**1. lntroduction**

The subject of digital communications pertains to transmission and reception of digital signals. The transmitter functions include periodically choosing a signal out of many possible, converting it into a waveform that suits the transmission media followed by its transmission. The functions of the receiver include reception of the transmitted signals, processing them using the statistical properties of the received waveforms and making decisions to recover the information signals with minimum probability of error. Because of its extensive use of probability and random processes, the study of digital communication is quite abstract.

The performance of digital communication heavily depends on the way the transmission medium affects the transmitted waveforms. The transmission medium alters the signal waveforms during their passage through it, therefore signal waveforms must be designed so that these are least affected by the propagation medi um and are easier to detect and reproduce the information signal with a minimum probability of error. It can be safely stated that formatting the information signals (operations at the transmitter) and making decisions at the receiver are mainly determined by the affect of the channel on the transmitted waveforms. 1n order to establish principles of signal transmission and its detection, we begin with the simplest of scenarios where the channel adds noise to the transmitted signal but does not alter the waveform; this type of channel is known as additive Gaussian noise (AWGN) channel. The principles thus established are later used to study the performances of several digital communication systems operating over channels that fade and disperse the signal waveforms. The channels may also be contaminated by interference - intelligent or otherwise.

It is interesting to note that the telegraphic system introduced in 1844 was an example of digital communication. The long distance telegraphy across Atlantic started in 1866. The search for a suitable code (signal design) to send digital signals resulted in Baudot code in 1875, which interestingly found application many decades later when teletypewriter was invented. After this, the status of digital communication did not make much progress primarily due to the invention of telephone, an analog device, by A. G. Bell in 1876. This invention led to rapid progress in analog communications with analog voice as the primary application until revival of digital communications in 1960 when IBM proposed an eight bit characters code called EBCDIC code; though in 1963 this code lost the standardization battle to a 7-bit code with an alphabet size of 128 characters called American Standard Code for Information Interchange (ASCII). Further improvement took place when ANSI Standard X3.16 introduced in 1976 and CCITT Standard V.4 added an additional bit as a "Parity Check" bit. These inventions were great but the problem that remained was non-availability of an efficient system which is able to convert analog waveforms into digitally encoded signals, although concepts of sampling and encoding were well established as far back as 1937 when Reeves conceived the so called pulse code modulation (PCM) [Reeves (1937)]. During the 1960s, the telecommunication network hierarchy for voice communications was defined on the basis of 64 kbits/sec pulse code modulation (PCM). The invention of high speed solid state switching devices in 1970 followed by development of very large scale integrated (VLSI) circuits resulted in early emergence of digital revolution. Though the TOMA was extensively used over wired network, its introduction into wireless public network in 1992 resulted in an explosive growth of digital communications. Currently, digital signaling is ubiquitous in modem communication systems.
