Introductory course textbook on signals and systems with numerous examples and code snippets implemented in Python Supported by code examples, Signals and Systems: Theory and Practical Explorations with Python is a textbook resource for a complete introductory course in systems and signals, enabling readers to run Python programs for convolution, discrete time Fourier transforms and series, sampling, and interpolation for a wide range of functions. Readers are guided step-by-step through basic differential equations, basic linear algebra, and calculus to ensure full comprehension of the exercises. This book is supported by a companion website, hosting interactive material to draw functions, and run programs in Python; it is enriched with audiovisual material via linking to related videos. Links to resources that provide a deeper explanation about the important concepts in the book, such as the systems approach, complex numbers, harmony, the Euler equation, and Hilbert spaces, are also included. Written by two highly qualified academics, topics covered include: Systems approach for modeling the natural and manmade systems and some application areas Representation of complex and real signals by basic functions, such as real and complex exponentials, unit step and unit impulse functions Properties of signals, such as symmetry, harmony, energy, power, continuity and discreteness Convolution and correlation operations for continuous time and discrete time signals and systems Representation of systems by impulse response, frequency response, transfer function, block diagram, differential and difference equations Properties of systems, such as linearity, time invariance, memory, invertibility, stability and causality Continuous time and discrete time Fourier analysis in Hilbert space and their extension to Laplaca transform and z-transform Filtering by Linear Time Invariant systems in time and frequency domains, covering low pass, high pass band pass and band reject filters Sampling theorems for continuous time and discrete time systems, covering A/D and D/A conversion, sampling and interpolation Signals and Systems is an ideal textbook resource for a one semester introductory course on signals and systems for upper level undergraduate and graduate students in computer science, electrical engineering and data science. It is also a useful reference for professionals working in bioinformatics, robotics, remote sensing, and related fields.

Table of Contents:
About the Authors xiii Preface xiv Acknowledgments xvii About the Companion Website xix 1 Introduction to Systems and Signals 1 1.1 Example Applications 2 1.1.1 Three-Dimensional World Models by LIDAR Signals 3 1.1.2 Modeling the Brain Networks from the Brain Signals 3 1.1.3 Detecting the Buildings from the Remote-Sensed Satellite Images 4 1.1.4 Noise Reduction in Old Records 4 1.2 Relationship Between Signals and Systems 4 1.3 Mathematical Representation of Signals and Systems 5 1.3.1 Signals Represented by Functions 6 1.3.2 Types of Signals 6 1.3.3 Energy of a Signal 9 1.3.4 Power of a Signal 10 1.4 Operations on the Time Variable of Signals 10 1.4.1 Time Shift 11 1.4.2 Time Reverse 12 1.4.3 Time Scale 13 1.4.4 Time Scale and Shift 15 1.5 Signals with Symmetry Properties 19 1.5.1 Periodic Signals 21 1.5.1.1 Continuous Time Periodic Signals 22 1.5.1.2 Discrete Time Periodic Signals 23 1.5.2 Even and Odd Signals 24 1.6 Complex Signals Represented by Complex Functions 28 1.6.1 Complex Numbers Represented in Cartesian Coordinate System 28 1.6.2 Complex Numbers Represented in Polar Coordinate System and Euler’s Number 30 1.6.3 Complex Functions 33 1.7 Chapter Summary 35 Problems 36 2 Basic Building Blocks of Signals 43 2.1 LEGO Functions of Signals 43 2.2 King of the Functions: Exponential Function 44 2.2.1 Real Exponential Function 44 2.2.1.1 Continuous Time Real Exponential Function 44 2.2.1.2 Discrete Time Real Exponential Function 46 2.2.2 Complex Exponential Function 47 2.2.2.1 Continuous Time Complex Exponential Functions 48 2.2.2.2 Harmonically Related Complex Exponential 49 2.2.2.3 Complex Exponential Function for Discrete Time Signals 53 2.3 Unit Impulse Function 55 2.3.1 Discrete Time Unit Impulse Function or Dirac-Delta Function 55 2.3.2 Continuous Time Unit Impulse Function 56 2.3.3 Comparison of