Simultaneous Code/Error-Trellis Reduction for Convolutional Codes Using Shifted Code/Error-Subsequences

0.1 MB·English

by  Masato Tajima

Simultaneous Code/Error-Trellis Reduction for
Convolutional Codes Using Shifted
Code/Error-Subsequences
Masato Tajima Koji Okino Takashi Miyagoshi
Graduate School of Sci. and Eng. Information Technology Center Graduate School of Sci. and Eng.
University of Toyama University of Toyama University of Toyama
3190 Gofuku, Toyama 930-8555, Japan 3190 Gofuku, Toyama 930-8555, Japan 3190 Gofuku, Toyama 930-8555, Japan
1 Email: [email protected] Email: [email protected] Email: [email protected]
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a Abstract—In this paper, we show that the code-trellis and the to think that the two trellises can be reduced simultaneously,
J error-trellis for a convolutional code can be reduced simulta- if reduction is possible. Here, consider the situation that the
neously, if reduction is possible. Assume that the error-trellis
1 identical shifts occur both in the components of y and in
can be reduced using shifted error-subsequences. In this case, k
3 thoseofe .Inthiscase,ifonetrellisisreduced,thentheother
if the identical shifts occur in the subsequences of each code k

path, then the code-trellis can also be reduced. First, we obtain trellis should be equally reduced. In this paper, based on this
]
T pairsof transformations which generate theidenticalshiftsboth idea, we discuss the simultaneous reduction of a code-trellis
I in the subsequences of the code-path and in those of the error- andthecorrespondingerror-trellis.First,weobtainthegeneral
. path. Next, by applying these transformations to the generator
s transformationswhichgeneratethe identicalshifts bothin the
c matrix and the parity-check matrix, we show that reduction of subsequencesofy andinthoseofe.Next,weshowthatthese
[ these matrices is accomplished simultaneously, if it is possible.
Moreover, it is shown that the two associated trellises are also transformations preserve the relation that one is a generator

1 reduced simultaneously. matrixandtheotheristhecorrespondingparity-checkmatrix.
v (In this paper, we call this relation the “GH Relation” and
8 I. INTRODUCTION
if G(D) and H(D) have this relation, then it is denoted as
5
8 In this paper,we alwaysassume thatthe underlyingfield is G(D) ⇔H(D)). Using this property, it is shown that G(D)
5 F =GF(2).LetG(D)andH(D)bethegeneratormatrixand andH(D)arereducedsimultaneously,ifreductionispossible.
. the parity-check matrix of an (n,n−m) convolutional code Moreover, it is shown that the corresponding two trellises
1
0 C,respectively.ArielandSnyders[1]presentedaconstruction are also reduced simultaneously. These results again imply
1 of error-trellises based on the scalar check matrix derived that a code/error-trellis construction using shifted code/error-
1 from H(D). They showed that when some (jth) “column” subsequences is very effective.
: of H(D) has a factor Dl, there is a possibility that state-
v
i space reduction can be realized. Being motivated by their II. TRELLISCONSTRUCTION USING SHIFTED
X work, we also examined the same case. The time-k error PATH-SUBSEQUENCES
ar ek = (e(k1),···,ek(n)) and syndrome ζk = (ζk(1),···,ζk(m)) A. Error-trellis construction using shifted error-subsequences
are connected with the relation ζ = e HT(D) (T means
k k Let H(D) be the parity-check matrix for an (n,n − m)
transpose). From this relation, we noticed [9] that the trans-
convolutional code C. Consider the error-trellis based on the
formation ek(j) →Dle(kj) = e(kj−)l is equivalent to dividing the syndrome former HT(D). In this case, the adjoint-obvious
jthcolumnofH(D)byDl.Thatis,reductioncanberealized realization of HT(D) is assumed unless otherwise specified.
