Communication Systems and Information Theory
ΠΡΡΠΎΡΠ½ΠΈΠΊ ΠΈ ΠΠΎΠ΄Π΅ΡΡ ΠΠ°Π½Π°Π»Π°. Π§ΡΠΎΠ±Ρ Π΄Π°Π»Π΅Π΅ ΡΠΏΡΠΎΡΡΠΈΡΡ ΠΈΠ·ΡΡΠ΅Π½ΠΈΠ΅ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠ° ΠΌΠΎΠ΄Π΅Π»ΠΈΡΡΠ΅Ρ ΠΈ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΊΠ°Π½Π°Π»Π°, ΠΏΠΎΠ»Π΅Π·Π½ΠΎ ΡΠ°ΡΡΠΈΡΠ½ΠΎ ΠΈΠ·ΠΎΠ»ΠΈΡΠΎΠ²Π°ΡΡ ΡΡΡΠ΅ΠΊΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠ° Π² ΡΠΈΡΡΠ΅ΠΌΠ΅ ΡΠ²ΡΠ·ΠΈ ΠΎΡ ΡΠ°ΠΊΠΎΠ²ΠΎΠ³ΠΎ ΠΊΠ°Π½Π°Π»Π°. ΠΡΠΎ ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΡΠ΄Π΅Π»Π°Π½ΠΎ, Π»ΠΎΠΌΠ°Ρ{*Π½Π°ΡΡΡΠ°Ρ*} ΠΊΠΎΠ΄Π΅Ρ ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅Ρ ΡΠΈΡ. 4.1. ΠΠ°ΠΆΠ΄ΡΠΉ Π² Π΄Π²Π΅ ΡΠ°ΡΡΠΈ ΠΊΠ°ΠΊ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ Π² ΡΠΈΡ. 4.2. Π¦Π΅Π»Ρ ΠΈΡΡ ΠΎΠ΄Π½ΠΎΠ³ΠΎ ΠΊΠΎΠ΄Π΅ΡΠ° ΡΠΎΡΡΠΎΠΈΡ Π² ΡΠΎΠΌ, ΡΡΠΎΠ±Ρ ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΠΈΡΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊ, Π²ΡΠ²ΠΎΠ΄ΠΈΠΌΡΠΉ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΡΡ Π΄Π²ΠΎΠΈΡΠ½ΡΡ … Π§ΠΈΡΠ°ΡΡ Π΅ΡΡ >
Communication Systems and Information Theory (ΡΠ΅ΡΠ΅ΡΠ°Ρ, ΠΊΡΡΡΠΎΠ²Π°Ρ, Π΄ΠΈΠΏΠ»ΠΎΠΌ, ΠΊΠΎΠ½ΡΡΠΎΠ»ΡΠ½Π°Ρ)
Communication Systems and Information Theory.
1. Communication Theory. Communication theory deals primarily with systems for transmitting information or data form one point to another. A rather general block diagram for visualizing the behavior of such systems is given in Fig. 4.1. The source output might represent, for example, a voice waveform, a sequence of binary digits form a magnetic tape, the output of a set of sensors in a space probe, a sensory input to a biological organism, or a target in a radar system. The channel might represent, for example, a telephone line, a high frequency radio link, a space communication link, a storage medium, or a biological organism (for the case where the source output is a sensory input to that organism). The channel is usually subjected to various types of noise disturbances, which on a telephone line, for example, might take the form of a time-varying frequency response, crosstalk from other lines, thermal noise, and impulsive switching noise. The encoder in Fig. 4.1. represents any processing of the source output performed prior to transmission. The processing might include, for example, any combination of modulation, data reduction, and insertion of redundancy to combat the channel noise. The decoder represents the processing of the channel output with the objective of producing at the destination an acceptable replica of (or response to) the source output.
2. Information Theory. In the early 1940 «s a mathematical theory, for dealing with the more fundamental aspects of communication systems, was developed. The distinguishing characteristics of this theory are, first, a great emphasis on probability theory and, second, a primary concern with the encoder and decoder, both in terms of their functional roles and in terms of the existence (or nonexistence) of encoders and decoders that achieve a given level of performance. In the past 20 years, information theory has been made more precise, has been extended, and brought to the point where it is being applied in practical communication systems.
As in any mathematical theory, the theory deals only with mathematical models and not with physical sources and physical channels.
One would think, therefore, that the appropriate way to begin the development of the theory would be with a discussion of how to construct appropriate mathematical models for physical sources and channels. This, however, is not the way that theories are constructed, primarily because physical reality is rarely simple enough to be precisely modeled by mathematically tractable models. The procedure will be rather to start by studying the simplest classes of mathematical models of sources and channels, using the insight and the results gained to study progressively more complicated classes of models. Naturally, the choice of classes of models to study will be influenced and motivated by the more important aspects of real sources and channels, but the view of what aspects are important will be modified by the theoretical results. Finally, after understanding the theory, it can be found to be useful in the study of real communication systems in two ways. First, it will provide a framework within which to construct detailed models of real sources and channels.
Second, and more important, the relationship established by the theory provide an indication of the types of tradeoffs that exist in constructing encoders and decoders for given systems. While the above comments can be applied to almost any mathematical theory, they are particularly necessary here.
Π‘ΠΈΡΡΠ΅ΠΌΡ ΡΠ²ΡΠ·ΠΈ ΠΈ Π’Π΅ΠΎΡΠΈΡ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ.
