Information Theory, IEEE Transactions on - Semantic Scholar

IEEE TRANSACTIONSON INFORMATIONTHEORY,VOL. 40,NO. 4, JULY 1994

1068

Algebraic Aspects of Two-Dimensional Convolutional Codes Ettore Fomasini and Maria Elena Valcher Abstnrct-Two-dimensional (2-D) codes are introduced as lin- Laurent polynomial encoders that produce B, and to find ear shift-invariant spaces of admissible signals on the discrete among them the most efficient ones. plane. Convolutional and, in particular, basic codes are characThe second major development is the introduction of terized both in ter” of their inte-1 properties and by means of their input-output representations. Tbe algebraic structure of 2-D finite memory systems 1121-[14], which constitute the the class of all encoders that correspond to a given co~~volu- natural state-model for realizing polynomial transfer mationat code is investigated and the possibftity of obtaining 2-D trices in two indeterminates and, therefore, for impledecoders, free from catastrophic errors,‘as well as efficient menting a code using digital hardware. syndrome decoders is considered. Some aspects of the state Seeking to make a contribution to the evolutionary space implementation of 2-D encoders and decoders via (finite trend described above, this paper outlines an algebraic memory) 2-D system are discussed. Index Terms-Encoders and decoders of 2-D sequences, dual codes, behaviors, 2-D state models, inverse systems.

1. INTRODUCITON

T

HE algebraic theory of one-dimensional (1-D) convolutional codes was originated by a noteworthy series of papers G. D. Fomey, Jr., published in the early 1970’s [1]-[3]. Employing the same polynomial matrix techniques utilized in research on multivariable linear systems, Forney laid on firm foundations the notions of equivalence, minimality, and duality of convolutional encoders and showed how one could apply the state-space realization methods for implementing a code in a transmission chain. Recently, the extension of the above techniques to polynomial matrices in two variables [41-[7], guaranteed a fairly good understanding of their algebraic properties and made possible two significant advances in 2-D signal modeling and realization, which seem very promising for applications in multidimensional data coding. The first such development is the behavioral approach, introduced by J. C. Willems and P. Rocha [S]-[ll] in the description of the admissible 2-D system trajectories. This approach, indeed, allows one to investigate the recursive structure of the codes without making any a priori assumption on the direction of the recursion and, consequently, on the specific kind of causality to which the encoding process refers. Moreover, once a convolutional code @? has been selected on the basis of some internal requirements (such as the reliability of the transmitted message, the distance between two distinct codewords, etc.), it is possible to provide a complete description of all Manuscript received January 11, 1993; revised October 22, 1993. The authors are with the Department of Electronics and Computer Science, University of Padova, 35131 Padova, Italy. IEEE Log Number 9403850. 1

theory of 2-D convolutional codes, which encompasses both a behavioral approach to the internal structure of the codes and a state-space procedure for synthesizing 2-D encoders and decoders. For these reasons, Willems theory constitutes the natural framework of our treatment, and the results of P. Kocha [9] provide a natural format for the analysis of many distinguishing features of 2-D convolutional codes. In the first portion of the paper, 2-D convolutional codes are introduced as modules of doubly indexed sequences. Several connections with the submodule of finite codewords are discussed, thus providing different characterizations of the convolutional property and a complete classification of all equivalent encoders. The next part deals with 2-D basic codes and injective encoders. Unlike the 1-D case, a 2-D convolutional code needs not admit an injective encoder. So “good” codes constitute only a proper subclass of the convolutional ones, and characterizing such a class requires the introduction of the notions of extendability and left zero-prime encoders. Most of the concepts introduced in the previous parts are revisited in the section devoted to the notion of duality. The different point of view adopted therein finds a very natural application in the synthesis of 2-D syndrome decoders. In the last section we concentrate on some aspects of the realization problem, considering finite memory 2-D systems as candidates for its solution. The quarter plane causality that underlies the state updating of these models requires the introduction of some restrictions on the support of the information signals to be encoded, and one must cope with standard polynomials, instead of Laurent polynomials, in representing encoders and decoders. Finally, in order to reduce the computational effort involved in designing the transmission chain, we investigate the possibility of realizing 2-D decoders as inverse state models of the corresponding encoders.

0018-9448/94$04.00 0 1994 IEEE

Authorized licensed use limited to: UNIVERSIDADE DE AVEIRO. Downloaded on February 4, 2009 at 16:04 from IEEE Xplore. Restrictions apply.

1069

FORNASINI AND VALCHER ALGEBRAIC ASPECTS OF 2-D CONVOLUTIONAL CODES

Due to the intrinsic complexity of the subject, some results still have a preliminary character and some topics remain rather unexplored. Nevertheless, it is hoped that the main features of the theory have been covered, and some directions for future developments are broadly visible from our exposition. In particular, looking to the future, a detailed analysis of the distance properties of 2-D behaviors would constitute an important goal in the subsequent development of the theory. As error-detecting and error-correcting features are likely related to the distance between the d e w o r d s , these properties provide important criteria when 2-D behaviors are used as errorcorrecting codes on a noisy channel. On the other hand, when considering the implementation of 2-D encoders and decoders, it would clearly be desirable to relate the properties of a 2-D polynomial matrix to the dimension of its minimal state-space realizations. This could eventually lead to an expression of the necessary conditions for obtaining an optimal encoder for a given code 8 as a constraint on the polynomial structure of the encoder itself. At the present time, however, little is known concerning the structure of 2-D minimal realizations and, as a c6nsequence, there is no way to single out, among the equivalent e n d e r s of a given 2-D code, those that exhibit the most economical realizations.

power series (2.1). The context will always make clear which object we are referring to. Sometimes, mostly when a power series v is obtained as the (Cauchy) product of a series and a polynomial, it will be useful to denote the coefficient of zi.4 in v as (v, zfzi). The main advantage in using formal power series is that many linear operators on can be represented by appropriate matrices, with elements in F,:= F[zl, z2,z;', z;'], the ring of 2-D Laurent polynomials ( L - polynomials). This way, several fundamental operator properties find an immediate counterpart in terms of the structure of the corresponding matrices and, in particular, of their factors. Definition: A matrix G(z,, z,) 0 0

E

ex"is

F*-unimodular, if k = n and det G is a unit in F+; left factor prime (k'FP), if for every factorization G = TC, with T E k, T is F+-unimodular; left zero prime (LZP), if the ideal generated by the maximal order minors of G is the ring 9,itself.

ex

Introducing a convolutional structure on 'ii? requires that we endow the set of its sequences with some closure properties, which constitute the mathematical formalization of very natural constraints of regularity. The most common requirements on 8 are linearity and shift invariance. 11.2-D CONVOLUTIONAL CODES AND THEIR a) Linearity: If w1and w2belong to 8,then a w l + pw2 ENCODERS A 2-D code 8 of length n over a finite field F can be belongs to S for every a and p in F. b) Shiji Invariance: w E 8 implies that v = z ~ z ~E 'w 27, viewed as a set of sequences indexed on the discrete plane Vh, k E Z, i.e. 8 is invariant with respect to the shifts in Z X Z and taking values in F". Thus, denoting the sequence space (FnIzxzas it follows that 8 is a subset Z x Z along the coordinate axes. of is naturally As the set of formal power series In 1-D coding theory, the natural order of Z is usually endowed with a module structure with respect to F+, associated with the time ordering and, therefore, with the codes that satisfy properties a) and b) can be charactersequential structure of the data flow. This motivates the ized as F+-submodules of They will be called admissi- , habit of considering 1-D codewords with left compact ble codes. support, and to represent them [151 as vectors with com* is an admissible Example I : Every submodule '27 of F ponents in the field F((z)) of formal power series with left code. Since F+ is an ST,-Noetherian module [161, 8 is compact support. When encoding two-dimensional data, there is no natu- finitely generated, i.e., there exists a finite set of row ral notion of causality inducing a particular ordering in vectors g,,g,,.--,g, in P*such that Z X Z and, consequently, some a pri~rirestrictions on the supports of the sequences in '27. So, adopting this point of 8 = ~ a , g l , aEF+ I = {aG,a E$} =: Im, G view, we will, in general, assume that the supports of the i=l elements of the code could extend indefinitely in all directions of the discrete plane. Special attention, how- where G denotes the polynomial matrix ever, will be given to the class of codes whose elements G = col{g1,g2,***,g,}. have finite supports and to the possibility of characterizing complete codes as the duals of the above class. Example 2: A sequence w has "past compact support" In the sequel, it will be convenient to represent the if, for every (1, m ) E Z X Z, the corresponding past cone signals of and, hence, the codewords of g,via formal { ( i ,j ) : i I I , j I m} intersects the support of w in a finite power series, by associating any sequence { w ( i , j ) } with number of points. the series The set W(i,j)ZiZ4. (2.1) i,J€Z '27 = {w E :supp (w)past compact} To avoid cumbersome notations, we will adopt the symbol w for denoting both the sequence and the associated is an admissible code.

=.

c,

c

=.

