1. Fourier Transforms and Their Use in Data Compression
By Joseph Gehring

2. What is a Fourier Transform?
From Simple Wikipedia:
“A Fourier transform is a math function that makes a sometimes less useful function into another more useful function.”
“A Fourier transform really just shows you what frequencies are in a signal.”

3. 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

4. Fourier Series

5. Fourier Transform Forward Inverse Symmetric Linear Transform
a=0, b=-2*pi for signal processing

6. 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

7. 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. Some Clarification
Fourier Series uses an infinite sum of sines and 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

9. Approximating a Square Wave

10. 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

11. 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

13. MP3
Input file is sampled, usually at 44.1 kHz, and the file 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 a 32-bit header containing sound quality (frame)
Multiple frames are combined to make a single mp3 file

14. JPG
The 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.

15. JPG (cont’d)
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 original values, a good approximation can be made of the initial image

16. JPG (Cont’d)
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 more frequently than those that occur more rarely.
The 64 encoded values are then converted to a linear sequence of values rather than an array

17. 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 original information

18. This cat has some serious periodic components.

19. Works Cited
“Fourier Transform.” Simple Wikipedia. Web. 04 April Various Articles. Wolfram MathWorld: The Web’s Most Extensive Mathematics Resource. Web. 03 April “MP3.” Wikipedia, The Free Encyclopedia. Web. 04 April Smith, Steven W. “JPEG (Transform Compression).” The Scientist and Engineer’s Guide to Digital Signal Processing. Web. 04 April Yoo, Yerin. “Tutorial on Fourier Theory.” Department of Computer Science. University of Otago. Web. 05 April Handley, Mark. “3: Fourier Transforms.” Department of Computer Science. Columbia University. Web. 05 April Munroe, Randall. “Fourier.” xkcd. Web. 05 April