A Comparative Study of Tsalli s and Kapur s Entropy in Communication Systems

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1 International Journal of omputer pplications (975 7) omparative Study of salli s and apur s in ommunication Systems Vijay umar, PhD. ssociate Professor, JRE School of Engineering, Greater oida. Y. Sharma, PhD. ssistant Professor(Sr. Scale), nsal University, Gurgaon. Prince Goyel Research Scholar, Maharishi Dayanand University, Rohtak. SR he total channel capacity of the system of channels composed of two independent subsystems of channel. On the basis of noise index and composability factor, we compared apur s entropy and salli s entropy and found some promising results, which helps in the further generalization of communication systems and makes them more stable. EYWORDS Generalized entropies, salli s entropy, apur s entropy.. IRODUIO In the process of transmission of signals, a communication system has to do with the amount of information added to the signals, having appearance of the errors in transmission of signals may be due to noise or perturbation. he communication system considered here is of statistical nature and is thus a bundle of different components. If one element of the system bundle is inappropriate to the setting, the communication system can under perform. ommunication is a continuous process and a channel is the pipe along which a information is conveyed. Shannon[9] realized that the noise added to one digital pulse would generally make the overall amplitude different from that of another pulse. Further, the noise amplitudes for two different pulses are independent in the channel as amplitude variations. From a thermodynamic viewpoint, the heat in atomic motions disturb the whole system. On the contrary, in communication systems, noise added in a single pulse of the channel disturbs the signal. omposability is an important issue in communication system development in which different sets of components from the system of channels may be composed into different sub-systems of channels. apur[,7, ] proposed a generalization of the oltzmann-gibbs (G) entropic measure to deal with the problems of non-extensivity in statistical mechanics. salli s[2] non-extensive statistical formalism is a useful framework for the analysis of many interesting properties of nonlinear systems. Suyari [] characterized these entropies by generalizing Shannon- hinchin s axiom. be [,2,3] described the pseudoadditivity of a thermodynamic system containing two subsystems by taking the composition of salli s entropy. allen [4] described the condition of equilibrium by the maximum of total entropy of the system. Gupta and umar[5] discuss a general composition law and show that the generalized apur s entropy is compatible with the composition law, defined consistently with the condition of the existence of equilibrium and satisfies pseudo-additivity. In this paper, comparison of salli s entropy and apur s entropy for the purpose of noise index and composability factor will discuss in section (2). hannels exhibit complex systems and attracting great attention. common feature of these systems is that they stay in non equilibrium stationary states for significantly long periods. In these systems, channels are generally inhomogeneous. Let X (, ) be the total channel capacity of the channel and is divided into two independent subsystems of channels, and. Let X (, ) satisfy additivity as: X (, ) X ( ) X ( ) ccording to oltzmann-gibbs-shannon entropy [9] ln( ) (.) i p i p i where, is the total no. of states at a given scale. For independent variables and, (, ) ( ) ( ) (.2) ccording to Landsberg and Vedral [], for any nonextensive entropy (, ) ( ) ( ) ( ) ( ). ( ) (.3) where, ( ) is a function of the entropic index. be [2], proved pseudo-additivity as: ( ) (, ) ( ) ( ) ( ) ( ) for sallis entropy is defined as: ( ) i ( ) ( ) ( p ) i ( ) where, and is called as non-extensive index. when, the measure (.5) reduces to (.). lso, be [2], proved pseudo-additivity as: ( ) (, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (.4) ( ) (.5) ( ) (.) ( ) ( ) 5

