The continuous Fourier transform is a linear operator which maps functions to other functions. Loosely, the Fourier transform decomposes a function into a continous spectrum of the frequencies that comprise that function. In mathematical physics, the Fourier transform of a signal f(t) can be thought of as that signal in the "frequency ___domain". This is similar to the basic idea of the various other Fourier transforms inclucing the Fourier series of a periodic function.
A number of slightly different but essentially equivalent definitions are used in the literature. Suppose f : R -> C is a Lebesgue integrable function. We then define its continuous Fourier transform F : R -> C as
- F(s) = ∫ f(t) e-2πist dt
for every real number s. (Here, π is pi and i is the imaginary unit). We think of s as a frequency and F(s) as the complex number which encodes amplitude and phase of the signal f(t) at that frequency.
The Fourier transform is close to self-inverse: if F(s) is defined as above, and f is sufficiently smooth, then
- f(t) = ∫ F(s) e2πist ds
for every real number t.
As a rule of thumb: the more concentrated f(t) is, the more spread out is F(s). The only functions which coincide with their own Fourier transforms are the constant multiples of the function f(t) = exp(-πt2). In a certain sense, this function therefore strikes the precise balance between being concentrated and being spread out.
Fourier transforms, and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if f(t) is a differentiable function with Fourier transform F(s), then the Fourier transform of its derivative is given by 2πis F(s). This can be used to transform differential equations into algebraic equations. Furthermore, the Fourier transform translates between convolution and multiplication of functions: if f(t) and g(t) are integrable functions with Fourier transforms F(s) and G(s) respectively, and if the convolution of f and g exists and is integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms F(s) G(s). If the the product f(t) g(t) is integrable, then the Fourier transform of this product is given by the convolution of F(s) and G(s).
The most general and natural context for studying the continuous Fourier transform is given by the tempered distributions; these include all the integrable functions mentioned above and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution and the rule for the inverse of the Fourier transform is universally valid. Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is the constant function 1. The above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.
The following table records some important Fourier transforms. F(s) and G(s) denote the Fourier transforms of f(t) and g(t), respectively. f and g may be integrable functions or tempered distributions.
| Signal | Fourier transform | Remarks |
|---|---|---|
| af(t) + bg(t) | aF(s) + bG(s) | Linearity |
| f(t - a) | e-2πias F(s) | Shift |
| e2πiatf(t) | F(s-a) | Shift |
| f(at) | 1/|a| F(s/a) | |
| f'(t) | 2πis F(s) | f'(t) is the derivative of f(t) |
| t f(t) | 1/(2πi) F'(-s) | |
| (f * g)(t) | F(s) G(s) | f * g denotes the convolution of f and g |
| f(t) g(t) | (F * G)(-s) | |
| δ(t) | 1 | δ(t) denotes the Dirac delta distribution |
| 1 | δ(s) | |
| δ(t-a) | e-2πias | |
| tn | 1/(-2πi)n &delta(n)(-s) | (needs to be checked) &delta(n)(s) is the n-th distribution derivative of the Dirac delta |
| e2πiat | δ(s-a) | |
| cos(2πat) | 1/2 ( δ (s - a) + δ(s + a) ) | |
| exp(-a t2) | (π/a)1/2 exp(-π2 s2 / a) |
See also: Fourier transform, Fourier series, Laplace transform, Discrete Fourier transform