Fourier Transforms and Their Use in Data Compression By Joseph Gehring

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.”

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

Fourier Series

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

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

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

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

Approximating a Square Wave

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

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

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 http://www.indiana.edu/~acoustic/s522/fourapdkp.html

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.

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

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

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

This cat has some serious periodic components.

Works Cited “Fourier Transform.” Simple Wikipedia. Web. 04 April 2011. http://simple.wikipedia.org/wiki/Fourier_transform. Various Articles. Wolfram MathWorld: The Web’s Most Extensive Mathematics Resource. 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 Engineer’s Guide to 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. Columbia University. 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.