{\printdate }
\contentsline {chapter}{\rm \em Preface}{IX}
\contentsline {chapter}{\rm \em Acknowledgments}{XI}
\contentsline {chapter}{\rm \em Glossary of Symbols}{XIII}
\contentsline {chapter}{\rm \em Acronyms and Abbreviations}{XVII}
\contentsline {part}{Part I\hspace {0.8em} Background}{}
\contentsline {chapter}{\numberline {1}\ignorespaces Introduction}{3}{chapter.1}
\contentsline {section}{\numberline {1.1}Definition of the Quantizer}{3}{section.1.1}
\contentsline {section}{\numberline {1.2}Sampling and Quantization (Analog-to-Digital Conversion)}{9}{section.1.2}
\contentsline {section}{\numberline {1.3}Exercises}{10}{section.1.3}
\contentsline {chapter}{\numberline {2}\ignorespaces Sampling Theory}{13}{chapter.2}
\contentsline {section}{\numberline {2.1}Linvill's Frequency Domain Description of Sampling}{14}{section.2.1}
\contentsline {section}{\numberline {2.2}The Sampling Theorem; Recovery of the Time Function from its Samples}{19}{section.2.2}
\contentsline {section}{\numberline {2.3}Anti-Alias Filtering}{22}{section.2.3}
\contentsline {section}{\numberline {2.4}A Statistical Description of Quantization, Based on Sampling\\ Theory}{24}{section.2.4}
\contentsline {section}{\numberline {2.5}Exercises}{27}{section.2.5}
\contentsline {chapter}{\numberline {3}Probability Density Functions, Characteristic Functions, Moments}{31}{chapter.3}
\contentsline {section}{\numberline {3.1}Probability Density Function}{31}{section.3.1}
\contentsline {section}{\numberline {3.2}Characteristic Function and Moments}{33}{section.3.2}
\contentsline {section}{\numberline {3.3}Joint Probability Density Functions}{35}{section.3.3}
\contentsline {section}{\numberline {3.4}Joint Characteristic Functions, Moments, and Correlation\\ Functions}{40}{section.3.4}
\contentsline {section}{\numberline {3.5}First-Order Statistical Description of the Effects of Memoryless Operations on Signals}{43}{section.3.5}
{\newpage }
\contentsline {section}{\numberline {3.6}Addition of Random Variables and Other Functions of Random Variables}{46}{section.3.6}
\contentsline {section}{\numberline {3.7}The Binomial Probability Density Function}{47}{section.3.7}
\contentsline {section}{\numberline {3.8}The Central Limit Theorem}{49}{section.3.8}
\contentsline {section}{\numberline {3.9}Exercises}{53}{section.3.9}
\contentsline {part}{Part II\hspace {0.8em} Uniform Quantization}{}
\contentsline {chapter}{\numberline {4}\ignorespaces Statistical Analysis of the Quantizer Output}{61}{chapter.4}
\contentsline {section}{\numberline {4.1}PDF and CF of the Quantizer Output}{61}{section.4.1}
\contentsline {section}{\numberline {4.2}Comparison of Quantization with the Addition of Independent Uniformly Distributed Noise, the PQN Model}{67}{section.4.2}
\contentsline {section}{\numberline {4.3}Quantizing Theorems I and II}{68}{section.4.3}
\contentsline {section}{\numberline {4.4}Recovery of the PDF of the Input Variable $x$ from the PDF of the Output Variable $x'$}{70}{section.4.4}
\contentsline {section}{\numberline {4.5}Recovery of Moments of the Input Variable $x$ from Moments of the Output Variable $x'$ when QT II is Satisfied; Sheppard's \\Corrections and the PQN Model}{79}{section.4.5}
\contentsline {section}{\numberline {4.6}General Expressions of the Moments of the Quantizer Output, and of the Errors of Sheppard's Corrections: Deviations from the PQN Model}{85}{section.4.6}
\contentsline {section}{\numberline {4.7}Sheppard's Corrections with a Gaussian Input}{85}{section.4.7}
\contentsline {section}{\numberline {4.8}Summary}{86}{section.4.8}
\contentsline {section}{\numberline {4.9}Exercises}{88}{section.4.9}
\contentsline {chapter}{\numberline {5}\ignorespaces Statistical Analysis of the Quantization Noise}{93}{chapter.5}
\contentsline {section}{\numberline {5.1}Analysis of the Quantization Noise and the PQN Model}{93}{section.5.1}
\contentsline {section}{\numberline {5.2}Satisfaction of Quantizing Theorems I and II}{99}{section.5.2}
\contentsline {section}{\numberline {5.3}Quantizing Theorem III/A}{99}{section.5.3}
\contentsline {section}{\numberline {5.4}General Expressions of the First- and Higher-Order Moments of the Quantization Noise: Deviations from the PQN Model}{102}{section.5.4}
\contentsline {section}{\numberline {5.5}Quantization Noise with Gaussian Inputs}{106}{section.5.5}
\contentsline {section}{\numberline {5.6}Summary}{107}{section.5.6}
\contentsline {section}{\numberline {5.7}Exercises}{108}{section.5.7}
\contentsline {chapter}{\numberline {6}Crosscorrelations between Quantization Noise, Quantizer Input, \\and Quantizer Output}{113}{chapter.6}
