Over the past century, advancements in computer science have consistently resulted from extensive mathematical work. Even today, innovations in the digital domain continue to be grounded in a strong mathematical foundation. To succeed in this profession, both today's students and tomorrow’s computer engineers need a solid mathematical background. The goal of this book series is to offer a solid foundation of the knowledge essential to working in the digital sector. Across three volumes, it explores fundamental principles, digital information, data analysis, and optimization. Whether the reader is pursuing initial training or looking to deepen their expertise, the Mathematics for Digital Science series revisits familiar concepts, helping them refresh and expand their knowledge while also introducing equally essential, newer topics.

Table of Contents:
Preface xi Chapter 1 Representation of Numbers 1 1.1 Representation of numbers 2 1.1.1 Position numbering 2 1.1.2 The binary system 3 1.1.3 Octal system and hexadecimal system 6 1.2 Representation of numbers in a machine 8 1.2.1 Machine representation of negative numbers 8 1.2.2 Representation of rational numbers 12 Chapter 2 Media Representation 17 2.1 Character coding 17 2.2 Image coding 23 2.2.1 Representation of digital colors 23 2.2.2 Scanning an image 24 2.2.3 Image quality 24 2.3 Sound coding 26 2.4 Video coding 28 2.5 Tagging codes 29 2.5.1 Figures 29 2.5.2 Bank cards 30 2.5.3 Barcodes 32 2.5.4 QR codes 34 Chapter 3 Signals and Systems 47 3.1 Characteristics and varieties 47 3.1.1 Introduction 47 3.1.2 Periodicity 49 3.1.3 Noise 50 3.2 Fourier analysis 51 3.2.1 Fourier series expansion 51 3.2.2 Examples 52 3.2.3 Special cases 55 3.2.4 Other development writing 55 3.2.5 Power 56 3.3 Dirac distribution 58 3.4 Convolution 62 3.4.1 Definition 62 3.4.2 Dirac distribution and convolution 64 Chapter 4 z-transforms Fourier Transforms and Laplace Transforms 67 4.1 z-transform 68 4.1.1 Definitions and main results 68 4.1.2 Application to discrete systems 70 4.2 Fourier transform 73 4.2.1 Periodic signals 73 4.2.2 Non-periodic signals 74 4.2.3 Main properties of Fourier transforms 79 4.2.4 Application to analog signals and systems 82 4.2.5 Transfer function 86 4.2.6 Autocorrelation and intercorrelation of signals 88 4.3 Discrete Fourier transform and fast Fourier transform 91 4.3.1 Discrete Fourier transform 91 4.3.2 Fast Fourier transform (FFT) 95 4.4 Laplace transform 101 4.4.1 Definition 101 4.4.2 Properties 102 4.4.3 Differential equation 103 4.4.4 Convolution product 104 Chapter 5 Digitizing an Analog Signal 107 5.1 Introduction 107 5.2 Sampling 108 5.3 Quantization 112 5.4 Coding 116 Chapter 6 Modulation 119 6.1 Types of modulation 119 6.2 Amplitude modulation 121 6.2.1 Principle 121 6.2.2 Frequency space 123 6.2.3 Signal strength 124 6.2.4 Overmodulation 125 6.2.5 Demodulation 126 6.2.6 Single sideband 127 6.2.7 Modulation of a binary signal 127 6.3 Frequency modulation 130 6.3.1 Principle 130 6.3.2 Case of a sinusoidal signal 132 6.3.3 Spectrum 133 6.3.4 Signal power 136 6.3.5 FSK modulation 136 6.4 Phase modulation 139 6.4.1 Principle 139 6.4.2 PSK modulation 141 Chapter 7 Filtering 145 7.1 Definitions and reminders 145 7.1.1 Discrete signals 146 7.1.2 Analog signals 147 7.2 Analog filtering 148 7.2.1 General information 148 7.2.2 Common filters 148 7.2.3 Differential equations and transfer functions 152 7.3 Digital filtering 160 7.3.1 General information 160 7.3.2 Difference equation 161 