Quadrature amplitude modulation

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QAM redirects here; for other uses of that abbreviation, see QAM (disambiguation).

Quadrature amplitude modulation (QAM) is a modulation scheme which conveys data by changing (modulating) the amplitude of two carrier waves. These two waves, usually sinusoids, are out of phase with each other by 90° and are thus called quadrature carriers — hence the name of the scheme.

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Overview

As with all modulation schemes, QAM conveys data by changing some aspect of a base signal, the carrier wave, (usually a sinusoid) in response to a data signal. In the case of QAM, the amplitude of two quadrature waves is changed (modulated or keyed) to represent the data signal.

Phase modulation (analogue PM) and phase-shift keying (digital PSK) can be regarded as a special case of QAM, where the amplitude of the modulating signal is constant, with only the phase varying. This can also be extended to frequency modulation (FM) and frequency-shift keying (FSK), as this can be regarded as a special case of phase modulation.

Although analogue QAM is possible, this article focuses on digital QAM. Analogue QAM is used in NTSC and PAL television systems, where the I- and Q-signals carry the components of chroma (colour) information. "Compatible QAM" or C-QUAM is used in AM stereo radio to carry the stereo difference information.

As for many digital modulation schemes, the constellation diagram is a useful representation and is relied upon in this article.

In QAM, the constellation points are usually arranged in a square grid with equal vertical and horizontal spacing, although other configurations are possible (see e.g. Cross-QAM). 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 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. 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.

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. 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.

64-QAM and 256-QAM are often used in digital cable television and cable modem applications. In the US, 64-QAM and 256-QAM are the mandated modulation schemes for digital cable, as standardised by the SCTE in the standard ANSI/SCTE 07 2000. 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).

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Ideal structure

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Transmitter

The following picture shows the ideal structure of a QAM transmitter:

Image:Qam transmitter.png

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 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:

<math> s(t) = \sum_{n=-\infty}^{\infty} 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</math>

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Receiver

The receiver simply performs the inverse process of the transmitter. Its ideal structure is shown in the picture below:

Image:Qam receiver diagram.png

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. Differently from the ASK, the frequency the A/D converter is working to is not twice the maximum frequency of the transmitted signal, but twice the maximum frequency of the signal itself. For example, if a 1 MHz-wide signal is transmitted at a central frequency of 100 MHz, the A/D converter has to work at 2 MHz (not at 202, like in ASK).

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

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Performance

The diagrams that were shown represent the so called rectangular QAM, and it can be seen like two different ASK-modulated channels. The performance for the QAM modulation can be easily deduced by the known results for the ASK modulation (see the related article). Let us introduce the following notation:

  • L is the number of possible levels of voltage to be used
  • hr (t) is the impulse response of the filter that is used at the receiver
  • G is the total gain introduced by the transmitter, the receiver and by the effect of the channel
  • A is the maximum value of the voltage that can be transmitted

The number of possible symbols to be transmitted is L2.Since the same channel is used twice, the noise it introduces is going to be sampled twice as well. The total power of the noise is doubled, and so is the variance of the probability density functions of receiving the right symbol:

<math>\sigma_N = 2 \int_{-\infty}^{+\infty} \Phi_N (f) \cdot |H_r (f)|^2 df</math>

where <math>\Phi_N (f)</math> is the spectral density of the noise within the band and Hr (f) is the continuous Fourier transform of the impulse response of the filter hr (t).

The possibility to make an error on a single ASK-modulated virtual channel is given by:

<math>P^{(ask)}_e = \left( 1 - \frac{1}{L} \right) \operatorname{erfc} \left( \frac{A G}{\sqrt{2} (L-1) \sigma_N} \right) </math>

The possibility to make an error in the whole transmission is given by the sum of the two Pe(ask) (because they are not correlated) minus the possibility of making an error on both virtual channels at the same time, that equals the square of Pe(ask), that means:

<math>P^{tot}_e = 2 P^{(ask)}_e - \left[ P^{(ask)}_e \right]^2 \approx 2 P^{(ask)}_e </math>
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Definitions

For determining error-rates we will need some definitions:

  • <math>M</math> = Number of symbols in modulation constellation
  • <math>E_b</math> = Energy-per-bit
  • <math>E_s</math> = Energy-per-symbol = <math>kE_b</math> with k bits per symbol
  • <math>N_0</math> = Noise power spectral density (W/Hz)
  • <math>P_b</math> = Probability of bit-error
  • <math>P_{bc}</math> = Probability of bit-error per carrier
  • <math>P_s</math> = Probability of symbol-error
  • <math>P_{sc}</math> = Probability of symbol-error per carrier
  • <math>Q(x) = \frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt,\ x\geq{}0

</math>.

<math>Q(x)</math> is related to the complementary Gaussian error function by: <math>Q(x) = \frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right)</math>, 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 non-normalised constellation. That is, if a particular mean average energy were required (e.g. unit average energy), the constellation would need to be linearly scaled.

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Rectangular QAM

Image:16QAM Gray Coded.png 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. 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. 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. 8-QAM presents problems in dividing an odd number of bits between the two carriers, and is rarely used since 8-PSK is considerably simpler.

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, <math>k</math>, exact expressions are available. They are most easily expressed in a per carrier sense:

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

so

<math>\,P_s = 1 - \left(1 - P_{sc}\right)^2</math>.

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:

<math>P_{bc} = \frac{4}{k}\left(1 - \frac{1}{\sqrt M}\right)Q\left(\sqrt{\frac{3k}{M-1}\frac{E_b}{N_0}}\right)</math>,

so

<math>\,P_b = 1 - \left(1 - P_{bc}\right)^2</math>.
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Odd-<math>k</math> QAM

Image:Rectangular 8QAM.png For odd <math>k</math>, such as 8-QAM (<math>k=3</math>) it is harder to obtain symbol-error rates, but a tight upper bound is:
<math>P_s \leq{} 4Q\left(\sqrt{\frac{3kE_b}{(M-1)N_0}}\right) </math>.

Two rectangular 8-QAM constellations are shown, without bit-assignments. These two both have the same minimum distance between symbol points and thus the same symbol-error rate.

The exact bit-error rate, <math>P_b</math> will depend on the bit-assignment.

Image:Rectangular 8QAM v2.png
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Non-rectangular QAM

Image:Circular 8QAM.png 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. 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):

<math>P_s < (M-1)Q\left(\sqrt{d_{min}^{2}/2N_0}\right)</math>.

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 <math>M</math>, they are not often used since the rectangular QAMs are much easier to modulate and demodulate.

Image:Circular 16QAM.png
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See also

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External links

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References

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

es:Modulación de amplitud en cuadratura fi:QAM fr:Modulation d'amplitude en quadrature hu:Kvadratúra amplitúdómoduláció it:QAM ja:デジタル変調 pl:Modulacja QAM zh:QAM

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