\contentsline {section}{\numberline {6.1}Crosscorrelations when Quantizing Theorem II is Satisfied}{113}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}Crosscorrelation between Quantization Noise and the \newline Quantizer Input}{113}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}Crosscorrelation between Quantization Noise and the \newline Quantizer Output}{115}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}Crosscorrelation between the Quantizer Input and the \newline Quantizer Output}{116}{subsection.6.1.3}
\contentsline {section}{\numberline {6.2}General Expressions of Crosscorrelations}{116}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}Crosscorrelation between Quantization Noise and the \newline Quantizer Input}{116}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}Crosscorrelation between Quantization Noise and the \newline Quantizer Output Signal}{119}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}Crosscorrelation between the Quantizer Input and Output Signals}{122}{subsection.6.2.3}
\contentsline {section}{\numberline {6.3}Correlation and Covariance between Gaussian Quantizer Input and Its Quantization Noise}{123}{section.6.3}
\contentsline {section}{\numberline {6.4}Conditions of Orthogonality of Input $x$ and Noise $\nu $: Quantizing Theorem III/B}{126}{section.6.4}
\contentsline {section}{\numberline {6.5}Conditions of Uncorrelatedness between $x$ and $\nu $: Quantizing \\Theorem IV/B}{127}{section.6.5}
\contentsline {section}{\numberline {6.6}Summary}{128}{section.6.6}
\contentsline {section}{\numberline {6.7}Exercises}{129}{section.6.7}
\contentsline {chapter}{\numberline {7}General Statistical Relations among the Quantization Noise, the \newline Quantizer Input, and the Quantizer Output}{131}{chapter.7}
\contentsline {section}{\numberline {7.1}Joint PDF and CF of the Quantizer Input and Output}{131}{section.7.1}
\contentsline {section}{\numberline {7.2}Quantizing Theorems for the Joint CF of the Quantizer Input and Output}{138}{section.7.2}
\contentsline {section}{\numberline {7.3}Joint PDF and CF of the Quantizer Input and the Quantization Noise: Application of the PQN Model}{140}{section.7.3}
\contentsline {section}{\numberline {7.4}Quantizing Theorems for the Joint CF of the Quantizer Input and the Quantization Noise: Application of the PQN Model}{146}{section.7.4}
\contentsline {section}{\numberline {7.5}Joint Moments of the Quantizer Input and the Quantization Noise: Quantizing Theorem\nobreakspace {}III}{149}{section.7.5}
\contentsline {subsection}{\numberline {7.5.1}General Expressions of Joint Moments when Quantizing Theorem III is not satisfied}{151}{subsection.7.5.1}
\contentsline {section}{\numberline {7.6}Joint Moments of the Centralized Quantizer Input and the \newline Quantization Noise: Quantizing Theorem\nobreakspace {}IV}{152}{section.7.6}
\contentsline {subsection}{\numberline {7.6.1}General Expressions of Joint Moments}{153}{subsection.7.6.1}
\contentsline {section}{\numberline {7.7}Joint PDF and CF of the Quantization Noise and the Quantizer Output}{154}{section.7.7}
\contentsline {section}{\numberline {7.8}Three-Dimensional Probability Density Function and \\ Characteristic Function}{158}{section.7.8}
\contentsline {subsection}{\numberline {7.8.1}Three-Dimensional Probability Density Function}{158}{subsection.7.8.1}
\contentsline {subsection}{\numberline {7.8.2}Three-Dimensional Characteristic Function}{159}{subsection.7.8.2}
\contentsline {section}{\numberline {7.9}General Relationship between Quantization and the PQN Model}{160}{section.7.9}
\contentsline {section}{\numberline {7.10}Overview of the Quantizing Theorems}{162}{section.7.10}
\contentsline {section}{\numberline {7.11}Examples of Probability Density Functions Satisfying Quantizing Theorems III/B or QT IV/B}{165}{section.7.11}
\contentsline {section}{\numberline {7.12}Summary}{170}{section.7.12}
\contentsline {section}{\numberline {7.13}Exercises}{171}{section.7.13}
\contentsline {chapter}{\numberline {8}Quantization of Two or More Variables: Statistical Analysis of \\the Quantizer Output}{173}{chapter.8}
\contentsline {section}{\numberline {8.1}Two-Dimensional Sampling Theory}{174}{section.8.1}
\contentsline {section}{\numberline {8.2}Statistical Analysis of the Quantizer Output for Two-Variable \\Quantization}{179}{section.8.2}
\contentsline {section}{\numberline {8.3}A Comparison of Multivariable Quantization with the Addition of Independent Quantization Noise (PQN)}{184}{section.8.3}
\contentsline {section}{\numberline {8.4}Quantizing Theorem I for Two and More Variables}{186}{section.8.4}