1

("

=


IEEE TRANSACTIONSON INFORMATION THEORY, VOL.40,NO. 4, JULY 1994

1070

Example 3: Let M , , M2 E F v x ” constitute a pair of commuting invertible matrices and let K be in F””. The set

B= w E C : w =

(

I

xM~MJKzjz~,xEF” t ,j E Z

is an admissible code. Moreover its dimension, as an F-vector space, is finite. It can be shown that all finitedimensional admissible codes have the above structure [IT]. When testing whether a sequence w belongs to a code 8 that includes codewordswith infinite support, the possibility of resorting to a finite set of autoregressive equations, applied at every point of Z X Z , constitutes a very favorable situation. Actually, in this case, we can recognize a codeword by using only a finite set of samples w ( i , j ) at each step of the testing procedure. Such a possibility clearly corresponds [71 to the existence of an L-polynomial matrix H T ( z 1 z,), , such that

B = ker W T ( z 12,) ,

:=

{w E

=

:w H T = 0 ) (2.2)

and it can be restated as a closure property of the code, as follows. c) Completeness: Let Pl CY, cP3,-**, be a sequence of nested subsets such that U , E NPm = Z X Z and let w E=. Then w E 8 if (and only if) for every nonnegative integer m there exists v, E B such that = WlY,

V,lYm

(2.3)

An equivalent way of stating the above property is the

following. Introduce in Z by assuming

X

Z a distance function d(.,

1,

- hl + I j - kl, W i ,j ) , ( h , k) E Z x Z and define distance A(-, ) between two sequences v and w in as follows: d[(i, j ) , ( h ,k)l

= li

(2.4) Then, X becomes a metric space, and property c) is exactly the completeness of 8 in the topology induced by the distance function A. This means that if a sequence v1,v2,***, in B converges to w, then w E 8. Proposition 2.1: Let 8G be an admissible code. Then 8 is a complete code, i.e., it satisfies condition c) if and only if (2.2) holds. Proof- The proposition above has been proved by P. Rocha in [7]. For an alternative proof see [MI. 0

As an immediate corollary, we have that properties a1-c) are equivalent to the possibility of representing 8 as the kernel of an L-polynomial matrix. Remark I: The codes considered in Examples 1 and 2 are not complete. On the other hand the code of Example

3 is complete, as it can be proved that it is the kernel of a polynomial matrix [18]. Given a finite window 9and a set of samples, obtained by restricting to 9a (possibly infinite) codeword w, it is interesting to investigate whether the data set, wl9, can be completed into an appropriate finite codeword v, whose support does not “exceed by too much” 9. If so, the values a codeword w assumes on the window 9, constrain only the samples w ( i , j ) in a finite neighborhood of it or, equivalently, do not provide any information on w at points that are far enough from 9. Therefore, if no additional information on w is available, we can always assume that the partial data at our disposal come from a finite codeword. The above property can be stated as follows. d) Conzrollabdity: There exists a positive integer S such that, for every finite set 9 c Z X Z and every v1 E 8, there is a codeword v, E 8,such that (see Fig. 1) V J 9 =

v,lY

and supp(v,)

c y 8:= ((i, j ) E z x Z : d ( i , j ) , P )< SI.

A fundamental objective of coding theory is the investigation of the intrinsic structure of codes, without taking into account the way codewords are generated, and the analysis of those features with which good’codes must be endowed. These should make them as efficient as possible with respect to design requirements, such as the distance among the codewords and the noise sensitivity. Under the hypothesis that a 2-D code 8 is complete, there are several equivalent formulations of the controllability property, which concern the internal structure of the codewords set. Some of them refer to the submodule of the finite codewords,

F,:={w EZ:supp(w)finite} = 8nF+, others to the possibility of obtaining the code by a sort of “patching” of appropriate finite codewords. (dl): There exists a positive integer p such that, given WOdisjoints subsets of Z X Z, PI and P, whose distance d is greater than p, and two codewords w1 and w2 in 8, there is a codeword v E S such that wlW1 = V I P l

and wZlP2= vI9,.

This result can be rephrased as the possibility of “concatenating” two portions of distinct codewords into a single codeword, provided their supports are far enough apart [6]-[8]. (d2): There exists a finite set 9of finite support codewords with the property that w E belongs to B if and only if w is represented as a locally finite sum of some, possibly shifted, elements of 9. So every codeword of B is obtained by resorting to an appropriate “covering” of the discrete plane with codewords and shifted codewords of 9. fdJ: The code 8can be completely reconstructed from g,,the F*-module of its finite codewords, by means of a

=


~

FORNASINI AND VALCHER: ALGEBRAIC ASPECTS OF 2-D CONVOLUTIONAL CODES

ts

1071

c.

S6

..codeword vz

codeword V I

Fig. 1. Controllability of the code Q.

quence which converges to an element in As a consequence of this property, one can show [18] that (3d) -. extends to the infinite subsets of Z X Z. Set p := S. There exist two codewords v1 and v2 such that vil,Y: = wilq, i = 1,2, and supp (vi) c q s .The signal v := v1 + v, belongs to B and satisfies VIZ= vil,Y: = WJX, i = 1,2, as required. (1) * (3d,): Consider the set 9:= {[a1a - 2 , * - - , ak]G, a, E F}, whose elements are the codewords corresponding to the “atomic” input signals [a1a,,--*, a k ]E

limit operation, namely, w belongs to $9 if and only if there is a sequence wl,wz,---, in Ff,converging to w in the sense of the pointwise topology. So $9 can be viewed as the closure of the module gf,that is, as the smallest (complete) code containing gf. On the other hand, a code is naturally understood as the result of an encoding process applied to the information signals. Therefore, many concepts in coding theory are connected with the existence of an input-output transformation, whose image is the code itself. In this perspective, if the information signals are sequences in Z X Z, with values in F k for some integer k, and the code G? is linear and shift invariant, it is natural to associate the transformation with a k x n L-polynomial matrix G(z,, z , ) and represent the code as $9=ImG:={w=uG,uEe}. (2.5) As we shall see, property (2.5) is equivalent to the “internal” properties a)-d). Consequently, the convolutional nature of %,’ i.e., the possibility of generating all codewords of %? by convolving the input sequences with the matrix G of the impulse response, has an exact counterpart in terms of the internal structure of the code, which can be characterized without any reference to the encoding process. We call convolutional a complete code satisfying condition d), or equivalently a code described as in

Fk. As F is a finite field, 9is finite too. Every codeword in $9

can be written as

Since all codewords wij(zl, z,) := [ul(i,j) uz(i7j ) - * * j)]G are elements of 9, (2.6) represents w as a locally finite sum of elements and shifted elements of 9. (3d,) (1): Let 9 := {cl(zl, z 2 ) , c ~ ( z 1 Z , Z), ,cp(zl,z,)}, with ci(zl, z,) E F* i, = 1,2,..-,p. By assumption, the codewords in 8 are the elements in % which can be expressed as U&,

(2.5).

Proposition 2.2, below, formalizes the main statements where St(-, takes values in {0,1). Letting G ( z l , 2,) = c o l ( c l ( z l , z 2 ) , concerning the controllability property. c2(z1,z z ) , - ~ ~ , c p (z,)}, z l , we have that the ST,-module genProposition 2.2 (Equivalent Characterizations of Convolu- erated by the rows of G is included in 8.We aim to prove tional Codes): Let $9 c c . The following are equivalent: that uG is in $9 for every U EP*. Actually, given any 1) $9 = Im G, G E sequence of finite nested sets 2Zl cY2c -..covering Z X 2) $9 = ker H T , H T E F k X P , H T right factor prime; Z, the sequence of input signals U, defined by 3d) 8 is complete and satisfies property ( d ) ; 3d,) %? is complete and satisfies property (di),i = 1,2,3. Proofi The equivalences (1) o (2) o (3d,) have been proved by P. Rocha in [9]. An independent proof, based converges to U. By the continuity of G [MI,o,G conon the notion of duality, will be provided in Section IV. verges to uG. Since @?is complete, u,G E g,tlv * uG E For the remaining equivalences we proceed by showing $9. Consequently, Im G E 8. On the other hand, by (2.7) every codeword can be that (3d1) (3d) and ( 3 4 ) (1) (3d,). with coeffi(3d1) (3d): Take S := p and apply (3d,) to w1 := vl, expressed as the G-image of a series in w2 = 0, Y1:=9 (finite), and 2Z2 := CY’, the comple- cients in {0,1}. Therefore 8 = Im G. (1) * (3d,): Let w be in 8= Im G and w = uG. Conmentary set of Y 8 .The finite codeword v obtained in sider an L-polynomial sequence {U,,}converging to U. ( 3 4 ) is the codeword v, we are looking for. (3d) (3d1): The space is sequentially compact, Because of the continuity of G, the sequence of L-polyi.e., every sequence of elements in contains a subse- nomial codewords {w,} := {u,G} converges to w = uG.

ex”;

