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Time/frequency distribution. The main application of the Gabor transform is used in time–frequency analysis.Take the following function as an example. The input signal has 1 Hz frequency component when t ≤ 0 and has 2 Hz frequency component when t > 0
While the ordinary DFT corresponds to a periodic signal in both time and frequency domains, = / produces a signal that is anti-periodic in frequency domain (+ =) and vice versa for = /. Thus, the specific case of a = b = 1 / 2 {\displaystyle a=b=1/2} is known as an odd-time odd-frequency discrete Fourier transform (or O 2 DFT).
Gaussian functions are the Green's function for the (homogeneous and isotropic) diffusion equation (and to the heat equation, which is the same thing), a partial differential equation that describes the time evolution of a mass-density under diffusion.
More generally, a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation [() + ()] = with constant and () being a non-constant even function remains invariant in form when applying the Fourier transform to both sides of the equation. The simplest example is provided by ...
The development of fast algorithms for DFT was prefigured in Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno.Gauss wanted to interpolate the orbits from sample observations; [6] [7] his method was very similar to the one that would be published in 1965 by James Cooley and John Tukey, who are generally credited for the invention of the modern generic FFT ...
As an example of propagation without dispersion, consider wave solutions to the following wave equation from classical physics =, where c is the speed of the wave's propagation in a given medium. Using the physics time convention, e − iωt , the wave equation has plane-wave solutions u ( x , t ) = e i ( k ⋅ x − ω ( k ) t ...
Examples of pulse shapes: (a) rectangular pulse, (b) cosine squared (raised cosine) pulse, (c) Dirac pulse, (d) sinc pulse, (e) Gaussian pulse. A pulse in signal processing is a rapid, transient change in the amplitude of a signal from a baseline value to a higher or lower value, followed by a rapid return to the baseline value. [1]
Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f : R → C {\displaystyle f:\mathbb {R} \to \mathbb {C} } satisfying certain conditions, and we use the convention for the Fourier ...