Quadrature amplitude modulation (QAM) is both an analog and a digital modulation scheme. It conveys two analog message signals, or two digital bit streams, by changing (modulating) the amplitudes of two carrier waves, using the amplitudeshift keying (ASK) digital modulation scheme or amplitude modulation (AM) analog modulation scheme. The two carrier waves, usually sinusoids, are out of phase with each other by 90° and are thus called quadrature carriers or quadrature components — hence the name of the scheme. The modulated waves are summed, and the final waveform is a combination of both phaseshift keying (PSK) and amplitudeshift keying (ASK), or (in the analog case) of phase modulation (PM) and amplitude modulation. In the digital QAM case, a finite number of at least two phases and at least two amplitudes are used. PSK modulators are often designed using the QAM principle, but are not considered as QAM since the amplitude of the modulated carrier signal is constant. QAM is used extensively as a modulation scheme for digital telecommunication systems. Arbitrarily high spectral efficiencies can be achieved with QAM by setting a suitable constellation size, limited only by the noise level and linearity of the communications channel.^{[1]}
QAM is being used in optical fiber systems as bit rates increase; QAM16 and QAM64 can be optically emulated with a 3path interferometer.^{[2]}
Digital QAM
Like all modulation schemes, QAM conveys data by changing some aspect of a carrier signal, or the carrier wave, (usually a sinusoid) in response to a data signal. In the case of QAM, the amplitude of two waves, 90° outofphase with each other (in quadrature) are changed (modulated or keyed) to represent the data signal. Amplitude modulating two carriers in quadrature can be equivalently viewed as both amplitude modulating and phase modulating a single carrier.
Phase modulation (analog PM) and phaseshift keying (digital PSK) can be regarded as a special case of QAM, where the magnitude of the modulating signal is a constant, with only the phase varying. This can also be extended to frequency modulation (FM) and frequencyshift keying (FSK), for these can be regarded as a special case of phase modulation.
Analog QAM
Analog QAM: measured PAL colour bar signal on a vector analyser screen.
When transmitting two signals by modulating them with QAM, the transmitted signal will be of the form:
\begin{align} s(t) &= \Re \left\{\left[I(t) + i Q(t)\right] e^{i 2 \pi f_0 t}\right\} \\ &= I(t) \cos(2 \pi f_0 t)  Q(t) \sin(2 \pi f_0 t) \end{align}
where \scriptstyle i^2 \;=\; 1, \scriptstyle I(t), and \scriptstyle Q(t) are the modulating signals, \scriptstyle f_0 is the carrier frequency and \Re\{\} is the real part.
At the receiver, these two modulating signals can be demodulated using a coherent demodulator. Such a receiver multiplies the received signal separately with both a cosine and sine signal to produce the received estimates of \scriptstyle I(t) and \scriptstyle Q(t) respectively. Because of the orthogonality property of the carrier signals, it is possible to detect the modulating signals independently.
In the ideal case \scriptstyle I(t) is demodulated by multiplying the transmitted signal with a cosine signal:
\begin{align} r(t) &= s(t) \cos (2 \pi f_0 t) \\ &= I(t) \cos (2 \pi f_0 t)\cos (2 \pi f_0 t)  Q(t) \sin (2 \pi f_0 t)\cos (2 \pi f_0 t) \end{align}
Using standard trigonometric identities, we can write it as:

\begin{align} r(t) &= \frac{1}{2} I(t) \left[1 + \cos (4 \pi f_0 t)\right]  \frac{1}{2} Q(t) \sin (4 \pi f_0 t) \\ &= \frac{1}{2} I(t) + \frac{1}{2} [I(t) \cos (4 \pi f_0 t)  Q(t) \sin (4 \pi f_0 t)] \end{align}
Lowpass filtering \scriptstyle r(t) removes the high frequency terms (containing \scriptstyle 4\pi f_0 t), leaving only the \scriptstyle I(t) term. This filtered signal is unaffected by \scriptstyle Q(t), showing that the inphase component can be received independently of the quadrature component. Similarly, we may multiply \scriptstyle s(t) by a sine wave and then lowpass filter to extract \scriptstyle Q(t).
The phase of the received signal is assumed to be known accurately at the receiver. If the demodulating phase is even a little off, it results in crosstalk between the modulated signals. This issue of carrier synchronization at the receiver must be handled somehow in QAM systems. The coherent demodulator needs to be exactly in phase with the received signal, or otherwise the modulated signals cannot be independently received. For example analog television systems transmit a burst of the transmitting colour subcarrier after each horizontal synchronization pulse for reference.
Analog QAM is used in NTSC and PAL television systems, where the I and Qsignals carry the components of chroma (colour) information. "Compatible QAM" or CQUAM is used in AM stereo radio to carry the stereo difference information.
Fourier analysis of QAM
In the frequency domain, QAM has a similar spectral pattern to DSBSC modulation. Using the properties of the Fourier transform, we find that:

S(f) = \frac{1}{2}\left[ M_I(f  f_0) + M_I(f + f_0) \right] + \frac{i}{2}\left[ M_Q(f  f_0)  M_Q(f + f_0) \right]
where S(f), M_{I}(f) and M_{Q}(f) are the Fourier transforms (frequencydomain representations) of s(t), I(t) and Q(t), respectively.
Quantized QAM
Digital 16QAM with example constellation points.
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. CrossQAM). Since in digital telecommunications the data are usually binary, the number of points in the grid is usually a power of 2 (2, 4, 8, …). Since QAM is usually square, some of these are rare—the most common forms are 16QAM, 64QAM and 256QAM. By moving to a higherorder constellation, it is possible to transmit more bits per 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 higherorder QAM can deliver more data less reliably than lowerorder QAM, for constant mean constellation energy. Using higherorder QAM without increasing the bit error rate requires a higher signaltonoise ratio (SNR) by increasing signal energy, reducing noise, or both.
If datarates beyond those offered by 8PSK are required, it is more usual to move to QAM since it achieves a greater distance between adjacent points in the IQ 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 and amplitude, rather than just phase.
64QAM and 256QAM are often used in digital cable television and cable modem applications. In the United States, 64QAM and 256QAM are the mandated modulation schemes for digital cable (see QAM tuner) as standardised by the SCTE in the standard ANSI/SCTE 07 2013. Note that many marketing people will refer to these as QAM64 and QAM256. In the UK, 64QAM is used for digital terrestrial television (Freeview) whilst 256QAM is used for FreeviewHD.
Communication systems designed to achieve very high levels of spectral efficiency usually employ very dense QAM constellations. For example current Homeplug AV2 500Mbit powerline Ethernet devices use 1024QAM and 4096QAM, as well as future devices using ITUT G.hn standard for networking over existing home wiring (coaxial cable, phone lines and power lines); 4096QAM provides 12 bits/symbol. Another example is VDSL2 technology for copper twisted pairs, whose constellation size goes up to 32768 points.
Ultrahigh capacity Microwave Backhaul Systems also use 1024QAM.^{[3]} With 1024QAM, Adaptive Coding and Modulation (ACM), and XPIC, Vendors can obtain Gigabit capacity in a single 56 MHz channel.
Ideal structure
Transmitter
The following picture shows the ideal structure of a QAM transmitter, with a carrier frequency \scriptstyle f_0 and the frequency response of the transmitter's filter \scriptstyle H_t:
First the flow of bits to be transmitted is split into two equal parts: this process generates two independent signals to be transmitted. They are encoded separately just like they were in an amplitudeshift keying (ASK) modulator. Then one channel (the one "in phase") is multiplied by a cosine, while the other channel (in "quadrature") is multiplied by a sine. This way there is a phase of 90° between them. They are simply added one to the other and sent through the real channel.
The sent signal can be expressed in the form:

s(t) = \sum_{n=\infty}^{\infty} \left[ v_c [n] \cdot h_t (t  n T_s) \cos (2 \pi f_0 t)  v_s[n] \cdot h_t (t  n T_s) \sin (2 \pi f_0 t) \right]
where \scriptstyle v_c[n] and \scriptstyle v_s[n] are the voltages applied in response to the \scriptstyle n^{th} symbol to the cosine and sine waves respectively.
Receiver
The receiver simply performs the inverse process of the transmitter. Its ideal structure is shown in the picture below with \scriptstyle H_r the receive filter's frequency response :
Multiplying by a cosine (or a sine) and by a lowpass filter it is possible to extract the component in phase (or in quadrature). Then there is only an ASK demodulator and the two flows of data are merged back.
In practice, there is an unknown phase delay between the transmitter and receiver that must be compensated by synchronization of the receivers local oscillator; i.e., the sine and cosine functions in the above figure. In mobile applications, there will often be an offset in the relative frequency as well, due to the possible presence of a Doppler shift proportional to the relative velocity of the transmitter and receiver. Both the phase and frequency variations introduced by the channel must be compensated by properly tuning the sine and cosine components, which requires a phase reference, and is typically accomplished using a PhaseLocked Loop (PLL).
In any application, the lowpass filter and the receive \scriptstyle H_r filter will be implemented as a single combined filter. Here they are shown as separate just to be clearer.
Quantized QAM performance
The following definitions are needed in determining error rates:

\scriptstyle M = Number of symbols in modulation constellation

\scriptstyle E_b = Energyperbit

\scriptstyle E_s = Energypersymbol = \scriptstyle kE_b with k bits per symbol

\scriptstyle N_0 = Noise power spectral density (W/Hz)

