EXIT chart analysis of nonlinear turbo coding over GF(4

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Published in IET CommunicationsReceived on 18th April 2007Revised on 15th November 2007doi: 10.1049/iet-com:20070178

ISSN 1751-8628

EXIT chart analysis of nonlinear turbo codingover GF(4)M.C. Liberatori D.M. Petruzzi J.Casti~neira MoreiraJ.C. BonaderoCommunications Laboratory, School of Engineering, Mar del Plata University, Mar del Plata, ArgentinaE-mail: [email protected]

Abstract: The behaviour of nonlinear turbo codes that use nonlinear finite state sequential machines (FSSMs),designed over the finite field GF(4), as constituent encoders is studied. These FSSMs show a nonzero outputfor the all-zero input case. The state-transitions structure and the randomness of the output for the all-zeroinput response of these FSSMs are studied for different initial state conditions. A relationship is foundbetween these two characteristics and the BER performance or, equivalently, the form of the extrinsicinformation transfer chart of the corresponding turbo code. It is found that FSSMs with responses of the formof long closed cycles, and good randomness properties of the corresponding output are the best options asconstituent encoders of nonlinear turbo codes designed over GF(4). A coefficient called the random behaviourcoefficient is defined, to provide a quantitative measure of the behaviour of these FSSMs as constituentencoders of a turbo code.

1 IntroductionFinite state sequential machines (FSSMs) are usually utilisedas constituent encoders of turbo codes, as it has beenproposed in the original paper of Berrou et al. [1–3]. Theycan operate, in general, over a Galois field GF(q), includingthe particular case of the binary field GF(2). In this paper,we analyse the performance of FSSMs designed overGF(4). These FSSMs are, in general, nonlinear, and theyare constructed as proposed in [4, 5]. Their structuresinclude a polynomial with coefficients over GF(4), whichdetermines their properties, and performances asconstituent encoders of a turbo code.

The behaviour of these FSSMs was studied in [4], andthey were found to show interesting random properties.Frey [6, 7] describes these properties, and defines what isconsidered an FSSM with quasi-chaotic (QC) behaviour.The QC properties are measured in [4] in order toquantitatively determine the randomness of the output ofthe FSSM. These QC properties do not necessarilyconstitute a measure of the cryptographic properties of theFSSM [6–8].

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These FSSMs operate as constituent encoders of a turbocode. A 1/2-rate systematic nonlinear turbo code overGF(4) is shown in Fig. 1.

On one hand, we analyse the behaviour of these FSSMsin terms of their state-transitions structures. A graphicrepresentation called the current state–next staterepresentation will be used to characterise this behaviour.On the other hand, we also analyse the QC behaviour ofthese FSSMs.

These two characteristics are related and define theperformance of the FSSM as constituent encoder of aturbo code. Bit error rate (BER) performance and extrinsicinformation transfer (EXIT) chart are selected as tools forthis analysis.

In this paper, we show that FSSMs with good QCbehaviour contribute to accelerate the slope of the waterfallregion of the BER performance curve of the code. FSSMsdefined over GF(4) are classified in terms of their state-transitions structures. A particular behaviour of a state-transition structure is a closed cycle, where the FSSM starts

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Figure 1 A 1/2-rate systematic nonlinear turbo code over GF(4)

at a given state, runs through a set of different states and endsat the initial state of this cycle.

It is found that FSSMs which have state-transitionstructures of the form of relatively long closed cycles, arethose with the best QC properties [4], and they result inthe best option as constituent encoders of a turbo code.

In this paper, the proposed turbo codes are nonlinear,systematic and of rate 1/2. The nonlinearity comes from theuse of the so-called encryption polynomial g(x), which hascoefficients taken from the corresponding Galois field GF(q)

g(x) ¼ a0 þ a1xþ � � � þ amxm (1)

where m is an integer and ai [ GF(4), i ¼ 0,1, . . . , m.

The set of coefficients fa0 a1 . . . amg determines the maincharacteristics of the proposed FSSM, either if we consideran FSSM as a generator of random sequences, or as a partof a turbo code.

The turbo code is always constructed using the sameFSSM in both constituent encoders. In particular, we willconsider all the possible turbo codes defined over GF(4),with s ¼ 2 memory units, and with encryption polynomialsof the form g(x) ¼ a0þ a1xþ a2x

2, that is, with m ¼ 2 (63cases). Thus, a given turbo code scheme defined overGF(4), will be referred to as a (a0 a1 a2) turbo code, andfor each of them

g1(x) ¼ g2(x) ¼ g(x) ¼ a0 þ a1xþ a2x2 (2)

2 Structure of the state-transitions of FSSMS over GF(4)The proposed FSSM can present a non-zero output for theall-zero input. This response is studied for different initialstate conditions, in order to characterise the FSSM.

