Minimum Distortion Quantizers

N A S A TECHNICAL NOTE

* Qo m 0s) n z c 4 wj 4 z

N A S A TN 0-8384 C*! __

MINIMUM DISTORTION QUANTIZERS

Harry We Jones, Jre

Ames Resedrch Center MofSett Field, Cali$ 94035

0 t-‘ W .I= 0 c3r n

N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N * W A S H I N G T O N , D. C. M A R C H 1977

TECH LIBRARY KAFB, NM

I 111111 lllll lllll lllll lllll IIIII lllll Ill Ill

NASA TN ~-8384 4. Title and Subtitle


1 7. Author(s1

2. Government Accession No. I 3. Recipient's Catalog No. 1 ~~

5. Report Date

March 1977 6. Performing Organization Code

i I I 8. Performing Organization Report No.

Harry W. Jones, Jr.* I A-6714 ~ ~

I 10. Work Unit No.

9. Performing Organization Name and Address

Ames Research Center Moffett Field, California 94035 i 656-11-02-01

11. Contract or Grant No.

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Technical Note 12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D.C. 20546 I 14. Sponsoring Agency Code

I 15. Supplementary Notes

I

"National Research Council Postdoctoral Research Associate.

~~ I 16. Abstract

The well-known algorithm of Max is used to determine the minimum distortion quantizers for normal, two-sided exponential, and specialized two-sided gamma input distributions and for mean- square, magnitude, and relative magnitude error distortion criteria; The optimum equally-spaced and unequally-spaced quantizers are found, with the resulting quantizer distortion and entropy. The quantizers, and the quantizers with entropy coding, are compared to the rate distortion bounds for mean-square and magnitude error.

17. Key Words (Suggested by AuthorW)

Quantizers, Minimum distortion Quantizers, Max Distortion measures Rate-distortion theory

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* Harry W. Jones, Jr.

Ames Research Center

SUMMARY

The well-known algorithm of Max is used to determine the minimum distortion quantizers for normal, two-sided exponential, and specialized two-sided gamma input distributions and for mean-square, magnitude, and relative magnitude error distortion criteria. spaced quantizers are found, with the resulting quantizer distortion and entropy. The quantizers, and the quantizers with entropy coding, are compared to the rate distortion bounds for mean-square and magnitude error.

The optimum equally-spaced and unequally-

INTRODUCTION

The well-known optimum quantizers and optimum, equally-spaced level quantizers of Max (ref. 1) have minimum mean-square error distortion for a given number of output levels, assuming a normal or Gaussian distribution of the input parameter. Paez and Glisson (ref. 2) used the numerical algorithm of Max to find optimum quantizers and optimum, equally-spaced level quantizers for minimum mean-square error distortion, assuming either the two-sided exponential (Laplacian) distribution, or McDonald's special form of the gama distribution. and extended to higher and intermediate numbers of levels for the mean-square error distortion and for the normal, exponential, and gamma distributions. In addition, the optimum quantizers and the optimum equally-spaced level quantizers were found for these three distributions using the magnitude error distortion and the relative magnitude error distortion criteria suggested by Andrews and Pratt (ref. 3 ) . The method of Max is reviewed, and the input distributions and distortion measures are defined. The quantizers are given and discussed, and their performance described. The new results are listed in the summary and conclusion section at the end of this report.

Here, the work of Max and of Paez and Glisson has been repeated

MAX ALGORITHM

If the input parameter distribution is p(x) and the distortion for input parameters x and representative value yi is d(x - yi), then for M representative values for the parameter, the total distortion is

*National Research Council Postdoctoral Research Associate

M rxi+l

where x i and are t h e c u t p o i n t s determining t h e range of x repre- s en ted by t h e va lue y i , and x1 and xM+1 are i n f i n i t e . The d i s t o r t i o n i s minimized by d i f f e r e n t i a t i n g D w i t h r e s p e c t t o t h e x i ’ s and y i ’ s and s e t t i n g t h e d e r i v a t i v e s equa l t o zero. For p(x) # 0 f o r a l l x and d(x) monotonically i n c r e a s i n g wi th x , Max ( r e f . 1, p. 268) shows t h a t s e t t i n g t h e d e r i v a t i v e w i t h r e s p e c t t o x i equa l t o zero r e q u i r e s t h a t

Thus, t h e c u t p o i n t x i i s halfway between t h e two r e p r e s e n t a t i v e va lues y i and ~ i - ~ . The d e r i v a t i v e wi th r e s p e c t t o y i i s a l s o se t t o ze ro , g iv ing

SXi” d ’ ( x - y i ) p ( x ) d x = 0 , i = 1, . . ., M (3 ) xi

Since p(x) i s symmetrical about x = 0, ze ro i s a c u t p o i n t f o r M-even and a r e p r e s e n t a t i v e va lue f o r M-odd, and t h e p o s i t i v e and n e g a t i v e r ep resen ta - t ive levels and c u t p o i n t s are symmetrical , having t h e s a m e magnitud’e and oppos i t e s i g n s . The indexing of t h e x i and y i i s t h e r e f o r e changed so t h a t x1 and y1 are t h e smallest nonnegative x i and y i , and x ~ / ~ , yfi12 f o r M-even, o r X ( S - ~ ) / ~ , f o r M-odd are t h e l a r g e s t p o s i t i v e non- i n f i n i t e x i and y i .

For unequal-level spacing and M-even, x1 i s ze ro and y1 is es t ima ted . Then, equa t ion ( 3 ) i s solved f o r on, u n t i l equa t ion (2) i s solved f o r yGl2. I f equat ion (3 ) i s n o t s a t i s f i e d by t h i s yk/2, e s t ima ted y1 i s ad jus t ed i n t h e s a m e d i r e c t i o n t h a t yfi/2 would be ad jus t ed t o s a t i s f y equat ion ( 3 ) . For unequal-level spacing and M-odd, y1 i s zero and x1 i s e s t ima ted . Equation ( 2 ) i s solved f o r y 2 , equa t ion ( 3 ) i s solved f o r x2, and s o on, u n t i l y fi+1)/2 i s found and

t h e est imated x1 is ad jus t ed i n t h e same d i r e c t i o n t h a t would be ad jus t ed t o s a t i s f y equat ion ( 3 ) . I f t h e ou tpu t levels are e q u a l l y spaced, t h e x i and y i are i n t e g r a l m u l t i p l e s of h a l f t h e level spacing, and t h e d i s t o r t i o n i s minimized wi th r e s p e c t t o a s i n g l e parameter, t h e level spacing. More d e t a i l i s given by Max ( r e f . 1 ) . Two computer programs w e r e developed t o f i n d t h e equal- and unequal-level spaced q u a n t i z e r s . The programs use t h e d i f f e r e n t i n p u t d i s t r i b u t i o n s and d i s t o r t i o n func t ions desc r ibed below.

x2, equa t ion (2) i s solved f o r y2 , and s o

t e s t e d i n equa t ion ( 3 ) . I f t h e i n t e g r a l i s n o t suf i i c i e n t l y c l o s e t o ze ro ,

INPUT DISTRIBUTIONS

Three i n p u t d i s t r i b u t i o n s - t h e normal, t h e s p e c i a l i z e d gamma d i s t r i b u t i o n - are considered. have zero mean and u n i t va r i ance , are p l o t t e d i n bu t ion is

two-sided exponen t i a l , and a

f i g u r e 1. The normal d i s t r i - The d i s t r i b u t i o n s , which

2

1 -x*/2 p w = - e &

( 4 )

The two-sided exponential or Laplacian distribution is (refs. 2 and 4 )

1 -filxI p(x) = - e fi

McDonald's specialized gamma distribution (refs. 2 and 5) is

(5)

These distributions are referred to below simply as the normal, exponential, and gamma distributions.

The normal distribution is used most frequently in rate distortion theory, but Paez and Glisson (ref. 2) refer to McDonald's (ref. 5) evidence that speech amplitude variations can be modeled by the gamma distribution, which is approximated by the mathematically simple exponential density. There is also evidence that two-dimensional Hadamard transform coefficients of image data have the exponential distribution (ref. 6). Experimental Hadamard transform coefficients (ref. 7) often have a gradually decreasing exponential slope as x increases, and can be modeled by the gamma distribution, or by an even more highly peaked distribution.

DISTORTION MEASURES

Three distortion measures - the mean-square error, the magnitude error, and the relative magnitude error - are considered. The distortion for representative values of 0 and 52 .0 and of cut points of k1.0 is plotted in figure 2. The mean-square error function, which defines distortion when sub- stituted in equation (l), is

d(x - y.) 1 = (x - y.)2 1

The magnitude error function is

The relative magnitude error function is

( 7 )

The mean-square error distortion is used most frequently in rate distortion theory and in image data compression. The magnitude distortion criterion and the relative magnitude distortion criterion were suggested for image data

3

by Andrews and Pratt (ref. 3 ) . The mean-square and magnitude distortion are monotonically increasing as not have this property; the property is used in the derivation of equation ( 2 ) .

I x - yi I increases , but relative distortion does

For example, consider a representative value of 2 , as shown in figure 2 ,

In fact, the limit, as x with input values of 1 and 4 . 14 - 21 > 11 - 2 1 , but becomes infinite, of Ix - 21/1x1 is 1. Another problem with relative distortion is that zero is a cut point for M-even, with the representative values symmetrical about zero. The distortion for small x is I x - y1 I / 1x1 which becomes infinite as x approaches zero. It is shown in the appendix that the relative distortion criteria require one representative value to be zero, so that the only sets of cut points and representative values that are symmetrical about zero have an odd number of representative values. Only the levels for odd numbers of M are computed, but M-even can be used. For example, to use four levels, the optimum quantizer would have representative values of 0, of the one negative level for M = 3 , and of the two positive levels for M = 5, or positive and negative levels could be interchanged. It is also shown in the appendix that, for M-odd, equation ( 2 ) is correct even though the monotonic distortion requirement is not satisfied.

Substituting in equation ( 9 ) , we have 14 - 2 ) / 1 4 1 < 11 - 2 1 / 1 1 1 .


The results of running the Max algorithm programs for the above input distributions and distortion measures are given in tables 2 through 17; these tables are described in table 1. The number of levels increases from the lowest to highest, as indicated in parentheses, by adding 1; by multiplying by 2 ; or by multiplying by 2 and adding 1. The arrangement of the tables follows Max (ref. 1) directly, and differs from Paez and Glisson (ref. 2). All the numbers in the tables have been rounded to four significant digits. The final infinite cut points have not been included in tables 9 through 17. The letter a next to most of the values for M = 1 and M = 2 indicates a value derived by direct computation, rather than by the Max algorithm programs.

The equal and unequal quantizers for normal input distribution and mean- square error for M = 1 (1+) 36 are given by Max (ref. 1). The corresponding values given here in tables 2 and 9 are usually identical, differing at most by 5 units in the fourth place. The equal and unequal quantizers for exponential and gamma input distributions and mean-square error for M = 2 ( 2 X ) 32 are given by Paez and Glisson (ref. 2 ) . The corresponding values given in tables 3, 10, and 11 differ slightly in most cases, and differ significantly for the unequal M = 16 and M = 32 gamma distribution quantizers. Since the same programs that obtained good agreement with Max were used, and since the mean-square error distortions obtained are in every case less than those given by Paez and Glisson (except for gamma, M = 2 , where directly computed values are used), it appears that the values given here are more optimal.

The results given in the tables are partially plotted in the figures discussed below. Figure 3 shows the optimum quantizer level spacing for normal, exponential, and gamma input distribution and for mean-square error.

4

This figure, like many of the other figures included here, shows intermediate values not listed in any table. The level spacings for the exponential and gamma distributions are wider, reflecting their higher probabilities for large x. This also appears in figure 4 , where the largest representative value is higher for the exponential and gamma distributions.

Figures 3 and 4 show also the similarity of the exponential and gamma quantizers, and indicate that the gamma quantizers have narrower spacing for small M-even than for small M-odd. This is due to the infinite value of the gamma density at x = 0 , which requires a zero or small positive representative value. This effect, much reduced, also occurs for the exponential distribution.

Figure 5 shows mean-square distortion for the optimum equal quantizers, and figure 6 shows the distortion for odd and even M, gamma input, equal quantizers. The distortion is significantly less for the normal input distribution because of its narrower spread and lower peak.

Figure 6 shows that, for the gamma distribution and equal spacing, 2* (integer)-1 levels provide better performance than 2* (integer) levels. Figure 5 also includes the distortion for the normal input, optimum unequal spacing quantizer, which is less than the distortion for the equal level quantizer. Figure 7 shows the distortion for all the minimum mean square error unequal quantizers. As in the normal input case, the distortions for the exponential and gamma inputs are less with unequal level spacing quantizers. Although the normal input distribution again has least distortion, the differences are smaller because the optimum unequal quantizers adjust to the input distribution shape. Figure 8 gives the largest representative value for unequal-spacing, minimum mean-square error quantizers. Compared to the largest representative values for equal spacing quantizers given in figure 4 , the largest representative value for unequal spacing quantizers increases more rapidly with M until a final value is approached. The largest representative values for odd and even M do not have different curves, as they did for equal spacing quantization. In all cases for unequal spacing quantizers, increasing the number of levels decreases distortion.

Figure 9 shows the optimum equal spacing for magnitude distortion and figure 11 shows the optimum equal spacing for relative distortion. The spacing is smaller for magnitude distortion, and smaller still for relative distortion, because of the reduced weighting of large magnitude errors, especially for the representative value interval that extends to infinity. The gamma distribution for magnitude distortion again has narrower spacing for an even number of levels. A s discussed above and in the appendix, only odd numbers of levels are used with relative distortion. As in the mean square error case, the normal input quantizers have the smallest spacing and the gama input quantizers have the largest spacing. Figure 10 gives the distortion for minimum magnitude error, equal-spacing quantizers. The distortion for the gamma distribution for odd and even numbers of levels is given in figure 6. A s in the mean-square error case, distortion is lower for odd numbers of levels. The distortion for minimum relative error equal spacing quantizers is given in figure 12. Again, as in the case of mean-square error, the equal quantizers have least distortion for a normal input and have most distortion for a gamma

5

i n p u t . The optimum unequal q u a n t i z e r s f o r magnitude and re la t ive magnitude d i s t o r t i o n are n o t s e p a r a t e l y p l o t t e d , b u t are d i scussed i n t h e next s e c t i o n .

Max ( r e f . 1) i n d i c a t e d t h a t , f o r M-large, i n c r e a s i n g t h e number of levels t o 2M would, as an approximation, cause each p rev ious r e p r e s e n t a t i v e level t o be d iv ided i n t o two equa l i n t e r v a l s ; f i g u r e s 3, 9 , and 11 confirm t h i s . For approximately cons t an t p r o b a b i l i t y d e n s i t y i n each i n t e r v a l , t h e mean squa re e r r o r is reduced by one-fourth; f i g u r e s 5 , 6 , and 7 confirm t h i s . S i m i l a r l y , doubling a h igh number of levels would reduce t h e magnitude e r r o r by one-half , as shown i n f i g u r e s 6 and 10. The relative d i s t o r t i o n ( f i g . 1 2 ) i s less w e l l behaved, a l though t h e d i s t r i b u t i o n s f o r normal and exponen t i a l i n p u t s n e a r l y fol low t h e one-half s lope .

Although t h e minimum d i s t o r t i o n q u a n t i z e r i s e x a c t l y de f ined , t h e d i s t o r - t i o n can approach t h e minimum f o r s i g n i f i c a n t l y d i f f e r e n t quan t i ze r s . For t h e M = 38 and M = 40 q u a n t i z e r s i n t a b l e 9 , d i s t o r t i o n i s 10 pe rcen t l a r g e r f o r M = 38. I f an M = 40 q u a n t i z e r is designed us ing t h e M = 38 q u a n t i z e r w i th any two a d d i t i o n a l r e p r e s e n t a t i v e v a l u e s , t h e d i s t o r t i o n must b e less than t h e M = 38 d i s t o r t i o n , and w i t h i n 10 pe rcen t of t h e minimum. This imp l i e s t h a t t h e form of t h e minimum d i s t o r t i o n q u a n t i z e r is less a c c u r a t e l y def ined than i t s performance, and t h a t convenient approximations t o t h e quan- t i z e r s , o r even g r e a t l y d i f f e r i n g q u a n t i z e r s , w i l l o f t e n perform acceptably.

PERFORMANCE OF THE MINIMLTM DISTORTION QUANTIZERS

It i s w e l l known t h a t , f o r a given d i s t o r t i o n , t h e normal inpu t d i s t r i b u - t i o n r e q u i r e s a h ighe r minimum t r ansmiss ion rate than any o t h e r ze ro mean, u n i t va r i ance d i s t r i b u t i o n ( r e f . 4 , pp. 101-102). However, t h i s rate d i s t o r - t i o n bound d e f i n e s only t h e minimum rate f o r a l l p o s s i b l e t r ansmiss ion methods. Simply us ing a q u a n t i z e r w i th an i n t e g e r number of e q u a l l y spaced levels f o r each sample i s n o t t h e b e s t t ransmission method, and i t is r e l a t i v e l y less e f f i c i e n t f o r t h e h igh ly peaked exponen t i a l and gamma d i s t r i b u t i o n s . Huffman coding ( r e f . 8, Ch. 2 ) can be used t o reduce t h e t r ansmiss ion ra te from log, M A p l o t of t h e equal level spacing q u a n t i z e r d i s t o r t i o n v e r s u s entropy ( f i g . 13) shows t h a t t h e normal d i s t r i b u t i o n does r e q u i r e h ighe r r a t e (except a t s m a l l M) f o r a t r ansmiss ion system c o n s i s t i n g of an e q u a l l y spaced quan t i ze r and a Huffman coder. The optimum unequal quan t i ze r has less d i s t o r t i o n than t h e equal quan- t i z e r . Although t h e unequal quan t i ze r f o r t h e normal inpu t d i s t r i b u t i o n aga in has lowest d i s t o r t i o n , t h e d i f f e r e n c e i n d i s t o r t i o n ( f i g . 7 ) is less than f o r t h e equa l q u a n t i z e r . A p l o t of t h e unequal level spacing quan t i ze r d i s t o r t i o n ve r sus entropy ( f i g . 14) shows t h a t t h e normal d i s t r i b u t i o n r e q u i r e s a t r a n s - mission ra te equa l t o o r g r e a t e r t han t h e o t h e r d i s t r i b u t i o n s , a t medium and l a r g e M.

t o t h e q u a n t i z e r entropy, a lower but v a r i a b l e ra te .

