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For limited distortion source coding, it is generally considered that the minimum value of the coding average distortion is 0, and the maximum value is the minimum distortion value of making
*R(D)* = 0. This is the definition domain of the information rate distortion function. In this paper, the upper and lower bounds of the information rate distortion function
*R(D)* are derived and computed for the typical sources. The results show that the lower bound of the coding average distortion D is related to the symbol distortion function, which can further improve the theory of limited distortion source coding.

From the point of view of information processing, no distortion source coding is entropy preserving. And the entropy preserving coding is not always necessary. Such as the human eyes to accept the visual signal, then there is no need to carry out the distotion-free entropy coding. But entropy preserving coding is not always possible. For example, when the continuous signal is subjected to digital processing, it is impossible to fundamentally remove the quantization error. At this point

Reducing the rate of information is beneficial to transmission and processing, so it is necessary to perform entropy compression coding. Therefore, it is necessary to introduce the source coding with limited distortion.

The information rate distortion function

The average distortion

The mathematical expectation of nonnegative real numbers

It can be seen by (3) formula, When each line of the distortion matrix has at least one zero element, the average distortion of the information source can reach the lower bound value, otherwise

If the formula (4) is established, each line in the distortion matrix has at least one zero, and each column can have at most one zero. Otherwise,

For the maximum average distortion of the information source

By Equation (5),

From the Equation (6) can be observed, In

that minimizes the

Formula (7) is for different , using the input probability distribution

Thus, the domain of definition

The information source is n element equal probability distribution, the distortion is Hamming distortion

Then the rate distortion function of the information source is [

By the Equation (9), the value of

For the information source of unequal probability distribution, the several typical information sources are considered, and the corresponding upper bound

In this paper, the upper and lower bounds of the definition domain of the information rate distortion function

Source | ||||
---|---|---|---|---|

Distortion matrix | ||||

pression of the information rate distortion function

This work was funded by National Natural Science Foundation of China―Re- search on no proportion coding cooperative transmission method based on dynamic antenna selection in large scale MIMO systems (61601170) and Henan Provincial Department of science and technology project―Study on wireless channel characteristics of bulk grain reactor (172102210230).

Zhu, C.H., Mi, H., Kan, M.F. and Song, Y. (2017) R(D) Definition Domain Deriving of Limited Distortion Source Coding With Different Sources. Int. J. Communications, Network and System Sciences, 10, 263-268. https://doi.org/10.4236/ijcns.2017.108B028