European Wirele, April 7-9,, Vienna, Autria ISBN 978--87-4-9 VE VERLAG GMBH Unavoidable Cycle in Polynomial-Baed Time-Invariant LPC Convolutional Code Hua Zhou and Norbert Goertz Intitute of Telecommunication Vienna Univerity of Technology Guhautrae 5/89, 4 Wien, Autria Email: {HuaZhou, NorbertGoertz}@nttuwienacat Abtract Low-enity Parity-Check convolutional code (L- PCcc) are very intereting for practical error-correction coding in wirele tranmiion a they have excellent performance and at the ame time they allow for variable block ize with low complexity encoding and decoding A for all LPC code that are decoded by the ub-optimal (but highly efficient) Sum Product Algorithm, the cycle in the code graph are very important for the practical performance of the coding cheme Time-invariant LPCcc can be defined by a polynomial yndrome former (tranpoed parity-check matrix in polynomial form), that can be derived from correponding Quai-Cyclic (QC) LPC block code Given the polynomial yndrome former with certain tructure, unavoidable cycle with length ranging from 6 to will be hown to exit We provide ome rule for deigning good code with repect to the hortet cycle in the code-graph, the girth of the code, which i a crucial parameter for it decoding performance I INTROUCTION LPC convolutional code (LPCcc) were firt propoed in [] Studie in [] have hown that LPC convolutional code are uitable for practical implementation with continuou tranmiion a well a block tranmiion in frame of arbitrary ize It ha been proved that, under pipeline decoding, LPC convolutional code have an error performance comparable to that of their block-code counterpart without an increae in computational complexity [] LPC convolutional code can be eparated into two categorie, time-invariant and time-variant LPCcc, with repect to the tructure of their yndrome former Time-invariant LPCcc can be derived from Quai-Cyclic (QC) LPC block code [4] and protograph [6], while time-variant one normally are obtained by unwrapping the parity check matrice of LPC block code [], [8] In term of the decoding performance of LPC convolutional code, free ditance and cycle propertie (girth and number of hort cycle) are two major iue, which are related to the exitence of an error floor and the convergence peed of the Sum Product Algorithm (SPA) [] In [5] and [9], cycle propertie of time-invariant and time-variant LPC convolutional code derived from QC LPC block code have been analyzed, repectively In thi paper, we will invetigate ome unavoidable cycle in time-invariant LPC convolutional code Given certain polynomial yndrome former tructure H T (), ome cycle can not be avoided, no matter what pecific monomial we pick when defining H T () Conequently, thee detructive tructure have to be eliminated in the code deign in order to achieve large girth The ret of the paper i organized a follow: In Section II, we review the definition and the yndrome former of an LPC convolutional code Section III review how to derive a timeinvariant LPC convolutional code from correponding QC LPC block code together with an example In Section IV, we dicu the definition and condition for cycle in a polynomial yndrome former of a time-invariant LPC convolutional code Finally, Section V preent ome unavoidable cycle under different polynomial entry weight II LPC CONVOLUTIONAL COES A rate R=b/c regular (m,j,k) LPC convolutional code i the et of equence v atifying the equation vh T =, where H T () Hm T (m ) H T = H T (t) Hm T (t + m ) () Thi emi-infinite tranpoed parity check matrix H T, called the yndrome former, i made up of a et of binary matrice Hi T (t + i) of ize c (c b) The yndrome former memory m together with the aociated contraint length, defined a v =(m +) c, i proportional to the decoding complexity However, to achieve capacity-approaching performance, a large value of m i required [] H T contain exactly J and K one in each row and column, repectively For time-invariant LPC convolutional code, the binary ub-matrice in H T are contant, ie, Hi T (i) =HT i (i+t), t =,,, while for periodic time-variant LPC convolutional code with period T we have Hi T (i) =HT i (i + T ) For a periodic time-variant LPC convolutional code with period T, given the time-domain yndrome former H T in () and yndrome former memory m, we define the correponding polynomial-domain yndrome former (briefly called Paper 5698959 9