1. Π’Π΅ΠΎΡΠΈΡ ΠΊΠΎΠΌΠΌΡΠ½ΠΈΠΊΠ°ΡΠΈΠΈ. ΠΠ΅Π»Π° Π’Π΅ΠΎΡΠΈΠΈ ΠΊΠΎΠΌΠΌΡΠ½ΠΈΠΊΠ°ΡΠΈΠΈ ΠΏΡΠ΅ΠΆΠ΄Π΅ Π²ΡΠ΅Π³ΠΎ Ρ ΡΠΈΡΡΠ΅ΠΌΠ°ΠΌΠΈ Π΄Π»Ρ ΠΏΠ΅ΡΠ΅Π΄Π°ΡΠΈ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ ΠΈΠ»ΠΈ Π΄Π°Π½Π½ΡΡ ΡΠΎΡΠΌΠΈΡΡΡΡ ΠΎΠ΄ΠΈΠ½ ΠΏΡΠ½ΠΊΡ{*ΡΠΎΡΠΊΡ*} ΠΊ Π΄ΡΡΠ³ΠΎΠΌΡ. ΠΠΎΠ²ΠΎΠ»ΡΠ½ΠΎ ΠΎΠ±ΡΠ°Ρ Π±Π»ΠΎΠΊ-ΡΡ Π΅ΠΌΠ° Π΄Π»Ρ Π²ΠΈΠ·ΡΠ°Π»ΠΈΠ·ΠΈΡΡΡΡΠ΅Π³ΠΎ ΠΏΠΎΠ²Π΅Π΄Π΅Π½ΠΈΠ΅ ΡΠ°ΠΊΠΈΡ ΡΠΈΡΡΠ΅ΠΌ Π΄Π°Π΅ΡΡΡ Π² ΡΠΈΡ. 4.1. ΠΡΡ ΠΎΠ΄Π½ΡΠΉ Π²ΡΠ²ΠΎΠ΄ ΠΌΠΎΠ³ Π±Ρ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΡΡ, Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, Π·Π²ΡΠΊΠΎΠ²Π°Ρ ΡΠΎΡΠΌΠ° Π²ΠΎΠ»Π½Ρ, ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΡ Π΄Π²ΠΎΠΈΡΠ½ΡΡ ΡΠΈΠΌΠ²ΠΎΠ»ΠΎΠ² ΡΠΎΡΠΌΠΈΡΡΠ΅Ρ ΠΌΠ°Π³Π½ΠΈΡΠ½ΡΡ Π»Π΅Π½ΡΡ, Π²ΡΠ²ΠΎΠ΄ Π½Π°Π±ΠΎΡΠ° Π΄Π°ΡΡΠΈΠΊΠΎΠ² Π² ΠΊΠΎΡΠΌΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΈ, ΡΠ΅Π½ΡΠΎΡΠ½ΡΠΉ Π²Π²ΠΎΠ΄{*Π²Ρ ΠΎΠ΄*} ΠΊ Π±ΠΈΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΎΠΌΡ ΠΎΡΠ³Π°Π½ΠΈΠ·ΠΌΡ, ΠΈΠ»ΠΈ ΡΠ΅Π»ΠΈ Π² ΡΠ°Π΄ΠΈΠΎΠ»ΠΎΠΊΠ°ΡΠΈΠΎΠ½Π½ΠΎΠΉ ΡΠ΅ΡΠΈ. ΠΠ°Π½Π°Π» ΠΌΠΎΠ³ Π±Ρ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΡΡ, Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, ΡΠ΅Π»Π΅ΡΠΎΠ½Π½Π°Ρ Π»ΠΈΠ½ΠΈΡ, Π²ΡΡΠΎΠΊΠΎΡΠ°ΡΡΠΎΡΠ½Π°Ρ ΡΠ°Π΄ΠΈΠΎΡΠ΅Π»Π΅ΠΉΠ½Π°Ρ Π»ΠΈΠ½ΠΈΡ, Π»ΠΈΠ½ΠΈΡ ΠΊΠΎΡΠΌΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ²ΡΠ·ΠΈ, Π½ΠΎΡΠΈΡΠ΅Π»Ρ Π΄Π°Π½Π½ΡΡ , ΠΈΠ»ΠΈ Π±ΠΈΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΠΉ ΠΎΡΠ³Π°Π½ΠΈΠ·ΠΌ (Π΄Π»Ρ ΡΠ»ΡΡΠ°Ρ, Π³Π΄Π΅ ΠΈΡΡ ΠΎΠ΄Π½ΡΠΉ Π²ΡΠ²ΠΎΠ΄ — ΡΠ΅Π½ΡΠΎΡΠ½ΡΠΉ Π²Π²ΠΎΠ΄{*Π²Ρ ΠΎΠ΄*} ΠΊ ΡΠΎΠΌΡ ΠΎΡΠ³Π°Π½ΠΈΠ·ΠΌΡ). ΠΠ°Π½Π°Π» ΠΎΠ±ΡΡΠ½ΠΎ ΠΏΠΎΠ΄Π²Π΅ΡΠ³Π°Π΅ΡΡΡ ΡΠ°Π·Π»ΠΈΡΠ½ΡΠΌ ΡΠΈΠΏΠ°ΠΌ ΡΡΠΌΠΎΠ²ΡΡ ΠΏΠΎΠΌΠ΅Ρ , ΠΊΠΎΡΠΎΡΡΠ΅ Π½Π° ΡΠ΅Π»Π΅ΡΠΎΠ½Π½ΠΎΠΉ Π»ΠΈΠ½ΠΈΠΈ, Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, ΠΌΠΎΠ³Π»ΠΈ Π±Ρ Π±ΡΠ°ΡΡ ΡΠΎΡΠΌΡ ΠΈΠ·ΠΌΠ΅Π½ΡΡΡΠ΅ΠΉΡΡ Π²ΠΎ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΡΠ°ΡΡΠΎΡΠ½ΠΎΠΉ Ρ Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ, ΠΏΠ΅ΡΠ΅ΠΊΡΠ΅ΡΡΠ½ΡΠ΅ ΠΏΠΎΠΌΠ΅Ρ ΠΈ ΠΎΡ Π΄ΡΡΠ³ΠΈΡ ΡΡΡΠΎΠΊ, ΡΠ΅ΠΏΠ»ΠΎΠ²ΡΡ ΠΏΠΎΠΌΠ΅Ρ , ΠΈ ΠΈΠΌΠΏΡΠ»ΡΡΠ½ΡΡ ΡΡΠΌΠΎΠ² ΠΏΠ΅ΡΠ΅ΠΊΠ»ΡΡΠ°ΡΠ΅Π»Ρ. ΠΠΎΠ΄Π΅Ρ Π² ΡΠΈΡ. 4.1. ΠΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΠ΅Ρ Π»ΡΠ±ΡΡ ΠΎΠ±ΡΠ°Π±ΠΎΡΠΊΡ ΠΈΡΡ ΠΎΠ΄Π½ΠΎΠ³ΠΎ Π²ΡΠ²ΠΎΠ΄Π°, Π²ΡΠΏΠΎΠ»Π½Π΅Π½Π½ΠΎΠ³ΠΎ Π΄ΠΎ ΠΏΠ΅ΡΠ΅Π΄Π°ΡΠΈ. ΠΠ±ΡΠ°Π±ΠΎΡΠΊΠ° ΠΌΠΎΠ³Π»Π° Π±Ρ Π²ΠΊΠ»ΡΡΠ°ΡΡ, Π½Π°ΠΏΡΠΈΠΌΠ΅Ρ, Π»ΡΠ±Π°Ρ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΈΡ ΠΌΠΎΠ΄ΡΠ»ΡΡΠΈΠΈ, ΠΎΠ±ΡΠ°Π±ΠΎΡΠΊΠΈ Π΄Π°Π½Π½ΡΡ , ΠΈ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ ΡΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ Π½Π΅ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΠΌΠΎΡΡΠΈ, ΡΡΠΎΠ±Ρ ΡΡΠ°Π·ΠΈΡΡΡΡ Ρ ΡΡΠΌΠΎΠΌ ΠΊΠ°Π½Π°Π»Π°. ΠΠ΅ΠΊΠΎΠ΄Π΅Ρ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΠ΅Ρ ΠΎΠ±ΡΠ°Π±ΠΎΡΠΊΡ Π²ΡΠ²ΠΎΠ΄Π° ΠΊΠ°Π½Π°Π»Π° Ρ ΡΠ΅Π»ΡΡ ΡΠΎΠ·Π΄Π°Π½ΠΈΡ Π² Π°Π΄ΡΠ΅ΡΠ°ΡΠ΅ ΠΏΡΠΈΠ΅ΠΌΠ»Π΅ΠΌΠ°Ρ ΡΠΎΡΠ½Π°Ρ ΠΊΠΎΠΏΠΈΡ (ΠΈΠ»ΠΈ ΠΎΡΠ²Π΅Ρ Π½Π°) ΠΈΡΡ ΠΎΠ΄Π½ΡΠΉ Π²ΡΠ²ΠΎΠ΄.
2. Π’Π΅ΠΎΡΠΈΡ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ. Π Π½Π°ΡΠ°Π»Π΅ 1940;ΡΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΡΠ΅ΠΎΡΠΈΡ, Π΄Π»Ρ ΠΈΠΌΠ΅ΡΡΠΈΠΉ Π΄Π΅Π»ΠΎ Ρ Π±ΠΎΠ»Π΅Π΅ ΡΡΠ½Π΄Π°ΠΌΠ΅Π½ΡΠ°Π»ΡΠ½ΡΠΌΠΈ Π°ΡΠΏΠ΅ΠΊΡΠ°ΠΌΠΈ ΡΠΈΡΡΠ΅ΠΌ ΡΠ²ΡΠ·ΠΈ, Π±ΡΠ»Π° ΡΠ°Π·ΡΠ°Π±ΠΎΡΠ°Π½Π°. Π Π°Π·Π»ΠΈΡΠ°ΡΡΠΈΠ΅ Ρ Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΡΡΠΈΠΊΠΈ ΡΡΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ, ΡΠ½Π°ΡΠ°Π»Π°, Π±ΠΎΠ»ΡΡΠΎΠΉ Π°ΠΊΡΠ΅Π½Ρ Π½Π° ΡΠ΅ΠΎΡΠΈΠΈ Π²Π΅ΡΠΎΡΡΠ½ΠΎΡΡΠΈ ΠΈ, Π²ΠΎ Π²ΡΠΎΡΡΡ , ΠΏΠ΅ΡΠ²ΠΈΡΠ½ΠΎΠ΅ ΠΏΡΠ΅Π΄ΠΏΡΠΈΡΡΠΈΠ΅{*Π±Π΅ΡΠΏΠΎΠΊΠΎΠΉΡΡΠ²ΠΎ*} Ρ ΠΊΠΎΠ΄Π΅ΡΠΎΠΌ ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠΎΠΌ, ΠΈ Π² ΡΠ΅ΡΠΌΠΈΠ½Π°Ρ ΠΈΡ ΡΡΠ½ΠΊΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΡ ΡΠΎΠ»Π΅ΠΉ ΠΈ Π² ΡΠ΅ΡΠΌΠΈΠ½Π°Ρ ΡΡΡΠ΅ΡΡΠ²ΠΎΠ²Π°Π½ΠΈΡ (ΠΈΠ»ΠΈ Π½Π΅ΡΡΡΠ΅ΡΡΠ²ΠΎΠ²Π°Π½ΠΈΠ΅) ΠΊΠΎΠ΄Π΅ΡΠΎΠ² ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠΎΠ², ΠΊΠΎΡΠΎΡΡΠ΅ Π΄ΠΎΡΡΠΈΠ³Π°ΡΡ Π΄Π°Π½Π½ΠΎΠ³ΠΎ ΡΡΠΎΠ²Π½Ρ Π²ΡΠΏΠΎΠ»Π½Π΅Π½ΠΈΡ{*ΡΠ°Π±ΠΎΡΡ*}. Π ΠΏΡΠΎΡΠ»ΡΡ 20 Π³ΠΎΠ΄Π°Ρ , ΡΠ΅ΠΎΡΠΈΡ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ Π±ΡΠ»Π° ΡΠ΄Π΅Π»Π°Π½Π° Π±ΠΎΠ»Π΅Π΅ ΡΠΎΡΠ½ΠΎΠΉ, Π±ΡΠ»Π° ΡΠ°ΡΡΠΈΡΠ΅Π½Π°{*ΠΏΡΠΎΠ΄Π»Π΅Π½Π°*}, ΠΈ ΠΏΡΠΈΠ½Π΅ΡΠ΅Π½Π° ΠΊ ΡΡΡΠΈ, Π³Π΄Π΅ ΡΡΠΎ ΠΎΠ±ΡΠ°ΡΠ°Π΅ΡΡΡ{*ΠΏΡΠΈΠΌΠ΅Π½ΡΠ΅ΡΡΡ*} Π² ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈΡ ΡΠΈΡΡΠ΅ΠΌΠ°Ρ ΡΠ²ΡΠ·ΠΈ.
ΠΠ°ΠΊ Π² Π»ΡΠ±ΠΎΠΉ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ, ΡΠ΅ΠΎΡΠΈΡ ΠΈΠΌΠ΅Π΅Ρ Π΄Π΅Π»ΠΎ ΡΠΎΠ»ΡΠΊΠΎ Ρ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΠΌΠΎΠ΄Π΅Π»ΡΠΌΠΈ, Π° Π½Π΅ Ρ ΡΠΈΠ·ΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠ°ΠΌΠΈ ΠΈ ΡΠΈΠ·ΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΠΊΠ°Π½Π°Π»Π°ΠΌΠΈ. ΠΠΎΠΆΠ½ΠΎ Π±ΡΠ»ΠΎ Π±Ρ Π΄ΡΠΌΠ°Π» Π±Ρ, ΠΏΠΎΡΡΠΎΠΌΡ, ΡΡΠΎ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΠΉ ΡΠΏΠΎΡΠΎΠ± Π½Π°ΡΠΈΠ½Π°ΡΡ ΡΠ°Π·Π²ΠΈΡΠΈΠ΅ ΡΠ΅ΠΎΡΠΈΠΈ Π±ΡΠ΄Π΅Ρ Ρ ΠΎΠ±ΡΡΠΆΠ΄Π΅Π½ΠΈΠ΅ΠΌ ΡΠΎΠ³ΠΎ, ΠΊΠ°ΠΊ ΡΠΎΠ·Π΄Π°ΡΡ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΠ΅ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈ Π΄Π»Ρ ΡΠΈΠ·ΠΈΡΠ΅ΡΠΊΠΈΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠΎΠ² ΠΈ ΠΊΠ°Π½Π°Π»ΠΎΠ². ΠΡΠΎ, ΠΎΠ΄Π½Π°ΠΊΠΎ, — Π½Π΅ ΠΏΡΡΡ, ΠΊΠΎΡΠΎΡΡΠΌ ΡΠ΅ΠΎΡΠΈΠΈ ΡΠΎΠ·Π΄Π°Π½Ρ, ΠΏΡΠ΅ΠΆΠ΄Π΅ Π²ΡΠ΅Π³ΠΎ, ΠΏΠΎΡΠΎΠΌΡ ΡΡΠΎ ΡΠΈΠ·ΠΈΡΠ΅ΡΠΊΠ°Ρ Π΄Π΅ΠΉΡΡΠ²ΠΈΡΠ΅Π»ΡΠ½ΠΎΡΡΡ ΡΠ΅Π΄ΠΊΠΎ Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΠΎ ΠΏΡΠΎΡΡΠ° Π±ΡΡΡ ΡΠΎΡΠ½ΠΎ ΡΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½Π½ΠΎΠΉ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈ ΠΏΠΎΡΠ»ΡΡΠ½ΡΠΌΠΈ ΠΌΠΎΠ΄Π΅Π»ΡΠΌΠΈ. ΠΡΠΎΡΠ΅Π΄ΡΡΠ° Π±ΡΠ΄Π΅Ρ Π΄ΠΎΠ»ΠΆΠ½Π° Π΄ΠΎΠ²ΠΎΠ»ΡΠ½ΠΎ Π·Π°ΠΏΡΡΡΠΈΡΡ, ΠΈΠ·ΡΡΠ°Ρ ΡΠ°ΠΌΡΠ΅ ΠΏΡΠΎΡΡΡΠ΅ ΠΊΠ»Π°ΡΡΡ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠΎΠ² ΠΈ ΠΊΠ°Π½Π°Π»ΠΎΠ², ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡ ΠΏΠΎΠ½ΠΈΠΌΠ°Π½ΠΈΠ΅ ΠΈ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΡ, ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΠ΅ ΠΈΠ·ΡΡΠ°ΡΡ ΠΏΡΠΎΠ³ΡΠ΅ΡΡΠΈΠ²Π½ΠΎ Π±ΠΎΠ»Π΅Π΅ ΡΠ»ΠΎΠΆΠ½ΡΠ΅ ΠΊΠ»Π°ΡΡΡ ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ. ΠΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎ, Π½Π° Π²ΡΠ±ΠΎΡ ΠΊΠ»Π°ΡΡΠΎΠ² ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ, ΡΡΠΎΠ±Ρ ΡΡΠΈΡΡΡΡ Π±ΡΠ΄ΡΡ ΠΏΠΎΠ²Π»ΠΈΡΡΡ ΠΈ ΠΌΠΎΡΠΈΠ²ΠΈΡΠΎΠ²Π°ΡΡΡΡ Π±ΠΎΠ»Π΅Π΅ Π²Π°ΠΆΠ½ΡΠΌΠΈ Π°ΡΠΏΠ΅ΠΊΡΠ°ΠΌΠΈ ΡΠ΅Π°Π»ΡΠ½ΡΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠΎΠ² ΠΈ ΠΊΠ°Π½Π°Π»ΠΎΠ², Π½ΠΎ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΠ΅{*Π²ΠΈΠ΄*} ΡΡΠΎ Π°ΡΠΏΠ΅ΠΊΡΡ Π²Π°ΠΆΠ½Ρ, Π±ΡΠ΄Π΅Ρ ΠΈΠ·ΠΌΠ΅Π½ΡΡΡΡΡ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ.
because quite an extensive theory must be developed before the more important implications for the design of communication systems will become apparent.
3. The source and Channel Encoders. In order to further simplify the study of source models and channel models, it is helpful to partly isolate the effect of the source in a communication system from that of the channel. This can be done by breaking the encoder and decoder of Fig. 4.1. each into two parts as shown in Fig. 4.2. The purpose of