A point-to-point communication system transfers a message from one point to another through a noisy environment called a communication channel. A familiar example of a communication channel is formed by the propagation of an electromagnetic wave from a transmitting antenna to a receiving antenna. The message is carried by the time-varying parameters of the electromagnetic wave. Another example of a communication channel is a waveform propagating through a coaxial cable that connects a jack mounted on an office wall to another such jack on another wall or to a central node. In these examples, the waveform as it appears at the receiver is contaminated by noise, by interference, and by other impairments. The transmitted message must be protected against such impairments and distortion in the channel. Early communication systems were designed to protect their messages from the environment by the simple expedient of transmitting at low data rates with high power. Later, message design techniques were introduced that led to the development of far more sophisticated communication systems with much better performance. Modern message design is the art of piecing together a number of waveform ideas in order to transmit as many bits per second as is practical within the available power and bandwidth. It is by the performance at low transmitted energy per bit that one judges the quality of a digital communication system. The purpose of this book is to develop modern waveform techniques for the digital transmission of information.

The field of telecommunication consists of the theory and the practice of communication at a distance, principally electronic communication. Many systems for telecommunication now take the form of large, complex, interconnected data networks with both wired and wireless segments, and the design of such systems is based on a rich theory. Communication theory studies methods for the design of signaling waveforms to transmit information from point to point, as within a telecommunication system. Communication theory is that part of information theory that is concerned with the explicit design of suitable waveforms to convey messages and with the performance of those waveforms when received in the presence of noise and other channel impairments. Digital telecommunication theory, or modem theory, is that part of communication theory in which digital modulation and demodulation techniques play a prominent role in the communication process, either because the information to be transmitted is digital or because the information is temporarily represented in digital form for the purpose of transmission.

Digital communication systems are in widespread use and are now in the process of sweeping away even the time-honored analog communication systems, such as those used in radio, television, and telephony. The main task of communication theory is the design of efficient waveforms for the transmission of information over band-limited or power-limited channels. The most sweeping conclusion of information theory is that all communication is essentially digital.

The function of a digital demodulator is to reconvert a waveform received in noise back into the stream of data symbols from the discrete data alphabet. We usually regard this datastream as a binary datastream. A demodulator is judged by its ability to recover the user datastream with low probability of bit error even when the received channel waveform is contaminated by distortion, interference, and noise. The probability of symbol error or bit error at the demodulator output is also called the symbol error rate or the bit error rate. The demodulator is designed to make these error rates small. We shall concentrate our discussion on optimum demodulation in the presence of additive noise because additive noise is the most fundamental disturbance in a communication system.

The energy per data bit is the primary physical quantity that determines the ability of the communication system to tolerate noise. For this purpose, energy (or power) always refers to that portion of the energy in the waveform that reaches the receiver. This will be only a small fraction of the energy sent by the transmitter.

The study of the optimum demodulation of a waveform in additive noise is an application of the statistical theory of hypothesis testing. The basic principles are surprisingly simple and concise. The most basic principle, and the heart of this chapter, is the principle of the matched filter. A very wide variety of modulation waveforms are demodulated by the same general method of passing a received signal through a matched filter and sampling the output of that matched filter.

The communication problem can be given a new dimension of complexity by the introduction of an adversary. The adversary may have a variety of goals. The goal may be to interrupt communication, to detect the occurrence of communication, to determine the specific message transmitted, or to determine the location or the identity of the transmitter. The communication problem now takes on aspects of the theory of games. The transmitter and receiver comprise one team while the adversary comprises the other team.

An adversary may try to interrupt communication by falsifying the messages or by inserting noise into the channel. In the former case, the adversary is called a spoofer while in the latter case, the adversary is called a jammer. An adversary who intends to read the specific message transmitted is called a cryptanalyst. An adversary who intends to determine the location or the identity of the transmitter or to detect the occurrence of communication is called a signal exploiter.

Waveform techniques to counter a jammer or an exploiter are similar; both try to spread the waveform over a wide bandwidth. Such waveforms are called antijam waveforms or antiexploitation waveforms. Techniques to counter a spoofer or a cryptanalyst tend to be similar: these may use a secret permutation on the set of messages to represent the actual message by a surrogate message formed in an agreed, invertible way based on a secret key.

The modulator and demodulator make a waveform channel into a discrete communication channel. Because of channel noise, the discrete communication channel is a noisy communication channel; there may be errors or other forms of lost data. A data transmission code is a code that makes a noisy discrete channel into a reliable channel. Despite noise or errors that may exist in the channel output, the output of the decoder for a good data-transmission code is virtually error-free. In this chapter, we shall study some practical codes for data transmission. These codes are designed for noisy channels that have no constraints on the sequence of transmitted symbols. Then a data transmission code can be used to make the noisy unconstrained channel into a reliable channel.

