Modulation techniques
Analog modulation
AM · SSB · FM · PM · QAM · SM
Digital modulation
OOK · FSK · ASK · PSK · QAM
MSK · CPM · PPM · TCM · OFDM
v  d  e
FHSS · DSSS

## Overview

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 Telecommunications, modulation is the process of varying a periodic Waveform, i Debt AIDS Trade in Africa (or DATA) is a Multinational non-government organization founded in January 2002 in London by U2 's In Telecommunications, a carrier wave, or carrier is a Waveform (usually Sinusoidal) that is modulated (modified with an input signal In the case of QAM, the amplitude of two waves, 90 degrees out-of-phase with each other (in quadrature) are changed (modulated or keyed) to represent the data signal.

Phase modulation (analog PM) and phase-shift 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. Phase modulation (PM is a form of Modulation that represents information as variations in the instantaneous phase of a Carrier wave. Phase-shift keying (PSK is a Digital Modulation scheme that conveys data by changing or modulating the phase of a reference signal This can also be extended to frequency modulation (FM) and frequency-shift keying (FSK), for these can be regarded a special case of phase modulation. Frequency-shift keying (FSK is a Frequency modulation scheme in which digital information is transmitted through discrete frequency changes of a Carrier wave

## 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:

$\ s(t) = I (t) \cos (2 \pi f_0 t) + Q (t) \sin (2 \pi f_0 t)$,

where I(t) and Q(t) are the modulating signals and f0 is the carrier frequency.

At the receiver, these two modulating signals can be demodulated using a coherent demodulator. A product detector is a type of Demodulator used for AM and SSB signals Such a receiver multiplies the received signal separately with both a cosine and sine signal to produce the received estimates of I(t) and Q(t) respectively. Because of the orthogonality property of the carrier signals, it is possible to detect the modulating signals independently. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

In the ideal case I(t) is demodulated by multiplying the transmitted signal with a cosine signal:

\begin{align}r_i(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_i(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}

Low-pass filtering ri(t) removes the high frequency terms (containing f0t), leaving only the I(t) term. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables A low-pass filter is a filter that passes low- Frequency signals but Attenuates (reduces the Amplitude of signals with frequencies This filtered signal is unaffected by Q(t), showing that the in-phase component can be received independently of the quadrature component. Similarly, we may multiply s(t) by a sine wave and then low-pass filter to extract Q(t).

It should be noted that here we assumed that the phase of the received signal is known at the receiver. If the demodulating phase is even a little off, it results in crosstalk between the modulated signals. In Electronics, the term crosstalk ( XT) refers to any phenomenon by which a signal transmitted on one circuit or channel of a Transmission system 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 (or analogue) television encodes Television picture and sound information and transmits it as an Analog signal: one in which the

Analog QAM is used in NTSC and PAL television systems, where the I- and Q-signals carry the components of chroma (colour) information. NTSC ( National Television System Committee) is the Analog television system used in the United States, Canada, Japan, Mexico PAL, short for Phase Alternating Line, is a colour -encoding system used in Broadcast television systems in large parts of the world "Compatible QAM" or C-QUAM is used in AM stereo radio to carry the stereo difference information. C-QUAM is the method of AM stereo broadcasting used in Canada, the United States and most other countries AM Stereo is a term given to a number of mutually incompatible techniques for Broadcasting stereo audio in the AM band in a manner Radio is the transmission of signals by Modulation of electromagnetic waves with frequencies below those of visible Light.

### Fourier analysis of QAM

In the frequency domain, QAM has a similar spectral pattern to DSB-SC modulation. Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency Double-sideband suppressed-carrier transmission (DSB-SC transmission in which (a frequencies produced by Amplitude modulation are symmetrically spaced above and Using the properties of the Fourier transform, we find that:

$S(f) = \frac{1}{2}\left[ M_I(f - f_0) + M_I(f + f_0) \right] + \frac{1}{2j}\left[ M_Q(f - f_0) - M_Q(f + f_0) \right]$

where S(f), MI(f) and MQ(f) are the Fourier transforms (frequency-domain representations) of s(t), I(t) and Q(t), respectively. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and