The equally-spaced and unequally-spaced minimum d i s t o r t i o n q u a n t i z e r s can each be used w i t h entropy coding, g iv ing fou r p o s s i b l e systems f o r each com- b i n a t i o n of i n p u t d i s t r i b u t i o n and d i s t o r t i o n measure. The performance of t h e systems l i s t e d i n t h e t a b l e s is shown i n f i g u r e s 15 through 23. The ra te d i s t o r t i o n bound, o r a lower bound on t h i s bound, is given f o r q u a n t i z e r s

6

designed for mean-square and magnitude error (ref. 4 , pp. 92-102, 141). In figure 15, the distortion is lower for entropy-coded, equal-spaced quantizers than for entropy-coded, unequally-spaced quantizers. This result was found by Wood (ref. 9), who used approximations for entropy and distortion based on the work of Ma.x (ref. 1). Goblick and Holsinger (ref. lo) noted earlier that the entropy-coded, equal-spaced quantizers were within 0.25 (here 0 . 3 ) bits of the rate distortion bound.

Figures 16 and 17 indicate that the cases of exponential and gamma input and mean-square error are similar. Entropy-coded, equal quantizers give the best performance, and they approach the rate distortion bound. Although these highly peaked distributions give poor performance with uncoded, equal-spaced quantizers, the low probability of the higher representative levels allows larger rate reduction with entropy coding. For the gamma input and mean-square error, entropy-coded (variable rate), equal quantizers are 0.5 to 1.5 bits better than unequal quantizers. The rate reduction is comparable to the reduction obtained using variable rate adaptive Hadamard image compression (ref. 11), and these two variable rate methods can be combined.

Figures 18, 19, and 20 give the rate distortion bounds and quantizer performance, with and without entropy coding, for the magnitude distortion measure. As in the case of mean-square distortion, the entropy-coded, equal- spaced quantizers have lowest distortion, approaching the bound, and are superior to uncoded, unequal quantizers, except at some small M.

Figures 21, 22, and 23 give the quantizer performance, with and without entropy coding, for the relative distortion measure. The lower bound on the rate distortion cannot be found using the method for difference distortion measures, because the relative distortion is a function of the sample value as well as the error (ref. 4 , p. 92). For the relative error, the unequally- spaced quantizers give significantly less distortion than entropy-coded, equal quantizers.

For all three distortion measures, the performance differences increase greatly for the exponential and gamma input distributions.

SUMMARY AND CONCLUSION

The results of the work include the structure of the new quantizers, the performance of the quantizers, and certain properties of the quantizers due to input distribution or distortion properties, as follows.

1. The minimum mean-square error, equal- and unequal-spaced quantizers for normal, exponential, and gamma input distributions, with the distortion and entropy, were found for new numbers of levels.

2. The magnitude error quantizers for normal, exponential, and gamma input distributions, and the distortion and entropy, were found.

7

3 . The relative magnitude error quantizers for normal, exponential, and gamma input distributions, and the distortion and entropy, were found.

4 . For exponential and gamma distributions and mean-square error, it was shown that equally-spaced quantizers with entropy coding are far superior to unequally-spaced quantizers with entropy coding, and that they approach the rate distortion bound. bution and mean-square error by Wood (ref. 9) and by Goblick and Holsinger (ref. l o ) , who used the work of Max (ref. 1)).

(These results were shown for the normal input distri-

5. For magnitude error and small M y unequal-spacing quantizers with entropy coding sometimes have slightly less distortion than entropy-coded, equally-spaced quantizers. The entropy-coded, equal-spaced quantizers are superior for medium and large M, and approach the rate distortion bound.

6. For relative magnitude error, unequal quantizers have significantly less distortion than entropy-coded, equally-spaced quantizers.

7. The rate reduction for entropy-coded, equally-spaced quantizers is significantly larger for exponential or gamma input distributions than for the normal input distribution.

8. Equally-spaced quantizers with odd numbers of levels are superior to equally-spaced quantizers with even numbers of levels for the gamma input distribution. Gamma distribution quantizers usually have a representative level either equal to, or very close to, zero.

9. The exponential input distribution has a similar but much smaller superiority of the equally-spaced quantizers with odd numbers of levels.

10. The relative magnitude distortion criterion forces one representative level to be zero.

Ames Research Center National Aeronautics and Space Administration

Moffett Field, California 95035, August 15, 1976

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APPENDIX

RELATIVE MAGNITUDE DISTORTION

Suppose t h a t M i s even; t h e c u t p o i n t x1 is zero and t h e d i s t o r t i o n f o r t h e f i r s t r e p r e s e n t a t i v e va lue , from equat ion (1) is:

D1 = 2 p ( 4 dx

x? i s g r e a t e r t han y1 and is i n f i n i t e f o r M = 2. To t i o n , t h e d e r i v a t i v e w i t h r e s p e c t t o is set t o zero h e r e , and i n equa t ions ( 2 ) , ( 4 ) , and ( 6 ) i n r e f e r e n c e (1 l u t e v a l u e by us ing two i n t e g r a l s i n t h e ranges 0 t o y1 using L e i b n i t z ’ s r u l e ,

y1 minimize t h e d i s t o r - as i n equat ion (3 ) . Removing t h e abso- and y1 t o x2 and

Suppose p(x) is monotonically decreasing as x i n c r e a s e s and i s con- t i nuous a t x = 0. For s m a l l p o s i t i v e E,

E X dD 1

dy 1 - = 2 x-’p(o)dx + 2 lY1 x-’p(x)dx - 2 $ x-’p(x)dx

Y 1

X

x-’p(x)dx - 2 x-lp(x)dx $ E = 2 p ( o ) ~ n X I ,

Y 1

The Rn E i s a l a r g e f i n i t e n e g a t i v e number, t h e second i n t e g r a l is p o s i t i v e , and t h e t h i r d is bounded as fo l lows :

W e t h e r e f o r e have

1 - > 2p(o)[m] + 2p(o)Rn E - - 1

dy 1 Y1 (A5 1

Since Rn E is f i n i t e , t h e d e r i v a t i v e of D1 w i th r e s p e c t t o y1 i s i n f i n i t e f o r a l l nonzero y l . D1 i s l a r g e f o r y1 n o t equal t o ze ro and i n c r e a s e s as y1 i n c r e a s e s , so t h a t y1 is fo rced t o zero by t h e behavior of

9

dDl/dyl. For the case where M is odd, y1 is zero and the relative distortion is always 1 or less, since y1 = 0 other values of yi are used when they are closer to x and reduce the distortion.

may be used in equation ( 9 ) . The

The assumption that p(x) gamma distribution, but if we take and, as in equation (A3)

is continuous at zero is not satisfied for the p(x) = p'(x)/w, p'(x) is continuous

E

- = 2 1: (o)dx + . . .

(A6

dD1

dyl

= 2p1(0)[-(2/5)x-2/51~l + . . . = (4/5)p1(0>[-e-2/5 + m ] + . . .

Bounding the third term of equation (A3) by derivative again forces y1 to zero.

y73/2, the infinite positive

Even for M-odd, the relative magnitude distortion is not a monotonically increasing function of Ix - yil , as shown in figure 2 and mentioned in the text above. In reference 1, Max uses the monotonically increasing property to prove equation (2) above, which is used in the algorithm. Equation (2) can be shown directly, in a manner parallel to that of Max (Max eqs. (1) and (5)). From equations (1) and (9 )

M rrxi+l Ix - Y- I D = C I (A7

Setting dD/Dxi equal to zero, by Leibnitz's rule, the i - 1 and i terms give

For p(xi) # 0 and ]xi1 # 0, which is true for M-odd because no cut point can equal the zero representative value,

Ixi - yi-i I = [Xi - Yil

Since ~ i - ~ , xi, and yi are increasing positive values,

which is equation (2).

We have shown that the relative magnitude distortion requires a representative value of zero, and that for M-odd, which implies a zero representative value, the relative magnitude distortion can be treated by the Max algorithm.

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REFERENCES

1. Max, Joel: Quantizing for Minimum Distortion, IRE Trans. Inform. Th., vol. IT-6, March 1960, pp. 7-12.

2. Paez, M. D.; and Glisson, T. H.: Minimum Mean-Squared-Error Quantization in Speech PCM and DPCM Systems. IEEE Trans. Comm., vol. COM-20, April 1972, pp. 225-230.

3. Andrews, H. C.; and Pratt, W.: Image Transforms, Ch. 6 in Computer Tech- niques in Image Processing, H. C. Andress, ed., Academic Press, 1970.

4 . Berger, T.: Rate Distortion Theory: a Mathematical Basis for Data Com- pression. Prentice-Hall, 1971.

5. McDonald, R. A.: Signal-to-Noise and Idle Channel Performance of Differ- ential Pulse Code Modulation Systems - Particular Applications to Voice Signals. Bell System Tech. J., vol. 45, Sept. 1966, pp. 1123-1151.

6. Fukinuki, Takahiko; and Miyata, Masachika: Intraframe Image Coding by Cascaded Hadamard Transforms. IEEE Trans. Comm., vo l . COM-21, Mar. 1973, pp. 175-180.

7. Cotton, M. C.: Image Processing Using Three-Dimensional Hadamard Trans- forms. Ph.D. Dissertation, Stanford Univ., 1977.

8. Ash, R.: Information Theory. Interscience, New York, 1965.

9. Wood, Roger C.: On Optimum Quantization. IEEE Trans. Inform. Th., vo l . IT-15, Mar. 1969, pp. 248-252.

10. Goblick, T. J., Jr.; and Holsinger, J. L.: Analog Source Digitization: A Comparison of Theory and Practice. IEEE Trans. Inform. Th., vo l . IT-13, April 1967, pp. 323-326.

11. Wintz, Paul A . : Transform Picture Coding. Proc. IEEE, vol. 60, July 1972, pp. 809-820.

11

TABLE 1.- LIST OF MINIMUM DISTORTION QUANTIZER TABLES

v

'ab 1 ea

2

3

4 5

6

7

8

9

10

11

12

13

14 15

16

17

Spacing Input distribution

Normal, exponential, g-

Gamma

Normal

Exponent ,a1

Gamma

Normal

Exponential

Gamma

Normal

Exponential

Gamma

. . . .

Distortion criterion

Mean square

Mean square

Magnitude

Magnitude

Re la t ive magnitude

magnitude

Mean square, magnitude

Mean square

Mean square

Mean square

Magnitude

Magnitude

Magnitude

Re la t ive

Relative

magni t ud e

magnitude

magnitude

Relative

Relative

. .~ ...

Number of levels, M

1(1+)40

2 (2x) 2048

1 (1+) 40 2 (2x) 2048

1 (1+) 40

2 (2x) 2048

1 (Zx, 1+) 51:

1(1+)40,

' 1(1+)40,

1 (1+) 40,

2(2x)16

2 (2x) 16 1 (2x ,1+) 15 1 (2x , 1+) 31

64 (2x) 251

64 (2x) 25(

64 (2x) 25(

1 (2x , 1+) 31

1 (2x , 1+) 31

a Tables 2-8 contain the spacing, distortion, and entropy for each number of representative values. Tables 9-17 contain the cut points and representative values, distortion, and entropy for each number of representative values. The probability and distortion for each representative interval are also given.

12

TABLE 2.- EQUIDISTANT QUANTIZATION LEVEL SPACING FOR MEAN SQUARE DISTORTION - Normal PDF Exponential PDF Gamma PDF

Spacing Distortion Entropy Spacing Distortion Entropy Spacing Distortion Entropy M I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

0.0000 1. 5960a 1.2240 .9957 .8430 .7334 .6508 .5860 .5338 .4908 .4546 .4238 .3972 .3739 .3534 .3352 .3189 .3042 .2909 .2788 .2678 .2576 .2482 .2396 .2315 .2241 .2171 .2105 .2044 .1987 .1935 .1881 .1833

0.1788 .1744 .1703 .1664 .1627 .1591 .1557

1. ooooa . 3634a .1902 .1188 .08218 .06066 .04686 .03744 .03070 .02569 .02186 .01885 .01645 .01450 .01289 .01154 .01040 .009434 .008598 .007873 .007239 .006682 .006189 .005751 .005360 .005008 .004692 .004406 .004146 .003909 .003693 .003495 .003313

0.003146 .002991 .002848 .002715 .0.02592 .002477 .002370

0. 0ooa 1. oooa 1.536 1.904 2.183 2.409 2.598 2.761 2.904 3.032 3.148 3.253 3.350 3.440 3.524 3.602 3.676 3.746 3.811 3.874 3.933 3.990 4.045 4.097 4.146 4.194 4.241 4.285 4.328 4.370 4.408 4.449

' 4.487 4.524 4.559

, 4.594 4.628 4.661 4.693 4.724

0.0000 1. 4140a 1.4140 1.0850 1.0240 .8688 .8209 .7295 .6921 .6309 .6034 .5594 .5362 .5028 .4837 .4573 .4431 .4218 .4083 .3906 .3790 .3640 .3554 .3423 .3337 .3226 .3147 .3050 .2980 .2892 .2842

.2708 0.2639 .2586 .2624 .2476 .2419 .2374 .2323

I .2751

1. OOOOa . 5000a .2634 .1931 .1322 .lo79 .08249 .07075 .05704 .05020 .04267 .03834 .03306 .03007

- .02645 .02428 .02206 .02044 .01848 .01723 .01573 .01474 .01380 .01278 .01205 .01140 .01062 .01007 .00943 .008969 .008592

I .008040 ' .007741 0.007405 .007012 .006721 .006382 .006128 .005834 .005611

0. oooa 1. oooa 1.324 1.752 1.872 2.127 2.213 2.394 2.468 2.608 2.670 2.783 2.842 2.937 2.991 3.074 3.118 3.191 3.236 3.301 3.344 3.402 3.437 3.491 3.527 3.576 3.612 3.657 3.690 3.733 3.759 3.805 3.829 3.866 3.895 3.930 3.958 3.991 4.018 4.049

0.0000 1. 1290a 1.8170 1.0790 1.3070 .9175

1.0420 .7926 .8757 .6990 .7593 .6245 .6726 .5649 .6052 .5318 .5509 .4837 .5062 .4565 .4687 .4192 .4366 .3939 .4089 .3716 .3847 .3519 .3633 .3342 .3442 .3182

' .3271 0.3033 .3118 .2899 .2978 .2774 .2851 .2663

1. ooooa . 6816a .2761 .2772 .1444 .1673 .09196 .11160 .06481 .08033 .04861 .06022 .03804 .04687 .03071 .03855 .02538 .03127 .02138 .02662 .01828 .02315 .01583 .02002 .01386 .01749 .01224 .01542 .01090 .01370 .009769 .012260 .008811

.007991

.009914

.007282

.008980

.006666

.008184

0.01099

0. oooa 1. oooa 1.337 1.754 1.846 2.058 2.133 2.268 2.339 2.433 2.504 2.571 2.642 2.697 2.763 2.779 2.870 2.890 2.967 2.964 3.055 3.063 3.137 3.139 3.214 3.211 3.285 3.278 3.352 3.341 3.416 3.402

3.461

3.517 3.588 3.558 3.640 3.617

3.476

, 3.533

aValue derived by direct computation, not by the Max algorithm.