byshiftingthe“subsequence”{e(kj)}oftheoriginalerror-path Assume that the jth column of H(D) has the form
e. It is stated [1] that their construction can be used also to
obtaincode-trellises.However,itisnotdescribedinthepaper. Dljh′ (D) Dljh′ (D) ... Dljh′ (D) T , (1)
1j 2j mj
On the other hand, our constructionis based on an equivalent (cid:0) (cid:1)
where l ≥ 1. Let H′(D) be the modified version of H(D)
modification of the relation ζ = e HT(D). Hence, our j
k k with the jth column being replaced by
method can be directly extended to code-trellises. That is,
in the case of code-trellises, the construction is based on the h′ (D) h′ (D) ... h′ (D) T . (2)
1j 2j mj
relationy =u G(D)anditsequivalentmodifications,where
k k (cid:0) (cid:1)
uk and yk are the time-k information and code symbols, re- Also, let e′k =△ (e(k1),···,e′k(j),···,ek(n)), where e′k(j) =△
sbpeetwcteiveenlyth.eNcootedeth-paattthhseirneaexciostdsea-troenllei-stoa-nodnethceoerrrreosrp-opnadthesncine Dlje(kj) =e(kj−)lj. Then we have
the corresponding error-trellis. Accordingly, it is reasonable ζ =e′H′T(D). (3)
k k
Hence, in the case where the jth column of H(D) has a by dividing the first “row” by D2. Hence, the dimension d
1
factor Dlj, there is a possibility that an error-trellis with can be reduced to 2.
reduced number of states can be constructed by shifting the
jth error-subsequence by l time units [9]. Assume that the C. Code-trellis construction using shifted code-subsequences
j
correspondingcode-trellisisterminatedinthe all-zerostate at Note that the relation y =u G(D) holds with respect to
t = N. Then ek′(j) = e(kj−)lj is modified as e′k(j) = e(<jk)−lj>, a generator matrix G(D),kwherke uk = (u(k1),···,uk(n−m))
where <t> denotes tmod(N +lj) (i.e., “cyclic shift”). and y = (y(1),···,y(n)) are the time-k information and
k k k
code symbols, respectively.In the same way as for H(D), by
B. Error-trellis construction using backward-shifted error-
dividing the jth column of G(D) by Dlj or by multiplying
subsequences
the jth column of G(D) by Dlj, reduction of G(D) can be
The construction using shifted error-subsequences is fur-
realized.We see thattheformercorrespondsto thebackward-
ther extended [9], [10]. That is, a reduced error-trellis shift y(j) → y(j) , whereas the latter corresponds to the
can be equally constructed using “backward-shifted” error- k k+lj
esu(jb)se.quWeencseese.Cthoantstihdiesritsheeqtruainvsafloernmtatotio“nmeu(kljt)ip→lyiDng−”ljteh(kej)jt=h rfeovrwerasredd-schoimftpyak(rje)d→toyHk(j−()lDj.).Note that the shift directions are
cok+lulmj n of H(D) by Dlj. Let H′(D) be the parity-check III. TRANSFORMATIONS GENERATING THEIDENTICAL
matrixaftermodification.IfH′(D)isreducedtoanequivalent SHIFTS BOTH INy AND INe
H′′(D) with overallconstraintlength less than that of H(D),
A. General case
thenreductioncanberealized.Weremarkthatthepowerl of
j
D has to be determined properly for each j. For the purpose, Consider the transformations which generate the identical
we can use the reciprocal dual encoder [6] H˜(D) associated shifts both in the componentsof y and in those of e . Now,
k k
with H(D). assume that the relation G(D) ⇔ H(D) holds. Consider a
Example1([9]):Considerthecanonicalparity-checkmatrix pair of transformations:
D2 D2 1 1) divide the jth column of G(D) by Dlj(d) and multiply
H1(D)=(cid:18) 1 1+D+D2 0 (cid:19). (4) the same column by Dlj(m),
Since all the columns of H (D) are delay free, any further 2) divide the jth column of H(D) by D˜lj(d) and multiply
reduction seems to be impo1ssible. In fact, it follows from the same column by D˜lj(m).
