Fourier Transform Applications
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Transcript of Fourier Transform Applications
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Fourier Transform and Applications
Fourier
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In this lecture..
Transforms Mathematical Introduction
Fourier Transform Time-Space Domain and Frequency Domain Discret Fourier Transform
Fast Fourier Transform Applications
Summary References
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Transforms
Example of a substitution: Original equation: x + 4x² – 8 = 0 Familiar form: ax² + bx + c = 0 Let: y = x² Solve for y x = ±√y
4
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Transforms
Transform: In mathematics, a function that results when a
given function is multiplied by a so-called kernel function, and the product is integrated between suitable limits.
Can be thought of as a substitution
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Transforms
Transforms are used in mathematics to solve differential equations: Original equation
Apply Laplace Transform:
Take inverse Transform: y = Lˉ¹(y)
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Fourier Transform Property of transforms:
A transform takes one function (or signal) and turns it into another function (or signal)
They convert a function from one domain to another with no loss of information
Fourier Transform:
converts a function from the time (or spatial) domain to the frequency domain
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Time Domain and Frequency Domain
Time Domain: Tells us how properties (air pressure in a sound function, for
example) change over time:
Amplitude = 100 Frequency = number of cycles in one second = 200 Hz
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Time Domain and Frequency Domain
Frequency domain: Tells us how properties (amplitudes) change over
frequencies:
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Time Domain and Frequency Domain
Example: Human ears do not hear wave-like oscilations,
but constant tone
Often it is easier to work in the frequency domain
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Time Domain and Frequency Domain
In 1807, Jean Baptiste Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions
In picture: the composition of the first two functions gives the bottom one
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Time Domain and Frequency Domain
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Fourier Transform
Because of the
property:
Fourier Transform takes us to the frequency domain:
The property of euler number is what gives the F transform the power to take us to freq. domain
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Discrete Fourier Transform
In practice, we often deal with discrete functions (digital signals, for example)
Discrete version of the Fourier Transform is much more useful in computer science:
O(n²) time complexity
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Applications of the DFT
Signal analysis Sound filtering Data compression Partial differential equations Multiplication of large integers
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Applications
In image processing: Instead of time domain: spatial domain (normal
image space) frequency domain: spatial to freq domain
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Summary
Transforms: Useful in mathematics (solving DE)
Fourier Transform: Lets us easily switch between time-space domain
and frequency domain so applicable in many other areas
Easy to pick out frequencies Many applications
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References
Concepts and the frequency domain http://www.spd.eee.strath.ac.uk/~interact/fourier/concepts.html
THE FREQUENCY DOMAIN Introduction http://www.netnam.vn/unescocourse/computervision/91.htm
JPNM Physics Fourier Transform http://www.med.harvard.edu/JPNM/physics/didactics/improc/intro/fourier2.html
Introduction to the Frequency Domain http://zone.ni.com/devzone/conceptd.nsf/webmain/
F814BEB1A040CDC68625684600508C88 Fourier Transform (Efunda)
http://www.efunda.com/math/fourier_transform/