Editing Quadrature amplitude modulation

{{short description|Family of digital modulation methods}}
{{Redirect|QAM|the digital television standard|QAM (television)|other uses|QAM (disambiguation)}}

{{Technical|date=June 2020}}
{{Modulation techniques}}

'''Quadrature amplitude modulation''' ('''QAM''') is the name of a family of [[digital modulation]] methods and a related family of [[analog modulation]] methods widely used in modern [[telecommunications]] to transmit information. It conveys two analog message signals, or two digital [[bit stream]]s, by changing (''modulating'') the [[amplitude]]s of two [[carrier wave]]s, using the [[amplitude-shift keying]] (ASK) digital modulation scheme or [[amplitude modulation]] (AM) analog modulation scheme. The two carrier waves are of the same frequency and are [[out of phase]] with each other by 90°, a condition known as [[orthogonality]] or [[Quadrature phase|quadrature]]. The transmitted signal is created by adding the two carrier waves together. At the receiver, the two waves can be coherently separated (demodulated) because of their orthogonality. Another key property is that the modulations are low-frequency/low-bandwidth waveforms compared to the carrier frequency, which is known as the [[In-phase and quadrature components#Narrowband signal model|narrowband assumption]].

[[Phase modulation]] (analog PM) and [[phase-shift keying]] (digital PSK) can be regarded as a special case of QAM, where the amplitude of the transmitted signal is a constant, but its phase varies. This can also be extended to [[frequency modulation]] (FM) and [[frequency-shift keying]] (FSK), for these can be regarded as a special case of phase modulation. {{citation needed|date=September 2024}}

QAM is used extensively as a modulation scheme for digital [[communications system]]s, such as in [[802.11]] Wi-Fi standards. Arbitrarily high [[Spectral efficiency|spectral efficiencies]] can be achieved with QAM by setting a suitable [[Constellation diagram|constellation]] size, limited only by the noise level and linearity of the communications channel.<ref>{{cite web|title=Digital Modulation Efficiencies|url=http://www.barnardmicrosystems.com/L4E_comms_2.htm|publisher=Barnard Microsystems|archive-url=https://web.archive.org/web/20110430132506/http://www.barnardmicrosystems.com/L4E_comms_2.htm|archive-date=2011-04-30}}</ref>&nbsp; QAM is being used in optical fiber systems as bit rates increase; QAM16 and QAM64 can be optically emulated with a three-path [[interferometer]].<ref>{{cite web
 | url = http://www.lightwaveonline.com/topics/16-qam.htm
 | title =Ciena tests 200G via 16-QAM with Japan-U.S. Cable Network
 | date = April 17, 2014
 | publisher = lightwave
 | access-date = 7 November 2016
}}</ref><ref>[http://kylia.com/QAM.html Kylia products] {{webarchive |url=https://web.archive.org/web/20110713175309/http://kylia.com/QAM.html |date=July 13, 2011 }}, dwdm mux demux, 90 degree optical hybrid, d(q) psk demodulatorssingle polarization</ref>

== Demodulation ==
{{unsourced|section|date=December 2018}}
[[File:PAL_Vector.png|200px|right|thumb|Analog QAM: PAL color bar signal on a [[vectorscope]]]]

In a QAM signal, one carrier lags the other by 90°, and its amplitude modulation is customarily referred to as the [[In-phase_and_quadrature_components|in-phase component]], denoted by {{math|''I''(''t'').}} The other modulating function is the [[quadrature component]], {{math|''Q''(''t'').}} So the composite waveform is mathematically modeled as:

:<math>s_s(t) \triangleq \sin(2\pi f_c t) I(t)\ +\ \underbrace{\sin\left(2\pi f_c t + \tfrac{\pi}{2} \right)}_{\cos\left(2\pi f_c t\right)}\; Q(t),</math> &nbsp; &nbsp; '''or:'''
{{NumBlk|:|<math>s_c(t) \triangleq \cos(2\pi f_c t) I(t)\ +\ \underbrace{\cos\left(2\pi f_c t + \tfrac{\pi}{2} \right)}_{-\sin\left(2\pi f_c t\right)}\; Q(t),</math>|{{EquationRef|Eq.1}}}}

where {{math|''f''{{sub|c}}}} is the carrier frequency.&nbsp; At the receiver, a [[product detector|coherent demodulator]] multiplies the received signal separately with both a [[cosine]] and [[sine]] signal to produce the received estimates of {{math|''I''(''t'')}} and {{math|''Q''(''t'')}}. For example:

:<math>r(t) \triangleq s_c(t) \cos (2 \pi f_c t) = I(t) \cos (2 \pi f_c t) \cos (2 \pi f_c t) - Q(t) \sin (2 \pi f_c t) \cos (2 \pi f_c t).</math>

Using standard [[list of trigonometric identities#Product-to-sum and sum-to-product identities|trigonometric identities]], we can write this as:

:<math>\begin{align}
  r(t) &= \tfrac{1}{2} I(t) \left[1 + \cos (4 \pi f_c t)\right] - \tfrac{1}{2} Q(t) \sin (4 \pi f_c t) \\
       &= \tfrac{1}{2} I(t) + \tfrac{1}{2} \left[I(t) \cos (4 \pi f_c t) - Q(t) \sin (4 \pi f_c t)\right].
\end{align}</math>

[[Low-pass filter]]ing {{math|''r''(''t'')}} removes the high frequency terms (containing {{math|4π''f''{{sub|c}}''t''}}), leaving only the {{math|''I''(''t'')}} term. This filtered signal is unaffected by {{math|''Q''(''t''),}} showing that the in-phase component can be received independently of the quadrature component.&nbsp; Similarly, we can multiply {{math|''s''{{sub|c}}(''t'')}} by a sine wave and then low-pass filter to extract {{math|''Q''(''t'').}}

[[File:Sine and Cosine.svg|thumb|180px|right|The graphs of the sine (solid red) and [[cosine]] (dotted blue) functions are sinusoids of different phases.]]

The addition of two sinusoids is a linear operation that creates no new frequency components. So the bandwidth of the composite signal is comparable to the bandwidth of the DSB (double-sideband) components. Effectively, the spectral redundancy of DSB enables a doubling of the information capacity using this technique. This comes at the expense of demodulation complexity. In particular, a DSB signal has zero-crossings at a regular frequency, which makes it easy to recover the phase of the carrier sinusoid. It is said to be [[self-clocking]]. But the sender and receiver of a quadrature-modulated signal must share a clock or otherwise send a clock signal. If the clock phases drift apart, the demodulated ''I'' and ''Q'' signals bleed into each other, yielding [[crosstalk]]. In this context, the clock signal is called a "phase reference". Clock synchronization is typically achieved by transmitting a burst [[subcarrier]] or a [[pilot signal]]. The phase reference for [[NTSC]], for example, is included within its [[colorburst]] signal.

Analog QAM is used in:
* [[NTSC]] and [[PAL]] analog [[color television]] systems, where the I- and Q-signals carry the components of chroma (colour) information. The QAM carrier phase is recovered from a special colorburst transmitted at the beginning of each scan line.
* [[C-QUAM]] ("Compatible QAM") is used in [[AM stereo]] radio to carry the stereo difference information.

== Fourier analysis ==

Applying [[Euler's formula]] to the sinusoids in {{EquationNote|Eq.1}}, the positive-frequency portion of {{math|''s''{{sub|c}}}} (or [[analytic representation]]) is:

:<math>
  s_c(t)_+ = \tfrac{1}{2} e^{i2\pi f_c t}[I(t) + i Q(t)]
  \quad\stackrel{\mathcal{F}}{\Longrightarrow}\quad
  \tfrac{1}{2}\left[\widehat{I\ }(f - f_c) + e^{i\pi/2} \widehat Q(f - f_c)\right],
</math>

where <math>\mathcal{F}</math> denotes the Fourier transform, and {{math|{{overset|︿|I}}}} and {{math|{{overset|︿|Q}}}} are the transforms of {{math|''I''(''t'')}} and {{math|''Q''(''t'').}} This result represents the sum of two DSB-SC signals with the same center frequency. The factor of {{math|1='''i''' (= ''e''{{sup|''iπ''/2}})}} represents the 90° phase shift that enables their individual demodulations.