TABLE 3.- EQUIDISTANT QUANTIZATION LEVEL SPACING FOR MEAN SQUARE DISTORTION

M

2 4 8 16 32 64 128 256 512 LO24 !048

Spacing

1. 5960a ,9957 .5860 .3352 .1881 .lo41 .05687 .03076 ,01650 .008785 .004650

Normal PDF

Distortion

0. 3634a .1188 .03744 .01154 .003495 .001040 .0003043 .00008769 ,00002492 .000006997 .000001944

Entropy

1. 0ooa 1.904 2.761 3.602 4.449 5.309 6.182 7.069 7.968 8.878 9.796

Exponential PDF

Spacing

1. 4140a 1.0850 .7295 .4573 .2751 ,1607 ,09018 .04876 .02569 .01323 .006724

Distortion

0. 5000a .1931 .07075 .02428 .008040 .002574 .0007638 ,0002145 .00005714 .00001491 .000003808

Entropy

1. 0ooa 1.752 2.394 3.074 3.805 4.580 5.413 6.300 7.225 8.182 9.159

Spacing

1. 1290a 1.0790 .7926 .5318 .3182 .1851 .09923 .05205 .02662 .01349 .006794

Gamma PDF

Distortion

0. 6816a .2772 .1116 .03855 .01226 .003574 .0009703 .0002476 .00006318 .00001586 .000003952

Entropy

1. 0ooa 1.754 2.268 2.779 3.402 4.085 4.897 5.763 6.682 7.628 8.593


TABLE 4.- EQUIDISTANT QUANTIZATION LEVEL SPACING FOR MAGNITUDE DISTORTION

Normal PDF Exponential PDF Gamma PDF

Spacing Distortion Entropy Spacing Distortion Entropy Spacing Distortion Entropy M

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 -

0.0000 1. 3490a 1.0270 .8338 .7044 .6116 .5416 .4869 .4428 .4065 .3760 .3500 .3276 .3081 .2909 .2756 .2619 .2497 .2385 .2284 .2192 .2107 .2029 .1957 .1890 .1827 .1769 .1715 .1664 .1616 .1571 ,

0.1529 .1489 .1451 .1415 .1381 .1349 .1318 .1289 .1261

0.7979a .4754 .3406 .2683 .2217 .1900 .1662 .1482 .1337 .1221 .1123 .lo42 .09703 .09100 .08558 .08092 .07666 .07294 .06950 .06646 .06362 .06109 .05870 .05655 .05451 .05267 .05091 .04931 .04778 .04638 .04502

0.04378 .04258 .04148 .04040 .03941 .03845 .03755 .03668 .03587

0. 0ooa 1. oooa 1.574 1.976 2.278 2.523 2.728 2.905 3.059 3.197 3.322 3.435 3.539 3.635 3.724 3.807 3.886 3.960 4.029 4.095 4.158 4.218 4.275 4.330 4.383 4.433 4.482 4.529 4.574

I 4.618 4.660 4.701 4.741 4.779 4.816 4.853 4.888 4.922 4.956 4.988

0.0000 . 9803a .9776 .7579 .7070 .6030 .5643 .5029 .4743 .4333 .4115 .3821 .3649 .3426 .3287 .3112 .2997 .2857 .2760 .2643 .2560 .2462 .2391 .2306 .2244 .2171 .2116 .2053 .2004 .1947 .1903

0.1853 .1814 .1769 .1733 .1692 .1659 .1623 .1593 .1559

0. 7071a . 4910a .3540 .2993 .2459 .2200 .1914 .1760 .1582 .1477 .1354 .1278 .1189 .1130 .lo62 .lo16 .09613 .09247 .08797 .08485 .08119 .07855 .07546 .07319 .07055 .06858 .06629 .06456 .06255 .06102 .05925

0.05788 .05630 .05507 .05361 .05250 .051.23 .05022 .04907 .04815

0. 0ooa 1. 0ooa 1.503 1.927 2.151 2.404 2.552 2.732 2.845 2.984 3.077 3.191 3.270 3.365 3.435 3.517 3.579 3.651 3.707 3.772 3.822 3.881 3.927 3.981 4.023 4.073 4.113 4.158 4.195 4.238 4.273 4.312 4.346 4.383 4.414 4.449 4.479 4.512 4.540 4.572

0.0000 . 5253a 1.1310 .6085 .8214 .5347 .6583 .4708 .5554 .4204 .4827 .3798 .4296 .3474 .3883 .3207 .3552 .2978 .3273 .2787 .3045 .2622 .2850 .2474 .2677 .2347 .2531 .2230 .2397 .2129 .2282

0.2035 .2175 .1953 .2083 .1875 .1995 .1806 .1917 .1740

0.5774a .4818 .3141 .3300 .2295 .2520 .1841 .2063 .1554 .1757 .1353 .1528 .1198 .1363 .lo83 .1233 .09917 .1127 .09107 .lo35 .08482 .09627 .07948 .08959 .07443 .08428 .07044 .07920 .06653 .07513 .06343

0.07113 .06031 .06791 .05782 .06467 .05525 .06206 .05321 .05937

0. oooa 1. 0ooa 1.449 1.923 2.023 2.313 2.363 2.576 2.608 2.777 2.803 2.942 2.964 3.082 3.102 3.204 3.222 3.313 3.331 3.411 3.428 3.499 3.516 3.582 3.599 3.657 3.674 3.729 3.746 3.795 3.811 3.858 3.874 3.916 3.932 3.972 3.989 4.025 4.042 4.076

r cn aValue derived by direct computation, not by the Max algorithm.

M

2 4 8

16 32 64

128 256 512

Spacing

1. 3490a ,8338 .4869 ,2756 ,1529 .08361 .04521 .02422 .01288

TABLE 5.- EQUIDISTANT QUANTIZATION LEVEL SPACING FOR MAGNITUDE DISTORTION

Distortion

0.4754 .2683 .1482 ,08092 .04378 .02351 .01254 .006646 .003505

Normal PDF

I 1024 j .006810 I .0018390

Entropy

1. 0ooa 1.973 2.905 3.807 4.701 5.596 6.496 7.404 8.320

Gamma PDF Exponential PDF I Spacing

0. 9803a .7579 ,5029 .3112 .1853 . lo75 ,06123 .03434 ,01900

Distortion

0. 4910a .2993 ,1760 . lo16 .05788 ,03253 ,01806 .009913 .005374

~

Entropy

1. 0ooa 1.927 2.732 3.517 4.312 5.124 5.951 6.795 7.653

~~ ~

Spacing

0. 5253a .6085 .4708 .3207 .2035 .1235 .0723 .04105 .02269

9.242 .. .010390 1 .002881 ~ 8.527 .012210 I 2048 ,003581 I .0009617 10.170 ,005611 ~ ,001524 9.418 .006408 L .I

Distort ion

0.4818 .3300 .2063 .1233 ,07113 .04028 .02220 .01200 .006343 .003289 .001683

Entropy

1. 0ooa 1.923 2.576 3.204 3.858 4.550 5.287 6.069 6.894 7.763 8.673


TABLE 6.- EQUIDISTANT QUANTIZATION LEVEL SPACING FOR RELATIVE DISTORTION

.3518

.3424

.3339

.3261

.3189

.3122

.3061

.3003

Normal PDF Exponential PDF G a m a PDF

Spacing Distortion Entropy Spacing Distortion Entropy Spacing Distortion Entropy M * I'

4.072 4.163 4.248 4.328 4.403 4.474 4.541 4.604

1 3 5 7 9

11 13 15 17 19

~ 21 23

0.0000 .6203 .4338 .3391 .2809 .2410 .2118 . la94 ,1717 .1572 .1452 .1350 .1262 .1186 .1119 .lo60 .loo8 .09602 .09174 .08786

1. OOOOa .5569 .4128 .3362 .2873 .2529 .2270 .2068 ,1904 .1768 .1653 .1555 .1469 .1394 .1328 .1268 .1215 .1166 .1122 . lo81

0. 0ooa 1.557 2.282 2.763 3.121 3.405 3.641 3.843 4.019 4.174 4.314 4.441 4.557 4.664 4.763 4.855 4.942 5.023 5.099 5.172

0.0000 ,4260 .3103 .2499 .2118 . la52 ,1653 .1498 .1374 .1271 ,1185 .1111 ,1047 .0991 ,09414 .08972 .08575 .08217 .07891 .07593

1. ooooa .6147 .4791 .4035 .3536 .3174 .2896 .2675 .2493 .2340 .2209 .2096 .1997 .1908 . la29 .1758 ,1694 .1635 ,1581 .1531

0. 0ooa 1.567 2.273 2.730 3.070 3.339 3.563 3.754 3.922 4.070 4.203 4.324 4.435 4.537 4.631 4.720 4.802 4.880 4.954 5.023

0.0000 ,2989 .2182 .1762 .1497 .1312 .1174 .lo66 .09790 .09073 .08469 07952

.07504

.07110 ,06762 .06452 .06172 ,05920 .05690 ,05480

aValue derived 'by direct computation, not by the Max algorithm.

M -

1 3 7

1 5 31 63

12 7 255

TABLE 7.- EQUIDISTANT QUANTIZATION LEVEL SPACING FOR RELATIVE DISTORTION

Spacing

0.0000 .6203 ,3391 .1894 . lo60 .05898 .03252 .01777

511 .009627 ,023 1 .005175 :047 .002763 -

Normal PDF

Distortion

1. OOOOa .5569 .3362 ,2068 ,1268 .07698 .04612 .02726 .01591 .009160 ,0052390

Entropy

0. 0ooa 1.557 2.763 3.843 4.855 5.829 6.782 7.721 8.653 9.581

I Exponential PDF I Spacing

0.0000 .4260 .2499 .1498 . C8972 .05321 .03117 .01802

Distortion

1. OOOOa .6147 .4035 .2675 .1758 ,1138 .07243 ,04530

Entropy

0. 0ooa 1.567 2.730 3.754 4.720 5.654 6.568 7.469

0.0000 .2989 .1762 . lo66 .06452 .03870 .02292 .01339

.02788 8.359 .00772

.01690 I 9.243 .004400 10.510 10.120 .002480

Gamma PDF

Distortion

1. OOOOa .7942 .5399 .4197 .3261 .2519 .1931 .1469 .1109 .08321 .06205

Entropj

0. oooa 1.585 2.595 3.472 4.328 5.189 6.063 6.949 7.844 8.745 9.650


TABLE 8.- EQUIDISTANT QUANTIZATION LEVEL SPACING FOR GAMMA PDF

M

1 3 7

1 5 31 63

127 255 511

Mean square e r r o r

Spacing

0.0000 1.8170 1.0420

,6052 .3442 .1882 . lo14 .05289 .02678

D i s t o r t i o n

1. OOOOa .2761 .09196 .03071 .009769 .002955 .0008294 ,0002221 .00005784

Entropy

0.000 1.337 2.133 2.763 3.416 4.147 4.933 5.789 6.708

Magnitude d i s t o r t i o n

Spacing

0.0000 1.1310

,6583 .3883 .2282 ,1322 .07533 .04205 .02300

D i s t o r t i o n

0. 5774a .3141 .1841 . lo83 .06343 .03636 .02043 ,01120 .006015

Entropy

0. oooa 1.449 2.363 3.102 3.811 4.535 5.288 6.078 6.907

aValue der ived by d i r e c t computation, no t by t h e Max algori thm.

19

TABLE 9.- OPTIMTJM QUANTIZATION LEVEL SPACING FOR NORMAL PDF W I T H

MEAN SQUARE DISTORTION

a Value derived by direct computation, not by the Max algorithm.

20

2 .9816(300 105500000 01631000 0 0 3 2 8 5 0 0

y E 5 X ( 1 ) Y ( I ) P t I ) D(I)

1 e3H23000 000000UO 0297800Q 0 0 ~ 4 2 2 0 0 2 1024400110 07646000 2 4 44000 00144300 3 001300000 10724000tl 0 1067000 0 184300

!.I f 6 X ( I Y ( I ) P ( I ) D(I)

1 000000u0 03177000 02450000 00087180 2 06569Cli10 1 ~ 0 0 0 0 0 0 U . i w m ~ o 00088770 s i . 4 4 7 r ~ ~ ~ ~ i .8940c100 00739600 01 13900

U I S T B R T I B N - e05798L)Cl

EhUTHOPY * 2,4630000

rl = 7 X ( I ) Y ( I ) P ( I ) n t r ,

1 *2c)03000 oOU00000 0 2 2 0 7 0 0 0 00057200 2 e 8 7 4 A O O O e5606000 0 iO87000 0057410 3 1.6110000 1.188000U .i373000 00058560 4 0 0 0 0 R O U O 200330000 00536200 00075430

U I S T U H T I B N = 0 U 4 4 0 Q 0 f l

E N I H B P Y = 2 .647OOOU

21

E N f e B P Y = 3.1250000

22

3.2540000

7 0 0 0 0 U U C ~ O 205650000 0 0 134000 CIO 13720

23

D I S ' I U R T I U I ~ = 0 140600

M = 14 X ( 1 ) Y ( I )

1 o0000000 e lA57000 2 o2935000 e44130013 3 e5959000 e7505000 4 e9180000 lo0860000 5 1e277WOUO 1.468000Q 6 1 0703UUOU 1 e93900O0 7 2 0 2 8 2 U U U O 2.6250000

D l S l R H T 1 C f l u = o n 122300

P ( 1 ) 1154000

a 1089000 90963300 00784200 a0566000 00330300 00212500

P ( 1 ) laHPOO0

e lU62000 00982800 e0855100 00686Y140 0049OQ00 00263000 e0095480

D(I) e0008263 e 0 0 0 8 2 6 7 e0008278

e0008360 00008553 00011140

. O O O A ~ Q ~

U ( I ) e0006790 000Gh791 00006795 oOC106805 e 0 0 0 6 8 2 5 00006873 00007033 a0009168

24

2 .2stj2n~0 .388ooon .09742ao .0QCl5650 J 52240UU . 6 5 6 8 0 U O .OAA7200 00005654

S 1 = O 9 Q O Q W 102S600LlO 00604800 nO005679 A 07995000 09023000 e 076 1600 .a005662

6 1 0 4 3 7 0 0 0 0 lo6180000 00427100 . ~ a 0 5 7 2 0 7 1 0843UOUC) 2-0690000 02445uu .o~as854 8 204f)lU000 207320000 oOO81810 a0007639

P l I ) .096700a 00948200 00632300 0080 1800 .06HC)700 a0535500 m O 3 7 A B U O 002 128UU 00070660

25

e0005464 9 205U30000 2082600C10 e0061510

UISTCrHTIBh = eUU75930

EhTRbPY = 3,9280000

M = 19 X ( I )

1 0 1092000 2 03294000 3 05551000 4 0 7 9 0 7 0 0 0 5 10C7420r100 6 1 0 3 1 8 0 0 0 0 7 1 0 6 3 4 0 0 ~ ~ A 200170000 9 20550LI000

10 0000C1000

ENThWPY = 4,0030000

V ( I ) 00000000 0 2 1 8 4 0 0 0

44040110 m6698000

9 1 17000 1 o 1730000 1 0 4 6 4 0 0 0 0 1 m8Q30000 20 23100QO 208680000

Y ( 1 ) 1030000 3 1280OO

052650110 e7485000 e9837000

1 a2380000 1 oS23OOOO 10857000O 2.27900130 209080000

P(1) e0869600 e0855900 e 0 8 15 100 e0748700 e0659000 e0549500 0 0 4 2 5 3 0 0 00293500 00564400 e0053890

P ( I ) e0824900 e01301300 . 0 7 5 4 w o 00666800 e0599500 e0496200 00381500 e 0 2 6 1 700 0 145700

e0047520

D(1) e 000345 1

0603452 e0003453 e 0003454 00003457 e0003463 e0003474 oQ003499 e0003582 00004682

d(11 e0082978 e0002978 eOU02979 e0002941 00002983 e0002988 e0002997 00003020

e0004043 . O U Q ~ O W

26

D i S T B H T I B h - 00062080

M a 2 1 X ( I )

1 e 0 9 9 1 A U O 2 02Q89000 3 e5026030 4 e7137000 5 e9360000 6 le1750000 7 1.4390000 0 1.7430000 9 2e1150000 10 206340nOQ 1 1 . ooooouo

M = 22

1 2 3 4 5 d 7 0 9

10 1 1

Y ( 1 ) rOOOOaU0

1984000 e3994000 a605900U e 82 14000

1e0510000

1 e57900OO

2e3230000 2 0 9460000

1.3OOOOOU

1 e9070000

P(1) e07900UO oQ779700 00749100 .0699000 e0630800 e05467tYQ e 0 4 4 9 6 0 0

a0234400 e0129900 e0042130

.a343700

U ( 1 ) eOQa2587 00002587 e 0 0 0 2 5 8 8 e0002588 00002590

00002596

e O O U Z ' 6 2 4 eOUU2687 e0003516

,0002592

.0a026m

27

P ( I ) e0723800 e 07 15900 e0692300 e 0 6 5 3 6 0 0 e0600600 00534800 cO45791)O e03725130 a0281900 e0190500 or3204600 en033630

U ( 1 ) QOOl 7 5 8

e0001758 e 0003759 e 0001 7 5 9 0 0001 760 e CJQ01761 c 0001762 e OQ01765 e 000 1771 e0001784 a flu0 1828 o O O U 2 3 9 7

EN'l i idPY = 4,3270000

28

ENTHCrPY - 6,3840000

Y ( 1 1 e0000000

1676000 e3368008 05093000 e 6 8 7 0 0 0 0 m8722000

1 e O 6 8 0 O O 0 1 e27900OO 1 c51ooooo 1~7720000 20013200ua 2 e 47900OO 3 e 0 7 8 0 O U O

P 4 I ) 00667900

06617013 e0643100 ~ 0 6 1 2 5 0 0 e0570600 e 05 18200 DO456500 00387200 e0312200 e0234200 0 157000 oO085550 e0027290

P ( I ) oObd17UO oO630600 D06086ClO e0576100 o0533700 00482300 e0422900 o03572OO o0286900 a 02 14500 * L ) l 4 3 3 U O . 0 0 7 7 7 8 0 o002.4750

D(1) e 0001562 e0001562 00001562 0 0001563 00001563

e0001564 e0001566 0000156Q e0001574 00001584 000 1624 00021 29

aoois63

!I( I ) 0 0001394 .OUG1394

OClO 1394 000 1394 00001395 oO001305 o000 1396 r0001398 ~ O ~ O l ~ O U e0001405 00 0 1 4 1 S

, U O 0 1 4 5 0 0 0 0 0 1 YO 1

29

M = 27

1 2 3 4 5 6 7 0 9

1 (1 1 1 1 2 13 1 4

U 1 S I U R T I

M = 2 8

1 2 3 4 5 6 7 t3 5)

1u 11 12 13

Y I ) e0620un0 e0615000 ~ 0 6 0 0 1 0 0 rU57560Q 054 1900

e 0 4 9 9 5 0 0 00449400

0 3 9 2 5 0 0 o0330200 .i)%64300 a 0 196900 . u 1 3 1 1 110 e0070960 00022490

II 0001249

e 000 1249 e 000 1249 eclr301249 .0001250 00001250 e OU(3125 1 e uoo 125 1 000 1253

00001255 e OOCI 1259 0 0001268 e oou 1300 e 0001705

2.5780000 e12064920

30

id L ~ ~ O L I O ~ I 3. ~.63r)oao e 0020510 e 0 0 0 1 5 3 4

E N l t i r t P Y - 4.S430000

Y ( I ) .oooooar~ .i45iocm 02913000 m4396000 05911000 *7474000 e9099000

1~0810000 1 e263000O 1.4610000 le67900OO 1 e 9 2 8 O O O C 1 2.2250000 2.6080000 3e18900OO

