Quadrature amplitude modulation

<h2 id="demodulation">Demodulation</h2>

In a QAM signal, one carrier lags the other by 90°, and its amplitude modulation is customarily referred to as the <a href="/facts/In-phase_and_quadrature_components/w2WgxUqg">in-phase component</a>, denoted by I(t). The other modulating function is the <a href="/facts/Quadrature_component/w2WgxUqg">quadrature component</a>, Q(t). So the composite waveform is mathematically modeled as:

s
 
 s
 
 
 (
 t
 )
 ≜
 sin
 ⁡
 (
 2
 π
 
 f
 
 c
 
 
 t
 )
 I
 (
 t
 )
  
 +
  
 
 
 
 
 sin
 ⁡
 
 (
 
 2
 π
 
 f
 
 c
 
 
 t
 +
 
 
 
 π
 2
 
 
 
 
 )
 
 
 ⏟
 
 
 
 cos
 ⁡
 
 (
 
 2
 π
 
 f
 
 c
 
 
 t
 
 )
 
 
 
 
 Q
 (
 t
 )
 ,
 
 
 {\displaystyle 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),}
 
     or:
<table><tbody><tr><td> s c ( t ) ≜ cos ⁡ ( 2 π f c t ) I ( t )   +   cos ⁡ ( 2 π f c t + π 2 ) ⏟ − sin ⁡ ( 2 π f c t ) Q ( t ) , {\displaystyle 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),} </td> <td></td> <td>Eq.1</td></tr></tbody></table>
where fc is the carrier frequency.  At the receiver, a <a href="/facts/Product_detector/wE72SMj0">coherent demodulator</a> multiplies the received signal separately with both a <a href="/facts/Cosine/czej7ur0">cosine</a> and <a href="/facts/Sine/gQ6LXK8z">sine</a> signal to produce the received estimates of I(t) and Q(t). For example:

r
        (
        t
        )
        ≜
        
          s
          
            c
          
        
        (
        t
        )
        cos
        ⁡
        (
        2
        π
        
          f
          
            c
          
        
        t
        )
        =
        I
        (
        t
        )
        cos
        ⁡
        (
        2
        π
        
          f
          
            c
          
        
        t
        )
        cos
        ⁡
        (
        2
        π
        
          f
          
            c
          
        
        t
        )
        −
        Q
        (
        t
        )
        sin
        ⁡
        (
        2
        π
        
          f
          
            c
          
        
        t
        )
        cos
        ⁡
        (
        2
        π
        
          f
          
            c
          
        
        t
        )
        .
      
    
    {\displaystyle 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).}

Using standard <a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">trigonometric identities</a>, we can write this as:

r
                (
                t
                )
              
              
                
                =
                
                  
                    
                      1
                      2
                    
                  
                
                I
                (
                t
                )
                
                  [
                  
                    1
                    +
                    cos
                    ⁡
                    (
                    4
                    π
                    
                      f
                      
                        c
                      
                    
                    t
                    )
                  
                  ]
                
                −
                
                  
                    
                      1
                      2
                    
                  
                
                Q
                (
                t
                )
                sin
                ⁡
                (
                4
                π
                
                  f
                  
                    c
                  
                
                t
                )
              
            
            
              
              
                
                =
                
                  
                    
                      1
                      2
                    
                  
                
                I
                (
                t
                )
                +
                
                  
                    
                      1
                      2
                    
                  
                
                
                  [
                  
                    I
                    (
                    t
                    )
                    cos
                    ⁡
                    (
                    4
                    π
                    
                      f
                      
                        c
                      
                    
                    t
                    )
                    −
                    Q
                    (
                    t
                    )
                    sin
                    ⁡
                    (
                    4
                    π
                    
                      f
                      
                        c
                      
                    
                    t
                    )
                  
                  ]
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}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{aligned}}}

<a href="/facts/Low-pass_filter/QFsUjw7h">Low-pass filtering</a> r(t) removes the high frequency terms (containing 4πfct), leaving only the I(t) term. This filtered signal is unaffected by Q(t), showing that the in-phase component can be received independently of the quadrature component.  Similarly, we can multiply sc(t) by a sine wave and then low-pass filter to extract Q(t).