\scriptstyle P_b = Probability of biterror

\scriptstyle P_{bc} = Probability of biterror per carrier

\scriptstyle P_s = Probability of symbolerror

\scriptstyle P_{sc} = Probability of symbolerror per carrier

\scriptstyle Q(x) \;=\; \frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{\frac{1}{2}t^{2}}dt,\ x \geq 0
\scriptstyle Q(x) is related to the complementary Gaussian error function by: \scriptstyle Q(x) \;=\; \frac{1}{2}\operatorname{erfc}\left(\frac{1}{\sqrt{2}}x\right), which is the probability that x will be under the tail of the Gaussian PDF towards positive infinity.
The error rates quoted here are those in additive white Gaussian noise (AWGN).
Where coordinates for constellation points are given in this article, note that they represent a nonnormalised constellation. That is, if a particular mean average energy were required (e.g. unit average energy), the constellation would need to be linearly scaled.
Rectangular QAM
Rectangular QAM constellations are, in general, suboptimal in the sense that they do not maximally space the constellation points for a given energy. However, they have the considerable advantage that they may be easily transmitted as two pulse amplitude modulation (PAM) signals on quadrature carriers, and can be easily demodulated. The nonsquare constellations, dealt with below, achieve marginally better biterror rate (BER) but are harder to modulate and demodulate.
The first rectangular QAM constellation usually encountered is 16QAM, the constellation diagram for which is shown here. A Gray coded bitassignment is also given. The reason that 16QAM is usually the first is that a brief consideration reveals that 2QAM and 4QAM are in fact binary phaseshift keying (BPSK) and quadrature phaseshift keying (QPSK), respectively. Also, the errorrate performance of 8QAM is close to that of 16QAM (only about 0.5 dB better), but its data rate is only threequarters that of 16QAM.
Expressions for the symbolerror rate of rectangular QAM are not hard to derive but yield rather unpleasant expressions. For an even number of bits per symbol, \scriptstyle k, exact expressions are available. They are most easily expressed in a per carrier sense:

P_{sc} = 2\left(1  \frac{1}{\sqrt M}\right)Q\left(\sqrt{\frac{3}{M1}\frac{E_s}{N_0}}\right)
so

\,P_s = 1  \left(1  P_{sc}\right)^2
The biterror rate depends on the bit to symbol mapping, but for \scriptstyle E_b/N_0 \gg 1 and a Graycoded assignment—so that we can assume each symbol error causes only one bit error—the biterror rate is approximately

P_{bc} \approx \frac{P_{sc}}{\frac{1}{2}k} = \frac{4}{k}\left(1  \frac{1}{\sqrt M}\right)Q\left(\sqrt{\frac{3k}{M1}\frac{E_b}{N_0}}\right).
Since the carriers are independent, the overall bit error rate is the same as the percarrier error rate, just like BPSK and QPSK.

\,P_b = P_{bc}
Oddk QAM
For odd \scriptstyle k, such as 8QAM (\scriptstyle k \;=\; 3) it is harder to obtain symbolerror rates, but a tight upper bound is:

P_s \leq{} 4Q\left(\sqrt{\frac{3kE_b}{(M  1)N_0}}\;\right)
Two rectangular 8QAM constellations are shown below without bit assignments. These both have the same minimum distance between symbol points, and thus the same symbolerror rate (to a first approximation).
The exact biterror rate, \scriptstyle P_b will depend on the bitassignment.
Note that both of these constellations are seldom used in practice, as the nonrectangular version of 8QAM is optimal. Example of second constellation's usage: LDPC and 8QAM.
Nonrectangular QAM
It is the nature of QAM that most orders of constellations can be constructed in many different ways and it is neither possible nor instructive to cover them all here. This article instead presents two, lowerorder constellations.
Two diagrams of circular QAM constellation are shown, for 8QAM and 16QAM. The circular 8QAM constellation is known to be the optimal 8QAM constellation in the sense of requiring the least mean power for a given minimum Euclidean distance. The 16QAM constellation is suboptimal although the optimal one may be constructed along the same lines as the 8QAM constellation. The circular constellation highlights the relationship between QAM and PSK. Other orders of constellation may be constructed along similar (or very different) lines. It is consequently hard to establish expressions for the error rates of nonrectangular QAM since it necessarily depends on the constellation. Nevertheless, an obvious upper bound to the rate is related to the minimum Euclidean distance of the constellation (the shortest straightline distance between two points):

P_s < (M1)Q\left(\sqrt{\frac{d_{min}^{2}}{2N_0}}\right)
Again, the biterror rate will depend on the assignment of bits to symbols.
Although, in general, there is a nonrectangular constellation that is optimal for a particular \scriptstyle M, they are not often used since the rectangular QAMs are much easier to modulate and demodulate.
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 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:^{[4]}
See also
References

^ UAS UAV communications links

^ Kylia products, dwdm mux demux, 90 degree optical hybrid, d(q) psk demodulatorssingle polarization

^ http://www.trangosys.com/products/pointtopointwirelessbackhaul/licensedwireless/trangolinkapexorion.shtml A Apex Orion

^ Howard Friedenberg and Sunil Naik. "Hitless Space Diversity STL Enables IP+Audio in Narrow STL Bands". 2005 National Association of Broadcasters Annual Convention. Retrieved April 17, 2005.
The notation used here has mainly (but not exclusively) been taken from

John G. Proakis, "Digital Communications, 3rd Edition",
External links

Interactive webdemo of QAM constellation with additive noise Institute of Telecommunicatons, University of Stuttgart

QAM bit error rate for AWGN channel – online experiment

How imperfections affect QAM constellation

Microwave Phase Shifters Overview by Herley General Microwave
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