Fourier transform
The Fourier transform (more precisely, the continuous Fourier transform; pronunciation: fuʁie ) is a method of Fourier analysis, which allows to decompose continuous, aperiodic signals in a continuous spectrum. The function describing this spectrum is also known as Fourier transform or spectral. This integral transform is named after the mathematician Jean Baptiste Joseph Fourier, who introduced the Fourier series in 1822, an analogue of the continuous Fourier transform for periodic signals.
- 4.1 Definition
- 4.2 Hausdorff -Young inequality
- 4.3 for differentiating
- 4.4 Unitary Figure
- 8.1 Square integrable functions
- 8.2 distributions
Definition
Be an integrable function. The (continuous ) Fourier transform of being defined by
Wherein an n-dimensional volume element, and the imaginary unit and is equipped with the standard scalar of the vectors and intended. The normalization constant is not uniform in the literature. In the theory of pseudo-differential operators and in signal processing, it is common to omit the factor so that the inverse transformation is given the pre-factor. The transformation is then:
This has the disadvantage that in the set of Parseval appears a pre-factor, which means that the Fourier transform is then no longer unitary map. In other words, the signal power is then changed by the Fourier transform. In the literature on signal processing and system theory can also be found following Convention, which requires no pre-factors:
The real form of the Fourier transform is known as the Hartley transform. For real functions, the Fourier transform may be substituted by the sine and cosine transform.
Example
As an example, the frequency spectrum of a damped oscillation is to be examined with sufficient low attenuation. These can be described by the following function:
Or in complex notation:
Here, the amplitude and the angular frequency of vibration, the period after which the amplitude is decreased to, and the step function. That is, the function is non-zero only for times positive.
Obtained
Properties
Linearity
The Fourier transform is a linear operator. That is, it is true.
Continuity
The Fourier transform is a continuous operator from the space of integrable functions in the space of functions that vanish at infinity. With the set of continuous functions is called, which disappear. The fact that the Fourier transforms vanish at infinity, is also known as the Riemann- Lebesgue lemma. Moreover, the inequality holds
Differentiation
Be a Schwartz function and a multi- index. Then we have
- And.
- .
Fixed point
The density function
With the ( dimensional ) Gaussian normal distribution is a fixed point of the Fourier transform. That is, it applies to all the equation
In particular, therefore is an eigenfunction of the Fourier transform to the eigenvalue. With the help of the residue theorem or using partial integration and solving an ordinary differential equation of the Fourier integral can be determined in this case.
Mirror symmetry
For the equation is valid for all
Back-transformation formula
Be an integrable function such that even applies. Then, the inverse transform applies
This is also called Fourier synthesis. On the Schwartz space, the Fourier transform is an automorphism.
Convolution theorem
The convolution of the Fourier transform means that the convolution of two functions is transformed by the Fourier transform in its image space into a real number multiplication. So for true
Indicates the inverse of the convolution theorem
Fourier transformation of L2 functions
Definition
For a function, the Fourier transform is defined by a tightness argument by
The convergence is understood in the sense of and the ball around the origin with radius. Functions for this definition is consistent with the first portion of the. Since the Fourier transform with respect to the scalar product is unitary (see below) and is located in close, it follows that the Fourier transform is an isometric of the automorphism. This is the statement of the theorem of Plancherel.
Hausdorff -Young inequality
Let and. For is and it is
Thus, the Fourier transform is a sequel to a continuous operator defined by
Will be described. The limit is to be understood here in the sense of.
Differentiation rule
If the function is weakly differentiable, there is a differentiation rule analogous to those for black features. So be a k- times weakly differentiable L2 - function and multi- index. Then we have
Unitary map
The Fourier transform with respect to the complex scalar product a unitary image, it is applicable
Fourier transform in the space of tempered distributions
Be a tempered distribution, the Fourier transform is defined for all by
Fourier transformation of measurements
The Fourier transform is generally defined for finite Borel measures on:
Is the inverse Fourier transform of the measurement. The characteristic feature is the inverse Fourier transform of a probability distribution.
Partial Differential Equations
In theory, the partial differential equations, the Fourier transform plays an important role. With their help one can find solutions of certain differential equations. The differentiation rule and the convolution theorem are essential. Using the example of the heat equation is then shown how to solve the Fourier transform of a partial differential equation. The initial value problem of the heat equation is
Herein, the Laplace operator, which acts only on the variables. Applying the Fourier transformation to two equations with respect to the variables and applying the results for differentiating
It now is an ordinary differential equation, the solution
Has. It follows and is due to the convolution theorem
It follows with
This is the fundamental solution of the heat equation. Therefore, the solution of the initial value problem considered here has the representation
Table important Fourier transform pairs
In this chapter, a compilation of important Fourier transform pairs.