CODING AND MODULATION

Coding and modulation provide the means of mapping information into waveforms such that the receiver (with an appropriate demodulator and decoder) can recover the information in a reliable manner. The simplest model for a communication system is that of an additive white Gaussian noise (AWGN) system. In this model a user transmits information by sending one of M possible waveforms in a given time, period T, with a given amount of energy. The rate of communication, R, in bits per second is log2(M)/T. The signal occupies a given bandwidth W Hz. The normalized rate of communications is R/W measured in bits/second/Hz. The received signal is the sum of the transmitted signal and white Gaussian noise (noise occupying all frequencies). The optimum receiver for deciding which of the M signals was transmitted filters the received waveform to remove as much noise as possible while retaining as much signal as possible. For a fixed amount of energy, the more waveforms (the larger M) the harder it is for the receiver to distinguish which waveform was transmitted. There is a fundamental tradeoff between the energy efficiency of a communication system and the bandwidth efficiency. This fundamental tradeoff is shown in Fig. 2.2. In this figure the possible normalized rate of transmission (measured in bits per second per Hz) is shown as a function of the received signal-to-noise ratio Eb/N0 for arbitrarily reliable communication. Here, Eb is the amount of energy received per information bit while N0 is the power spectral density of the noise. The curves labeled AWGN place no restrictions on the type of transmitted waveform except that the average energy must be constrained so that the received signal energy per bit is Eb. The curve labeled BPSK restricts the modulation (but not the coding) to binary phase shift keying. The curve labeled QPSK is for quaternary phase shift keying and the 8-PSK curve is for 8-ary phase shift keying. Clearly at low rates and low Eb/N0 there is virtually no loss in using QPSK modulation with the best coding compared to the best modulation and coding. Also shown in the figure is what can be achieved with certain coding schemes. While these curves show the best possible transmission rate for a given energy, no restrictions are placed on the amount of delay incurred and on the complexity of implementation. It has been the goal of communication researchers and engineers to achieve performance close to the fundamental limits with small complexity and delay.

Fig. 2.2. Possible transmission rates versus signal-to-noise ratios for an additive white Gaussian noise channel.

For a wireless communications system, the AWGN model is much too simplistic. In a wireless communication system the transmitted signal typically propagates over several distinct paths before reaching the receiving antenna. Depending on the relative phases of the received signal the multiple signals could interfere in a destructive manner or in a constructive manner. The result of the multiple paths is that the received signal amplitude is sometimes attenuated severely when the signals from different paths cancel destructively, while sometimes the signal amplitude becomes relatively large because of constructive interference. The nature of the interference is, in general, time varying and frequency dependent. This is generally called time and frequency selective fading. A typical time response for a multipath fading channel is shown in Fig. 2.3.

The received signal varies more quickly as the vehicle speed increases. In the original analog cellular systems in order to compensate for the multipath fading, the transmitter increased or decreased the amount of transmitted power. As with the additive white Gaussian noise channel, there are fundamental limits on the rate of transmission for a given average received energy-to-noise ratio (Eb/N0). In the simplest model the received signal energy is modeled as a Rayleigh distributed random variable, independent from symbol to symbol. With this assumption the transmissions rates possible, as a function of the average received signal-to-noise ratio, are shown in Fig. 2.4. The gray curves represent the performance possible in an additive white Gaussian noise channel while the dark curves represent the performance with Rayleigh fading. The assumption in this figure is that the channel bandwidth is very narrow and so the result of fading is to only change the amplitude of the signal and not distort the signal in any other way. This is clearly not valid for many communication systems (especially wide bandwidth systems like direct-sequence CDMA).

Fig. 2.3. Received signal strength as a function of time for vehicle velocity 10 mph.

Fig. 2.4. Possible transmission rates versus signal-to-noise ratios for a Rayleigh fading channel.

A key observation from this figure is that there is not a significant loss in performance between what is possible in an additive white Gaussian noise channel and what is possible in a fading channel. For example for transmission rates less than 1/2 bps/Hz the loss in performance due to fading is less than 2 dB with the optimal coding and with BPSK modulation. However, for BPSK alone (without coding), the loss in performance compared to white Gaussian noise channels is on the order of 40 dB when the desired error probability is 10-5. This is a huge loss and is due to the fluctuations of the signal amplitude. Basically the fading process sometimes attenuates the signal so that the conditional error probability is close to 1/2. Sometimes the fading accentuates the signal so that the conditional error probability is virtually zero. The average error probability then is dominated by the probability that the fading level is small. This can be overcome with one or a combination of several techniques. Antenna arrays whereby the received signal at different antennas fades independently is one such technique (discussed in Chapter 6). Another technique is time diversity through coding. In the simplest realization of this, information is transmitted multiple times spaced far enough apart (in time) so that the fading is independent. At the receiver the signals are combined appropriately. In this manner the probability of error is dominated by the probability that the fading processes attenuate all the transmissions of a single bit. The probability of this event is much smaller than the probability that during a single time instant the fading process will cause significant attenuation. This is essentially a simple form of coding. Another simple form of coding is through frequency diversity. In this case the same information is transmitted over several different frequencies simultaneously. Because the channel is frequency selective not all the frequencies fade simultaneously. In this way diversity is achieved as long as the frequencies used are sufficiently separated (separation larger than the coherence bandwidth of the channel).