Y = 30 X ( I ) Y ( 1 )

1 a OUnOUOO a70 I 600 2 I) 1 AllSCIOU a2109000 3 02H2UOOU 3531 000 4 e4254090 04977000 5 e S 7 1 8 U Q O o6459000 6 a7224000 e 7 9 8 9 0 0 0 7 e6767000 .95a5oor3 H 1 e O 4 3 U O O O 1 e 1270000

P ( 1 ) e0578500 e0574400

e0542400 e0514900

e0438900 e0391800 e0339800 e 0 2 8 3 9 0 0 e0225800 e0167300 e0110700 e0059570 e 00 18760

.05623oa

. 048omo

P ( 1 ) e0558900 ~0552500 e0537000 e 135 15400 e0487100 e0452400 e0412000 e0366600

U t i ) e OR01 015 e 0001015 00010 15

0 0001015 0 000 1015 e0001015 eO00t015 e Q001016 00001 0 16 e 000101 8 0 0001019 .0001023 e0001031 oOOUlOS6 e0001386

D( I ) e000U919 .UOOU919 e 00009 19 e0000919 .0000920 e 0 0 0 0 9 2 0 e O O O r 3 9 2 0 e0000920

31

1 2 3 0

5 6 7 8 Y

10 1 1 12 13 1 4 15 i6

y = 32

32

riY131000 0 331 4000 04667000 06049000 07471000 08946000

1.0490000 1 0 2 12uooo 1.3860O~ 1 0576Q00O 1 e7870000 2e0290000 2.3180000 2.6Y10000 3 026100UO

D I S T B R T I W N = .0025050

.OSlt)900

.OS061100 00468980 e0465300 00436400 00402600 00364300 e0322200 oO276900 on229600 .0181100

0 133200 e0087490 e0046720 00 14610

0000761

00000761 0 0 0 0 0 7 6 2 00000762 00000762 00000762

.OD00763 00000764

a0800768

000076 i

,0000762

.oa007~s

, ,0a0~)774 oO000793 000 1042

33

M = 34

1 2 3 4 5 6 7 8 9

1u 11 12 13 I 4 15 16 17

M = 35

34

a0340000 v O 3 0 5 3 0 0 o 0 2 6 A O O O e0228700 a 01 8 8 J O O a 0 1 4 7 7 0 0 a 0 9 0 7 Y U O a 0 0 7 0 4 4 0 .0037380 ~ O U l l b l O

35

M = 37

1 2 3 4 5 6 7 8 9

1 u 11 12 13 14 15 16 57 l e 19

E N ' l Q 6 P Y =

M r 38

X ( 1 ) 00572300 0 17 19000 02874000 040d2000 0522t3000 06440000 076840UO e8969000

100310000 1 171 0000 1031900UQ 1.47SUOLiO 1 . 6 ~ O O O U O 1 . B I O O O U O 2005700L)O 2 e 3 i 1 00(.10 206310OflO 3e9880000 00000000

X ( 1 ) 000u0u00

1 11 6 O O O e223605)O 03566000 0 6510000 05675000 e 6 13 6 6 0 0 0 o80980E30 e9356000

Y ( I 1 00000000 . i i4sooc1 02294000 m3454000 04629000 058270013 07052000 08315000 e9623000

1 0 0 9 9 0 0 0 0 1024300O0 i.39500ao 105600000 3.7400000 1.9410090 201720000 2.4500080 2.8~10000 3036500r30

Y ( 1 ) 013557300 0 id74000 02798000 e3934000 05087OOU 06263000 07468000 0871100O

1 e0600000

P ( 1 ) ,0456400 . 0454400 m0d48400 e0438600 , 0424900 e0407590 00386600 00362400 s 0335100 00305100 00272700 00238400 00202700 0 0 166300 0 0 129900 001394550

0061510 e0032520 00010060

P ( 1 ) 00644200 00440500 .0433200 00422300 e0407900 m0390100 90369200 00345200 e0318600 00289400

D ( I ) , 0000498

*0000498 oOQ00498

00000498

00000498 00000498 e0000498 r0000498 e0000499

.oa00498

. O O O C J ~ ~ ~

000a0498

.oa00499 ~ 0 ~ 0 0 ~ 0 0 0 0 0 0 0 5 0 0 , 0000502 00000506 m0000519 rOOU0682

D ( I ) oQOC10461 00000461 00000461 oOOOO461 *0000461 e QUO0461

00000461 oOQ00461 00000461

. o m o 4 6 i

36

11 12 13 14 15 16 17 18 19

ll I S 1'8 i3 T i U N

E N 7 R O P Y =

M u 39

1 2 3 4 5 6 7 0 9

10 11 12 13 14 15 16 1 7 18 19 20

102060Cl00 1 * 2 7 7 0 0 0 0 103520000 5a429f l f l00 105C)9000Q 1 a5910000 1 0 6 8 Q O U 0 0 P o 7 6 9 U U O O 108690000 1*9690001) 200830000 201980000 2e3360000 204740@0O 2.6540000 2.8330000 3010900i10 3038400130

I 0017920

4,9720000

o 0 2 5 8 2 0 8 e0225380 00191200 .01566i30 00 122100 .OBt)tr)720 a0057620 0 0 3 0 4 i U

.00U93&9

P ( I ) .043351)0 , 0 4 3 1 H 0 0 mO4.26700 0418200 o0406500 13 3 9 1 6 0 0

*03736C)O oO352700 *0339100 *0303000 o0274700 00244600 00% 13100 001 Y O 5 0 0 * [ I147600 aUll49UQ oOCJ83370 0 0 0 5 ~ 0 6 0 .0028480 al!OO8778

37

M r 4 0

1 2 3 4 S 6 7 8 9 10 1 1 12 13 1 4 15 16 17 18 19 20

Y ( I ) 0053QOOO 0 1592000 m2660000 e3738060 e4831000 05944000 * 7 0 8 2 0 0 6 e8251000 rP4590OO

100710000 1 0 2u30000 1 m3420001) 1 m 4 9 0 O O Q C l 1 065OOOOO 1 0825OOOO 2.02 10000 2o267000n 2 5 2 O t ~ O O U 2m8750000 30 421000(1

P(1) 00422500 so419300 e0413000 004037Q0 0391 200 00375900 00357800 e0337000 0 0313800 OO288400 00261000 o023200C7 m0201800 mU170700 0 0 139300 0 108300 q01370450 -0fl50790

00008219 00267 20

D ( I ) mOQOO396 m0000396 ~0000396 * 0 0 0 0 3 9 6 e0000396 r0000396 00000396 e0000396 e0000396 m0000396 0 0000397 00009397 mOOO(3397 00000397 mOOOOJY8 rOOU039B mO000400 eOO0Q403 e OQOO413 mOODO543

38

M = 6 4

1 2 3 4 5 6 7 6 9

10 1 1 12 13 1 4 15 16 17 18 J. 9 20 2 1 22 23 24 2 5 a h 2 7 28 2 9 3 U 3 s. 52

E N T t i O P Y =

M = 128

39

40

I

42 43 44 45 46 4 7 4 8 4 9 5 0 5 1 52 5 3 s 4 5 5 56 57 58 59 60 61 62 6 3 54

a .ooo 17 13

E N T R O P Y - 6.7200000

H = 256

1 2 3 4 5 6 7 B 9 10

X ( I ) e0000000 ~0165900 e0331800 e0497800 00663800 e0829900 e0996000 e 1162000 e 1329000

1495000

1.5670000 l e 6 1 6 0 0 0 0 1 A 7 0 0 0 0 0 1072400OL) 1 *7790000 1 o8360000 1 e89600OU 1.9570000 2. f2210000 2.0Bc)0000 2. 159OUUO 2.2330000 2-31 10000 203950000 204640000 2.5860000 2.6860000 2.8030000 2.9330000 5.0830000 3 26 10000 3.4600000 3.77700UO

Y C I ) 00082950 0 0 2 4 8 9 0 0 e0414600 .0580AU@ .a746800 00912900 lfl79000 1245000

014120QO 1578000

e0058380 00055330 e0052260 e00491 80 oOO46100 o0043030 a0039970 e0036930 a0033920 e0030940 .0028010 00251 30

00022310 e0019570 000 16920 o0014370 e0011940 o0009650 oOOO7515 e0005562 e0003821 oOOO2328 o O O O 1 4 2 4

e0000012 .0000012 * 0000012

.0000012

.00000J2 eo000012 .0000012 .OU00012 .0000012 e0000012 . OOOOO 12 .0000012 a OOOOO 12 e0000012 .OO00012 ~ 0 0 0 0 0 1 2 e0000012 00000012 e 00000 12 .0000012 e 00000 12 .C)000090

.a000012

t ( 1 ) .00000rl2 .ouoooo2 eo000002 .0000002 ac1000002 . 0000002 e c1000002 .00000L32 .0000002 r0000002

41

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4 1 42 4 3 4 4 4 5 46 47 4 8 49 50 S t 52

0 1662000 01828000 01995000 02162000 02330060 02497000 02665000 02833000 03001000 0 3 170000 e3339000 e 3 5 0 8 0 6 0 e3677000 038470UO o4018000 04188000 m4359000 04531000 04703000 04875009 05048OQO 05221000 e5395000 05569000 05744000 05920090 06096090 06273000 06451000 06629000 06808000 06987000 0 7 1 6 8 O O O 07349000 07531000 077 13OOO 07897000 e 8 8 8 2 0 0 0 08267000 08453000 08641000 08829000

42

I1 I. 111111 111 l 1 1 . 1 11111 I

ll

53 54 5 5 5 6 57 5 8 5 9 68 61 6 2 63 6 4 6 5 46 67 68 69 70 7 1 7 2 7 3 74 75 76 77 78 7 9 80 8 1 82 83 8 4 85 86 87 88 8 9 90 9 1 92 9 3 9 4

o9019000 o9209000 o9401000 o9594000 .97880(30 .9983000

I 0 l8@00(3 1 oO380OUO 1 o O 5 8 O O O C ) 1 0 0 7 R O O @ 0 1 mQ980000 1 1 1 8 0 0 D O 1 13900OO 1 15900OO J. 1 8 O O O Q O 1 .20 ia000 i .2230000 1 .2d40OQCl 1 o2650000 1 e2870OQO 1 r 3 6 9 O O O O 1 o33100OO i .3540000 1 d76DOOO I o399OOOO 0.422OatIr3 1 o 4 4 6 0 0 O O 1 046900OO 1 o493OOOO 1.5180000 1 o542OOOO 1 e5670000 1 oS92OQOO 1.62AOOQO 1 o643OOQO 1 o670OOQO 1 06968000 M ~ ~ Q Q O 1.75 10000 P o77800QCl 1.887QQ00 108360R00

4 3

95 96 97 98 99

100 101 102 103 104 105 106 107 108 109 110 111 112 113 1 1 4 115 116 117 118 119 120 121 122 123 124 125 126 127 128

1.8650000 1.8950000 1.9250000 1.9560000 1 o9880000 2.0210000 2.0540000 2.0880080 2. 1220000 2.1580000 2,1950000 2.2320000 2.2710000 2.3100000 2.3520000 2.3940000 2.4380090 2.4830000 2 0 5 3 10000 2.5800000 2r6310000 2.6850000 2.7420000 2.8020000 2.8650000 2.9320000 3.0040000 3.0820000 3.1660000 3.25900[30 303630000 3.4790000 3 * 6 1 5 8 O O O 3.7770000

ENTRUPY = 7.7200000

44

1 0 8800000 1.9 I oc)ooo 1.9410000 1 .Q72OOOQ 2e00400QO 2.0370000

2.1050060 2 1 4OOOOO 2017600Q0 2 02 1300OO 2.2510000 2.29@0600 2.3310000 2.3720000 2.415QOO0 2.4000000 2.5070000 2.5550000 2 a 605OOOO 2.6580000 2.71 30000 2 77 1 OO(30 2.8320000 2 a 8 9 7 0 0 0 0 20967GQOU

2. a7 io000

3.~42ooao 3. 1220000 3.2110080 3.3~380000

3 5420OOO

3.9570000

3.417aom

3 . 6 m m o

TABLE 10.- OPTIMUM QUANTIZATION LEVEL SPACING FOR EXPONENTIAL PDF

W I T H MEAN SQUARE ERROR

M = 1 X ( I ) Y ( 1 ) P ( 1 ) D ( I )

1 . O O O O O O O a , O o O O O O O a 1 .OOOOOOOa .5000a

D I S T B R T I B N = 1 .OOoa

ENTRBPY = 0 0000030a

M = 2 X ( 1 ) Y ( I )

1 . O O O O O O O a .7071a P ( I ) D ( I )

.5O0000Oa .2500a

D I S T B R T I B N = . 5000a

EWTRBPY = 1.0000030

? I= 3 X ( 1 ) Y ( I ) P ( I ) D ( I )

1 .7062000 . 0 0 0 0 0 0 0 .6279000 . 0800R00 2 . o o o o o o o 1.4120000 .1860000 ,0910800

D I S T B R T I B N = -2622030

ENTRBPY = 1.3240030

M = 4 X ( I ) Y ( I ) P(I) D( I )

1 .onooooo .4193000 .3981000 . 0 3 7 1 703 2 1.125OOUO 1.8310000 0 10 19000 - 0 5 0 0 8 0 0


45

r l I S T B H T I O I L = ,1745030

ENTROPY = 1.72Y0030

M = 5

1 * 4 1 9 0 3 0 0 .oooooon .4457000 ,0205000 2 1.5430000 .8379000 .2200000

3 . O O O O O O O 2 - 2 4 8 0 0 0 0 .OS70800 ,0274800

X ( I ) Y ( I ) P ( 1 ) D( I ) *0223800

D I S T B R T I B N = . 1 1 8 4 0 3 0

E N T R B P Y = 1.9520030

' * 1 = 6 X ( 1 ) Y ( 1 ) P ( 1 ) D ( I )

1 .ooooooo ,2992000 3 188000 0 1 3 0 1 00 2 o7177000 1 - 1 3 6 0 0 0 0 .1441000 .0133900 3 1.8400300 205430000 .0370700 * 0 1 7 8 8 0 0

ENTRBPY = 2.2100030

Y = 7 X ( 1 ) Y ( 1 ) P ( I ) D ( I )

1 . 2 9 8 9 0 0 0 oooooc)o .3441000 .no92040 2 1 - 0 1 6 0 0 0 0 . 5 9 7 9 0 0 0 . 2 0 8 8 0 0 0 - 0 0 8 5 0 4 0

4 . O O O O O O O 208390000 .0247000 ,0116000 3 2.136OOOO 1.4340003 - 0 9 4 4 7 0 0 oOO9756O

D I S T B H T I B N = , 0 6 6 3 2 3 0

46

ENTRBPY = 2 . 3 8 0 0 0 3 0

M = 9

X ( I ) Y ( I ) 1 .ooooooo ,2326000 2 .5311000 ,8296000 3 1,247OOOO 1.6640000 4 2.3640000 3,0640000

D I S T B H T I B N = ,0533330

E N T R U P Y = 2,5700050

M = 9

1 ,232Si300 ,0000000 2 7632000 ,4650000 3 1,4780000 1.0610000 A 2.5930000 1,8950000 5 . 0 0 0 0 0 0 0 3,2920000

X ( I ) Y ( I )

P(I) D(1) ,2641000 ,0060370 1502000 ,0060960

,0680700 ,0062750 ,0176700 ,0062600

P(I) D ( I ) ,2798000 ,0046360 0 1900000 ,0043370

,0490400 ,0045070 ,0058810 ,0129700

,1061000 ,0043790

D I S T B H T I B N = ,0420530

E N T R B P Y = 2.7080030

M = 10 X ( I ) Y ( I ) P ( 1 ) D ( I )

1 ,0000000 ,1903000 .2249000 ,0032860

3 ,9523300 1,2500000 ,0826200 ,0033320 4 1,6660000 2.0810000 ,0375600 ,0034290 5 2.7770300 3.4720000 *0090540 .0044550

2 -4225000 ,6546000 ,1451000 .oo33oro

D I S T B R T I R N =

E N T R B P Y =

M = 1 1

1 2

,0356130

2.5580030

X ( I ) Y ( I ) P ( I ) D ( I ) ,2355000 ,0026520 .1901000 .ooooooo

c 6 121000 ,3803000 ,1717000 ,0025060

47

3 1.1410000 .8440000 . 1108000 .OG25150 4 1 .R530000 1.4380000 .0631800 .0025390 5 2.9610000 2.2680000 .0287700 .0026120 6 .OOOOOOO 3.5530000 .0077320 .0033590

D I S T B R T I U N = ,0297130

E N T R B P Y = 2.9740030

M = 12 X ( I ) Y ( I ) P ( I ) d(11

1 ,0000000 . 1610000 .1956000 ,0019843 2 .3510000 .5409000 . 1367000 oOfl19890 3 .7725000 1,0040000 ,0882600 . 00 19970 4 1.3010300 1,5980000 .0503500 .0020160 5 2.0120000 2.4250000 .0229700 .0020740 6 3,1160000 3.8060000 , 0061 020 ,0026640

DI S T B R T I 3 N = .0254530

ENTRFIPY = 3,0960030

M = 13 X ( I )

1 . 3608000 2 , 51 14000 3 ,9323300 4 1.4600000 5 2.1680300 6 3.2670000 7 . 0 0 0 0 0 0 0

