Page 297 - Electrical Engineering Dictionary

where H(f ) is the Hilbert transform of f . engineering applications have a Fourier se-
See Fourier amplitude, Fourier phase, Hilbert ries.
transform.
Fourier transform a linear mathematical
Fourier plane a plane in an optical system transform from the domain of time or space
where the exact Fourier transform of an input functions to the frequency domain. The dis-
image is generated. crete version of the transform (DFT) can be
implemented with a particularly efficient al-
Fourier series Let f(t) be a continuous gorithm (fast Fourier transform or FFT). The
time periodic signal with fundamental period discrete Fourier transform of a digital image
T such that represents the image as a linear combination
of complex exponentials.
∞
X jω 0 t The Fourier transform of a continuous
f(t) = c n e , −∞ <t < ∞
time period signal f(t) is given by
n=−∞
Z
∞
where ω 0 is the fundamental frequency, c 0 −jωt
F(ω) = f(t)e dt, −∞ <ω < ∞.
is a real number, and c n ,n 6= 0 are com- −∞
plex numbers. This representation of f(t) is
If the signal f(t) is absolutely integrable and
called the exponential Fourier series of f(t).
is well behaved, then its Fourier transform
The coefficients c n are called the Fourier co-
exists. For example, the rectangular pulse
efficients and are given by
signal
1 Z −jω 0 t
c n = f(t)e dt, n = 0, ±1, 1, − T ≤ t< T
T T f(t) = 2 2
±2,... 0, otherwise
2 ωT
The function f(t) can also be expressed as has Fourier transform F(w) = ω sin 2 .
The inverse Fourier transform of a signal
∞ is given by
X
f(t) = c 0 + (a n cos(nω 0 t)+b n sin(nω 0 t))
Z
1 ∞
n=1 jωt
f(t) = F(ω)e dω
2π −∞
with Fourier coefficients a n and b n deter-
mined from See also optical Fourier transform, two-di-
mensional Fourier transform.
Z
2
a n = f(t) cos(nω 0 tdt), n = 1,
T T Fourier, Jean Baptiste Joseph (1768–
2, 3,... 1830) Born: Auxerre, France
Z
2 Fourier is best known for the development
b n = f(t) sin(nω 0 tdt), n = 1,
T T of new mathematical tools. The Fourier se-
2, 3,... ries, which describes complex periodic func-
tions, andtheFourierintegraltheorem, which
This representation of f(t) is called the allows complex equations to be broken into
trigonometric Fourier series of f(t). A sig- simplertrigonometricequationsforeasierso-
nal f(t) has a Fourier series if it satisfies lution, are named in honor of their discoverer.
Dirichlet conditions, given by (i) f(t) is ab- Fourier was an assistant lecturer under the
solutely integrable over any period, (ii) f(t) great mathematicians Joseph Lagrange and
is piecewise continuous over any period, and Gaspard Monge at the Ecole Polytechnique
(iii) d f(t) is piecewise continuous over any in Paris. He also served in a number of posi-
dt
period. Virtually all periodic signals used in tions in Napoleon’s government, eventually

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