\contentsline {section}{\numberline {8.5}Quantizing Theorem II for Two and More Variables}{187}{section.8.5}
\contentsline {section}{\numberline {8.6}Recovery of the Joint PDF of the Inputs $x_1, x_2$ from the Joint PDF of the Outputs $x_1^{\prime }, x_2^{\prime }$}{187}{section.8.6}
\contentsline {section}{\numberline {8.7}Recovery of the Joint Moments of the Inputs $x_1,x_2$ from the Joint \unhbox \voidb@x \hbox {Moments} of the Outputs $x'_1,x'_2$: Sheppard's Corrections}{190}{section.8.7}
\contentsline {section}{\numberline {8.8}Summary}{192}{section.8.8}
\contentsline {section}{\numberline {8.9}Exercises}{193}{section.8.9}
\contentsline {chapter}{\numberline {9}Quantization of Two or More Variables: Statistical Analysis of \\Quantization Noise}{197}{chapter.9}
\contentsline {section}{\numberline {9.1}Analysis of Quantization Noise, Validity of the PQN Model}{197}{section.9.1}
\contentsline {section}{\numberline {9.2}Joint Moments of the Quantization Noise}{200}{section.9.2}
\contentsline {section}{\numberline {9.3}Satisfaction of Quantizing Theorems I and II}{203}{section.9.3}
\contentsline {section}{\numberline {9.4}Quantizing Theorem III/A for $N$ Variables}{204}{section.9.4}
\contentsline {section}{\numberline {9.5}Quantization Noise with Multiple Gaussian Inputs}{206}{section.9.5}
\contentsline {section}{\numberline {9.6}Summary}{207}{section.9.6}
\contentsline {section}{\numberline {9.7}Exercises}{207}{section.9.7}
\contentsline {chapter}{\numberline {10}Quantization of Two or More Variables: General Statistical \\Relations between the Quantization Noises, and the Quantizer \\Inputs and Outputs}{209}{chapter.10}
\contentsline {section}{\numberline {10.1}Joint PDF and CF of the Quantizer Inputs and Outputs}{209}{section.10.1}
\contentsline {section}{\numberline {10.2}Joint PDF and CF of the Quantizer Inputs and the Quantization Noises}{210}{section.10.2}
\contentsline {section}{\numberline {10.3}Joint PDF, CF, and Moments of the Quantizer Inputs and Noises when Quantizing Theorem\nobreakspace {}I or II is Satisfied}{211}{section.10.3}
\contentsline {section}{\numberline {10.4}General Expressions for the Covariances between Quantizer \\Inputs and Noises}{213}{section.10.4}
\contentsline {section}{\numberline {10.5}Joint PDF, CF, and Moments of the Quantizer Inputs and Noises when Quantizing Theorem IV/B is Satisfied}{214}{section.10.5}
\contentsline {section}{\numberline {10.6}Joint Moments of Quantizer Inputs and Noises with Quantizing Theorem III Satisfied}{216}{section.10.6}
\contentsline {section}{\numberline {10.7}Joint Moments of the Quantizer Inputs and Noises with \\Quantizing Theorem IV Satisfied}{217}{section.10.7}
\contentsline {section}{\numberline {10.8}Some Thoughts about the Quantizing Theorems}{217}{section.10.8}
\contentsline {section}{\numberline {10.9}Joint PDF and CF of Quantization Noises and Quantizer Outputs under General Conditions}{218}{section.10.9}
\contentsline {section}{\numberline {10.10}Joint PDF and CF of Quantizer Inputs, Quantization Noises, and Quantizer Outputs}{219}{section.10.10}
\contentsline {section}{\numberline {10.11}Summary}{221}{section.10.11}
\contentsline {section}{\numberline {10.12}Exercises}{222}{section.10.12}
\contentsline {chapter}{\numberline {11}Calculation of the Moments and Correlation Functions of Quantized Gaussian Variables}{223}{chapter.11}
\contentsline {section}{\numberline {11.1}The Moments of the Quantizer Output}{223}{section.11.1}
\contentsline {section}{\numberline {11.2}Moments of the Quantization Noise, Validity of the PQN Model}{231}{section.11.2}
\contentsline {section}{\numberline {11.3}Covariance of the Input $x$ and Noise $\nu $}{235}{section.11.3}
\contentsline {section}{\numberline {11.4}Joint Moments of Centralized Input $\mathaccent "0251\relax x$ and Noise $\nu $}{238}{section.11.4}
\contentsline {section}{\numberline {11.5}Quantization of Two Gaussian Variables}{240}{section.11.5}
\contentsline {section}{\numberline {11.6}Quantization of Samples of a Gaussian Time Series}{247}{section.11.6}
\contentsline {section}{\numberline {11.7}Summary}{250}{section.11.7}
\contentsline {section}{\numberline {11.8}Exercises}{251}{section.11.8}
\contentsline {part}{Part III\hspace {0.8em} Floating-Point Quantization}{}
\contentsline {chapter}{\numberline {12}\ignorespaces Basics of Floating-Point Quantization}{255}{chapter.12}
\contentsline {section}{\numberline {12.1}The Floating-Point Quantizer}{255}{section.12.1}
\contentsline {section}{\numberline {12.2}Floating-Point Quantization Noise}{258}{section.12.2}