Discrete Time and Continuous Time Unit Impulse Functions 57 2.4 Unit Step Function 58 2.4.1 Discrete Time Unit Step Function 58 2.4.2 Relationship Between the Discrete Time Unit Step and Unit Impulse Functions 58 2.4.3 Continuous Time Unit Step Function 60 2.4.4 Comparison of Discrete Time and Continuous Time Unit Step functions 61 2.4.4.1 Relationship Between the Continuous Time Unit Step and Unit Impulse Functions 61 2.5 Chapter Summary 65 Problems 65 3 Basic Building Blocks and Properties of Systems 69 3.1 Representation of Systems by Equations 69 3.2 Interconnection of Basic Systems: Series, Parallel, Hybrid, and Feedback Control Systems 70 3.2.1 Series Systems 70 3.2.2 Parallel Systems 71 3.2.3 Hybrid Systems 71 3.2.3.1 Feedback Control Systems 72 3.2.3.2 An Example of System Modeling: Neurons as a Subsystem of Human Brain 73 3.3 Properties of Systems 74 3.3.1 Memory 75 3.3.2 Causality 76 3.3.3 Invertibility 77 3.3.4 Stability 79 3.3.5 Time Invariance 80 3.3.6 Linearity and Superposition Property 81 3.4 Basic Building Blocks of Systems and Their Properties 85 3.4.1 Scalar Multiplier 85 3.4.2 Adder 85 3.4.3 Multiplier 85 3.4.4 Integrator 86 3.4.5 Differentiator 86 3.4.6 Unit Delay Operator 87 3.4.7 Unit Advance Operator 87 3.5 Chapter Summary 89 Problems 89 4 Representation of Linear Time-Invariant Systems by Impulse Response and Convolution Operation 95 4.1 Representation of LTI Systems by Impulse Response 96 4.1.1 Representation of Discrete Time Linear Time-Invariant Systems by Impulse Response 97 4.1.2 Representation of Continuous Time Linear Time-Invariant System 98 4.1.3 Convolution Operation in Continuous Time 101 4.1.4 Convolution Operation in Discrete Time Systems 105 4.1.5 Cross-correlation and Autocorrelation Operations 107 4.2 Properties of Impulse Response for LTI Systems 110 4.2.1 Impulse Response of Memoryless LTI Systems 110 4.2.2 Impulse Response of Causal LTI Systems 110 4.2.3 Inverse of Impulse Response for LTI Systems 111 4.2.4 Impulse Response of Stable LTI Systems 114 4.2.5 Unit Step Response 115 4.3 An Application of Convolution in Machine Learning 116 4.4 Chapter Summary 118 Problems 118 5 Representation of LTI Systems by Differential and Difference Equations 123 5.1 Linear Constant-Coefficient Differential Equations 123 5.2 Representation of a Continuous Time LTI System by Differential Equations 124 5.3 Solving the Linear Constant Coefficient Differential Equations That Represent LTI Systems 126 5.3.1 Finding the Particular Solution 127 5.3.2 Finding the Homogeneous Solution 128 5.3.3 Finding the General Solution 129 5.3.4 Transfer Function of a Continuous Time LTI System 134 5.4 Linear Constant Coefficient Difference Equations 136 5.4.1 Representation of a Discrete Time LTI Systems by Difference Equations 136 5.4.2 Solution to Linear Constant Coefficient Difference Equations 137 5.4.3 Transfer Function of a Discrete Time LTI System 139 5.5 Relationship Between the Impulse Response and Difference or Differential Equations 140 5.6 Block Diagram Representation of Differential Equations for LTI Systems 144 5.7 Chapter Summary 147 Problems 148 6 Fourier Series Representation of Continuous Time Periodic Signals 155 6.1 History 156 6.2 Mathematical Representation of Waves and Harmony 157 6.3 Dirichlet Conditions 160 6.4 Fourier Theorem 162 6.4.1 Proof Sketch for the Fourier Theorem 162 6.4.2 Terminology 163 6.5 Frequency Domain and Hilbert Spaces 164 6.6 Response of a Linear Time-Invariant System to the Continuous Time Complex Exponential Input Signal 170 6.6.1 Eigenfunctions and Eigenvalues of a Continuous Time LTI Systems 171 6.7 Convergence of the Fourier Series and Gibbs Phenomenon 173 6.8 Properties of Fourier Series for Continuous Time Functions 174 6.8.1 Linearity Property 174 6.8.2 Time Shifting Property 174 6.8.3 Time Scale Property 175 6.8.4 Time Reversal Property 175 6.8.5 Convolution Property 175 6.8.6 Multiplication Property 176 6.8.7 Conjugate Symmetry 176 6.8.8 Parseval’s Equality 177 6.8.9 Differentiation Property 177 6.9 Trigonometric Fourier Series for Continuous Time Functions 180 6.10 Trigonometric Fourier Series for Continuous Time Even and Odd Functions 182 6.11 Chapter Summary 185 Problems 186 7 Fourier Series Representation of Discrete Time Periodic Signals 191 7.1 Fourier Series Theorem for Discrete Time