## Quantized QAM

As with many digital modulation schemes, the constellation diagram is a useful representation. A constellation diagram is a representation of a signal modulated by a digital Modulation scheme such as Quadrature amplitude modulation or Phase-shift keying 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. Cross-QAM). Since in digital telecommunications the data is usually binary, the number of points in the grid is usually a power of 2 (2, 4, 8 …). The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. Since QAM is usually square, some of these are rare—the most common forms are 16-QAM, 64-QAM, 128-QAM and 256-QAM. By moving to a higher-order constellation, it is possible to transmit more bits per symbol. A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication 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. is a one volume manga created by Tsutomu Nihei as a prequel to his ten-volume work Blame!. In Telecommunication, an error ratio is the Ratio of the number of Bits elements, characters, or blocks incorrectly received

If data-rates beyond those offered by 8-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. Phase-shift keying (PSK is a Digital Modulation scheme that conveys data by changing or modulating the phase of a reference signal 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. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system

64-QAM and 256-QAM are often used in digital cable television and cable modem applications. Digital cable is a type of Cable television distribution using Digital Video compression. A cable modem is a type of Modem that provides access to a data signal sent over the Cable television infrastructure In the US, 64-QAM and 256-QAM are the mandated modulation schemes for digital cable (see QAM tuner) as standardised by the SCTE in the standard ANSI/SCTE 07 2000. The United States of America —commonly referred to as the Digital cable is a type of Cable television distribution using Digital Video compression. In North American Digital video, a QAM tuner is a device present in some Digital televisions and similar devices which enables direct reception of The Society of Cable Telecommunications Engineers or SCTE is a Non-profit Professional association for the advancement of technology related to cable telecommunications Note that many marketing people will refer to these as QAM-64 and QAM-256. In the UK, 16-QAM and 64-QAM are currently used for digital terrestrial television (Freeview and Top Up TV). The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located Digital Terrestrial Television ( DTTV or DTT) is an implementation of digital technology to provide a greater number of channels and/or better quality of picture and Freeview is an operator of free Digital terrestrial television in the United Kingdom, using the DVB-T standard Top Up TV is a digital Terrestrial Pay TV service offering content from 23 subscription Channels such as Living, G

### Ideal structure

#### Transmitter

The following picture shows the ideal structure of a QAM transmitter, with a carrier frequency f0 and Ht the frequency response of the transmitter's filter:

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 amplitude-shift keying (ASK) modulator. Amplitude-shift keying ( ASK) is a form of Modulation that represents Digital Data as variations in the Amplitude of a Carrier 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 vc[n] and vs[n] are the voltages applied in response to the nth symbol to the cosine and sine waves respectively.

The receiver simply performs the inverse process of the transmitter. Its ideal structure is shown in the picture below with Hr the receive filter's frequency response:

Multiplying by a cosine (or a sine) and by a low-pass 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. Amplitude-shift keying ( ASK) is a form of Modulation that represents Digital Data as variations in the Amplitude of a Carrier

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 Phase-Locked Loop (PLL). A phase-locked loop or phase lock loop (PLL is a Control system that generates a signal that has a fixed relation to the phase of a "reference"

In any application, the low-pass filter will be within hr (t): here it was shown just to be clearer.

## Quantized QAM performance

The following definitions are needed in determining error rates:

• M = Number of symbols in modulation constellation
• Eb = Energy-per-bit
• Es = Energy-per-symbol = kEb with k bits per symbol
• N0 = Noise power spectral density (W/Hz)
• Pb = Probability of bit-error
• Pbc = Probability of bit-error per carrier
• Ps = Probability of symbol-error
• Psc = Probability of symbol-error per carrier
• $Q(x) = \frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt,\ x\geq{}0$. A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication In Science, and especially in Physics and Telecommunication, noise is fluctuations in and the addition of external factors to the stream of target In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( The watt (symbol W) is the SI derived unit of power, equal to one Joule of energy per Second. The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. Probability is the likelihood or chance that something is the case or will happen

Q(x) is related to the complementary Gaussian error function by: $Q(x) = \frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right)$, which is the probability that x will be under the tail of the Gaussian PDF towards positive infinity. In Mathematics, the error function (also called the Gauss error function) is a Special function (non- elementary) which occurs in Probability In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

The error-rates quoted here are those in additive white Gaussian noise (AWGN). Explanation In communications, the additive white Gaussian noise ( AWGN) channel model is one in which the only impairment is the linear addition of White noise is a random signal (or process with a flat Power spectral density. Gaussian noise is noise that has a Probability density function (abbreviated pdf of the Normal distribution (also known as Gaussian distribution Explanation In communications, the additive white Gaussian noise ( AWGN) channel model is one in which the only impairment is the linear addition of

Where coordinates for constellation points are given in this article, note that they represent a non-normalised constellation. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point 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

Constellation diagram for rectangular 16-QAM. A constellation diagram is a representation of a signal modulated by a digital Modulation scheme such as Quadrature amplitude modulation or Phase-shift keying

Rectangular QAM constellations are, in general, sub-optimal 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. Pulse-amplitude modulation, acronym PAM, is a form of signal Modulation where the message information is encoded in the Amplitude of a series of signal The non-square constellations, dealt with below, achieve marginally better bit-error rate (BER) but are harder to modulate and demodulate.

