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The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable. Envelope for a modulated sine wave.
Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale. Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
Since the integral of ρ t is constant while the width is becoming narrow at small times, this function approaches a delta function at t=0, = again only in the sense of distributions, so that () = for any test function f. The time-varying Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity ...
The Morlet wavelet filtering process involves transforming the sensor's output signal into the frequency domain. By convolving the signal with the Morlet wavelet, which is a complex sinusoidal wave with a Gaussian envelope, the technique allows for the extraction of relevant frequency components from the signal.
Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. [6] If we expect that for "near-by" input points x {\displaystyle x} and x ′ {\displaystyle x'} their corresponding output points y {\displaystyle y} and y ′ {\displaystyle y'} to be "near-by ...
Donsker's theorem states that the empirical distribution function, when properly normalized, converges weakly to a Brownian bridge—a continuous Gaussian process. This is significant as it assures that results analogous to the central limit theorem hold for empirical processes, thereby enabling asymptotic inference for a wide range of ...
Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [ 1 ] [ 2 ] A stationary Gauss–Markov process is unique [ citation needed ] up to rescaling; such a process is also known as an Ornstein–Uhlenbeck ...
A non-trivial way to mix the latent functions is by convolving a base process with a smoothing kernel. If the base process is a Gaussian process, the convolved process is Gaussian as well. We can therefore exploit convolutions to construct covariance functions. [20] This method of producing non-separable kernels is known as process convolution.