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...− log p − log n≥− p log p ∑ ∑ ∑ D D i D i D i n i= 1 n i= 1 i= 1 or n n 1 − p log p + log p ≤0 ∑ ∑ i D i D i i= 1 n i= 1 Thus, we have nH ( P) + ϕ1 ( p) ≤ 0 which gives the relation between Shannon’s =-=[13]-=- measure of entropy and Burg’s [4] entropy. C. Kapur’s [9] measure of directed divergence is given by n α ⎛ ⎛ α −1 ⎞ ⎞ ⎜ 1 ⎜∑pq i i ⎟ ⎟ i= 1 Dα ( PQ , ) = log ⎜ ⎝ ⎠ ⎟ D ≥ 0 (25) 1 1 n α − −α ⎜ n ⎟ ⎛ α...

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...= 1 ⎦ and showed that its lower bound lies between Rα (P) and Rα ( P) + 1 where Rα (P) is expressed as: n 1 R ( P) (1 ) log D pi ; 0, 1 i 1 α ⎡ ⎤ − α = − α ⎢∑ ⎥ α > α ≠ (5) ⎣ = ⎦ The above is Renyi’s =-=[12]-=- measure of entropy of order α . As α → 1, it is easily shown that L → L and ( ) R P approaches H ( P ). Guiasu and Picard [6] defined the weighted average length for a uniquely decipherable code as ⎛...

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...een proved that Shannon’s [13] entropy, Renyi’s [12] entropy of order , Kapur’s [8] entropy of order and type all provide lower bounds for different mean codeword lengths, while Havrada and Charvat’s =-=[7]-=-, Arimoto’s [1] and Behara and Chawla’s [3] measures of entropy provide lower bounds for some monotonic increasing functions of mean codeword lengths but not for mean codeword length’s themselves. Bel...

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... i= 1 or n n 1 − p log p + log p ≤0 ∑ ∑ i D i D i i= 1 n i= 1 Thus, we have nH ( P) + ϕ1 ( p) ≤ 0 which gives the relation between Shannon’s [13] measure of entropy and Burg’s [4] entropy. C. Kapur’s =-=[9]-=- measure of directed divergence is given by n α ⎛ ⎛ α −1 ⎞ ⎞ ⎜ 1 ⎜∑pq i i ⎟ ⎟ i= 1 Dα ( PQ , ) = log ⎜ ⎝ ⎠ ⎟ D ≥ 0 (25) 1 1 n α − −α ⎜ n ⎟ ⎛ α ⎞ α qi p ⎜⎜∑ ⎟ ∑ i ⎟ ⎝⎝ i= 1 ⎠ i= 1 ⎠ −li D Putting qi = ...

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...t is called code alphabet or set of code characters and sequence assigned to each x i , i = 1, 2, ....., n is called code word. Let ni be the length of code word associated with xi satisfying Kraft’s =-=[10]-=- inequality given by the following mathematical expression: l l l − − I 1 2 n D D ... D 1 − + + + ≤ (1) Where, D is the size of alphabet. In calculating the long run efficiency of communications, we c...

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... codes, Shannon’s [13] noiseless coding theorem which states that H( P) H( P) ≤ L < + 1 (3) log D log D Determines the lower and upper bounds on L in terms of Shannon’s [13] entropy H ( P ). Campbell =-=[5]-=- for the first R.K.Tili is with the Department of Mathematics, S.S.M.College, Dinanagar (India) r_kumartuli@yahoo.co.in. time introduced the idea of exponentiated mean codeword length for uniquely dec...

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...HEORY APPROACH In this section, we generate the following inequalities by using well known measures of directed divergence: 1000World Academy of Science, Engineering and Technology 51 2011 A. Burg’s =-=[4]-=- measure of entropy is given by ϕ 1 ( p) n = ∑ log p (20) i= 1 D i Also, Burg’s [4] measure of directed divergence is given by n ⎛ pi p ⎞ i D( P, Q) = ∑⎜ −logD−1⎟ ≥ 0 (21) i= 1 ⎝ qi qi ⎠ Putting q = o...

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...ntroduced in equation is a genuine mean codeword length as it satisfies the essential i i properties of being a mean codeword length. B. For the development of second mean, we consider Bhattacharya’s =-=[2]-=- measure of directed divergence is given by ( , ) n ∑ D PQ ( ) 2 = p − q or Putting i= 1 i n i= 1 i D( P, Q) = 1−∑ pi qi ≥0 (10) q i = n D ∑ i= 1 −li D 1 −li in equation (10), we get 1 ⎛ ⎞ ⎜ ≥ ⎝ i= 1 ...

