Carrier and Timing Recovery

The carrier at the receive side is necessary for coherent demodulation. It could be either received from the transmitter (for example, as a pilot tone) or recovered from the received data signal. Both options are used in practical implementations, although carrier recovery is usually more attractive because it does not require any additional signals to be transmitted.

However, it is also more complex.

The required high accuracy of carrier phase recovery was shown during the description of the demodulation process. Proper symbol timing is a key requirement for successful signal processing and decision making. The accuracy of symbol timing is characterized by possible fluctuations of the symbol period (jitter), usually estimated as a percentage.

Obviously, these fluctuations influence equalization and reduce accuracy of the decision instants. Typically, jitter in symbol timing is maintained to be below a few percent.

Methods of carrier recovery and timing recovery, including acquisition and precise track-ing of phase, are discussed in Chapter 12, and additional information can be found in [Gitlin 1995]. High precision of the carrier and symbol timing recovery is achieved by using adjusting algorithms directed by decisions on the received data. In many implementations, a joint carrier and timing recovery mechanism is used. This is especially convenient for systems like VDSL1 where the carrier frequency period and symbol period are multiples of the same timing reference; thus, the carrier recovery circuit may source the timing recovery circuit and vice versa.

6.2.2 PAM

PAM uses pulses with duration T as a signal waveform. Different PAM symbols are distin-guished by their amplitudes. In particular, a symbol of 2^M-PAM can have 2^Mpossible values of amplitude (also called “levels”) representing 2^Mpossible values of an M-bit group. An example of a 4-PAM (also known as 2B1Q) signal is presented in Figure 6.11. Sometimes, a passband PAM is also considered. Passband PAM refers to a modulation technique that uses a sine waveform of a particular frequency (carrier frequency) with different amplitudes. All conclusions regarding PAM technology obtained in this section hold for passband PAM.

The functional diagrams of the PAM transmitter and receiver are presented in Figure 6.12.

They are very similar to those of QAM and include the same functional elements, except the multiplication by the carrier. Referring to Figure 6.2 and Figure 6.8, the PAM transceiver may be represented as a result of multiplication of the QAM in-phase component by 1 (cos(2π fct) = 1 if fc = 0) and the QAM quadrature component by 0 (sin(2π fct) = 0 if fc = 0). Thus, PAM can be directly interpreted as QAM with a zero carrier frequency.

Consequently, all the main components of the PAM transceiver shown in Figure 6.12 operate in the same way as those described for QAM.

The constellation diagram of PAM is one-dimensional, as shown in Figure 6.11b. The number of possible settings of In(kI) equals the number of transmit signal levels. Gray mapping is typically used, so that groups of bits mapped into adjacent levels of amplitude differ by one bit only. A simple rule providing Gray mapping for 2^M-PAM is similar to the one in Table 6.1 and is presented in Table 6.3.

Differential encoding allows the PAM decoder to be invariant to crossing of the wires in the twisted pair. Similar to QAM, the first bit of the encoded group of bits (bit b₁missing in

n-symbol

(b). PAM constellation diagrams and encoding examples

−3 −1 +1 +3

Examples of PAM signal and constellation diagrams.

the second column of Table 6.3) defines whether the polarity of the current symbol coincides with or differs from the polarity of the previous symbol. An example of bit mapping for 8-PAM when differential encoding is used is presented in Figure 6.11. In practical applica-tions, however, a “trial and error” method is also used in the decoder to identify crossing of the wires instead of differential encoding.

Low-pass shaping filters, equalizers, and decision circuits of a PAM transceiver operate in the same way as in a QAM transceiver and have similar characteristics; decoding is a

Transmitter

Functional diagram of a PAM transceiver.

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Fundamentals of Single-Carrier Modulation 159

TABLE 6.3

Gray Mapping for PAM

Direct encoding Differential encoding

In(binary) = [X1X2. . . XM1]−2^M(binary) In(binary) = [X1X2. . . X_M−11]

X1= b1 X1= b2

X2= X1+ b2 X2= X1+ b3

X3= X2+ b3 X3= X2+ b4

. . . . . . . .

