8/13/2019 11 Gehring

1/19

Fourier Transformsand Their Use in

Data Compression

By Joseph Gehring

8/13/2019 11 Gehring

2/19

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.

8/13/2019 11 Gehring

3/19

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

8/13/2019 11 Gehring

4/19

Fourier Series

8/13/2019 11 Gehring

5/19

Fourier Transform

Forward

Inverse

Symmetric Linear Transform

a=0, b=-2*pi for signal processing

8/13/2019 11 Gehring

6/19

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

8/13/2019 11 Gehring

7/19

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

8/13/2019 11 Gehring

8/19

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

8/13/2019 11 Gehring

9/19

Approximating a Square Wave

8/13/2019 11 Gehring

10/19

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

8/13/2019 11 Gehring

11/19

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

8/13/2019 11 Gehring

12/19

8/13/2019 11 Gehring

13/19

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.html

8/13/2019 11 Gehring

14/19

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.

8/13/2019 11 Gehring

15/19

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

8/13/2019 11 Gehring

16/19

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

8/13/2019 11 Gehring

17/19

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

8/13/2019 11 Gehring

18/19

This cat has some serious periodic components.

8/13/2019 11 Gehring

19/19

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