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Signals occur in a wide range of physical phenomenon. They might be human speech, blood pressure variations with time, seismic waves,radar and sonar signals, pictures or images, stress and strain signals in a building structure, stock market prices, a city'spopulation, or temperature across a plate. These signals are often modeled or represented by a real or complex valued mathematicalfunction of one or more variables. For example, speech is modeled by a function representing air pressure varying with time. Thefunction is acting as a mathematical analogy to the speech signal and, therefore, is called an analog signal. For these signals, the independent variable is time and it changescontinuously so that the term continuous-time signal is also used. In our discussion, we talk of the mathematical function asthe signal even though it is really a model or representation of the physical signal.
The description of signals in terms of their sinusoidal frequency content has proven to be one of the most powerful tools ofcontinuous and discrete-time signal description, analysis, and processing. For that reason, we will start the discussion ofsignals with a development of Fourier transform methods. We will first review the continuous-time methods of the Fourier series (FS),the Fourier transform or integral (FT), and the Laplace transform (LT). Next the discrete-time methods will be developed in moredetail with the discrete Fourier transform (DFT) applied to finite length signals followed by the discrete-time Fourier transform(DTFT) for infinitely long signals and ending with the Z-transform which allows the powerful tools of complex variable theory to beapplied.
More recently, a new tool has been developed for the analysis of signals. Wavelets and wavelet transforms [link] , [link] , [link] , [link] , [link] are another more flexible expansion system that also can describe continuousand discrete-time, finite or infinite duration signals. We will very briefly introduce the ideas behind wavelet-based signal analysis.
The problem of expanding a finite length signal in a trigonometric series was posed and studied in the late 1700's by renowned mathematicians suchas Bernoulli, d'Alembert, Euler, Lagrange, and Gauss. Indeed, what we now call the Fourier series and the formulas for the coefficients were used byEuler in 1780. However, it was the presentation in 1807 and the paper in 1822 by Fourier stating that an arbitrary function could be represented bya series of sines and cosines that brought the problem to everyone's attention and started serious theoretical investigations and practicalapplications that continue to this day [link] , [link] , [link] , [link] , [link] , [link] . The theoretical work has been at the center of analysis and the practical applications havebeen of major significance in virtually every field of quantitative science and technology. For these reasons and others, the Fourier seriesis worth our serious attention in a study of signal processing.
We assume that the signal $x\left(t\right)$ to be analyzed is well described by a real or complex valued function of a real variable $t$ defined over a finite interval $\{0\le t\le T\}$ . The trigonometric series expansion of $x\left(t\right)$ is given by
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