A codeword weight lower bound for a class of tail-biting turbo codes

(1)

A

Codeword

Weight

Lower Bound

for a Class of

Tail-Biting Turbo

Codes

Zheng Yan-Xiu and Yu T. Su Department of Communications Engineering National Chiao Tung University, Hsinchu, TAIWAN [email protected] [email protected] Abstract- Thispaper presents anachievablecodewordweight

lower bound associated with weight-2 input sequences of a class

ofturbo codes. The classofcodes has aninterleaverstructurethat encompasses most practical interleavers usedbyturbo codes. It

partitions the incoming information sequenceinto blocks ofthe same size and the interleaver performs intra-block and inter-block permutations. Both pre- and post-permuted blocks are

individually tail-biting encoded. Following [4], we refer to the codewordassociated withaweight-2 inputsequence as a

weight-2 error event. We apply a special permutation function that

incorporates the separate encoding concept to derive a lower bound of the weight-2 error event. This lower bound reveals that(i)alargercomponent codeperiod givesbetterdistancefor the weight-2 error events, and (ii) separate encoding results in

improved distance if the block length issuitably chosen andis

large enough.

I. INTRODUCTION

Considerareasonablegood interleaver of size N. Partition-ing an N-bitgroup into L = FN

W]

or LN Wj-bitblocks,

wefind theinterleaving rule rendersaninter-blockpermutation

structure like that shown in Fig. 1. Such a structure can be foundinother codes suchasproduct codes (block turbo codes, BTCs). Hence both classic convolutional turbo codes (CTCs) and BTCs can be considered as subclasses of the recently proposed inter-block permuted (IBP) turbo codes (IBPTCs) [3] whose interleaver performs consecutive intra- and then inter-block permutations.

However, an interleaver used in a classic CTC, after the above virtual partition, usually yields a non-regular local interleaving structure, i.e., the interleaving relation between a

block and other blocks inthesamegroupdoes notfollow the

same permutation rule. In contrast, product codes and some

IBPTCs have muchmoreregular local interleaving structures. An appropriate regular local interleaving (and deinterleaving)

structure makes implementation easier and offers properties that are useful for parallel decoding, e.g., (memory access) contention-free andsimpler routing requirement.

Another distinction between classic CTCs and other sub-classes of IBPTCs is that, for a classic CTC with an

in-terleaving size of N bits (in L virtual blocks), encoding of consecutive blocks is often continuous. On the otherhand, a

product codearrangesNinformation bitsin a twodimensional

array and encodes each row and column separately (discon-tinuously). The class of IBP turbo codes (IBPTCs) can also encodes each blockseparately.

Betweenthetwo separate(discontinuous) encoding options, thetail-biting encoding scheme, since it can do without

tail-bits, givesahigher spectral efficiency. Moreover, itwasshown that [1], [2], as a tail-biting CTC can eliminate some error events acrossneighboring blocks, improved distanceproperties

can be obtained. Weiss et al. [2] proposed a product code (without the check-on-check part) whose column and row vectors are tail-biting encoded convolutional codewords and

derived somedistance properties.

1 P2r e - P e r m u t a t3 i o n4

1 2 3 4

P o s t - P e r m u t a t i o n

Fig.1. The inherent inter-blockinterleavingstructure canbe foundin most

practical interleavers.

The codeword associated with a weight-2 input sequence was called a weight-2 error event by Breiling [4] for an

obvious reason. Most CTC interleaver designs [6], [3] take this class of errorevents into account,trying tomaximize the minimumweight of theseerror events.Breiling [4] suggesteda

novelpartitionstrategy toderiveupperbounds for the

weight-2 error events. Although the upper bound is not as tight as moregeneralupperbounds [4], [5] which consider othererror events as well, weight-2 error event remains an important

design concern.

As mentionedbefore, ageneralIBPinterleaver [3] encom-passes many existing interleavers as special subclasses. It is built on smaller interleavers and uses some re-permutation

across these interleavers to construct a larger interleaver. By usingasuitableIBPrule,anIBPTCcanpossessgood distance properties.Itis therefore reasonableto conjecture that the

dis-tancespectrumofaCTCusinganIBPinterleaver andseparate

encoding would offersomedesiredproperties. Thepurposeof thispaperisto validateapartof this conjecture. Wederivea

general lower bound for the weight-2 error events associated withgeneral IBP-interleaved CTCs. By analyzing the effects

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ISIT2007, Nice, France,June 24-June29,2007

of selectedparticularsystem parameters onthisgeneralbound

we obtain someusefuldesign guidelines. Weuse a

simplified

partition rule presentedin[4] andapplyaregularpermutation function to derive the bound. We also examine some

special

casesand evaluate distance lower bounds of the

weight-2

error

events for different block lengths.

