What Is Modulation?

Data bits are mathematical entities that have no physical attributes. To send them over a channel, one needs to first map them into some physical signal, which is then “fed” into a channel to produce a physical signal at the channel's output. For example, when we send data over a telephone line, the data bits are first converted to an electrical signal, which then influences the voltage measured at the other end of the line. (We use the term “influences” because the signal measured at the other end of the line is usually not identical to the channel input: it is typically attenuated and also corrupted by thermal noise and other distortions introduced by various conversions in the telephone exchange system.) Similarly, in a wireless system, the data bits are mapped to an electromagnetic wave that then influences the electromagnetic field measured at the receiver antenna. In magnetic recording, data bits are written onto a magnetic medium by a mapping that maps them to a magnetization pattern, which is then measured (with some distortion and some noise) by the magnetic head at some later time when the data are read.

In the first example the bits are mapped to continuous-time waveforms corresponding to the voltage across an impedance, whereas in the last example the bits are mapped to a spatial waveform corresponding to different magnetizations at different locations across the magnetic medium. While some of the theory we shall develop holds for both cases, we shall focus here mainly on channels of the former type, where the channel input signal is some function of time rather than space.

Claude Shannon, the father of Information Theory, described the fundamental problem of point-to-point communications in his classic 1948 paper as “that of reproducing at one point either exactly or approximately a message selected at another point.” How engineers solve this problem is the subject of this book. But unlike Shannon's general problem, where the message can be an image, a sound clip, or a movie, here we restrict ourselves to bits. We thus envision that the original message is either a binary sequence to start with, or else that it was described using bits by a device outside our control and that our job is to reproduce the describing bits with high reliability. The issue of how images or text files are converted efficiently into bits is the subject of lossy and lossless data compression and is addressed in texts on information theory and on quantization.

The engineering solutions to the point-to-point communication problem greatly depend on the available resources and on the channel between the points. They typically bring together beautiful techniques from Fourier Analysis, Hilbert Spaces, Probability Theory, and Decision Theory. The purpose of this book is to introduce the reader to these techniques and to their interplay.

The book is intended for advanced undergraduates and beginning graduate students. The key prerequisites are basic courses in Calculus, Linear Algebra, and Probability Theory. A course in Linear Systems is a plus but not a must, because all the results from Linear Systems that are needed for this book are summarized in Chapters 5 and 6.

In Digital Communications the task of the receiver is to observe the channel outputs and to use these observations to accurately guess the data bits that were sent by the transmitter, i.e., the data bits that were fed to the modulator. Ideally, the guessing would be perfect, i.e., the receiver would make no errors. This, alas, is typically impossible because of the distortions and noise that the channel introduces. Indeed, while one can usually recover the data bits from the transmitted waveform (provided that the modulator is a one-to-one mapping), the receiver has no access to the transmitted waveform but only to the received waveform. And since the latter is typically a noisy version of the former, some errors are usually unavoidable.

In this chapter we shall begin our study of how to guess intelligently, i.e., how, given the channel output, one should guess the data bits with as low a probability of error as possible. This study will help us not only in the design of receivers but also in the design of modulators that allow for reliable decoding from the channel's output.

In the engineering literature the process of guessing the data bits based on the channel output is called “decoding.” In the statistics literature this process is called “hypothesis testing.” We like “guessing” because it demystifies the process.

In most applications the channel output is a continuous-time waveform and we seek to decode a large number of bits. Nevertheless, for pedagogical reasons, we shall begin our study with the simpler case where we wish to decode only a single data bit.