=$

-

-

c

c


1072

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40,NO. 4, JULY 1994

(3d,) =, (1): The finite codewords of P constitute an that (A2) holds. We therefore have Y+-module gf, which is finitely generated as a submodule WX = uGX = u h ( z , ) , W Y= uGY = u k ( z , ) of F* Let . g, ET,, v = 1,2,..-,k, constitute a set of generators for gf and G := col{g],g2,**',gk}. Clearly, and, by multiplying the first equation by k(z,) and the second one by h(z,), we get wXk(2,) = wYh(z,). Since ImGcg. h ( z , ) and k ( z , ) are coprime, h(z,) is a common factor of To prove the opposite inclusion, consider any codeword w in 8 and a sequence of finite codewords, w,, E Ff, all the components of wX,that is there exists an L-polyv = 1,2,---,converging to w. Since all finite codewords in nomial vector p(zl, z,) E @* such that WX = ph(z,). @ are linear combinations over Siof the rows of G , Thus, h(z,)w = h(z,XuG) = h(z,XpG) and, consethere is a sequence {U,,),U, E such that {u,G) = {w,}. quently, h(z,)(w - p G ) = 0 . (2.8) By the sequential compactness of [181, we can extract from {U,) a subsequence {U,} that converges to some U in Since all entries in (2.8) are in S,, it follows that w = pG, &.So, by the continuity of the operator G, we have and therefore (Im G)f is included in Im G. ii) i): Assume that G is not LFP. As a consew.= j lim U G U,,, G = uG. 0 +m( ) = ( j lim +m quence of Corollary A.2 in the Appendix, G can be is LFP and T E It can be easily realized that, while a k X n polynomial rewritten as G = TG, where E is a full rank nonunimodular matrix. Thus there matrix G(z,, 2,) uniquely identifies the convolutional code flkxk Im G = {w = u G : u E&), the converse does not hold, exists a vector p(zl, 2,) E sk, such that equation as the same code S can be described as the image of uT=p (2.9) different L-polynomial matrices. Two matrices G,(z,,2,) has no solution in 9,. However, as T is a full-rank square and G,(z,, z,), with elements in Fiand the same nummatrix, (2.9) admits a unique solution in F ( z , , z , ) given ber of columns, are equiualent encoders if Im G I = Im G,. b Since each convolutional code biuniquely corresponds to Y Adj T a class of equivalent encoders, the natural problems arise U = pT-' = p ziFto investigate what conditions guarantee that two matrices belong to the same class and to find out in every equiva- The entries of U can be viewed as series in and hence lence class the most efficient encoders. U is an infinite input sequence in We aim to show that To answer these questions we need some preliminary w = uG is a finite codeword that does not belong to results, concerning the relationships between the F+Im * G. Actually, w = uG = (uT)c = is finite. module On the other hand, assume that there is a finite input seque_nce v such that w = vG. Then we have w = 1 G = Im, G := {uG:u E$} (vT)G, which implies ( p - V T )=~0. Since Im G is a and the F*-submodule gf of the finite codewords of free module, we have vT = p, and (2.9) has an L-poly8 = Im G , that will be also denoted as (Im GIf. nomial solution, a contradiction. Lemma 2.3 [171: Let z 2 ) be a k X n LFP L-polyi) e iii): ker G := {U E & : uG = 0) can be viewed as nomial matrix. Then an autoregressive description of a complete behavior. It i) the F+-module Im * is free; has been proved [9], [17] that a necessary and sufficient ii) if T ( z , ,2,) is a k' X k L - polynomial matrix, of condition for a behavior being finite dimensional is that G rank k over F(z1,z2), then (ImTGIf coincides with is / F P . 0 We are now in a position for introducing the basic (Im GIf U results about the equivalence of two encoders. Lemma 2.4: Let G ( z 1 2,,) be a k X n L-polynomial Proposition 2.5 (EquivalentEncoders): Let G,(z,,z , ) and matrix, with full row rank over F(zl,z2). The following G 2 ( z , , z 2 )be two matrices with elements in F, and properties are equivalent. dimensions k, X n and k , X n, respectively. G , and G , i) G ( z , ,2,) is LFP; ii) the module (ImG& of the finite codewords in are equivalent encoders if and only if i) under the assumption that both G , ( z , , z , ) and Im G coincides with Im G, i.e., every finite codeword w G2(z1,2,) are LFP, we have of Im G is the image of a finite input sequence; k , = k , and G , ( z , ,2,) = U ( z , ,z 2 ) G 1 ( z ,z, 2 ) , iii) ker G , the F-vector space of all information sequences in & that produce the zero codeword, is finite with U(z1,2, unimodular dimensional. ii) under the assumption that G 1 ( z 1 , z 2is) LFP, Proof i) * ii): Clearly Im + G is included in (Im G ) f , there is a k, X k, full-column-rank L-polynomial matrix, since every linear combination in F+ - of the rows of G is P,(z,, z 2 ) ,such that G , = P,G,; a finite codeword of S. iii) in the general case, there exist two full column rank We aim to prove the opposite inclusion. The LFP condition implies C51 the existence of two matrices L-polynomial matrices Pl(zl,2,) and P2(z1,z2), of suitX(z,,z,) and Y ( z , , z 2 ) with , elements in FA, and two able dimensions, such that

e,

)

J'

-

c ex"

e.

z,

pc

,

&,,

,


1

1073


with T eXk,

Prooj? Given a full-column-rank matrix P E the map P :&' + :U * U P is onto. Consequently the convolutional codes Im G and Im PG coincide for any G E @*xn. Thus (2.10) in iii), and in particular G , = UG, and G , = P,G, in i) and ii), imply that G , and G , are equivalent encoders. Conversely, assume first that G , and G, are PFP equivalent encoders. Then, by property (d,) and Lemma 2.4 we have Im G , = (Im G , ) f = (Im G , ) f = Im * G , . (2.11)

As each row of G , (of G , ) is an F,-linear combination of the rows of G , (of G,), there exist L-polynomial matrices P, and P, such that P,G, = G, and P2G2= G , . We have then G , = P,P,G, and G , = P,P,G,. The PFP property yields = I k , , p 2 p 1 = Ik, showing that k, = k, and both P , and P, are unimodular. Assume next that G , and G , are equivalent encoders, and only G , is LFP. By Corollary A.2, G, can be factorized as G , = T c , , where is PFP and T full-column rank. Thus, Im G , = Im and, consequently, G , and are / F P equivalent encoders. It follows that 5, = UG,, for a suitable unimodular matrix U, and, letting P, = TU, one gets G , = P,G,, as required. Finally, case iii), suppose that G I and G , are equivalent encoders, and neither G , nor-G, are / F P . Clearly, we have G, = Cc,,i = 1,2, with T, full-column-rank and E, / F P matrices. Moreover, = UCz for some S*-unimodular matrix U.So, letting T, = T,U and T2 = T, we get G , = T I C 2 , G , = T,G, Consider any pair of L-polynomial matrices X, and X, with the property that A ; := [Ti X i ] ,i = 1,2, is a nonsingular k , X k, L-polynomial matrix, and assume k , 2 k,. Then we have

e, e,

cl

e,

full-column ran@ and [FP. By the above proposition, z , ) is a PFP encoder of 8.

&,,

111. INJECTIVITY AND DECODING The purpose of an encoding scheme is to associate every input sequence with a specific codeword that preserves the information message, but is less sensitive to the action of noise. So, in order to make possible the retrieval of the original message at the decoding stage, it is quite obvious that every codeword has to be the image of a unique information sequence, which amounts to assuming that the map from the input space &' to the codewords space 5F is injective. As proved in the previous section, a convolutional code can be expressed as the image or the kernel of appropriate Laurent polynomial matrices. The following proposition shows that the injectivity requirements reduces to a zero primeness condition on the above matrices. This entails some relevant consequences on both the internal properties of the code 8 and the classes of encoders and decoders of 8. Proposition 3.1 (Znjectiue Encoders): Let 8 be a convolutional code of length n and rank k. The following are equivalent: i) '8 admits an injective encoder; ii) 8 = Im G(z,, z,), G E [ZP; iii) '8 = ker H T ( z , ,z,), H T E qx(n-k) rZP. Proofi) ii): If G is LZP, by Proposition A 3 there exists an n X k matrix, K ( z 1 , z 2 ) ,with elements in S,, such that GK = Zk. So, UG = 0 implies 0 = (uG)K = u(GK) = U , which means that G defines an injective input-output map. Conversely, we aim to prove that, if G is not [ZP, it is not an injective encoder. If rank G < k, the result is trivial, so we confine ourselves to the case rank G = k. Consider first the case when G is not /FP. B y Corollary A.2, there exist two L-polynomialmatrices, G ( z , ,z,), k X n and PFP, and T ( z , ,z,), k x k with det T f 0 and not a unit in .F+ such , that G = TC.

