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In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency.
For a two dimensional phase retrieval problem, there is a degeneracy of solutions as () and its conjugate () have the same Fourier modulus. This leads to "image twinning" in which the phase retrieval algorithm stagnates producing an image with features of both the object and its conjugate. [3]
This generalizes the Fourier transform to all spaces of the form L 2 (G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the [− π , π ] case.
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...
The two amplitude-modulated sinusoids are known as the in-phase (I) and quadrature (Q) components, which describes their relationships with the amplitude- and phase-modulated carrier. [ A ] [ 2 ] Or in other words, it is possible to create an arbitrarily phase-shifted sine wave, by mixing together two sine waves that are 90° out of phase in ...
Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform. In applications of the Fourier transform the Fourier inversion theorem often plays a critical role. In many situations the basic ...
The modulus of the number is the amplitude of that component, and the argument is the relative phase of the wave. For example, using the Fourier transform, a sound wave, such as human speech, can be broken down into its component tones of different frequencies, each represented by a sine wave of a different amplitude and phase. The response of ...
The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which changes the relative magnitudes and/or angles ( phase ) of the non-zero values of S(f).