7.3.3 Transfer function 163 7.3.4 Filter stability 166 7.3.5 Frequency behavior 168 7.3.6 FIR filters 170 7.3.7 IIR filters 173 Chapter 8 The Digital Image 177 8.1 Raster and vector images 177 8.1.1 Raster images 177 8.1.2 Vector images 179 8.2 Notions of colorimetry 179 8.2.1 Grayscale 180 8.2.2 Colors 184 8.2.3 True color and indexed color 188 8.4 Image display modes 190 8.4.1 Matrix coding 190 8.4.2 Vector coding 192 8.4.3 Fractal curves 193 8.5 Compression and compaction 193 8.6 Image formats 195 8.6.1 Raster image formats 195 8.6.2 Vector image formats 196 Chapter 9 2D Computer Graphics 197 9.1 Basic graphics processing 197 9.1.1 Drawing a segment 197 9.1.2 Drawing a circle 201 9.1.3 Windowing 202 9.1.4 Filling and coloring 204 9.2 2D geometric transformations 205 9.2.1 Homogeneous coordinates 205 9.2.2 Translation 206 9.2.3 Rotation around the origin 207 9.2.4 Dilation 208 9.2.5 Symmetries 209 9.2.6 Composition of transformations 210 9.2.7 Object representation 211 9.3 2D parametric curves 212 9.3.1 Using cubic curves 212 9.3.2 Hermite curves 213 9.3.3 Bézier curves 214 9.3.4 B-spline curves 216 Chapter 10 Concepts in Image Processing and Analysis 217 10.1 Image display 218 10.1.1 Simple correspondence 218 10.1.2 Random threshold display 218 10.1.3 Threshold matrix display 220 10.2 Basic image analysis tools 222 10.2.1 Histogram 222 10.2.2 Profiles 224 10.2.3 Level search 224 10.2.4 Information contained in an image 224 10.3 Basic processing 226 10.3.1 Histogram transformation 226 10.3.2 Changing the shape of the histogram: equalization 230 10.3.3 Image subtraction and averaging 233 10.4 Filtering 233 10.4.1 Filtering in the spatial domain 233 10.4.2 Frequency domain filtering 241 10.5 Binary images 245 10.5.1 Morphological operators 246 10.6 Segmentation 247 10.6.1 Outline extraction 248 10.6.2 Regional segmentation 252 Chapter 11 Basics of Image Compression 257 11.1 General information 258 11.1.1 Coding redundancy 258 11.1.2 Interpixel redundancy 259 11.1.3 Psychovisual redundancy 260 11.1.4 Confidence criteria 261 11.1.5 Modeling image compression 262 11.2 Lossless compression or compaction 263 11.2.1 Variable-length coding 263 11.2.2 Bit-plane coding 266 11.2.3 Predictive coding 268 11.3 Lossy compression 269 11.3.1 Predictive coding 269 11.3.2 Transform coding 270 11.4 An image compression standard: JPEG 272 Chapter 12 Elements of Numerical Analysis 277 12.1 Numerical solution of a linear system 278 12.1.1 Exact solution of a linear Cramerian system 278 12.1.2 Principle of iterative methods 280 12.1.3 Diagonal iteration and Gauss–Seidel iteration 282 12.1.4 Direct methods 283 12.2 Numerical solution of fx = 0 287 12.2.1 Introduction 287 12.2.2 General methods 288 12.2.3 Methods applicable to polynomial equations 294 12.3 Numerical integration 297 12.3.1 Introduction 297 12.3.2 Classic methods 297 12.3.3 Polynomial interpolation 302 12.3.4 Quadrature formulas 304 12.3.5 Monte Carlo method 307 12.4 Numerical solution of differential equations 310 12.4.1 Introduction 310 12.4.2 Separate-step algorithms 311 12.4.3 Linked-step methods 318 12.5 Numerical solution of partial differential equations 320 12.5.1 Definitions 320 12.5.2 Finite difference method 321 12.5.3 Resolution examples 325 12.6 Appendices 331 12.6.1 Dichotomy method 331 12.6.2 Iterative method 332 12.6.3 Secant method 332 12.6.4 Tangent method 333 12.6.5 Monte Carlo method Example 12.7 334 12.6.6 Monte Carlo method Example 12.8 336 References 339 List of Authors 343 Index 345