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A Class of Multirate Convolutional Codes by Dummy Bit InsertionWen Xu and Jac Romme

Department of Mobile Phone Development, Siemens AG,Hofmannstr. 51,81359 Munich, Germany

Abstract - We present a method to construct lower rate codesfrom a known high rate code by inserting known dummy bitsinto the information bit sequence before encoding. Like othermultirate codes based on puncturing or repetition, this new classof codes can adjus t the amount of protectio n given to the infor-mation bits, and can be decoded using the same decoder as forthe mo ther codes. Specifically, we focused on the recu rsive sys-tematic insertion convolutional (RSIC) codes. Our analysis andsimulation results show that the proposed RSIC code achieves aperformance which is comparable to the correspo nding optimalrecursive systematic repetition convolutional(RSRC) ode.I. INTRODUCT~ON

Flexible multirate encoding and ad aptive decoding are oftenneeded since different data to be transmitted have usually dif-ferent error protection requirements, and/or the channel istime varying and/or unknown to some extent. The most com-mon way to desig n a multirate channel coding schem e on thebasis of a fixed code with a certain rate and co rrection capa-bility is puncturing or/and repetition, i.e. puncturing to o btainhigh rate codes and repetition to obtain low rate c odes. Such amultirate scheme has for instance been standardized as therate-matching algorithm in 3GPP (third generation partnerproject) mobile communicationsystem [I ] .Here we focus on convolu tional codes, which are widelyused in practice. Specifically. a subclass of them, the so-called recursive systematic convolutional (RSC) codes havegained great attention recently, since compared to theirequivalent non-systematic counterparts they can provide abetter error protection in bad chann el conditions and/or whenthe code rate is relatively high [2]. For instance the unequalerror protection (UEP) channel coding schemes based onRSC codes were standardized for the newly developed adap-tive multirate (Ah4R) speech codec in the GSM (global sys-tem for mobile com munica tions), where the UEP matching todifferent classes of speech encoded bits is realized by punc-turing and repeating the RSC m other codes [3]. Moreover, theRSC c odes as component codes have been successfully usedto construct the very powerful "Turbo" codes, which canachieve a coding gain near the Sh annon limit [4].

Punctured convolutional (PC) cod es were first introduced in151 mainly for the purpose of obtaining simple Viterbi algo-rithm (VA) decoding for rate kJn codes with two branchesarriving at each node instead of 2' branches. By addin g therate comp atibility restriction to th e pun cturing tables, Hagen-auer [61 generated a family of rate-compatible puncturedconvolutional (RCPC) codes. The notation of repetition con -volutional (RC) codes was introduced by Kallel and Haccoun

[7], where a method was proposed to construct rate-compatible repetition convolutional (RCRC) codes fromknown high-rate convolutional codes, obtained from best rate1/2 codes. In [8], n alternative algorithm was presented tofind large families of the optim um RCRC cod es from the bestrate 1/2, 1/3 and 1/4 codes with memory m =2...10.A significant advantage of the puncture d and repetition con-volutional codes is that the decoder (e.g. Viterbi algorithm)operates always with the mother codes and can thereforeremain unchanged, which greatly reduces the receiver com-plexity. For decoding of punctured convolutional codes, thechannel soft values of the punctured code bits can be simplyset to zero and in repetition convolutional decoding the re-peated code bits are first comb ined (code com bining) beforeVA decoding.In this study, we present a new class of multirate convolu-tional codes called the insertion convolutional (IC) codeswhich are lower rate codes constructed from a known highrate code. Like the other multirate codes based on puncturingor repetition, this new class of codes are able to adjust theamount of protection given to the information bits accordingto their importance, and can be decod ed by using the samedecoder as well. Concretely, the known dummy bits are in-serted into the information bit sequence before encoding(thereafter the insertion convolutional codes). The principleof IC codes has a lot of resemb lance with the termination ofconvolutional codes, where m (= code memory) dummy bitsare attached to the information sequence to force the encoderto end in a given state. For decoding of IC codes, the a prioriinform ation about the known dummy bits can be utilized, e.g.by emplo ying the Apri-Viterbi algorithm (APRI-VA) [9][ 101.For an RSC code, the dummy bits are contained in the codebits and thus do not need to be transmitted (or equivalentlythey will be punctured ). This class of IC codes is denotedhere by recursive systematic insertion convolution al (RSIC)codes. To decode the RSIC codes, the channel soft values ofthe punctured (systematic) dummy bits can be simply set tohave the maximum reliability (+- or --) correspond ing to thedummy bits (+1 or -1). Then, the normal V A ecoder withoutany modification, similar to the punctured convolutionalcodes, can directly be used for RSIC d ecoding. In comparison

to the repetition codes, the I C codes have the advantag e thatno code com bining is required. Our analysis and simulationresults show that the proposed RSIC code achieves a per-formance which is comparable to the corresponding optimalrecursive systematic repetition convolutiona l (RSRC) code.

