FTS chp 1.pdf

1 Chapter 1 Introduction 1.1 Signals and Systems Mathematics has a very broad scope, encompassing many areas such as: linear algebra, calculus, probability and statistics, geometry, differential equations, and numerical methods. For engineers, however, an area of mathematics of particular imp...

1 Chapter 1 Introduction 1.1 Signals and Systems Mathematics has a very broad scope, encompassing many areas such as: linear algebra, calculus, probability and statistics, geometry, differential equations, and numerical methods. For engineers, however, an area of mathematics of particular importance is the one that pertains to signals and systems (which is, loosely speaking, the branch of mathematics known as functional analysis). It is this area of mathematics that is the focus of this book. Before we can treat this topic in any meaningful way, however, we must first explain precisely what signals and systems are. This is what we do next. 1.2 Signals A signal is a function of one or more variables that conveys information about some (usually physical) phenomenon. Some examples of signals include: a human voice a voltage in an electronic circuit the temperature of a room controlled by a thermostat system the position, velocity, and acceleration of an aircraft the acceleration measured by an accelerometer in a cell phone the force measured by a force sensor in a robotic system the electromagnetic waves used to transmit information in wireless computer networks a digitized photograph a digitized music recording the evolution of a stock market index over time 1.2.1 Classification of Signals Signals can be classified based on the number of independent variables with which they are associated. A signal that is a function of only one variable is said to be one dimensional. Similarly, a signal that is a function of two or more variables is said to be multi-dimensional. Human speech is an example of a one-dimensional signal. In this case, we have a signal associated with fluctuations in air pressure as a function of time. An example of a two-dimensional signal is a monochromatic image. In this case, we have a signal that corresponds to a measure of light intensity as a function of horizontal and vertical position. A signal can also be classified on the basis of whether it is a function of continuous or discrete variables. A signal that is a function of continuous variables (e.g., a real variable) is said to be continuous time. Similarly, a signal that is a function of discrete variables (e.g., an integer variable) is said to be discrete time. Although the independent variable need not represent time, for matters of convenience, much of the terminology is chosen as if this were so. Edition 3.0 Copyright © 2012–2020 Michael D. Adams 2 CHAPTER 1. INTRODUCTION x(t) x[n] 3 3 2 2 1 1 t n −30 −20 −10 0 10 20 30 −3 −2 −1 0 1 2 3 (a) (b) Figure 1.1: Graphical representations of (a) continuous-time and (b) discrete-time signals. For example, a digital image (which consists of a rectangular array of pixels) would be referred to as a discrete-time signal, even though the independent variables (i.e., horizontal and vertical position) do not actually correspond to time. If a signal is a function of discrete variables (i.e., discrete-time) and the value of the function itself is also discrete, the signal is said to be digital. Similarly, if a signal is a function of continuous variables, and the value of the function itself is also continuous, the signal is said to be analog. Many phenomena in our physical world can be described in terms of continuous-time signals. Some examples of continuous-time signals include: voltage or current waveforms in an electronic circuit; electrocardiograms, speech, and music recordings; position, velocity, and acceleration of a moving body; forces and torques in a mechanical system; and flow rates of liquids or gases in a chemical process. Any signals processed by digital computers (or other digital devices) are discrete-time in nature. Some examples of discrete-time signals include digital video, digital photographs, and digital audio data. A discrete-time signal may be inherently discrete or correspond to a sampled version of a continuous-time sig- nal. An example of the former would be a signal corresponding to the Dow Jones Industrial Average stock market index (which is only defined on daily intervals), while an example of the latter would be the sampled version of a (continuous-time) speech signal. 1.2.2 Notation and Graphical Representation of Signals In the case of discrete-time signals, we sometimes refer to the signal as a sequence. The nth element of a sequence x is denoted as either x(n) or xn. Figure 1.1 shows how continuous-time and discrete-time signals are represented graphically. 1.2.3 Examples of Signals A number of examples of signals have been suggested previously. Here, we provide some graphical representations of signals for illustrative purposes. Figure 1.2 depicts a digitized speech signal. Figure 1.3 shows an example of a monochromatic image. In this case, the signal represents light intensity as a function of two variables (i.e., horizontal and vertical position). 1.3 Systems A system is an entity that processes one or more input signals in order to produce one or more output signals, as shown in Figure 1.4. Such an entity is represented mathematically by a system of one or more equations. In a communication system, the input might represent the message to be sent, and the output might represent the received message. In a robotics system, the input might represent the desired position of the end effector (e.g., gripper), while the output could represent the actual position. Copyright © 2012–2020 Michael D. Adams Edition 3.0 1.3. SYSTEMS 3 Figure 1.2: Segment of digitized human speech. Figure 1.3: A monochromatic image. x0 y0 x1 y1 x2 System y2............ xM yN |{z} |{z} Input Signals Output Signals Figure 1.4: System with one or more inputs and one or more outputs. Edition 3.0 Copyright © 2012–2020 Michael D. Adams 4 CHAPTER 1. INTRODUCTION R + vs (t) − i(t) vc (t) Figure 1.5: A simple RC network. 1.3.1 Classification of Systems A system can be classified based on the number of inputs and outputs it has. A system with only one input is described as single input, while a system with multiple inputs is described as multi-input. Similarly, a system with only one output is said to be single output, while a system with multiple outputs is said to be multi-output. Two commonly occurring types of systems are single-input single-output (SISO) and multi-input multi-output (MIMO). A system can also be classified based on the types of signals with which it interacts. A system that deals with continuous-time signals is called a continuous-time system. Similarly, a system that deals with discrete-time signals is said to be a discrete-time system. A system that handles both continuous- and discrete-time signals, is sometimes referred to as a hybrid system (or sampled-data system). Similarly, systems that deal with digital signals are referred to as digital, while systems that handle analog signals are referred to as analog. If a system interacts with one- dimensional signals, the system is referred to as one-dimensional. Likewise, if a system handles multi-dimensional signals, the system is said to be multi-dimensional. 