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 <a href="/facts/Self-clocking/VRaK2817">self-clocking</a>. 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 <a href="/facts/Crosstalk/IT2LIwth">crosstalk</a>. In this context, the clock signal is called a "phase reference". Clock synchronization is typically achieved by transmitting a burst <a href="/facts/Subcarrier/eWqxg2h2">subcarrier</a> or a <a href="/facts/Pilot_signal/FFb2peWz">pilot signal</a>. The phase reference for <a href="/facts/NTSC/HYWIvxCR">NTSC</a>, for example, is included within its <a href="/facts/Colorburst/Ft9WqAKq">colorburst</a> signal.
Analog QAM is used in:

<ul><li><a href="/facts/NTSC/HYWIvxCR">NTSC</a> and <a href="/facts/PAL/t8TugcxK">PAL</a> analog <a href="/facts/Color_television/Un5dDuF7">color television</a> 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.</li>
<li><a href="/facts/C-QUAM/xQda6pAG">C-QUAM</a> ("Compatible QAM") is used in <a href="/facts/AM_stereo/7exL93Tz">AM stereo</a> radio to carry the stereo difference information.</li></ul>
<h2 id="fourier-analysis">Fourier analysis</h2>
Applying <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a> to the sinusoids in Eq.1, the positive-frequency portion of sc (or <a href="/facts/Analytic_representation/8CB4Cyul">analytic representation</a>) is:

s
          
            c
          
        
        (
        t
        
          )
          
            +
          
        
        =
        
          
            
              1
              2
            
          
        
        
          e
          
            i
            2
            π
            
              f
              
                c
              
            
            t
          
        
        [
        I
        (
        t
        )
        +
        i
        Q
        (
        t
        )
        ]
        
        
          
            
              
                ⟹
              
              
                
                  F
                
              
            
          
        
        
        
          
            
              1
              2
            
          
        
        
          [
          
            
              
                
                  
                    I
                     
                  
                  ^
                
              
            
            (
            f
            −
            
              f
              
                c
              
            
            )
            +
            
              e
              
                i
                π
                
                  /
                
                2
              
            
            
              
                
                  Q
                  ^
                
              
            
            (
            f
            −
            
              f
              
                c
              
            
            )
          
          ]
        
        ,
      
    
    {\displaystyle s_{c}(t)_{+}={\tfrac {1}{2}}e^{i2\pi f_{c}t}[I(t)+iQ(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],}

where 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 denotes the Fourier transform, and ︿I and ︿Q are the transforms of I(t) and Q(t). This result represents the sum of two DSB-SC signals with the same center frequency. The factor of i (= eiπ/2) represents the 90° phase shift that enables their individual demodulations.

<h2 id="digital-qam">Digital QAM</h2>

As in many digital modulation schemes, the <a href="/facts/Constellation_diagram/aCtPloV9">constellation diagram</a> 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 <a href="/facts/Telecommunications/vxoj1cxR">telecommunications</a> the data is usually <a href="/facts/Binary_numeral_system/a6JDSRPs">binary</a>, 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 <a href="/facts/Bit/S972NSaD">bits</a> per <a href="/facts/Symbol_(data)/YlQ3JHVr">symbol</a>. 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 <a href="/facts/Noise/aUstHDqU">noise</a> and other corruption; this results in a higher <a href="/facts/Bit_error_rate/ovAAPFjV">bit error rate</a> 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 <a href="/facts/Signal-to-noise_ratio/qohClhyG">signal-to-noise ratio</a> (SNR) by increasing signal energy, reducing noise, or both.
If data rates beyond those offered by 8-<a href="/facts/Phase-shift_keying/TFnpYe64">PSK</a> 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 <a href="/facts/Demodulator/GFLGoxFl">demodulator</a> must now correctly detect both <a href="/facts/Phase_(waves)/cpGtuVll">phase</a> and <a href="/facts/Amplitude/aEckuXJM">amplitude</a>, rather than just phase.
64-QAM and 256-QAM are often used in <a href="/facts/Digital_cable/hUzy2eqW">digital cable</a> television and <a href="/facts/Cable_modem/YsXWhXhK">cable modem</a> applications. In the United States, 64-QAM and 256-QAM are the mandated modulation schemes for <a href="/facts/Digital_cable/hUzy2eqW">digital cable</a> (see <a href="/facts/QAM_tuner/H18I0cCP">QAM tuner</a>) as standardised by the <a href="/facts/SCTE/XSzjlTXb">SCTE</a> in the standard <a href="https://web.archive.org/web/20140817034950/http://www.scte.org/FileDownload.aspx?A=3445">ANSI/SCTE 07 2013</a>. In the UK, 64-QAM is used for <a href="/facts/Digital_terrestrial_television/9UoV0vV1">digital terrestrial television</a> (<a href="/facts/Freeview_(UK)/ezPeNWGY">Freeview</a>) whilst 256-QAM is used for Freeview-HD.