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...quation (17) is Kapur’s [8] measure of entropy of orderα and type β but R.H.S. is neither a mean codeword length nor a monotonic increasing unction of mean codeword length. i ⎞ B. Sharma and Mittal’s =-=[14]-=- measure of entropy as a possible lower bound Sharma and Mittal’s [14] measure of entropy is given by α −1 ⎡ n ⎤ 1 ⎛ β 1 β ⎞ − φ ( P) = ⎢ p 1 1−α i − ⎥ 2 1⎢⎜∑ ⎟ , α ≠ 1, β ≠ 1 − ⎝ ⎥ i= 1 ⎢ ⎠ ⎣ ⎥⎦ α > ...

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...ade in section IV. α α 997World Academy of Science, Engineering and Technology 51 2011 II. DEVELOPMENT OF TWO NEW MEAN CODEWORDS A. For the development of first mean, we consider Sharma and Mittal’s =-=[15]-=- measure of directed divergence given by s r ( ) D P Q Putting q = i ⎡ ⎤ 1 ⎛ = ⎢ − ≥ s −1 ⎢⎜ ⎢ ⎝ ⎠ ⎣ ⎥⎦ n D ∑ i= 1 −li D −li s−1 n r−1 r 1−r⎞ ∑ pi qi ⎟ i= 1 1⎥ ⎥ 0 (7) in equation (7), we get s−1 n ( ...

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... L and ( ) R P approaches H ( P ). Guiasu and Picard [6] defined the weighted average length for a uniquely decipherable code as ⎛ ⎞ n ⎜ ⎟ ui ni pi L = ∑ ⎜ ⎟ (6) n i = 1 ⎜ ⎟ ⎜∑ui pi ⎟ ⎝ i = 1 ⎠ Longo =-=[11]-=- interpreted (6) as the average cost of transmitting letters xi with probability pi and utilityui and provided some practical interpretation of this length and also derived the lower and upper bounds ...

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... Shannon’s [13] entropy, Renyi’s [12] entropy of order , Kapur’s [8] entropy of order and type all provide lower bounds for different mean codeword lengths, while Havrada and Charvat’s [7], Arimoto’s =-=[1]-=- and Behara and Chawla’s [3] measures of entropy provide lower bounds for some monotonic increasing functions of mean codeword lengths but not for mean codeword length’s themselves. Below, we discuss ...

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...H.S is a genuine mean codeword length as it satisfies the essential properties of being a mean codeword length. It has been proved that Shannon’s [13] entropy, Renyi’s [12] entropy of order , Kapur’s =-=[8]-=- entropy of order and type all provide lower bounds for different mean codeword lengths, while Havrada and Charvat’s [7], Arimoto’s [1] and Behara and Chawla’s [3] measures of entropy provide lower bo...

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...yi’s [12] entropy of order , Kapur’s [8] entropy of order and type all provide lower bounds for different mean codeword lengths, while Havrada and Charvat’s [7], Arimoto’s [1] and Behara and Chawla’s =-=[3]-=- measures of entropy provide lower bounds for some monotonic increasing functions of mean codeword lengths but not for mean codeword length’s themselves. Below, we discuss the correspondence between s...

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...i ; 0, 1 i 1 α ⎡ ⎤ − α = − α ⎢∑ ⎥ α > α ≠ (5) ⎣ = ⎦ The above is Renyi’s [12] measure of entropy of order α . As α → 1, it is easily shown that L → L and ( ) R P approaches H ( P ). Guiasu and Picard =-=[6]-=- defined the weighted average length for a uniquely decipherable code as ⎛ ⎞ n ⎜ ⎟ ui ni pi L = ∑ ⎜ ⎟ (6) n i = 1 ⎜ ⎟ ⎜∑ui pi ⎟ ⎝ i = 1 ⎠ Longo [11] interpreted (6) as the average cost of transmitting...

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... ⎠ lies between n 1 ⎛ β− α+ 1 ⎞ log D pi α β ⎜∑⎟and − ⎝ i= 1 ⎠ n 1 ⎛ β− α+ 1 ⎞ log D pi 1 α β ⎜∑ ⎟+ , − ⎝ i= 1 ⎠ α − 1 < β < α, α ≥ 1 where n 1 ⎛ β− α+ 1 ⎞ log D pi α β ⎜∑ − ⎟ ⎝ i= 1 ⎠ is the Varma’s =-=[16]-=- measure of entropy. Proof: Here we use Holder’s inequality 1 1 n n n ⎛ p q p⎞ ⎛ q⎞ xy i i ≥ ⎜ xi ⎟ ⎜ yi ⎟ i= 1 i= 1 i= 1 ∑ ∑ ∑ , ⎝ ⎠ ⎝ ⎠ 1 1 + = 1, p q porq< 1 (13) Substituting ⎛β− α+ 1⎞ −⎜ ⎟ ⎝ α−β ...