Ìàòåìàòèêà / 5. Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå

 

Ivanov Y.Y.

Vinnytsia National Technical University, Ukraine

Expansion for the Mathematical Apparatus of the Algebra Logarithm of the Likelihood Function for the Turbo-Codes in Distributed Computer Systems of Different Functional Purposes

 

Introduction. At the time of the latest technology the information became object of the automated processing. The data transfer process in information systems is susceptible to errors, because any error in material breach of the calculation. To combat interferences in data transfer systems at all the stages is used the error-correction coding, that provides reliability and credibility of information transmitted. The analysis of the methods allowed us to determine, that the most prominent achievement in the theory of error-correction coding in recent years is turbo-code. They are used to encode large volume information messages at the high speed with high error-correcting [1]. Turbo-codes are used in practice in the most important areas, such as space satellite communications (standards DVB-S, DVB-RCS), digital television (ViaSat, HDTV), mobile communication systems of the 3rd generation [2]. The practical widespread have decoding algorithms: SOVA (Soft-Output Viterbi Algorithm), MAP (decoding algorithm for the Maximum A posteriori Probability) and modifications to decrease the computational complexity (Log-MAP,                  Max-Log-MAP) [3].

A significant drawback of the turbo-codes is comparatively high decoding complexity and high delay, which sometimes make them unattractive. But for using in satellite channels this shortcoming is not essential, because the length of the communication channel introduces a significant delay. Another drawback of the turbo-codes (fig. 1) is a comparatively small code distance. This leads to the fact, that even when a large input error probability, the turbo-code performance is high, but at the low input error probability the turbo-code performance is limited [1].

An important advantage of the turbo-codes is the decoding complexity independence from the length of the information block, which reduces the decoding error probability by increasing its length [1].

Figure 1 − BER − Bit Error Rate (word length 27 bits and 37 bits)

 

The main problems are the lack of freeware distribution and complexity of the decoding algorithms.

The work purpose is to improve computing efficiency of the turbo-code decoding procedure through improvement of the mathematical algorithms of their work.

Main Part. To achieve the purpose should be to build a model of error-corection turbo-decoder, that uses the new mathematical principles of the algebra logarithm of the likelihood function [4] for N statistically independent information bits with the transition to trigonometric and hyperbolic functions without using complex numbers.

         (1)

where  − statistically independent information bit, .

Equation (1) can be used to simplify calculations, when solving complex problems, when given a lot of statistically independent information bits. For the given formula is possible to perform a connection between trigonometric (L. Euler,                       F. Viete, N. Copernicus, Arabic mathematics) and hyperbolic (A. de Moivre,                     V. Riccati, J.H. Lambert) functions without using complex numbers (C. Gudermann  function) [5, 6] to simplify the calculation.

Therefore, it is necessary to find Gudermannian [5] from some argument .

 

                (2)

 

Now we use the transformation

 

                               (3)

 

Comparing formulas (2) and (3), we can obtain such Gudermannian property

 

                                             (4)

 

Applying this property to the expression (1), and using the Gudermannian             form (2), it is possible to obtain                            

 

                       (5)

 

Now it is necessary to perform numeric calculations in the environment of the mathematical modeling MathCad (fig. 2) to calculate the values sum of the logarithm of the likelihood function for example in the case of 2 and 6 statistically independent information bits.

Figure 2 − Calculations ( bits)

 

Conclusion. So, has been derived a new formula for calculating the LLR sum S for many bits. It can be used in computing devices, for optimizing the approximation in distributed computer systems of different functional purposes.

 

Literature:

1.          Êóëèê À.ß. Äåêîäóâàííÿ òà ðåàë³çàö³ÿ àëãîðèòìó BCJR äëÿ òóðáî-êîäó ñòàíäàðòèçîâàíîãî â DVB-RCS / À.ß. Êóëèê, Ä.Ñ. Êðèâîãóá÷åíêî, Þ.Þ. ²âàíîâ // ³ñíèê Ñóìñüêîãî äåðæàâíîãî óí³âåðñèòåòó. Ñåð³ÿ: Òåõí³÷í³ íàóêè. − Ñóìè: ÑÓÌÄÓ, 2012. − Ò. 4. − ¹ 1. − Ñ. 84-93.

2.          Sripimanwat K. Turbo-Code Applications: A Journey from a Paper to Realization / K. Sripimanwat. − New-York: Springer, 2005. − 386 p.

3.          Morelos-Zaragoza R. The Art of Error-Correcting Coding, 2nd Edition /                 R. Morelos-Zaragoza. − Chippenham: John Wiley & Sons, Ltd, 2006. − 278 p.

4.          Moon T.K. Error-Correction Coding: Mathematical Methods and Algorithms / T.K. Moon. − John Wiley & Sons, Ltd, 2005. − 750 p.

5.          Ãðàäøòåéí È.Ñ. Òàáëèöû èíòåãðàëîâ, ñóìì, ðÿäîâ è ïðîèçâåäåíèé               (4-å èçä.) / È.Ñ. Ãðàäøòåéí, È.Ì. Ðûæèê. − Ì.: Íàóêà, 1963. − 1100 ñ.

6.          Êîðí Ã.À. Ñïðàâî÷íèê ïî ìàòåìàòèêå äëÿ íàó÷íûõ ðàáîòíèêîâ è èíæåíåðîâ (4-å èçä.) / Ã.À. Êîðí, Ò.Ì. Êîðí. − Ì.: Íàóêà, 1978. − 832 ñ.