OFDM Wireless LANs: A Theoretical and Practical Guide

OFDM Wireless LANs: A Theoretical and Practical Guide

by Juha Heiskala, John Terry Ph.D.

OFDM Wireless LANs: A Theoretical and Practical Guide explores OFDM WLAN basics, including components of OFDM and multi-carrier WLAN standards. This book also provides a practical approach to OFDM by including software and hardware examples and detailed implementation explanations. And it defines and explains the mathematical concepts behind OFDM

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OFDM Wireless LANs: A Theoretical and Practical Guide explores OFDM WLAN basics, including components of OFDM and multi-carrier WLAN standards. This book also provides a practical approach to OFDM by including software and hardware examples and detailed implementation explanations. And it defines and explains the mathematical concepts behind OFDM necessary for successful OFDM WLAN implementations.

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Kaleidoscope Series
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6.90(w) x 9.00(h) x 0.80(d)

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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].


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 (dmin) is the smallest distance between any two points in the constellation. Therefore dmin determines the least amount of noise that is needed to generate a decision error. Actual BER or Pb of a constellation can in many cases be calculated using the Q-function defined in Equation 3.2. The value of the Q-function is equal to the area under the tail of the Probability Density Function (PDF) of a zero mean and unit variance normal random variable.

Pb 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 Es=kEb. 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 Pb equations for several important constellations in terms of the Q-function is shown in Equation 3.3.

The value of the Q-function gets smaller for larger arguments, hence Equation 3.3 shows that large implies better performance. Different constellations have different dmin values for the same. The constellation with the largest dmin for a given has the best performance. In Pb equations like 3.3, this effect results in scaling of the value and the argument of the Q-function.

Minimum distance depends on several factors: number of points M, average power Pave 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 Pave scales the constellation smaller or larger, depending on the transmitted power. Consequently, to make comparisons between different constellations fair, Pave is usually normalized to one when dmin of a constellation is calculated. Average power of a constellation is evaluated simply by averaging the power of all the M points ck of the constellation as shown in Equation 3.4.

Another important factor that influences in dmin 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 dmin. This is an optimization problem that tries to place all the points such that dmin is maximized for a finite Pave. 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 dmin 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 non-coherent 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 Ak term does the actual modulation by multiplying the carrier wave by the selected amplitude level. The transmitted bits determine which of the possible symbols {A1,...,AM} is selected.

Figure 3.1 shows several different ASK modulations; (a) is a 2-ASK or BPSK, (b) is a 4-ASK, and (c) is a 8-ASK. In the figure, all the modulations have the same minimum distance d2min=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 d2min and the required increase in SNR to maintain the same BER when changing to a one-step-larger 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 2-ASK modulation does not have an SNR increase value, because it is the smallest possible constellation. The IEEE 802.11a system uses only the smallest 2-ASK 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) 2-ASK, (b) 4-ASK, (c) 8-ASK

Table 3.1 Distance Properties of ASK Modulations



d2min Normalized

SNR Increase







6.99 dB



6.23 dB

Bit error rate Pb of binary ASK is the most basic error probability (see Equation 3.6). The Pb of 2-ASK is plotted in Figure 3.2. It is useful to memorize a reference point from this curve, like Pb=2*10-4 at 8.0 dB or Pb=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 Pb equations become quite complicated, therefore either formulas for symbol error rate Ps or approximations to Pb are more commonly used.

BPSK bit error rate.

Symbol error rate Ps of M-ary ASK modulations is equal to Equation 3.7. Bit error rate Pb can be approximated by dividing Ps by the number of bits per symbol as in Equation 3.8. The formula for Pb 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 peak-to-average 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 2-PSK or binary PSK (BPSK), (b) is 4-PSK or quadrature PSK (QPSK), and (c) is 8-PSK. Note that BPSK is identical to 2-ASK. 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 d2min as the number of points in the constellation increases.

The three smallest PSK modulations, (a) BPSK, (b) QPSK, (c) 8-PSK.

Table 3.2 shows the required increase in SNR for 2- to 16-point PSK constellations. Note particularly that the increase from BPSK to QPSK is only 3 dB, compared to 2-ASK to 4-ASK 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 two-dimensional space to locate the points of the constellation. Therefore larger than 8-point PSK modulations are not commonly used.

Table 3.2 Distance Properties of PSK Modulations



d2min Normalized

SNR Increase








3.00 dB




5.33 dB




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 Ps expression is somewhat more complicated than BPSK as shown in Equation 3.10.

Higher order PSK modulation Ps can be approximated by Equation 3.11. Bit error rate can again be approximated by dividing Ps by the number of bits k=log2M 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 so-called IQ-form that presents the modulation of both the I- and Q-carriers. 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) 4-QAM or QPSK, (b) 8-QAM, and (c) 16-QAM. Especially QPSK and 16-QAM are very common modulations. On the other hand, 8-QAM is not used as often, probably due to its somewhat inconvenient shape; instead 8-PSK is commonly used when 8-point constellations are required. Figure 3.4 (b) is not the only possible shape for a 8-QAM constellation.

QAM constellations, (a) QPSK, (b) 8-QAM, (c) 16-QAM

Table 3.3 shows how the d2min of QAM behaves as a function of the constellation size. The step from QPSK to 8-QAM is somewhat anomalous, and it is actually possible to improve the minimum distance of our example 8-QAM modulation. However, we have used the present example because of its regular form. The step from 16-QAM to 32-QAM 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 Q-carriers. This is shown in Equation 3.16.

Table 3.3 Distance Properties of QAM Modulators



d2min Normalized

SNR Increase








4.77 dB




2.22 dB




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 16-QAM 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....

Meet the Author

John Terry, Ph.D., is a senior research engineer at Nokia Research Center in Dallas, TX. He is currently managing the OFDM modulation and coding project in the High Speed Access (HSA) group. Dr. Terry has published several conference and journal papers, given numerous presentations on wireless communications, and generated six pending patents related to OFDM wireless LANs. He has 12 years of experience working in wireless communications, including tenures at NASA Glen Research Center and Texas Instruments, Inc. In addition, Dr. Terry is the vice-chair of the IEEE 802.11 Task Group and serves as a technical reviewer for several conference and journal publications of the IEEE in wireless communications.

Juha Heiskala is a senior research engineer at Nokia Research Center in Dallas, TX. He is active in the IEEE 802.11 standards bodies and has been tasked with developing the 802.11a system simulation on several software platforms. He is the inventor/co-inventor of three pending patents in the area of OFDM LANs and co-designed with Dr. John Terry the modulation and coding scheme for achieving 100 Mbps speeds within currently allocated bandwidth specifications for OFDM wireless LANs.

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