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Chapter 3: Modulation and Coding
In this chapter
 Modulation
 Interleaving
 Channel Codes
 Bibliography
Modulation and channel coding are fundamental components of a digital communication system. Modulation is the process of mapping the digital information to analog form so it can be transmitted over the channel. Consequently every digital communication system has a modulator that performs this task. Closely related to modulation is the inverse process, called demodulation, done by the receiver to recover the transmitted digital information. The design of optimal demodulators is called detection theory. An OFDM system performs modulation and demodulation for each subcarrier separately, although usually in a serial fashion, to reduce complexity.
Channel coding, although not strictly speaking mandatory for digital communication, is essential for good performance. After Claude Shannon published his groundbreaking paper [16] that started information theory as a scientific discipline and established the fundamental law equation 3.1 of the capacity C of an AWGN communication channel with bandwidth W and signal to noise ratio SNR, communication scientists and engineers have been searching for methods on how to reach the capacity of the channel.
Shannon's proof of the channel capacity formula was nonconstructive, and thus did not provide a method to actually reach the capacity C. As a consequence, coding theory got started as a discipline that research methods to design a communication system that has performance close to the fundamental limit C. The methods developed under coding theory have different names; channel coding, forward error correction (FEC), and error control coding are the most usual. Channel coding and basic types of FEC codes were introduced in "Channel Coding" in Chapter 1. Note that channel coding and source coding, introduced in the "Source Coding" section in Chapter 1, although related, are quite different topics.
Our goal in this chapter is to discuss modulation, detection, channel coding and interleaving from an OFDM system of point of view. Channel coding is such an important component of OFDM systems that the term Coded Orthogonal Frequency Division Multiplexing (COFDM) is sometimes used. The reasons for this emphasis on channel coding for OFDM systems will become apparent in the channel coding section of this chapter. Modulation and demodulation or detection are discussed in depth in several books [11, 17, 18]. Channel coding is the topic of a wide variety of books [10, 20] and also books that concentrate on specific types of FEC codes [7, 9].
Modulation
As was discussed in "Modulation" in Chapter 1, modulation can be done by changing the amplitude, phase or frequency of transmitted Radio Frequency (RF) signal. For an OFDM system, the first two methods can be used. Frequency modulation cannot be used because subcarriers are orthogonal in frequency and carry independent information. Modulating subcarrier frequencies would destroy the orthogonality property of the subcarriers; this makes frequency modulation unusable for OFDM systems.
The main design issue of the modulator is the constellation used. Constellation is the set of points that can be transmitted on a single symbol. The used constellation affects several important properties of a communication system; for example, bit error rate (BER), peak to average power ratio (PAPR), and RF spectrum shape. The single most important parameter of a constellation is minimum distance. Minimum distance (d_{min}) is the smallest distance between any two points in the constellation. Therefore d_{min} determines the least amount of noise that is needed to generate a decision error. Actual BER or P_{b} of a constellation can in many cases be calculated using the Qfunction defined in Equation 3.2. The value of the Qfunction is equal to the area under the tail of the Probability Density Function (PDF) of a zero mean and unit variance normal random variable.
P_{b} is a function of energy per bit to noise ratio or signal energy to noise ratio ; these ratios are related by a simple scaling factor that depends on the number of bits k transmitted per symbol E_{s}=kE_{b}. The usefulness of as a measure of the signal quality will became apparent when different coding schemes are discussed in "Channel Codes" later in this chapter. Then a general form of P_{b} equations for several important constellations in terms of the Qfunction is shown in Equation 3.3.
The value of the Qfunction gets smaller for larger arguments, hence Equation 3.3 shows that large implies better performance. Different constellations have different d_{min} values for the same. The constellation with the largest d_{min} for a given has the best performance. In P_{b} equations like 3.3, this effect results in scaling of the value and the argument of the Qfunction.
Minimum distance depends on several factors: number of points M, average power P_{ave} and shape of the constellation. The most important is the number of points in the constellation, which is directly dependent on the number of bits k transmitted in one symbol M=2k. Average power P_{ave} scales the constellation smaller or larger, depending on the transmitted power. Consequently, to make comparisons between different constellations fair, P_{ave} is usually normalized to one when d_{min} of a constellation is calculated. Average power of a constellation is evaluated simply by averaging the power of all the M points c_{k} of the constellation as shown in Equation 3.4.
Another important factor that influences in d_{min} is the shape of the constellation. A constellation is basically a set of points located in one dimensional or larger space spanned by the transmitted symbols. Traditional constellations are either one or two dimensional. In principle, these points can be located in any shape whatsoever. Naturally, for communications purposes care must taken to ensure good performance and practical implementation. There are two main goals when constellation shapes are designed. The first is constant amplitude of the transmitted signal. In this case, all the points in the constellation are required to have equal amplitude. The second goal is to improve d_{min}. This is an optimization problem that tries to place all the points such that d_{min} is maximized for a finite P_{ave}. For the usual case of a two dimensional constellation with large number of points, this results in a circular constellation. Forney and Wei [4] give a through analysis of the behavior of d_{min} for different constellation types.
Coherent Modulations
Coherent modulation can be used by a communication system that maintains a phase lock between the transmitter and receiver RF carrier waves. Coherent modulation improves performance, but requires more complex receiver structure compared to noncoherent systems that are discussed later in this chapter. The performance gain of coherent modulation is significant when the system uses large constellations. High speed communication systems, like IEEE 802.11a, are usually coherent. The following sections describe the most common coherent modulations and their performance.
Amplitude Shift Keying
Amplitude Shift Keying (ASK) modulation transmits information by changing the amplitude of the carrier. Equation 3.5 shows the carrier waveform of ASK modulation. The A_{k} term does the actual modulation by multiplying the carrier wave by the selected amplitude level. The transmitted bits determine which of the possible symbols {A_{1},...,A_{M}} is selected.
Figure 3.1 shows several different ASK modulations; (a) is a 2ASK or BPSK, (b) is a 4ASK, and (c) is a 8ASK. In the figure, all the modulations have the same minimum distance d^{2}_{min}=4, however the average power of the constellations is not equal. To perform fair comparison of the ASK constellations, Table 3.1 shows the normalized d^{2}_{min} and the required increase in SNR to maintain the same BER when changing to a onesteplarger constellation. The SNR increase converges to about 6 dB for each additional bit in the constellation; for this reason, large ASK modulations are rarely used in practice. The 2ASK modulation does not have an SNR increase value, because it is the smallest possible constellation. The IEEE 802.11a system uses only the smallest 2ASK modulation; for larger constellations, Phase Shift Keying and Quadrature Amplitude Modulations are used in practice. These two methods are the topics of the next sections.
ASK modulations, (a) 2ASK, (b) 4ASK, (c) 8ASK
Table 3.1 Distance Properties of ASK Modulations
Modulation 
P_{ave} 
d^{2}_{min} Normalized 
SNR Increase 

2ASK 
1 
4 
 
4ASK 
5 

6.99 dB 
8ASK 
21 

6.23 dB 
Bit error rate P_{b} of binary ASK is the most basic error probability (see Equation 3.6). The P_{b} of 2ASK is plotted in Figure 3.2. It is useful to memorize a reference point from this curve, like P_{b}=2*10^{4} at 8.0 dB or P_{b}=10^{5} at 9.6 dB. These points can serve as a quick check to validate simulation results. As the number of points in the constellation increases, exact analytic P_{b} equations become quite complicated, therefore either formulas for symbol error rate P_{s} or approximations to P_{b} are more commonly used.
BPSK bit error rate.
Symbol error rate P_{s} of Mary ASK modulations is equal to Equation 3.7. Bit error rate P_{b} can be approximated by dividing P_{s} by the number of bits per symbol as in Equation 3.8. The formula for P_{b} is an approximation, because generally the number of bit errors that occur for each symbol error can be more than one. Equation 3.8 assumes that each symbol error causes only one bit error. However, with the help of Gray coding, which is discussed in the section "Labeling Constellation Points," for most symbol errors the number of bit errors is equal to one. The quality of this approximation improves at high signal to noise ratios.
In Equation 3.7, A is the amplitude difference between symbol levels. For example, in the constellation in Figure 3.1, A=2.
Phase Shift Keying
Phase Shift Keying (PSK) modulations transmit information by changing the phase of the carrier, the amplitude is kept constant; hence PSK modulations are also called constant amplitude modulations. Equation 3.9 shows the carrier waveform of PSK signal. Modulation is done by the term.
The main benefit of constant amplitude modulation is the peaktoaverage power ratio (PAPR) that is practically equal to one. However, this is only true for a single carrier systems. An OFDM signal that is a sum of several modulated subcarriers does not have a constant amplitude, even if the individual subcarriers do. The main benefit of constant amplitude modulation is simplified RF chain design for both the transmitter and receiver. An OFDM signal can never have a constant amplitude, hence having a constant amplitude constellation does not benefit OFDM systems. Several techniques to mitigate the large amplitude variations of OFDM signals are discussed in Chapter 5.
Figure 3.3 shows the three smallest PSK constellations, (a) is 2PSK or binary PSK (BPSK), (b) is 4PSK or quadrature PSK (QPSK), and (c) is 8PSK. Note that BPSK is identical to 2ASK. The figure shows how PSK modulations use both the Inphase (I) and Quadrature (Q) carrier waves, hence the modulation is two dimensional. This increase in dimensionality of the constellation improves the behavior of d^{2}_{min} as the number of points in the constellation increases.
The three smallest PSK modulations, (a) BPSK, (b) QPSK, (c) 8PSK.
Table 3.2 shows the required increase in SNR for 2 to 16point PSK constellations. Note particularly that the increase from BPSK to QPSK is only 3 dB, compared to 2ASK to 4ASK increase of 6.99 dB. However, as the number of points is increased to 16, SNR increase for one additional bit per symbol converges towards the 6 dB figure as for ASK modulation. This behavior can be attributed to the fact that although PSK constellation is two dimensional, it still has only one degree of freedom: the phase of the carrier. Thus PSK does not fully use the twodimensional space to locate the points of the constellation. Therefore larger than 8point PSK modulations are not commonly used.
Table 3.2 Distance Properties of PSK Modulations
Modulation 
P_{ave} 
d^{2}_{min} Normalized 
SNR Increase 