1.3.2 Examples of Systems Systems can manipulate signals in many different ways and serve many useful purposes. Sometimes systems serve to extract information from their input signals. For example, in the case of speech signals, systems can be used in order to perform speaker identification or voice recognition. A system might analyze electrocardiogram signals in order to detect heart abnormalities. Amplification and noise reduction are other functionalities that systems could offer. One very basic system is the resistor-capacitor (RC) network shown in Figure 1.5. Here, the input would be the source voltage vs and the output would be the capacitor voltage vc. Consider the signal-processing systems shown in Figure 1.6. The system in Figure 1.6(a) uses a discrete-time system (such as a digital computer) to process a continuous-time signal. The system in Figure 1.6(b) uses a continuous- time system (such as an analog computer) to process a discrete-time signal. The first of these types of systems is ubiquitous in the world today. Consider the communication system shown in Figure 1.7. This system takes a message at one location and reproduces this message at another location. In this case, the system input is the message to be sent, and the output is the estimate of the original message. Usually, we want the message reproduced at the receiver to be as close as possible to the original message sent by the transmitter. A system of the general form shown in Figure 1.8 frequently appears in control applications. Often, in such applications, we would like an output to track some reference input as closely as possible. Consider, for example, a robotics application. The reference input might represent the desired position of the end effector, while the output represents the actual position. 1.4 Why Study Signals and Systems? As can be seen from the earlier examples, there are many practical applications in which we need to develop systems that manipulate signals. In order to do this, we need a formal mathematical framework for the study of such systems. Such a framework can be used to guide the design of new systems as well as to analyze the behavior of already existing systems. Over time, the complexity of systems designed by engineers has continued to grow. Today, most systems Copyright © 2012–2020 Michael D. Adams Edition 3.0 1.4. WHY STUDY SIGNALS AND SYSTEMS? 5 Discrete-Time Discrete-Time Input Signal Signal Output Continuous-Time Before After Continuous-Time Signal Continuous-to- Processing Processing Discrete-to- Signal Discrete-Time Discrete-Time Continuous-Time System (C/D) Converter (D/C) Converter (a) Continuous-Time Continuous-Time Input Signal Signal Output Discrete-Time Before After Discrete-Time Signal Discrete-to- Processing Processing Continuous-to- Signal Continuous-Time Continuous-Time Discrete-Time System (D/C) Converter (C/D) Converter (b) Figure 1.6: Signal processing systems. (a) Processing a continuous-time signal with a discrete-time system. (b) Pro- cessing a discrete-time signal with a continuous-time system. Estimate of Message Transmitted Received Message Signal Signal Signal Signal Transmitter Channel Receiver Figure 1.7: Communication system. Reference Input Error Output + Controller Plant − Sensor Feedback Signal Figure 1.8: Feedback control system. Edition 3.0 Copyright © 2012–2020 Michael D. Adams 6 CHAPTER 1. INTRODUCTION of practical interest are highly complex. For this reason, a formal mathematical framework to guide the design of systems is absolutely critical. Without such a framework, there is little hope that any system that we design would operate as desired, meeting all of the required specifications. 1.5 Overview of This Book This book presents the mathematical tools used for studying both signals and systems. Although most systems consid- ered herein are SISO, the mathematics extends in a very straightforward manner to systems that have multiple inputs and/or multiple outputs. Only the one-dimensional case is considered herein, however, as the multi-dimensional case is considerably more complicated and beyond the scope of this book. The remainder of this book is organized as follows. Chapter 2 presents some mathematical preliminaries that are essential for both the continuous-time and discrete-time cases. Then, Chapters 3, 4, 5, 6, and 7 cover material that relates primarily to the continuous-time case. Then, Chapters 8, 9, 10, 11, and 12 cover material that relates primarily to the discrete-time case. The material on continuous-time signals and systems consists of the following. Chapter 3 examines signals and systems in more depth than covered in earlier chapters. A particular class of systems known as (continuous-time) linear time-invariant (LTI) systems is introduced. Chapter 4 then studies continuous-time LTI systems in depth. Finally, Chapters 5, 6, and 7 introduce continuous-time Fourier series, the continuous-time Fourier transform, and the Laplace transform, respectively, which are important mathematical tools for studying continuous-time signals and systems. The material on discrete-time signals and systems consists of the following. Chapter 8 examines discrete-time signals and systems in more depth than earlier chapters. A particular class of systems known as (discrete-time) linear time-invariant (LTI) systems is introduced. Chapter 9 then studies discrete-time LTI systems in depth. Finally, Chapters 10, 11, and 12 introduce discrete-time Fourier series, the discrete-time Fourier transform, and the z transform, respectively, which are important mathematical tools for studying discrete-time signals and systems. This book also includes several appendices, which contain supplemental material related to the topics covered in the main chapters of the book. Appendix A provides a review of complex analysis. Appendix B introduces partial fraction expansions. Appendix C presents time-domain methods for solving for differential equations. Appendix D presents a detailed introduction to MATLAB. Appendix E provides some additional exercises. Appendix F offers a list of some useful mathematical formulas. Finally, Appendix G presents some information about video lectures available for this book. Copyright © 2012–2020 Michael D. Adams Edition 3.0