In-phase and quadrature components

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:[1]^{: eqs.(4.45)&(7.64)}

${\displaystyle {\begin{aligned}A(t)\cdot \cos[2\pi ft+\phi (t)]\ &=\cos(2\pi ft)\cdot A(t)\cos[\phi (t)]\ +\ \cos \left(2\pi ft+{\tfrac {\pi }{2}}\right)\cdot A(t)\sin[\phi (t)]\\&=\underbrace {\cos(2\pi ft)\cdot A(t)\cos[\phi (t)]} _{\text{in-phase}}\ \underbrace {\ -\ \sin(2\pi ft)\cdot A(t)\sin[\phi (t)]} _{\text{quadrature}}.\end{aligned}}}$

(Eq.1)

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.[upper-alpha 2] Because of the modulation, the components are no longer completely orthogonal functions. But when A(t) and φ(t) are slowly varying functions compared to 2πft, the assumption of orthogonality is a common one.[upper-alpha 3] Authors often call it a narrowband assumption, or a narrowband signal model.[4][5]

A stream of information about how to modulate the I and Q phases of a sine wave is known as the I/Q data.[6] IQ data has extensive use in many signal processing contexts, including for radio modulation, software-defined radio , audio signal processing and electrical engineering.

When called "I/Q data" the information is likely digital. I/Q may be represented as analogue signals.

The data rate of I/Q is largely independent to the frequency of the signal being modulated. I/Q data can be generated at a relatively slow rate (e.g. millions of bits per second), perhaps generated by software in part of the physical layer of a protocol stack. I/Q data is used to modulate a carrier frequency, which may be faster (e.g. Gigahertz, perhaps an intermediate frequency).[7]

As well as within a transmitter, I/Q data is also a common means to transfer the data from some receiver. Designs such as the Digital down converter allow the input signal to be represented as streams of IQ data, likely for futher processing and symbol extraction in a DSP.

This technique of using I/Q data to represent the modulations of a signal separate to the signal's frequency is known as equivalent baseband signal, supported by the § Narrowband signal model.

I/Q data may also be used as a means to capture and store data used in spectrum monitoring.[3]

A vector signal generator will typically use I/Q data alongside some programmed frequency to generate its signal.[7] And similarly a vector signal analyser can provide a stream of I/Q data in its output. Many modulation schemes, e.g. quadrature amplitude modulation rely heavily on I/Q.

The term alternating current applies to a voltage vs. time function that is sinusoidal with a frequency f. When it is applied to a typical (linear time-invariant) circuit or device, it causes a current that is also sinusoidal. In general there is a constant phase difference, φ, between any two sinusoids. The input sinusoidal voltage is usually defined to have zero phase, meaning that it is arbitrarily chosen as a convenient time reference. So the phase difference is attributed to the current function, e.g. sin(2πft + φ), whose orthogonal components are sin(2πft) cos(φ) and sin(2πft + π/2) sin(φ), as we have seen. When φ happens to be such that the in-phase component is zero, the current and voltage sinusoids are said to be in quadrature, which means they are orthogonal to each other. In that case, no average (active) electrical power is consumed. Rather power is temporarily stored by the device and given back, once every 1/2f seconds. Note that the term in quadrature only implies that two sinusoids are orthogonal, not that they are components of another sinusoid.

• IQ imbalance
• Constellation diagram
• Phasor
• Polar modulation
• Quadrature amplitude modulation
• Single-sideband modulation

• Steinmetz, Charles Proteus (2003-02-20). Lectures on Electrical Engineering. Vol. 3 (1 ed.). Mineola,NY: Dover Publications. ISBN 0486495388.
• Steinmetz, Charles Proteus (1917). Theory and Calculations of Electrical Apparatus 6 (1 ed.). New York: McGraw-Hill Book Company. B004G3ZGTM.

• I/Q Data for Dummies