Theorem1of[1]thatthedimensiond1ofthestatespaceofthe Then
error-trellis based on HT(D) is 4. However, a corresponding
1 1) the jth component of y becomes
generator matrix is given by G (D)=(1+D+D2,1,D3+ k
1
DD24.).(ROebmsearrvke:tIhtastutfhfiecetshitrodd“icvoidluemthne”tohfirGd1c(oDlu)mhnasbyaDfa2ctoinr yk(j) →yk(j+)lj(d)−lj(m), (8)
ordertoobtaina reducedcode-trellis.)Thisfactimpliesthata 2) the jth component of e becomes
k
reducederror-trelliscanbeconstructed[1],[9].Thenconsider
the reciprocal dual encoder e(j) →e(j) . (9)
k k−˜l(d)+˜l(m)
j j
1 1 D2
H˜1(D)=(cid:18) D2 1+D+D2 0 (cid:19). (5) After shifting e(kj−)˜l(d)+˜l(m) by l time units (l is independent
j j
Note that the third column of H˜1(D) has a factor D2. of j), compare the time-index of e(kj+)l−˜l(d)+˜l(m) and that of
Accordingly, dividing the third column of H˜1(D) by D2, we y(j) . If the two time-indices coinjcidej, then y(j) and
can construct an error-trellis with 4 states (i.e., d˜1 = 2) [1], k+lj(d)−lj(m) k
[9]. Here, notice that each error-path in the error-trellis based e(j) have “relatively” the identical shift. This condition is
k
on HT(D) can be represented in time-reversed order using written as
1
the error-trellis based on H˜T(D). Hence, a factor D2 in the
column of H˜ (D) correspo1nds to backward-shifting by two l=(lj(d)+˜lj(d))−(lj(m)+˜lj(m)) (1≤j ≤n), (10)
1
time units (i.e., D−2) in terms of the original H1(D). Hence, where l is a constant independent of j (1 ≤ j ≤ n). (In the
multiply the third column H1(D) by D2. Then we have following, this condition is denoted as “C ”.)
SR
′ D2 D2 D2
H (D)= . (6) B. Special cases
1 (cid:18) 1 1+D+D2 0 (cid:19)
Case 1: Only division is applied both to the columns of
We see that this matrix can be reduced to an equivalent
G(D) and to those of H(D).
canonical parity-check matrix
From the assumption, l(m) =˜l(m) =0. Hence, we have
j j
′′ 1 1 1
H1(D)=(cid:18) 1 1+D+D2 0 (cid:19) (7) l=l(d)+˜l(d). (11)
j j
Here, assume that either l(d) or ˜l(d) is 0. Define the sets L C. Property of transformations
j j G
and L as
H ObservethatinExample2andExample3,theGHRelation
L =△ {j :l(d) =l}={j :˜l(d) =0} (12) is preserved after type-1 and type-2 transformations. It is
G j j shown that this property holds in general. Assume that the
L =△ {j :˜l(d) =l}={j :l(d) =0}. (13) relation G(D) ⇔ H(D) holds. Also, assume that a pair of
H j j
transformations which satisfies the condition C is applied
In words, LG is the set of columns of G(D) from which Dl to G(D) and H(D). Let G′(D) and H′(D) beStRhe resulting
is factoring out, whereas L is the set of columns of H(D)
H matrices, respectively. Then we have the following.
from which Dl is factoring out. Note that LG and LH are Proposition 1: The relation G′(D)⇔H′(D) holds.
disjoint and the relation
Proof:Fix p, q (1≤p≤n−m, 1≤q ≤m)arbitrarily.
