In mathematics, the Fourier transform (often abbreviated FT) is an operation that transforms one complex-valued function of a real variable into another. In such applications as signal processing, the domain of the original function is typically time and is accordingly called the time domain. That of the new function is frequency, and so the Fourier transform is often called the frequency domain representation of the original function. It describes which frequencies are present in the original function.
For example, a note played on a flute is reasonably pure: a single frequency with only minor overtones, which are additional, related frequencies. It may be envisaged as a simple repeating wave; a sine wave, because of the relative absence of such overtones. But the same note played on, say, an oboe or guitar would have a somewhat shriller sound, caused by a higher level of overtones. It would appear visually more complex, because of the significant presence of such overtones. In this example, a Fourier transform is the representation of the frequencies (and their amplitudes) present in that note of that instrument. For the flute note it would be approximately a single vertical line at the single frequency; for the oboe note there would be additional, smaller (lower amplitude) vertical lines representing the overtones.
The term Fourier transform refers both to the frequency domain representation of a function and to the process or formula that "transforms" one function into the other.
The Fourier transform and its generalizations are the subject of Fourier analysis. In this specific case, both the time and frequency domains are unbounded linear continua. It is possible to define the Fourier transform of a function of several variables, which is important for instance in the physical study of wave motion and optics. It is also possible to generalize the Fourier transform on discrete structures such as finite groups, efficient computation of which through a fast Fourier transform is essential for high-speed computing.
| Fourier transforms |
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| Continuous Fourier transform |
| Fourier series |
| Discrete Fourier transform |
| Discrete-time Fourier transform |
| Related transforms |
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Q. Does it involve convolution? Can you explain it in plain english?
Asked by Ejsenstejn - Thu Jan 25 19:37:51 2007 - - 2 Answers - 0 Comments
A.
Answered by amphora001 - Thu Jan 25 19:47:16 2007
