In common working the coder and decoder name as codec.

In common working the modulator and demodulator name as the modem.

The set the transmitter, receiver, communication line with interference sources forms the communication channel.

A source of the messages, communication channel and addressee of the messages form single-channel system of information transmission (SCSIT) (See Fig 1.2).

The SCSIT is a base element of all more complex information systems. Therefore large attention is given to its study.

For mathematical modelling of work of information systems use language of the theory of functional analysis (the operators theory). In this theory the certain operator attributes to each information technology, which describes transformation of the message, signal or interference. As a result of consideration of all sequence of transformations of the messages, signals and handicaps in information system they receive the operator equation, which describes functioning this system. The operator equations serve a theoretical basis for the decision of tasks of the analysis, synthesis and optimisation of information systems.

Example 1.1. Let's consider a conclusion of the operator equation for elementary SCSIT (Fig. 1.1), which is represented by system of transmission of the telegraphic messages.

We shall consider consistently concrete operations of transformation of the messages and signals, which take place by transmission of the telegraphic text on wire channels of communication. The message A(t) as the telegram acts on an input of the transmitter, which carries out operation of coding K. At coding a sequence of the letters and figures will be transformed to a sequence of electrical pulses S(t). The character S(t) is defined by a kind of a code. Hence, as a result of coding receives an electrical video signal

S(t) = К [A(t), S_K],

where S_K is vector, which components are the elementary code signals.

By transmission of a sequence of a video of pulses S(t) on the communication line they are deformed because of influence of a interference (t). The operator of influence of interferences on a useful signal we shall designate through L, then the signal on an input of the receiver can be presented so:

Z (t) =L {K [A (t), S_K], (t)}.

After decoding by the receiver of the accepted signals there receive a copy of the message

B (t) =D [Z (t)] =D {L [K {[A (t), S_K], (t)]}, (1.1)

where D is operator of decoding.

The equation (1.1) describes functioning of telegraphic simplex (unilateral) SCSIT in the operator’s form. For the further construction of mathematical model of functioning of this system it is necessary to construct mathematical models of formation of the telegraphic messages, code of video pulses, interferences in the communication line, errors by transmission of pulses and letters of the messages, operators of coding and decoding. Received after that the mathematical model of functioning of telegraphic system allows to simulate bilateral (duplex system), to decide tasks of the analysis, synthesis and optimisation by the chosen criteria of quality of transmission of the telegraphic messages.

A quality of information system defines set of properties, which characterize its suitability for use on assignment.

To number of such properties usually carry speed of information transmission, throughput, confidentiality, noise stability, faultlessness, reliability, accuracy, profitability, dimensions of devices and others. The quantitative characteristics of these properties name as criteria of quality. The quality criteria used together with the operator’s equations, allow to analyse work of existing systems and to create (to synthesize) systems with required properties.

The efficiency of information system represents the characteristic, how use available resources for reception of required quality. The parameters of efficiency are the quantitative characteristics of efficiency.