\contentsline {section}{\numberline {12.3}An Exact Model of the Floating-Point Quantizer}{259}{section.12.3}
\contentsline {section}{\numberline {12.4}How Good is the PQN Model for the Hidden Quantizer?}{264}{section.12.4}
\contentsline {section}{\numberline {12.5}Analysis of Floating-Point Quantization Noise}{270}{section.12.5}
\contentsline {section}{\numberline {12.6}How Good is the PQN Model for the Exponent Quantizer?}{278}{section.12.6}
\contentsline {subsection}{\numberline {12.6.1}Gaussian Input}{278}{subsection.12.6.1}
\contentsline {subsection}{\numberline {12.6.2}Input with Triangular Distribution}{283}{subsection.12.6.2}
\contentsline {subsection}{\numberline {12.6.3}Input with Uniform Distribution}{284}{subsection.12.6.3}
\contentsline {subsection}{\numberline {12.6.4}Sinusoidal Input}{288}{subsection.12.6.4}
\contentsline {section}{\numberline {12.7}A Floating-Point PQN Model}{300}{section.12.7}
\contentsline {section}{\numberline {12.8}Summary}{301}{section.12.8}
\contentsline {section}{\numberline {12.9}Exercises}{302}{section.12.9}
\contentsline {chapter}{\numberline {13}\ignorespaces More on Floating-Point Quantization}{305}{chapter.13}
\contentsline {section}{\numberline {13.1}Small Deviations from the Floating-Point PQN Model}{305}{section.13.1}
\contentsline {section}{\numberline {13.2}Quantization of Small Input Signals with High Bias}{309}{section.13.2}
\contentsline {section}{\numberline {13.3}Floating-Point Quantization of Two or More Variables}{311}{section.13.3}
\contentsline {subsection}{\numberline {13.3.1}Relationship between Correlation Coefficients $\rho _{\nu _1,\nu _2}$ and $\rho _{\nu _{{\scriptscriptstyle \rm FL}_1},\nu _{{\scriptscriptstyle \rm FL}_2}}$ for Floating-Point Quantization}{322}{subsection.13.3.1}
\contentsline {section}{\numberline {13.4}A Simplified Model of the Floating-Point Quantizer}{323}{section.13.4}
\contentsline {section}{\numberline {13.5}A Comparison of Exact and Simplified Models of the Floating-Point Quantizer}{329}{section.13.5}
\contentsline {section}{\numberline {13.6}Digital Communication with Signal Compression and Expansion: ``$\mu $-law'' and ``$A$-law''}{330}{section.13.6}
\contentsline {section}{\numberline {13.7}Testing for PQN}{331}{section.13.7}
\contentsline {section}{\numberline {13.8}Practical Number Systems: The IEEE Standard}{341}{section.13.8}
\contentsline {subsection}{\numberline {13.8.1}Representation of Very Small Numbers}{341}{subsection.13.8.1}
\contentsline {subsection}{\numberline {13.8.2}Binary Point}{342}{subsection.13.8.2}
\contentsline {subsection}{\numberline {13.8.3}Underflow, Overflow, Dynamic Range, and SNR}{342}{subsection.13.8.3}
\contentsline {subsection}{\numberline {13.8.4}The IEEE Standard}{344}{subsection.13.8.4}
\contentsline {section}{\numberline {13.9}Summary}{346}{section.13.9}
\contentsline {section}{\numberline {13.10}Exercises}{349}{section.13.10}
\contentsline {chapter}{\numberline {14}Cascades of Fixed-Point and Floating-Point Quantizers}{353}{chapter.14}
\contentsline {section}{\numberline {14.1}A Floating-Point Compact Disc}{353}{section.14.1}
\contentsline {section}{\numberline {14.2}A Cascade of Fixed-Point and Floating-Point Quantizers}{354}{section.14.2}
\contentsline {section}{\numberline {14.3}More on the Cascade of Fixed-Point and Floating-Point Quantizers}{358}{section.14.3}
\contentsline {section}{\numberline {14.4}Connecting an Analog-to-Digital Converter to a Floating-Point Computer: Another Cascade of Fixed- and Floating-Point \\Quantization}{365}{section.14.4}
\contentsline {section}{\numberline {14.5}Connecting the Output of a Floating-Point Computer to a Digital-to-Analog Converter: a Cascade of Floating-Point and Fixed-Point Quantization}{366}{section.14.5}
\contentsline {section}{\numberline {14.6}Summary}{367}{section.14.6}
\contentsline {section}{\numberline {14.7}Exercises}{367}{section.14.7}
\contentsline {part}{Part IV\hspace {0.8em} Quantization in Signal Processing, Feedback \\Control, and Computations}{}
\contentsline {chapter}{\numberline {15}\ignorespaces Roundoff Noise in FIR Digital Filters and in FFT Calculations}{371}{chapter.15}
\contentsline {section}{\numberline {15.1}The FIR Digital Filter}{371}{section.15.1}
\contentsline {section}{\numberline {15.2}Calculation of the Output Signal of an FIR Filter}{372}{section.15.2}
\contentsline {section}{\numberline {15.3}PQN Analysis of Roundoff Noise at the Output of an FIR Filter}{374}{section.15.3}
\contentsline {section}{\numberline {15.4}Roundoff Noise with Fixed-Point Quantization}{375}{section.15.4}
