978 - 1 - 107 - 03860 - 8 - Foundations of Signal Processing

In the late thirteenth century, Theodoric of Freiberg, a Dominican monk, theologian, and physicist, performed a simple experiment: with his back to the sun, he held a spherical bottle filled with water in the sunlight. By following the trajectory of the refracted and reflected light and having the bottle play the same role as a single water drop, he gave a scientific explanation of rainbows, including secondary rainbows with weaker, reversed colors. His geometric analysis, described in his famous treatise De iride (On the Rainbow, c. 1310), was “perhaps the most dramatic development of fourteenthand fifteenth-century optics” [60]. Theodoric of Freiberg fell short of a complete understanding of the rainbow phenomenon because, like many of his contemporaries, he believed that colors were simply intensities between black and white. A full understanding emerged three hundred years later when René Descartes and Isaac Newton explained that dispersion decomposes white light into spectral components of different wavelengths – the colors of the rainbow. In 1730, Newton, in his landmark book on optics [70], describes what is often called the experimentum crucis (crucial experiment) to prove that white light can be decomposed into constituent colors and then recombined into white light. This experiment is a physical implementation of decomposing light into its Fourier components – pure frequencies or colors of the rainbow, followed by a synthesis to recover the original; the cover photograph of the book depicts this experiment. This bit of history evokes two central themes of this book: geometric thinking