The rest paper is organized as follows. The next section

presents our derivation of the achievable weight-2 lower bound.InsectionIII,weexaminesomespecial

codes,

evaluate the corresponding distance bounds and discuss the

resulting

design constraints. The last section contains some

concluding

remarks.

II. THE ACHIEVABLEWEIGHT-2 INPUT LOWER BOUND

Forconvenience ofsubsequentdiscourse, weneedtodefine

somenotations tobeginwith. Definition 1:

lxly

X modY.

(1)

Definition 2:

IIXIIY

= { X

lx

XX0

=

(2)

Definition 3:

scrbjLb(u)

is the weight of a

length-L

tail-biting convolutional code outputfor a

input

sequence u.

Definition 4:

W21(L)

= min scrb

L(u

J),

i,j,ji-jjT,AO,|L-i+jlT,- tU

The simplified partition rule for the (k = 0) and post-permutation (k = 1) is givenby

F(k)

j

{+ C: °

<ji

<

FTL

'

i +

Tcj

:

°

<

j

<

LT }7

ith pre-permutation sets

F(

, k = 0,1 0 <

i

<

LI

T

ILIT,

<i<Tc

(5)

An exemplary partition of (5) is shown in Fig. 2 where the integersrepresentthe coordinates of eitheranpre-permutation

orpost-permutation sequence. Each row represents an index

set

Fk)

and is of size 8 or7.

Tc=

(3)

where

u'j

isaweight-2inputsequencewithnonzeroelements

at coordinatesi andj. Definition 5: 9 1 )2 i3 64 j5 11

41

2 3 4

56

211 2 O2 3

45

" 6 31

i22

3 O3 i4 15 ; 4 1i2 i3 4 4 G5 / 51 42 i3 44 15 O5 41

32

3

445

6 7 426 j3 44 5 ;6 1 1 2

3

45

5 W1

(L)

=minscrb (u

'),

(4)

where

u'

is a weight-I input sequence with the nonzero

element located atcoordinate i.

scrbfL

(uij)

islower-boundedbya i

+H13

ora

(L

Tij)

+

13

[4], where

T,

is the period of the convolutional code used. Moreover

scrbjLj(uij)

>

W2(L)

if i-

j

Ir,

0

and L i -i

jT,

:y7

0; otherwise scrbfL

(uij)

= a

min

},

(L -) +

13. Furthermore,

if no

puncturing

is applied, the linearity of the convolutional code

implies

a

(L-TC)

+ 3<

W2(L)

<

(L+T+)

+1. A. Partition rule

Systematic recursive convolutional code used in a CTC is equivalentto anIIR scrambler whose

period

hasagreat

impact

onthe distancepropertyof the associated CTC.Afinite

weight

codeword can be generated

by

a

weight-k input

sequence,

k > 2. Ifk= 2,the distance of thesetwo nonzerocoordinates

must be divisible by the period.

Breiling

[4]

applies

this

property to partition the coordinates of

input

sequences into

some equivalence classes in which any two coordinates is associated with afinite

weight

codeword. He concluded that

a larger component period

implies

a smaller

probability

in generating low weightcodewords.

Fig. 2. Partition ofequivalenceclasses; L=66, T,=9.

B. Main Theorem

In this section we establish our main result whose

proof

needs thefollowing two lemmas.

Lemma 1: Foreach integer set

Sx

=

{f0,

1,2, ..., X

-1},

there existsapermutation ruleHx such that

minj#AjESx

(iS

-jix+w1-x(i)

-wx(j)Ix,

li

-jX

+x-1FX(i)

w-X(j)lX,X-xi-

Ix

+

FX(i)

-

Fx(j)

x,2X -ji -

jlx

-

I7FX(i)

-x(j)|x)

> r +

1,

where r =

Xa

-1. A

permutation

satisfying these constraints is

qilq

_{gcd(X, r)}

x

(6)

Proof: Itis obvious that the

inequality

holdsif i

-jIx

> r andX -i

-jlx

> r. Hence weconsider i

-jlx

< r or X -i

-jlx

<r only.

When i > j and 0 < i-j < r,

gcd(X,

r)

< r and r =

X-]

1< X

implies

that q =

X/gcd(X, r)

> x >

(3)

X > rwhile 0 <i-j < rleads to i

-j+

jlq

-

lilq

q

{

i-j+lq+j-ilq q if

ljlq

i-j-li-jl q if

ljlq

i-j+q+(j -i) q -

lilq

>0, 1-j-(i-j) q -

lilq

< 0-Itfollows that

7iX(t)

-

lx(i)lx

q x q

r(i

-j)+i

-ij

+

ljlq -liq

qan

and

Proof: We place the elements in the jth n-element set in a cycle by j + i(NI+ N2 -L2

2]),

where 0 < i < n and 0 < j < N1. The elements of the jth (n- 1)-element set are placed at positions indexed

(

[iN2+±j-N] (N1+Al)+N1+

iN2

N1+jl,1,

0

~(7)

by

NlN2n+Al-+LNN2+JMN

iN2

+-i

<Al3,

N|N|2

In

+ 13+ - 3(N Al+) +N

+

iN2

+

j-N1

- 3

A3M2, otherwise,

whereO<i<n-1,N <j<Ni+N2,Ml =N2

L-N2],

M2 = N2

N2]

and M3 = Ml N2n It is easy to see

that such an arrangement achieve the bounds and no larger

minimumseparationcanbe found. U

lxIx >r X 7Fj-(i) wx(ji)x X- ri+ q rj+ V x x q -x = q X r(r )+lIX X r2+r > r2+1- r2+r- =r.

Therefore,

minj,j,s.(i

j+ 7FX(i)

-wx(j)

Ix

i

1FX(i) -wx(j)lx)

>r+ 1.

Thispermutation function is q-invariantinthat

7Fx(li

-

qlx)

-

wx(Ij

-

qlx)lx

N9+

=

9

j+X (i q)+

(i-q)-i-qlq

q

-r(j

q)

+

(ji-q)-

Ij-

qlq

q = ri + t

iq

rj

.+

xl

x q x q x 7Fx(i) -

wx(j)Ix

Wenow show that both the remaining cases canbe converted

into the above case.

(A) For thecasei <j andO <j-i < r, wehave

li-jlx

i+X

-jlx

= i +X-mq-(j

-mq)lx

= i'-

j'l

and

w-Fx(Ii+X-mqIx)--wx(jr-mqIx)Ix

= 7FX(i)- wx(j)|x,

X >i' = i +X

-mqlx

>j' =

Ij-mqlx

>O for some

m >0.

(B)Ifi>j,X -i

-jlx

= X+j

-ilx

=

X+j-mq-(i-mq)l

=

lj'-i'l

and

7-Fx(1i-mqIx)--wx(IX+j-mqlx)Ix

= 17x(i)- wrx(j)Ix, x >V =

Ii

+X >mqx>

j' =

Ij-mqlx

>Ofor somem>0.

Lemma 2: GivenN1 distinct n-elementsetsandN2 distinct

(n -)-element sets, where n> 1.Ifwearrange all elements

in these N1 +N2 setsinto acycle, the minimum separation

among elements in the same set is lower-bounded by N1 + N2-FN21 for the n-element sets, and N1+ N2-L]N2 for

the (n-1)-element sets. Moreover,there areat most

IN2 In

element pairs with separation N1 +N2-F N2] for these

n-element sets. n= 8 /0 9 1 82 63 44 25 0 \ 1 1 0 1 92 73 54 35-5 1'. 2 1 12 02 83 64 45 2> -G1

5N2=

6

111z=-==~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1

\8

4 '41=N2=

Fig.3. Setmapping; N1 =3, N2=6andn 8.

Fig. 3 shows a exemplary placementfor N1 3, N2 = 6

andn=8. The minimumseptationintheseN1 8-element and

N2 7-element setsis atleast 7 and 8,

respectively.

Moreover,

there are only

IN2

8 = 6 element pair with

separation

7 for

these 8 elementsets.