\contentsline {section}{\numberline {15.5}Roundoff Noise with Floating-Point Quantization}{379}{section.15.5}
\contentsline {section}{\numberline {15.6}Roundoff Noise in DFT and FFT Calculations}{381}{section.15.6}
\contentsline {subsection}{\numberline {15.6.1}Multiplication of Complex Numbers}{383}{subsection.15.6.1}
\contentsline {subsection}{\numberline {15.6.2}Number Representations in Digital Signal Processing \\Algorithms, and Roundoff}{384}{subsection.15.6.2}
\contentsline {subsection}{\numberline {15.6.3}Growing of the Maximum Value in a Sequence Resulting from the DFT}{385}{subsection.15.6.3}
\contentsline {section}{\numberline {15.7}A Fixed-Point FFT Error Analysis}{386}{section.15.7}
\contentsline {subsection}{\numberline {15.7.1}Quantization Noise with Direct Calculation of the DFT}{386}{subsection.15.7.1}
\contentsline {subsection}{\numberline {15.7.2}Sources of Quantization Noise in the FFT}{387}{subsection.15.7.2}
\contentsline {subsection}{\numberline {15.7.3}FFT with Fixed-Point Number Representation}{390}{subsection.15.7.3}
\contentsline {section}{\numberline {15.8}Some Noise Analysis Results for Block Floating-Point and \\Floating-Point FFT}{392}{section.15.8}
\contentsline {subsection}{\numberline {15.8.1}FFT with Block Floating-Point Number Representation}{392}{subsection.15.8.1}
\contentsline {subsection}{\numberline {15.8.2}FFT with Floating-Point Number Representation}{392}{subsection.15.8.2}
\contentsline {section}{\numberline {15.9}Summary}{395}{section.15.9}
\contentsline {section}{\numberline {15.10}Exercises}{395}{section.15.10}
\contentsline {chapter}{\numberline {16}\ignorespaces Roundoff Noise in IIR Digital Filters}{401}{chapter.16}
\contentsline {section}{\numberline {16.1}A One-Pole Digital Filter}{401}{section.16.1}
\contentsline {section}{\numberline {16.2}Quantization in a One-Pole Digital Filter}{402}{section.16.2}
\contentsline {section}{\numberline {16.3}PQN Modeling and Moments with FIR and IIR Systems}{404}{section.16.3}
\contentsline {section}{\numberline {16.4}Roundoff in a One-Pole Digital Filter with Fixed-Point \\Computation}{405}{section.16.4}
\contentsline {section}{\numberline {16.5}Roundoff in a One-Pole Digital Filter with Floating-Point \\Computation}{412}{section.16.5}
\contentsline {section}{\numberline {16.6}Simulation of Floating-point IIR Digital Filters}{414}{section.16.6}
\contentsline {section}{\numberline {16.7}Strange Cases: Exceptions to PQN Behavior in Digital Filters with Floating-Point Computation}{416}{section.16.7}
\contentsline {section}{\numberline {16.8}Testing the PQN Model for Quantization Within Feedback Loops}{417}{section.16.8}
\contentsline {section}{\numberline {16.9}Summary}{423}{section.16.9}
\contentsline {section}{\numberline {16.10}Exercises}{425}{section.16.10}
\contentsline {chapter}{\numberline {17}\ignorespaces Roundoff Noise in Digital Feedback Control Systems}{429}{chapter.17}
\contentsline {section}{\numberline {17.1}The Analog-to-Digital Converter}{430}{section.17.1}
\contentsline {section}{\numberline {17.2}The Digital-to-Analog Converter}{430}{section.17.2}
\contentsline {section}{\numberline {17.3}A Control System Example}{432}{section.17.3}
\contentsline {section}{\numberline {17.4}Signal Scaling Within the Feedback Loop}{440}{section.17.4}
\contentsline {section}{\numberline {17.5}Mean Square of the Total Quantization Noise at the Plant Output}{445}{section.17.5}
\contentsline {section}{\numberline {17.6}Satisfaction of QT\nobreakspace {}II at the Quantizer Inputs}{447}{section.17.6}
\contentsline {section}{\numberline {17.7}The Bertram Bound}{453}{section.17.7}
\contentsline {section}{\numberline {17.8}Summary}{458}{section.17.8}
\contentsline {section}{\numberline {17.9}Exercises}{459}{section.17.9}
\contentsline {chapter}{\numberline {18}Roundoff Errors in Nonlinear Dynamic Systems -- A Chaotic Example}{463}{chapter.18}
\contentsline {section}{\numberline {18.1}Roundoff Noise}{463}{section.18.1}
\contentsline {section}{\numberline {18.2}Experiments with a Linear System}{465}{section.18.2}
\contentsline {section}{\numberline {18.3}Experiments with a Chaotic System}{468}{section.18.3}
\contentsline {subsection}{\numberline {18.3.1}Study of the Logistic Map}{468}{subsection.18.3.1}
\contentsline {subsection}{\numberline {18.3.2}Logistic Map with External Driving Function}{476}{subsection.18.3.2}
\contentsline {section}{\numberline {18.4}Summary}{479}{section.18.4}
\contentsline {section}{\numberline {18.5}Exercises}{479}{section.18.5}
\contentsline {part}{Part V\hspace {0.8em} Applications of Quantization Noise Theory}{}