Communication systems designed to achieve very high levels of <a href="/facts/Spectral_efficiency/GaePAvv9">spectral efficiency</a> usually employ very dense QAM constellations. For example is <a href="/facts/ADSL/Y5wsglVy">ADSL</a> 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).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
Ultra-high capacity microwave backhaul systems also use 1024-QAM.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> With 1024-QAM, <a href="/facts/Adaptive_coding_and_modulation/vegwIxN8">adaptive coding and modulation</a> (ACM) and <a href="/facts/XPIC/Iftodxre">XPIC</a>, vendors can obtain gigabit capacity in a single 56 MHz channel.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="interference-and-noise">Interference and noise</h2>
In moving to a higher order QAM constellation (higher data rate and mode) in hostile <a href="/facts/RF/aWku7h4R">RF</a>/<a href="/facts/Microwave/HmVSN31Y">microwave</a> QAM application environments, such as in <a href="/facts/Broadcasting/AV1f3dGy">broadcasting</a> or <a href="/facts/Telecommunications/vxoj1cxR">telecommunications</a>, <a href="/facts/Multipath_interference/dJFXvAbW">multipath interference</a> 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 <a href="/facts/Noise/aUstHDqU">noise</a> 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:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<ul><li><a href="/facts/Carrier_signal/ae98pBlY">Carrier</a>/interference ratio</li>
<li><a href="/facts/Carrier-to-noise_ratio/DP42mbjT">Carrier-to-noise ratio</a></li>
<li>Threshold-to-noise ratio</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Amplitude_and_phase-shift_keying/moXM9HMB">Amplitude and phase-shift keying</a> or <a href="/facts/Asymmetric_phase-shift_keying/moXM9HMB">asymmetric phase-shift keying</a> (APSK)</li>
<li><a href="/facts/Carrierless_amplitude_phase_modulation/9JJj1I8d">Carrierless amplitude phase modulation</a> (CAP)</li>
<li><a href="/facts/Circle_packing/nKFWD2ug">Circle packing § Applications</a></li>
<li><a href="/facts/In-phase_and_quadrature_components/w2WgxUqg">In-phase and quadrature components</a></li>
<li><a href="/facts/Modulation/aAXPjINv">Modulation</a> for other examples of modulation techniques</li>
<li><a href="/facts/Phase-shift_keying/TFnpYe64">Phase-shift keying</a></li>
<li><a href="/facts/QAM_tuner/H18I0cCP">QAM tuner</a> for HDTV</li>
<li><a href="/facts/Random_modulation/GekpBAaQ">Random modulation</a></li></ul>

<h2 id="further-reading">Further reading</h2>

<ul><li>Sun, Jonqyin (May 2014). "Linear diversity analysis for QAM in Rician fading channels". 2014 23rd Wireless and Optical Communication Conference (WOCC). pp. 1–3. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FWOCC.2014.6839960">10.1109/WOCC.2014.6839960</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4799-5249-6.</li>
<li>Proakis, John G. (1995). Digital Communications (3rd ed.). New York: McGraw-Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780070517264.</li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Quadrature amplitude modulation.

