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Introduction of Fourier analysis

In mathematics, Fourier analysis is the study of the way common functions might be signified or approximated by sums of simpler trigonometric functions. As from the study of Fourier series, Fourier analysis grow and is named after Joseph Fourier who showed that representing a function as a sum of trigonometric functions really simplify the study of heat propagation.

Nowadays, the subject of Fourier analysis encompasses a huge spectrum of mathematics. In engineering and sciences, the process of decomposing a function into simpler pieces is frequently known as Fourier analysis, whereas the procedure of rebuilding the function from these pieces is termed as Fourier synthesis. The word Fourier analysis in mathematics frequently refers to the study of both operations.

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The Fourier analysis has many scientific applications - in diffraction, oceanography, signal processing, partial differential equations, combinatorics, number theory, physics, sonar, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, optics, protein structure analysis etc.

Introduction of Wavelets

A wavelet is a wave-like oscillation with amplitude which begins at zero then increases and then decreases back to zero. It can imagined as a "brief oscillation" like one may see recorded by a seismograph or heart monitor. In general wavelets are purposefully crafted to have particular properties which make them helpful for signal processing. The Wavelets can combine by using a "sum, shift, reverse, & multiply" method known as convolution, with portions of signal that is not known, to extract the information from unknown signal.

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                                                                                  Different Types of Wavelets

As a mathematical tool, the wavelets can be used to extract information from numerous different types of data, involving- but certainly not restricted to -images and audio signals. The sets of wavelets are generally required to analyze data fully. The set of complementary wavelets will deconstruct data without gaps or overlap therefore the deconstruction procedure is mathematically reversible. Hence sets of complementary wavelets are helpful in wavelet based compression or decompression algorithms where it is desirable to recover the original information with minimal loss.

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