the source encoder is to represent the source output by a sequence of binary digits and one of the major questions of concern is to determine how many binary digits per unit time are required to represents the output of any given source model. The purpose of the channel encoder and decoder is to allow the binary data sequences to be reliably reproduced at the output of the channel decoder, and one of course, whether restricting the encoder and decoder to the form of Fig. 4.2. imposes any fundamental limitations on the performance of the communication system. One of the most important results of the theory, however, is that under very broad conditions no such limitations are imposed (this does not say, however, that encoder and decoder of the form in Fig. 4.2. is always the most economical way to achieve a given performance). From a practical standpoint, the splitting of encoder and decoder is particularly convenient since it makes the design of the channel encoder and decoder virtually independent of the source encoder and decoder, using binary data as an interface. This, of course, facilitates the use of different sources on the same channel.
4. Conclusion. Much of modern communication theory stems from the works of communication systems and also the desirability of modeling both signal and noise as random processes. Wiener was interested in finding the best linear filter to separate the signal from additive noise with a prescribed delay and his work had an important influence on subsequent research in modulation theory. Also Wiener «s interest in reception with negative delay (that, is, prediction) along with Kolmogorov «s work on prediction if the absence of noise have had an important impact on control theory. Similarly, Kotelnikov was interested in the detection and estimation of signals at the receiver. ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ°ΠΌΠΈ. ΠΠ°ΠΊΠΎΠ½Π΅Ρ, ΠΏΠΎΡΠ»Π΅ ΠΏΠΎΠ½ΠΈΠΌΠ°Π½ΠΈΡ ΡΠ΅ΠΎΡΠΈΠΈ, ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ Π½Π°ΠΉΠ΄Π΅Π½ΠΎ Π±ΡΡΡ ΠΏΠΎΠ»Π΅Π·Π½ΡΠΌ Π² ΠΈΠ·ΡΡΠ΅Π½ΠΈΠΈ ΡΠ΅Π°Π»ΡΠ½ΡΡ ΡΠΈΡΡΠ΅ΠΌ ΡΠ²ΡΠ·ΠΈ Π² Π΄Π²ΡΡ ΠΏΡΡΡΡ . Π‘Π½Π°ΡΠ°Π»Π°, ΡΡΠΎ ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠΈΡ ΡΡΡΡΠΊΡΡΡΡ, ΡΡΠΎΠ±Ρ ΡΠΎΠ·Π΄Π°ΡΡ Π΄Π΅ΡΠ°Π»ΡΠ½ΡΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΈΠ· ΡΠ΅Π°Π»ΡΠ½ΡΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠΎΠ² ΠΈ ΠΊΠ°Π½Π°Π»ΠΎΠ². ΠΠΎ Π²ΡΠΎΡΡΡ , ΠΈ ΡΡΠΎ Π±ΠΎΠ»Π΅Π΅ Π²Π°ΠΆΠ½ΠΎ, ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡ, ΡΡΡΠ°Π½ΠΎΠ²Π»Π΅Π½Π½ΡΠ΅ Π² ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΠΈΠΈ Ρ ΡΠ΅ΠΎΡΠΈΠ΅ΠΉ ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠΈΠ²Π°ΡΡ ΠΈΠ½Π΄ΠΈΠΊΠ°ΡΠΈΡ ΠΎΡΠ½ΠΎΡΠΈΡΠ΅Π»ΡΠ½ΠΎ ΡΠΈΠΏΠΎΠ² ΡΠ΄Π΅Π»ΠΎΠΊ, ΠΊΠΎΡΠΎΡΡΠ΅ ΡΡΡΠ΅ΡΡΠ²ΡΡΡ Π² ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΠΈ ΠΊΠΎΠ΄Π΅ΡΠΎΠ² ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠΎΠ² Π΄Π»Ρ Π΄Π°Π½Π½ΡΡ ΡΠΈΡΡΠ΅ΠΌ. Π ΡΠΎ Π²ΡΠ΅ΠΌΡ ΠΊΠ°ΠΊ Π²ΡΡΠ΅ΡΠΏΠΎΠΌΡΠ½ΡΡΡΠ΅ ΠΊΠΎΠΌΠΌΠ΅Π½ΡΠ°ΡΠΈΠΈ ΠΌΠΎΠ³ΡΡ ΠΏΡΠΈΠΌΠ΅Π½ΡΡΡΡΡ ΠΊ ΠΏΠΎΡΡΠΈ Π»ΡΠ±ΠΎΠΉ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ, ΠΎΠ½ΠΈ ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎ Π½Π΅ΠΎΠ±Ρ ΠΎΠ΄ΠΈΠΌΡ Π·Π΄Π΅ΡΡ, ΠΏΠΎΡΠΎΠΌΡ ΡΡΠΎ Π²Π΅ΡΡΠΌΠ° ΠΎΠ±ΡΠΈΡΠ½Π°Ρ ΡΠ΅ΠΎΡΠΈΡ Π΄ΠΎΠ»ΠΆΠ½Π° Π±ΡΡΡ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠ°Π½Π° ΠΏΡΠ΅ΠΆΠ΄Π΅, ΡΠ΅ΠΌ Π±ΠΎΠ»Π΅Π΅ Π²Π°ΠΆΠ½ΡΠ΅ Π·Π½Π°ΡΠ΅Π½ΠΈΡ Π΄Π»Ρ ΠΏΡΠΎΠ΅ΠΊΡΠ° ΡΠΈΡΡΠ΅ΠΌ ΡΠ²ΡΠ·ΠΈ ΡΡΠ°Π½ΡΡ ΠΎΡΠ΅Π²ΠΈΠ΄Π½ΡΠΌΠΈ.
3. ΠΡΡΠΎΡΠ½ΠΈΠΊ ΠΈ ΠΠΎΠ΄Π΅ΡΡ ΠΠ°Π½Π°Π»Π°. Π§ΡΠΎΠ±Ρ Π΄Π°Π»Π΅Π΅ ΡΠΏΡΠΎΡΡΠΈΡΡ ΠΈΠ·ΡΡΠ΅Π½ΠΈΠ΅ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠ° ΠΌΠΎΠ΄Π΅Π»ΠΈΡΡΠ΅Ρ ΠΈ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΊΠ°Π½Π°Π»Π°, ΠΏΠΎΠ»Π΅Π·Π½ΠΎ ΡΠ°ΡΡΠΈΡΠ½ΠΎ ΠΈΠ·ΠΎΠ»ΠΈΡΠΎΠ²Π°ΡΡ ΡΡΡΠ΅ΠΊΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠ° Π² ΡΠΈΡΡΠ΅ΠΌΠ΅ ΡΠ²ΡΠ·ΠΈ ΠΎΡ ΡΠ°ΠΊΠΎΠ²ΠΎΠ³ΠΎ ΠΊΠ°Π½Π°Π»Π°. ΠΡΠΎ ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΡΠ΄Π΅Π»Π°Π½ΠΎ, Π»ΠΎΠΌΠ°Ρ{*Π½Π°ΡΡΡΠ°Ρ*} ΠΊΠΎΠ΄Π΅Ρ ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅Ρ ΡΠΈΡ. 4.1. ΠΠ°ΠΆΠ΄ΡΠΉ Π² Π΄Π²Π΅ ΡΠ°ΡΡΠΈ ΠΊΠ°ΠΊ ΠΏΠΎΠΊΠ°Π·Π°Π½ΠΎ Π² ΡΠΈΡ. 4.2. Π¦Π΅Π»Ρ ΠΈΡΡ ΠΎΠ΄Π½ΠΎΠ³ΠΎ ΠΊΠΎΠ΄Π΅ΡΠ° ΡΠΎΡΡΠΎΠΈΡ Π² ΡΠΎΠΌ, ΡΡΠΎΠ±Ρ ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΠΈΡΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊ, Π²ΡΠ²ΠΎΠ΄ΠΈΠΌΡΠΉ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΡΡ Π΄Π²ΠΎΠΈΡΠ½ΡΡ ΡΠΈΠΌΠ²ΠΎΠ»ΠΎΠ², ΠΈ ΠΎΠ΄ΠΈΠ½ ΠΈΠ· Π³Π»Π°Π²Π½ΡΡ Π²ΠΎΠΏΡΠΎΡΠΎΠ² ΠΏΡΠ΅Π΄ΠΏΡΠΈΡΡΠΈΡ{*Π±Π΅ΡΠΏΠΎΠΊΠΎΠΉΡΡΠ²Π°*} Π΄ΠΎΠ»ΠΆΠ΅Π½ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΡΡ, ΡΠΊΠΎΠ»ΡΠΊΠΎ Π΄Π²ΠΎΠΈΡΠ½ΡΡ ΡΠΈΠΌΠ²ΠΎΠ»ΠΎΠ² Π² Π΅Π΄ΠΈΠ½ΠΈΡΡ Π²ΡΠ΅ΠΌΠ΅Π½ΠΈ ΡΡΠ΅Π±ΡΡΡΡΡ ΠΊ, ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΠ΅Ρ Π²ΡΠ²ΠΎΠ΄ Π»ΡΠ±ΠΎΠΉ Π΄Π°Π½Π½ΠΎΠΉ ΠΈΡΡ ΠΎΠ΄Π½ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ. Π¦Π΅Π»Ρ ΠΊΠΎΠ΄Π΅ΡΠ° ΠΊΠ°Π½Π°Π»Π° ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠ° ΡΠΎΡΡΠΎΠΈΡ Π² ΡΠΎΠΌ, ΡΡΠΎΠ±Ρ ΠΏΠΎΠ·Π²ΠΎΠ»ΠΈΡΡ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΡΠΌ Π΄Π°Π½Π½ΡΡ Π² Π΄Π²ΠΎΠΈΡΠ½ΠΎΠΌ ΠΊΠΎΠ΄Π΅ Π±ΡΡΡ Π½Π°Π΄Π΅ΠΆΠ½ΠΎ Π²ΠΎΡΠΏΡΠΎΠΈΠ·Π²Π΅Π΄Π΅Π½Π½ΠΎΠΉ ΠΏΡΠΈ Π²ΡΠ²ΠΎΠ΄Π΅ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠ° ΠΊΠ°Π½Π°Π»Π°, ΠΈ ΠΎΠ΄ΠΈΠ½ ΠΊΠΎΠ½Π΅ΡΠ½ΠΎ, Π»ΠΈ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΠ΅ ΠΊΠΎΠ΄Π΅ΡΠ° ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠ° ΠΊ ΡΠΎΡΠΌΠ΅ ΡΠΈΡ. 4.2. ΠΠ°Π»Π°Π³Π°Π΅Ρ Π»ΡΠ±ΡΠ΅ ΡΡΠ½Π΄Π°ΠΌΠ΅Π½ΡΠ°Π»ΡΠ½ΡΠ΅ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡ Π½Π° Π²ΡΠΏΠΎΠ»Π½Π΅Π½ΠΈΠ΅{*ΡΠ°Π±ΠΎΡΡ*} ΡΠΈΡΡΠ΅ΠΌΡ ΡΠ²ΡΠ·ΠΈ. ΠΠ΄ΠΈΠ½ ΠΈΠ· Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ Π²Π°ΠΆΠ½ΡΡ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠΎΠ² ΡΠ΅ΠΎΡΠΈΠΈ, ΠΎΠ΄Π½Π°ΠΊΠΎ, — ΡΠΎ, ΡΡΠΎ ΠΏΡΠΈ ΠΎΡΠ΅Π½Ρ ΡΠΈΡΠΎΠΊΠΈΡ ΡΡΠ»ΠΎΠ²ΠΈΡΡ {*ΡΠΎΡΡΠΎΡΠ½ΠΈΡΡ *} Π½ΠΈΠΊΠ°ΠΊΠΈΠ΅ ΡΠ°ΠΊΠΈΠ΅ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΡ Π½Π΅ Π½Π°Π»ΠΎΠΆΠ΅Π½Ρ (ΡΡΠΎ Π½Π΅ Π³ΠΎΠ²ΠΎΡΠΈΡ, ΠΎΠ΄Π½Π°ΠΊΠΎ, ΡΡΠΎ ΠΊΠΎΠ΄Π΅Ρ ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅Ρ ΡΠΎΡΠΌΡ Π² ΡΠΈΡ. 4.2. Π―Π²Π»ΡΠ΅ΡΡΡ Π²ΡΠ΅Π³Π΄Π° Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΠΈΠΌ ΡΠΏΠΎΡΠΎΠ±ΠΎΠΌ Π΄ΠΎΡΡΠΈΡΡ Π΄Π°Π½Π½ΠΎΠ³ΠΎ Π²ΡΠΏΠΎΠ»Π½Π΅Π½ΠΈΡ{*ΡΠ°Π±ΠΎΡΡ*}). Π‘ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΎΡΠΊΠΈ Π·ΡΠ΅Π½ΠΈΡ, ΡΠ°Π·Π±ΠΈΠ²Π°Π½ΠΈΠ΅ ΠΊΠΎΠ΄Π΅ΡΠ° ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠ° ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎ ΡΠ΄ΠΎΠ±Π½ΠΎ, ΡΠ°ΠΊ ΠΊΠ°ΠΊ ΡΡΠΎ Π΄Π΅Π»Π°Π΅Ρ ΠΏΡΠΎΠ΅ΠΊΡ ΠΈΠ· ΠΊΠΎΠ΄Π΅ΡΠ° ΠΊΠ°Π½Π°Π»Π° ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠ°, ΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈ Π½Π΅Π·Π°Π²ΠΈΡΠΈΠΌΠΎΠ³ΠΎ ΠΎΡ ΠΈΡΡ ΠΎΠ΄Π½ΠΎΠ³ΠΎ ΠΊΠΎΠ΄Π΅ΡΠ° ΠΈ Π΄Π΅ΠΊΠΎΠ΄Π΅ΡΠ°, ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡ Π΄Π°Π½Π½ΡΠ΅ Π² Π΄Π²ΠΎΠΈΡΠ½ΠΎΠΌ ΠΊΠΎΠ΄Π΅ ΠΊΠ°ΠΊ ΠΈΠ½ΡΠ΅ΡΡΠ΅ΠΉΡ. ΠΡΠΎ, ΠΊΠΎΠ½Π΅ΡΠ½ΠΎ, ΠΎΠ±Π»Π΅Π³ΡΠ°Π΅Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ ΡΠ°Π·Π»ΠΈΡΠ½ΡΡ ΠΈΡΡΠΎΡΠ½ΠΈΠΊΠΎΠ² Π½Π° ΡΠΎΠΌ ΠΆΠ΅ ΡΠ°ΠΌΠΎΠΌ ΠΊΠ°Π½Π°Π»Π΅.
