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Concept
In-phase and quadrature components
Summary
A sinusoid with modulation can be decomposed into, or synthesized from, two amplitude-modulated sinusoids that are offset in phase by one-quarter cycle (90 degrees or pi/2 radians). All three sinusoids have the same center frequency. The two amplitude-modulated sinusoids are known as the in-phase (I) and quadrature (Q) components, which describes their relationships with the amplitude- and phase-modulated carrier.
Or in other words, it is possible to create an arbitrarily phase-shifted sine wave, by mixing together two sine waves that are 90° out of phase in different proportions.
The implication is that the modulations in some signal can be treated separately from the carrier wave of the signal. This has extensive use in many radio and signal processing applications. I/Q data is used to represent the modulations of some carrier, independent of that carrier's frequency.
List of trigonometric identities#Linear combinations
In vector analysis, a vector with polar coordinates A, φ and Cartesian coordinates x = A cos(φ), y = A sin(φ), can be represented as the sum of orthogonal components: [x, 0] + [0, y]. Similarly in trigonometry, the angle sum identity expresses:
sin(x + φ) = sin(x) cos(φ) + sin(x + π/2) sin(φ).
And in functional analysis, when x is a linear function of some variable, such as time, these components are sinusoids, and they are orthogonal functions. A phase-shift of x → x + π/2 changes the identity to:
cos(x + φ) = cos(x) cos(φ) + cos(x + π/2) sin(φ),
in which case cos(x) cos(φ) is the in-phase component. In both conventions cos(φ) is the in-phase amplitude modulation, which explains why some authors refer to it as the actual in-phase component.
In an angle modulation application, with carrier frequency f, φ is also a time-variant function, giving:
When all three terms above are multiplied by an optional amplitude function, A(t) > 0, the left-hand side of the equality is known as the amplitude/phase form, and the right-hand side is the quadrature-carrier or IQ form.
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