-

eXn

0 If det T is not a unit in Nz1)[z2, z,'], in the (renormalized) Hermite form [2], [5] of T with respect to Introduce the following left matrix fraction description Hzl, ~ ; ~ 1 [ 2 , ,zi'l, we have Sk,,2,) = L(z,, zz)T(zl,z,), is upper triangular, and L E has where S E pkx determinant in Nz,,z;']. As det S = det T det L, the assumption on det T implies that at least one diagonal element in S is a nonunit with N full-column rank, and rewrite A,L-' as a left polynomial in NzlNz2,zil].Let Sli(zl,2,) be the first MFD, A , L - ' = Q - ' B . AS G , = A , L - ' N G , = element with this property, and consider uI(z1,z2),a Q-lBNG,, we end up with QG, = BNG,, which corre- series in S, such that sponds to (2.101, upon assuming P , = Q and P, = BN. 0

ex

Remark ZZ: It is worthwhile to underline that every visii = 0 and ui det L # 0. (3.1) convolutional code 5F can be represented as the image of a / F P matrix. Actually, given any encoder G ( z , , z , ) of 5F, by Corollary A 2 we can extract a greatest left factor, Then there exists a vector v E@, with the first i - 1 entries identically zero, such that VS = 0. On the other obtaining hand VL is not zero, otherwise 0 = VLAdj L = vdet L G ( z , , z , ) = T(zl, z,)&,, z , ) would imply ui det L = 0, which contradicts (3.1). So


*

1074

IEEE TRANSACTIONS ON INFORMATIONTHEORY, VOL. 40,NO. 4, JULY 1994

vL(z,, 2,) is a nonzero element in ker T, and hence, in ker G. If det T is a unit in F(z,)[z,, 2i1], it cannot be also a unit in F(z2)[z1,z;'], otherwise det T would be a unit in 9,. So we can resort to the Hermite form of T with respect to f i z z , 2;' Hzl,z ; ' ] to prove that ker G is nontrivial. When G is LFP (but not LZP), kerG is a finite dimensional vector space [141, and G is not an injective encoder. ii) w iii): By Proposition 2.2 and Remark 11, the equivalence of ii) and iii) holds for factor prime matrices. Since for every U E&, (uG)HT = d G H T ) = 0, it follows that GHT = 0, and therefore, by Proposition A.4, G 0 is LZP if and only if H T is rZP. Given a convolutional code g = Im G, of length n and rank k, it is natural to wonder whether it admits injective encoders. Clearly injective (i.e., LZP) encoders, if any, have to be looked for among the LFP ones. On the other hand, by Proposition 2.5, if E(z1, z 2 ) is a k X-n LFP encoder of S,any other LFP encoder is given by G = UG, U unimodular. Since the premultiplication by F,-unimodular matrices preserves the LZP property, the existence of a P Z P encoder implies that all LFP encoders are LZP. Unlike the 1-D case, left factor-primeness does not imply left zero-primeness, and examples can be given of 2-D convolutional codes devoid of injective encoders. Example 3: Let F = GF(2). It is easy to check that the following L-polynomial matrix

checks of the code in ( r , s) E Z (VHT, zaz2y) = 0, V( p, v )

X

Z if

E

( r , s)

+ supp ( H ? , (3.2)

0

is LZP, as the only input sequence producing the zero codeword is U = 0. On the contrary,

is PFP but not LZP, since all maximal order minors have a common zero in (1,l). Therefore, G, is not an injective encoder. It can be easily realized that U = [O Ci,izit4]is the unique nonzero input sequence in ker G,. According to the above discussion, we can single out among 2-D convolutional codes those that admit a LZP encoder. They will be called basic, in analogy with the 1-D case [l], and will be characterized by the possibility of extending somehow finite sequences into codewords. A complete code g,and, a fortiori, a convolutional one, is a submodule of C whose elements satisfy a finite set of autoregressive equations, the parity checks of the code. By associating each equation with an L-polynomial column vector h:(z,, z2),we have that w E 2' ? if and only if whr = 0, i = 1,2,-.-,p . So, by juxtaposing the columns hr into a matrix H T = col {hy, h;,..., hz}, we get the usual kernel representation 59 = ker H T = {w E C :w H T = 0). De$nition: A sequence v E C satisfies the parity

where ( r , s) + supp ( H T ) := { ( r + i , s + j ) : ( i , j ) E supp(HT)}. More generally, if T i s an arbitrary subset of Z X Z, v satisfies the parity checks of the code on 7,if it satisfies them in every,point ( r , s) E S; that is (VHT, zaz2y) = 0, V( p , v )

E T+supp ( H T )

(3.3)

where 9+supp(HT) := U (r,S)Eg{(r, s) + supp(HT)}. Letting H T ( z l ,z ,) = CijHJzfzi, condition (3.2) reduces to the following system of linear equations:

C

u ( p - i, v - j ) H i r = 0,

(i, j ) E supp (H')

V( p, U )

E

( r , s)

+ supp ( H T > ,

(3.4)

and hence to the system of all difference equations which involve the sample U( r, s). Analogously, v meets condition (3.3) if all difference equations involving the samples u(r, s), with ( r , s) in S;are satisfied. In Fig. 2, each dashed polygon intersecting T represents the coordinates ( p i , v - j ) of the samples which appear in a system like (3.4). Clearly, when verifying whether v satisfies the parity checks of the code on S; we take into account not only the samples on S; but also those which belong to an appropriate set 7 2 9 ; The remaining tests we have to perform, when deciding whether v is a codeword, are represented by systems of difference equations that involve only the samples of v on CT-Some of them, however, utilize again the samples on s\S; as suggested by the checkered polygon in Fig. 2. So, it could happen that the data on F satisfy the_ parity checks on S; yet no selection of the data on ?makes possible the fulfillment of the parity checks on c9; Otherwise-stated, the specific assignment of the values of v on S compromises any possibility of extending the data on 9into a legal codeword. In these situations, the natural question arises whether such an extension could be made possible by changing only the data which are "close" to the border of 9;More precisely, we wonder whether there is a positive integer 8, such that any sequence v, satisfying the parity checks of the code on 9, can be,modified into a codeword w, which coincides with v on the window 9; A positive answer is very important from the syndrome decoder point of view. Actually, when the parity checks of the code are verified in 9,we can assume the restriction v p as correct and, whenever the parity checks fail in some point ( r , s) 9, we have to modify only the values ofvonc9; Generally, neither the completeness assumption nor the more restrictive hypothesis that 59 is a convolutional code, imply that the code 2'7 exhibits the aforementioned features. As we shall see, these constitute the exact coun-


1075


.......... ..........

........... ........... .......... ......

......

............ ...........

other. To further highlight the strict connection between the two notions, we will also show that, for a complete code, property e) is equivalent to the following property e,), (which represents the natural counterpart of (d,), and hence is called “strong controllability” in [9]). e,) Let S = ker H T. There exists a positive integer p such that for every pair of subsets 9, and Y,of Z X 2, with d(Y,, 9’’)> 2p, and for every pair of sequences v,, v2 E C, which satisjj the parity checks of the code on -F;” and Yf respective&, a codeword w E ‘ii? exists, such that wI9,

= vl9,

and wl9, = vlY,.

.....

(3.6)

The difference between properties d) and e) might at first appear somewhat elusive, because of the mathematiFig. 2. Parity checks locations. cally involved character of their definitions. It can be best understood, however, when one looks for a coding interterpart, from an internal point of view, of the condition pretation of both properties. As a consequence of the for the existence of an injective encoder (stated in Propo- proof of Proposition 3.3, e) implies that, given any sesition 3.11, and provide an equivalent definition of 2-D quence v E F*that satisfies the parity checks of the code basic codes. The formal definition of these properties will in Y’, there is a codeword w that coincides with v in Y be assumed as a further constraint on the structure of ‘8’. and whose support does not exceed Y2*. Note that no e) Ejrtendability: Let %? = ker H’. There exists a posi- a priori information is needed, guaranteeing that the data tive integer 6 such that, for every finite subset 9’c Z X Z on the window 9’are the restriction of a legal codeword. and every v E which satisfies on 9” the parity checks This information is needed, on the contrary, in the case of the code, there exists a codeword w E F such that of simple controllability, when trying to extend the available data into a finite codeword. As in practice one wlY= VlY. (3.5) expects that only a finite set of data be available and, Lemma 3.2: Let %? = ker H T ( z , ,2 , ) be a code satisfy- anyhow, only a finite number of parity checks could be ing the extendability property. Then property e) holds for performed on them, in most cases the above information is beyond our reach. all (not necessarily finite) subsets of Z X Z. The proof of the equivalence between e) and e,) is Proofi Assume that the sequence v E z satisfies the parity checks of the code on an infinite set 9’*, and let included in the following summarizing proposition, which 9, c9, CY’ c be a sequence of finite nested sets in provides a complete picture of the connections between Z X Z such that =9 . Since v satisfies the parity basic codes, introduced from an “external” point of view checks on L?’, i = 1,2,3;.., there exists a sequence of in Proposition 3.1, and codes endowed with the extendcodewords w,,i = 1,2,3,..., such that w , l q = v l q . As ability property. Proposition 3.4 (Equivalent Characterizations of Basic C is sequentially compact, there is a subsequence {wv,} of The following are equivalent: {wJ converging to w E=. The proposition is proved by Codes): Let 8E 1) 8 =ImG, G k‘ZP; observing that 2) g = ker H T , HTE 9 ” x ( n -rZP; k) as w, E g for every j, by the completeness of %?, 3e) 8 satisfies property t k w E &; 3e,) %? satisfies property e,). since wlYv,= wvJIYvJ = v I q J for every v,, it follows Proof (I) (2): See Proposition 3.1. U that w l 9 = VIZ (2) * (3e): Assume that H T EST"+'("-^) is rZP and w E @+n-k)xn is a polynomial left inverse of H .’ Proposition 3.3: Extendability Implies Controllability Let 6, = max{lil + ljl:(HT, z i z i ) # O}, 6, = max{(lil Proofi Let w E 8,with 8 a code satisfying property zfzi) # O}, and 6 = 6, + 6,. If v E C satisfies e), and consider a finite set Y. Define v E C as follows: + I jl: (W, the parity checks of the code on a finite set 9’ c Z X Z, the series a = v H T satisfies ( a , zp, z,Y) = 0 for every for every ( h , k ) €9’ v ( h ,k ) = ( p , v) € Y 8 + supp(HT). Since supp(HT) ~ 9 ’ 1 , we w ( h,k ), otherwise have where 6 is like in e). 9 ’ 2 EY’ + supp(HT) c C[supp(a)l. Since v satisfies the parity checks in Y = Y U C ( Y 2 ’ ) , by the previous lemma there is a codeword V E %? such Introduce next the series x = aW. As supp (x) c supp (a) + supp (W) and supp (a) E C9’’2, it follows that that F l s = vcS; Clearly (w - VI is in 8, and (w - V)l9’= w l 9 . Moreover, (w - V)IC(9’**) = 0, implies that supp(w =C Y supp(x) E c9’2 + supp(W) E 0 - V) cY2*, so B satisfies property d). Extendability and controllability are very close to each and therefore x l Y = 0.

e,

,.e-,

e.

