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Let each curve C t in the family be given as the solution of an equation f t (x, y)=0 (see implicit curve), where t is a parameter. Write F(t, x, y)=f t (x, y) and assume F is differentiable. The envelope of the family C t is then defined as the set of points (x,y) for which, simultaneously,
The upper envelope or pointwise maximum is defined symmetrically. For an infinite set of functions, the same notions may be defined using the infimum in place of the minimum, and the supremum in place of the maximum. [1] For continuous functions from a given class, the lower or upper envelope is a piecewise function whose pieces are from the ...
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.
In mathematics, the lower convex envelope ˘ of a function defined on an interval [,] is defined at each point of the interval as the supremum of all ...
In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space , or equivalently as the set of all convex combinations of points in the subset.
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By Hopf–Lax formula, the Moreau envelope is a viscosity solution to a Hamilton–Jacobi equation. [3] Stanley Osher and co-authors used this property and Cole–Hopf transformation to derive an algorithm to compute approximations to the proximal operator of a function. [4]
R(t) is called the envelope of the signal. Hence an AM signal is given by the function = (+ ()) with m(t) representing the original audio frequency message, C the carrier amplitude and R(t) equal to C + m(t). So, if the envelope of the AM signal can be extracted, the original message can be recovered.