Search results
Results from the WOW.Com Content Network
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. [note 1] For example, many relatively simple applications use the Dirac delta function, which can be treated formally as ...
The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT). The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform: [b]
The DFT is (or can be, through appropriate selection of scaling) a unitary transform, i.e., one that preserves energy. The appropriate choice of scaling to achieve unitarity is /, so that the energy in the physical domain will be the same as the energy in the Fourier domain, i.e., to satisfy Parseval's theorem.
The 2D Z-transform, similar to the Z-transform, is used in multidimensional signal processing to relate a two-dimensional discrete-time signal to the complex frequency domain in which the 2D surface in 4D space that the Fourier transform lies on is known as the unit surface or unit bicircle.
Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces.
Analogously to the classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph Fourier transform is important in spectral graph theory. It is widely applied in the recent study of graph structured learning algorithms, such as the widely employed convolutional networks.
The Fourier transform can therefore be seen to relate the coefficients and the values of a polynomial: the coefficients are in the time-domain, and the values are in the frequency domain. Here, of course, it is important that the polynomial is evaluated at the n th roots of unity, which are exactly the powers of α {\displaystyle \alpha } .
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain).