The
Fourier
transform,
named after Jean
Baptiste Joseph Fourier (French mathematician who lived
between 1768 and 1830), is an almost magical mathematical tool that
decomposes any periodic
function of time (or periodic in space) into a sum of sinusoidal basis
functions (frequency dependent), similarly to how a
musical chord can be expressed as the amplitude (loudness) of its
constituent notes.
The
term Fourier transform
refers to both the
frequency domain representation and the mathematical operation that
associates the frequency domain representation to the corresponding
function of time (or space).
C
major chord (=C+E+G)
A good example is what the human
ear makes as
it receives an audio waveform and transforms it, breaking it down into
different frequencies (which is what ultimately is heard). The human
ear will perceive different frequencies as time passes, however, the Fourier transform
contains all frequencies in the time during which the signal exists;
i.e., through the Fourier transform of a function of time we get a
unique frequency spectrum for the whole function. In short,
the
Fourier transform of a periodic function over time, is
basically
the frequency spectrum of that function.
Fourier
transform between two functions.
Function f(x),
(1), is a time dependent
function (red line). It is
a sum of six sine functions of different amplitudes and harmonically
related frequencies. Their summation is called a Fourier
series.
The Fourier transform,ˆf(ω), (2), (blue
line), which depicts amplitude vs. frequency,
reveals the 6 frequencies and their amplitudes.
(1)
(2)
Function (2)
is the Fourier
transform of (1)
Function (1) is the inverse Fourier transform
of (2)
Each of these basis functions
into which a function can decompose, is a
complex exponential of a different frequency (ω). The Fourier transform
therefore gives us a unique way of viewing any function as the
sum of simple sinusoids.