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 /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. Orthogonality In vector analysis, a vector with polar coordinates and Cartesian coordinates can be represented as the sum of orthogonal components: Similarly in trigonometry, the angle sum identity expresses: And in functional analysis, when is a linear function of some variable, such as time, these components are sinusoids, and they are orthogonal functions. A phase-shift of changes the identity to: , in which case is the in-phase component. In both conventions is the in-phase amplitude modulation, which explains why some authors refer to it as the actual in-phase component. Narrowband signal model In an angle modulation application, with carrier frequency φ is also a time-variant function, giving: When all three terms above are multiplied by an optional amplitude function, 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. Because of the modulation, the components are no longer completely orthogonal functions. But when and are slowly varying functions compared to the assumption of orthogonality is a common one. Authors often call it a narrowband assumption, or a narrowband signal model. I/Q data A stream of information about how to amplitude-modulate the I and Q phases of a sine wave is known as the I/Q data. By just amplitude-modulating these two 90°-out-of-phase sine waves and adding them, it is possible to produce the effect of arbitrarily modulating some carrier: amplitude and phase. 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