€exn

-



1076

Finally, let w := v - x. As a consequence of x H T = a W T= a = vHT,we have wHT = (v - x)HT = 0, which implies w E 8.Moreover w l 9 = (v - x ) l 9 = VIZ Oe) * (2): k t 0 satisfy the extension property. Then 8 is a convolutional code, and can be described as 8= kerHT, with H T rFP. To prove that H T is rZP, by Proposition A5 it is sufficient to show that the equation x H T= a

(3.7)

Fig. 3. Set inclusions in Proposition 3.4.

admits an L-polynomial solution for all vectors a in Fk-k. As HT has full-column rank over Hz,,z2),(3.7) admits a solution v E=. To complete the proof we will show Indeed, if G is a left zero prime L-polynomial matrix, it that there is a codeword w, differing from v on a finite set admits at least one L-polynomialright inverse G-'(z,, z 2 ) , 9:Actually, in this case we have and therefore every finite codeword e in i 9 is generated by one (and only one) finite input sequence U, = eG-'. An (v - w ) = a, ~ supp(v ~ - w) G F interesting consequence of the above reasoning is that, and, consequently, (v - w) is an L-polynomial solution of when achieving the injectivity of a convolutional encoder, one also guarantees the existence of a polynomial de(3.7). By assumption, v satisfies the parity checks of 8 on the coder, thus ruling out the possibility of catastrophic errors. set Remark W: As G defines an injective input-output map, X = { ( r ,S) : ( ( r , s) + supp(HT)) n supp(a) = 0 ) a decoder represented by a rational right inverse G-'(z1, z 2 )of G would associate to any finite codeword e whose complement is a finite set. Clearly, (see Fig. 3), a set 9 exists such that C Y is in 8, the same input sequence U, as the polynomial finite and Y* c&4As v satisfies the parity checks of the decoder. In this case, however, expression eG-' would be code in 9', by 3e) there is a codeword w such that meaningless when e is not polynomial, and we should w l 9 = v l 9 . Therefore v and w differ only on a (finite) restrict our attention to codewords whose supports do not extend to the whole discrete plane. subset of C y . (3e) (3e,): The proof can be performed along the same lines followed in showing the equivalence (3d) 0 (3d,) in Proposition 2.2. 0 Once an encoder converts an information message into the corresponding codeword, the encoded message w is transmitted over a noisy channel. Generally, the received sequence r not only differs from the original message w, but also does not belong to the code 8. So we need to project r on the space of codewords, in order to find out the best approximation to r in 8, namely a codeword i whose distance from r is minimal. Often i differs from the transmitted codeword w by a nonzero reconstruction error

e=i-w (3.8) which is a codeword too. Having no possibility of finding e, all we can do is to reconstruct B, the input sequence corresponding to t, and assume it as an approximation of the correct information sequence. This step is performed by a decoder, namely a right inverse of the encoder matrix G(z,, z2), which produces the sequence 8, when receiving i = BG as its input. It can be easily realized from (3.8) that if there exist finite codewords generated by infinite information sequences, then a finite error e in the reconstruction of the codeword w could produce an infinite error when decoding 2 instead of w. Such catastrophic errors, however, are avoided when, to preserve the injectivity property, we confine ourselves to the class of basic encoders and, therefore, to codes that satisfy the extendability property.

IV. DUALCODESAND SYNDROME DECODERS The structure of 2-D codes, as discussed in Section 11, can be clarified further through the duality relation between finite and complete codes. When referring to a finite code G? of length n, we mean (see Example 1 of Section 11) a submodule of Sy defined as 8 = Im G = (w = uG :U E s",},where G denotes an arbitrary matrix in @+xn. On the other hand, a complete (and, in particular, a convolutional) code B of length n is defined as 9 = ker HT= {w E :wHT = 01, with H T an arbitrary matrix in F+"'. The two are dual concepts that interact in the encoding and decoding processes. In most cases, it is quite reasonable to assume that 2-D information signals are finite support, and, therefore, finite codes are easily regarded as the result of an encoding operation. Even if complete and convolutianal codes can be introduced by simply extending this point of view to infinite information signals, it is very convenient to give them a different interpretation in algebraic terms. A complete code is more naturally viewed as a family of F-valued linear functions on the space of finite sequences, via the canonical algebraic duality [171 between F+ and the space of linear functionals L(9",). So our philosophy will be to characterize a finite code 8 as the set of codewords that are in the kernel of a suitable space of linear functionals, and conversely, a (complete) dual code 9 as the set of linear parity checks necessary to decide whether a finite sequence is a legal codeword.


1077

FORNASIM A N D VALCHER ALGEBRAIC ASPECTS OF 2-D CONVOLUTIONAL CODES

The duality properties find an obvious application in the synthesis of syndrome decoders. Indeed, a complete characterization of the syndrome decoders of 8 can be achieved by resorting to a systematic analysis of the class of its dual codes. Introduce in X the following nondegenerate bilinear form:

e e

( -, . ) m : c X c

+

F

defined by (u,v), = (uvT,1) = C i , , E Z ~ (j)uT(-i, i, -j). Two vectors U E and v E are called orthogonal if (u,v>, = 0. Given any submodule A of its orthogonal complement A ' , comprises all the vectors of which are orthogonal to A. Similarly, every submodule JT of identifies an orthogonal complement M' in can be viewed as L ( c ) , the algebraic The space dual of In fact, we can associate with every v E = the linear functional on - defined by

e

c,

e.

e

e.

f"(.) = ( ,v)m

(4.1)

and, conversely, every linear functional on s;l can be represented as in (4.1) for an appropriate choice of v E E .The identification of with L ( c ) makes it possible to use some results, not valid for arbitrary pairs of dual spaces [19]. Let 8 be a finite code, described as the image of the map G:@,+F, :U uG

e

-

and consider the map GT:c+&:v*vGT.

The mappings G and GT are dual, since (uG,v), = (uGvT,1) = ( u ( v G ~ )1) ~ , = (U,vGT)k. By resorting to the well-known relations (A)

,

(Im G)'

=

ker GT (ker GT)'

=

,

Im G

we induce a bijective correspondence between finite codes of length n, represented as images of appropriate L-polynomial matrices, and complete codes of the same length, described as kernels of L-polynomial matrices. This correspondence associates a finite code Im + G, with its dual, namely the F+-module ker GT G Zof all the parity checks of the code. Conversely, the dual of a complete code ker GT is the module Im G zP+ of its parity checks. As a straightforward consequence of (A), one gets

,

'= Im ,G

matrix, can be represented as the image of an L-polynomial matrix (see [9] and Proposition 2.2). Lemma 4.1, below, shows that for finite codes a dual situation holds. Actually they are always the images of L-polynomid matrices, but only the images of /FP matrices can be expressed as kernels. Lemma 4.1: Let 8 be a submodule of Y,. Then 8 is the kernel of an L-polynomial matrix if and only if @re exists an LFP matrix, G(zLz2), suck that 8 = Im, G. tho$-Let 8 =Im, G with G E @ * ' " /FP, and consider a full-column rank matrix, H T E q"_1"-", such that c H T = 0. Clearly, if w E 8,then w = uG for some U E @ , , and w H T = (uG)HT= u(GHT) = 0, SO w E Ker H T . On the other hand, if w E F ,is in ker H T , it belongs to the subspace of Hzl, z2)n orthogonal- to the columns of H T , and spanned by the rcws of G. %en there exists f E Rzl,z , ) ~ such that f G = w. As G is LFP, by Lemma A.l f can be chosen in sk,. So w belongs to Im + G. Convefsely, let 8 = ker, H T and consider any /FP matrix G(zl, 2,) E sr(JLn-P)xp such that = 0. Using the same arguments as in the first part of the proof, one 0 shows that 8 = Im G. By completing the duality relations (A) with (C) (ImGT)' = ker, G (ker, G)'= ImGT one immediately proves the proposition below.

,

,

eHT

,

Proposition 4.2: Let G : @ , + q and G T : Z be dual mappings. The following are equivalent: 1) the finite code Im, G can be represented as the kernel of an L-polynomial matrix, 2) the complete code ker GT is convolutional, i.e., it can be described as the image of an L-polynomial matrix. Remark V;. Properties (A) and (C) together with Lemma 4.1 allow us to obtain an alternative proof of the equivalence between 1) and 2) in Proposition 2.2. .Actually, if 9 = Im GT is a convolutional code, as a consequence of (C) 9 is the dual of the finite code 8 := ker, G. Then, by Lemma 4.1, there exists a /FP matrix H ( z , , z z ) such that 8 = Im H and hence, by (A), 9 = (Im H)' = ker H T . So 9 is the kernel of a rFP L-polynomial matrix. Conversely, if 9 = ker H T is a complete code and HT(z1,z2)is rFP, by (A) 9 is the dual of the finite code 8 := Im H , with H LFP. By Lemma 4.1, thete exists a matrix G(z,,z,) such that 8= ker * G, and therefore, by (C), 9 = Im GT is a convolutional code.