== Digital QAM ==
[[File:16-QAM Demonstration 3.gif|alt=Digital 16-QAM with example symbols|thumb|Digital 16-QAM with example symbols]]
[[File:Rectangular constellation for QAM.svg|thumb|Constellation points for 4-QAM, 16-QAM, 32-QAM, and 64-QAM overlapped]]
As in many digital modulation schemes, the [[constellation diagram]] is useful for QAM. In QAM, the constellation points are usually arranged in a square grid with equal vertical and horizontal spacing, although other configurations are possible (e.g. a hexagonal or triangular grid). In digital [[telecommunications]] the data is usually [[Binary numeral system|binary]], so the number of points in the grid is typically a power of 2 (2, 4, 8, …), corresponding to the number of bits per symbol. The simplest and most commonly used QAM constellations consist of points arranged in a square, i.e. 16-QAM, 64-QAM and 256-QAM (even powers of two). Non-square constellations, such as Cross-QAM, can offer greater efficiency but are rarely used because of the cost of increased modem complexity.

By moving to a higher-order constellation, it is possible to transmit more [[bit]]s per [[Symbol (data)|symbol]]. However, if the mean energy of the constellation is to remain the same (by way of making a fair comparison), the points must be closer together and are thus more susceptible to [[noise]] and other corruption; this results in a higher [[bit error rate]] and so higher-order QAM can deliver more data less reliably than lower-order QAM, for constant mean constellation energy. Using higher-order QAM without increasing the bit error rate requires a higher [[signal-to-noise ratio]] (SNR) by increasing signal energy, reducing noise, or both.

If data rates beyond those offered by 8-[[Phase-shift keying|PSK]] are required, it is more usual to move to QAM since it achieves a greater distance between adjacent points in the I-Q plane by distributing the points more evenly. The complicating factor is that the points are no longer all the same amplitude and so the [[demodulator]] must now correctly detect both [[Phase (waves)|phase]] and [[amplitude]], rather than just phase.

64-QAM and 256-QAM are often used in [[digital cable]] television and [[cable modem]] applications. In the United States, 64-QAM and 256-QAM are the mandated modulation schemes for [[digital cable]] (see [[QAM tuner]]) as standardised by the [[SCTE]] in the standard [https://web.archive.org/web/20140817034950/http://www.scte.org/FileDownload.aspx?A=3445 ANSI/SCTE 07 2013]. In the UK, 64-QAM is used for [[digital terrestrial television]] ([[Freeview (UK)|Freeview]]) whilst 256-QAM is used for Freeview-HD.

[[File:ADSL spectrum Fritz Box Fon WLAN.png|thumb|Bit-loading (bits per QAM constellation) on an ADSL line]]

Communication systems designed to achieve very high levels of [[spectral efficiency]] usually employ very dense QAM constellations. For example is [[ADSL]] technology for copper twisted pairs, whose constellation size goes up to 32768-QAM (in ADSL terminology this is referred to as bit-loading, or bit per tone, 32768-QAM being equivalent to 15 bits per tone).<ref>{{Cite web |others=Constellation mapper - maximum number of bits per constellation BIMAX ≤ 15 |title=G.992.3 : Asymmetric digital subscriber line transceivers 2 (ADSL2) |url=https://www.itu.int/rec/T-REC-G.992.3-200904-I |access-date=2024-10-09 |website=www.itu.int}}</ref>