For the kinds of discrete channels formed by the demodulators of Chapter 3, the output is simply a regenerated stream of channel input symbols, some of which may be in error. Such channels are called hard-decision channels. The data transmission code is then called an error-control code or an error-correcting code. More generally, however, the demodulator may be designed to qualify its output in some way. Viewed from modulator input to demodulator output, we may find a channel output that is less specific than a hard-decision channel, perhaps including erasures or other forms of tentative data such as likelihood data on the set of possible output symbols.

The demodulation of a passband waveform or of a complex baseband waveform uses methods similar to those used to demodulate baseband signals. However, there are many new details that emerge in the larger setting of passband or complex baseband demodulation. This is because a complex baseband function (or a passband function) can be expressed either in terms of real and imaginary components or in terms of amplitude and phase. It is obvious that phase is meaningful only if there is an absolute phase reference. A new set of topics arises when the modulator and demodulator do not share a common phase reference. This is the distinction between coherent and noncoherent demodulation. When the phase reference is known to the demodulator, the demodulator is called a coherent demodulator. When the phase reference is not known to the demodulator, that demodulator is called a noncoherent demodulator.

We begin the chapter with a study of the matched filter at passband. Then we use the matched filter as a central component in the development of a variety of demodulators, both coherent and noncoherent, for the passband waveforms that were introduced in Chapter 5.

The methods for the demodulation of baseband sequences that were described in Chapter 4 can be restated in the setting of passband waveforms. We shall prefer, however, the equivalent formulation in terms of complex baseband waveforms. It becomes obvious immediately how to generalize methods of demodulation from sequences of real numbers to sequences of complex numbers, so the chapter starts out with a straightforward reformulation of the topic of demodulation.

Rather than modulate one data symbol at a time into a channel waveform, it is possible to modulate the entire datastream as an interlocked unit into the channel waveform. The resulting waveform may exhibit symbol interdependence that is created intentionally to improve the performance of the demodulator. Although the symbol interdependence does have some similarity to intersymbol interference, in this situation it is designed deliberately to improve the minimum euclidean distance between sequences, and so to reduce the probability of demodulation error.

The methods developed in Chapter 4 for demodulating interdependent sequences led us to a positive view of intersymbol interdependence. This gives us the incentive to introduce intersymbol interdependence intentionally into a waveform to make sequences more distinguishable. The digital modulation codes that result are a form of data transmission code combined with the modulation waveform. The output of the data encoder is immediately in the form of an input to the waveform channel. The modulator only needs to apply the proper pulse shape to the symbols of the code sequence.

In this chapter, we shall study trellis-coded modulation waveforms, partial-response signaling waveforms, and continuous-phase modulation waveforms. Of these various methods, trellis-coded modulation is the more developed, and is in widespread use at the present time.

Partial-response signaling

The simplest coded-modulation waveforms are called partial-response signaling waveforms. These coded waveforms can be motivated by recalling the method of decision feedback equalization.

A channel may introduce unpredictable changes into the waveform passing through it. In a passband channel, such as a radio frequency channel, unpredictable phase shifts of the carrier may occur in the atmosphere, in antennas, and in other system elements or because of uncertainty in the time of propagation. In order to demodulate a digital waveform coherently, a coherent replica of the carrier is needed in the receiver. Because the receiver does not know the carrier phase independently of the received signal, the receiver must locally regenerate a coherent replica of the carrier. Uncertainty in the phase of the received waveform introduces the task of phase synchronization in the receiver.

Uncertainty in the time of propagation also introduces problems of time synchronization. The local clock must be synchronized with the incoming datastream so that incoming symbols and words can be correctly framed and assigned their proper indices. Time synchronization may be subdivided into two tasks: symbol synchronization, and block or frame synchronization. These two kinds of time synchronization are quite different. Symbol synchronization is a fine time adjustment that adjusts the sampling instants to their correct value. It exploits the shape of the individual pulses making up the waveform to adjust the time reference. The content of the datastream itself plays no role in symbol synchronization. Block synchronization takes place on a much longer time scale. It looks for special patterns embedded in the datastream so that it can find the start of a message or break the message into constituent parts.

This chapter approaches the theory of modem design starting from well-accepted basic principles of inference. In particular, we will study the maximum-likelihood principle and the maximum-posterior principle. By studying the optimum demodulation of passband sequences, we shall develop an understanding of the maximum-likelihood principle and its application. In Chapter 8, we will also treat the topic of synchronization as applied to both carrier recovery and time recovery by using the maximum-likelihood principle.

It is appropriate at this point to develop demodulation methods based on the likelihood function. The maximum-likelihood principle will enable us to derive optimal methods of demodulating in the presence of intersymbol interdependence and, as a side benefit, to establish the optimality of the demodulators already discussed in Chapter 3.

