Fundamentals of Single-Carrier Modulation
CONTENTS 6.1 Overview
6.2 Basics of QAM, PAM, and CAP Modulation
6.2.1.3 Spectrum Shaping Filters
X(M−5)/2= X(M−5)/2−1+ b(M−5)/2+5 Y(M−5)/2= Y(M−5)/2−1+ bM
points will cause one bit error (9 points with I ,Q > 1 and point {1, 1}). For the other 6 points, misdetection to any of 3 closest points will cause 1 bit error. However, for points {1, 3}, {1, 7}, {3, 1}, {7, 1} misdetection to the fourth closest point will cause 3 bit errors, and for points{1, 5} and {5, 1} misdetection to the fourth closest point will cause 5 bit errors.
Therefore, if all symbols are equally probable, the average probability of a 1-bit error per symbol error is (10× 4 + 6 × 3)/(16 × 4) = 0.906. The probabilities of a 3-bit error and a 5-bit error, respectively, are 4/64 = 0.063, and 2/64 = 0.031. Neither 2-bit nor 4-bit errors will occur.
In DSL, additional data encoding schemes, such as forward error-correction (FEC) coding or trellis coding, are usually used prior to the constellation encoding. Referring to the functional diagram in Figure 6.2, these types of encoding are assumed to have been applied already to the incoming data stream. These and other encoding techniques are described in Chapters 8, 9, and 10 of this book.
6.2.1.3 Spectrum Shaping Filters
Two low-pass filters are intended for limiting and shaping of the transmit signal spectrum.
If the passband of the filter is much wider than T1, the amplitude and phase of the carrier changes almost instantly at the transition from one symbol to another (Figure 6.1). The
Qn
In 000XXX..
no flip 100XXX..
horizontal flip 110XXX..
horizontal flip
010XXX..
vertical and horizontal flip
101XXX..
vertical and horizontal flip
001XXX..
vertical flip
011XXX..
vertical flip 111XXX..
vertical and horizontal flip
FIGURE 6.5
Mapping sections for crossed constellations.
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Fundamentals of Single-Carrier Modulation 151
Normalized PSD, dBm/Hz
0
2 3 4 5
Frequency, MHz
6 7 8 9
−20
−10
−15
−25
−30
−35
−40
−5
FIGURE 6.6
Spectrum of QAM signal with 1/T = 1 Mbaud, fc = 5.5 MHz (wideband shaping—dashed line, square-root raised cosine shaping withα = 0.2—solid line).
normalized spectral magnitude function of the signal may be expressed as:
|SQAM( f )| ≈
sin(πT( f − fc)) πT( f − fc)
. (6.8)
Usage of this sinc-type spectrum (see Figure 6.6) is inefficient, because its slowly decaying side-lobes cause significant crosstalk into signals occupying neighboring frequency bands, except those that use narrow frequency bands spaced by 1/T from the carrier frequency
fc. The latter advantage is used in multi-carrier modulation, described in Chapter 7.
Shaping improves the efficiency of the spectrum usage. A popular example is a square-root raised cosine shaping filter having a spectral magnitude function given by
|G( f )| =
1 ,| f | ≤ f1
cosπ·T
2α[| f | − f1]
, f1≤ | f | ≤ f2
0 , elsewhere
, f1=1− α
2T , f2=1+ α
2T . (6.9)
Accordingly, the spectral magnitude function of the transmit signal is given by
|SQAM( f )| = |G( f − fc)|. (6.10) Parameter 0≤ α ≤ 1 in Equation 6.9 is called the excess bandwidth. The frequency bound-aries of the QAM signal spectrum with excess bandwidthα are:
fmin= fc−(1 + α)
2T , fmax= fc+(1 + α)
2T . (6.11)
Normalised power spectral density 1
1
0.9 1.2 1.4 1.6 1.8 2 2.1
Frequency, MHz 0.8
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.9
3 dB bandwidth = 1/T (MHz) = 1 MHz
Excess bandwidth = 0.1 MHz (20% in total)
FIGURE 6.7
Spectrum of a QAM signal with square-root raised cosine shaping (1/T = 1 Mbaud, fc= 1.5 MHz, α = 0.2).
