Fourier Transfrom

Jean Baptiste Fourier (1768-1830) is a french mathematician and physician. To put it simply, Fourier discovered that the a sound could be discomposed into series, or sums of vibrations, described as periodic functions.

In physics, a phenomenon is considered to be periodic when it exhibits a regular, recurring pattern, repeating itself in a strictly identic way after a given time interval. This time interval, or period, is notated T, and measured in seconds. The number of times, or cycles, the phenomenon is repeated in a second is called frequency, and is expressed in Hertz (Hz).

Sounds, or signals, can be periodic – with a simple flute tone for instance – , or nonperiodic – a percussive sound –. In the first case, the vibrations pattern will repeat identically over a given laps of time. In the second case, this pattern will evolve.

When a signal is periodic, all the vibrations that constitute the signal are synchronized. We have a fundamental frequency, Mathematically, this means that

Fourier analysis was originally concerned with representing and analyzing periodic phenomena, via Fourier series, and laterwith extending those insights to nonperiodic phenomena, via the Fourier transform. In fact, one way of getting from Fourier series to the Fourier transform is to consider nonperiodic phenomena (and thus justabout any general function) as a limiting case of periodic phenomena as the period tends to infinity. Adiscrete set of frequencies in the periodic case becomes a continuum of frequencies in the nonperiodic case, the spectrum is born, and with it comes the most important principle of the subject. Every signal has a spectrum and is determined by its spectrum. You can analyze the signal either in the time (or spatial) domain or in the frequency domain