View source for In-phase and quadrature components

{{Short description|Mathematical technique for manipulating signals etc.}}
{{distinguish|text = [[phase quadrature]] between different sinusoidal signals (such as voltage and current)}}
{{for|the colourspace components|YIQ}}
[[File:In-phase and quadrature components of angle modulation.gif|thumb|right|400px|Graphic example of the formula &nbsp;<math>{\color{Green}\cos(2\pi ft + \varphi(t))}\ =</math><br><math>{\color{Blue}\cos(2\pi ft)\cos(\varphi(t))}\ +\ {\color{Red}\cos(2\pi ft + \pi/2)\sin(\varphi(t))}.</math><br>
The phase modulation (''φ''(''t''), not shown) is a non-linearly increasing function from 0 to {{pi}}/2 over the interval 0 < t < 16.  The two amplitude-modulated components are known as the in-phase component (I, thin blue, decreasing) and the quadrature component (Q, thin red, increasing).]]

A [[sinusoid]] with [[modulation]] can be decomposed into, or synthesized from, two [[Amplitude modulation|amplitude-modulated]] sinusoids that are in [[quadrature phase]], i.e., with a [[Phase (waves)|phase offset]] of 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.{{efn-ua
|The low-frequency modulating waveforms are also referred to as the I and Q ''components''.<ref name="Franks"/>{{rp|p.82}}
}}<ref name="Gast"/>

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.<ref name=ITU /> [[#I/Q data|I/Q data]] is used to represent the modulations of some carrier, independent of that carrier's frequency.

==Orthogonality==
{{see also|List of trigonometric identities#Linear combinations}}
In vector analysis, a vector with polar coordinates {{math|''A'', ''φ''}} and Cartesian coordinates {{math|1=''x'' = ''A'' cos(''φ''), ''y'' = ''A'' sin(''φ''),}} can be represented as the sum of orthogonal components: {{math|[''x'', 0] + [0, ''y''].}} Similarly in trigonometry, the [[angle sum identity]] expresses:
: {{math|1=sin(''x'' + ''φ'') = sin(''x'') cos(''φ'') + sin(''x'' + {{pi}}/2) sin(''φ'').}}

And in functional analysis, when {{mvar|x}} is a linear function of some variable, such as time, these components are [[sinusoid]]s, and they are [[orthogonal functions]]. A [[phase-shift]] of {{math|''x'' → ''x'' + {{pi}}/2}} changes the identity to:
: {{math|1=cos(''x'' + ''φ'') = cos(''x'') cos(''φ'') + cos(''x'' + {{pi}}/2) sin(''φ'')}},

in which case {{math|cos(''x'') cos(''φ'')}} is the in-phase component.  In both conventions {{math|cos(''φ'')}} 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]] {{mvar|f,}} ''φ'' is also a time-variant function, giving''':'''<ref name="Franks"/>{{rp|eqs.(4.45)&(7.64)}}

{{Equation box 1
|indent= |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk|:|
<math>
\begin{align}
A(t)\cdot\cos[2\pi ft + \varphi(t)]\ &= \cos(2\pi ft)\cdot A(t)\cos[\varphi(t)] \ 
+\ \cos\left(2\pi ft + \tfrac \pi 2 \right) \cdot A(t)\sin[\varphi(t)]\\[8pt]
&=\underbrace{ \cos(2\pi ft)\cdot A(t)\cos[\varphi(t)]}_{\text {in-phase}} \ 
  \underbrace{\ -\ \sin(2\pi ft)\cdot A(t)\sin[\varphi(t)]}_{\text {quadrature}}.
\end{align}</math> &nbsp; &nbsp;
|{{EquationRef|Eq.1}} }} }}