Respectively, the bandwidth occupied2 by the QAM signal equals (1+α)T . In practical ap-plications, the PSD of the signal at frequencies below fminand above fmaxdrops at least 20 dB relative to its value at frequency fcdue to the square-root raised-cosine function. The 3-dB bandwidth, W3dB, representing the width of the QAM spectrum, is equal to the QAM symbol rateT1:
fmax,3dB= fc+ 1/2T,
fmin,3dB= fc− 1/2T, (6.12)
W3dB= fmax,3dB− fmin,3dB= 1 T.
Typically, the value ofα varies from 0.1 to 0.2. Some systems targeted for transmission media with poor and barely predictable characteristics, such as home wiring, may operate withα = 1 [G.989.1 2001]. The normalized PSD of a QAM signal using square-root raised-cosine shaping withα = 0.2 is presented in Figure 6.7. The frequency boundaries are: fmin= 1.5 − 1 × (1 + 0.2)/2 = 0.9 MHz, fmax= 1.5 + 1 × (1 + 0.2)/2 = 2.1 MHz.
6.2.1.4 Receiver
The functional diagram of a QAM receiver is presented in Figure 6.8a. It includes a demod-ulator with low-pass shaping filters, an equalizer and decision circuits (slicers) for both in-phase and quadrature components, and a constellation decoder. The timing recovery
2As was mentioned above, the symbol rate of a QAM signal must be less than 2 fcto avoid modulation distortion. In the case that 1/2T < fc≤ (1+α)/2T, the signal bandwidth is slightly less than (1+α)/T and equals fc+(1+α)/2T.
The 3-dB bandwidth remains the same.
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Fundamentals of Single-Carrier Modulation 153
FIGURE 6.8
(a) Functional diagram of a QAM receiver and (b) its complex-analytic representation.
circuit extracts the carrier frequency for demodulation purposes and symbol timing for the rest of the processing. The fact that after demodulation the signal has a baseband format is sometimes an advantage, because it simplifies implementation of QAM transceivers op-erating with high carrier frequencies. A complex-analytic representation of the receiver is presented in Figure 6.8b.
6.2.1.5 Demodulator
The demodulation process can be described by the following equations for the demodulated in-phase and quadrature signal components.
Sinph(t) = SQAM(t)cos(2π fct+ θ), Squad(t) = SQAM(t)sin(2π fct+ θ), (6.13) whereθ is the phase shift in the recovered carrier. Substituting Equation 6.6 and elimi-nating the high-frequency components, which are filtered out by the low-pass filters (see Figure 6.8), yields
Sinph(t) =
n
Ing(t − nT)
cosθ
2 −
n
Qng(t − nT)
sinθ 2 ,
(6.14) Squad(t) =
n
Ing(t − nT)
sinθ
2 −
n
Qng(t − nT)
cosθ
2 .
Equation 6.14 shows that demodulation requires a very accurate phase adjustment of the recovered carrier, and the demodulation is referred to in the industry as coherent demodu-lation. Ifθ is nonzero, the signal components are not orthogonal, and interference from the quadrature component affects the detection of the in-phase component, and vice versa.
The required accuracy of carrier phase recovery for QAM transmission may be derived directly from Equation 6.14. Assume that operation with the desired constellation requires a signal-to-noise ratio (SNR) of a dB. The inaccuracy of phase recovery will not cause sig-nificant SNR reduction if interference from the quadrature component is at least 12 dB below the level of noise corresponding to the required SNR.3 Because the used constel-lation diagrams are symmetric (kI = kQ), the average power of in-phase and quadrature components is usually the same,4and the maximum allowed value ofθ may be obtained from the following equation.
20 log10 cosθ
sinθ
= a + 12. (6.15)
Consider, for instance, that one uses 256-QAM transmission with a required BER of 10−7 and a noise margin of 6 dB. The minimum required SNR is a = 31.7 dB + 6 dB = 37.7 dB.
