Search results
Results from the WOW.Com Content Network
Download as PDF; Printable version; ... a DFT matrix is an expression of a discrete Fourier transform ... For other properties of the DFT matrix, ...
For example, several lossy image and sound compression methods employ the discrete Fourier transform: the signal is cut into short segments, each is transformed, and then the Fourier coefficients of high frequencies, which are assumed to be unnoticeable, are discarded. The decompressor computes the inverse transform based on this reduced number ...
The periodic summation, , along with an -length DFT, can also be used to sample the DTFT at intervals of /. Those samples are also real-valued and do exactly match the DTFT (example: File:Sampling the Discrete-time Fourier transform.svg).
The discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix, where the are chosen to be n th roots of unity. The Fast Fourier transform computes the product of this matrix with a vector in O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} time. [ 9 ]
Many of the standard properties of the Fourier transform are immediate consequences of this more general framework. [33] For example, the square of the Fourier transform, W 2, is an intertwiner associated with J 2 = −I, and so we have (W 2 f)(x) = f (−x) is the reflection of the original function f.
An circulant matrix takes the form = [] or the transpose of this form (by choice of notation). If each is a square matrix, then the matrix is called a block-circulant matrix.. A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of .
Over the complex numbers, it is often customary to normalize the formulas for the DFT and inverse DFT by using the scalar factor in both formulas, rather than in the formula for the DFT and in the formula for the inverse DFT. With this normalization, the DFT matrix is then unitary.
In quantum mechanics, a density matrix (or density operator) is a matrix that describes an ensemble [1] of physical systems as quantum states (even if the ensemble contains only one system). It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule .