Multicarrier Modulation and OFDM

This tutorial is part of the National Instruments Signal Generator Tutorial series. Each tutorial in this series, will teach you a specific topic of common measurement applications, by explaining the theory and giving practical examples. This tutorial covers multicarrier modulation and OFDM. For additional signal generator concepts, refer to the Signal Generator Fundamentals main page.

Multicarrier Modulation and OFDM

In the preceding sections, we considered digital transmission through nonideal channels and observed that such channels cause intersymbol interference when the reciprocal of the system rate is significantly smaller than the time dispersion (duration of the impulse response) of the nonideal channel. In such a case, a channel equalizer is employed at the receiver to compensate for the channel distortion. If the channel is a bandpass channel with a specified bandwidth, the information-bearing signal may be generated at the baseband and then translated in frequency to the passband of the channel. Thus, the information-bearing signal is transmitted on a single carrier. We also observed that intersymbol interference usually results in some performance degradation, even in the case where the optimum detector is used to recover the information symbols at the receiver.

An alternative approach to the design of a bandwidth-efficient communication system in the presence of channel distortion is to subdivide the available channel bandwidth into a number of equal-bandwidth subchannels, where the bandwidth of each subchannel is sufficiently narrow so that the frequency response characteristics of the subchannels are nearly ideal. Such a subdivision of the overall bandwidth into smaller subchannels is illustrated in Figure 8.49. Thus, we create K = W/Df subchannels, where different information symbols can be transmitted simultaneously in the K subchannels. Consequently, the data is transmitted by frequency-division multiplexing (FDM).

With each subchannel, we associate a carrier

where f_k is the mid-frequency in the kth subchannel. By selecting the symbol rate 1/T on each of the subchannels to be equal to the separation Df of adjacent subcarriers, the subcarriers are orthogonal over the symbol interval T, independent of the relative phase relationship between subcarriers; i.e.,
where f_k – f_j = n/T, n = 1, 2, ..., independent of the values of the phases f_k and f_j . In this case, we have orthogonal frequency-division multiplexing (OFDM).

With an OFDM system having K subchannels, the symbol rate on each subcarrier is reduced by a factor of N relative to the symbol rate on a single carrier system that employs the entire bandwidth W and transmits data at the same rate as OFDM. Hence, the symbol interval in the OFDM system is T = KT_s, where T_s is the symbol interval in the single-carrier system. By selecting K to be sufficiently large, the symbol interval T can be made significantly larger than the time duration of the channel-time dispersion. Thus, intersymbol interference can be made arbitrarily small by selection of K. In other words, each subchannel appears to have a fixed frequency response C(f_k), k = 0, 1, ..., K – 1.

As long as we maintain time synchronization among the subcarriers, OFDM allows us to transmit a different number of bits/symbol on each subcarrier. Hence, subcarriers that yield a higher SNR due to a lower attenuation can be modulated to carry more bits/symbol than subchannels that yield a lower SNR (high attenuation). For example, QAM with different constellation sizes may be used in an OFDM system.

The modulator and demodulator in an OFDM system can be implemented by use of a parallel bank of filters based on the discrete Fourier transform (DFT). When the number of subchannels is large, say K > 25, the modulator and demodulator in an OFDM system are efficiently implemented by use of the fast Fourier transform algorithm (FFT) to compute the DFT. Next, we describe an OFDM system in which the modulator and demodulator are implemented based on the DFT.

A major problem with the multicarrier modulation in general and OFDM system in particular is the high peak-to-average power ratio (PAR) that is inherent in the transmitted signal. Large signal peaks occur in the transmitted signal when the signals in the K subchannels add constructively in phase. Such large signal peaks may saturate the power amplifier at the transmitter and, thus, cause intermodulation distortion in the transmitted signal. Intermodulation distortion can be reduced and, generally, avoided by reducing the power in the transmitted signal and, thus, operating the power amplifier at the transmitter in the linear range. Such a power reduction results in inefficient operation of the OFDM system.

A variety of methods have been devised to reduce PAR in multicarrier systems. A relatively simple method is to insert different phase shifts in each of the subcarriers, where the phase shifts are selected pseudorandomly, or by means of some algorithm, to reduce the PAR. Additional methods are cited in the references cited in Section 8.8.

An OFDM System Implemented via the FFT Algorithm

In this section, we describe an OFDM system in which QAM is used for data transmission on each of the subcarriers and the FFT algorithm is used in the implementation of the modulator and demodulator.

The basic block diagram of the OFDM is illustrated in Figure 8.50. A serial-to-parallel buffer subdivides the information sequence into frames of B_f bits. The B_f bits in each frame are parsed into K groups, where the i th group is assigned b_i bits.

Hence,

We may view the multicarrier modulator as generating K independent QAM subchannels, where the symbol rate for each subchannel is 1/T and the the signal in each subchannel has a distinct QAM constellation. Hence, the number of signal points for the i th subchannel is . Let us denote the complex-valued signal points corresponding the information signals on the K subchannels by X_k , k = 0, 1, ..., K – 1. These information symbols {X_k} represent the values of the discrete Fourier transform (DFT) of a multicarrier OFDM signal x(t), where the modulation on each subcarrier is QAM. Since x(t) must be a real-valued signal, its N-point DFT {X_k} must satisfy the symmetry property . Therefore, we create N = 2K symbols from K information symbols by defining

Note that the information symbol X₀ is split into two parts, both of which are real. If we denote the new sequence of symbols as , the N-point inverse DFT (IDFT) yields the real-valued sequence

where is simply a scale factor. This sequence {x_n, 0 £ n £ N – 1} corresponds to samples of the multicarrier OFDM signal x(t), consisting of K subcarriers, which may be expressed as

where T is the signal duration and x_n = x(nT/N), n = 0, 1, ..., N – 1. The subcarrier frequencies are f_k = k/T, k = 0, 1, ..., K – 1. The signal samples {x_n} generated by computing the IDFT are passed through a digital-to-analog (D/A) converter, where output, ideally, is the OFDM signal waveform x(t).

With x(t) as the input to the channel, the channel output at the receiver may be expressed as

where c(t) is the impulse response of the channel and denotes convolution. Since the bandwidth Df of each subchannel is selected to be very small relative to the overall channel bandwidth W = KDf, the symbol duration T = 1/Df is large compared to the duration of the channel impulse response. To be specific, suppose that the channel impulse response spans m + 1 signal samples, where . A simple way to completely avoid intersymbol interference (ISI) is to insert a time guard of duration mT/N between transmission of successive data blocks. This allows the response of the channel to die out before the next block of K symbols are transmitted.

An alternative method to avoid ISI is to append a so-called cyclic prefix to each block of N signal samples {x_n , 0 £ n £ N – 1}. The cyclic prefix for the block of samples consists of the samples X_{N – m} , X_{N – m +1}, ..., X_{N – 1}. These samples are appended to the beginning of the block, thus, creating a signal sequence of length N + m samples, which may be indexed from n = –m to n = N – 1, where the first m samples constitute the cyclic prefix. Then, if the sample values of the channel response are {c_n, 0 £ n £ m}, the convolution of {c_n} with {x_n, –m £ n £ N – 1} produce the received signal {r_n}. Since the ISI in any pair of successive signal transmission blocks affects the first m signal samples, we discard the first m samples of {r_n} and demodulate the signal based on the received signal samples {r_n, 0 £ n £ N – 1}.

If we view the channel characteristics in the frequency domain, the channel frequency response at the subcarrier frequencies f_k = k/T is

Since the ISI is eliminated by the use of either the cyclic prefix or the time guard band, the demodulated sequence of symbols may be expressed as

where is the output of the N-point DFT computed by the demodulator and {h_k) is the additive noise corrupting the signal.