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11. PROPERTIES OF IC CODESA. Encoder Structure

An IC enco der first inserts i dumm y bits into an informationbit sequence u ( D) of length n , which results in a bit sequ encew ( D ) of length n + i. The bit sequence w ( D ) is fed to theconvolutional (mother) encoder of rate Re , where the infor-mation bits together with dummy bits are woven to a code-word. It is clear that the rate R of the overall code can beadjusted by varying the amoun t of dummy bits inserted.

I P) F(,) + IWfD) 1 0 ) ; I

U(D ) =2 I D II

it holdsF ( u ( D ) )= u I D D (3 )

Iwhere il is the time delay for bit U,.for any u(D)and U() ( D ) , t holds

It is easy to see that the delay ope ration F(. ) is linear since

(4)= F ( u ( D ) )+ ( u ( ) ( D ) )As an example, we considerU ( D ) = ~ U , D u o + u I D + u 2 D 2 u 3D 3 ( 5 )

Iand two dummy bits & and 4 which will be inserted after uoand u2 . Then we havew(D ) =2 wI D = u I D Dit + z ( D )

I I= u o + $ , D + ~ , D ~u 2 0 3 + U ~ Dwith 0 for tSO

(7)2 for t > 2

andz ( D )=@ O D $ I D 4 . ( 8 )

B. LinearityThe IC e ncoding schem e is linear, if both the insertion op-eration and the convolutional encoder are linear. Since G ( D )is linear, to verify whether the insertion operation W(u(D))slinear, we need to check if th e following equality holds forany uI(D)and u((D)

Substitute (1 ) into (9). (9) becomesw ( u ( ) D )+ U (2) ( D ) )= W ( U (I ) ( D ) )+ W ( U (2) ( D ) ) (9 )

(10)F(u)(D )+U() ( 0 ) ) z (D )= F ( u (D ) ) z ( D )+F ( u ) ( D ) ) z ( D )Using (4) we can conclude that the IC code is linear if andonly if (iff,) he inserted dummy bits sequ encei.e. dummy bits must have a value of zero. T he codewo rds ofthe IC codes in this case becom e

z ( D )=0 (1 )

v ( D )= ( F ( u ( D ) ) + ( D ) ) G ( D )= F ( u ( D ) ) G ( D ) . (12)C. Dummy Bits

Here, we want to show that the choice of the dummy bitvalues has no impact on the mutual Hamming distance, thusno impact on their error correcting capabilities. In fact, ac,-cording to (l), we obtain the Hamming distance d betweenany codewords v(D) and V ~ ( D) .hich correspond to theinformation sequences u(D) and u*)(D).espectivelyd ( v ( D ) , v * ( D ) )= d ( ( F ( u ( D ) ) + ( D ) )G ( D ) . F ( u ( ~ ) ( D ) ) +( D ) ) G ( D ) )

(13)The linearity of G ( D ) mpliesd (J1)0 ) . ( ~ ( ~ )D ) )= ~ ( F ( u ( ~ ) ( D ) ) G ( D )z(D)G(D),F(u(~)(D))c(D)z ( D ) G ( D ) )

(14)Since the Hamming distance between two codewords doesnot change when the same sequence is added to both code-words, we haved(v()D ) ,d2) ( ) )=d (F (u( )D))G( ) ,F(u( D))G(0 ) )Thus, the distance spectra depend on the convolutional en-coder G ( D ) and on th e delay F(.) which determines theamount and positions of the inserted dummy bits. However,they are independ ent of the va lue of the dummy bits inserted.