Ultra-high capacity microwave backhaul systems also use 1024-QAM.<ref name="auto">{{Cite web |date= |title=TrangoLink Apex Orion - Trango Systems |url=http://www.trangosys.com/products/point-to-point-wireless-backhaul/licensed-wireless/trangolink-apex-orion.shtml |archive-url=https://web.archive.org/web/20120315235455/http://www.trangosys.com/products/point-to-point-wireless-backhaul/licensed-wireless/trangolink-apex-orion.shtml |archive-date=2012-03-15 |access-date= |website=www.trangosys.com}}</ref> With 1024-QAM, [[adaptive coding and modulation]] (ACM) and [[XPIC]], vendors can obtain gigabit capacity in a single 56&nbsp;MHz channel.<ref name="auto"/>

== Interference and noise ==

In moving to a higher order QAM constellation (higher data rate and mode) in hostile [[RF]]/[[microwave]] QAM application environments, such as in [[broadcasting]] or [[telecommunications]], [[multipath interference]] typically increases. There is a spreading of the spots in the constellation, decreasing the separation between adjacent states, making it difficult for the receiver to decode the signal appropriately. In other words, there is reduced [[Noise#Electronic noise|noise]] immunity. There are several test parameter measurements which help determine an optimal QAM mode for a specific operating environment. The following three are most significant:<ref>{{cite web| title = Hitless Space Diversity STL Enables IP+Audio in Narrow STL Bands| url = http://www.moseleysb.com/mb/whitepapers/friedenberg.pdf| work = 2005 National Association of Broadcasters Annual Convention| author = Howard Friedenberg and Sunil Naik| access-date = April 17, 2005| archive-url = https://web.archive.org/web/20060323141431/http://www.moseleysb.com/mb/whitepapers/friedenberg.pdf| archive-date = March 23, 2006| url-status = dead}}</ref>
* [[Carrier signal|Carrier]]/interference ratio
* [[Carrier-to-noise ratio]]
* Threshold-to-noise ratio

== See also ==
* [[Amplitude and phase-shift keying]] or [[asymmetric phase-shift keying]] (APSK)
* [[Carrierless amplitude phase modulation]] (CAP)
* {{section link|Circle packing|Applications}}
* [[In-phase and quadrature components]]
* [[Modulation]] for other examples of modulation techniques
* [[Phase-shift keying]]
* [[QAM tuner]] for HDTV
* [[Random modulation]]

== References ==
{{Reflist}}

== Further reading ==
{{refbegin}}
* {{cite book |last1=Sun |first1=Jonqyin |chapter=Linear diversity analysis for QAM in Rician fading channels |title=2014 23rd Wireless and Optical Communication Conference (WOCC) |date=May 2014 |pages=1–3 |doi=10.1109/WOCC.2014.6839960|isbn=978-1-4799-5249-6 }}
* {{cite book |last1=Proakis |first1=John G. |title=Digital Communications |date=1995 |publisher=McGraw-Hill |location=New York |isbn=9780070517264 |edition=3rd}}
{{refend}}

== External links ==
{{Commons category|Quadrature amplitude modulation}}
* [http://www.wirelesscommunication.nl/pdfandps/qam.pdf QAM Demodulation]
* [http://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_1/qam_constellation_diagram_from_snr/ Interactive webdemo of QAM constellation with additive noise]{{cbignore|bot=medic}}  Institute of Telecommunicatons, University of Stuttgart
* [http://www.etti.unibw.de/labalive/experiment/qam/ QAM bit error rate for AWGN channel – online experiment]
* [https://web.archive.org/web/20041112232234/http://www.blondertongue.com/QAM-Transmodulator/QAM_defined.php How imperfections affect QAM constellation] 
* [https://web.archive.org/web/20030327163207/http://www.herley.com/index.cfm?act=app_notes&notes=iqv_phaseshift Microwave Phase Shifters] Overview by [[Herley Industries|Herley General Microwave]]
* [http://www.vpiphotonics.com/Applications/TransmissionSystems/ModFormat_PolMuxQPSK.php Simulation of dual-polarization QPSK (DP-QPSK) for 100G optical transmission]

{{Analogue TV transmitter topics}}
{{Telecommunications}}

{{DEFAULTSORT:Quadrature Amplitude Modulation}}
[[Category:Radio modulation modes]]
[[Category:Data transmission]]