<ul><li><a href="http://www.wirelesscommunication.nl/pdfandps/qam.pdf">QAM Demodulation</a></li>
<li><a href="http://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_1/qam_constellation_diagram_from_snr/">Interactive webdemo of QAM constellation with additive noise</a> Institute of Telecommunicatons, University of Stuttgart</li>
<li><a href="http://www.etti.unibw.de/labalive/experiment/qam/">QAM bit error rate for AWGN channel – online experiment</a></li>
<li><a href="https://web.archive.org/web/20041112232234/http://www.blondertongue.com/QAM-Transmodulator/QAM_defined.php">How imperfections affect QAM constellation</a></li>
<li><a href="https://web.archive.org/web/20030327163207/http://www.herley.com/index.cfm?act=app_notes¬es=iqv_phaseshift">Microwave Phase Shifters</a> Overview by <a href="/facts/Herley_Industries/PxVA4DNS">Herley General Microwave</a></li>
<li><a href="http://www.vpiphotonics.com/Applications/TransmissionSystems/ModFormat_PolMuxQPSK.php">Simulation of dual-polarization QPSK (DP-QPSK) for 100G optical transmission</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Digital Modulation Efficiencies". Barnard Microsystems. Archived from the original on 2011-04-30. <a href="https://web.archive.org/web/20110430132506/http://www.barnardmicrosystems.com/L4E_comms_2.htm" target="_blank">https://web.archive.org/web/20110430132506/http://www.barnardmicrosystems.com/L4E_comms_2.htm</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Ciena tests 200G via 16-QAM with Japan-U.S. Cable Network". lightwave. April 17, 2014. Retrieved 7 November 2016. <a href="http://www.lightwaveonline.com/topics/16-qam.htm" target="_blank">http://www.lightwaveonline.com/topics/16-qam.htm</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kylia products Archived July 13, 2011, at the Wayback Machine, dwdm mux demux, 90 degree optical hybrid, d(q) psk demodulatorssingle polarization <a href="http://kylia.com/QAM.html" target="_blank">http://kylia.com/QAM.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"G.992.3 : Asymmetric digital subscriber line transceivers 2 (ADSL2)". www.itu.int. Constellation mapper - maximum number of bits per constellation BIMAX ≤ 15. Retrieved 2024-10-09.{{cite web}}: CS1 maint: others (link) <a href="https://www.itu.int/rec/T-REC-G.992.3-200904-I" target="_blank">https://www.itu.int/rec/T-REC-G.992.3-200904-I</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"TrangoLink Apex Orion - Trango Systems". www.trangosys.com. Archived from the original on 2012-03-15. <a href="https://web.archive.org/web/20120315235455/http://www.trangosys.com/products/point-to-point-wireless-backhaul/licensed-wireless/trangolink-apex-orion.shtml" target="_blank">https://web.archive.org/web/20120315235455/http://www.trangosys.com/products/point-to-point-wireless-backhaul/licensed-wireless/trangolink-apex-orion.shtml</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">"TrangoLink Apex Orion - Trango Systems". www.trangosys.com. Archived from the original on 2012-03-15. <a href="https://web.archive.org/web/20120315235455/http://www.trangosys.com/products/point-to-point-wireless-backhaul/licensed-wireless/trangolink-apex-orion.shtml" target="_blank">https://web.archive.org/web/20120315235455/http://www.trangosys.com/products/point-to-point-wireless-backhaul/licensed-wireless/trangolink-apex-orion.shtml</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Howard Friedenberg and Sunil Naik. "Hitless Space Diversity STL Enables IP+Audio in Narrow STL Bands" (PDF). 2005 National Association of Broadcasters Annual Convention. Archived from the original (PDF) on March 23, 2006. Retrieved April 17, 2005. <a href="https://web.archive.org/web/20060323141431/http://www.moseleysb.com/mb/whitepapers/friedenberg.pdf" target="_blank">https://web.archive.org/web/20060323141431/http://www.moseleysb.com/mb/whitepapers/friedenberg.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Quadrature amplitude modulation open-in-new

Quadrature amplitude modulation