For this reason, the dummy bit value is always set to zero inthis study. This preserves the linearity of th e IC code and theresults obtained will also be valid for other dummy bit value.D.Trellis Structure and Goodness Criteria

The insertion of dummy bits into the information sequenceincreases the length in branches of a ll possible paths from the

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initial state to the ending state, but the number of possiblepaths remains unchanged.For any linear code, its error correcting capability is usuallyupper bounded by the so-called Viterbi's union bound [111for the average event error probability

pc x a d p d (16)-d=dfrtc

and bit error probabilityca

pb x c d P d (17)d = d reewhere dfree s the free distance of the code and P d s the prob-ability that the wrong path at distance d is selected. ( a d } and

( c d ) which depend only on the code, are the so-called dis-tance spectra and should be as small as possible.When i dummy bits are inserted in every n information bits,we have an insertion period T = n + i. It can be easily shownthat the Viterbi's union bounds can be written as

111. NON-RECURSIVE IC CODESBy ignoring the termination effect, the overall rate R of a(recursive or non-recursive) non-systematic IC code can bewritten as

Fig. 2 shows the trellis of a non -recursive IC code of mem-ory 2. We assume, unless otherwise stated, that the all zerocodeword is transmitted. The trellis is time variant with aperiod T = 3 and i = 1, which results in R = 113 for R, = 112.Notice that a path can only diverge from the all zero path atan information bit (denoted as x) state transition, not at adummy bit (denoted as 4) transition. In contrary to the PC orRC codes where a branch weight depends on time and statetransition, the branch weights of IC codes are time invariant.In order to show the goodness of IC codes, the distancespectra are calculated by a trellis search. The code used is amother code of memory 4, rate 112 with gene rator matrix

Fig.2. Trellis ofa non-recursiveICencoder.where the mother code has amemory of2.

G(D) [Go G I . (21)Here Go = 1 + D3 + D4 nd GI = 1 + D + D3 + D4 are thegenerator polynomials used in the GSM [3]. After each in-formation bit, i dummy bits are inserted.

Tab. I shows the distance spectra of this cod e depending onthe amount of dummy bits per period. From the table we seethat the free distance dfrec oes not increase as the number ofcodewords at a certain Hamming distance decreases. Theinsertion of dummy bits causes a decrease in the number ofcodewords, and thus leads to a decrease of the bit error rate(BER) or Pb. However, more dummy bits per period will notautomatically improve the error co rrecting capabilities of anIC code. As shown in Tab. I, a decrease of the IC code rate Rfrom 114 to 116 leads to even worse distance spectra. Whenmore than m dummy bits are inserted at a time, the perform-ance of the IC codes remains unchanged since insertion of mdummy bits terminates the encoder already.

114 I 7

I16 2 7

< I 6 >3 7

442 1015 23343504 a849 n1sO

1 3 44 I I 18I l l8 9 IO0 0 1n' n 4

IV.RECURSIVEC CODESIn the following, it can be shown that this is not the case forIC codes generated using a recursive convolutional mothercode. Specifically, the recursive systematic convolutional(RSC) codes have good distance characteristics similar totheir non-recursive counterparts and dummy bits, which arecontained as systematic bits in the codewords, can be punc-tured without any influence on error correcting capab ilities. Ingeneral, th e overall code rate R of a (recursive or non-recursive) systematic IC cod e is higher in comparison to theequivalent non-systematic IC code. For systematic convolu-tional codes in which the same information bits are mappedinto codewords only once, it holdsR = R, nn + (1- R,)iWe use the same RSC code as before but in the recursiveform, i.e.G ( D ) = [ I G o I G I ] . (23)

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The distance spectra depending on the number of dummy bitsi per period T = 1 + i are given in Tab. 11. Notice that the freedistance and distance spectra, contrarily to th e non-recursivecodes, vary even if m ore than m dummy bits are inserted. Thetrellis of a recursive IC code is different from that of the cor-responding non-recursive IC code (see Fig. 3).

R i dh. d = d b . , , k = O . l . 2 ..... 9-0-

I-2-3-4-5-6

I 13: 18; I 1 1 1 1 474 206 502 1242 3100

6 IO 16 22 362 0 0 4 0

~

00 0 0

2 0 0 2 0

10157620I 2640

---0-

Fig.3. Trellis of a recursive IC encoder, where the mother code has a mem-ory of 2 and feedback polynomial I + D + DZ.It can be noted that the paths in the trellis do not merge aftera non-information bit (@) ransition and therefore there is noreduction in the number of states. This is contrary to non-recursive IC codes.In what follows, we show that two paths in an IC trellis.generated using a recursive convolutional encoder, will notmerge after a dummy bit transition iff the last memory posi-

tion is connected to the feedback loop. Without loss of gener-ality, we consider a recursive convolutional encoder as shownin Fig. 4, where the possible output polynomials have beenleft out for simplicity.