XM= X_M−1+ bM X_M−1= X_M−2+ bM

simple inversion of the mapping rule (as in Table 6.3, for instance). Figure 6.13 shows some examples of PAM signal spectra. The eye-diagrams of 4-QAM and 8/16-QAM presented in Figure 6.10 are also relevant for 2-PAM (commonly known as NRZ) and 4-PAM (2B1Q), respectively.

6.2.3 CAP

The carrierless amplitude-phase modulation [Im 1995b], [Haykin 1998] was proposed as an alternative to QAM. CAP creates a passband transmit signal with characteristics very similar to QAM but uses digital filtering instead of multiplication by the carrier frequency.

PSD, dBm/Hz

0 0.1 0.2 0.3 0.4 0.5

Frequency, MHz

0.6 0.7 0.8 0.9 1

−70

−40

−50

−60

−30

−80

−90

−100

−110

4-PAM (2B1Q), 4-th order shaping filter Spectrum of PAM signals

16-PAM, 4-th order shaping filter 16-PAM, 6-th order shaping filter

FIGURE 6.13

Spectrum of 4-PAM (1/T = 396 kBaud) and 16-PAM (1/T = 198 kBaud); (α ≈ 0.8 for the 4th order shaping filter, andα ≈ 0.45 for the 6th order shaping filter).

Transmitter

Functional diagram of a CAP transceiver.

This sometimes simplifies the implementation and avoids carrier recovery in the receiver.

The penalty is a relatively high sampling rate, which makes CAP inconvenient for appli-cations when the ratio of the carrier frequency to the symbol rate is small. Thus, CAP was used in pre-standard ADSL and in HDSL but is not used in VDSL1, which utilizes several rather narrow frequency bands (although some standards such as [ETSI TS 101 270-2] allow QAM/CAP dual mode operation). The rest of the properties of CAP are mostly the same as for QAM.

The functional diagrams of a CAP transmitter and receiver are presented in Figure 6.14.

The transmitter is similar to the one in Figure 6.2a, and creates a passband signal by the in-phase and quadrature filters with impulse responses

f(t) = g(t) cos(2π fct)q(t) = g(t) sin(2π fct), (6.18) where fcis the “virtual carrier,” which is simply the center frequency of the transmit signal spectrum.⁵The CAP transmit signal, accordingly, equals

SCAP(t) =!

5In practical implementations, the actual time span where the channel impulse responses accurately follow Equation 6.18 should be at least 8T (±4T from the center lobe of the impulse response).

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Fundamentals of Single-Carrier Modulation 161

where I_nand Q_nare outputs from the constellation encoder, and ˜p denotes a Hilbert trans-formation of p.

Equation 6.19 shows that the CAP signal is actually very similar to a QAM signal. It oc-cupies the same spectrum (determined by the shaping component g(t) of the in-phase and quadrature filters (see Figure 6.7)), may use the same constellation diagrams, but experiences additional rotation by a fixed phase increment of 2π fcT in each symbol pe-riod. This rotation is the same in magnitude but opposite in sign to the rotation of the QAM carrier during the symbol period. Therefore, a CAP signal looks like a QAM sig-nal in which the phase of the carrier is set to zero at the beginning of each symbol pe-riod. In the case when fc × T is equal to any integer, CAP and QAM signals are exactly the same, and thus the same receiver (either QAM or CAP) works for both signals. This feature is used in some combined CAP/QAM SCM transceivers to simplify the startup process.

The receiver (see Figure 6.14) contains two adaptive filters, a decision circuit, and a con-stellation decoder. Adaptive filters are intended to combat the ISI and to provide a channel response that satisfies the Nyquist criterion. They are built as an inverted Hilbert pair with impulse responses g_a1(t) = −˜ga2(t), respectively. As the paths between the transmitter and the adaptive filters are linear, the signals a₁, a₂on the outputs of the adaptive filters are:

a₁=! the convolution operator), including the shaping filters g(t), the loop h(t), and the adaptive filters ga(t). Because the channel response x(t) is equalized to satisfy the Nyquist criterion, i.e., x(kT) = 1 and ˜x(kT) = 0, the decision circuit recovers the values of Inand Qnusing output signals of the adaptive filters at the decision instants.