As illustrated in Figure 8.50, the received signal is demodulated by computing the DFT of the received signal after it has been passed through an analog-to-digital (A/D) converter. As in the case of the OFDM modulator, the DFT computation at the demodulator is performed efficiently by use of the FFT algorithm.

In order to recover the information symbols from the values of the computed DFT, it is necessary to estimate and compensate for the channel factors {C_k}. The channel measurement can be accomplished by initially transmitting either a known modulated sequence on each of the subcarriers or, simply, transmitting the unmodulated subcarriers. If the channel characteristics vary slowly with time, the time variations can be tracked by using the decisions at the output of the detector in a decision-directed manner. Thus, the multicarrier OFDM system can be made to operate adaptively. The transmission rate on each of the subcarriers can be optimized by properly allocating the average transmitted power and the number of bits that are transmitted by each subcarrier. The SNR per subchannel may be defined as

where T is the symbol duration, P_k is the average transmitted power allocated to the kth subchannel, is the squared magnitude of the frequency response of the kth subchannel, and is the corresponding noise variance. In subchannels with high SNR, we transmit more bits/symbol by using a larger QAM constellation compared to subchannels with low SNR. Thus, the bit rate on each subchannel can be optimized in such a way that the error-rate performance among the subchannels is equalized to satisfy the desired specifications.

Multicarrier OFDM using QAM modulation on each of the subcarriers as described above has been implemented for a variety of applications, including high-speed transmission over telephone lines, such as digital subcarrier lines. This type of multicarrier OFDM modulation has also been called discrete-multitone (DMT) modulation. Multicarrier OFDM is also used in digital audio broadcasting in Europe and other parts of the world and in digital cellular communication systems.

Further Reading

The pioneering work on signal design for bandwidth-constrained channels was done by Nyquist (1928). The use of binary partial response signals was originally proposed in the paper by Lender (1963) and was later generalized by Kretzmer (1966). The problem of optimum transmitter and receiver filter design was investigated by Gerst and Diamond (1961), Tufts (1965), Smith (1965), and Berger and Tufts (1967).

Adaptive equalization for digital communication was introduced by Lucky (1965, 1966). Widrow (1966) devised the LMS algorithm for adaptively adjusting the equalizer coefficients.

The Viterbi algorithm was devised by Viterbi (1967) for the purpose of decoding convolutional codes, which are described in Chapter 9. Its use as the ML sequence detector for partial response signals and, more generally, for symbols corrupted by intersymbol interference, was proposed and analyzed by Forney (1972) and Omura (1971). A comprehensive treatment of adaptive equalization algorithms is given in the book by Proakis (2001).

There is a large amount of literature on multicarrier digital communication systems. One of the earliest systems, described by Doeltz et al. (1957) and called Kineplex, was used for digital transmission in the high-frequency radio band. Other early work on the multicarrier system design is described in the papers by Chang (1966) and Saltzberg (1967). The use of DFT for modulation and demodulation of multicarrier OFDM systems was proposed by Weinstein and Ebert (1971). More recent references on applications of OFDM in practical systems are the papers by Chow et al. (1995) and Bingham (1990). The recent book by Bahai and Saltzberg (1999) provides a comprehensive treatment of OFDM.

The problem of PAR reduction in multicarrier systems has been investigated by many people. The interested reader may refer to the papers by Boyd (1986), Popovic (1991), Jones et al. (1994), Wilkinson and Jones (1995), Wulich (1996), Tellado and Cioffi (1998), and Tarokh and Jafarkhani (2000).

Relevant NI products

Customers interested in this topic were also interested in the following NI products:

• Function, Arbitrary, and RF Signal Generators
• Other Modular Instruments (digital multimeters, digitizers, switching, etc...)
• LabVIEW Graphical Programming Environment
• SignalExpress Interactive Software Environment

For the complete list of tutorials, return to the NI Signal Generator Fundamentals Main page.

Reader Comments | Submit a comment »