Signals and Systems : Fundamentals

The concepts of signals and systems arise in a wide variety of areas, ranging from home-oriented consumer electronics and multimedia entertainment products to sophisticated communications, aeronautics and astronautics, and control. The ideas and approaches associated with these concepts have great effects on our life in one way or another. Although the signals and systems which arise across those fields are naturally different in their physical make-up and application, the principles and tools for analyzing signals and systems are the same and hence applicable to all of them. Therefore, an introductory course on signals and systems is fundamental and compulsory for an engineering under-graduate curriculumin anywell-established tertiary (education) institutions. Such a course is commonly designed in one of the two forms below: – a one-semester subject that intends to provide students with a rich set of concepts and tools for analyzing deterministic signals and an important class of systems known as linear time-invariant systems; – a two-semester subject that expands on the one-semester course by adding more detailed treatment of signal processing and systems for specified applications such as communications, multimedia signal processing and control engineering. This book takes the first form and assumes that the students have a background in calculus and introductory physics. Contents Preface 1 Introduction 1.1 Overview of signals and systems 1.1.1 What is a signal? 1.1.2 What is a system? 1.2 Description and classification of signals 1.2.1 Continuous-time signals and discrete-time signals 1.2.2 Energy signals and power signals 1.2.3 Periodic signals and nonperiodic signals 1.2.4 Deterministic signals and random signals 1.2.5 Elementary signals 1.3 Description of systems 1.3.1 Elementary systems 1.3.2 System modelling 1.4 Properties of systems 1.4.1 Memoryless and with memory 1.4.2 Causality 1.4.3 Invertibility 1.4.4 Stability 1.4.5 Time-invariance 1.4.6 Linearity 1.5 Summary 1.6 Problems 2 Time-domain analysis of LTI systems 2.1 Introduction 2.2 The unit impulse response and convolutions 2.2.1 The convolution sum 2.2.2 The convolution integral 2.3 Properties of convolutions and equivalent systems 2.4 Causality and stability of LTI systems 2.5 Systems constrained with LCCDEs 2.5.1 Continuous-time systems constrained with LCCDEs 2.5.2 Discrete-time systems characterized by LCCDEs 2.6 Summary 2.7 Problems 3 Fourier analysis of signals 3.1 Introduction 3.2 Fourier series for continuous-time periodic signals 3.3 Fourier series for discrete-time periodic signals 3.4 Why should a signal be transformed? 3.5 Fourier transform for continuous-time signals 3.5.1 Properties of Fourier transform 3.5.2 Inverse Fourier transform 3.6 The discrete-time Fourier transform 3.6.1 Properties of DTFT 3.6.2 Inverse DTFT 3.7 Fourier series and Fourier transforms 3.8 Summary 3.9 Problems 4 Frequency-domain approach to LTI systems 4.1 Introduction 4.2 Frequency response of LTI systems 4.3 Bode plots for continuous-time LTI systems 4.4 Frequency response of LTIs described with LCCDEs 4.5 Frequency domain approach to system outputs 4.6 Some typical LTI systems 4.6.1 All-pass systems 4.6.2 Linear phase response systems 4.6.3 Ideal filters 4.6.4 Ideal transmission channels 4.7 Summary 4.8 Problems 5 Discrete processing of analog signals 5.1 Introduction 5.2 Sampling of a continuous-time signal 5.3 Spectral relationship and sampling theorem 5.4 Reconstruction of continuous-time signals 5.5 Hybrid systems for discrete processing 5.6 Discrete Fourier transform 5.7 Compressed sensing 5.8 Summary 5.9 Problems 6 Transform-domain approaches 6.1 Motivation 6.2 The Laplace transform 6.2.1 Derivation of the transform 6.2.2 Region of convergence 6.2.3 Inverse Laplace transform 6.2.4 Properties of Laplace transform 6.3 The z-transform 6.3.1 Region of convergence 6.3.2 Properties of the z-transform 6.3.3 Inverse z-transform 6.4 Transform-domain approach to LTI systems 6.4.1 Transfer function of LTI systems 6.4.2 Inverse systems of LTIs and deconvolutions 6.4.3 Revisit of LTI system’s stability and causality 6.4.4 Transfer function of LTI systems by LCCDEs 6.5 Transform domain approach to LCCDEs 6.6 Decomposition of LTI system responses 6.7 Unilateral transforms 6.7.1 Unilateral Laplace transform 6.7.2 Unilateral z-transform 6.8 Summary 6.9 Problems 7 Structures and state-space realizations 7.1 Block-diagram representation 7.2 Structures of LTIs with a rational transfer function 7.3 State-space variable representation 7.3.1 State model and state-space realizations 7.3.2 Construction of an equivalent state-space realization 7.3.3 Similarity transformations 7.4 Discretizing a continuous-time state model 7.5 Summary 7.6 Problems 8 Comprehensive problems 8.1 Motivation 8.2 Problems Appendices A. Proof of the condition of initial rest (1.49) B. Proof of Theorem 2.5 C. Orthogonality principle D. Residue theorem and inverse transforms E. Partial-fraction expansion Bibliography Index

Signals and systems enjoy wide application in industry and daily life, and understanding basic concepts of the subject area is of importance to undergraduates majoring in engineering. With rigorous mathematical deduction, this introductory text book is helpful for students who study communications engineering, electrical and electronic engineering, and control engineering. Additionally, supplementary materials are provided for self-learners.

Signals and systems enjoy wide application in industry and daily life, and understanding basic concepts of the subject area is of importance to undergraduates majoring in engineering. With rigorous mathematical deduction, this introductory text book is helpful for students who study communications engineering, electrical and electronic engineering, and control engineering. Additionally, supplementary materials are provided for self-learners. Illustrates basic concepts in signals and systems in a concise way. Demonstrates mathematical nature of signals and systems. Designed to facilitate self-learning with PPTs and solutions to tutorials.