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By applying the Fourier transform and using these formulas, some ordinary differential equations can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb " f ( x ) is smooth if and only if f̂ ( ξ ) quickly falls to 0 for | ξ | → ∞ ."
As was our expectation, the frequency distribution can be separated into two parts. One is t ≤ 0 and the other is t > 0. The white part is the frequency band occupied by x(t) and the black part is not used. Note that for each point in time there is both a negative (upper white part) and a positive (lower white part) frequency component.
Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
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).
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 ...
The response value of the Gaussian filter at this cut-off frequency equals exp(−0.5) ≈ 0.607. However, it is more common to define the cut-off frequency as the half power point: where the filter response is reduced to 0.5 (−3 dB) in the power spectrum, or 1/ √ 2 ≈ 0.707 in the amplitude spectrum (see e.g. Butterworth filter).
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum based on a least-squares fit of sinusoids to data samples, similar to Fourier analysis. [1] [2] Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in the long and gapped records; LSSA mitigates such problems. [3]
Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. [1]: 13–15 Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered.