Fourier Analysis example essay topic

Fourier analysis of spatial and temporal visual stimuli has become common in the last 35 years. For many people interested in vision but not trained in mathematics this causes some confusion. The use of these Fourier methods does not mean that the visual system performs a Fourier analysis. At present it should be understood that this approach is a convenient way to analyze visual stimuli. Jean Baptiste Fourier, a mathematician, showed that any repetitive waveform can be broken down into a series of sine waves at appropriate amplitudes and phases.

Fourier analysis may be performed mathematically if the expression f (t) describing the waveform or complex tone. One of the important ideas for sound reproduction, which arises from Fourier analysis, is that it takes a high quality audio reproduction system to reproduce percussive sounds or sounds with fast transients. The sustained sound of a trombone can be reproduced with a limited range of frequencies because most of the sound energy is in the first few harmonics of the fundamental pitch. A sine wave is a wave of a single frequency. It has a given frequency, amplitude and phase. Fourier discovery can be illustrated through taking an appropriately chosen set of sine waves and add them together to produce a square wave.

Clearly, if it is possible to construct a wave of a particular pattern by adding together appropriately chosen sine waves then the reverse is true as well. The building of complex waves by combining appropriately chosen sine waves is called Fourier synthesis. The breaking apart of a complex wave into its component sine waves is called Fourier analysis. Fourier analysis studies approximations and decompositions of functions using trigonometric polynomials.

Of incalculable value in many applications of analysis, this field has grown to include many specific and powerful results, including convergence criteria, estimates and inequalities, and existence and uniqueness results. Extensions include the theory of singular integrals, Fourier transforms, and the study of the appropriate function spaces. The Fourier approach to analyzing visual stimuli actually comes under the heading of Linear Systems Analysis. Linear transforms, especially Fourier and Laplace transforms, are widely used in solving problems in science and engineering. The Fourier transform is used in linear systems analysis, antenna studies, optics, random process modeling, probability theory, quantum physics, and boundary-value problems (Brigham, 2-3) and has been very successfully applied to restoration of astronomical data (Br ault and White).

The Fourier transform, a pervasive and versatile tool, is used in many fields of science as a mathematical or physical tool to alter a problem into one that can be more easily solved. Some scientists understand Fourier theory as a physical phenomenon, not simply as a mathematical tool. In some branches of science, the Fourier transform of one function may yield another physical function (Brace well, 1-2). The Fourier transform, in essence, decomposes or separates a waveform or function into sinusoids of different frequency which sum to the original waveform. It identifies or distinguishes the different frequency sinusoids and their respective amplitudes (Brigham, 4).

The Fourier transform of f (x) is defined as F (s) = f (x) exp (-i 2 xs) dx. Applying the same transform to F (s) gives f (w) = F (s) exp (-i 2 ws) ds. If f (x) is an even function of x, that is f (x) = f (-x), then f (w) = f (x). If f (x) is an odd function of x, that is f (x) = -f (-x), then f (w) = f (-x).

When f (x) is neither even nor odd, it can often be split into even or odd parts.