Digital Signal Processing Reference
In-Depth Information
4
FOURIER SERIES
Jean Baptiste Joseph Fourier
(born March 21, 1768, in Auxerre,
Bourgogne, France; died May 16, 1830.) A mathematician known also as an
Egyptologist and administrator, he exerted strong influence on mathematical
physics through his
Théorie analytique de la chaleur
(1822;
The Analytical
Theory of Heat
). He showed how the conduction of heat in solid bodies may
be analyzed in terms of infinite mathematical series now called by his name,
the Fourier series. Far transcending the particular subject of heat conduction,
his work stimulated research in mathematical physics.
Reference:
“Fourier, Joseph, Baron.” Encyclopedia Britannica, 2007. Encyclopedia
Britannica Online, January 6, 2007:
http://www.britannica.com/eb/article-9035044
A common engineering analysis technique is the partitioning of complex problems
into simpler ones. The simpler problems are then solved, and the total solution be-
comes the sum of the simpler solutions. One example is the use of a Taylor's series
expansion, in which a function is expressed as a constant, plus a ramp function,
plus a parabolic function, and so on:
f(t)
f(t) = f(0) + f¿(0)t + f-(0)
t
2
2!
+
Á
.
(4.1)
In this equation,
d
2
f(t)
dt
2
df(t)
dt
`
`
f¿(0) =
;
f¿(0) =
.
t = 0
t = 0
We solve the problem involving by considering only the constant, then considering
only the ramp function, and so on. The final solution is the sum of these solutions.
f(t)
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