Introduction

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. This corresponds in the statistics literature to “binary hypothesis testing,” where the term “binary” reminds us that in this guessing problem there are only two alternatives. Moreover, we shall assume that the observation, rather than being a continuous-time waveform, is a vector or a scalar. In fact, we shall begin our study with the simplest case where there are no observations at all.