The conclusion from the previous discussion is that, for a fading channel, coding and/or diversity techniques are essential in providing performance close to optimal. As before, there are underlying assumptions that the delay is not a major constraint. In the time diversity system, identical data are transmitted but spread out in time. In order to achieve good performance the time separation needs to be sufficiently large so that the fading is nearly independent. In the frequency diversity case, a similar argument is made with the frequencies used. So a large time-bandwidth-space product is needed in order to achieve reasonable performance. In a wireless system, error control, coding and modulation are used to protect the data not only against the effects of fading but interference as well. Interference will be discussed in the multiple access section.

In 1993 a new coding technique (known as turbo codes) was shown to have exceptional performance in an additive white Gaussian noise environment, coming within 0.7 dB of the fundamental limit for a Gaussian channel with a code with block length on the order of 65,000 bits (Berrou et al. 1993). Since that discovery was made, considerable effort has begun on investigating these codes on other channels and with different block lengths. For an ideal Rayleigh fading channel (independent fades for each symbol) turbo codes with block length 50,000 approach within about 1.5 dB of the fundamental limit when the channel is known perfectly. For the white Gaussian noise channel, low density parity check codes are within 0.01 dB of the fundamental limit when the block length is very large. When the block length is shorter (as required by delay constraints in many applications) then the performance of turbo codes deteriorates to the point that traditional convolutional codes perform better. Third generation cellular systems will employ turbo codes for relatively long (e.g., larger than 300 bits) block length messages.

Many different modulation schemes are used in current wireless systems, among these binary phase shift keying (BPSK), Gaussian-filtered minimum shift keying (GMSK), pi/4 DPSK, offset quadrature phase shift keying (OQPSK), and orthogonal frequency division multiplexing (OFDM) (multicarrier). There are a couple key issues when designing a modulation technique. One of these issues is whether the technique uses a constant envelope or a nonconstant envelope. Constant envelope modulation techniques can cope with amplifier nonlinearities but have larger bandwidth than nonconstant envelope modulation techniques. On the other hand, a power amplifier is most energy efficient when operating in the nonlinear region. Nonconstant envelope techniques have smaller bandwidth but need a very linear amplifier to avoid generating both in-band distortion and adjacent channel power. The goal is to have bandwidth efficiency and power efficiency simultaneously. However, with current amplifier designs there is a tradeoff between these two conflicting objectives.

Another key issue when dealing with modulation is intersymbol interference. A wireless channel generally has multipath fading, which causes intersymbol interference if the data symbol duration is the same magnitude or smaller than the delay spread of the channel. As the data rate increases, the amount of (number of symbols affected by) intersymbol interference increases. This generally increases the complexity of the receiver. One method to avoid this is to transmit information on many different carrier frequencies simultaneously. This makes the symbol duration on each carrier much longer (by a factor equal to the number of carriers) and thus decreases the amount of intersymbol interference. However, multicarrier modulation techniques have a particularly high fluctuation of the signal envelope; and thus to avoid generating unwanted signals (in-band or adjacent channel) an amplifier with high backoff (low input drive level) is required, which means that the energy efficiency will be very small.

Another approach to dealing with multipath fading is to use wide bandwidth modulation techniques, generally referred to as spread-spectrum techniques. Because of the frequency and time selective nature of the wireless channel, a narrowband signal might experience a deep fade if the phases from multiple paths add up in a destructive manner at the receiver. These deep fades generally need extra protection to prevent errors by either increasing the power or adding additional redundancy for error control coding. On the other hand, if the signal has a wide bandwidth (relative to the inverse of the delay spread) then not all the frequencies in a given band will simultaneously be in a deep fade. As such the signal from the part of the spectrum that is not faded can still be recovered. One realization of this idea is that of a direct-sequence system that uses a Rake receiver to "collect" the energy from several paths (at different delays). The probability of all of the paths fading simultaneously becomes much smaller than the probability of one of the paths fading. Because of this the performance is significantly improved compared to a narrow-band system. However, the performance is limited by the bandwidth available.


Published: July 2000; WTEC Hyper-Librarian