L ∪L ={1,2,···,n} (14) Let
G H
(g (D),···,g (D),···,g (D)) (22)
p1 pj pn
holds. In the following, we call this kind of transformations
“type-1”. be the pth row of G(D). Then the (p,j) elementof G′(D) is
Example 2: Consider the relation given by
G2(D) = (D+D2,D2,11+D0) D gpj(D)DDllj(j(md)). (23)
⇔ H (D)= . (15)
2 (cid:18) D 1+D 0 (cid:19) Similarly, defining the qth row of H(D) as
Choosing l =1, LG ={1,2}, and LH ={3}, we have (hq1(D),···,hqj(D),···,hqn(D)), (24)
G′2(D) = (1+D,D,1+D) the (q,j) element of H′(D) is given by
⇔ H2′(D)=(cid:18) D1 1+0D 10 (cid:19). (16) h (D)D˜lj(m). (25)
Case 2: Division and multiplication are separately applied qj D˜lj(d)
either to the columns of G(D) or to the columns of H(D). Then the (p,q) element h′ of G′(D)H′T(D) is given by
pq
Without loss of generality, assume that division is applied
to the columns of G(D), whereas multiplication is applied to ′ n Dlj(m) D˜lj(m)
the columnsof H(D). From the assumption,lj(m) =˜lj(d) =0. hpq = Xj=1gpj(D)Dlj(d) hqj(D)D˜lj(d)
Hence, we have
n
l =lj(d)−˜lj(m). (17) = gpj(D)hqj(D)D(lj(m)+˜lj(m))−(lj(d)+˜lj(d))
In particular, set l =0. Then we have Xj=1
n
1
lj(d) =˜lj(m) (=△ lj). (18) = Dl gpj(D)hqj(D). (26)
Xj=1
This is equivalent to dividing the jth column of G(D) by
Dlj and multiplying the jth column of H(D) by Dlj. In the Since G(D) ⇔H(D), nj=1gpj(D)hqj(D) =0. Hence, we
following, we call this kind of transformations “type-2”. have h′pq =0. P
Example 3: Consider the relation
IV. SIMULTANEOUS REDUCTION OFG(D)ANDH(D)
G (D) = (1+D,1,D+D2) The discussion in the previous section implies that G(D)
3
D 0 1 and H(D) can be reduced simultaneously, if reduction is
⇔ H (D)= . (19)
3 (cid:18) 1 1+D 0 (cid:19) possible. Assume that the relation G(D)⇔H(D) holds. Let
ν andν⊥ betheoverallconstraintlengthsofG(D)andH(D),
Choosing l(d) =˜l(m) =1, we have respectively. If both G(D) and H(D) are canonical [4], [5],
3 3
′ then we have ν = ν⊥. Here, apply a pair of transformations
G (D) = (1+D,1,1+D)
3 whichsatisfiestheconditionC toG(D)andH(D).Denote
SR
⇔ H′(D)= D 0 D . (20) by ν′ and ν′⊥ the overall constraint lengths of the modified
3 (cid:18) 1 1+D 0 (cid:19) matricesG′(D)andH′(D),respectively.Notethattherelation
Note that H′(D) can be reduced to G′(D) ⇔ H′(D) still holds from Proposition 1. Hence, if
3 necessary, by modifying equivalently, we have ν′ = ν′⊥.
H′′(D)= 1 0 1 . (21) Therefore, if the strict inequality ν′ < ν (ν′⊥ < ν⊥) holds,
3 (cid:18) 1 1+D 0 (cid:19)
thenG(D)andH(D)arereducedsimultaneously.Thatis,we
Type-1 and type-2 transformations form a subclass of gen- have the following.
eral transformations defined in Section III-A. However, these Proposition 2: Assume that the relation G(D) ⇔ H(D)
transformations are quite effective. holds. Also, assume that a pair of transformations which
satisfies the condition CSR is applied to G(D) and H(D). In (00)t=0000 t=1000 t=2000 t=3000 t=4
this case, if G(D) is reduced, then H(D) is equally reduced, 001 001 001 001
and vice versa.
Example 4: Assume that
[phone] 101 101
G (D) = (1+D+D2,D,D4+D5) 100 100 100 100
4
D3 D2 1 110 110 110 110
⇔H4(D) = (cid:18) D 1+D+D2 0 (cid:19). (27) (10) 111 111 111 111
Note that both G (D) and H (D) are canonical and the
4 4
equality ν = ν⊥ = 5 holds. Choosing l = 1, L = {2,3}, 011 011 011 011
G (11)
010 010 010 010
andL ={1},let usapplya type-1transformation.Thenwe
H
have
Fig.1. Examplecode-trellis associated withG2(D).
