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A generalization of the predictable degree property to rational convolutional encodingmatrices

Johannesson, Rolf; Wan, Zhe-Xian

Published in:[Host publication title missing]

DOI:10.1109/ISIT.1994.394954

1994

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Citation for published version (APA):Johannesson, R., & Wan, Z-X. (1994). A generalization of the predictable degree property to rationalconvolutional encoding matrices. In [Host publication title missing] (pp. 17)https://doi.org/10.1109/ISIT.1994.394954

Total number of authors:2

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A Generalization of the Predictable Degree Property to Rational Convolutional Encoding Matrices'

Rolf Johannesson, Zhe-xian Wan Dept. of Information Theory, Lund University, P.O. Box 118, S-221 00 LUND, Sweden.

Abstract - The predictable degree property was in- troduced by Forney [l] for polynomial convolutional encoding matrices. In this paper two generalizations to rational convolutional encoding matrices are dis- cussed.

I . INTRODUCTION The predictable degree property, introduced by Forney [l], is a useful analytic tool when we s tudy the structural properties of convolutional encoding matrices.

Let G ( D ) be a rate R = b /c binary polynomial encoding matrix with U, as the constraint length of the i-th row. For any polynomial input g ( D ) the output v ( D ) = g ( D ) G ( D ) is also polynomial. We have

Definition 1 A polynomial encoding matrix G ( D ) is said t o have the predictable degree property if for all polynomial inputs g ( D ) we have equality in (1).

Let [G(D)]t, be the (O,l)-matrix with 1 in the position ( i , j ) where deg g V ( D ) = U, and 0, otherwise. Then we have

Theorem 1 Let G ( D ) be a polynomial encoding matriz . Then G ( D ) has the predictable degree property if and only i f [G(D)]h is o f f u l l rank.

Since a basic encoding matrix is minimal-basic if and only if [C(D)]h is of full rank ([l] [4]) we have the following theorem which is due t o Forney [l]:

Theorem 2 Let G ( D ) be a basic encoding ma t r i r . Then G ( D ) has the predictable degree property if and only af it IS

minimal-basic.

In [4] we gave a n example of a basic encoding matrix that is minimal but not minimal-basic. T h a t minimal encoding matrix does no t have the predictable degree property.

11. THE PREDICTABLE DEGREE PROPERTY FOR

Let g ( D ) = ( g i ( D ) , . . . , g c ( D ) ) , where g i ( D ) , . . . , g c ( D ) E F z ( D ) . Denote by

RATIONAL ENCODING MATRICES

P' = ( p ( D ) E &[D) 1 p ( D ) is irreducible) U { D - I ) . (2 )

For any p E P' we define

ep(g(D)) = min{ep(gl(D)), ' ' 9 e p ( g c ( D ) ) ) , ( 3 )

For any rational input u ( D ) the output g ( D ) is also ratio- where e p ( g , ( D ) ) is an exponential valuation of g , ( D ) [?I [3].

nal. We have

Deflnition 2 A rational encoding matrix G ( D ) is said t o have the predictable degree property if for p = D-' and all rational inputs E(D) we have equality in (4).

As a counter- par t t o [G(D)]t, for polynomial encoding matrices, for any p E P' weintroduce the b x c matrix [G(D)]h(p) t o be ama t r ix whose element in the position ( i , ] ) is equal t o the coefficient of the lowest term of g , , (D) , written as a Laurent series of p , if e p ( g v ( D ) ) = e p ( g , ( D ) ) , and equal to 0. otherwise.

Let G(D) be a rational encoding matrix.

Then we can prove

Theorem 3 Let G ( D ) be a rational encoding matriz . Then G ( D ) has the predictable degree property if and only if [G(D)]h(D-') has full m n k .

1 1 1 . THE PREDICTABLE E X P O N E N T I A L VALUATION PROPERTY

Deflnition 3 A rational encoding ma t r i s G ( D ) is said to have the predictable crponcntial t d u n t t o n property if we have equality in (4) for all p E P'

Theorem 4 Let G( D ) be a rattonnl encoding matriz . Then C( D ) has the predictable erponenttol ruluation property if and only rf[G(D)]h(p) mod p has f u l l rank f o r all p E P'.

A rational encoding matrix is said to he canonicalif it can he realized with a minimal numher of dclav elements in controller canonical form [5].

Theorem 5 Let G ( D ) be a rntionnl encoding matrir and as- sume that e p ( x t ( D ) ) < 0 , 1 < 7 < h . f o r all p E P * . Then G( D ) has the predictable erpontntinl tnltiotton property If and only if G( D ) is canonical.

T h e predictable exponmtial valuation property is no t equiva- lent to being canonical.

RE FE REN c ES [l] Forney, G.D.. Jr. ( 1 9 7 0 ) . Convolutinnalcodes I: Algebraicstruc-

ture. IEEE Trans. Inform. T h e o r y . IT-l6:i20-738.

[Z] Jacobson, N . ( 1 9 8 9 ) , Bas ic A i g f b r a 11. 2nd ed. Freeman. New York.

[3] Fomey, G.D., Jr. ( 1 9 9 1 ) . Algebraic structure of convolutional codes. and algebraic sys tem theory. In M a t h e m a t t c a l System Theory, A.C. Antoolas, Ed., Springer-Verlag, Berlin, 527-558.

[4] Johannesson, R. and Wan, Z. ( 1 9 9 3 ) , A linear algebraapproach to minimal convolutional encoders, IEEE Trans. Inform. The- ory, IT-39:1219-1233.

[ 5 ] Johannesson, R. and Wan, 2. ( 1 9 9 4 ) . On canonical encoding matrices and the generalizedconstraint lengths of convolutional codes. To appear in a book published by Kluwer on the occasion of J.L. Massey's 60ih birthday.

'This work was supported 111 part by lhe Swedish Research Council for Engineering Sciences under Grant 91-91 .

0 - 7803-2015-8/94/$4.00 01994 IEEE - 1 7 -