13.1 Fourier transforms:

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Chapter 13 Integral transforms
13.1 Fourier transforms
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Chapter 13 Integral transforms
The Fourier transform of f(t)
Inverse Fourier transform of f(t)
Ex Find the Fourier transform of the exponential
decay function and
Sol
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Chapter 13 Integral transforms
Properties of distribution
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Chapter 13 Integral transforms

• The uncertainty principle

Gaussian distribution probability density
function
(1) is symmetric about the point
the standard deviation describes the width of a
curve (2) at falls to
of the peak value, these
points are points of inflection
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Chapter 13 Integral transforms
Ex Find the Fourier transform of the normalized
Gaussian distribution.
Sol the Gaussian distribution is centered on
t0, and has a root mean square
deviation
1

• is a Gaussian distribution centered on
  zero and with a root
• mean square deviation
  is a constant.

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Chapter 13 Integral transforms
Applications of Fourier transforms
(1) Fraunhofer diffractionWhen the cross-section
of the object is small compared with the distance
at which the light is observed the pattern is
known as a Fraunhofer diffraction pattern.
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Chapter 13 Integral transforms
Ex Evaluate for an aperture consisting of
two long slits each of width 2b whose centers are
separated by a distance 2a, agtb the slits
illuminated by light of wavelength .
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Chapter 13 Integral transforms

• The Diracd-function

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Chapter 13 Integral transforms
Ex Prove that
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Chapter 13 Integral transforms

• consider an integral
  to obtain

Proof

• Define the derivative of

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Chapter 13 Integral transforms

• Physical examples for d-function

• an impulse of magnitude
  applied at time
• a point charge at a point
• (3) total charge in volume V
  

• unit step (Heviside) function H(t)

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Chapter 13 Integral transforms

Proof

• Relation of the d-function to Fourier transforms

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Chapter 13 Integral transforms

• for large becomes very
• large at t0 and also very narrow
• about t0
• as

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Chapter 13 Integral transforms

• Properties of Fourier transforms
• denote the Fourier transform of by
  or

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Chapter 13 Integral transforms
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Chapter 13 Integral transforms
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Chapter 13 Integral transforms
Consider an amplitude-modulated radio wave
initial, a message is represent by ,
then add a constant signal
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Chapter 13 Integral transforms

• Convolution and deconvolution

Note x, y, z are the same physical variable
(length or angle), but each of them appears three
different roles in the analysis.
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Chapter 13 Integral transforms
Ex Find the convolution of the function
with the
function in the above figure.
Sol
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Chapter 13 Integral transforms
The Fourier transform of the convolution
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Chapter 13 Integral transforms
The Fourier transform of the product
is given by
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Chapter 13 Integral transforms
Ex Find the Fourier transform of the function
representing two wide slits by considering the
Fourier transforms of (i) two d-functions, at
, (ii) a rectangular function of height 1
and width 2b centered on x0
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Chapter 13 Integral transforms
Deconvolution is the inverse of convolution,
allows us to find a true distribution f(x) given
an observed distribution h(z) and a resolution
unction g(y).
Ex An experimental quantity f(x) is measured
using apparatus with a known resolution function
g(y) to give an observed distribution h(z). How
may f(x) be extracted from the measured
distribution.
the Fourier transform of the measured distribution
extract the true distribution
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Chapter 13 Integral transforms

• Correlation functions and energy spectra

The cross-correlation of two functions and
is defined by
It provides a quantitative measurement of the
similarity of two functions and as one is
displaced through a distances relative
to the other.
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Chapter 13 Integral transforms
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Chapter 13 Integral transforms
Parseval