2 International Journal of omputer pplications (975 7) ( ) for the normalized salli s as: ( ) ( ) = ( pi ) i ( p i ) i Gupta and umar[5], proved pseudo-addditivity as: and is defined (.7) (, ) ( ) ( ) ( ) ( ) ( ) (.) for the generalized apur entropy of order and is defined as: = n / p k k when, reduces to, (.9) For sallis, ormalized sallis and apur s, the entropic index ( ) is ( ), ( ) and ( ) respectively and is known as the frequency of the noise. lso, it is seen that has different values for different entropy measures. Since, the entropic index ( ) of sallis entropy and apur s entropy is same. herefore, the noise in subsystem depends on the noise index ( ). ( ) ( ) and composability factor ( ) ( ) aking, ( ). ( ) ( ), in case of salli s entropy and ( ). ( ) ( ), in case of apur s entropy. he composability factor for individual entropies are as follows: 2., in case of sallis entropy and ( ) ( ), in case of apur s entropy ( ) ( ) OMPRISO OF SLLI S D PUR S EROPY S PER E OISE IDEX D OMPOSILIY FOR In this section, we will study the comparison between the oise index and omposability factor for different values of parameter. onsider an example of the binomial distribution of two sets and for the given values of X are as follows: X P (x) P (x) he value of the noise index and composability factor is shown in the following two cases as: 2. ase I: For 2.. oise Index : able Value of the noise index ) ( ( ) and ) ( ( ) for ( ).5.72 ( ) ) 2. ( )..7 (.5 ( ) ( ) omposability factor : able Value of the composability factor for ) ( ( ) ) ( an ( ).5 2 ( ) ) 2. ( ( ) ) (.5.993

3 International Journal of omputer pplications (975 7) ( ) ( ) From the tables (2..) and (2..2), we plot graphs for different values of parameter and are as shown: Fig omparison of salli s and apur s for as per omparison of salli's and apur's as per oise index Index of sallis Index of apur's Fig omparison of salli s and apur s for as per omparison of sallis and apur's as per composability factor omposability Factor for sallis composability Factor for apur's From the fig. (2..3) and (2..4), it is seen that noise index and composability factor in case of apur s entropy is very less as compared to salli s entropy, when (,). lso, when the value of comes closer to one, noise index and composability factor for both the entropies is almost same. 2.2 ase II: For 2.2. oise Index : able Value of the noise index. and for ).22 ( ) ) ).93 ( ) ) ) )

4 International Journal of omputer pplications (975 7) omposability factor : able omposability factor and for ).22 ( ) ) ( ).43 ) ) ) ) From the tables (2.2.) and (2.2.2), noise index and composability factor, we plot graphs for the parameter. Fig omparison of salli s and apur s for as per omparison of salli's and apur's as per oise index. 2 4 Index of salli's Index of apur's Fig omparison of salli s and apur s for as per omparison of sallis and apur's as per omposability factor composability factor for sallis omposability factor for apur's From the graphs (2.2.3) and (2.2.4), it conclude that, for, noise index and composability factor in case of sallis entropy is very less as compared to apur s entropy 3. OLUSIO he concept of composability puts a stringent constraint on possible forms of entropies. lso, on the basis of noise index and composability factor, we compared apur s entropy and salli s entropy and found that, for,

5 International Journal of omputer pplications (975 7) apur s entropy gives more promising results than salli s entropy and hence used for further generalization in communication systems. For, sallis entropy finds suitable than apur s entropy for the purpose of noise. 4. REFEREES [] be, S. (2), xioms and uniqueness theorem for sallis entropy, Physics Letters, 27, [2] be, S. (2), General pseudo-additivity of composable entropy prescribed by the existence of equilibrium, Physical Review E 3, 5. [3] be, S. (24), Generalized non-additive information theory and quantum entanglement in non-extensive - Interdisciplinary pplications, eds. M. Gell- Mann and. sallis, Oxford University Press, ew York. [4] allen,.. (95), hermodynamics and an introduction to thermostatistics, 2 nd edition, wiley ew York. [5] Gupta, Priti and umar, Vijay(2), General Pseudoadditivity of apur s prescribed by the existence of equilibrium, International Journal of Scientific & Engineering Research, Volume (3). [] apur, J.. (99), Maximum entropy models in science and engineering, Wiley, Eastern limited. [7] apur, J.. and esavan,.. (992), Optimisation Principles with pplications, cademic Press I. [] apur, J.. (994), Measures of Information and their pplications, Wiley, ew York. [9] Shannon,.E. (94), he Mathematical heory of ommunication, ell s system echnical Journal, 27, [] Landsberg, P.. and Vedral, V. (99), Distributions and channel capacities in generalized statistical mechanics, Physics Letters 247, [] Suyari,. (24), Generalization of Shannon-hinchin axioms to non- extensive systems and the uniqueness theorem for the non- extensive entropy, IEEE ransactions on Information theory, 5, [2] sallis,. (9), Possible generalization of olzmann- Gibbs statistics, Journal of Statistical Physics, 52,