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Similarly, the spectral energy density of signal x(t) is = | | where X(f) is the Fourier transform of x(t).. For example, if x(t) represents the magnitude of the electric field component (in volts per meter) of an optical signal propagating through free space, then the dimensions of X(f) would become volt·seconds per meter and () would represent the signal's spectral energy density (in volts ...
The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function.
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
where ν is the frequency of the wave, λ is the wavelength, ω = 2πν is the angular frequency of the wave, and v p is the phase velocity of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a dispersion relation.
In 1900, Max Planck postulated the proportionality between the frequency of a photon and its energy , =, [11] [12] and in 1916 the corresponding relation between a photon's momentum and wavelength, =, [13] where is the Planck constant.
A common technique in signal processing is to consider the squared amplitude, or power; in this case the resulting plot is referred to as a power spectrum. Because of reversibility, the Fourier transform is called a representation of the function, in terms of frequency instead of time; thus, it is a frequency domain representation. Linear ...
Since the amplitude to travel from x to y after a time t + t ' can be considered in two steps, the propagator obeys the composition identity, (;) (; ′) = (; + ′) , which can be interpreted as follows: the amplitude to travel from x to z in time t + t ' is the sum of the amplitude to travel from x to y in time t, multiplied by the amplitude ...
A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transform , which converts a time function into a complex valued sum or integral of sine waves of different frequencies, with amplitudes and phases, each of which represents a ...
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