FOURIER THEORY

INTRODUCTION Fourier series started life as a method to solve problems about the flow of heat through ordinary materials. Now It has grown so far that It is a tool in abstract analysis and electromagnetism and statistics and radio communication and . . . . People have even tried to use it to analyze the stock market. (It didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible example. It provides an indispensible tool in solving partial differential equations

The power series or Taylor series is based on the idea that you can write a general function as an infinite series of powers. The idea of Fourier series is that you can write a function as an infinite series of sines and cosines. You can also use functions other than trigonometric ones

In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.

We can understand some problems in a better way if we look at look at them in a different manner so we use Fourier series to change signal from time domain to frequency domain But we can find Fourier series for only periodic signals to change aperiodic signal from frequency domain we use another tool named as Fourier transform.

APPILICATIONS OF FOURIER SERIES: It can be used to approximate any signal

For example we can decompose our voice signal into sum of orthogonal signals It turns out that (almost) any kind of a wave can be written as a sum of sines and cosines. So for example, if I was to record your voice for one second saying something, I can find its Fourier series which may look something like this for example voice=sin(x)+110sin(2x)+1100sin(3x)+...voice=sin⁡(x)+110sin⁡(2x)+1100sin⁡(3x)+...

and this interactive module shows you how when you add sines and/or cosines the graph of cosines and sines becomes closer and closer to the original graph we are trying to approximate. The really cool thing about Fourier series is that first, almost any kind of a wave can be approximated. Second, when Fourier series converge, they converge very fast.

So one of many applications is compression So one of many applications is compression. Everyone's favorite MP3 format uses this for audio compression. You take a sound, expand its Fourier series. It'll most likely be an infinite series BUT it converges so fast that taking the first few terms is enough to reproduce the original sound. The rest of the terms can be ignored because they add so little that a human ear can likely tell no difference. So I just save the first few terms and then use them to reproduce the sound whenever I want to listen to it and it takes much less memory. JPEG for pictures is the same idea.

Signal Processing: It may be the best application of Fourier analysis. Approximation Theory: We use Fourier series to write a function as a trigonometric polynomial. Control Theory: The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution. Partial Differential equation: We use it to solve higher order partial differential equations by the method of separation of variables.

Another variation of the Fourier Series to compare DNA sequences is A Novel Method for Comparative Analysis of DNA Sequences which used Ramanujan-Fourier series. The idea is the same as the Fourier series, but with a different orthogonal basis (Fourier has a basis of trig functions, R-F uses Ramanujan sums). Other orthogonal basis are Walsh–Hadamard functions, Legendre polynomials, Chebyshev polynomial, etc

Fourier series is broadly used in telecommunications system, for modulation and demodulation of voice signals, also the input, output and calculation of pulse and their sine or cosine graph. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution.

Hiss and pop in sound recordings can be cleaned up using Fourier analysis. What is static but a super-high frequency sound--higher than most sounds that normally appear in music and speech, etc. When a time-domain signal is represented in the frequency domain, i.e., as a sum of sine waves, you can cure the static by simply erasing all the highest frequencies and then reconstituting the sound. exactly the same trick works in removing speckles from photographs. The boundaries between the photo and the speckles are the highest frequency components of the image. Using Fourier analysis to you can drop all the highest frequency components. Then reconstitute the picture, and like magic, speckles are gone. Some minor features you might want will be gone too, but in practice it works very well.

Applications of Fourier transform: Designing and using antennas · Image Processing and filters · Transformation, representation, and encoding · Smoothing and sharpening · Restoration, blur removal, and Wiener filter · Data Processing and Analysis · Seismic arrays and streamers · Multibeam echo sounder and side scan sonar · Interferometers — VLBI — GPS · Synthetic Aperture Radar (SAR) and Interferometric SAR (InSAR) · High-pass, low-pass, and band-pass filters · Cross correlation — transfer functions — Coherence · Signal and noise estimation — encoding time series.

The Fourier transform has many applications, in fact any field of physical science that uses sinusoidal signals, such as engineering, physics, applied mathematics, and chemistry, will make use of Fourier series and Fourier transforms. It would be impossible to give examples of all the areas where the Fourier transform is involved, but here are some examples from physics, engineering, and signal processing.

Communications In communications theory the signal is usually a voltage, and Fourier theory is essential to understanding how a signal behaves when it passes through filters, amplifiers and communications channels. Even discrete digital communications which use 0's or 1's to send information still have frequency contents. This is perhaps easiest to grasp in the case of trying to send a single square pulse down a channel. The field of communications spans a range of applications from high-level network management down to sending individual bits down a channel. The Fourier transform is usually associated with these low level aspects of communications.

ASTRONOMY: The Fourier transform is not just limited to simple lab examples. When used in real situations it can have far reaching implications about the world around us. Take for example the field of astronomy. Some times it isn't possible to get all the information you need from a normal telescope and you need to use radio waves or radar instead of light. These radar signals are treated just like any other ordinary time varying voltage signal and can be processed digitally.

GEOLOGY: The FFT and Nuclear Explosions Sesmic research has always been a common user for the Discrete Fourier Transform (and the FFT). If you look at the history of the FFT you will find that one of the original uses for the FFT was to distinguish between natural seismic events and nuclear test explosions because they generate different frequency spectra.

OPTICS: In electromagnetic theory, the intensity of light is proportional to the square of the oscillating electric field which exists at any point in space. The Fourier transform of this signal is the equivalent of breaking the light into it's component parts of the spectrum, a mathematical spectrometer. One simple example application of the Fourier transform in optics is the diffraction of light when it passes through narrow slits. The ideas represented here can be equally applied to acoustic, x-ray, and microwave diffraction, or any other form of wave diffraction.

SUBMITTED BY Y.PRAVEEN-16075A0405 D.YOGESH-16075A0411 A POWER POINT PRESENTATION ON HISTORY AND APLLICATIONS OF FOURIER ANALYSIS SUBMITTED BY Y.PRAVEEN-16075A0405 D.YOGESH-16075A0411

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