D I S T B H T I B N = .021R030

E N T R B P Y = 3.1990030

Y = 14

Y ( 1 ) .O000000 .3217000 ,7011000

1 . lh30000 1 . 7b600UO 2.5810000 3.9S400O0

P( I ) .2033000 1557000 . 1088000

.0703200

.0401700

.0183700 . 0050 190

D ( 1 ) .0016560 .0015750 .0015780 . 001 5850 .0016000 .0016460 .0020860

D( I )

48

1 . 3 0 3 0 0 o c 0 1 3 9 5 0 0 0 , 1 7 2 9 0 0 0 - 0 0 1 2 8 7 3 2 0 3 0 3 1 3 0 0 .4608000 1 2 7 7 0 0 0 o0012890 3 ,65020130 . 8 3 9 7 0 0 0 ,0893200 - 0 0 1 2 9 2 0 4 t .n700C)00 1 .3010000 , 0 5 7 7 6 0 0 oOO12970 5 1 .T9700O0 1,892i)OUO , 0 3 3 0 3 0 0 ~ 0 0 1 3 0 9 0 6 2 .3040300 2.71SOOOO , 01 51 400 oOO13460 7 3 . 3 9 3 0 0 0 0 6 , 0 8 1 0 0 0 0 , 0 0 4 0 9 2 0 , 0 0 1 7 0 5 0

!7ISTbHTI4N = , 0 1 9 0 5 3 0

ENTRBPY = 3 . 3 0 5 0 0 3 0

Y = I 5 X ( 1 ) Y ( I )

1 . I 3 9 4 0 0 0 . 0 0 0 0 0 0 0 2 , 4 3 9 4 3 0 0 - 2 7 8 8 0 0 0 3 , 7 9 9 2 2 0 0 , 5 9 9 9 0 0 0 4 1.20903UO . 9 7 5 s o u o 5 1.7350;300 1 , 4 4 3 0 0 0 0 6 2.44103UO 2 , 0 3 0 0 0 0 0 7 3 . 5 3 3 0 0 0 0 2 .8510000 Y .0000CIOO 4 .214000n

D I S T B H T I Y r \ : = , 0 1 6 6 9 3 0

E N T R O P Y = '3 .3930030

?I= 1 6 X ( I ) Y ( 1 )

1 , 0 0 3 0 0 0 0 . 1 % 3 1 0 0 0 2 . .2623200 , 4 0 1 3 0 0 0 3 , 5 6 1 8 3 0 0 . 7 2 2 1 0 0 0 4 . 9 1 1 1 3 0 0 1 .1000000 9 1.33O0300 1 , 5 h 0 0 0 0 0 6 1 . 9 5 5 0 3 0 0 2 .1490000 7 2.5580000 2 , 9 6 7 0 0 0 0 9 3 . 6 4 3 0 0 0 0 4 .3200000

P ( I ) 1 7 8 8 0 0 0 , 1 4 1 Y O 0 0

1 0 4 8 0 0 0 , 0 7 3 5 3 0 0 . 0 4 7 4 4 0 0 , 0 2 7 1 5 0 0 . 0 1 2 4 6 0 0 ,0034560

P ( I ) , 1 5 4 9 0 0 0

,0880500 06 1 6 3 0 0 , 0 3 9 9 1 00 , 0 2 2 8 8 0 0 ,0105300 . 0 0 2 8 9 2 0

o1192000

D ( I ) 00 1 1030

, 0 0 1 0 5 5 0

.0(310590 , 0 0 1 0 6 3 0 , 0 0 1 0 7 3 0 .01311030 - 0 0 1 3 8 4 0

.DO10560

D ( I ) , 0 0 0 8 8 2 1 , 0 0 3 8 8 3 0 ,0038842

, 0 0 0 6 9 0 1 .0008984 . 0 0 0 9 2 3 1 ,00115YO

, 0 0 0 8 8 6 3

T l I S T 8 H T I 3 N = 0 1 481 30

49

E N T R 4 P Y = 3,4850000

M = 17 X ( I ) Y ( I )

1 1229000 ,0000000 2 .3847000 ,2457000 3 o6837000 ,5237000 4 1.0320000 ,8437000

6 1,9730000 1,5800000 7 2.6730000 2,2660000 8 3,7510000 3.0800000 9 .0000000 4.4210000

5 1 *4500000 102210000

DISTDRTISN = . 01 3 1200 ENTROPY = 3,5650030

v = 18 X ( I ) Y ( I )

1 , 0000300 0 1100000 2 ,2327000 ,3355000 3 .A343300 , 6331 000 4 .7929300 ,9527000 5 1.1410000 1,3290000 6 1.5580000 1,7870003 7 2.0800000 2.3720000 9 2.7780300 3.18300GO 9 3 . 8 5 0 0 3 0 0 4.5170000

D I S T B R T I 4 N = ,0117930

F N T R S P Y = 3.6470030

P(1) 1594000

.1301000

.1001000 ,0739800 ,0518300

-01 93000 ,0089240 ,0025420

*0336100

P ( I ) 1402000

0 1 1 12000 ,0856100 ,0633200 .0443900 , 0 2 6 8 0 0 0 .0165700 ,0076800 .0021590

D ( I ) ,0007679 ,0007380 ,0007387 ,0007398 .0007415

.0007515 ,00377 18 .0009507

,0007446

D( I ) ,0006295 ,0006299 ,0006305 ,0006313

.0006355 ,0006413 .0006584 ,0008041

,0006328

y = 19 X ( 1 ) Y ( 1 ) P ( I ) r ; ( I )

1 , 1009300 ,0000000 , 1439000 ,0005571 2 03424300 021 98000 1199000 ,0005375

50

3 .6037500 ,4650000 4 .9020000 ,7424000 5 1,2490000 1.0620000 6 1,6660000 1.4370000 7 2,1860000 f ,8950000 8 2.8830500 2.4780000 9 3.9500000 3.2870000 10 ,0000000 4.6130000

D I S T B R T I B N = ,0 106030

ENTRBPY = .3 , 7 1800 30

M r : 20 X ( I )

1 ,noooooo 2 ,2093000 3 -44 15000 4 .7025300 5 1,0000000 6 1.3470000 7 1.7630300 8 2.2520000 9 2.9760000 10 4.0370300

D I S T r 3 R T I J N = ,0396550

F N T R B P Y = 3.7920000

Y ( I ) .0994800 ,3190000 .5640000 . 94 10000

1,1600000 1.5350000 1,9910000 2.5730000 3 , 3780000 4,6960000

Y = 21 X ( I ) Y ( I )

1 . 0994 100 . o o o o o o o 2 ,3085000 . 1988UUU 3 ,5406000 . 4 182000 4 .8013000 .hh30000 5 1,0990000 .9397000 6 1.4450000 1,2580000 7 1,8600000 1.6330000 8 2.3790000 2.oa8oooo

.0951800

.0542200

.0380200 ,0246900 ,0142200 .0066080 .00 19200

a0732700

P ( I ) ,1281000 ,1041000 ,0826400 .0636400 -047 1200 ,0330700 ,0214900 . 0 124000 ,0057790 .0016570

P( I ) .1311000 .1112000 .0904300 007 17900 ,0552900 .0409500 .0287500 , 0 197000

.0005379 ,0005383 .0005391 .0005403 ,0005426 ,0005475 .0005620 .0036777

m r ) .0004653 ,0004655 .0004658 ,0004663 .0004669 ,0004680 .0004699 .0004741 .0004865 ,0005994

D ( I ) ,0004 169 ,0004035 .0004037 .0004039 ,0004043 .0004049 .0004058 .0004075

51

9 3.0710000 2.6690000 .0108000 . 0004 1 1 1 10 4.1280000 3.4720000 oOO50440 .0004217 1 1 ,0000300 4.7840000 ,0014940 *0005103

I I I S T 9 R T I B N = ,0087730

FNTRBPY = 3.8560030

M = 22 X ( I )

1 .ooooooo 2 .1899000 3 .3986000 4 .6301300

6 l.lR70i300 7 1.5320000 8 1.9450000 9 2.0610000

10 3.1480000 1 1 4.1930000

5 o8902000

nISTBRTI4N = .0080150

E N T R B P Y = 3.9250050

y = 23 X ( I )

1 .0906500 2 .2804000 3 .A889000 4 .7202300 5 . 9 8 0 0 0 0 0 6 1,2760000 7 1.6210000 R 2.0340000 9 2.5480000

10 3.2330000 1 1 4.2740000 12 . 0000000

Y ( I ) .0907200 . 2891 000 .5081000 . 7522000

1.02800OO 1.345OOOO 1.71900OO 2.1720000 2,7500000 3.5460000 4.8400000

Y ( I ) .ooooouo .1813OOO .3796000 .5983000 .8422000

1.1180000 1.A350000 1.8070000 2.2600000 2.5360000 3.6300000 4.9170000

P ( 1 ) . 1178000 .0976800 .0794500 .0631100 .0486500 .0360600 .0253500 -01 65200

.0044990 -0095720

*0013290

P ( 1 ) . 1203000 1035000

.0858800

.0698700

.0555100

.042C)OOO 0031 7400 e0223300 0145600 e0084470 .0039800 .0012170

D ( I ) .0003528 .0003529 .0003531 .0003533

. 000354 1

.0003549

,0003594

.0004493

oOO33536

oOOO3563

.0003685

D ( I ) 00031 91

.0003096

.0003097

.0003099

.000310 1

. 0003108 ,0003115 .0003127 . 0003155 .0003233

*0003103

oOOO3859

52

n I S T O H T I D N = .00733!30

E Y T R R P Y = 3.9840030

M = 24 X ( I )

1 . 0 0 0 0 0 0 0 2 .1740000 3 .3636000 4 057 19300 5 . 8 0 3 0 0 0 0 6 1.0620300 7 1.3580000 8 1.7030000 9 2.1140000 10 2.6270000 1 1 3.3100000 12 4.3450000

DISTORTIdN = .0067840

E N T R R P Y = A.0450030

Y ( 1 ) .08344UO .2646000 .A627000 .h812000 .9248000

1.2000000 1.5160000 1.8-890000 2.3400000 2.9150000 3.7060000 4.9840000

M = 25 X ( I ) Y ( I )

1 .OR33800 .ooooooo 2 .2573000 . 1668000 3 .4467000 .3478000 4 ,6548000 .5457000 5 .8857000 .7640000 6 1.1450300 1.0070000 7 1.4400000 1.2820000 8 1.78400OO 1.5980000 9 2.1950000 1.9700000 10 2.7070000 2.4210000 1 1 3.3880000 2.9940000 12 4.4180000 3.7820000 13 . O O O O O O O 5.0540000

P ( I ) D ( 1 ) . 1091000 .0002744 .0919500 ,0002745 ,0762800 .0002745 .0620800 .0002747 .0493300 .0002749 .0380500 ,0002751

,0002755 .0282300 .0198700 .0002761 .0129700 ,0002772 .0075360 .0002796 ,0035610 .0002865 . 00 10720 .0003489

P ( I ) . 1 1 12000 .0968800 . 081 6800 .OS51600 ,0438500 .033&300 .0251100 - 0 176800 .0115500 . 00672 10 -003 1840 .0009935

oO677700

D ( 1 ) .0002502 .0002434 .0002434 .0002435 .0002436 .0002438 .0002440 .0002443 .0002449 .0002459 .0002480 .0002540 .0003017

5 3

n I S T U R T I B N = .0062510

E N T R B P Y = 4.1000000

M = 26

1 2 3 4 5 6 7 8 9

10 1 1 12 13

FNTR6PY =

M = 27

1 2 3 4 5 5 7 8 9

10 1 1 12 13 14

X ( I ) .ooooooo . 1503000 .3337000 .5227300 .7301300 .9602000

1.2180000 1.51 30000 1.855OCJOO 2.2630000 2.77 10000 3,4450000 4.4580300

4.1590030

X ( I ) .0771800 .2376000

4 112000 . 6 0 0 3 0 0 0 o R 0 8 0 0 0 0

1.03801)00 1.2970OOO 6.5910000 1.934OOOO 2.3430000 2.8530000 3.5280000 4.5470000 .0000000

Y ( 1 ) . 077 1300 .243SOOO .4240000 .~213000 .R389000

1.0810000 1.3550000 1.670000O 2.0390000 2.4870000 3.0550000 3.8340000 5.0d20000

Y ( I ) . ooooooo .1544000 .3209000 .5015000 .h991000 .9169000

1 160O003 1.43AOOOO 1 . 7490000 2.1190000 2.5680000

3 . 9190000 5.1740000

3.1370000

P ( I ) ~1014000 ,0867000 . 073 1300 .0607 100 .0494500 .0395400 .0303800 ,0225800 .0159300 - 0 104400 . 006 1010 .0029160 .0009 139

P(I) .1034000 . 0910000 .0777800 .0655900 .0544400 .044.3300 . 0 3 5 2 6 0 0 .0272200 . 0202200 .0142600 .0093370 .0054480 .0025970 .0008283

D ( I ) ,0002166 ,0002167 .00021 67

,0002169 .0002 170 00032172 .0002175 .0002180 .0002188 .0002207 .0002258 ,0002722

0002 168

D ( I ) . 0001999 .0001947 .0001?47 . 0001 9 4 8 .0001945 .0001949 .0001950 .0001952 .0001955 .0001959 ,0001967 oOOO1983 00002030 .0002400

54

D I S T B R T I B N = ,0053850

ENTROPY = 6.2080030

M = 28 X ( I )

1 .ooooooo 2 -1487OOr3 3 -3087000 4 48 18000 5 -6703000 6 -8772000

8 1.3640000 9 1.6570000 10 1.9980000 1 1 2.4040000 12 2.9090000 13 3.5770000 14 4.5760000

7 1.1070000

D I S T B R T I B N = . 0 3 5 0 0 5 0

EhTROPY = 4.2620030

Y ( I ) -0717700 -2257000 . 39 17000 ,5719000 .7687000 .9857000

1.2270000 1.5000000 1.8140000 2.1810000 2.6270000 3.1910000 3.9630000 5.19oooon

Y = 29 X ( I ) Y ( I )

1 . 07 17800 . oooooon 2 -2205000 - 1436000 3 * 3 8 0 5 0 0 0 -2975000 4 * 5 5 3 6 0 0 0 .4636000 5 .7422000 ,6437000 6 .9491000 .84060i10 7 1.1790000 1.0580000 8 1.4.360000 1,2990000 9 1.7290000 1.5720000

1 1 2.4760000 2.2540000 12 2.9810000 2.6990000

10 2.0700000 1.8860000

P ( I ) .0948500 ,0820400 -0701500 .0591900 -0491600 .0400600 .0318900 .0246500 -0183400

.0085150 -0049950 .0024050 .0007731

01 29700

P ( 1 ) -0964900 .0857100

.0633800

.0534800

.0444200

.036190G -0288100 .0222700 . 01 65700 .0117100 .0076900

-0741200

D ( I ) 0001 745 -0001745 -0001746

-0001747 -0001747 -0001749 .0001750 -0001 752 .0001756 -0001763 .0001777 .0001817 -0002190

0001 746

D ( 1 ) -0UOl616 -0001577 - 0 0 0 1577 .0001578 000 1578 . 0001579 ,0001579 -0001580 .0001582 .0001584 .0001587 . 000 1593

55

13 .6500000 3.2640000 ,0045100 .0001606 14 4.6500000 4,.0360000 .0021700 ,0001643 IS .OOOOOOO 5,2640000 ,0007160 .0001866

DISTBRTISN = .0046630

E N T R B P Y = 4.3050030

y = 3 0 X ( I )

1 .ooooooo 2 .1387000 3 . 2871300 4 .4468000 5 ,6294000 5 .R074000 7 1,0140000 8 1.2420000 9 1.4990000

10 1.7910000 1 1 2.1300300 12 2.5340000

14 3.6960000 15 4.6810000

13 3.0350300

nISTBRTI8N = ,0843740

E N T R B P Y = 4.3600030

?l= 31 X ( I )

1 ,067 1000 2 .2058000 3 ,3543300 4 .5139000 5 ,6866000 6 .E3746309 7 1.3810000 P 1,3100000

Y ( I ) ,0670900 . 2 103000 ,3640000 .5296000 ,7093000 .9056000

1.1220000 1.3630000 1.6350000 1.Y470000 2.3120000 2.7350000 3 3 150000 4.0780000 5.2840000

Y ( I ) . ooooooo .1342000 ,2775000 .4311000 -5968000 .7765000 ,9728000

1 . 1890000

P ( I ) ,0890600 .0778ioa oO673200 .0575800 .O4861OO ,0403900 oO329400 ,0262600 . 02031 00 0 151300

- 0 107100

.0041560

.0020170 ,0006669

,0070540

P ( I ) .0905000 = 08 10100 .0707700

,0523700 ,04421 00 .0367400 .0299500

06 12300

nc 1 1 .0001425 .0001426 .0001426

.0001426

-0001 427 ,0001428

0001 426

*0001427

oOOO1430 *0001431 0001 434 0091 440 0001451 . I

.0001483 ,0001787

D( I ) .0001326

,0001297 ,0091297 .0001297 .0001298 .0001299 , 0001 299

*0001297

56

9 1 . 5 6 6 0 0 0 0 1 . 4 3 0 0 0 0 0 1 0 1 . 8 5 8 0 0 0 0 1 . 7 0 2 0 0 0 0 1 1 2 . 1 9 7 0 0 0 0 2 .0140000 1 2 2 . 6 0 1 0 0 0 0 2 . 3 8 0 0 0 0 0 13 3 . 1 0 3 0 0 0 0 2 .8230000 14 3 . 7 6 5 0 3 0 0 3 . 3 8 3 0 0 0 0 1 5 4 . 7 5 0 0 0 0 0 4 .1460000 1 6 . O O O O O O O 5 .3540000

D I S T B R T I B N = . 0 0 4 0 9 10

E N T R O P Y = 4 . 4 0 3 0 0 0 0

Y = 32 X ( 1 )

