In this chapter we consider a class of codes known as trellis codes. Unlike block codes, trellis codes can encode an arbitrary length sequence of information symbols to produce a sequence of coded symbols. So the notion of block length is not directly applicable. A subclass of codes known as convolutional codes, which are linear trellis codes, has found widespread applications in many communication systems.

In this chapter we consider a communication system that transmits a single bit of information using one of two signals. The receiver filters the received signal, samples the filter output, and then makes a decision about which of the two signals was transmitted. We first consider an example in which the two signals are just rectangular pulses with opposite sign. For those signals in additive white Gaussian noise (AWGN) we analyze the probability of error for a receiver that uses a filter matched to the transmitted signal. Second, we consider optimizing the system over all possible filters, signals, and decision rules. The optimal filter and signals are derived for binary modulation in which one of two signals is transmitted. Finally, the effect of imperfect receivers is considered. Approaches to analyzing a system with intersymbol interference (ISI) are discussed.

Digital communication systems are ubiquitous. Examples of digital communication systems include cell phones, Bluetooth, WiFi, and cable modems. This book explores in depth how these communication systems work and the fundamental limits on the performance of digital communication systems. We begin in this chapter with a high-level description of digital communication systems so as to understand the trade-offs in designing a communication system. We also explore the fundamental limits that can be achieved in terms of the data rate possible for a given bandwidth and the energy needed for a given level of noise.

The optimal receiver finds the most likely noise signal corresponding to a transmitted signal.

Noise is an important aspect of what limits the performance of communication systems. As such, it is important to understand the statistical properties of noise. Noise at the input of a receiver will affect the performance of a communication system. The received signal consists of the desired signal plus noise. Because receivers filter the received signal, it is important to be able to characterize the noise out of a linear system (i.e. a filter).

These modulation techniques are widely used in practice. We quantify the trade-off between data rate and energy for these techniques and compare performance with the capacity limits discussed in Chapter 1. We begin by discussing MPSK, where the information determines the phase of a sinusoidal signal. Second, we discuss PAM and QAM, in which the amplitude in one and two dimensions, respectively, are varied depending on the data. Third, we discuss orthogonal modulation in which the bandwidth efficiency is very low, but the required energy is also very low.

In this chapter we first discuss the relationship between transmitted signals and received signals. There are three effects of the propagation medium on the transmitted signals: path loss, shadowing, and multipath fading. Path loss refers to the relation between the average received power and the transmitted power as a function of distance. Shadowing refers to the situation where buildings or other objects might block the line of sight between the transmitter and receiver.