BPSK 
1 
4.00 
 
QPSK 
1 
2.00 
3.00 dB 
8PSK 
1 
0.5858 
5.33 dB 
16PSK 
1 
0.1522 
5.85 dB 
Symbol error rate of PSK modulations can be derived exactly for BPSK and QPSK; the former case was already shown in Equation 3.6. For QPSK, the P_{s} expression is somewhat more complicated than BPSK as shown in Equation 3.10.
Higher order PSK modulation P_{s} can be approximated by Equation 3.11. Bit error rate can again be approximated by dividing P_{s} by the number of bits k=log_{2}M in the constellation.
Quadrature Amplitude Modulation
Quadrature Amplitude Modulation (QAM) changes both the amplitude and phase of the carrier, thus it is a combination of both ASK and PSK. Equation 3.12 shows the QAM signal in socalled IQform that presents the modulation of both the I and Qcarriers. QAM can also be described by Equation 3.13 that shows how amplitude and phase modulations are combined in QAM.
The amplitude and phase terms of Equation 3.13 are calculated from Equations 3.14 and 3.15.
Figure 3.4 shows three QAM constellations: (a) 4QAM or QPSK, (b) 8QAM, and (c) 16QAM. Especially QPSK and 16QAM are very common modulations. On the other hand, 8QAM is not used as often, probably due to its somewhat inconvenient shape; instead 8PSK is commonly used when 8point constellations are required. Figure 3.4 (b) is not the only possible shape for a 8QAM constellation.
QAM constellations, (a) QPSK, (b) 8QAM, (c) 16QAM
Table 3.3 shows how the d^{2}_{min} of QAM behaves as a function of the constellation size. The step from QPSK to 8QAM is somewhat anomalous, and it is actually possible to improve the minimum distance of our example 8QAM modulation. However, we have used the present example because of its regular form. The step from 16QAM to 32QAM shows a 3 dB increase in required SNR to maintain constant BER. This 3 dB SNR increase for each additional bit per symbol is a general rule for QAM constellations.
Symbol error rate of QAM modulation is approximated by considering QAM as two independent ASK modulation on both I and Qcarriers. This is shown in Equation 3.16.
Table 3.3 Distance Properties of QAM Modulators
Modulation 
P_{ave} 
d^{2}_{min} Normalized 
SNR Increase 

QPSK 
1 
2.00 
 
8QAM 
6 
0.67 
4.77 dB 
16QAM 
10 
0.40 
2.22 dB 
32QAM 
20 
0.20 
3.01 dB 
Labeling Constellation Points
Labeling constellation points means assigning a bit pattern to all the points. The selected labeling scheme has an impact on the performance and thus must be optimized. Figure 3.5 shows two common ways to assign bit patterns to 16QAM constellation points: (a) is called natural order, and (b) is called Gray coding. Figure 3.5 shows the decimal value of the label above the point and binary value below the point. Natural ordering has appeal in its straightforward assignment of labels using decimal numbers from 0 to 15 to the points. The disadvantage of natural ordering is in the actual bit patterns representing constellation points....