1 .0O00000 2 . 1 2 9 9 0 0 0 3 , 2 6 8 3 3 0 0 4 . 4 1 6 4 0 0 0 5 . 5 7 5 6 0 0 0 6 . 7 4 7 8 0 0 0 7 , 9 3 5 2 0 0 0 8 1 . 1 4 1 0 0 0 0 ’ 9 1 . 3 6 9 0 0 0 0

l G 1 . 6 2 4 0 0 0 3 1 1 1 .9150000 1 2 2 . 2 5 2 0 0 0 0 1 3 2 . 6 5 3 0 0 0 0 1 4 3 . 1 5 0 0 0 0 0 15 3 . 8 0 5 0 0 0 0 1 6 4 . 7 7 3 0 0 0 0

n I S T O R T I r j N = -00385 10

E N T R O P Y = 4 . 4 5 1 0 0 0 0

Y ( I ) , 0 6 2 9 7 0 0 . 1 9 6 9 0 0 0 . 3 3 9 8 0 0 0 , 4 9 3 0 0 0 0 . 6 5 8 2 0 0 0

1.033001)O 1.249000O 1 , 48900GO 1.75900OO 2 .0700000 2 . 4 3 3 0 0 0 0 2 . 8 7 3 0 0 0 0 3 . 4 2 8 0 0 0 0 4 . 1 8 1 0 0 0 0 5 . 3 6 4 0 0 0 0

. a 3 7 4 0 0 0

. 0 2 3 a 6 0 0

. 0 1 8 4 6 0 0

. 0 0 9 7 3 9 0

. 0 0 3 7 7 7 0

- 0 0 0 6 2 1 6

01 3 7 6 0 0

oOO64120

0 0 1 8 3 2 0

P ( I ) . 0 8 3 9 2 0 0 - 0 7 3 9 7 0 0 . 0 6 4 6 4 0 0

, 0 4 7 8 7 0 0

. 0 3 3 6 2 0 0

. 0 2 1 8 8 0 0

. 0 1 6 9 5 0 0

. 0 1 2 6 4 0 0

. 0 0 8 9 7 0 0 , 0 0 5 9 2 5 0

e 0 5 5 9 4 0 0

- 0 4 0 4 3 0 0

oO274300

. 0 0 3 5 0 7 0 oOO17170 a0005856

.OOO 1 2 9 9 , 0 0 0 1 3 0 1 . 0 0 0 1 3 0 2 .OOOl 3 0 5 . 0 0 0 1 3 1 0 . O O O l 320 , 0 0 0 1 3 4 9 .0001524

D( I ) . 0 0 0 1 1 7 8 . 0 0 0 1 1 7 9 . 0 0 0 1 1 7 9 . 0 0 0 1 1 7 9 . 0 0 0 1 1 7 9 . 0 0 0 1 1 7 9 .0001180 . 0 0 0 1 1 8 0 . 0 0 0 1 1 8 1 . 0 0 0 1 1 8 2 . 0 0 0 1 1 8 3 . 0 0 0 1 1 8 6 . 0 0 0 1 1 9 0 . 0 0 0 1 2 0 0 . 000 1 2 2 5 . 0 0 0 1 4 7 7

! q = 53 . X ( I ) Y ( I ) P(I) D ( I )

1 .01530100 .ooooooo .0852300 . 0 0 0 1 1 0 3 2 1 9 3 0 0 0 0 . 1 2 6 0 0 0 0 , 0 7 6 8 1 0 0 , 0 0 0 1 0 8 0

57

3 .3515000 4 .4797000 5 . 6 3 9 1 000 6 .8114000 7 .9990000 8 1.2050000 9 1.4330000

10 1.6890000 11 1.9800000 1 2 2.3170000 13 2.7190000 1 4 3.2180000 15 3.8740000 16 4.8480000 1 7 .ooooooo

DIST8RTIBN = .0036350

F N T H B P Y = 4.4910030

Y = 34 X ( I )

1 .ooooooo 2 .1221000 3 . 2518500 4 .3898000 5 .5375000 6 .6963000 7 .8679000 8 1.0’550000 9 1.2600000

10 1.48600017 11 1.7410000 12 2.0300000 13 2,3650000 1 4 2,7630000 15 3.2560000 16 3.9020000 17 4.8530000

,2600000 .4030000 .5564000 . 7 2 1 7 0 0 0 . 90 10000

1.0970000 1.3130000 1.5ii30OUO 1 .R240000 2.1350000 2.4990000 2.9400000 3.4960000 4.2530000 5.4420000

Y ( I ) .Ob93200 .1850000 .3185000 0461 1 0 0 0 .5139000 ,7786000 .9572000

1. 1 5 2 0 0 0 0 1,3670000 1.6060008 1.8750000 2,1840000 2.5450000 2 . 9.8 10000 3.5310000 4.2740000 5 . 4310000

.0677000

.0591500 . 05 1 1900

.0438000

.0369900

.0307500 ,0250800 ~ 0 2 0 0 0 0 0 .0154900 .0115500 .0081880 .0054030

.0015600 ,0031 940

,00054 1 7

? ( I ) .0793200 .0704600 ,0621200 . 0 5 4 3 1 0 0

.0402500 * 0 4 7 0 2 0 0

.0340200 *0283000 .0231100 - 0 1 8 4 5 0 0 .0143100 . 0 1 0 6 9 0 0

.0050380

.0029980

.0014820

* 0 0 7 6 0 3 0

-0005229

.0001080

.0001080 0 0 0 1 0 8 0

- 0 0 0 1 0 8 1 .0001081 .0001081 , 0 0 0 1 082 . 0 0 0 1 0 8 2 .0031083 .0001085 . 0 0 0 1 0 8 7 . 0 0 0 1 0 9 1 . 0 0 0 1 0 9 9 .0001123 ,0001328

D( 1 ) .0000985 .0000985 ,0000985 .0000985 .0000985 .0000985 .OOCJO986 .0000986 .0000986 ,0030987 .0000988 .0000989 .0000991 .0000994 . 0 0 0 1 0 0 2 . 0 0 0 1 0 2 2 .0001234

DISTeHTI4r\i = .0034150

58

ENTRUPY = 4 , 5 3 7 0 0 3 0

Y = 35 X ( 1 )

1 . 0 5 9 3 5 0 0 2 . 1 8 1 6 0 0 0 3 . 3 1 P 3 0 0 0 4 . 4 4 9 4 0 0 0

6 , 7 5 6 1 0 0 0 7 . 9 2 7 9 0 0 0 8 1 , 1 1 5 0 0 0 0 9 1 . 3 2 0 0 0 0 0

1 0 1 . 5 4 7 0 0 0 0 1 1 1 . 8 0 2 0 0 0 0 1 2 2 , 0 9 1 0 0 0 0 1 3 2 .4270000 14 '2 .8260000 1 5 3 . 3 2 0 0 0 0 0 16 3 . 9 6 8 0 3 0 0 17 4 , 9 2 4 0 0 0 0 1 8 . o o o o o o o

5 , 5 9 7 2 0 0 0

D I S T B R T I BIN = , 0 0 3 2 3 4 0

E N T R B P Y = 4 . 5 7 5 0 0 3 0

Y = 3 6 X ( I )

1 OQOO0OO 2 1 1 5 4 3 0 0 3 - 2 3 7 4 0 0 0 4 . 3 6 6 9 0 0 0 5 ,5048CIOO 6 , 6 5 2 3 0 0 0 7 . 8 1 0 9 0 0 0 8 . 9 8 2 3 0 0 0 9 1 . 1 6 9 0 0 0 0

10 1 . 3 7 3 0 0 0 0 1 1 1 . 5 0 0 0 0 0 0

Y ( I ) .ooooooo . 1 1 8 7 0 0 0 , 2 4 4 4 0 0 0 , 3 7 8 1 0 0 0 , 5 2 0 7 0 0 0 . 6 7 3 7 0 0 0 . 8 3 8 5 0 0 0

1 . 0 1 7 0 0 0 0 1 . 2 1 2 0 0 0 0 1 . 4 2 7 0 0 0 0 1 , 6 6 7 0 0 0 0 1 . 9 3 6 0 0 0 0 2 . 2 4 6 0 0 0 0 2 , 6 0 8 0 0 0 0 3 . 0 4 4 0 0 0 0 3.5950000 4 . 3 4 2 0 0 0 0 5 . 5 0 6 0 0 0 0

Y ( I ) , 0 5 6 1 2 0 0 . 1 7 4 6 0 0 0 , 3 0 0 2 0 0 0 .4336000 , 5 7 6 0 0 0 0 , 7 2 8 7 0 0 0 . 8 9 3 2 0 0 0

1 , 0 7 1 0 0 0 0 1 .2hh0000 1 . 4 8 1 0 0 0 0 1 , 7 1 9 0 0 0 0

P ( I ) . 0 8 0 4 8 0 0

- 0 6 4 8 2 0 0 . 0 5 7 1 4 0 0 , 0 4 9 9 5 0 0 . 0 4 3 2 4 0 0

0 3 1 2 7 0 0 , 0 2 6 0 2 0 0 -02 1 2 4 0 0 . 0 1 6 9 5 0 0 , 0 1 3 1 4 0 0 , 0 0 9 8 1 0 0

. 0 0 4 6 1 9 0 , 0 0 2 7 4 5 0 . 0 0 1 3 5 4 0 . 0 0 0 4 8 6 4

, 0 7 2 9 7 0 0

oO370200

, 0 0 6 9 7 7 0

P ( I ) . 0 7 5 2 8 0 0 . 0 6 7 5 2 0 0

. 0 5 2 7 3 0 0 , 0 4 6 1 1 0 0 .a399300 oO341900 . 0 2 8 9 0 0 0 . 0 2 4 0 5 0 0 ,0196500

, 0 5 9 8 0 0 0

00 1 5 6 9 0 0

D ( I ) , 0 0 0 0 9 2 5 ,0000907 , 0 0 0 0 9 0 7 , 0 0 0 0 9 0 7 ,0000907 , 0 0 0 0 9 0 7 . 0 0 0 0 9 0 8 , 0 0 0 0 9 0 8 , 0 0 0 0 9 0 8 , 0 0 0 0 9 0 8 , 0 0 0 0 9 0 9 , 0 0 0 0 9 1 0 . 0 0 0 0 9 1 1 . 0 0 0 0 9 1 3 , 0 0 0 0 9 1 6 . 0 0 0 0 9 2 3

. 0 0 0 1 1 1 6 , 0 0 0 0 9 4 2

D( I ) . 0 0 0 0 8 3 4 . 0 0 0 0 8 3 4 .OOC)O834 , 0 0 0 0 8 3 4 . 0 0 0 0 8 3 4 . 0 0 0 0 8 3 4 . 0 0 0 0 8 3 5 ,0000835

. 0 0 0 0 8 3 5

. 0 0 0 0 8 3 6

, 0 0 0 0 8 3 5

59

12 1,5540000 1.9880000 13 2.1420000 2.2960000 14 2.4760000 2.6560000 15 2.8730000 3.0910001) 15 3.3640000 3.6380000 17 4.0070000 4.3770000 18 4.Q500000 5,5240000

D I S T B R T I B N = . 0030530

E N T R O P Y = 4-61 70030

M = 37 X ( I )

1 .0560900 2 .1714000 3 ,2933000 4 .4227ClOO 5 . 5 6 0 5 0 0 0 6 ,7079300 7 .5663300 8 l.r)380000 9 1.2240000

10 1.4280300 1 1 1.6540000 12 1.90800i)O 13 2.1950300 14 2.5290000 15 2.9250000 16 3.4150000 17 4.0550000 1 6 4.9920000 19 . ooooooo

D I S T O R T I S N = .0328940

Y I I ) .ooooooo 1122000

.2306000 ,3560000 ,4894000 .6316000 ,7842000 .9485000

1 . 1270000 1.3210000 1.5350000 1.7730000 2.0420000 2.3490000 2.7090000 3.1420000 3.6870000 4.4230000 5.5620000

. 01 2 1700 . 0091 050

.0064800 ,0043000 -0025640 00 12730 00004556

P ( I ) oO7623OG 00694900 062 1500 -0552200 ,0486900 .0425800 .0368800 ,0315800 oO267000 -0222300 *0181600 oO145100 .0112600 .0084270

.0039870

.0011860

.0004411

oOO60020

,0023820

.0000837

.0000838

.0000939

.0000842

.0000849 ,0000865 .0000999

D ( I ) ,0000784 .Q000769 .0000769 .0000769 . 0 0 0 0 7 6 9 .0000769 .0000769 .0030769 .0000770 .0000770 .0000770

.0000771 ,0000772 ,0000774 .0000776 .0000782 .0000797 .0000941

.0000771

E N T R B P Y = 4,6550000

60

Y = 38

1 2 3 4 5 6 7 3 9

10 l i 1 2 1 3 1 4 15 1 6 17 19 1 3

DISTSKTI3N

E V T R B P Y =

X ( I ) . O ~ I O O O O D 1 0 ~ 2 0 0 0

. 7 2 4 4 3 0 0

.3462900

. A 7 5 A 3 0 0 . 6 130300

.76iI2200 - 9 1 8 3 0 0 0

1 0 8 9 0 3 U O 1 . 2 7 5 0 3 0 0 1.47900OO 1.7O50C10O 1 . 9 5 7 0 0 0 0 2 . 3 4 4 0 0 0 0 2. Fj77OOOO 2.57 10000 J .d580000 4 . 0 9 4 0 0 0 0 5.0220i100

.11037570

4 . 5 9 4 0 0 3 0

- -

Y ( I ) 0 5 3 2 1 OCJ

. 1652000

.2d35000

. 4088000 . 5 4 2 u o o o

. h 8 4 0 0 0 0

.836301jO 1 . 0 0 0 0 0 0 0 1 .1780000 1 .3720000 1.58600OO 1 .8230000 2 , 0 9 i 0 0 0 0 2.337 00013 2.756000G 3 .1870000 3 . 7 2 9 0 0 0 0

5 .5840000 4 . 4 5 9 0 0 ~ 0

P ( I ) . 0 7 1 5 7 0 0 . 0 6 4 3 9 0 0 . 0 5 7 6 0 0 0 . 0 5 1 1 8 0 0 .i1451400 , 0 3 9 4 8 0 0 , 0 3 4 2 0 0 0 . 0 2 9 3 0 0 0 , 0 2 4 7 8 0 0 , 0 2 0 6 3 0 0 - 0 1 6 9 7 0 0 . 0 1 3 4 8 0 0 -0 1 0 4 7 0 0 , 0 0 7 8 4 4 0 , 0 0 5 5 9 4 0 . 0 0 3 7 2 3 0 , 0 0 2 2 3 1 0 , 0 0 1 1 1 7 0 , 0 0 0 4 1 1 8

D(I) , 0 0 0 0 7 1 1

. 0 0 0 0 7 1 1 , 0 0 3 0 7 1 i . 0090711

0 0 0 0 7 1 1

oOO30711 , 0 0 0 0 7 11 . o n 0 0 7 1 1 .0000711 ,003072 2 , 0 0 0 0 7 1 2 , 0 0 0 0 7 2 2 oOi300713 .0000714 . 0 0 0 0 7 1 5 o O O O O 7 1 R , 0 0 0 0 7 2 3 . 0 0 0 0 7 3 6 oO030904

Y = 39 X ( I ) y ( I ) ? ( I ) !I( I )

*Or300670 1 .L75317UO .0001)000 . 0 7 2 4 2 0 0 z . 1 B 2 3 i ) O O . 10630130 . 0 5 6 J 4 0 0 . 0 0 0 0 6 5 8 3 . ? 7 7 A L l d O . 2 1 5 3 0 0 0 o 0 5 9 6 3 0 0 . 0 0 3 0 6 5 8 4 , 3 9 9 1 3 0 0 . 3 5 6 s 0 0 3 oO534000 . 0 0 0 0 6 5 8

6 1

5 . 5 2 8 2 3 0 0 6 .66;570013 7 .5127000 8 .9707000 9 1.1410000 10 1.3270000 1 1 1.5310000 12 1.7560000 13 2.iln80300 14 2,2950000 15 2.6260200 16 3.0200300 17 3.5050300

19 5 .06303GO 20 ~00000130

18 4.m1030n

D I S T U R T I B N = ,0026040

E N T R D P Y = 4.7300030

M = 4 0 X ( I )

1 .0000000 2 . 1036000 3 - 2 1 26000 A .3274500 5 .4489000 6 .5777300 7 -7 1 4 8 0 u O €3 . 8 6 15000 9 1.0190000

10 l . l R 9 0 0 0 0 1 1 1.3740000 12 1.5770300 13 1.802C)300 14 2.17530300 15 2.3390000 16 2.6680300 17 3.0590300 1 5 S.5430000 19 4.1560000 20 5.0730000

-46 17000 .ti947009 ,7366000 .BR87000

1.0530000 1.2300000 1,4240000 1,637000O 1.87500OO 2.1420000 2.44800130 2.8050000 3.2350000 3.7750000 4.5020OO0 5.6180000

Y ( I ) . O S 0 5 5 0 0 . 1567000 -2684000 ,3864000 .5113000 . 644 1000 .7856000 .9374000

1.10100(30 1.2780000 1.4710000 1.5840000 1.9200000 2.1860000 2 . 4 9 O c ) O O O 2.8450000 3.2720000 3.8080000 4.52500i)O 5 . 6210000

.OA74500 . 0 4 18600 -03661 00 . 0317200 .0271800 .0329900 - 0 191500 - 0 156600 . 0 125200 .0097300 .0072920 .0052040 ,0034680 , 002081 0 .001O460 .0004007