Functions 191 7.2 Discrete Time Fourier Series Representation in Hilbert Space 193 7.3 Properties of Discrete Time Fourier Series 199 7.3.1 Difference Property 203 7.3.2 Convolution Property 205 7.3.3 Multiplication Property 208 7.4 Discrete Time LTI Systems with Periodic Input and Output Pairs 211 7.4.1 Eigenfunctions, Eigenvalues, and Transfer Functions of a Discrete Time LTI Systems 212 7.4.2 Relationship Between the Fourier Series of Periodic Input and Output Pairs of Discrete Time LTI Systems 213 7.5 Chapter Summary 215 Problems 215 8 Continuous Time Fourier Transform and Its Extension to Laplace Transform 221 8.1 Fourier Series Extension to Aperiodic Functions 222 8.2 Existence and Convergence of the Fourier Transforms: Dirichlet Conditions 224 8.3 Fourier Transforms 225 8.4 Comparison of Fourier Series and Fourier Transform 226 8.5 Frequency Content of Fourier Transform 227 8.6 Representation of LTI Systems in Frequency Domain by Frequency Response 232 8.7 Relationship Between the Fourier Series and Fourier Transform of Periodic Functions 235 8.8 Properties of Fourier Transform: For Continuous Time Signals and Systems 238 8.8.1 Basic Properties of Continuous Time Fourier Transform 239 8.8.2 Continuous Time Linear Time-Invariant Systems in Frequency Domain 256 8.9 Laplace Transforms as an Extension of Continuous Time Fourier Transforms 259 8.9.1 One-Sided Laplace Transform 260 8.9.2 Region of Convergence in Laplace Transforms 261 8.10 Inverse of Laplace Transform 265 8.11 Continuous Time Linear Time-Invariant Systems in Laplace Domain 268 8.11.1 Eigenvalues and Transfer Functions in s-Domain 269 8.12 Chapter Summary 272 Problems 273 9 Discrete Time Fourier Transform and Its Extension to z-Transforms 281 9.1 Fourier Series Extension to Discrete Time Aperiodic Functions 281 9.1.1 Discrete Time Fourier Transform 282 9.2 Dirichlet Conditions Are Relaxed for the Existence of Discrete Time Fourier Transform 284 9.3 Fourier Transform of Discrete Time Periodic Functions 293 9.4 Properties of Fourier Transforms for Discrete Time Signals and Systems 297 9.4.1 Basic Properties of Discrete Time Fourier Transform 297 9.5 Discrete Time Linear Time-Invariant Systems in Frequency Domain 307 9.6 Representation of Discrete Time LTI Systems 310 9.6.1 Impulse Response 311 9.6.2 Unit Step Response 311 9.6.3 Frequency Response 312 9.6.4 Difference Equation 313 9.6.5 Block Diagram Representation 314 9.7 z-Transforms as an Extension of Discrete Time Fourier Transforms 317 9.7.1 One-Sided z-Transform 319 9.7.2 Region of Convergence in z-Transforms 320 9.8 Inverse of z-Transform 325 9.9 Discrete Time Linear Time-Invariant Systems in z-Domain 329 9.9.1 Eigenvalues and Transfer Functions in z-Domain 329 9.10 Chapter Summary 333 Problems 334 10 Linear Time-Invariant Systems as Filters 343 10.1 Filtering the Periodic Signals by Frequency Response 344 10.2 Filtering the Aperiodic Signals by Frequency Response 345 10.3 Frequency Ranges of Frequency Response 347 10.4 Filtering with LTI Systems 347 10.5 Ideal Filters for Discrete Time and Continuous Time LTI Systems 348 10.5.1 Ideal Low-Pass Filters 349 10.5.2 Ideal High-Pass Filters 349 10.5.3 Ideal Band-Pass and Band-Reject Filters 350 10.6 Discrete Time Real Filters 358 10.6.1 Discrete Time Low-Pass and High-Pass Real Filters 358 10.6.2 Band-Stop Filters for Filtering Well-Defined Frequency Bandwidths 363 10.7 Continuous Time Real Filters 365 10.8 Chapter Summary 370 Problems 370 11 Continuous Time Sampling 375 11.1 Sampling 376 11.2 Properties of the Sampled Signal in Time and Frequency Domains 377 11.3 Reconstruction 382 11.4 Aliasing 385 11.5 Sampling Theorem 388 11.6 Sampling with Zero-Order Hold 389 11.7 Reconstruction with Zero-Order Hold 391 11.8 Sampling and Reconstruction with First-Order Hold 393 11.9 Chapter Summary 394 Problems 395 12 Discrete Time Sampling and Processing 403 12.1 Time Normalization 404 12.2 C/D Conversion: x(t) → x[n] 405 12.3 D/C Conversion 407 12.3.1 Band-Limited Digital Differentiator 409 12.3.2 Digital Time Shift 413 12.4 Sampling the Discrete Time Signals 415 12.4.1 Discrete Time Impulse Train Sampling 415 12.5 Reconstruction of Discrete Time Signal from Its Sampled Counterpart 418 12.6 Discrete Time Decimation and Interpolation 419 12.7 Chapter Summary 421 Problems 421 Bibliography 425 Index 427