When all three terms above are multiplied by an optional amplitude function, {{math|''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.{{efn-ua
|The negative sign in {{EquationNote|Eq.1}} can be associated with either the quadrature carrier or its amplitude modulation.  In the former case, Q-carrier leads I-carrier by <math>\tfrac{1}{4}</math> cycle.  Otherwise, it lags by <math>\tfrac{1}{4}</math> cycle.  The distinction is not important, but it can be confusing.
}}
Because of the modulation, the components are no longer completely orthogonal functions.  But when {{math|''A''(''t'')}} and {{math|''φ''(''t'')}} are slowly varying functions compared to {{math|2{{pi}}''ft'',}} the assumption of orthogonality is a common one.{{efn-ua
|Orthogonality is important in many applications, including demodulation, direction-finding, and bandpass sampling.
}}
Authors often call it a ''narrowband assumption'', or a '''narrowband signal model'''.<ref name="Wade" /><ref name="Naidu" />

==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'''.<ref>{{cite web |last1=Bryan |first1=Peter Barrett |title=Mind your I's and Q's: The Basics of I/Q data |url=https://towardsdatascience.com/mind-your-is-and-q-s-the-basics-of-i-q-data-d1f2b0dd81f4 |website=Medium |access-date=2023-02-15 |language=en |date=15 January 2022}}</ref> 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. [[File:Phase shifter using IQ modulator.gif|thumb|right|400px|A [[phasor]] for I/Q, and the resultant wave which is continually phase shifting, according to the phasor's frequency.  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: amplitude, phase and frequency.

For received signals, by determining how much in-phase carrier and how much quadrature carrier is present in the signal it is possible to represent that signal using in-phase and quadrature components, so IQ data can get generated from a signal with reference to a carrier sine wave.  
[[File:IQ Mod Demod block 2.svg|thumb|right|400px|IQ modulation and demodulation. <br />'''LO''' is the [[local oscillator]] - the carrier sine wave being modulated <br />'''I(t)''' and '''Q(t)''' are the time-series data for the in-phase and quadrature components. <br />'''S''' is the signal ]]

IQ data has extensive use in many signal processing contexts, including for [[radio]] [[modulation]], [[software-defined radio]], [[audio signal processing]] and [[electrical engineering]].

I/Q data is a two-dimensional stream.  Some sources treat I/Q as a [[complex number]];<ref name=Franks /> with the I and Q components corresponding to the real and imaginary parts.  Others treat it as distinct pairs of values, as a 2D [[Vector (mathematics and physics)|vector]], or as separate streams.

When called "I/Q data" the information is likely digital.  However, I/Q may be represented as analog signals.<ref>{{cite web |title=Quad Demodulators Arm Direct-Conversion Receivers |url=https://www.mwrf.com/technologies/components/article/21843541/quad-demodulators-arm-directconversion-receivers |website=www.mwrf.com |date=26 January 1998 |access-date=2023-02-17}}</ref> The concepts are applicable to both the analog and digital representations of IQ.

This technique of using I/Q data to represent the modulations of a signal separate to the signal's frequency is known as ''[[Baseband#Equivalent baseband signal|equivalent baseband signal]]'', supported by the {{slink||Narrowband signal model}}.  It is sometimes referred to as ''vector modulation''.

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]]).<ref name=tek>{{cite web |title=What's Your IQ – About Quadrature Signals… |url=https://www.tek.com/en/blog/quadrature-iq-signals-explained |website=www.tek.com |access-date=2023-02-15 |language=en}}</ref>

As well as within a transmitter, I/Q data is also a common means to represent the signal from some receiver.  Designs such as the [[Digital down converter]] allow the input signal to be represented as streams of IQ data, likely for further processing and symbol extraction in a [[digital signal processor|DSP]].  Analog systems may suffer from issues, such as [[IQ imbalance]].

