18-07-2013, 04:28 PM
Review of Transforms
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Introduction
Every thing in the world can be described via a waveform - a function of time, space or some other variable. For instance, sound waves, electromagnetic fields, the elevation of a hill versus location, the price of favorite stock versus time, etc. The Fourier Transform gives a unique and powerful way of viewing these waveforms.
The Fourer Transform decomposes a waveform - basically any real world waveform, into sinusoids. That is, the Fourier Transform gives another way to represent a waveform. Any space or time varying data can be transformed into a different domain called the frequency space. A fellow called Joseph Fourier first came up with the idea in the 19th century, and it was proven to be useful in various applications, mainly in signal processing.
Frequency domain offers some attractive advantages for image processing. It makes large filtering operations much faster, and it collects information together in different ways that can sometimes separate signal from noise or allow measurements that would be very difficult in spatial domain. Furthermore, the Fourier transform makes it easy to go forwards and backwards from the spatial domain to the frequency space. For example, consider an image with some periodic noise that need to be eliminated. Just imagine a photocopied image with some dirty gray-ish spots in a regular pattern. If we convert the image data into the frequency space, any periodic noise in the original image will show up as bright spots. If we “block out” those points and apply the inverse Fourier transform to get the original image, we can remove most of the noise and improve visibility of that image.
List of Fourier-related transforms
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. These transforms are generally designed to be invertible. In the case of the Fourier transform, each basis function corresponds to a single frequency component.
Fourier series
In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a 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.
The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called Eigen solutions. Fourier's idea was to model a complicated heat source as a superposition of simple sine and cosine waves, and to write the solution as a superposition of the corresponding Eigen solutions. This superposition or linear combination is called the Fourier series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality
Fourier transform
In mathematics, the Fourier transform is the operation that decomposes a signal into its constituent frequencies. Thus the Fourier transform of a musical chord is a mathematical representation of the individual notes that make it up. The original signal depends on time, and therefore is called the time domain representation of the signal, whereas the Fourier transform depends on frequency and is called the frequency domain representation of the signal. The term Fourier transform refers both to the frequency domain representation of the signal and the process that transforms the signal to its frequency domain representation.
Discrete Time Fourier Transform
In mathematics, the discrete-time Fourier transform (DTFT) is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the "DTFT", of the original function. But the DTFT requires an input function that is discrete. Such inputs are often created by sampling a continuous function, like a person's voice.