Solving Equation 6.15 yieldsθ < 0.188o, which is remarkably accurate. Usage of higher-order constellations requires even higher accuracy. To reach this high accuracy, special techniques of carrier frequency and phase recovery are used. Some of them are mentioned below.
6.2.1.6 Equalizer
The equalizer attempts to reduce intersymbol interference (ISI) in the receive signal and to maximize the SNR at the input of the decision circuit. Equalization techniques used in QAM receivers are described in Chapter 11. Typically, a fractionally spaced linear equalizer (FSLE) or decision feedback equalizer (DFE) is used. Both are usually built using finite impulse response (FIR) digital filters with adjustable complex coefficients, which allow simulta-neous processing of the in-phase and quadrature components. In addition, the equalizer can automatically adapt its impulse response to the characteristics of the particular loop.
The equalizer is usually adjusted to the loop during the link initialization. During normal operation, the filter coefficients are finely tuned in response to environmental changes.
The main impairments complicating DSL signal detection are ISI and noise (mostly crosstalk) accumulated in the line. ISI is caused by reflections and bandwidth limitation introduced in the loop and shaping filters. As the equalizer attempts to reduce ISI by am-plifying the signal in the suppressed frequency ranges (sometimes this procedure is called
“inverting the channel”), the noise tends to grow as a result of the amplification. Therefore, in the aim to increase the SNR, the equalizer attempts to find a compromise that allows low ISI without significant noise enhancement. This compromise is usually reached when the impulse response of the transmission channel is close to satisfying the Nyquist criterion, resulting in zero ISI at the instants t = nT. If a square-root raised-cosine shaping filter is used in the transmitter, and the rest of the transmission channel (from the input of the mod-ulator in Figure 6.2 to the output of the equalizer) has the frequency/phase response that is the same square-root raised-cosine, then the transmission channel has the well-known
3Assume the noise due to the interference of the quadrature component is additive, uncorrelated with the other noise components, and is below the level of the power sum of all other noise components by 12 dB. Then, impact of the interference from the quadrature component can be estimated as: 10× log10(1 + 10−1.2) = 0.26 dB, which is usually considered insignificant.
4To be more precise, the mentioned average power assumes averaging over the whole set of symbols. For any particular symbol, the value of quadrature noise may be either above the averaged (if its quadrature component power is larger than the average) or below the average (if its quadrature component power is smaller than the average). The peak value of noise is above the average by 20× log10(Qmax/Imin); the minimum value is below the average by the same value.
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Fundamentals of Single-Carrier Modulation 155
Normalised amplitude
1
0
t/T
Impulse response of a raised-cosine channel
0.8
0.6
0.4
0.2
−0.2
−6 −4 −2 0 2 4 6
10% excess bandwidth 20% excess bandwidth
FIGURE 6.9
Impulse response of a channel with raised-cosine transfer function.
raised-cosine transfer function with the impulse response g(t) = sinπ·t
T
· cosπα·t
T
π ·t
T
·
1− 4α·t
T
2. (6.16)
The impulse response in Equation 6.16 is presented in Figure 6.9. It obviously satisfies the Nyquist criterion because g(0) = 1 and g(nT) = 0, the latter due to the sin component in the equation. The figure shows that higher values ofα result in impulse responses that decay faster and have lower ISI between the decision points nT, which relaxes restrictions on jitter in symbol timing. On the other hand, higher values ofα obviously reduce the effi-ciency of QAM transmission by introducing additional bandwidth overhead. The optimal compromise is usually found in the rangeα = 0.1 to 0.2.
A superposition of traces of different waveforms appearing at the output of the equal-izer is usually referred to as an eye-diagram. An example of an eye-diagram for either the in-phase or quadrature component of 4-QAM, 8-QAM, and 16-QAM is presented in Figure 6.10. The traces show some residual ISI, which is close to zero at the instants of decision (sampling points).