polynomial yndrome former) a: H T () H T () H T () = Ht T (), () HT T () where each polynomial ub-matrix i m Ht T () = Hn T (t + n) n () n= with H T n (t + n) given in () Note that, in () the upercript T mean tranpoe while the ubcript T refer to the period Finally, H T () conit of T ditinct polynomial ubmatrice, and each i of ize c (c b) It i clear that, for time-invariant LPC convolutional code, ie, T =, there i only one polynomial ub-matrix in the polynomial yndrome former In thi paper, we will focu on time-invariant cae III ERIVATION OF LPC CONVOLUTIONAL COES A Time-invariant LPCcc A particular category of time-invariant LPC convolutional code, decribed by polynomial matrice, i derived from Quai Cyclic (QC) LPC block code A decribed in [4], in the Galoi Field GF(m), with m prime, we aume that a and b are two nonzero element with multiplicative order o(a) =k and o(b) =j, repectively With the (, t)th element P,t =b a t mod m, =,,,j- and t=,,,k-, we form the j k matrix P of element from GF(m): a a a k P = b ab a b a k b (4) b j ab j a b j a k b j (all product in the matrix are modulo m ) The parity-check matrix H QC of a QC LPC block code i generated from (4) a follow: I I a I a I a k I b I ab I a b I a k b H QC = I b j I ab j I a b j I a k b j (5) The matrice I P,t are m m identity matrice cyclically hifted to the left by P,t - poition Note that, therefore, I correpond to an unhifted tandard identity matrix A explained in [4], LPC convolutional code can be formed by a matrix that conit of polynomial entrie, with each polynomial entry decribing the firt column of the correponding hifted identity ub-matrix in H QC For intance, if the firt column of a horizontally left-hifted identity matrix i [] T (thi mean three horizontal hift to the left), it i repreented by the polynomial Therefore, according to the parity-check matrix in (5), the aociated polynomial matrix for the LPC convolutional code i given by a a ak H() = b ab a b ak b bj abj a b j ak b j (6) A polynomial matrix generated in thi way will produce a time-invariant LPC convolutional code, a the polynomial parity-check matrix i contant H() i the parity-check matrix of a regular (j, k) LPC convolutional code in polynomial form; it ha exactly j monomial in each column and k monomial in each row The notation can be interpreted a a delay a common in convolutional code The degree/weight of the polynomial entrie in (6) are all one In literature, uch a matrix with all degree-one polynomial a it element i alo called a monomial matrix B Code example of a time-invariant LPCcc We ue the method ummarized above to generate a LPC convolutional code that will be ued to illutrate cycle formation; the code wa alo ued a an example in [4], [7] We pick the field GF(7) and the element a= and b=6 Ak= i the mallet poitive integer k>for which a k mod 7=, the order of a i k =o(a)= ; imilarly, the order of b i j =o(b)= From thi we obtain by (5) [ ] I I H QC = I 4 (7) I 6 I 5 I 7 7 for the parity-check matrix of the QC LPC block code By (6) we obtain the parity-check matrix [ ] [ ] H() = 5 4 (8) in polynomial form for the LPC convolutional code The code ha a code rate of R=-j/k= /=/ We have removed the common factor from the lower row of H() Removing uch common factor will lead to maller yndrome former memory m =(compared to m =5in the original form of H()) and i, therefore, favorable for a practical implementation [7] IV CYCLE PROPERTIES OF LPCCCS It i well known that hort cycle in the factor graph of LPC block code affect the decoding performance The larger the girth the hortet cycle in a factor graph the better the decoding performance will be A good code, however, will have to have cycle in the code graph to achieve good propertie uch a large minimum ditance Hence, we are The degree or weight of a polynomial i defined a the number of additive term with different power of that are involved For intance, the degree of + + i three, while the degree of + i two When the degree of a polynomial i one it i alo called a monomial the yndrome former memory m i equal to the larget power among all the polynomial entrie l in H() 4
intereted to analyze cycle propertie of LPC code In thi ection, we will briefly review how cycle are formed in timeinvariant LPCcc regarding the example in (8) Without lo of generality, we will below conider the tranpoe of the H() matrix, which i called polynomial yndrome former A cycle of length i hown in the polynomial yndrome former H T () in Fig H T ( ) = n 5 4 9 Fig : Polynomial domain tranpoed parity-check matrix H T () n th edge in a cycle tarting monomial of a cycle 4 Monomial from the ame row or column in the polynomial yndrome former H T () connect to each other with or without delay: the delay incurred equal the difference of the power of the two monomial that are connected Thi i illutrated for our code example in Fig : the element ij are the power of from the yndrome former matrix at the right-hand ide of Fig The formal definition of the delay = 8 5 7 = = = delay: Δ( ij, kl ) = = Fig : elay for code example derived from polynomial yndrome former matrix i given by Δ( ij, kl ) =Δ( ij, il )+Δ( il, kl ) (9) where it ha decompoed a poible move from