Fast Fourier Transform (FFT)
The impact of this in 1965 invented linear operation is unbelievable big. Each laptop calculates for example 250 thousand FFTs per second! And if you would like to multiply two large numbers you also use FFT. JPEG uses FFT as well.
With Fourier you write the signal as sum of everlasting harmonic functions. That is quite strange, when the signal (e.g. a piece of music) is finitely long. You can look at wavelets as an extension of Fourier, but then one that is closer to the musical notation: the building blocks are of finite length, but we still have limited frequencies. We illustrate wavelets for images and we shall see that spectacular compression ratios can be achieved.
This is maybe the most beautiful example of the force of mathematical modeling. A file with only zeros is easy to compress and if the zeros and ones are alternating often, the compressing becomes more difficult. But how can we understand this mathematically? In this part we provide the basis of the information theory of Claude Shannon. We will see that there is a natural measure for the lack of structure called entropy and that this entropy is equal to the optimal compression ratio.
You would also like to send files, say, from your router to a laptop and you have a limited amount of information that you can send per unit time. That is what we call (channel)capacity. How do you model this and optimize it? This are we going to cover too and we make the connection with entropy.
This course will take place in the second quartile of your second year. It will be given by Gjerrit Meinsma.
Gjerrit Meinsma was born on the 29th of January 1965 in Opeinde, a little village in the northern part of The Netherlands. There he attended the nursery school ``De Kindertuin'' and elementary school. In the off-school hours he spent many hours making and throwing boomerangs at and with his friends, and in the weekends he used to play korfball. From the age of 12 till the age of 18 he attended the grammar-school ``Het Drachtster Lyceum'' in the nearby town Drachten. Somewhere in the middle of this period the fascinating world of music caught his attention. Listening to music and playing the piano have been his major hobbies since. At the same time Gjerrit developed a keen interest in mathematics, and he decided to pursue his career in this direction. In 1983 he went to Enschede to study applied mathematics at the University of Twente. At the end of 1988 he finished his master's thesis entitled ``Chebyshev approximation by free knot splines'' and in March 1989 he received his master's degree. After a break of five weeks he returned to the faculty he graduated with, only this time as a research assistant (AiO) with the Systems and Control Group. Half a year later he bought himself a brand new Klug & Sperl upright piano, and a few years later, in 1993, he finished his thesis on H-infinity control. Not long thereafter he dumped his winter coat in the garbage bin and left for sunny Australia were he worked as a post-doc for three years with the Electrical Engineering Department at the University of Newcastle. In January 1997 he returned to Enschede to take up a two-year post-doc position with his former group. He quickly acquired a much needed winter coat. Since 1999 he holds a permanent position at this group.
Gjerrit likes all sorts of music, but he is fanatic only about classical music. His favorite composers are Bach, Beethoven, Schubert, Chopin, Grieg, Ravel, Skriabin and Janacek. He is a fan of the pianist Sviatoslav Richter, and also likes very much Sofronitsky, Lipatti, Rubinstein, Oborin and Henkemans and many others.