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Operators given by ^ = (^ † + ^) and ^ = (^ † ^) are called the quadratures and they represent the real and imaginary parts of the complex amplitude represented by ^. [1] The commutation relation between the two quadratures can easily be calculated:
Note that since this resultant wave is continuously phase shifting at a steady rate, effectively the frequency has been changed: it has been frequency modulated. And if the IQ data itself has some frequency (e.g. a phasor) then the carrier also can be frequency modulated. So I/Q data is a complete representation of how a carrier is modulated ...
It is assumed that the value of a function f defined on [,] is known at + equally spaced points: < < <.There are two classes of Newton–Cotes quadrature: they are called "closed" when = and =, i.e. they use the function values at the interval endpoints, and "open" when > and <, i.e. they do not use the function values at the endpoints.
Quadrature amplitude modulation (QAM), a modulation method of using both an (in-phase) carrier wave and a 'quadrature' carrier wave that is 90° out of phase with the main, or in-phase, carrier Quadrature phase-shift keying (QPSK), a phase-shift keying of using four quadrate points on the constellation diagram, equispaced around a circle
Also known as Lobatto quadrature, [7] named after Dutch mathematician Rehuel Lobatto. It is similar to Gaussian quadrature with the following differences: The integration points include the end points of the integration interval. It is accurate for polynomials up to degree 2n – 3, where n is the number of integration points. [8]
In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics, ^ = (^ † ^), where is the amount of displacement in optical phase space, is the complex conjugate of that displacement, and ^ and ^ † are the lowering and raising operators, respectively.
Routines for Gauss–Kronrod quadrature are provided by the QUADPACK library, the GNU Scientific Library, the NAG Numerical Libraries, R, [2] the C++ library Boost., [3] as well as the Julia package QuadGK.jl [4] (which can compute Gauss–Kronrod formulas to arbitrary precision).
The measured electric field strengths at the wave's phase are the eigenvalues of the normalized quadrature operator , defined as [5] ^ = [^ + ^ †] = ^ + ^ where ^ and ^ † are the annihilation and creation operators, respectively, of the oscillator representing the photon.