In-phase and Quadrature Components

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## Concept

### Alternating current (AC) circuits

### Narrowband signal model

### IQ phase convention

## See also

## Notes

## References

## Further reading

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

In-phase and Quadrature Components

In electrical engineering, a sinusoid with angle modulation can be decomposed into, or synthesized from, two amplitude-modulated sinusoids that are offset in phase by one-quarter cycle (?/2 radians). All three functions have the same center frequency. The amplitude modulated sinusoids are known as the **in-phase** and **quadrature** components.^{[1]}
In some contexts it is more convenient to refer to only the amplitude modulation (*baseband*) itself by those terms.^{[2]}

In vector analysis, a vector with polar coordinates *A,?* and Cartesian coordinates
*x* = *A* cos(*?*), *y* = *A* sin(*?*), can be represented as the sum of orthogonal "components":
[*x*,0] + [0,*y*].
Similarly in trigonometry, the expression **sin( x + ?)** can be represented by

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 circuit or device, it causes a current that is also sinusoidal. In general there is a constant phase difference, ?, 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(2?*ft* + ?), whose orthogonal components are
sin(2?*ft*) cos(?) and sin(2?*ft* + ?/2) sin(?), as we have seen. When ? 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 electrical power is consumed. Rather it is temporarily stored by the device and given back, once every
seconds. Note that the term

In an angle modulation application, with carrier frequency f, ? 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. Because of the modulation, the components are no longer completely orthogonal functions. But when *A*(*t*) and ?(*t*) are slowly varying functions compared to 2?*ft*, the assumption of orthogonality is a common one.^{[A]}
Authors often call it a *narrowband assumption*, or a **narrowband signal model**.^{[3]}^{[4]}

The terms *I-component* and *Q-component* are common ways of referring to the in-phase and quadrature signals. Both signals comprise a high-frequency sinusoid (or *carrier*) that is amplitude-modulated by a relatively low-frequency function, usually conveying some sort of information. The two carriers are orthogonal, with I lagging Q by ¼ cycle, or equivalently leading Q by ¾ cycle. The physical distinction can also be characterized in terms of :

- : The composite signal reduces to just the I-component, which accounts for the term
*in-phase*. - : The composite signal reduces to just the Q-component.
- : The amplitude modulations are orthogonal sinusoids, I leading Q by ¼ cycle.
- : The amplitude modulations are orthogonal sinusoids, Q leading I by ¼ cycle.

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

**^**Orthogonality is important in many applications, including demodulation, direction-finding, and bandpass sampling.

**^**Gast, Matthew (2005-05-02).*802.11 Wireless Networks: The Definitive Guide*.**1**(2 ed.). Sebastopol,CA: O'Reilly Media. p. 284. ISBN 0596100523.**^**Franks, L.E. (September 1969).*Signal Theory*. Information theory. Englewood Cliffs, NJ: Prentice Hall. p. 82. ISBN 0138100772.**^**Wade, Graham (1994-09-30).*Signal Coding and Processing*.**1**(2 ed.). Cambridge University Press. p. 10. ISBN 0521412307.**^**Naidu, Prabhakar S. (November 2003).*Modern Digital Signal Processing: An Introduction*. Pangbourne RG8 8UT, UK: Alpha Science Intl Ltd. pp. 29-31. ISBN 1842651331.CS1 maint: location (link)

- Steinmetz, Charles Proteus (2003-02-20).
*Lectures on Electrical Engineering*.**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.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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