-..-.Fig. 4. A general convolutional encoder structure.

Assume that the path j (j= 1, 2) goes through state s at,W = ( I ) ( j ) (1)

time t, where the state s,(" is defined as(24)(25)( 26)

W,-1WI-2 ...w,, .The paths 1 and 2 merge at time t + 1 iff

(1) = (2)SI+I SIt1.0) 0 ) 0) (2 ) (2) ( 2 )w, WI-I a . . W,.+-l) = w, WI-l e . . W+ *

Substituting (24) into (25) gives

Note that w?' up to ~ ~ . ( ~ - ~ y 'emaining in the shift register arethe same for both pat hs j= 1.2, i.e.Therefore it holds wf.,,,(I) w,.,,,(~), therwise both paths wouldalready have merged. From Fig. 4we havew:!: = w:!: for i = 0,1, ...,m - 1 . (27)

k = l k = Iwhere $f is the value of the inserted dummy bit, which is thesame for all paths through the trellis. From (28) we obtain

(29)2)q,w:', = qmw,-, .As a result, two paths can only m erge iff qm= 0 since wf.,, ,(I)

# wf.,,,(*). learly, the conclusion is also valid for any kJn re-cursive codes since a k-input convolutional encoder can beconsidered as k 1-input shift registers. To ensure that thepaths do not merge, at least one of the k 1-input shift registersfed with a dummy bit must fulfil th e condition qm= 1. This isindeed the case for almost all recursive codes used.Furthermore, it can be shown that in most practical situa-tions the free distance of a recursive IC code can be increasedto infinity by inserting infinite number of dummy bits. This isfor instance the case for the polynomials shown in (23).

V. A COMPARISON OF RSIC AN D RSRC CODESWe see that recursive IC codes can be used as an alternativemethod to RC codes in order to construct lower rate codefrom higher rate codes. Since we are in particular interested inRSC codes, we would like to compare the lower rate codesgenerated by repetition (RSRC code) and insertion (RSICcode). The mother code used has the polynomials

(30)11 l + D + D 2 + D 3 + D 5 t D 7 + D s

1+D2+ D' t D4t DaThe non-recursive form of this code is also standardized for3GPP data transmission [I]. The comparison is conducted atrate R = 215 and R = 113. The RSRC code uses an optimaldistance spectra (ODS) repetition matrices [ 8 ] , namely

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[:] for rate 2 / 5 , [ for rate 1/3, respectively. TheRSIC cod e uses a dummy bit insertion pattern [x x $1 (periodT = 3, i = 1) for rate R = 215, and a pattern [x $1 ( T=2, = 1)for rate R = 1/3, respectively.Fig. 5 and Fig. 6 how simulation results for AWGN chan-nel. Here, both Viterbis union bound (Vit) and simulationresults (sim) are provided. It can be seen that the RSICcode can provide even better performance than the ODSRSRC code for a BER ranging from 10 to lo5.

o 05 I 1 5 2 25 3 a5 4Ed10-

Fig. 5 . Comparison between recursive systematic IC and RC codes (R = 2 5 ) .

0 0 5 1 I5 2 2 5 3 3 5 4W O

Fig.6. Comp arison between recursive systematic IC and RC codes (R = 113).VI . DISCUSSIONS

Some simple searches were carried out for patterns of in-formation and dummy bits, which can provide best perform-ance. Tab. 111. lists the best IC codes with different dumm ybits in the sense of the ODS criteria. In the p atterns shown, xstands for an information bit and $ for a dummy bit. Other

experiments have shown that the free distance and distancespectra of the IC code also depends on the feedback polyno-mial (see Tab. IV). In addition, it is conceivable that betterperformance can be obtained using a recursive convolutionalmother code by combining insertion and puncturing. This,together with other aspe cts, such as sea rching for good recur-sive IC codes, is under current study.

TAB. 11. FREE DISTANCE DEPENDINGN TH E NSERTION PATIERN FORMOTHER ODE I cdC11.

TAB.Iv.FREE DISTANCE EPENDINGN THE FEEDBACK POLYNOMIAL ORMOTHER CODE I Gi/Go].

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