G′(D) = (1+D+D2,1,D3+D4)
4
′ D2 D2 1 t=0 t=1 t=2 t=3 t=4 t=5
⇔H4(D) = (cid:18) 1 1+D+D2 0 (cid:19). (28) [phone] 100 010 100 010
101 011 101 011
Also, let us apply a type-2 transformation with l(d) =˜l(m) =
3 3
2. Then we have [phone] 000 110 0 00
G′′(D) = (1+D+D2,1,D+D2) 011 111 001 111 001
4
′′ D2 D2 D2 000 110 000 110
⇔H (D) = . (29) 100
4 (cid:18) 1 1+D+D2 0 (cid:19) (10)
101 001 111 001 111
Since H′′(D) is reduced to
4
110 010 100 010 100
H4′′′(D)=(cid:18) 11 1+D1+D2 10 (cid:19), (30) (11) 111 011 101 011 101
ζ(1)ζ(2)=00 ζ(1)ζ(2)=10 ζ(1)ζ(2)=01 ζ(1)ζ(2)=10 ζ(1)ζ(2)=01
1 1 2 2 3 3 4 4 5 5
we finally have
G′′(D) = (1+D+D2,1,D+D2) Fig.2. Exampleerror-trellis basedonH2T(D).
4
′′′ 1 1 1
⇔H (D) = . (31)
4 (cid:18) 1 1+D+D2 0 (cid:19)
Proposition 2, it is reasonable to think that T and T are
c e
Inthisexample,theoverallconstraintlengthsarereducedfrom reduced simultaneously. In fact, we have the following.
ν =ν⊥ =5 to ν′ =ν′⊥ =2.
Proposition 3: Assumethatapairoftransformationswhich
Remark: The reduction process is not unique. In the above
satisfies the condition C is applied to G(D) and H(D). In
SR
example, if a type-2 transformation is applied to G (D) and
4 this case, if the code-trellis associated with G(D) is reduced,
H4(D) with l3(d) =˜l3(m) =3, then we have then the error-trellisbased on HT(D) is equally reduced,and
G∗(D) = (1+D+D2,D,D+D2) vice versa.
4 Proof: Denote by e′ the shifted version of e. Assume
⇔H∗(D) = D3 D2 D3 that the set of shifted error-paths {e′} is represented using
4 (cid:18) D 1+D+D2 0 (cid:19) the reducederror-trellisT′ based on H′T(D). Note that there
e
≃H∗∗(D) = D 1 D , (32) exists a one-to-one correspondence between the code-paths
4 (cid:18) D 1+D+D2 0 (cid:19) {y} and the error-paths {e}. Also, from the assumption of
where “≃” means equivalent. Here, choosing l = 1, L = the transformations, the identical shifts are generated both
G
{2}, and L = {1,3}, let us apply a type-1 transformation. in the subsequences of a code-path y and in those of the
H
Then we have G′′(D)⇔H′′′(D). corresponding error-path e. Hence, the set of shifted code-
4 4 paths {y′} is also represented using the reduced code-trellis
V. SIMULTANEOUSCODE/ERROR-TRELLIS REDUCTION T′ associated with G′(D). That is, if one trellis is reduced,
c
Assume that the relation G(D) ⇔ H(D) holds. Let Tc then the other trellis is equally reduced.
be the code-trellis associated with G(D). It is assumed that Example 5: Consider the relation G (D)⇔H (D). Fig.1
2 2
Tc is terminated in the all-zero state at t = N. Denote by shows the code-trellis associated with G2(D). Note that the
Te the correspondingerror-trellis. Note that each code-pathy trellis is terminated in the all-zero state (00) at t = 4. The
in Tc corresponds to the unique error-path e in Te by way correspondingerror-trellisbasedonHT(D)isshowninFig.2.
