Signal Processing and Linear Systems-B.P.Lathi copy

41 is called t he g eneralized f ourier s eries of f

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Unformatted text preview: sed only one t erm ( N = 1) o r two t erms ( N = 2) or any number of terms, t he o ptimum value of t he coefficient C I would be t he same [as given by Eq. (3.39)]. T he a dvantage of t his a pproximation of a signal f (t) by a set of mutually orthogonal signals is t hat we can continue t o a dd terms t o t he a pproximation without disturbing t he previous t erms. T his property of f inality of t he values of t he coefficients is very tNote that the energy of a signal e x(t) is c2 E x. 3.3 Signal r epresentation by Orthogonal Signal Set 187 i mportant from a practical point of view.:j: S ome Examples o f Generalized Fourier Series Signals are vectors in every sense. Like a vector, a signal can be represents as a s um o f i ts components in a variety of ways. J ust as vector coordinate systems are formed by mutually orthogonal vectors (rectangular, cylindrical, spherical), we also have signal coordinate systems (basis signals) formed by a variety of sets of lJlutually orthogonal signals. There exist a large number of orthogonal signal sets which can be used as basis signals for generalized Fourier series. Some well-known signal sets are trigonometric (sinusoid) functions, exponential functions, Walsh functions, Bessel functions, Legendre polynomials, Laguerre functions, Jacobi polynomials, Hermite polynomials, and Chebyshev polynomials. T he functions t hat concern us most in this book are the trigonometric a nd t he e xponential sets discussed in the rest o f t he c hapter. A Historical Note: Baron Jean-Baptiste-Joseph Fourier ( 1768-1830) T he Fourier series a nd i ntegral is a most beautiful a nd fruitful development, which serves as a n indispensable instrument in t he t reatment of many problems in mathematics, science, and engineering. Maxwell was so taken by the beauty of the Fourier series t hat he called it a great mathematical poem. In electrical engineering, it is c entral to t he a reas of communication, signal processing a nd several other fields, including antennas, b ut i t was not received enthusiastically by t he scientific world when i t was presented. In fact, Fourier could n ot g et his results published as a paper. Fourier, a tailor's son, was orphaned a t age 8 and educated a t a local military college (run by Benedictine monks), where he excelled in mathematics. T he Benedictines prevailed upon t he young genius t o choose t he p riesthood as his vocation, b ut t he revolution broke o ut before he could take his vows. Fourier joined t he people's party. B ut in its early days, t he French Revolution, like most revolutions of its kind, liquidated a large segment of t he intelligentsia, including prominent scientists such as Lavosier. This persecution caused many intellectuals to leave France to save themselves from a rapidly rising tide of barbarism. Fourier, who was an early enthusiast of t he Revolution, narrowly escaped t he guillotine twice. I t was t o t he everlasting credit of Napoleon t hat he stopped t he p ersecution of the intelligentsia a nd found...
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