3. Tasks for independent work

Task 3.1. The alphabet consists of four letters x₁, x₂, x₃ and x₄. Probabilities of occurrence of symbols P (x_i) are set by a vector - column

Between the next symbols there are correlation communications which are described at a matrix of conditional probabilities P(x_i/x_j) = a following kind

It is required to define redundancy of a source at statistical independence of symbols and at the dependence account between them.

The answer. Redundancy of a source at statistical independence of symbols and at _Statistical dependence of symbols.

Task 3.2. For message transfer the code consisting of three symbols which probabilities of occurrence are equal 0.8, 0.1 and 0.1 is used. Correlation between symbols is absent. To define redundancy of a code.

The answer. Redundancy of code R=0.42.

Task 3.3. The measuring device develops at среднеквадратическом a deviation and parameter random signal U (t) with normal density of probability and correlation function

To define the redundancy of a signal caused by presence of correlation.

The answer. At a step of quantization redundancy of a signal

or

Task 3.4. On a communication channel input the ensemble of signals {x_i}, with duration and frequency of following arrives. The signal x₁ considerably differs from others and always is accepted correctly. Aprioristic probabilities P(x_i) and probabilities of transitions P (y_j/x_i), have at both parameters; and the values specified in corresponding matrixes

; .

It is required to find Bandwidth of a communication channel and to establish, whether it is possible to use the given channel for information transfer from the set source.

The answer. At factor introduction

Bandwidth of channel

; .

Productivity of a source

; .

Task 3.5. On a communication channel input the ensemble of signals {x_i}, with duration and frequency of following arrives . Aprioristic probabilities P (x_i) and conditional probabilities P(y_j/x_i), have at and probabilities of an error the values specified in corresponding matrixes

; .

To calculate Bandwidth of a communication channel and to establish, whether enough the given channel for information transfer from a source.

The answer. The bandwidth of signal

; .

Productivity of a source

; .

Task 3.6. To define Bandwidth of a communication channel on which signals {x_i}, with duration and frequency of following are transmitted . Influence of hindrances is characterized , , and probabilities of an error a matrix of conditional probabilities

The answer. The bandwidth of signal

;.

Task 3.7. To define a pass - band of the channel of transfer of the television black - and - white image with number of elements , number of shots in time and number of gradation of brightness for the relation a signal - disturbance provided that the image can assume the most chaotic air, namely a kind of white noise.

The answer. The bandwidth of channel

; .

The note. The image becomes white noise if all its elements both in one shot, and in various shots are independent.

Task 3.8. To define Bandwidth of the symmetric channel with a matrix of conditional probabilities

.

The answer. The bandwidth C=0.09 bit symbol ^{- 1} _.

Task 3.9. The binary source with equiprobable elements has productivity . By transfer on the channel on the average one of the transferred 100 symbols is deformed. To define speed of information transfer on the given channel.

The answer. At probability of an error speed of information transfer on the channel

;

.

Task 3.10. On a radio channel with the minimum time of a signal transmission and a band of frequencies in which operates white Gaussian noise with spectral density of capacity , the signal u (t), having boundary frequency and average capacity is transmitted. The signal transmission on the given channel what is the time will occupy.

The answer. Signal transmission time

;