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Fourier Transformsand Their Use in
Data Compression
By Joseph Gehring
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What is a Fourier Transform?
From Simple Wikipedia:
AFourier transformis a math
function that makes a sometimes lessuseful function into another more
useful function.
A Fourier transform really just showsyou what frequencies are in a signal.
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The Math
The Fourier Transform is a generalization of
the Fourier Series
Any periodic function can be represented as
an infinite sum of sines and cosines
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Fourier Series
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Fourier Transform
Forward
Inverse
Symmetric Linear Transform
a=0, b=-2*pi for signal processing
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Fourier Transform
Every function f(x) has a forward and inverse
Fourier Transform such that
Given:
Integral of f(x) exists Discontinuous at a finite number of points
Function has a bounded variation
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Discrete Fourier Transform
For given input data:
Reveals periodic elements
Shows the relative strength of those periodic
elements
Input sequence of real numbers results in
Fourier Transform output of complex numbers
Efficiently computed using Fast Fourier
Transform
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Some Clarification
Fourier Series uses an infinite sum of sinesand cosines
Fourier Transform uses an integral over an
infinite range to develop an approximation Discrete Fourier Transform uses a finite sum of
sines and cosines over a given range, based on
sampling rates and sample length In music, the sample rate is usually set to 44,100
samples/second based on CD quality
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Approximating a Square Wave
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Fast Fourier Transform
Efficient algorithm reducing the number of
computations required to determine the
discrete Fourier Transform of a function from
O(n^2) to O(n*log2(n))
Has been used in mp3 and JPG compression
Ultimately, even the FFT could not compete
with the Discrete Cosine Transform, which is
the cosine portion of the Fourier Transform,
and uses only real values
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Compression
The compression ratio offered by use of the
Fourier Transform is dependent on the quality
required by the application
The higher quality the result needs to be, the
lower the compression ratio will be
To create a more accurate output, more
coefficients are needed and the data cannot
be compressed as significantly
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MP3
Input file is sampled, usually at 44.1 kHz, and thefile is split into chunks of 576 samples each(~.013 seconds)
FFT or DCT is performed to convert time domain
to frequency domain Frequencies outside range of human hearing are
removed
Coefficient data is stored in conjunction with a32-bit header containing sound quality (frame)
Multiple frames are combined to make a singlemp3 file
http://www.indiana.edu/~acoustic/s522/fourapdkp.html
http://www.indiana.edu/~acoustic/s522/fourapdkp.htmlhttp://www.indiana.edu/~acoustic/s522/fourapdkp.html8/13/2019 11 Gehring
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JPGThe original image is broken up into
64 pixel blocks, each 8x8 pixels.
The DCT is taken of each 8x8 group
using a set of 64 basis functions.
Each numerical value in the group is
replaced with a new, smaller
number representing a coefficient
for a basis function.
Because these numbers are smaller,
the number of bits required to
represent them can be reduced. So,
each value in the group is truncated
to a lower number of bits.
By storing this lower number of bits
instead, the total amount of
information is compressed.
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JPG (contd)
In this image, we see how many coefficients
are required to achieve an approximation of
the original image. Using 15 coefficients for
the Fourier Transform instead of 64 originalvalues, a good
approximation
can be madeof the initial
image
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JPG (Contd)
The last step of JPG compression involves the
use of Huffman Encoding, which is a form of
variable bit length encoding that uses fewer
bits to represent values that occur morefrequently than those that occur more rarely.
The 64 encoded values are then converted to
a linear sequence of values rather than anarray
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In Conclusion
All these methods have undergone periodic
updates depending on the complexity of input
data and the computing power available to
perform the tasks.
As storage space becomes cheaper,
compression ratios can become less strict to
create closer approximations to originalinformation
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This cat has some serious periodic components.
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Works CitedFourier Transform. Simple Wikipedia. Web. 04 April 2011.
http://simple.wikipedia.org/wiki/Fourier_transform.
Various Articles.Wolfram MathWorld: The Webs Most Extensive MathematicsResource. Web. 03 April 2011. http://mathworld.wolfram.com.
MP3. Wikipedia, The Free Encyclopedia. Web. 04 April 2011.http://en.wikipedia.org/wiki/MP3.
Smith, Steven W. JPEG (Transform Compression). The Scientist and Engineers Guideto Digital Signal Processing. Web. 04 April 2011.http://www.dspguide.com/ch27/6.htm.
Yoo, Yerin. Tutorial on Fourier Theory. Department of Computer Science. University
of Otago. Web. 05 April 2011.http://www.cs.otago.ac.nz/cosc453/student_tutorials/fourier_analysis.pdf
Handley, Mark. 3: Fourier Transforms. Department of Computer Science. ColumbiaUniversity. Web. 05 April 2011.http://www.cs.columbia.edu/~hgs/teaching/ais/slides/03-fourier.pdf
Munroe, Randall. Fourier. xkcd. Web. 05 April 2011. http://xkcd.com/26.
http://www.dspguide.com/ch27/6.htmhttp://www.dspguide.com/ch27/6.htmhttp://www.dspguide.com/ch27/6.htmhttp://www.cs.otago.ac.nz/cosc453/student_tutorials/fourier_analysis.pdfhttp://www.cs.otago.ac.nz/cosc453/student_tutorials/fourier_analysis.pdfhttp://www.cs.columbia.edu/~hgs/teaching/ais/slides/03-fourier.pdfhttp://www.cs.columbia.edu/~hgs/teaching/ais/slides/03-fourier.pdfhttp://www.cs.columbia.edu/~hgs/teaching/ais/slides/03-fourier.pdfhttp://www.cs.columbia.edu/~hgs/teaching/ais/slides/03-fourier.pdfhttp://www.cs.otago.ac.nz/cosc453/student_tutorials/fourier_analysis.pdfhttp://www.dspguide.com/ch27/6.htm