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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 wellknown 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 JeanBaptisteJoseph Fourier ( 17681830)
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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 Spring '13
 Bayliss
 Signal Processing, The Land

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