Chap2 - Chapter 2 CT and DT Signal Representations • 1...

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Unformatted text preview: Chapter 2 CT and DT Signal Representations • 1 Overview • 2 Fourier Series • 3 Fourier Transform • 4 Discrete-Fourier Series • 5 Discrete-Time Fourier Transform • 6 Applications GOALS 1. Development of representations of CT and DT signals in the fre- quency domain. 2. Familiarization with Fourier Series and trasform representations for CT and DT signals. 9 10 CHAPTER 2. CT AND DT SIGNAL REPRESENTATIONS Since the signals we encounter in engineering, science, and everyday life are as varied as the applications in which we engage them, it is often helpful to rst study these applications in the presence of simpli ed versions of these signals. Much like a child learning to play an instrument for the rst time, it is easier to start by attempting to play a single note before an entire musical score. Then, after learning many notes, the child becomes a musician and can synthesize a much broader class of music, building up from many notes. This approach of building-up our understanding of complex concepts by rst understanding their basic building blocks is a fundamental precept of engineering and one that we will use frequently throughout this book. In this chapter, we will explore signals in both continuous time and discrete time, together with a number of ways in which these signals can be built-up from simpler signals. Simplicity is in the eye of the beholder and what makes a signal appear simple in one context may not shed much light in another context. Many of the concepts we will develop throughout this text arise from studying large classes of signals, one building block at a time, and extrapolating system (or application) level behavior by considering the whole as a sum of its parts. In this chapter, we will focus speci cally on sinusoidal signals as our basic building blocks as we consider both periodic and aperiodic signals in continuous and discrete time. Along this path, we will encounter the Fourier series representations of periodic signals as well as Fourier transform representations of aperiodic, in nite-length signals. In later chapters, we will nd that so-called time-domain representations of signals sometimes prove more fruitful, and for discrete-time signals there is a natural way to construct signals one sample at a time. Historical comments on the Fourier series and the Fourier transform. 2.1 Fourier Series representation of nite-length and periodic CT signals In many applications in science and engineering, we often work with signals that are periodic in time. That is, the signal repeats itself over and over again with a given period of repetition. Examples of periodic signals might include the acoustic signal that emenates from a musical instrument, such as a trumpet when a single sustained note is played, or the vertical displacement of a mass in a frictionless spring-mass oscillator set into motion, or the horizontal displacement of a pendulum swaying to and fro in the absence of friction....
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