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The pseudocode below performs the GS algorithm to obtain a phase distribution for the plane "Source", such that its Fourier transform would have the amplitude distribution of the plane "Target". The Gerchberg-Saxton algorithm is one of the most prevalent methods used to create computer-generated holograms .
By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation .
When a function () is a function of time and represents a physical signal, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude of the resulting complex-valued function S ( f ) {\displaystyle S(f)} at frequency f {\displaystyle f} represents the amplitude of a frequency component whose initial phase ...
Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition. In general, ξ {\displaystyle \xi } must always be taken to be a linear form on the space of its domain, which is to say that the second real line is the dual space of the first real line.
Otherwise it is called unwrapped phase, which is a continuous function of argument t, assuming s a (t) is a continuous function of t. Unless otherwise indicated, the continuous form should be inferred. Instantaneous phase vs time. The function has two true discontinuities of 180° at times 21 and 59, indicative of amplitude zero-crossings.
Left: A continuous function (top) and its Fourier transform (bottom). Center-left: Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right: Original function is discretized (multiplied by a Dirac comb) (top).
All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation about the sawtooth is called the Gibbs phenomenon. There are many known sufficient conditions for the Fourier series of a function to converge at a given point x, for example if the function is differentiable at x.
The Fourier inversion theorem holds for all Schwartz functions (roughly speaking, smooth functions that decay quickly and whose derivatives all decay quickly). This condition has the benefit that it is an elementary direct statement about the function (as opposed to imposing a condition on its Fourier transform), and the integral that defines ...