one element ij in the matrix in Fig to any other element kl into a horizontal move ij il with zero delay, ie, Δ( ij, il ) = () 6 and a vertical move il kl with the delay Δ( il, kl ) = kl il () Note that the delay Δ( il, kl ) for the vertical move in Fig ha a ign, which identifie whether we move forward or backward in time when progreing along a vertical edge A path P which i a equence of pair of element from the polynomial yndrome former form a cycle of length L, when Δ(, )= () with the path given by tart {, } P P = { {}}{ { jk, jk }, { jk }{{}, jk }, { jk }{{}, jk }, }{{} horizontal move vertical horizontal end {}}{ } { jl k L, jl k L } { jl k L = jl k, jk } () }{{}}{{} horizontal vertical with j x {,, j} and k y {,, k} and x {,, L} and y {,, L} It i important to note that the firt and the lat element in the path have to be the ame in order for P to form a cycle Thi condition i, however, only neceary but not ufficient, a in addition the um in () mut be zero! We will illutrate the concept by an example: the -cycle i decribed by a cloed path in the polynomial yndrome former a depicted in Fig We apply the path from Fig to Fig and calculate the um of the delay which are accumulated along the path: {, } P Δ(, )= horizontal move {}}{ Δ(, ) + vertical move {}}{ Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) + Δ(, ) (4) With () we have zero delay for all horizontal move and we ue () for the vertical move We obtain {, } P Δ(, )= + ( ) + + ( ) + + ( ) + + ( ) + + ( ) + + ( ) =, (5) which confirm the condition in (): thi -cycle doe indeed exit V UNAVOIABLE CYCLES According to the decription in Section IV, it eem that cycle can be eliminated by chooing uitable power index of each monomial entry in H T () that will not enable any path to atify the condition in () However, we find that there are ome unavoidable cycle, no matter what the power of the 4
monomial in the polynomial yndrome former are To better undertand the exitence of thee cycle, we will, a a novelty, illutrate the unavoidable cycle with length ranging from 6 to from a geometric perpective A Unavoidable -cycle Similar to the -cycle path in Fig, in Fig and 4 we demontrate by imple graphical mean that any LPC convolutional code that can be decribed by a time-invariant monomial yndrome former matrix ha to have -cycle, if the underlying QC code ha j and k (which will be true for any ueful code) We ue (in Fig ) and element ij have, the path delay um i alway zero, ie, + ( ) + + ( ) + Δ(, + ( )= ) + =, + ( ) + {, } P + ( ) + + ( ) (6) therefore, we have a cycle The ame applie to the econd cae depicted in Fig 4 The ituation i even impler than in the firt cae, a all vertical branche (which incur a delay) are travered twice and in oppoite direction: therefore, the um of the delay i again zero In Fig 5 we how a full lit of all uch ubmatrice for a (, 5) LPC convolutional code From Fig 5 we conclude Fig : Unavoidable -Cycle in a Monomial Syndrome Former Fig 4: Unavoidable -Cycle in a Monomial Syndrome Former (in Fig 4) ubmatrice of a poible yndrome former to demontrate how -cycle are formed Thoe cycle appear, regardle of the power of the monomial in the matrix: it i jut a tructural property, common to all time-invariant LPC convolutional code baed on a QC block code contruction by circulant matrice In Fig we only need to form the um of the delay along the path tarting at element : no matter which value the Fig 5: -Cycle-Submatrice in the yndrome former of a (, 5) LPC convolutional code that a lower bound for the number of tructurally different nonrepetitive -cycle of an (j, k) LPC convolutional code with j and k can be computed analytically according to ( ) ( ) ( ) ( ) j k j k N -Cycle + (7) For ( a (, 5) LPC convolutional code we obtain N -Cycle ) ( 5 ) ( + ) ( 5 ) =4which i confirmed by the full lit in Fig 5 It hould be noted, however, that depending on the choice of the polynomial in the yndrome former matrix the number of -cycle may be much larger than the lower bound In Table I in [5], it how the cycle calculation of a couple of code example The (,5)-code with m/m = /5 i a particularly intereting example: thi code fulfill the lower bound (7) and it doe not have any 4,6,8,-cycle Therefore thi time-invariant LPC convolutional code i perfect with repect to it cycle propertie and cannot be improved Thi, however, doe not make a tatement about the free ditance or weight ditribution of thi code 4
