The transmission of bandpass signals and the corresponding channel effects are introduced. Basic single-carrier bandpass modulation schemes – namely, bandpass pulse amplitude modulation, phase-shift keying, and quadrature amplitude modulation – are studied. Lowpass equivalents of bandpass signals are introduced, and the in-phase and quadrature components of a bandpass signal are described. It is shown that bandpass signals and systems can be studied through their lowpass equivalents. The π/4-QPSK and offset QPSK are presented as two practically motivated variations of quadrature phase-shift keying. Coherent, differentially coherent, and non-coherent receivers are described. Differential phase-shift keying is studied in some depth. Finally, carrier phase-synchronization methods, including the use of phase-locked loops, are described.

Deterministic signals and linear time-invariant systems are studied. The Fourier transform is introduced, and its properties are reviewed. The concepts of probability and random variables are developed. Conditional probability is defined, and the total probability theorem and Bayes’ rule are given. Random variables are studied through their cumulative distribution functions and probability density functions, and statistical averages, including the mean and variance, are defined. These concepts are extended to random vectors. In addition, the concept of random processes is covered in depth. The autocorrelation function, stationarity, and power spectral density are studied, along with extensions to multiple random processes. Particular attention is paid to wide-sense stationary processes, and the concept of power spectral density is introduced. Also explored is the filtering of wide-sense stationary random processes, including the essential properties of their autocorrelation function and power spectral density. Due to their significance in modeling noise in a communication system, Gaussian random processes are also covered.

Several issues in communication system design are highlighted. Specifically, the effects of transmission losses in a communication system and ways of addressing the related challenges are reviewed. A basic link budget analysis is performed. The effects of non-ideal amplifiers to combat transmission losses are demonstrated, and the loss in the signal-to-noise ratio at the amplifier output is quantified. The use of analog and regenerative repeaters for transmission over long distances is explored. Furthermore, time-division, frequency-division, and code-division multiple-access techniques are described.

A brief coverage of amplitude modulation (AM) and angle modulation techniques is provided. The basic principles of conventional AM, double-sideband suppressed carrier AM, single-sideband AM, and vestigial sideband AM are described both through time-domain and frequency-domain techniques. Frequency and phase modulation are described and their equivalence is argued. A comparison of different analog modulation techniques in terms of complexity, power, and bandwidth requirements is made. Conversion of analog signals into a digital form through sampling and quantization is studied. Proof of the sampling theorem is given. Scalar and vector quantizers are described. Uniform and non-uniform scalar quantizer designs are studied. The Lloyd-Max quantizer design algorithm is detailed. The amount of loss introduced by a quantizer is quantified by computing the mean square distortion, and the resulting signal-to-quantization noise ratio. Pulse code modulation (PCM) as a waveform coding technique, along with its variants – including differential PCM and delta modulation – is also studied.

The main reason for the possibility of data compression is the experimental (empirical) law: Real-world sources produce very restricted sets of sequences. How do we model these restrictions? Chapter 10 looks at the first of three compression types that we will consider: variable-length lossless compression.

In this chapter our goal is to determine the achievable region of the exponent pairs for the type-I and type-II error probabilities. Our strategy is to apply the achievability and (strong) converse bounds from Chapter 14 in conjunction with the large-deviations theory developed in Chapter 15. After characterizing the full tradeoff we will discuss an adaptive setting of hypothesis testing where, instead of committing ahead of time to testing on the basis of n samples, one can decide adaptively whether to request more samples or stop. We will find out that adaptivity greatly increases the region of achievable error exponents and will learn about the sequential probability ratio test (SPRT) of Wald. In the closing sections we will discuss relations to more complicated settings in hypothesis testing: one with composite hypotheses and one with communication constraints.

The operation of mapping (naturally occurring) continuous time/analog signals into (electronics-friendly) discrete/digital signals is known as quantization, which is an important subject in signal processing in its own right. In information theory, the study of optimal quantization is called rate-distortion theory, introduced by Shannon in 1959. To start, in Chapter 24 we will take a closer look at quantization, followed by the information-theoretic formulation. A simple (and tight) converse bound is then given, with the matching achievability bound deferred to Chapter 25.

In Chapter 4 we collect some results on variational characterizations of information measures. It is a well-known method in analysis to study a functional by proving variational characterizations representing it as a supremum or infimum of some other, simpler (often linear) functionals. Such representations can be useful for multiple purposes:

• Convexity: the pointwise supremum of convex functions is convex.

• Regularity: the pointwise supremum of lower semicontinuous (lsc) functions is lsc.

• Bounds: the upper and lower bounds on the functional follow by choosing good solutions in the optimization problem.

So far we have been focusing on the paradigm for one-way communication: data are mapped to codewords and transmitted, and later decoded based on the received noisy observations. Chapter 23 looks at the more practical setting (except for storage), where the communication frequently goes in both ways so that the receiver can provide certain feedback to the transmitter. As a motivating example, consider the communication channel of the downlink transmission from a satellite to earth. Downlink transmission is very expensive (power constraint at the satellite), but the uplink from earth to the satellite is cheap which makes virtually noiseless feedback readily available at the transmitter (satellite). In general, channel with noiseless feedback is interesting when such asymmetry exists between uplink and downlink. Even in less ideal settings, noisy or partial feedbacks are commonly available that can potentially improve the reliability or complexity of communication. In the first half of our discussion, we shall follow Shannon to show that even with noiseless feedback “nothing” can be gained in the conventional setup. In the process, we will also introduce the concept of Massey’s directed information. In the second half of the Chapter we examine situations where feedback is extremely helpful: low probability of error, variable transmission length and variable transmission power.

In Chapter 21 we will consider an interesting variation of the channel coding problem. Instead of constraining the blocklength (i.e., the number of channel uses), we will constrain the total cost incurred by the codewords. The motivation is the following. Consider a deep-space probe that has a k-bit message that needs to be delivered to Earth (or a satellite orbiting it). The duration of transmission is of little worry for the probe, but what is really limited is the amount of energy it has stored in its battery. In this chapter we will learn how to study this question abstractly and how this fundamental limit is related to communication over continuous-time channels.

In Chapter 25 we present the hard direction of the rate-distortion theorem: the random coding construction of a quantizer. This method is extended to the development of a covering lemma and soft-covering lemma, which lead to the sharp result of Cuff showing that the fundamental limit of channel simulation is given by Wyner’s common information. We also derive (a strengthened form of) Han and Verdú’s results on approximating output distributions in Kullback–Leibler.