P ( I ) .0681600 . 06 16700 . 0 5 5 5 3 0 0 .0496600 .0441400 .0389500 .0340800 .0295400 ,0253200 . 02 14200 -01 78600 ,0146100 .0116900 ,0090980

,0048830 ,0032630

.0009973

. 0 0 0 3 8 3 0

.006a280

oOOl9680

.0000659 ,0000655 ,0000658 ,0000658 ,0000658 .0000658 .COO0659 .0000659 ,0000659 ,0000660 .0000661 ,0000662 .OO00664 ,0000669 . 003068 1 .OD00789

D ( 1 ) ,0000609 .0000609 .0000609 ,0000609 .0030609 .0030609 . 0 0 0 0 6 0 9 . 000061 0 . 0000610 .0000610 ,0000613 . 00006 10 .0000611 .0000611 .0000612 . 0000613 . O c ) O O 6 1 5 .0000619 .0000630 .0000776

rlISTSRTIBN = . 0224810

62

1 2 3 4 5 6 7 8 Y

111 f l 1 2 1 3 1 4 15 I b 17 IS 1 (2 23 2 1 2% 2 3 24 2 5 2 h 2 7 4 d h 9 33 31 J2

63

M = 128

X ( 1 ) 1 .oaoooau 2 0 03 1851)O 3 e 0 6 4 1900 4 00978300 5 0 13040119 6 0 1643080 7 0 1987000 8 02337000 9 e 2 6 9 3 0 0 0

10 03055000 11 03423@00 12 03799000 13 04180000 l a 0 4 5 6 6 U O O 15 e 4 9 6 A O L 3 0 $6 r5567000 f 7 057780tJO $ 8 e6146000 19 e6625600 20 0 7f162UCI0 2 1 07!308000 22 r7863CJUO 23 08428000 2 4 08904000 25 09390000 26 e 9 8 8 6 O C l R 27 1 0 ~ 4 ~ ~ o ~ ~ 28 100920QUO 29 I 1460OOO 30 l"2QlLrOljG 31 10257r3c~!20 32 J 0 3 1 5 0 0 0 0 33 10375OOO0 34 1 . 4 x o o a o 35 1 4 9 9 Q U U O 36 1 0 5 6 4 0 O O O 37 1 o 6 3 1 0 O O G 38 lo7010005 39 lr7720000 40 i *84600WO 41 1e923OOUO

P ( I ) 00220200 002 13700 e 0 2 0 7 2 0 0 en200600 o G 5 9 4 6 0 0

0188400 oOlA2300 00176400

CJ 1.7O5UO 00164700 0 C) 1 59 1 00 .Ol53500 e0148000 0 0 I42680 00137400 00132200 0 (3 1 2 7 1 0 0 e0122200 e0117300 001 12500 e0107800 e 01 03200 oC11396770 00094390 0110901 10 00085930 00082850 , 0 0 7 7 8 7 0 00073980 00070200 e0066520 00062930 .0059458 o O U f i 6 0 6 0 00052770 00049590 00046500 .on43510

. o u 3 7 ~ 3 0 00040620

00035140

5ci) e0000019 .000Q010 c oc300019 00000019 0 00000 19 00000019 0 0~3000 19 .0000019 0 oocloo 19 e0000019 .000001P

,ooano19 0 D O G 0 0 19 013000919 00000019 00000019 e0000019 * 0uOno 19 00[300019 0 000ou 19 0 U O O C l O l 9

roa00019 e ouooo 19 0ciD00019 0 OOGOO 19 000000J 9 rOrlQ0019 0 00000 1 0 oaaooo19 e QOOOOl9 0 000001 Y e 0000cl19 eo000019 e 0 a O O O 19 e 0 0 G 0 (J 1 9 .00000J 0 O O U O c J 0 1 9 00000019 00000019

00aoooi9

oOb00019

,

64

4 2 200030000 4 3 2.0860000 44 2e1720000 4 5 2.2620000 4 6 2.3560000 47 2.454Uf lU0 4 0 2.5570U01) 4 9 2.6663000 50 2.700UUUO 5 1 2.9uouooa 5 2 ’3r02800BO 53 3.165U000 54 ~.w” 55 3.4660000 56 3*635001)0 57 3r8180000 5 8 40019GQOG 59 d r 2 4 1 0 0 0 Q 60 4o4680000 61 4.769U000 62 5.0930000 63 5.475UOOO 6 4 5.9440000

M - 256

1 2 3 4 5 6 7 8 9

10

65

11 e l 6 4 3 0 U U 12 18 1 .clGOCI 13 e 1987000 1 4 e 2 1 6 1 i l U 0 15 ,23370L30 16 e 25 14086 1 7 .2693OUl? IS e2873000 19 eJr35SU00 20 e 3 2 3 8 0 0 0 2 1 e3423000 2 2 .361!JOn0 23 e 3 7 5 8 C! 8 0 2 4 , 3988OC10 25 4 14GUQO 26 e 4d73021U 2 7 e4Yh9000 26 e4765UBO 29 4964OQO 40 e5165!lL:O 3 1 e 5 3 6 7 0 0 C 3 2 e 5 5 7 2 U U U 53 e 5 7 7 8 O U O 3 4 e 5 9 I! 7 U fi 0 35 e61970OcI 36 e 6 4 lOU00 37 efi6250(30 38 e6d42f l00 3 3 , 7 0 6 2 0 ~ 0 40 e 7 2 8 3 0 0 0 4 1 e 7 5 0 7 0 0 0

4 3 e79630QO 4 4 e 61 94000 4 5 e A d 2 9 O O Q 4 6 e8664000 4 7 eP8113U00 d B 9 14501JO 4 9 ,93913000 5 0 e9637OUt. l 5 1 e9888G00 32 l e 0 1 4 0 0 U U

42 ~ 7 3 4 ~ ~

66

53 lrO4OOOOO 54 1.0660000 55 1.062000O 56 1.1i90000 57 1.1A60000 58 1.173Q000 59 1.2010000 60 1,2290000 6 1 1.2570000 62 1.2860000 63 1~3250000 64 1.345000CI 65 1.3750000

6 7 1.4360000 68 1.4670000 69 lr49900UO 70 1.5310000 7 1 1.564UOUT) 72 1.5970300 7 3 1.6310000 7 4 1.6660000 75 1.7010000 7 6 1.73600C0 77 1*7720000 76 1.HL1900QO 7 9 1.9460000 B O 1.98AClOC1C 81 1.923C'~OO d 2 1.9630000 63 2.0030000 8 4 2.0440000 85 2.046UOCCI 136 2.1290000 87 2*1720OCO 86 2 * 2 1 7 L ' O C ; O 69 2 .262~?ClG0 qCj 2 3 C l d i u 0 5 Y 1 2 3 5 6 r l c l l ~ c ~

6 6 1.4050000

$ 2 2 A i j 4 (li C I cj I: $43 2. 45Li . Js l )Ll QLi 2, 5l.l~l-lL~i. I:

7

67

95 96 97 Y 8 99

100 i o 1 102 103 104 105 106

1 DH lU9 110 111 112 1 1 3 1 2 4 1 1 s 116 117’ 118 IS9 120 121 122 123 124 125 126 1 2 7 128

i n 7

68

I ’

TABLE 11.- OPTIMUM QUANTIZATION LEVEL SPACING FOR GAMMA PDF

W I T H MEAN SQUARE ERROR

U I S T U R T I B i ~ - 1 .OOoa


69

70

71

D(1) tOU33780 0 0 0 2 6 5 2 0 mOO26800 oQ027050 001327660 moo31260

D(I) e O U 2 1 1 4 0 eCJ022750 00022920 00025120 e0023620 00026690

72

73

P t I ) .2213000 0 1rl53clQfJ e0668100 -0432900 e 0 2 7 0 5 0 0 + 01 55400 e0076110 00 131300

D(1) -0009067 00009766

OUO9816 e 0 0 0 9 8 4 4 000098C)l 00009949 e 0010 120 oOO11330

7 4

D X S T W R T I O Y *0135500

ENTFBPY = 3.39Yf.1000

75

76

77

i

P ( I ) * 2 5 & 4 0 U 0

11 O 7 O m

.OS20800 ~ 0 3 7 4 6 0 0 ~0269000 .O 166100 * 0122200 oOO73780 *OU36660 0 028JJL10

.073aooo

B ( 1 ) 0005202

00004143 00004178 0004 187 0004 192

a 0004197 qOOO4205 *000422 7 e0004241 mOOQ4297 e0004702

79

80

81

M = 27

1 2 3 A

5 6 7 H 9

1 il 1 1 12 1 J 1 4

a2

83

D( OOCI I 39 1

-0001502 c OOG 1509 00001511

9001512 OOG 1513

c 000 15 13 @OO 15 14

. [ ~ a o i 5 i 5 L 0001516 000 1518 000 1522

*(3001528 a 000 1 5 4 1 e UOG 1 7 3 4

84

M = 31

1 2 3 4 5 6 7 8 9

1Li 1 1 12 13 1 4 15 1 6

85

86

87

88

M = 35

1 2 3 4 5 h 7 8 9 10 11 12 13

15 1 6 1 7 i - 5

1.4

u1sr Fi7 f 3 N = 0 iJ0.33370

Y ( 1 .oflo0000 0 0 9 9 9 8 0 0 -2251 O O f l o3689000 o5298OtlCl e 7U8 1000 e9047000

1.12200@0 1.36100013 1.627C)OXl 1.9230000

2oh370000 3.07SiJOOO 3 5920000 4.2 I6OOOO 5.0830000 6.0600000

2.2570000

U I 1 e@OO 1138 o0000912 o0000920 a0000922 oOUOO922

N l 0 0 9 2 2 a0000923 00000923 .a000923 00000924 00000924 o0800924 e0000925 oOOOO926 0000928

r0000931 e0000937 .01303280

89

I

M E 36

1 2 3 4 5 6 7 8 9

10 11 12 13 1 4 15 16 17 i a

90

91

92

M = 39

1 2 3 A 5 6 7 13 9

10 11 12 13 1 4 15 16 17 i t 3 19 20

93

1 2 3 4 5 6 7 t3 9

1 0 I 1 1% 13 1 4 15 16 1. 7 18 19 2 (1

94

M = 64

1 2 3 4 5 6 7 8 9 10 1 1 12 13 i A 15 14 17

19 211 21 22 23 24 2 5 26 27 28 29 30 31 32

i a

= . UO 10070

Y ( 1 ) . 0139400 0 0 6 6 9 5 0 0 o 1 2 9 3 0 0 0

o2736000 0 3 S 4 1 0 0 13

0530E)OQn 06270000

08360000 0 9 4 9 2 0 0 0

1 oQS9000O f e lO5OOOO 1 e 32dOOOO 10470OOQQ

10779000O lo4480000 2 .1290000 2 a 3 2 3 9 0 0 0 205310000 2 756OOOO

. i 9 ~ o a o

. 4 3 9 ~ 0 o a

0 7 2 8 7 a a o

i . t w o o a o

w m o o o o o 3.2660000 30559000Q 3 e 8H4OCIOf.l 4.24PL1000 4 64300OO 5.1410000 5.7060000 6a3950000

P(I) .0999900

. a 4 2 9 9 0 0 e 0 5 5 5 5 0 0

mO358060 oO308000 o0269800 .023wmo

.a i90500 00212900

e O f 7 0 8 0 0 e 0 1 5 3 4 0 0 e o 1 3 7 7 0 0 e0123600 e 0 1 1 0 7 O O 00098910 . Clct88 150 0 0 0 7 8 2 7 0 .013369210 mOO60890 oUO53240 a 0 0 4 6 2 5 0 .OU39850 e 0 0 3 4 0 2 0 oO028720 . 00 2 3 9 3 il o O 0 1 9 6 4 0

000 12430 OOO09489 a 0 0 0 6 9 6 8 00004 t i54 o0049820

a m 5 8 1 o

D ( I ) 0 00001 4 3 r o 0 ~ o i 5 4 r0000155 00000155 00000155

e0000155 00000155 0 0000155 * 0 0 0 0 1 5 5 rOO00155 e0000155 e0000155 r0000155 0 0000 155

0 000n155 e 0000 155

0 0 0 0 0 1 5 5 0 Ocl00155 e0000155 eOU00155 e0000155 e0000155 o0ROOS55 r0000156 cOOOOi56 0 oOnOl56 e0000556 00000156 0 0 0 0 0 2 2 9

aoao 155

. oona I 55

.ooaoim

95

M = 1 2 8

1 2 3 A 5 6 7 a 9

10 1 1 12 13 14 15 16 17

19 20 21 22 23 24 25 26 27 29 2 9 30 31 32 33 34 3 5 36

ia

X ( 1 ) .c1000000 *c)175500 .0~237no .070360C 1007000 1329GCO 1669COO

.202&0CG

.277d000 2 3 C d O C i O

o3175000 o35860 i jO 40 1 coon 04A47000 a49G7000 . 5 3 6 @ 0 0 @ .58;360130 .6326000 . 6@30000 .73A7OOC

. 6621000

.7waooo

.Eiq85000 e9562000

1.0150000 1 *07600OO 1. 1390300 1 o 2 O 3 O O O O 1 * 2 6 9 0 0 0 0 1.3370OQO 1.4O70OOO

1.5"l 1.62C"O 1 * 7 0 8 0 0 0 0 1. . 799000@

1 *47900@0

Y l I ) . cc60eoo .029020c .C5572111; . 0 8 5 0 0 0 0

1 l6400C id95000 lP43000

o22050b0 .3582000 e2473000 m3377OOC o3795000 .A225000 .4€+69000 *5125@00 *5595000 .607!30170 . 6575@00 .7085000 .7609030 .HldA000 e701 O G O

* 9 7 7 0 0 0 0 *98S4000

1 .Od5@0C)O 1.1070000 1 1 7 0 O O O O 1 J 3 6 0 0 0 C 1.303OOOO 1 ,372OOOO 1 . 4 4 2 0 0 0 0 1.51500Q!o 1 ~ 5 Y 1 f l G O G 1.6660000 1 *7CR000@ 1.8J10000

D ( I ) .@5632@0 oO375600 e0297300 . a 2 s w o o .0224500 oO2026GO 00 185200 0 170800 .fl1586CO 01 47900

o0138500 .c1130000 01 22400

.0109900 c! 1030[30 .0097A20 .UG92220 .00873AC .0082740 .0078AOC . 0 0 7 A 2 9 0 .OO70390

.006316C;

.0@59790

.nii5~00

,0066680

.00565ea oU053520 . 0 0 5 0 5 9 0 .OOA7790 .OOAEllC) ,c10d25dO a c j 0 A O O B n ,0037720 ,0035460 oCG33300

D I I ) 00000 18 c)oooo19 .0000019 .OOG0010 .000@019 .nOGOOJ9 .0000019

O O G O C 19 .oocuo19 .0000@19 0000019 .flO0@029 @OOOO 19

m 00000 19

.OO@@C19

.nooooi9

.on00019

.oaooo19 0 0 0 0 0 J 9 O@OC1O 10 .GOO0019 o O O O O G 1 9 O G O O G 19

.eocooig

. nocIoo I 9

. C O C : O G 1 9

.OOGOCl9 ooooc 19

. O C O Q C l 9

.OOOOGl9 mOOOOO19

OOOOC? 19 G O O O @ 19

oC!GCi0019 O O r 2 c 1 0 1 9 *C!OCc)Ol9

96

37 108730000 3s 1.9600000 39 2 m O ~ 9 0 0 0 0 40 20142GOCI0 41 2 - 2 3 7 0 0 0 0 42 2-5370000 A3 2-4390000 A4 2 - 5 4 6 0 0 0 0 ~5 2 .~?570000 46 2 0 7 7 3 0 0 0 0 47 2mR9300CO 48 3m0180000 49 3.15ouoco 50 302870000 51 3m43100C0 52 305830000 53 3o7d20000 54 309110G@O 55 d m O A Q D C f i 0 56 A m 2 7 9 0 0 0 0 57 6-4810000 5 8 A-6980000 5 9 ~ m ~ 3 1 i l O O O 550 5.18300UO 61 5m45700GO 62 5 m 7 5 P O O O O 63 6 n Y O O O O c 66 6.4620OOil

1.9 ldCOO@ 2oC'OAOOOO 2m0950000 2.1Y9000O 2-2860000 2.38700GO 2oA920000 2-60 ioonn 2.7 I 4noa0 2.e320000 209540000 3m0830000 3.317OOCO 3.3570000 3 - 5 0 5 0 0 0 0 3 - 6 6 0 0 0 0 0 3 0 8 2 d 0 0 0 0 30998(30ofl A.181COOO 4.3770000 4 . 5 ~ 6 0 0 o n

S . O E ; ~ ~ O G O 4 - P i O O C O O

5.3'140000 5 . h 0 0 0 0 0 0 5 m W 5 O f l O G 6.2h5000C hm659000C

-0031220 00029240 00027330 0 00255 10 - 0 0 2 3 7 7 0 0022100

0 002051 0 m0018990 * 0 0 1 7 5 4 0 00 16150 0001 4840 00013580 -0012390 -0011270 00@102@c) .0009 189 oO00823!? 00007345 -00065137 .@GO5724 mOOOA995 -000AJ19 . 0 0 0 5 6 9 5 -0Cfl3123

L l C O 2 6 0 1 . coo2129 .00017@7 oOC32950

00000019

mOGOOO19 00000019 00000019 00000 19

.0000019 00000019 0 O O O D O 19 0 00000 19 .ouooo19 00000 19 @ 0 0 0 0 1.9 oQ0000l.9 *cofl0019 .0000019 -0000019 oG000019