,

,

It is quite clear that every code described as the kernel of an L-polynomial matrix H T ( z l , z 2 )admits H T as a which means that every code can be exactly reconstructed syndrome decoder, since a sequence v belongs to the code from the space of its parity checks. if and only if vHT = 0. Hence, every complete code The duality between complete and finite codes can be admits a syndrome decoder, while, among finite codes, better understood by analyzing the correspondence be- only those that are the image of an /FP matrix have this tween convolutional codes and a particular subclass of property. When a (finite or infinite) code, assigned through finite codes. As we have seen, every complete code can be the encoder G, can be represented as the kernel of an described as the kernel, in of an L-polynomial matrix, L-polynomial matrix, the following proposition provides while only a convolutional code, i.e., the kernel of a rFP an algorithm to find a syndrome decoder.

(B) (Im G)'

(ker GT)'

=

ker GT

c,



1078

,e

Proposiition 4.3: (i) If 8= Im is a finite code with 2,) E Pkxn LFP, every L-plynomial matrix H T ( z , ,z , ) of rank n - k, satisfying GH‘ = 0, is a syndrome decoder of 8; (ii) If 8 = Im G is a convolutional code of rank k, then every rFP LTolynomial matrix PT(z1, z , ) of rank n - k satisfying GHT = 0 is a syndrome decoder of 8. Rmfi (i) It is obvious that, if w belongs to 8,then w = uG,U E satisfies W H =~UGH~= 0. Conversely, every w ET,satisfying w H T = 0, belongs to the subspace of N z , , z2)n orthogonal to the columns of H T ( z 1 , z 2 )which , is spanned by the rows of As G is LFP, by Lemma A 1 w is a linear combination over S,of the rows of and therefore w E 8. (ii) After factorizing G into G = L e , with G /FP, the convolutional code 8can be equivalently represented as @ = Im 5 and condition GHT = 0 is equivalent to GHT= 0. So we need only to prove

e(zl,

as legal codewords even sequences in @ \ 8 that are not elements of the code.

V. STATE-SPACE REALIZATION OF ENCODERS AND DECODERS Assigning an encoder via an L-polynomial matrix G ( z , , z , ) corresponds to describing the algorithm that transforms an input information sequence into an output codeword, and hence to specifying only what happens at the terminals of an encoding device. The realization problem consists of obtaining a mathematical model of some “machine” that implements the input-output map. In other words, a state-space realization shows how the encoding algorithm proceeds, by explicitly displaying the corresponding evolution of the memory function. In general there is not a unique way to fmd an algorithm that produces the input-output map of a convolutional encoder. So we have to introduce some a p k r i assumptions, as in the 1-D case, on the class of the Im = ker H T . (4.2) mathematical models to use for this purpose. Moreover, Since H T is rFP, kerHT is convolutional. As convolu- as there is no natural notion of causality in the discrete tional codes can be uniquely reconstructed from the sub- plane, we need also to specify the partial ordering which module of the finite codewords [see (d3)], it will be suffi- underlies the recursive data processing. The class of 2-D models more extensively investigated in the literature is cient to show that (4.2) holds when restricted to F ,, that of 2-D systems, for which the state equation updates namely (Im = Im = ker BT. But that is just what has been shown in part (i). 0 according to a quarter plane causality notion. In this Note that the number of the parity checks we have to section we shall analyze to what extent 2-D systems can apply to a sequence v does not exceed the number of the be used for realizing 2-D encoders and decoders. A (first-order quarter-plane causal) 2-D system Z = columns of H T ( z l ,2,). Implementing a parity check, however, generally involves an infinite number of steps, unless ( A , , A , , B,, B,, C, D ) is given by the following equations 8 is a finite code, described as ker, H T . In this case only [141: a finite number of steps is required to decide whether v x ( i + 1 , J + 1) = x ( i , j + l ) A l + x ( i + l , J ) A , belongs to 8, if an upper bound on the diameter of its + u ( i , j + OB, + u(i + 1,j)B2 support is a priori known.

e,

c.

e,

e

c)f

,e

,

We conclude this section by focusing our attention on the problem of obtaining syndrome decoders for a finite code 8 that cannot be represented as the kernel of an L-polynomial matrix. As a general result, we already know that, if 8 = Im G is a finite code, the dual code 9 = ker GT allows us to identify 8 as 9 . The differences from case i) in Proposition 4.3 come from the fact that a representation of 9 as the image of an L-polynomial matrix is no longer available. As the module of the parity checks cannot be generated by the columns of an L-polynomial matrix, the best we can do is to extract from

,

9 = ker GT = {v E

:vGT = 0)

the submodule gf:= ker, GT = {v ET+ :vGT= 0)

and to represent it as the module generated by the rows of an n X p /FP matrix H ( z l ,z,), i.e., gf= Im, P. Clearly, any codeword w E 8 satisfies w H T = 0. Letting G = L c , with LFP, we have that the %-module of the finite sequences in ker RT is given by 8 = Im 2 8 where the inclusion is proper because of the a_ssumpQon on 8.This means that the syndrome decoder H T accepts

e

e*

w ( i ,j )

=x(i,j)C

+ u(i,j ) D

(5.1)

where the local state x ( i , j ) is a v-dimensional vector over F, input and output functions take values in F k and F” respectively, and A,, A,, B,, B,, C,and D are matrices of suitable dimensions, with entries in F. When txying to eliminate the state variables of a 2-D system, in order to make explicit the input-output relation it produces, some restrictive hypotheses have to be introduced on both the supports of the input signals and the initial conditions of the system, which are formalized as follows. (i) (Past Finite Support of the Input U): For every ( I , m ) E Z x Z, the corresponding past cone K i , j ) : i I 1, j Im} intersects the support of U in a finite number of points. (it) (Zero Initial Conditions): For every (1, m) E Z X Z, supp (U) n { ( i , j ) : i I2, j Im} = 0 implies x(l, m ) = 0. Under assumptions (i) and (ii), the system output w corresponding to the input sequence U is given by w = uG where Gz(z1,2,)

=D

+ (Blz, + B 2 z 2 ) ( I- A 1 z l - A,z,)-’C (5.2)

is the transfer matrix of

2.



It is a well-known result [201 that every proper rational 2-D matrix G(z,,z,) is the transfer matrix of a suitable 2-D-state model Z. In other words, when considering an input-output map y : U w = uG on the space of sequences with past compact support, there exist 2-D-state models (5.1) that produce uG as an output, when feeded with the input U. Since a convolutional (in particular, basic) code %? can always be thought of as the image of a k X n matrix G with elements in HZ,, z,], the submodule of the codewords of B with past finite support can be generated by an appropriate-state model Z as given in (5.Q whose transfer matrix G, coincides with G. Indeed, when condition (ii) is met, such codewords of %? are obtained by applying to Z all input sequences with past finite supports, and the correspondence between inputs and outputs of Z is exactly the map between information sequences and codewords, generated by the encoder G. Clearly, Z can be synthesized by resorting to suitable connections of adders, multipliers, and shift registers in both directions of the discrete plane, and therefore will be called a “realization” (or a state model) of the encoder G. Given an encoder G, there are infinitely many models (5.1) that realize G, i.e., infinitely many sextuples of matrices Z = ( A , , A,, B,, B,, C , D ) that satisfy the equation G(zl, z z >= D

+ ( B 1 z l + B,z,)(Z

-A,z,

-A,z,)-’C.

These realizations, however, are by no means equivalent to each other. In particular, the local state-spaces of different realizations of G need not have the same dimension, and realizations with minimal dimension are not necessarily related by a change of basis in the local-state space. So, the problem arises of characterizing in some way what realizations are the most reliable ones. To that purpose, various additional aspects have to be taken into account. a) If the state-output transfer matrix ( I - A , z , A,z,)-’C is not polynomial, local states x exist, which give rise to free output evolutions x(Z - Alzl - A,x,)-’C with infinite supports. Clearly, such local states, when induced by noise, generate infinite error sequences in the encoding process. b) If the input-state transfer matrix (B1zl + B,z,XZ A,z, -A,z,)-l is not polynomial, finite support input sequences possibly produce infinite support sequences in the state space. Therefore, Z could remain indefinitely excited by a finite signal, even through the corresponding output dies out in a finite number of steps. Both the previous drawbacks can be avoided if the inverse matrix (I - A , Z , - A,z,)-’ is polynomial or, equivalently [131, if the characteristic polynomial of the system is unitary, that is

