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FOURIER ANALYSIS
Fourier analysis can be defined as the study of the way general functions may be represented or approximated by sums of simpler Trigonometric functions and so it grew from the study of the Fourier series. This is used to characterize a function as a sum of trigonometric functions which greatly simplifies the study of heat propagation.
This subject encompasses a vast spectrum of mathematics as well as the sciences and engineering. The process of simplifying a function into simpler pieces is often called Fourier analysis and the operation of rebuilding the function from these pieces is known as Fourier synthesis.
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In mathematics, it is often called the study of both operations where the decomposition process itself is called a Fourier transform where the transform is often given a more specific name. It depends on the domain as well as other properties of the function that is being transformed and the original concept of these has been extended over time to apply to more and more abstract and general situations.
The general field is often called Harmonic analysis and each transform is used for analysis that has a corresponding inverse function that is used to transform which can be used for synthesis.
The Fourier transform of a periodic function, sP(t), with period P is converted into a Dirac comb function. This is altered by a sequence of complex coefficients which is denoted as follows
for all integer values of k. is the integral over any interval of length P.
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The inverse transform is also called as Fourier series which is a representation of sP(t) in terms of a summation of a potentially infinite number of harmonically related sinusoids each with an amplitude and phase specified by one of the coefficients as denoted below
Where sP(t), is expressed as a periodic summation of another function, s(t) which is as follows
And the coefficients are in relation to the samples of S(ƒ) at discrete intervals which are denoted as
A sufficient condition for recovering s(t) from just these samples is that the non-zero portion of s(t) be confined to a known interval of duration P. This forms the frequency domain dual of the Nyquist–Shannon sampling theorem.
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