\contentsline {chapter}{\numberline {19}\ignorespaces Dither}{483}{chapter.19}
\contentsline {section}{\numberline {19.1}Dither: Anti-alias Filtering of the Quantizer Input CF}{483}{section.19.1}
\contentsline {section}{\numberline {19.2}Moment Relations when QT II is Satisfied}{486}{section.19.2}
\contentsline {section}{\numberline {19.3}Conditions for Statistical Independence of \lowercase {$x$} and $\nu $, and \lowercase {$d$} and $\nu $}{487}{section.19.3}
\contentsline {section}{\numberline {19.4}Moment Relations and Quantization Noise PDF when QT III or QT IV is Satisfied}{490}{section.19.4}
\contentsline {section}{\numberline {19.5}Statistical Analysis of the Total Quantization Error $\xi =\lowercase {d}+\nu $}{491}{section.19.5}
\contentsline {section}{\numberline {19.6}Important Dither Types}{495}{section.19.6}
\contentsline {subsection}{\numberline {19.6.1}Uniform Dither}{495}{subsection.19.6.1}
\contentsline {subsection}{\numberline {19.6.2}Triangular Dither}{498}{subsection.19.6.2}
\contentsline {subsection}{\numberline {19.6.3}Triangular plus Uniform Dither}{499}{subsection.19.6.3}
\contentsline {subsection}{\numberline {19.6.4}Triangular plus Triangular Dither}{500}{subsection.19.6.4}
\contentsline {subsection}{\numberline {19.6.5}Gaussian Dither}{500}{subsection.19.6.5}
\contentsline {subsection}{\numberline {19.6.6}Sinusoidal Dither}{501}{subsection.19.6.6}
\contentsline {subsection}{\numberline {19.6.7}The Use of Dither in the Arithmetic Processor}{501}{subsection.19.6.7}
\contentsline {section}{\numberline {19.7}The Use of Dither for Quantization of Two or More Variables}{502}{section.19.7}
\contentsline {section}{\numberline {19.8}Subtractive Dither}{504}{section.19.8}
\contentsline {subsection}{\numberline {19.8.1}Analog-to-Digital Conversion with Subtractive Dither}{506}{subsection.19.8.1}
\contentsline {section}{\numberline {19.9}Dither with Floating-Point}{510}{section.19.9}
\contentsline {subsection}{\numberline {19.9.1}Dither with Floating-Point Analog-to-Digital Conversion}{510}{subsection.19.9.1}
\contentsline {subsection}{\numberline {19.9.2}Floating-Point Quantization with Subtractive Dither}{513}{subsection.19.9.2}
\contentsline {subsection}{\numberline {19.9.3}Dithered Roundoff with Floating-Point Computation}{514}{subsection.19.9.3}
\contentsline {section}{\numberline {19.10}The Use of Dither in Nonlinear Control Systems}{518}{section.19.10}
\contentsline {section}{\numberline {19.11}Summary}{518}{section.19.11}
\contentsline {section}{\numberline {19.12}Exercises}{520}{section.19.12}
\contentsline {chapter}{\numberline {20}\ignorespaces Spectrum of Quantization Noise and Conditions of Whiteness}{527}{chapter.20}
\contentsline {section}{\numberline {20.1}Quantization of Gaussian and Sine-Wave Signals}{528}{section.20.1}
\contentsline {section}{\numberline {20.2}Calculation of Continuous-Time Correlation Functions and Spectra}{530}{section.20.2}
\contentsline {subsection}{\numberline {20.2.1}General Considerations}{530}{subsection.20.2.1}
\contentsline {subsection}{\numberline {20.2.2}Direct Numerical Evaluation of the Expectations}{533}{subsection.20.2.2}
\contentsline {subsection}{\numberline {20.2.3}Approximation Methods}{534}{subsection.20.2.3}
\contentsline {subsubsection}{Approximation Based on the Characteristic Function}{535}{section*.6}
\contentsline {subsubsection}{Approximation Based on the Modulation Principle}{536}{section*.7}
\contentsline {subsection}{\numberline {20.2.4}Correlation Function and Spectrum of Quantized Gaussian Signals}{536}{subsection.20.2.4}
\contentsline {subsubsection}{Bennett's Direct Analysis}{536}{section*.8}
\contentsline {subsubsection}{Application of the Characteristic Function Method}{539}{section*.9}
\contentsline {subsubsection}{Analysis of the Autocorrelation Function of the Noise}{539}{section*.10}
\contentsline {subsubsection}{Series Representation of the Joint Normal PDF}{540}{section*.11}
\contentsline {subsubsection}{Use of the Modulation Principle}{541}{section*.12}
\contentsline {subsection}{\numberline {20.2.5}Spectrum of the Quantization Noise of a Quantized Sine Wave}{542}{subsection.20.2.5}
\contentsline {section}{\numberline {20.3}Conditions of Whiteness for the Sampled Quantization Noise}{546}{section.20.3}
\contentsline {subsection}{\numberline {20.3.1}Bandlimited Gaussian Noise}{548}{subsection.20.3.1}
\contentsline {subsubsection}{A Condition Based on Approximate Correlation}{548}{section*.13}
\contentsline {subsubsection}{A Condition Based on the Bandwidth of the Spectrum}{550}{section*.14}