I/Q data may also be used as a means to capture and store data used in spectrum monitoring.<ref name=ITU>{{cite web |title=Data format definition for exchanging stored I/Q data for the purpose of spectrum monitoring |url=https://www.itu.int/dms_pubrec/itu-r/rec/sm/R-REC-SM.2117-0-201809-I!!PDF-E.pdf |publisher=International Telecommunication Union (ITU) |access-date=2023-02-15}}</ref>  Since I/Q allows the representation of the modulation separate to the actual carrier frequency, it is possible to represent a capture of all the radio traffic in some [[RF band]] or section thereof, with a reasonable amount of data, irrespective of the frequency being monitored.  E.g. if there is a capture of 100&nbsp;MHz of [[Wi-Fi]] channels within the [[U-NII|5 GHz U-NII band]], that IQ capture can be [[Sampling (signal processing)|sampled]] at 200 million samples per second (according to [[Nyquist rate|Nyquist]]) as opposed to the 10,000 million samples per second required to sample directly at 5&nbsp;GHz.

A [[signal generator#Vector signal generator|vector signal generator]] will typically use I/Q data alongside some programmed frequency to generate its signal.<ref name=tek />  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.

==Alternating current (AC) circuits==
The term ''alternating current'' applies to a voltage vs. time function that is sinusoidal with a [[frequency]] {{mvar|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. 
{{math|sin(2{{pi}}''ft'' + ''φ''),}} whose orthogonal components are 
{{math|sin(2{{pi}}''ft'') cos(''φ'')}} and {{math|sin(2{{pi}}''ft'' + {{pi}}/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 
{{math|{{sfrac|1|2''f''}}}} seconds. Note that the term ''in quadrature'' only implies that two sinusoids are orthogonal, not that they are ''components'' of another sinusoid.

{{Gallery
|title=
|width=200
|height=200
|align=center
|File:VI phase.svg|When a sinusoidal voltage is applied to either a simple capacitor or inductor, the resultant current that flows is "in quadrature" with the voltage.
|File:IQ phasor diagram.svg|IQ phasor diagram
}}

==See also==
* [[Analytic signal]]
* [[IQ imbalance]]
* [[Constellation diagram]]
* [[Negative frequency]]
* [[Phasor]]
* [[Polar modulation]]
* [[Quadrature amplitude modulation]]
* [[Single-sideband modulation]]

==Notes==
{{notelist-ua}}

==References==
{{reflist|1|refs=
<ref name=Wade>
{{cite book
 | last =Wade
 | first =Graham
 | title =Signal Coding and Processing
 | publisher =Cambridge University Press
 | volume =1
 | edition =2
 | date =1994-09-30
 | page =10
 | isbn =0521412307 }}
</ref>

<ref name=Naidu>
{{cite book
 | last =Naidu
 | first =Prabhakar S.
 | title =Modern Digital Signal Processing: An Introduction
 | publisher =Alpha Science Intl Ltd
 | date =November 2003
 | location =Pangbourne RG8 8UT, UK
 | pages =29–31
 | isbn =1842651331 }}
</ref>

<ref name = Gast>
{{cite book
 | last =Gast
 | first =Matthew
 | title =802.11 Wireless Networks: The Definitive Guide
 | publisher =O'Reilly Media
 | location =Sebastopol, CA
 | volume =1
 | edition =2
 | date =2005-05-02
 | page =284
 | isbn =0596100523 }}
</ref>

<ref name = Franks>
{{cite book
 | last =Franks
 | first =L.E.
 | title =Signal Theory
 | publisher =Prentice Hall
 | series =Information theory
 | date =September 1969
 | location =Englewood Cliffs, NJ
 | isbn =0138100772 }}
</ref>
}}

==Further reading==
*{{cite book
 | last =Steinmetz
 | first =Charles Proteus
 | title =Lectures on Electrical Engineering
 | publisher =Dover Publications
 | volume =3
 | edition =1
 | date =2003-02-20
 | location =Mineola, NY
 | isbn =0486495388 }}
*Steinmetz, Charles Proteus (1917). ''Theory and Calculations of Electrical Apparatus'' '''6''' (1 ed.). New York: McGraw-Hill Book Company. [https://web.archive.org/web/20160313042102/http://lookupbyisbn.com/search/book/b004g3zgtm/1 B004G3ZGTM].

== External links ==
* [http://whiteboard.ping.se/SDR/IQ I/Q Data for Dummies]

{{DSP}}

[[Category:Signal processing]]
[[Category:Radio electronics]]