.0oooc19

.0000G19

.O@CC010 O E O O C 19 .c)c00c19 0 00000 1 rj *@oof l020 m O G O O 0 2 0 0 O C O O O 3 3

onooo i 9

. onono i 9

97

1 2 3 0 5 6 7 8 3

10 11 12

iJ( I 1 o O A 3 9 5 0 0 0251000

.020(36@0

.0173100

. 0 1 5 A 8 0 0 * 0 1 4 1 A O O 0130900 .@122300 -0115100 oC!06913 [? -01 03500 a03987113

!J( I ) s C G 0 0 0 0 2 - G O 0 0 0 0 2 0 r3000002 .oooooc;2 0 0 0 0 0 0 0 2 .0000002 00000002 . 0 0 @ 0 0 0 2 o O G G 0 0 0 2 . 0 0 0 0 0 0 2

G O D 0 0 0 2 mCO00002

98

.0000007

. 0 0 0 0 0 0 2 OL!O0002

.OOO00@2

. 0 0 0 0 0 0 2

0 T2000002 . 0 8 0 0 0 @ 2

0000003 . 0 0 0 0 0 0 2

.0COOO02

. 0 0 0 0 0 0 2

. O G O @ O D 2 @000002 .0000002 .ooGnoo2 .O000002 .Or300@fl2 . O @ O O O f l 2

0 0 0 0 0 0 2 .0r;O@002 .0OG13002

. O Q C ~ 0 0 2

.@ooooc2

.1)001?002

.OQ00002

.OC0@002

.00000G2 *000@GO2 .00@Qn02

.ont i0002

. O Q ~ O O O ~

. i)c,oao02

. nnoooc2

.OUI?I-ICQ~

. 0 0 @ 0 3 @ 2

. c 0 0 0 0 0 2

.Cr300002

. 0 @ 0 0 0 O 2

. 0 0 0 0 0 @ 2

.000GO@2

. O B O O @ r 2

.cacnorj2

99

55 SB 5 7 s 0 59 60 61 6 2 63 6 A 6 5 66 67 6 8 6 9 70 7 1 7 2 7 3 74 7 5 7 6 77 78 7 9 YO 8 1 H 2 8 3 C l A k! 5 '36 8 7 a 8 59 9 0 9 1 92 a 3 9 A 95 9 6

0 00370 1 0 .GO26260 00035530 0 0 0 2 4 8 IO . 0 0 2 4 1 1 0 e 0 0 2 3 4 2 0 oOO22750

0 0 2 2 100 0 0 2 1 4 5 0

o0020830 . 0 0 2 0 2 1 0 00019610 . Q 0 1 9 0 2 0

00 1 8 4 4 0 * 0 0 1 7 8 8 0 oC017320 oOO16780 00016250 . 0 0 1 5 7 3 0 oOO15230 or j014730 .OC 1 4 2 6 0 0 0 1 3 7 7 0 GO13300 o0012850 . 0 0 1 2 4 0 0 .OO11970 .0011SAO *GG11130 * @ 0 1 0 7 2 0 o001032Q .0@09935 oOOO9555 . 0 0 0 9 1 8 5 .OOO8823 0 0 0 0 8 4 6 9 oOC108124 e 0 0 0 7 7 8 8 oOOO7459

0 0 0 7 1 3 9 . 0 0 0 0 8 2 7 .COO6523

.OGOOO02 00000002 -.09@c1002 00000002 oG00fl002 00000002 00000002 00000002 .0000002 .000@002 0c000002 00000002 .OOOfl002 oG000002 0 0001?002 000fl0002 0 0 0 c 0 0 0 2 .ouoooo2 .nOoooo2 000Q0002 .OflO00@2 00000002 00000062 .OOOOOC2. 00000002 00000002 . O O C i O 0 0 2 .0000002 . 0O00002 00000002 0~@00002 .@0000@2 . 0 0 0 0 0 0 2 .@OOC002 * O O G C O O 2 00000002 . 0 0 0 0 0 0 2 000(30002 . 0Of lO0~2 . 000O002 00000002 0n000002

100

97 9 8 09

100 IC1 1G2 1G3 1 0 4 105 106 107 108 109 110 1 1 1 112 113 114 1 J. 5 115 117 1 1 8 119 123 1 2 1 122 123 124 125 126 127 128

D I S T 8 R T I b Y

- 0 0 0 6 2 2 7 .0005939 aG005658 a0005395 .oan5izc) so004862 a0004611 .000434A .ClO04132 ,0003904 a O D 0 3 6 r ? 2 a0'303458 a0003260 .oon3060 m0002866 - 0 0 0 2 6 7 9 00002499 .0602326? .0002159 .005199? o f l 0 0 l R ~ 6

0001 6 9 9 .@O91559 a 0 0 0 1 4 2 4 .Ob01295 * @ 0 0 1 1 7 4 .oc161059 s0@00950 00000847 -0On0751 00000660 .0032081!

E Q T R B P Y = 7.G650300

101

TABLE 12.- OPTIMUM QUANTIZATION LEVEL SPACING FOR NORMAL PDF

WITH MAGNITUDE DISTORTION

I X(I)

1 0.0000

M

2

4

8

- L !

Y(I) 1 I D(I)

0.6745a I 0.5000a I 0.2377a

__

1 . 0000 .3371 2 .7921 1.247

j: 8

.05511

.07733 .2859 .2141

. 0000

. lo29

.2367

.4173

.6610 ,9900

1.4340 2.0330

,0460 .1599 ,3136 .5210 .8010

1.1790 1.6890 2.3780

.04099

.05257

.06820

.08393

.09321

.08531

.05478

.02100

Distortion = .08162 Entropy = 3.8900

.001062

.001768

.003086 ,005097 .007563 .009181 .007765 .005289


102

TABLE 13.- OPTIMUM QUANTIZATION LEVEL SPACING FOR EXPONENTIAL PDF WITH MAGNITUDE DISTORTION

~~~

1 I X(I) I YO) I D (1)

.3083 .3563 ,07441

8

16


.oooo .2159 .02096 ,3996 ,1516 .01969 .9390 1.2490 .08519 .Oi478

1.6670 2.0850 .04732 .02322 Distortion = .1573 Entropy = 2.8020

.oooo

.1272

.2984

.5295

.8414 1.2620 1.8310 2.5980

.0542

.2001

.3968

.6622 1.0200 1.5040 2.1570 3.0390

.08229

.08987

.09139

.08433

.06825

.04633

.02487

.01269 Distortion = .08401 Entropy = 3.8150

,002602 .003813 .005201 .006433 .006970 .006326 .004538 .006124


103

TABLE 14.- OPTIMUM QUANTIZATION LEVEL SPACING FOR GAMMA PDF

M I

1

~ ___ .. --

X U ) I Y O ) I P(I)

0.0000 I 0.0000 I l . O O O O a

-

, 2

.2586 .oooo

.8664 .5172 2.0990 1.2150

2,9820 7

1.1430 .2265

15

,4077 .1414 .0820 ,07278

1.5440 2.2710 3.2520


.oooo

.1863

.4377

.7771 1.2350 1.8540 2.6880 3.8150

~~

.2589

.1128

.07862

.05912

.04243

.02730

.01485

.03540


D (1)

0. 2887a

.1134

.09965

.04032

.02073

.02309 ,01913

.009482

.006029

.005663

.005734

.005528

.004770

.003475

.005371


104

TABLE 15.- OPTIMUM QUANTIZATION LEVEL SPACING FOR NORMAL PDF

WITH RELATIVE DISTORTION

-

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 -_.

'!

.00237

.01065

.02455

.04333

.06866

. lo28

.1489

.2110

.2947

.4077

.5601 ,7659

1.0440 1.4190 1.9250

-.

-. .

M

1

3

7

-.

15

3 1

q- x(1)- -

1 1 0.0000 0.0000

Distortion Entropy

1 1 .3160 I .OOOO .6320 2

Distortion Entropy

- - - --_

- - _.

1 .07174 .OOOO 2 1- J::: 1 .1435 3 .5064 4 .9964

1 2 3 4 5 6 7 8

~~ _ _ _ ~ p (1)

1. OOOOa

= l . O O O o a = .0000a 1 .2480 ' 1

.3760 - = .5562 = 1.5600

.05719

.0988

.1464

.2262


.01912

.1692 .OS588

.3329 .07213 .08771 .09495 .08263

1.7980 .06347

.2510

.4433

.7029


.oooo

.00474 -01656 .03254 .05411 .0832 .1224 .1753 .2466 .3428 .4726 .6477 .8841

1.2030 1.6340 2.2160

.00189

.00330

.00554

.00748

.01009

.01358

.01823

.02437

.03235

.04235

.05406

.06583

.07354

.07031

.05089

.02713

0 . 5000a

.2480

.1541

.05719

.03522

.03000

.05763 - ~-

.01912

.01203 - ,ELL141

.01003

.009887

.009369

.007438

.008352

.001891

.001176

.001128

.001040

.001140

.001349

.001662

.002095

.002666

.003388

.004229

.005060

.005569 ,005250 .003753 .002761 _ _



105

TABLE 16.- OPTIMUM QUANTIZATION LEVEL SPACING FOR EXPONENTIAL PDF VITH RELATIVE DISTORTION

. ..

1 .05709 2 .2586 3 .7684 4

~

M

.OOOO .07754

.1142 .1144

.4030 .1782 1.1340 .1687

1

3

7

15

-

31

8

- - -

.0199

.0901

.2086

.3689

.5854

.8775 1.2720

_ -

.oooo

.0398 ,1404 .2768 .4610 .7097

1.0450 1.4990

.02775

.04594

.06792

.07551

.07825

.07396

.06180

.08274 .- .


10 11 12

16

. 0000

.00528

.01860

.03662

.06096

.09379

.1381 ,1979 .2785 .3874 .5343 .7327

1.0000 1.3620 1.8500 2.5080

.003727

.006507

.01077

.01419

.01849

.02379

.03011

.03727

.04478

.05167

.05644

.05731

.05272

.04248

.02867

.02294 . . -.


.-

D (I) 0 . 5000a

.2632

.1754

.07757

.04050

.04570

.04720

.02775

.01636

.01397

.01053

.008835

.007317

.005597

.01807 . - . ~-

_ _ .003727 .002324 .002202 .001976 .002092 .002364 .002744 .003201 .003686 .004123 .004402 .004391 .003984 .003176 .002128 .003388 - _ _

_ _ - aValue derived by direct computation, not by

the Max algorithm.

106

TABLE 17.- OPTIMUM QUANTIZATION LEVEL SPACING FOR GAL4MA P D F WITH RELATIVE DISTORTION

1 .0220 .oooo .1290 2 .1592 .0440 .1229 3 .5660 .2744 .1387 4 .8575 .1740

.1483

.0549

.04134

.05744

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

.0020

.00906

.02692

.0721

.1864

.4756 1.2070

.oooo

.0040

.01412

.03972

. lo45

.2683 ,6828

1.7310

2.7930

.03916

.02638

.03564

.05262

.07695

. l o 2 9

. lo80

.07798

.oooo

.001044

.004236

.01229

.03253

.06044

.09815

.1491

.2178

.3105

.4358

.6048

.8330 1.1410 1.5570 2.1180

Distortion = ,2743 Entropy = 3.7620

.02002

.01496

.02066

.03048

.03360

.03283

.03368

.03529

.03711

.03867

.03948

.03899 ,03671 .03233 .02598 .03920

.000522

.002640

.008263

.02241

.04648

.07929

.1236

.1834

.2642

.3732

.5203

.7189

.9870 1.3490 1.8370

.04499

.009456

.009333

.01256

.01769

.02307

.02342

.01910

.02299

.00579

.005652

.007378

.006070

.004304

.003666

.003416

.003320

.003275

.003216

.003087

.002846

.002466

.001958

.007932



107

- X - n

0 X , o = l

Figure 1.- Normal, exponential, and gamma probability distributions.

Magnitude of quantizer input

Figure 2.- The mean square, magnitude, and relative magnitude distortion measures.

10 9

0 Normal 0 Exponential 0 Gamma, M even

Gamma, M odd

+ In W e 4

3 -

I I I I I I I I I I I I 2 3 4 5 6 7 8 9 IO I I

Rate, log2 M

2 -

Figure 3.- Optimum equal spacing for the mean- square error quantizers.

9

8

7

6 0) =l

0 > W

0 c W In

p. W

-

g 5

?! L 4

c

I 1 I I I I I I I I 1 0 1 2 3 4 5 6 7 8 9 1 0 1 1

Rate, log2 M

Figure 4.- Largest representative values for the equal spacing, minimum mean-square error quantizers.

p! s U In c 0

E .. c 0

0 In

.- e L

c .- n

1.0

. I

.o

.oo

,000

.oooo

.ooooc

Figure 5.- Distortion for the equal spacing, minimum mean-square error quantizers.

111

c 0 .- L

b .o i - u) ._ 0

,001 - Magnitude distortion

Mean square distortion

.0001 I I I I I I I I 0 I 2 3 4 5 6 7 8 9 1 0 1 1

Rate, log M

Figure 6.- Mean square and magnitude distortions for the gamma input equal-spacing quantizers.

5 U in

C 0

E c 0 .- c

b n c .-

1.0

. I

.a1

001

.0001

0 Norm( 0 Exponential 0 Gamma, M even

Gamma, M odd

IO -

8 -

7-

6 -

0)

5- 0)

0 .- e

c

1 4 - e e a

0 I 2 3 4 5 6 7 8 9 1 0 Rate, log2 M

1.0 -


Gommo, M odd


Gommo, M odd

0 1 2 3 4 5 6 7 8 9 1 0 1 1 Rate, log2M

.OOl

Figure 8.- Largest representative values for the Figure 9.- Optimum equal-spacing for the magni- I-' I-' unequal-spacing, minimum mean square error tude error quantizers. W quantizers.

I

,001 1 0 Normal 0 Exponential 0 Gamma, M even only

(See figure 6 for M odd)

I I I I I I I 1 0 1 2 3 4 5 6 7 8 9 l O 1 1

Rate, log2 M

1.0 I-

0 Normal 0 Exponential 0 Gomma

.01-

.OOl0 I I I 1 I " ' I 1 2 3 4 5 6 7 8 9 1 0 1 1

Rate, log2 M

Figure 10.- Distortion for the equal spacing, Figure 11.- Optimum equal spacing for the mini- minimum magnitude error quantizers. mum relative error quantizers for M odd

only.

I .o,

. I

Q) > 0

.- t - 2 c- .01 0

0 v)

0

.- + L 4- _-

,001

.0001

0 Normal I7 Exponential 0 Gamma

1 1 1 I l l 1 I 1 I I I 2 3 4 5 6 7 8 9 IO I I

Rate, log2 M

Figure 12.- Distortion for the equal spacing, minimum relative error quantizers for M odd only.

115

I

1.0'

.I

.01

E 0 1 0- v)

c 0 0)

E ,001 .. c 0

0 v)

.- c L

+ .- n

.0001

.00001

1 1 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 1 0 1 1

Entropy, bits

.000001

Figure 13.- Mean-square distortion vs. entropy for equal-spacing quantizers.

116

. I

.01

.OOl

.ooo I

.oooo

0 Gamma

I 1 1 - 1 2 3 4 5 6 7 8

Entropy, bits

Figure 14.- Mean-square distortion vs. entropy for unequal-spacing quantizers.

117

I

7

6

5

Figure 15.

- Rate distortion bound 0 Equal spaced quantizers [7 Unequol spaced quantizers 0 Equal spaced, entropy coding A Unequal spaced, entropy coding

... Distortion, mean square

- Rate-distortion curves for normal input and mean-square distortion.

118

-Rate distortion bound (lower bound) 0 Equal spaced quantizers

Unequal spaced quantizers 0 Equal spaced, entropy coding A Unequal spaced, entropy coding

I I I ,001 .01 .I IO .OOC)I

Distortion, mean square

Figure 16.- Rate-distortion curves for exponential input and mean-square distortion.

7

6

5

I 4 N ol - 0) c

z 3

2

I

C .(

- Rate distortion bound (lower bound) \

(M 64, 128 used for unequal)

1- I 01 ,001 .01 .I 1.0

Distortion, mean square

Figure 17.- Rate-distortion curves for gamma input and mean-square distortion, M odd.

119

-Rate distortion bound (lower bound) 0 Equal spaced quantizers 0 Unequal spaced quantizers 0 Equal quantizer, entropy coded A Unequal quantizer, entropy coded

0 .01 .03 _I .3

Distortion, magnitude

Figure 18.- Rate-distortion curves for normal input and magnitude distortion.

.. Distortion, magnitude

Figure 19..- Rate-distortion curves for exponential input and magnitude distortion.

120

7

6

5

5 4 N 0 - d L

z 3

2

I

C

- Rote distortion bound (lower bound) 0 Equal spaced quantizer 0 Unequal spaced quantizer 0 Equal quantizer, entropy coded A Unequal quantizer, entropy coded

I .03

I .I

Distortion, magnitude .3 I

Figure 20.- Rate-distortion curves for gamma input and magnitude distortion, M odd.

0 0

A 0

Equal spaced quantizer Unequal spaced quantizer Equal quantizer, entropy coding Unequal quantizer, entropy coding

I I I 0

Distortion, relative

Figure 21.- Rate-distortion curves for normal input and relative distortion, M odd.

121

0 Equal spaced quantizer 0 Unequal spaced quantizer 0 Equal quantizer, entropy coded A Unequal quantizer, entropy coded

\

Figure 22.- Ra te -d i s to r t ion curves €o r exponen t i a l i n p u t and re la t ive d i s t o r - t i o n , M odd.

0 Equal spaced quantizer 0 Unequal spaced quantizer 0 Equal quantizer, entropy coded A Unequal quantizer, entropy coded

Figure

122

23.- D i s t o r t i o n curves €or

I _I

Distortion,

gamma

I .3 I

re lot ive

i n p u t and re la t ive d i s t o r t i o n , M odd.

NASA-Langley, 1977 A-6714

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Minimum Distortion Quantizers

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