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from Part I - Finite Abelian Groups
Published online by Cambridge University Press: 06 July 2010
We come now to reality. The truth is that the digital computer has totally defeated the analog computer. The input is a sequence of numbers and not a continuous function. The output is another sequence of numbers, whether it comes from a digital filter or a finite element stress analysis or an image processor. The question is whether the special ideas of Fourier analysis still have a part to play, and the answer is absolutely yes.
G. Strang [1986, p. 290]First we consider the easiest kind of Fourier analysis – that on the additive group ℤ/nℤ, the integers modulo n. This is an abelian group of order n and it is cyclic (generated by the congruence class 1 mod n). Thus it is the simplest possible group for Fourier analysis. Yet it seems to have the most applications. As we saw in the last chapter, it may be viewed as the multiplicative group of nth roots of unity. This can be drawn as n equally spaced points on a circle of radius 1. Thus ℤ/nℤ is a finite analogue of the circle (or even of the real line).
The discrete Fourier transform on ℤ/nℤ, or DFT, arises whenever anyone needs to compute the classical Fourier series and integrals of sines and cosines. In fact, the first application of the discrete Fourier transform was perhaps A.-C. Clairaut's use of it in 1754 to compute an orbit, which can be considered as a finite Fourier series of cosines.
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