4.
ΠΠ°ΠΊΠ»ΡΡΠ΅Π½ΠΈΠ΅
ΠΠ½ΠΎΠ³ΠΎΠ΅ ΠΈΠ· ΡΠΎΠ²ΡΠ΅ΠΌΠ΅Π½Π½ΡΡ ΠΎΡΠ½ΠΎΠ² ΡΠ΅ΠΎΡΠΈΠΈ ΠΊΠΎΠΌΠΌΡΠ½ΠΈΠΊΠ°ΡΠΈΠΈ ΠΎΡ ΡΠ°Π±ΠΎΡ ΡΠΈΡΡΠ΅ΠΌ ΡΠ²ΡΠ·ΠΈ ΠΈ ΡΠ°ΠΊΠΆΠ΅ ΠΆΠ΅Π»Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΡ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΈ ΡΠΎΠΎΠ±ΡΠ°Π΅Ρ ΠΈ ΡΡΠΌ ΠΊΠ°ΠΊ ΡΠ»ΡΡΠ°ΠΉΠ½ΡΠ΅ ΠΏΡΠΎΡΠ΅ΡΡΡ. ΠΠΈΠ½Π΅Ρ Π±ΡΠ» Π·Π°ΠΈΠ½ΡΠ΅ΡΠ΅ΡΠΎΠ²Π°Π½ ΠΎΠ±Π½Π°ΡΡΠΆΠ΅Π½ΠΈΠ΅ΠΌ Π»ΡΡΡΠ΅Π³ΠΎ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ³ΠΎ ΡΠΈΠ»ΡΡΡΠ°, ΡΡΠΎΠ±Ρ ΠΎΡΠ΄Π΅Π»ΠΈΡΡ ΡΠΈΠ³Π½Π°Π» ΠΎΡ Π°Π΄Π΄ΠΈΡΠΈΠ²Π½ΠΎΠ³ΠΎ ΡΡΠΌΠ° Ρ ΠΏΡΠ΅Π΄ΠΏΠΈΡΠ°Π½Π½ΠΎΠΉ Π·Π°Π΄Π΅ΡΠΆΠΊΠΎΠΉ, ΠΈ Π΅Π³ΠΎ ΡΠ°Π±ΠΎΡΠ° ΠΈΠΌΠ΅Π»Π° Π²Π°ΠΆΠ½ΠΎΠ΅ Π²Π»ΠΈΡΠ½ΠΈΠ΅ Π½Π° ΠΏΠΎΡΠ»Π΅Π΄ΡΡΡΠ΅Π΅ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠ΅ Π² ΡΠ΅ΠΎΡΠΈΠΈ ΠΌΠΎΠ΄ΡΠ»ΡΡΠΈΠΈ. Π’Π°ΠΊΠΆΠ΅ ΠΈΠ½ΡΠ΅ΡΠ΅Ρ{*ΠΏΡΠΎΡΠ΅Π½Ρ*} ΠΠΈΠ½Π΅ΡΠ° Π½Π° ΠΏΡΠΈΠ΅ΠΌΠ΅ Ρ ΠΎΡΡΠΈΡΠ°ΡΠ΅Π»ΡΠ½ΠΎΠΉ Π·Π°Π΄Π΅ΡΠΆΠΊΠΎΠΉ (ΡΡΠΎ, ΠΏΡΠ΅Π΄ΡΠΊΠ°Π·Π°Π½ΠΈΠ΅) Π½Π°ΡΡΠ΄Ρ Ρ ΡΠ°Π±ΠΎΡΠΎΠΉ ΠΠΎΠ»ΠΌΠΎΠ³ΠΎΡΠΎΠ²Π° Π½Π° ΠΏΡΠ΅Π΄ΡΠΊΠ°Π·Π°Π½ΠΈΠΈ Π΅ΡΠ»ΠΈ ΠΎΡΡΡΡΡΡΠ²ΠΈΠ΅ ΡΡΠΌ ΠΈΠΌΠ΅Π» Π²Π°ΠΆΠ½ΠΎΠ΅ Π²ΠΎΠ·Π΄Π΅ΠΉΡΡΠ²ΠΈΠ΅{*ΡΡΠΎΠ»ΠΊΠ½ΠΎΠ²Π΅Π½ΠΈΠ΅*} Π½Π° ΡΠ΅ΠΎΡΠΈΡ ΠΊΠΎΠ½ΡΡΠΎΠ»Ρ{*ΡΠΏΡΠ°Π²Π»Π΅Π½ΠΈΡ*}. Π’ΠΎΡΠ½ΠΎ ΡΠ°ΠΊ ΠΆΠ΅ Kotelnikov Π±ΡΠ» Π·Π°ΠΈΠ½ΡΠ΅ΡΠ΅ΡΠΎΠ²Π°Π½ ΠΎΠ±Π½Π°ΡΡΠΆΠ΅Π½ΠΈΠ΅ΠΌ ΠΈ ΠΎΡΠ΅Π½ΠΊΠΎΠΉ ΡΠΈΠ³Π½Π°Π»ΠΎΠ² Π² ΠΏΡΠΈΠ΅ΠΌΠ½ΠΈΠΊΠ΅.