LINE CODING AND MODULATION

Line coding is the conversion of abstract symbols into real, temporal waveforms to be transmitted in the baseband while modulation is the process where the message infor-mation is added to a carrier according to radio terminology [1]. However, for wired digital transmission, e.g, for xDSL (x stands for different types of digital subscriber line), people tend to mix the terminologies of line coding and modulation. For xDSL, we sometimes refer to modulation as the conversion of bit streams into equivalent ana-log signals that are suitable for the transmission line [2]. The reason is that modulation is usually done digitally for xDSL while modulation is done at radio frequencies (RF) for radio communication.

In this book, we do not confuse readers with different jargons. We specify line codes as baseband or passband codes. The baseband codes are referred to the codes that have energy at DC while the passband codes are referred to the codes that do not have energy at DC, although baseband codes can be modulated to a carrier as well.

2.2.1 Baseband Codes

One of the widely used baseband codes is the phase amplitude modulation (PAM) code. A b-bit PAM has equally spaced levels symmetrically placed about zero A b-bit PAM is often referred to PAM-M or M-ary PAM. In Fig. 2-1 we

show a 3-bit or 8 PAM. The assignment of the b-bit information to the possible signal amplitudes can be done in different ways. The preferred mapping or assignment is called Gray coding in which the adjacent signal amplitudes differ by only one binary digit [3]. If a PAM modulates a carrier, it is usually called amplitude-shift keying (ASK) in digital communication.

As discussed in Chapter 1, the peak-to-average ratio (PAR) has a strong impact on the data converter requirements. Assume that the distance between two adjacent levels is d, the peak amplitude of an M-ary PAM is then d • Under the assumption of equal probability of occurrence for all the levels, the rms amplitude is given by (if M is even)

2.2 Line Coding and Modulation 29

Therefore, the peak-to-average ratio is given by

The simulated PAR for PAM is shown in Fig. 2-2. The PAR for PAM approaches 1.73

when the number of levels increases. The 4-PAM, or two bits per quaternary (2B1Q) used in integrated service digital network (ISDN) and high-bit-rate digital subscriber line (HDSL) has a PAR of about 1.34.

Binary PAM or 2-PAM is widely used in both radio communication and xDSL due to its simplicity. There are two major classes of binary line codes: level codes and tran-sition codes. Level codes carries information in the voltage level while trantran-sition

30 Chapter 2. Data Converter Requirements for Communications

codes carry information in the change in level appearing in the line code waveform.

There are many variations based on 2-PAM. Interested readers are referred to [1,2,3]-It is not unusual that PAM codes can have an odd number of levels. [1,2,3]-It can include a level having a value of 0. For instance, PAM-3 is used in Ethernet 100BASE-TX and PAM-5 is used in Ethernet 100BASE-T2 and 1000BASE-T [4]. In this case, the code mapping is more complex [4]

Pulse code modulation (PCM) codes are widely used in communication for transmit-ting voice signals. A PCM signal is obtained from the PAM signal by encoding each value into a digital word. A PCM signal can be thought of as a serial representation of a PAM signal. The actual transmission of PCM codes is binary. In order to reduce the data converter requirement, compression such as A law or law is used for voice communication. Interested readers are referred to [5].

2.2.2 Passband Codes

Passband codes have no energy at DC. The widely used passband code is the quadra-ture amplitude modulation (QAM) code. A QAM signal is constructed by the summa-tion of an in-phase signal (I) and a quadrature signal (Q), given by

where is the carrier frequency and is a real-time pulse like a sinc or a square-root raised cosine pulse that is determined by the digital data stream. The multiplica-tion of the pulse by a cosine and sine moves the energy away from the DC to the carrier frequency and different pulse shapes have different properties such as bandwidth requirement, inter-symbol interference, etc.

The QAM codes are two-dimensional. With a b-bit QAM there are symbols in the constellation. A 6-bit or 64-QAM constellation is shown in Fig. 2-3. Assume

that the distance between two neighboring symbols on the x- or the y-axis is d, an QAM constellation have the following magnitudes if b is even

2.2 Line Coding and Modulation 31

Under the assumption of equal probability of occurrence for all the levels, the rms magnitude is given by

The maximum signal magnitude is

Therefore, the peak-to-average ratio is given by

where is the peak-to-average ratio of the carrier. If the carrier is a sine wave, is equal to 1.4.

In Fig. 2-4 we show the peak-to-average ratio as a function of the number of levels M.

When the number of levels M increases to infinity, the PAR approaches the maximum value of 2.45.

Carrierless amplitude and phase (CAP) modulation is considered as a special case of QAM. If the carrier frequency is not significantly larger than the bandwidth, the car-rier modulation in QAM is superficial because a judicious choice of two DC-free func-tions can realize the same function. Compared with QAM, CAP simplifies the transmitter implementation [6].

32 Chapter 2. Data Converter Requirements for Communications

If we keep the signal amplitude constant and only change the phase according to the digital bit stream, we have MPSK, or M-ary phase-shift-keying. Binary PSK (BPSK, M = 2) and quadrature PSK (QPSK, M = 4) are widely used in radio communications.

BPSK is equivalent to 2 PAM (if modulated) or 2 QAM, and QSPK is equivalent to 4 QAM.

We can also use the bit information to modulate the carrier frequencies. It is called fre-quency-shifting keying (FSK). Binary FSK (BFSK) is widely used in radio communi-cations.

2.2.3 Excess Bandwidth

When we transmit data successively, interferences between the successive symbols deteriorate the performance. To minimize the interferences between successive sym-bols (intersymbol interferences), we can use Nyquist pulses that are orthogonal to one another. Another possibility to minimize the intersymbol interferences is to introduce a controlled amount of interference at the transmitter, which can be removed at the receiver. This technique is called partial-response signalling. Interested readers are referred to [1].

There are many Nyquist pulses, and the best known are the raised-cosine pulses, which decay with instead of 1/t as in sinc pulses. The fast decay in time is crucial since it reduces the timing-phase errors in the sampling clock of the receiver.

Suppose that the symbol period is T, or the symbol rate is The excess