Quadrature amplitude modulation

"QAM" redirects here. For other uses, see QAM (disambiguation).

Quadrature amplitude modulation (QAM) (Pronounced kwaːm or kăm) 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.

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 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. 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.

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 $f_{0}$ is the carrier frequency.

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 $I(t)$ and $Q(t)$ respectively. Because of the orthogonality property of the carrier signals, it is possible to detect the modulating signals independently.

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

${\begin{aligned}r_{i}(t)=&s(t)\cos(2\pi f_{0}t)\\=&I(t)\cos(2\pi f_{0}t)\cos(2\pi f_{0}t)+Q(t)\sin(2\pi f_{0}t)\cos(2\pi f_{0}t)\end{aligned}}$

Using standard trigonometric identities, we can write it as:

${\begin{aligned}r_{i}(t)=&{\frac {1}{2}}I(t)\left[1+\cos(4\pi f_{0}t)\right]+{\frac {1}{2}}Q(t)\sin(4\pi f_{0}t)\\=&{\frac {1}{2}}I(t)+{\frac {1}{2}}[I(t)\cos(4\pi f_{0}t)+Q(t)\sin(4\pi f_{0}t)]\end{aligned}}$

Low-pass filtering $r_{i}(t)$ removes the high frequency terms (containing $4\pi f_{0}t$), 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 may multiply $s(t)$ by a sine wave and then low-pass filter to extract $Q(t)$.

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 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.

Fourier analysis of QAM

In the frequency domain, QAM has a similar spectral pattern to DSB-SC modulation. Using the properties of the Fourier transform, we find that:

$S(f)={\frac {1}{2}}\left[M_{I}(f-f_{0})+M_{I}(f+f_{0})\right]+{\frac {1}{2j}}\left[M_{Q}(f-f_{0})-M_{Q}(f+f_{0})\right]$

where S(f), M_I(f) and M_Q(f) are the Fourier transforms (frequency-domain representations) of s(t), I(t) and Q(t), respectively.

Quantized QAM

As with many digital modulation schemes, the constellation diagram is a useful representation. 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 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, for constant mean constellation energy.

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 (see QAM tuner) 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).

Ideal structure

Transmitter

The following picture shows the ideal structure of a QAM transmitter, with a carrier frequency $f_{0}$ and $H_{t}$ 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. 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:

Failed to parse (syntax error): {\displaystyle 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 $v_{c}[n]$ and $v_{s}[n]$ are the voltages applied in response to the $n$^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 $H_{r}$ 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.

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).

In any application, the low-pass filter will be within h_r (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
• $E_{b}$ = Energy-per-bit
• $E_{s}$ = Energy-per-symbol = $kE_{b}$ with k bits per symbol
• $N_{0}$ = Noise power spectral density (W/Hz)
• $P_{b}$ = Probability of bit-error
• $P_{bc}$ = Probability of bit-error per carrier
• $P_{s}$ = Probability of symbol-error
• $P_{sc}$ = Probability of symbol-error per carrier
• $Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }e^{-t^{2}/2}dt,\ x\geq {}0$.

$Q(x)$ 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.

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.

Rectangular QAM

Constellation diagram for rectangular 16-QAM.

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. 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.

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, $P_{b}$ will depend on the bit-assignmentde.

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

• 
  Constellation diagram for rectangular 8-QAM.
• 
  Alternative constellation diagram for rectangular 8-QAM.

Non-rectangular QAM

Constellation diagram for circular 8-QAM.

Constellation diagram for circular 16-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, 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):

$P_{s}<(M-1)Q\left({\sqrt {d_{min}^{2}/2N_{0}}}\right)$.

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

See also

• Modulation for other examples of modulation techniques
• Phase-shift keying
• Amplitude and phase-shift keying or Asymmetric phase-shift keying (APSK)
• Carrierless Amplitude Phase Modulation (CAP)
• QAM tuner for HDTV

External links

• How imperfections affect QAM constellation