2
of the received data z. Here, apply a pair of transformations A received data z is assumed to be
which satisfies the condition C to G(D) and H(D). (Let
SR
G′(D) and H′(D) be the resulting matrices.) Then from z =z z z z z =001 000 011 010 000, (33)
1 2 3 4 5
t=-1 t=0 t=1 t=2 t=3 t=4 00. Similarly, the third code-bitof the branchfrom t=−1 to
(0) 000 000 000 000 000
t = 0 must be 0. Here, to each of admissible error-paths in
101 101 101 101 101
Fig.4, we add the modified received data z′. Then we have
111 111 111 111 111 y′ = 000 000 000 000 000
(1) p1
010 010 010 010 010 ′
y = 000 000 101 111 000
p2
′
Fig.3. Reduced code-trellis associated withG′2(D). yp3 = 000 101 111 000 000
′
y = 000 101 010 111 000.
p4
t=0 t=1 t=2 t=3 t=4 t=5
(0) 000 001 111 001 111 We observe that the obtained paths completely coincide with
101 100 010 100 010 thoseinFig.3.Thatis,thetwotrellisesassociatedwithG2(D)
and HT(D) have been reduced simultaneously.
2
111 110 000 110 000 VI. CONCLUSION
(1)
010 011 101 011 101
We have shown that the code-trellis and the error-trellis
ζ(11)ζ(12)=00 ζ(21)ζ(22)=10 ζ(31)ζ(32)=01 ζ(41)ζ(42)=10 ζ(51)ζ(52)=01 for a convolutional code can be reduced simultaneously. The
proposedmethodisbasedonthefactthatiftheidenticalshifts
Fig.4. Reduced error-trellis basedonH2′T(D). occurbothinthe componentsofyk andinthe componentsof
e ,thenthetwotrellisesarereducedsimultaneously,ifreduc-
k
tionispossible.We haveobtainedthe generaltransformations
where z5 = 000 is the “imaginary” received data at t = 5. whichgeneratetheidenticalshiftsbothinthesubsequencesof
The syndrome sequence is given as y andinthoseofe.Wehaveshownthatthesetransformations
preservetheGHRelation.Usingthisproperty,wehaveshown
ζ =ζ ζ ζ ζ ζ =00 10 01 10 01. (34)
1 2 3 4 5
that reduction of G(D) and H(D) is accomplished simulta-
As we have already seen in Example 2, if the first and neously, if it is possible. Moreover, we have shown that the
second components of yk are shifted left by the unit time corresponding two trellises are also reduced simultaneously.
and if the third component of ek is shifted right by the unit Theseresultsagainimplythata code/error-trellisconstruction
time, then G2(D) and H2(D) are reduced simultaneously. using shifted code/error-subsequences is very effective. We
Denote by G′(D) and H′(D) the modified generator and remark that a parity-check matrix with the form described in
2 2
parity-check matrices after transformation, respectively. The the paper appearsin [11] in connectionwith a class of LDPC
correspondingcode and error-trellises are shown in Fig.3 and convolutional codes. We think [10] that the proposed method
Fig.4, respectively. is useful for reducing the state complexity of the code/error-
First, consider the reduced error-trellis in Fig.4. In this trellis for such an LDPC convolutionalcode.
example,itisdefinedase′(3) =△ e(3) ,where<t>denotes
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= 000 001 010 011 000. (35) [9] ,“Error-trellisconstructionforconvolutionalcodesusingshifted
error/syndrome-subsequences,”IEICETrans.Fundamentals,vol.E92-A,
Notethatifz′ isinputtedtoH2′T(D),thenthesamesyndrome no.8,pp.2086–2096,Aug.2009.
sequence ζ =00 10 01 10 01 as for HT(D) is obtained. [10] , “Error-trellis state complexity of LDPC convolutional codes
2
basedoncirculantmatrices,”inProc.ISITA2010,pp.19–24,Oct.2010.
Next,considerthereducedcode-trellisinFig.3.Sincey =
000, we have y′(i) =y(i) =y(i) =0 (i=1,2). That is,0the [11] RJr..,M“L.DTaPnCnebrl,oDck.Sanriddhcaornav,oAlu.tSiornidahlacroadne,sTb.aEs.edFuojna,cainrcdulDan.tJ.mCaotrsitceellso,”,
4 <5> 0
first two code-bits of the branch from t=3 to t=4 must be IEEETrans.Inform.Theory,vol.50,no.12,pp.2966–2984,Dec.2004.