\contentsline {subsubsection}{A Condition Based on Computer Evaluation of Spectra}{551}{section*.15}
\contentsline {subsection}{\numberline {20.3.2}Sine Wave}{552}{subsection.20.3.2}
\contentsline {subsection}{\numberline {20.3.3}A Uniform Condition for White Noise Spectrum}{554}{subsection.20.3.3}
\contentsline {section}{\numberline {20.4}Summary}{558}{section.20.4}
\contentsline {section}{\numberline {20.5}Exercises}{560}{section.20.5}
\contentsline {part}{Part VI\hspace {0.8em} Quantization of System Parameters}{}
\contentsline {chapter}{\numberline {21}\ignorespaces Coefficient Quantization}{563}{chapter.21}
\contentsline {section}{\numberline {21.1}Coefficient Quantization in Linear Digital Filters}{564}{section.21.1}
\contentsline {section}{\numberline {21.2}An Example of Coefficient Quantization}{567}{section.21.2}
\contentsline {section}{\numberline {21.3}Floating-Point Coefficient Quantization}{570}{section.21.3}
\contentsline {section}{\numberline {21.4}Analysis of Coefficient Quantization Effects by Computer \\Simulation}{572}{section.21.4}
\contentsline {section}{\numberline {21.5}Coefficient Quantization in Nonlinear Systems}{574}{section.21.5}
\contentsline {section}{\numberline {21.6}Summary}{576}{section.21.6}
\contentsline {section}{\numberline {21.7}Exercises}{577}{section.21.7}
\contentsline {part}{APPENDICES}{ }{Item.556}
\contentsline {chapter}{\numberline {A}\ignorespaces Perfectly Bandlimited Characteristic Functions}{587}{appendix.A}
\contentsline {section}{\numberline {A.1}Examples of Bandlimited Characteristic Functions}{587}{section.A.1}
\contentsline {section}{\numberline {A.2}A Bandlimited Characteristic Function Cannot Be Analytic}{592}{section.A.2}
\contentsline {subsection}{\numberline {A.2.1}Characteristic Functions that Satisfy QT\nobreakspace {}I or QT\nobreakspace {}II}{593}{subsection.A.2.1}
\contentsline {subsection}{\numberline {A.2.2}Impossibility of Reconstruction of the Input PDF when QT\nobreakspace {}II is Satisfied but QT\nobreakspace {}I is not}{593}{subsection.A.2.2}
\contentsline {chapter}{\numberline {B}General Expressions of the Moments of the Quantizer Output, \\and of the Errors of Sheppard's Corrections}{595}{appendix.B}
\contentsline {section}{\numberline {B.1}General Expressions of the Moments of the Quantizer Output}{595}{section.B.1}
\contentsline {section}{\numberline {B.2}General Expressions of the Errors of Sheppard's Corrections}{600}{section.B.2}
\contentsline {section}{\numberline {B.3}General Expressions for the Quantizer Output Joint Moments}{605}{section.B.3}
\contentsline {chapter}{\numberline {C}\ignorespaces Derivatives of the Sinc Function}{611}{appendix.C}
{\newpage }
\contentsline {chapter}{\numberline {D}\ignorespaces Proofs of Quantizing Theorems III and IV}{615}{appendix.D}
\contentsline {section}{\numberline {D.1}Proof of QT III}{615}{section.D.1}
\contentsline {section}{\numberline {D.2}Proof of QT IV}{616}{section.D.2}
\contentsline {chapter}{\numberline {E}\ignorespaces Limits of Applicability of the Theory -- Caveat Reader}{619}{appendix.E}
\contentsline {section}{\numberline {E.1}Long-time vs.\ Short-time Properties of Quantization}{619}{section.E.1}
\contentsline {subsection}{\numberline {E.1.1}Mathematical Analysis}{622}{subsection.E.1.1}
\contentsline {section}{\numberline {E.2}Saturation effects}{624}{section.E.2}
\contentsline {section}{\numberline {E.3}Analog-to-Digital Conversion: Non-ideal Realization of Uniform Quantization}{626}{section.E.3}
\contentsline {chapter}{\numberline {F}\ignorespaces Some Properties of the Gaussian PDF and CF}{631}{appendix.F}
\contentsline {section}{\numberline {F.1}Approximate Expressions for the Gaussian Characteristic Function}{632}{section.F.1}
\contentsline {section}{\numberline {F.2}Derivatives of the CF with ${\rm E}\{{\lowercase {x}}\}\not =0$}{633}{section.F.2}
\contentsline {section}{\numberline {F.3}Two-Dimensional CF}{634}{section.F.3}
\contentsline {chapter}{\numberline {G}\ignorespaces Quantization of a Sinusoidal Input}{635}{appendix.G}
\contentsline {section}{\numberline {G.1}Study of the Residual Error of Sheppard's First Correction}{636}{section.G.1}
\contentsline {section}{\numberline {G.2}Approximate Upper Bounds for the Residual Errors of Higher Moments}{638}{section.G.2}
\contentsline {subsection}{\numberline {G.2.1}Examples}{640}{subsection.G.2.1}
\contentsline {section}{\numberline {G.3}Correlation between Quantizer Input and Quantization Noise}{641}{section.G.3}
\contentsline {section}{\numberline {G.4}Time Series Analysis of a Sine Wave}{643}{section.G.4}
