What - in plain english - is a Fourier Transform? No Wikipedia please.?

Answer 1

INTUITIVE ANSWER:

If f(x) is the plot of a sound wave on the oscilloscope, then the Fourier transform F(w) is the plot of this same wave on the display of an equalizer: the wave is analyzed as the sum of waves with different frequencies.


MORE DETAIL:

The Fourier transform of a function f(x) with real domain and complex codomain describes how the function can be approximated as the sum of infinitely many (complex) sine curves.

Technically, the Fourier transform of f(x) is a function F(w). For every "frequency" w, F(w) describes the complex amplitude of the wave exp(iwx). Adding all these waves together (or rather, taking the integral), gives the original function:

f(x) = INT F(w) exp(iwx) dx

The marvelous thing is that the Fourier transform can be calculated with a similar formula!

F(w) = (1/2pi) * INT f(x) exp(-iwx) dx

Fourier Transforms are applied in many fields. For instance, in speech analysis, we are not interested in the precise shape of the sound wave, but we want to know which frequencies are present. This information is explicity stored in the Fourier transform.

Answer 2

Suppose that you have a CD with a song on it. The CD gives the amplitude of the signal at various instants in time. Now suppose that you would like to know the frequencies of the notes of the song -- the F sharps, B flats, et cetera. A Fourier transform is a mathematical procedure for converting data in time space to data in frequency space. Or vice versa -- you can go either way. A very important use of the technique is in Computer Aided Tomography, when the x-ray signals are transformed to give a picture. The technique became much easier to use back about 1960 when someone figured out a trick to drastically reduce the amount of computation needed to do it. With today's much faster processors, the amount of computation is no longer an issue.

Answer 3

The Fourier tranform is an operation that takes a complex wave or signal and converts it into a set of simpler pieces that, when added together, give you the original signal back.

Sometimes it take an infinite series or number of steps to do it, each getting a little closer to the ultimately correct answer. We can describe it as the sum of all the harmonics that recreate the waveform of interest. Using simple sine waves as a starting point, if we select the proper frequencies and amplitudes, we can reconstruct any waveform, no matter how complex. Fourier can yield the values for the frequencies and harmonics.

Answer 4

Essentially an image is analyzed by the geometric/trigonometric relationships of the color frequencies it contains and the resulting image is a visual representation of those relationships.

The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent. In the Fourier domain image, each point represents a particular frequency contained in the spatial domain image.

The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression.

Answer 5

A Fourier transform is a mapping from one vector space of functions into another. Certain classes of problems can be solved by mapping the function via Fourier transform, solving the problem, and then mapping the solution back.

It's the same idea as switching from rectangular to spherical coordinates to simplify a problem.

Answer 6

taking a function and representing it by the sum of a series of sine waves with ever increasing frequencies. each individual sine wave will have a different magnitude coefficient depending on the strength of that particular frequency in the original function.

fourier transforms are used in a variety of ways in signal analysis, surface metrology, and the solution of differential equations, among others