Since the scrambler output weight of the weight-2 error

events is lower-boundedbythe difference ofan

(i,

j)

coordi-nate pair, the weight ofa tail-biting encoded CTC is lower-bounded by

min(2+W(i,j,L)+

W(w(i),

w(j), L))

l,j

wherewFis alengthL permutation function and

{

W(iL+

othei-rT

=0

W(i,

j,

L)

= a

Lc

-ii

+

13

IL|-

Ii-jIIT,

W2

(L)

, otherwise

(8)

O

(9)

Basedonthe aboveresults, we canprove

Theorem 1: There existsaseparatetail-bitingencoded CTC of blocklength L whose minimum codeword

weight

W2,min

I+N I

1 2

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ISIT2007, Nice, France, June 24-June29, 2007

for weight-2 inputsequences is lower-boundedby W2,min > 2 +2p3+ min(W2(L) +

aDmin

13,

aTDmin

min ( Nmax ]2N2

IN,,,)

+

oDmax max ( Nmax]

-2IN22

IN

))), (10)

where 2W14(L) > 2+

CTDmin

+W2(L)+/3, Dmin

dT,-UN 1

_N

Dmax

=

dTs

dax

-

LN

Nil'ax

N2

=

dTs

_l L_d _dT,'

d

Nmax

=

d2T%

1,

d=gcd(|

LT,

7Tc)

and T, is the number of blocks involvedin encoding.

Proof: Tail-biting encoding results in low-weight code-words whose nonzero coordinates are confined to the tail and the head parts of two consecutive sets. This hap-pens if one nonzero coordinate of a weight-2 input

se-quence belongs to

Fk)

and the other one belongs to

(li+T-ILITk)

IT,

One can then place the set F(k)

-(k)

right

after the set Fi so that

they

form a

cycle.

If

gcd(ILIT,

Tc)= d,wehave dcycles with the mthcycle being

(k)

_

I{(k)

F(k)

(k)

F~m - lFm7

lrlm+T,-ILITJIT,'

|m+2(T,-ILIT9)IT,'

|(k

-1)(TC LT)},where 0<m <d.

(k)

Mapping the coordinatesin

FmV

sequentiallytotheintegers in the interval [0 m 1 = i], we obtain the set

SlF(k)l

=

{0,1,2,.

- 1}. We further partition

SjF(k)

into dT, sets

{Si},

where

ISi

Nmax

[d2ii

1 for

0< i < N dT and

I,Si

Nmin

LdiTJ

for dT -

N2

= dT < i < dT,.

According

to Lemma

2,

we can maximize the minimum separation of

Si

to

Dmin

=

dT _ [N>] and Dmax=

dTi,

NL

] for 0 < i < N

anddlS N2 <i < dT, respectively.

We can construct anIBPrule such thatp C

Si

and q C

Si

are permuted to the same block iff

Ii -jIT

= 0. Since all

blocks canapply thesamepartitionrule forpermutation, such

an IBPrule does exist.

Incorporating separateencoding results inthattwo indexes

in twodifferent blocksproduceacodewordweight largerthan the bound, either the pre-permuted orthepost-permuted pair

makes the codeword weight

2W1(L).

Therefore we consider the case two indexes are permutedto the sameblock.

There are d sets

Si

and d sets SI (2) All

Si

C SF(1)l canbepermutedtodifferent

SIF(2)

Iftwo indexes areintwo

different

Si's,

either the pre-permuted or the post-permuted

pair makes the codeword weight >

W2(L),

which is larger

than the bound. Therefore we onlyhave to consider the case

whenacoordinatepairbelongstothesame

Si

before and after permutation.

According to Lemma 1, the separation sum of pre-permutationandpost-permutationfor

Si

withNmaxand

Nmin

elements can be F Nmax] and

F/Nmin]

respectively.

Ac-cording toLemma2,the minimumseparationoftwoadjacent

indexes is

Dmin

and there are at most

IN21Nmax

pairs with such a separation. The minimum codeword weight is thus

lower-bounded by 2 +avDminmin(2 N2

N,,

F

Nmax])

+

avDmax

max(FNm-ax]

-2 N2

Nm,as

x 0) +

2/3.

Finally,wenotice that smallweighterror event occurswhen thetwocoordinate pair (i,j) CFm is such that i-j #4 0

and

IL

-i-j

I

T

7y

0and the separation of thepermuted pair (w-F l(i), (jF

l(j))

is greater than

TcDmin.

