Fourier transform
Category:Digital signal processing\nCategory:Numerical analysis\n
The
Fourier transform, named for
Jean Baptiste Joseph Fourier, is an
integral transform that re-expresses a function in terms of
sinusoidal basis functions, i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). There are many closely-related variations of this transform, summarized below, depending upon the type of function being transformed. See also:
List of Fourier-related transforms.
Fourier transforms have many scientific applications — in
physics,
number theory,
combinatorics,
signal processing,
probability theory,
statistics,
cryptography,
acoustics,
oceanography,
optics,
geometry, and other areas. (In signal processing and related fields, the Fourier transform is typically thought of as decomposing a signal into its component
frequencies and their amplitudes.) This wide applicability stems from several useful properties of the transforms:
- The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).
- The transforms are invertible, and in fact the inverse transform has almost the same form as the forward transform.
- The sinusoidal basis functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. (For example, in a linear time-invariant physical system, frequency is a conserved quantity, so the behavior at each frequency can be solved independently.)
- By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.
- Fast algorithms, based on the fast Fourier transform (FFT), exist to evaluate Fourier transforms on computers.
Variants of the Fourier transform
Most often, the unqualified term "Fourier transform" refers to the continuous Fourier transform, representing any square-integrable function f(t) as a sum of complex exponentials with angular frequencies ω and complex amplitudes F(ω):
-
This is actually the
inverse continuous Fourier transform, whereas the Fourier transform expresses
F(ω) in terms of
f(
t); the original function and its transform are sometimes called a
transform pair. See
continuous Fourier transform for more information, including a table of transforms, discussion of the transform properties, the various conventions, etcetera.
The continuous transform is actually a generalization of an earlier concept, a
Fourier series, which was specific to periodic (or finite-domain) functions
f(
x) (with period 2π), and represents these functions as a
series of sinusoids:
-
where is the (complex) amplitude. Or, for
real-valued functions, the Fourier series is often written:
-
where
an and
bn are the (real) Fourier series amplitudes.
For use on computers, both for scientific computation and
digital signal processing, one must have functions
xk that are defined over
discrete instead of continuous domains, again finite or periodic. In this case, one uses the
discrete Fourier transform (DFT), which represents
xk as the sum of sinusoids:
-
where
fj are the Fourier amplitudes. (The discrete Fourier transform can be viewed as a special case of the
Z-transform, evaluated on the unit circle in the complex plane.) Although applying this formula directly would require O(
n2) operations, it can be computed in only O(
n log
n) operations using a
fast Fourier transform (FFT) algorithm (see
Big O notation), which makes Fourier transformation a practical and important operation on computers.
These Fourier variants can also be generalized to Fourier transforms on arbitrary
locally compact abelian topological groups, which are studied in
harmonic analysis; there, one transforms from a group to its
dual group. This treatment also allows a general formulation of the
convolution theorem, which relates Fourier transforms and convolutions. See also the
Pontryagin duality for the generalized underpinnings of the Fourier transform.
Interpretation in terms of time and frequency
In terms of signal processing, the transform takes a
time series representation of a signal function and maps it into a
frequency spectrum, where ω is
angular frequency. That is, it takes a function in the
time domain into the
frequency domain; it is a decomposition of a function into
harmonics of different frequencies.
When the function
f is a function of time and represents a physical
signal,\nthe transform has a standard interpretation as the
spectrum of the signal. The
magnitude of the resulting complex-valued function
F represents the amplitudes of the respective frequencies (ω), while the
phase shiftss are given by
arctan(imaginary parts/real parts).
However, it is important to realize that Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze
spatial frequencies, and indeed for nearly any function domain.
See also