The first rectangular QAM constellation usually encountered is 16-QAM, the constellation diagram for which is shown here. A Gray coded bit-assignment is also given. Name Bell Labs researcher Frank Gray introduced the term reflected binary code in his 1947 patent application remarking that the code had "as The reason that 16-QAM is usually the first is that a brief consideration reveals that 2-QAM and 4-QAM are in fact binary phase-shift keying (BPSK) and quadrature phase-shift keying (QPSK), respectively. Phase-shift keying (PSK is a Digital Modulation scheme that conveys data by changing or modulating the phase of a reference signal Phase-shift keying (PSK is a Digital Modulation scheme that conveys data by changing or modulating the phase of a reference signal Also, the error-rate performance of 8-QAM is close to that of 16-QAM (only about 0. 5 dB better), but its data rate is only three-quarters that of 16-QAM. The decibel ( dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power or intensity relative to

Expressions for the symbol error-rate of rectangular QAM are not hard to derive but yield rather unpleasant expressions. For an even number of bits per symbol, k, exact expressions are available. They are most easily expressed in a per carrier sense:

$P_{sc} = 2\left(1 - \frac{1}{\sqrt M}\right)Q\left(\sqrt{\frac{3}{M-1}\frac{E_s}{N_0}}\right)$,

so

$\,P_s = 1 - \left(1 - P_{sc}\right)^2$.

The bit-error rate will depend on the exact assignment of bits to symbols, but for a Gray-coded assignment with equal bits per carrier:

$P_{bc} = \frac{2}{k}\left(1 - \frac{1}{\sqrt M}\right)Q\left(\sqrt{\frac{3k}{M-1}\frac{E_b}{N_0}}\right)$,

so

$\,P_b = 1 - \left(1 - P_{bc}\right)^2$.

#### Odd-k QAM

For odd k, such as 8-QAM (k = 3) it is harder to obtain symbol-error rates, but a tight upper bound is:

$P_s \leq{} 4Q\left(\sqrt{\frac{3kE_b}{(M-1)N_0}}\right)$.

Two rectangular 8-QAM constellations are shown below without bit assignments. These both have the same minimum distance between symbol points, and thus the same symbol-error rate (to a first approximation).

The exact bit-error rate, Pb will depend on the bit-assignment.

Note that neither of these constellations are used in practice, as the non-rectangular version of 8-QAM is optimal.

### Non-rectangular QAM

Constellation diagram for circular 8-QAM. A constellation diagram is a representation of a signal modulated by a digital Modulation scheme such as Quadrature amplitude modulation or Phase-shift keying
Constellation diagram for circular 16-QAM. A constellation diagram is a representation of a signal modulated by a digital Modulation scheme such as Quadrature amplitude modulation or Phase-shift keying

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, lower-order constellations.

Two diagrams of circular QAM constellation are shown, for 8-QAM and 16-QAM. The circular 8-QAM constellation is known to be the optimal 8-QAM constellation in the sense of requiring the least mean power for a given minimum Euclidean distance. The 16-QAM constellation is suboptimal although the optimal one may be constructed along the same lines as the 8-QAM constellation. The circular constellation highlights the relationship between QAM and PSK. Phase-shift keying (PSK is a Digital Modulation scheme that conveys data by changing or modulating the phase of a reference signal 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 non-rectangular 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 straight-line distance between two points):

$P_s < (M-1)Q\left(\sqrt{d_{min}^{2}/2N_0}\right)$. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler

Again, the bit-error rate will depend on the assignment of bits to symbols.

Although, in general, there is a non-rectangular constellation that is optimal for a particular M, they are not often used since the rectangular QAMs are much easier to modulate and demodulate.

## References

These results can be found in any good communications textbook, but the notation used here has mainly (but not exclusively) been taken from:

• John G. Proakis, "Digital Communications, 3rd Edition", McGraw-Hill Book Co. , 1995. ISBN 0-07-113814-5
• Leon W. Couch III, "Digital and Analog Communication Systems, 6th Edition", Prentice-Hall, Inc. , 2001. ISBN 0-13-081223-4