The correspond-ing codewordweight will beatleast2+W2 (L)+

aDmmi

+/3. Therefore, wehave

wt(X'j)

> 2+

2/3

+min(W2(L) +

CaDmin

-3, ovDminmin (2N2

Nmas,,

oaDmaxmax( Nmax

[1

-2Nf2

NI,])+

ma7)) (1 1)

U

If L> (TC+2d)M, wehave

TlcDmin

min(2 N2

NNm,,

+TcDmax

max(V_Nmax

2 N2

Nmasc

7,

U)

< Tc

dTJs

dJ < Md( d

lT

+1+1) < lsL +d2M2 + dM <+ (L-2dlili)L+d2M2+dM <

V'L_

2dML+d2M2+dM=L, (12) whereM =TlT,. Then

aDmin

min (2N2

1N|,

Nmax])

+ oDmaxmax

VNmax

2N21N,

°) +

13

< aL +1 < W2(L)+a,

Tc

c (13)

if nopuncturing is applied for the scrambler.

Corollary 1: If the block length L is greater than

(TC

+

2d)Mandnopuncturing isapplied,then there existsaseparate

encoding tail-biting turbo code whose minimum codeword weightW2,min forweight-2inputsequencesis lower-bounded by

W2,min

>2 +

aDDmin

min(2

N2

IN,,

[ Nmax

+ozDmax

max ( Nmax- 2

N2

N ,0) +

2/3,

(14)

where 2W1(L) > 2 +

avDmin

+

W2(L)

+

13,

M TcTs,

Dmin

=

dTS

-

[N2]

Dmax

=dT

[J

N2

=

dsl,

dlldT '

Nmax

=dTJ

d=gcd(LIT,lTc)

and Ti is

the number of blocks involvedinencoding.

III. NUMERICAL EXAMPLES

Weevaluates lower bounds for the scramblers

given

inTable I.Figs. 4-6plot the lower bounds for various interleaverlength

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ISIT2007, Nice, France, June 24 - June 29, 2007

TABLE I

(a, B) FOR SOME SCRAMBLERS.

Scramblers

TT

(a, B)

1+D2

1+D+D2 3 (2, 2)

1

+D+D2

7 (4,2)

1+D

+D4+D

15 (8,2)

T, L. Larger component code periods generally give better

bounds, asindicatedby these curves.

Separate encoding improves the lower bounds for some

interleaver lengths but also imposes constraints on interleaver lengths. These figures shows 10-50 weight improvements on

the lower bound for long interleaver lengths but W2(L) is small for short interleaver lengths. Fig. 6 indicates that, the lower bound is a decreasing function of

T,

for short block length. Corollary] saysthat W2(L) is not a dominant factor of the lower boundifthe blocklength constraintL >

(TC+2d)M

is satisfied.

Fig. 4 compares theupperbound [4] and the lower bound

we derived. The large "gap" between the upper and lower bounds is duetothe fact that[4] doesnotconsider theweight-2

error eventsresulted fromadjacent partitions butourderivation does. The gap would be much reduced if these events were

taken into account.

co N 140 - 120- 100-80 -60 -40 -L *Ts-T=1

--T=2

---&---T =4 ---T=6 ---T=8 Upperbound[4] 150 -120 -m aT) o 90- 60- s-_Tr=a ---T=2 ---T =3 ---4---T=4 Fill=5 400 800 1200 Interleaver Length i+D+D3

Fig.5. Theweight2lower bound for theScrambling function 1D2 DT3

360 -200 0) 120-40 co Fig. 6. The 1 D2+D3 +D4 1+D+D4 800 InterleaverLength 1600

weight 2 lower bound for the Scrambling function

201

-200 400

InterleaverLength

600

Fig.4. Theweight2lower boundforthe Scramblingfunction 1++DD2

IV. CONCLUSION

This paper derives a general achievable codeword weight lower bound for theweight-2erroreventswhenaseparate tail-biting encoded CTCuses twoidentical scramblers (component codes) and an IBP interleaver. The bound implies separate

encoding stands a better chance to obtain a weight-2 lower boundlarger than that of the conventional continuous encoding scheme ifthe block length is not too small and is properly chosen. The relationships between these two parameters and

the lower bound provide useful design guideline for the

800 separate tail-biting encoded CTCs.

REFERENCES

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parallel concatenated tail-biting codes,"IEEE Trans. Inform., vol. 47,

no. 1,pp.366-386 Jan.2001.

[3] Y-X.Zheng,Y T.Su,"Onthe inter-block permutation and turbocodes,"

in Proc.3ndInter Sympo. Turbocodes, Brest, France,pp.107-110, Sep.

2003.

[4] M. Breiling, J. B. Huber, "Combinatorial analysis ofthe minimum

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