1079

number of steps after zeroing the input signal. Since every matrix G ( z , ,z , ) E F [ Z ~ , Z , ] ~ ~admits “ a finite memory realization [191, it follows that every polynomial encoder, and a fortwri every basic polynomial encoder, can be synthesized by resorting to a 2-D system with finite memory. c) When implementing a complete transmission system, we have to realize both the encoder and the corresponding decoder via finite memory 2-D state models ( 5 . 0 , and hence to use an encoder-decoder pair with elements in Hz,,z , ] . So, the code B has to be the image of a matrix G that is /ZP not only over S, (which amounts to assuming that 5Z is basic) but also over Hz,,z , ] . Namely, the ideal generated in F[z1,z2]by the maximal order minors of G is the ring Ff.zl,z2]itself. This condition guarantees the existence of a right inverse G-’ with elements in Hzl, z , ] , so that both G and G-’ have finite memory realizations. The following proposition characterizes 2-D basic codes that admit an encoder, /ZP in Hzl, z , ] . Proposition 5.1 [Basic Encoders in F(z,, z2)]: Let %? be a basic code of length n and rank k. The following are equivalent: (i) there exists a basic encoder G+(z,, z,) E Hzl, z Z I k x nthat is /‘ZP in Hq,z,]; (ii) for any basic encoder G(z,, 2,) €fix”, with maximal order minors mi(G), i = 1,2,.--,N = there is a pair (1, m) E Z X Z such that

k),

N

U supp(m,(G)zfzT) 5 N X N

(5.4)

i= 1

and the ideal in Hz,, z , ] generated by ml(G)z:z?, i 1,2,---,N is the whole ring, that is

=

(m,(G)z:z,”,m,(G)z:zF,.--,m,(G)z:z,”) = F[zl, z21 (5.5)

(iii) given a basic encoder G+(zl, z,) any factorization

E

Ff.zl,z Z I k x nin,

G + ( z l , 2,) = T + ( z , ,z , ) c + ( z l , z , ) , where the matrix T + ( z , ,2,) is a greatest left factor (GLF) of G+(z1,z2) over F[zl,z2], c + ( z l , z , ) is /ZP over F[q, 221. (ii): If G+(z,, z,) E Hzl, z2Ikxnis a baProof (i) sic encoder, LZP in Hzl, z,], any equivalent basic encoder G(z,, 2,) differs from G + ( z , ,2,) in a unimodular . matrix U(zl, 2,) E Pkxk, namely G = UG+. As det U is a unit in S,, that is det U = z;’zp, it follows that the ith maximal order minor of G is obtained from the corresponding ith minor of G, as m,(G) = z;lzPmi(G+). So, assuming 1 = - v l and m = - v 2 , we satisfy both (5.4) and (5.5). (ii) * (iii): Let G + be a basic encoder with elements z,], and consider the pair (1, m) such that the det(Z - A l z l - A , z , ) = 1. (5.3) in Hz,, maximal order minors mi(G+), i = 1,2;.., N , satisfy (5.4) 2-D systems satisfying condition (5.3) are called “finite and (5.5). Clearly, 1 and m are nonpositive integers, and memory,” since they reach the zero state in a finite (5.5) implies that m,(G+), i = 1,2,--.,N , generate in


,

1080

IEEE TRANSACIIONS ON INFORMATION THEORY, VOL. 40,NO. 4, JULY 1994

Flz1,z21the principal ideal ( z ; ' ~ ; ~that ) , is

(mi(G+),m2(G+)

m,(d+))

I

(ZT'Z;").

By extracting from _G+ a greatest left factor T + ( z , ,z 2 ) , we obtain G + = T+G+.As the determinant of a GLF of G+ is the greatest common divisor of its maximal order minors mi(G+), i = 1,2,..-, N , then det T + ( z l ,2,) = 2;'z;m.

c+

Obviously, the maximal order minors of are m,(G,)zfz,", m,(G+)zfz,",---,m,(G+)zfz,", and therefore G+ is f'ZP in n z , , z , ] . (iii) 3 6): It is immediate that G+ and cL are equivalent encoders of 8.Being /ZP in n z , , z 2 ] ,G, is /ZP also in S, and therefore basic. 0 Example 4: Let F = GF(2). The following encoder G(Z1,z2) =

0 1

2 '

+1 0

2,

0

I

has maximal order minors z1 + 1, 2, and 0. Since z2 is a unit in S*,G is /ZP in S, and the code 8generated by G is basic. Neither G, nor (by the above proposition) any other equivalent encoder of 8,is /ZP in n z l , z,]. Using a basic code 8 with the aforementioned properties in a transmission system, requires that we first design an encoder G that is /ZP in Hz,, z,], and then compute a decoder G-' with elements in Hzl,z21.Correspondingly, two finite memory realizations for both G and G-' have to be constructed, by resorting to 2-D realization algorithms available in the literature [21]. Most of the computational effort of the above procedure is devoted to obtaining the transfer matrix of the decoder, and to realizing it as a state model. On the other hand, when relaxing the requirement that G-' has to be a polynomial matrix, a considerable simplification is achieved by resorting to the inverse system technique. Actually, if Z = ( A , , A , , B,, B,, C, D)is a realization of G, then D = G(0,O) is right invertible, and for each right inverse of D,the corresponding inverse system

Z-'(D-')

=

( A , - CD-lB,, A , - CD-'B2,

Fig. 4. Syndrome decoder realization.

(i) the inverse system Z - ' ( D - ' ) is finite memory; (ii) G(z,, 2,)D-l is a unimodular matrix in Hz,,z,]; (iii) G can be row bordered into an n x n matrix

unimodular in ETzl,z21, by any constant full-row rank ( n - k) X n matrix K such that KD-' = 0. 0 When 8 admits an encoder G that satisfies the above equivalent conditions, the inverse system technique can be applied to any other basic encoder G o,f 8, /ZP in F[zl,z2]. In fact, by Lemma A.6, G and G differ in an n z , , z,l-unimodular matrix V(q,z,), that is G = VG. Therefore, if G meets condition (ii) of the above proposition, &z,,

2,>6-'

=

is unimodular too. Before concluding this section, we aim to mention the problem of finding, and realizing through a finite memory system, a syndrome decoder of a given encoder G, /ZP in Hz,, z , ] . In the general case, this requires the construction of a /ZP polynomial matrix H T ( z 1 , z 2 ) of , suitable dimensions, such that GHT = 0, and the implementation of a finite memory realization of it. When the encoder G fulfills the equivalent requirements of Proposition 5.2, the problem becomes considerably simpler. Actually, let G-'(z,, 2,) denote the decoder realized by the inverse system Z - ' ( D - ' ) and [D-'LI the inverse matrix of , K as in (iii) of Proposition 5.2. It is quite easy to prove that the polynomial matrix

I[:

H T ( z 1 2, 2 )

D - l B , , D-'B,, - c D - ' , D-1)

[ V ( z , ,z,)G(z,, z2)l[D-'V(0,0)-'l

:=

[ Z - G-'(z1, z,)G(z~,z,)]L

(5.6)

is a syndrome decoder of H G ) . Moreover, the block is a realization of a proper rational inverse of G. So, an scheme of Fig. 4 immediately suggests how to obtain from interesting question is to investigate what conditions on Z Z and Z - ' ( D - ' ) the following finite memory realization and G guarantee that the inverse system Z - ' ( D - ' ) is of H T : finite memory, and therefore realizes a decoder of 8.The A , - CD-'B, -CD-'B, following proposition shows how the fact that Z - ' ( D - ' ) inherits the finite memory property of Z only depends on A, the encoder G and possibly on the constant matrix D-', [ A 2- Y - ' B 2 -CD-'B, whereas the particular structure of the state-space realization does not play any role. A2 Proposition 5.2 [22]-[231: Let Z = ( A l , A,, B , , B,, C, D)be a finite memory 2-D system which realizes a [ -D-'B, - D - ' B , ] , [ -D-'B2 - D - ' B 2 ] , k X n encoder G(z,, z,), /ZP over Hz,, z,]. For every right inverse D-' of D,the following statements are equivalent:


1081


Proposition A.4: Let G(z,, 1,)

APPENDIX

Lemma A.1: (i) Let G(z,, z,) be a k X n LFP matrix with elements in S,. If v E 9 y nis a linear combination over Nz,,z , ) of the rows of G ( z l ,z,), that is

v = aG, a E F(z,, z Z ) l X k (Al) then a can be chosen in (ii) The same statement holds when Siis replaced by

Srk.

E

ex” and

H T ( z l ,z , )

E

Fix(n-k) be /FP and rFP matrices, respectively, satisfying GHT = 0. The corresponding maximal order minors of G and H T are equal, modulo a unit of the ring 9,. Proofi Since G is LFP, there exist [5]two matrices, X , and X,, with elements in S,, and two polynomials, gl(zl) E Rz,,z;’] and g,(z,) E F[zz,z;’], such that GlX,

= gl(zl)zk

and GzXz

= gz(zz)zk.

F[Zl,ZZl.

Proofi (i) Since G is /FP, there exist [7] two polynomials h(z,) E 4 z 1 ,z;’] and k(z,) E Nz,, z;’], and two L-polynomial matrices X ( z , , z,) and Y ( z l ,z,), such that

Consider for instance m,(G), the maximal order minor of G corresponding to the selection of the first k columns of G . Complete G into a square matrix by resorting to a (n - k ) X n matrix, whose columns are all zero except for the last n - k, G x = h(z1)zk and G Y = k(Zz)zk. (A2) which constitute the identity matrix. Thus It entails no loss of generality supposing that the row vector a has irreducible entries, a,. So, letting Bo the least common multiple (LCM) of the denominators of a,, (Al) can be rewritten

as pov = [ p1 &lG p, E Si, i = 1,2 ... . (A3) where M , ( H T ) is the (n - k ) X (n - k ) submatrix of H T obtained by selecting the last n - k rows. Assuming Rl(zl, 2,) := Postmultiplying both members of (A3)by X ( z , , z , ) and Y ( z , ,z,), [ X , H T ]and pl := det Ml(HT),we get we obtain m,(G) det R , = ( g l ( z l ) ) k p , ( H T ) . &Vx= [ ’ * ’ &lGX = * ‘ * PkIh(Z1) Now, replace X , with X , in (As)and let R , := [ X , H T ] .We ‘.’ &lk(Z,). &VU= [ & *‘. & l G Y = obtain As Po, pl,---,pk have no common factors, it follows that m,(G) det R , = [ g , ( z , ) l k p l ( H T ) . Po(z,,zz)lh(zl)and &(zl, z,)lk(z,), and therefore po(zl, z,) is a unit in S,. So, m,(G)l [ g , ( ~ ~ ) l $ ~ (and H ~m,(G)I ) [g2(z2)1$,(HT).Since (ii) Obvious. 0 [gl(zl)lkand [g2(z,)lkare coprime, then CorollaryA.2: Let G(z,, z,) be in Cxn, with row rank k over m,(G)I p l ( H T ) . (A6) NZ,,2,). There exist two L-polynomial matrices, G(zl, z,), k x n LFP, and T(z,, z,), k x k with full-column rank, such that Dually, as H T is rFP, there exist two matrices, Yl and Y,, and two polynomials, h1(zl)E F[z1,z;’] G ( z , , z , ) = T ( z , , ~,)G(Z,, ~ 2 ) . (A4) with elements in Si, and h,(z,) E Hz,,z;’], such that Z’rmfi Let G’(zl,z,) be a matrix obtained by selecting in Y1HT=h,(Z1)zn-k and Y z H T = hz(zz)Z,-k. G(z,,z,) rows linearly independent ?er N z , , ~ , ) ,and a z , , z,) a GLF of G’(zl,z,). Then G’ = QG. Every row in G is We can proceed as before, getting a linear combination over N z , , z,) of the rows of G(zl, z,), and, by Lemma Al, the coefficients of the combination can be chosen in Si. Therefore G = TG, T(zL,z,) being the k X k matrix of the combinators. As rank G is k, rank T cannot be less than k. 0 where M,(G) is the k X k submatrix of G obtained by selecting Proposition A.3: Let G(z,, z,) be in k I n . G has an ,we get det S, p l ( H T ) its first k columns. Assuming S, := L-polynomial right inverse if and only if G is LZP. ~ , analogously, det S, p l ( H T ) = Proofi If G is [ZP, the ideal generated by its maximal = m , ( G ) [ h , ( ~ , ) ] ” -and, , coincides with the whole ml(G)[h,(z,)l’-k,where S , := order minors mi(G), i = 1,2;.-, Therefore, p l ( H T ) is a common factor of ml(G)[h,(z,)l”-k ring S*. It follows that there exists a, ES*,i = 1,2;-*,( and ml(G)[hZ(zZ)l’-k, and then such that Elalml(G)= 1. Consider the identity, m,(G)Zk= GS, Adj (GS,),where S, depl(HT)lml(G>. (A71 notes the selection matrix corresponding to the minor m,(G). (A6) and (A7) together imply that m,(G) and p l ( H T )differ in a Then we have unit of Fk. aim,(G)Zk Similarly, we can show that the same result holds for any I 0 other pair of corresponding minors in G and in H T .

ex‘,

IG1

[ g].

i),

Clearly, t h e L-polynomial matrix K ( z , , z , ) := C i a i ( z l ,z,)Si Adj(G(z,, z,)S,) is a right inverse of G . The converse is a direct consequence of the Binet-Cauchy 0 formula.

-

Lemma A.5: Let H T ( z , ,1,) E Fkx(n-k). The map H T :F* :w w H T is onto if and only H T is rZP. Proofi Assume that H T is onto. Then there exist Wl,W2;*’,wn-k € q such that WiHT= ei [o...o 1 0 * * * 0 ] . Letting w = coI{wl,wz,***,wn-k}, we have W H T = z n - k . So, by Proposition A.3, H T ( z l ,z , ) is rZP. +


1082

IEEE TRANSACTIONS ON INFORMATION THEORY,VOL. 40,NO.4, JULY 1994

[81 J. C. Willems, “Models for dynamics,” @”. Rep. vol. 2, pp. 171-267, 1989. 191 p. Rocha, “struCt,ure and representation Of 2D Systems,” Ph.D. dissertation, Rijksuniversiteit, Groningen, 1990. [lo] P. Rocha and J. C. Willems, “Controllability of 2D systems,” IEEE Trans. Automat. Control, vol. AC-36, pp. 413-423, April 1991. Lemma A.6: Suppose that cl(zl, z,) and z2) are k x n and Puzzles in the theory Of dynamical J. c. willems, polynomial matrices, /Fp in Nzl, z2]. If there e&ts an s,-dsystems,” IEEE Trans. Automat. Conml, vol. AC-36, pp. 259-294, modular matrix U(zl,z2) such that March 1991. C,(Z1,2,) = mz,, z,)c2(z1, 22)’ (A@ [12] E.Fomasini and S. Zampieri, “A note on the state space realization of 2D FIR transfer functions,” Syst. Contr. Letters, vol. 16, then pp. 17-22, 1990. G(z1, z,) = Uq, z,)G,(z,, ZJ (A9) 1131 M. Bisiacco, “State and output feedback stabilizability of 2D systems,” IEEE Trans. Circ. Syst., vol. CAS-32, pp. 1246-54, for some Rzl, z,]-unimodular matrix V. December 1985. hmfi By assumption (A8), every row of is a linear [14] E. Fomasini and G. Marchesini, “Properties of pairs of matrices and state models for 2D systems, Pt. I: State dynamia and geomecombination over F(zl, z2) of the rows of By Lemma A.l, try of the pairs, Pt. I 1 Models structure and realization problems,” there exists 5 matrix V(zl, z,) E Nzl, zZIkxksuch that = in Multivariate Analysk Future Direcrions, C . R. Rao, Ed., North Vc,. Since Gl is PFP, det V is a nonzero constant and Y is Holland Series in probability and vol. 5. Amsterdam, q z l , z,]-unimodular. 0 The Netherlands: North Holland, 1993, pp. 131-180. [15] Ph. Piret, Convolutional Codes. Cambridge, MA: MIT, 1981. REFERENCES I161 S. Lang, Algebra. Reading, MA: Addison-Wesley, 1967. G. D. Fomey, Jr., “Convolutional codes I Algebraic structure,” [17] E. Fomaini, P. Rocha, and S. Zampieri, “State space realization IEEE Trans. Inform. i’hory, vol. IT-16, pp. 720-738, November of 2D finite dimensional behaviours,” SLAM J. Contr. opfim.,vol. 1970. 31, no. 6, pp. 1502-1517, Nov. 1993.. -, “Structural analysis of convolutional codes via dual codes,” [18] E. Fomasini and M. E. Valcher, “An algebraic approach to 2D IEEE Trans. Inform. Theory, vol. lT-19, pp. 512-518, May 1973. convolutional codes. Pt.I,” Tech. Rep., 1992. -, “Minimal bases of rational vector spaces, with applications to multivariable linear systems,” SLAM J . Contr., vol. 13, [19] W. Greub, Linear algebra. New York Springer-Verlag, 1975. [20] E. Fomasini and G. Marchesini, “State space realization of twopp. 493-520, May 1975. dimensional filters,” IEEE Trans. Automat. Contr., vol. AC-21, B. C. Uvy, “2D polynomial and rational matrices, and their pp. 484-492, August 1976. applications for the modeling of 2D dynamical systems,” Tech. [21] M. Bisiacco, E. Fomasini, and Marchesini, “Dynamic regulation of Rep. M735-11, Stanford Electron. Lab., Standford Univ., 1981. 2D systems: A state-space approach,” Lin. Alg. Appl., vol. 122/124, D. C. Youla and P. F.Pickel, “The Quillen-Suslin theorem,” IEEE pp. 195-218, 1989. Trans. C k . Svst.. vol. CAS-31. DD. 513-518. June 1984. D. C. Youla &d G. Gnavi; ‘“Notes m’n-dimensional system [22] E. Fomasini and M. E. Valcher, “Polynomial inverses of 2D transfer matrices and finite memory realizations via inverse systheory,” IEEE Trans. C k . Syst., vol. CAS-26, pp. 105-111, Februtems,” Multidim. Syst. Signal Proc., vol. 4, pp. 269-294, 1993. ary 1979. M. Morf, B. C. Uvy, S. Y. Kung, and T. Kailath, “New results in [23] E. Fomasini, G. Marchesini, and M. E. Valcher, “On the structure 2D systems theory: Part I and 11,”F”.IEEE, vol. 65, pp. 861-872 of finite memory and separable 2D systems,” Automatica, vol. 30, and 945-961, June 1977. no. 2, Feb. 1994.

Conversely, if H T is rZP, it admits a left inverse W E So, for every p EF*-~ we have p = P Z , , - ~= ( p ~ ) ~ which T , implies that is the image, under HT, of an L-polynomial vector.

@’-k)xn.

cz(zl,

cl

cz.

cl
