Before the 1960s, Fourier transforms for newly discovered functions that were based on experimental data or not found in reference tables necessitated an intimidating and tedious amount of arithmetic calculations: for every n data points (usually well over 1,000 for many studies), approximately n additions and n 2 multiplications were needed to perform the transform.
Fourier transform equation series#
In these cases, a Fourier series is calculated by the sum of a series of sine or cosine functions at discrete frequencies.Įven though Fourier derived this method of analysis in the early 1800s, its general applicability to solving scientific problems in an efficient manner was greatly limited until the advent of modern electronic computation. A second type of function that the Fourier transform can be applied is one consisting of numerous discrete values, a common form of data obtained from scientific experimentation. In these cases, the original function is deconstructed into component sinusoidal functions at every frequency, and these are combined by the Fourier integral operation. First, it can be used to describe continuous functions – functions providing values for every real number. Today, the Fourier transform can be applied in two different ways. In fact, during one of Fourier’s research presentations, a contemporary French mathematician Joseph Louis Lagrange reportedly exclaimed that his ideas were “nothing short of impossible.”Īlthough Lagrange himself made a variety of mathematical contributions during the 1800s that greatly aided modern studies of astronomy and economics, the ubiquitous power of the Fourier transform in the modern mathematical world indicates that his doubts were misguided. Not surprisingly, these claims were met with heavy scrutiny. He proposed that discontinuous functions, such as temperature distributions, could be described by combining many continuous functions for example, an infinite number of sinusoids could represent any function, including one with multiple jumps or discontinuities. Professor Coifman explains, “The time vibrations of any mechanical system is a combination of sines and cosines.” However, during the early 1800s, Fourier’s claims were radical. With today’s conceptions of mathematics and physics, these claims seem natural. His premise was that the maximum temperature and position of the harmonics of these sinusoidal components could be derived via the Fourier transform of the originally irregular distribution of temperatures. He proposed that the irregular distribution of temperature could be described by the frequencies of many component sinusoidal waves around the ring. The question itself was complicated Fourier wanted to solve his equation to describe the flow of heat around an iron ring that attaches a ship’s anchor to its chain. Within seven years, he invented the Fourier transform to solve this equation. During this time, his obsession with heat transfer drove him to derive an equation describing the conduction of heat in solid bodies. During Napoleon’s expansion campaigns, Fourier served on the Institute of Egypt’s scientific body in 1800, and after the French left Egypt, he set his efforts on repairing France from the devastation of the 1789 French Revolution. In fact, many of his guests often complained that he kept his home uncomfortably warm. After years of research, French Baron Jean-Baptiste-Joseph Fourier uncovered this powerful tool in the early 1800s, naming it the Fourier transform.įourier, a French military scientist, became interested in heat transfer in the late 1790s. Hence it was only a matter of time until the human intellect caught up to our internal processing systems and was able to functionally describe this process. This tool is truly ubiquitous in nature, as our eyes and ears have subconsciously performed the Fourier transform to interpret sound and light waves for millions of years. “To form an image on your retina, the lens in your eye performs Fourier transformations on the light that enters it,” he explains. Professor Peter Moore, a Yale structural biologist and professor of biophysics, agrees.
Described as “nature’s way of analyzing data” by Yale professor Ronald Coifman, the Fourier Transform is arguably the most powerful analytical tool in modern mathematics.
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