The maximum-likelihood principle is a general method of inference applying to many problems of decision and estimation besides those of digital communications. The development will proceed as follows. First we will introduce the maximum-likelihood principle as a general method to form a decision under the criterion of minimum probability of decision error when given a finite set of measurements. Then we will approximate the continuous-time waveform v(t) by a finite set of discrete-time samples to which we apply the maximum-likelihood principle. Finally, we will take the limit as the number of samples of v(t) goes to infinity to obtain the maximum-likelihood principle for the waveform measurement v(t).

The likelihood function

We begin with the general decision problem, not necessarily the problem of demodulation, of deciding between M hypotheses when given a measurement.

A waveform channel is a channel whose inputs are continuous functions of time. A passband channel is a waveform channel suitable for an input waveform that has a spectrum confined to an appropriately narrow interval of frequencies centered about a nonzero reference frequency, f0. A complex baseband channel is a waveform channel whose input waveform is a complex function of time that has a spectrum confined to an interval of frequencies containing the zero frequency. We shall see that every passband channel can be converted to or from a complex baseband channel by using standard techniques in the modulator and demodulator.

The function of a digital modulator for a passband channel is to convert a digital datastream into a waveform representation of the data that can be accepted by the passband channel. The waveform from the modulator is designed to accommodate the spectral characteristics of the channel, to obtain high rates of data transmission, to minimize transmitted power, and to keep the bit error rate small.

A passband modulation waveform cannot be judged independently of the performance of the demodulator. To understand how a modem works, it is necessary to study both the passband modulation techniques of this chapter and the passband demodulation techniques of Chapter 6. The final test of a modem is in the ability of the demodulator to recover the input datastream from the signal received by the demodulator in the presence of noise, interference, distortion, and other impairments.

We have studied in great detail the effect of additive gaussian noise in a linear system because of its fundamental importance. Usually the ultimate limit on the performance of a digital communication system is set by its performance in gaussian noise. For this and other reasons, the demodulators studied in Chapter 3 presume that the received waveform has been contaminated only by additive gaussian noise. However, there are other important disturbances that should be understood. The demodulators studied in Chapter 4 extend the methods to include intersymbol interference in the received waveform. While additive gaussian noise and intersymbol interference are the most important channel impairments, the demodulator designer must be wary of other impairments that may affect the received signal. The demodulator must not be so rigid in its structure that unexpected impairments cause an undue loss of performance. This chapter describes a variety of channel impairments and methods to make the demodulator robust so that the performance will not collapse if the channel model is imperfect.

Most of the impairments in a system arise for reasons that are not practical to control, and so the waveform must be designed to be tolerant of them. Such impairments include both interference and nonlinearities. Sometimes nonlinearities may be introduced intentionally into the front end of the receiver because of a known, desired outcome. Then we must understand the effect of the nonlinearity in all its ramifications in order to anticipate undesirable side effects.

A waveform channel is a channel whose inputs are continuous functions of time. A baseband channel is a waveform channel suitable for an input waveform that has a spectrum confined to an interval of frequencies centered about the zero frequency. In this chapter, we shall study the design of waveforms and modulators for the baseband channel.

The function of a digital modulator is to convert a digital datastream into a waveform representation of the datastream that can be accepted by the waveform channel. The waveform formed by the modulator is designed to accommodate the spectral characteristics of the channel, to obtain high rates of data transmission, to minimize transmitted power, and to keep the bit error rate small.

A modulation waveform cannot be judged independently of the performance of the demodulator. To understand how a baseband communication system works, it is necessary to study both the baseband modulation techniques of this chapter and the baseband demodulation techniques of Chapter 3. The final test of a modem is in the ability of the demodulator to recover the symbols of the input datastream from the channel output signal when received in the presence of noise, interference, distortion, and other impairments.

Baseband and passband channels

A waveform channel is a channel whose input is a continuous function of time, here denoted c(t), and whose output is another function of time, here denoted v(t).

Communication waveforms in which the received pulses, after filtering, are not Nyquist pulses cannot be optimally demodulated one symbol at a time. The pulses will overlap and the samples will interact. This interaction is called intersymbol interference. Rather than use a Nyquist pulse to prevent intersymbol interference, one may prefer to allow intersymbol interference to occur and to compensate for it in the demodulation process.

In this chapter, we shall study ways to demodulate in the presence of intersymbol interference, ways to remove intersymbol interference, and in Chapter 9, ways to precode so that the intersymbol interference seems to disappear. We will start out in this chapter thinking of the interdependence in a sequence of symbols as undesirable, but once we have developed good methods for demodulating sequences with intersymbol interference, we will be comfortable in Chapter 9 with intentionally introducing some kinds of controlled symbol interdependence in order to improve performance.

The function of modifying a channel response to obtain a required pulse shape is known as equalization. If the channel is not predictable, or changes slowly with time, then the equalization may be designed to slowly adjust itself by observing its own channel output; in this case, it is called adaptive equalization.

This chapter studies such interacting symbol sequences, both unintentional and intentional. It begins with the study of intersymbol interference and ends with the subject of adaptive equalization.