B Unavoidable 6-8- and -cycle The time-invariant LPC convolutional code derived from QC LPC block code in thi paper only conit of monomial entrie (ie, their weight i one) in the polynomial yndrome former H T () In thi ubection, cycle propertie of LPCcc with entry weight larger than one will be conidered In [7], two girth theorem have been arithmetically proved for the cae that the weight of any polynomial entry i larger than one Similar to the unavoidable -cylce, we will focu on the tructure of the polynomial yndrome former to invetigate hort unavoidable cycle Property : If the weight of any polynomial entry in a polynomial yndrome former i larger than two then the correponding LPC convolutional code will have girth le than or equal to ix + + S S S S S Fig 6: Unavoidable 6-cycle in a polynomial matrix with an entry weight of three An unavoidable 6-cylce i hown in Fig 6: the mall circle arranged in the block repreent the polynomial entry with weight, ie, = + + In other word, thi polynomial entry conit of three monomial ue to the property that monomial in the ame row or column connect to each other with delay () or (), thi applie to the weight-three polynomial entry a well For convenience in the graphical repreentation of the connection, i arranged in a matrix form Filled circle can be viewed a vertice in a cycle After umming up all the delay along the path, power involved in the path are cancelled by themelve, and, according to (), we hence have a cycle Fig 6, therefore, how that 6-cycle alway exit in a polynomial yndrome former with any polynomial entry with weight larger than two Property : If the weight of any two polynomial entrie in the ame column or row in a polynomial yndrome former i larger than one, then the correponding LPC convolutional code have girth le than or equal to eight Similar a above, we how in Fig 7 and Fig 8 that, if we accumulate the delay along the path, the reult i zero and we now obtain unavoidable 8-cycle Note that thi property i only valid when thee two polynomial entrie are placed + + S S S S Fig 7: Unavoidable 8-cycle in a polynomial matrix with two polynomial entrie with weight two in the ame row + + S S S S Fig 8: Unavoidable 8-cycle in a polynomial matrix with two polynomial entrie with weight two in the ame column in the ame row or column A unique weight-two polynomial entry in H T () will caue unavoidable -cycle a hown in Fig 9; the formal tatement i a follow: Property : If the weight of any polynomial entry in a polynomial yndrome former i larger than one, then the correponding LPC convolutional code ha girth le than or equal to ten + S S S S S S S S S S S Fig 9: Unavoidable -cycle in a polynomial matrix with a weight of two of any polynomial entry VI CONCLUSION In thi paper, the cycle formation in polynomial-domain yndrome former of time-invariant LPC convolutional code, derived from correponding QC LPC block code, ha been 4
dicued Moreover, we invetigated ome unavoidable cycle with length ranging from 6 to that are caued by certain polynomial tructure, no matter what the pecific choice of the matrix coefficient i Baed on the analyi we conjecture that, to obtain a good LPC convolutional code with repect to it (large) girth, rather than polynomial entrie with large weight, monomial are preferred in the polynomial yndrome former H T () ince thi will avoid mall cycle and, hence, allow for larger girth Furthermore, in order to get an LPC convolutional code with girth larger than, the polynomial yndrome former hould not contain any ubmatrice of ize or a in Fig 5 REFERENCES [] A J Feltröm and K Sh Zigangirov, Time-varying periodic convolutional code with low-denity parity-check matrice, IEEE Tran Information Theory, VOL 45, NO 6, September 999 [] S Bate, G Elliot, and R Swamy, Termination equence generation circuit for low-denity parity-check convolutional code, in IEEE Tran Circuit and Sytem, VOL 5, NO 9, September 6 [] J Cotello, Jr, A E Puane, S Bate, and K Sh Zigangirov, A comparion between LPC block and convolutional code, in Proc Information Theory and Application Workhop, San iego, USA, 6 [4] R M Tanner, Sridhara, A Sridharan, T E Fuja, and J Cotello Jr LPC block and convolutional code baed on circulant matrice, IEEE Tran Information Theory, VOL 5, NO, ecember 4 [5] Hua Zhou, Norbert Goertz, Cycle analyi of time-invariant LPC convolutional code, 7th International Conference on Telecommunication, oha, Qatar, April, [6] G M Mitchell, A E Puane, K Sh Zigangirov, and J Cotello, Aymptoticallly good LPC convolutional code baed on protograph, IEEE Intern Symp on Inform Theory, Toronto, Canada, July 6, 8 [7] A Sridharan, eign and analyi of LPC convolutional code, Phd iertation, Univerity of Notre ame, Indiana, USA, Feb 5 [8] A E Puane, R Smarandache, P O Vontobel, J Cotello On deriving good LPC convolutional code from QC LPC block code, IEEE Intern Symp on Inform Theory, Nice, France, June, 7 [9] Hua Zhou, Norbert Goertz, Cycle analyi of time-variant LPC convolutional code, 6th International Sympoium on Tubo Code & Iterative Information Proceing, Bret, France, September, [] S Lin and J Cotello, Error Control Coding, Pearon Prentice Hall, 4 44