\contentsline {section}{\numberline {G.5}Exact Finite-sum Expressions for Moments of the Quantization Noise}{646}{section.G.5}
\contentsline {section}{\numberline {G.6}Joint PDF and CF of Two Quantized Samples of a Sine Wave}{651}{section.G.6}
\contentsline {subsection}{\numberline {G.6.1}The Signal Model}{651}{subsection.G.6.1}
\contentsline {subsection}{\numberline {G.6.2}Derivation of the Joint PDF}{652}{subsection.G.6.2}
\contentsline {subsection}{\numberline {G.6.3}Derivation of the Joint CF}{655}{subsection.G.6.3}
\contentsline {section}{\numberline {G.7}Some Properties of the Bessel Functions of the First Kind}{658}{section.G.7}
\contentsline {subsection}{\numberline {G.7.1}Derivatives}{658}{subsection.G.7.1}
\contentsline {subsection}{\numberline {G.7.2}Approximations and Limits}{659}{subsection.G.7.2}
\contentsline {chapter}{\numberline {H}Application of the Methods of Appendix\nobreakspace {}\ref {ch:quantization of sinusoidal variables} to Distributions other than Sinusoidal}{661}{appendix.H}
\contentsline {chapter}{\numberline {I}\ignorespaces A Few Properties of Selected Distributions}{665}{appendix.I}
\contentsline {section}{\numberline {I.1}Chi-Square Distribution}{665}{section.I.1}
\contentsline {section}{\numberline {I.2}Exponential Distribution}{668}{section.I.2}
\contentsline {section}{\numberline {I.3}Gamma Distribution}{670}{section.I.3}
\contentsline {section}{\numberline {I.4}Laplacian Distribution}{672}{section.I.4}
\contentsline {section}{\numberline {I.5}Rayleigh Distribution}{674}{section.I.5}
\contentsline {section}{\numberline {I.6}Sinusoidal Distribution}{675}{section.I.6}
\contentsline {section}{\numberline {I.7}Uniform Distribution}{677}{section.I.7}
\contentsline {section}{\numberline {I.8}Triangular Distribution}{678}{section.I.8}
\contentsline {section}{\numberline {I.9}``House'' Distribution}{680}{section.I.9}
\contentsline {chapter}{\numberline {J}\ignorespaces Digital Dither}{683}{appendix.J}
\contentsline {section}{\numberline {J.1}Quantization of Representable Samples}{684}{section.J.1}
\contentsline {subsection}{\numberline {J.1.1}Dirac Delta Functions at $q/2+kq$}{686}{subsection.J.1.1}
\contentsline {section}{\numberline {J.2}Digital Dither with Approximately Normal Distribution}{687}{section.J.2}
\contentsline {section}{\numberline {J.3}Generation of Digital Dither}{687}{section.J.3}
\contentsline {subsection}{\numberline {J.3.1}Uniformly Distributed Digital Dither}{688}{subsection.J.3.1}
\contentsline {subsection}{\numberline {J.3.2}Triangularly Distributed Digital Dither}{691}{subsection.J.3.2}
\contentsline {chapter}{\numberline {K}\ignorespaces Roundoff Noise in Scientific Computations}{695}{appendix.K}
\contentsline {section}{\numberline {K.1}Comparison to Reference Values}{695}{section.K.1}
\contentsline {subsection}{\numberline {K.1.1}Comparison to Manually Calculable Results}{695}{subsection.K.1.1}
\contentsline {subsection}{\numberline {K.1.2}Increased Precision}{696}{subsection.K.1.2}
\contentsline {subsection}{\numberline {K.1.3}Ambiguities of IEEE Double-Precision Calculations}{696}{subsection.K.1.3}
\contentsline {subsection}{\numberline {K.1.4}Decreased-Precision Calculations}{698}{subsection.K.1.4}
\contentsline {subsection}{\numberline {K.1.5}Different Ways of Computation}{698}{subsection.K.1.5}
\contentsline {subsection}{\numberline {K.1.6}The Use of the Inverse of the Algorithm}{700}{subsection.K.1.6}
\contentsline {section}{\numberline {K.2}The Condition Number}{701}{section.K.2}
\contentsline {section}{\numberline {K.3}Upper Limits of Errors}{703}{section.K.3}
\contentsline {section}{\numberline {K.4}The Effect of Nonlinearities}{705}{section.K.4}
\contentsline {chapter}{\numberline {L}Simulating Arbitrary-Precision Fixed-Point and Floating-Point \\Roundoff in Matlab}{709}{appendix.L}
\contentsline {section}{\numberline {L.1}Straightforward Programming}{710}{section.L.1}
\contentsline {subsection}{\numberline {L.1.1}Fixed-point roundoff}{710}{subsection.L.1.1}
\contentsline {subsection}{\numberline {L.1.2}Floating-Point Roundoff}{710}{subsection.L.1.2}
\contentsline {section}{\numberline {L.2}The Use of More Advanced Quantizers}{711}{section.L.2}
\contentsline {section}{\numberline {L.3}Quantized DSP Simulation Toolbox (QDSP)}{714}{section.L.3}
\contentsline {section}{\numberline {L.4}Fixed-Point Toolbox}{716}{section.L.4}
\contentsline {chapter}{\numberline {M}\ignorespaces The First Paper on Sampling-Related Quantization Theory}{719}{appendix.M}
\contentsline {chapter}{\rm \em Bibliography}{{\rm \em 732}}
\contentsline {chapter}{\rm {\em Index}}{{\rm \em 742}}