and the Quantizer Output
from Part II - Uniform Quantization
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Glossary of Symbols
- Acronyms and Abbreviations
- Part I Background
- Part II Uniform Quantization
- 4 Statistical Analysis of the Quantizer Output
- 5 Statistical Analysis of the Quantization Noise
- 6 Crosscorrelations between Quantization Noise, Quantizer Input, and Quantizer Output
- 7 General Statistical Relations among the Quantization Noise, the Quantizer Input, and the Quantizer Output
- 8 Quantization of Two or More Variables: Statistical Analysis of the Quantizer Output
- 9 Quantization of Two or More Variables: Statistical Analysis of Quantization Noise
- 10 Quantization of Two or More Variables: General Statistical Relations among the Quantization Noises, and the Quantizer Inputs and Outputs
- 11 Calculation of Moments and Correlation Functions of Quantized Gaussian Variables
- Part III Floating–Point Quantization
- Part IV Quantization in Signal Processing, Feedback Control, and Computations
- Part V Applications of Quantization Noise Theory
- Part VI Quantization of System Parameters
- APPENDICES
- Bibliography
- Index
Summary
In the previous chapter, certain relations between the quantization noise and the quantizer input and output were established. These relations are in the form of moments, particularly crosscorrelations and covariances. These moments are extremely useful for the analysis of quantization and for the analysis of control and signal processing systems containing quantizers. The similarities between quantization and the addition of independent uniformly distributed noise when certain quantizing theorems are satisfied are very useful for analysis of quantization.
The properties of these moments are not totally descriptive of the behavior of the quantizer however, because the quantization noise is deterministically related to the quantizer input, and the moments do not show this. To gain a complete and deeper understanding, we dedicate this chapter to the derivation of joint PDFs and CFs of quantization noise and the quantizer input and output.
JOINT PDF AND CF OF THE QUANTIZER INPUT AND OUTPUT
In this section, we will derive the joint PDF and CF of x and x', i.e. fx,x' (x, x'), and Φx,x' (ux, ux'). The statistical relationship between x and x' is like a statistical version of an input–output “transfer function” for the quantizer.
The derivation will proceed in the following manner. Refer to Fig. 7.1. This figure compares quantization (the addition of actual quantization noise ν to x) with the PQN model (the addition of independent noise n to x).
- Type
- Chapter
- Information
- Quantization NoiseRoundoff Error in Digital Computation, Signal Processing, Control, and Communications, pp. 131 - 172Publisher: Cambridge University PressPrint publication year: 2008