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The state-transitions structure of the all-zero inputresponse can be obtained by depicting the next state as afunction of the current state, a representation that will bereferred to as the current state–next state representation.States can be described by its decimal numericrepresentation, which is the decimal representation of thebinary form of each element of the corresponding Galoisfield. This graphic representation adopts different forms,and usually one of them is the form of a cycle.

For each FSSM, we have analysed the all-zero inputresponse for the 16 possible initial state conditions.Responses of FSSMs under analysis can be classified intodifferent groups. FSSMs can run a cycle, that is, a sequenceof states that start and end at the same state. This cycle canstart from any of its states, or either be reached after agiven transient response. Since this difference in thebehaviour of the FSSM is relevant to its performance as aconstituent encoder of a turbo code, we want todifferentiate these two types of cycles. A cycle that includesalways a given set of states and is run starting from any ofthese states, will be referred to as a closed cycle. Thenumber of states that the cycle runs is the length of thecycle, and it will be denoted as LC. A closed cycle is a cyclethat starts and ends at the initial state. When the cycle isreached after a sequence of states that do not form thecycle, showing a transient response before entering into thecycle, the response will be referred to as a cycle withtransients, and the number of states of the cycle (excludingthe number of states of the transient), is the length of thecycle, denoted as LT.

Figs. 2a and 2b show examples of the next state–currentstate representation of these two types of cycles. FSSMscan also show responses without cycles, and also stationaryresponses. Stationary responses are seen as a point in thenext state–current state representation, which can beestablished at the all-zero state, or at other non-zero state.

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3 Random behaviour coefficientAs described in the previous section, FSSMs under analysiscan be initialised at q2 ¼ 42 ¼ 16 different states.

In [4], two coefficients are defined to provide a quantitativemeasure of the randomness (QC) properties of these FSSMs.These two coefficients can be calculated for each response ofthe FSSM under analysis, and they can be different fordifferent initial conditions. The first coefficient is called thedelta function coefficient hd. It measures how similar tothe ideal delta function is the autocorrelation of the outputsequence for the all-zero input [4]. Another coefficientproposed is the noise-like behaviour coefficient sN.

These two coefficients are useful to measure therandomness of the output of these FSSMs. However, thereare FSSMs responses with constant or quasi constantoutputs that sometimes result in good values of the

Figure 2 Examples of the next state– current staterepresentation of a cycle

a Next state–current state representation of a closed cycle oflength 4b Next state–current state representation of a cycle withtransients of length 4


described coefficients, and should not be considered asresponses with good QC properties. Since these twocoefficients are somehow equivalent, we select the deltafunction coefficient hd to characterise a given all-zero inputresponse.

Since the presence and length of long closed cycles, andthe randomness of the response are importantcharacteristics of the constituent FSSM, we have definedthe coefficient DR able to take into account not only thevalue of the delta function coefficient hd for each response,but also the number and length of closed cycles of thecorresponding FSSM.

This coefficient is evaluated in our case over the 16responses of the FSSM by calculating a weighted average ofthe delta function coefficient hd for each response. Theweight is the length of the closed cycle (if any) present atthat response. The random behaviour coefficient DR for theFSSM is

DR ¼1

qsLc max

Xqsi¼1

LC(i)hd(i) (3)

This coefficient measures the random behaviour of theFSSM with more accuracy than that of the mean or peakvalues of the delta function coefficient hd evaluated overthe 16 responses.

LC(i) is the length of the closed cycle present at theresponse for the initial condition i, hd(i) is the deltafunction coefficient measured for the all-zero input case,and for the initial condition i, and Lc_max is the maximumlength of a closed cycle verified for the correspondingFSSM and evaluated over all the responses. If the responsehas cycles with transients, or responses without cycles forthe initial condition i, then LC(i) ¼ 1. When the responseis a closed cycle, LC(i) can adopt any value in the setf2,3,4,6,15g. The addition in (3) excludes stationaryresponses and all-zero outputs.

Another classic parameter to be used for classifying theseFSSMs is the binary Hamming minimum free distancedhfree of each constituent encoder. The Hamming distanceis evaluated over the binary representation of each elementof the corresponding Galois field, taking into account thatthe constituent encoders are nonlinear.

4 Classification of FSSMS in termsof their state-transitions structuresThere are essentially two groups of FSSMs, which will becalled groups A and B. These two groups differ not only inthe length of the cycle but also in the type of cycle of theresponse for each initial condition. In each group, there are


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also subgroups, which are classified by their differences in thestate-transitions structure.

4.1 Group A

The main characteristic of FSSMs of this group is that 15responses for different initial conditions are always closedcycles, in some cases of the maximum length, in other casesof considerable long length. There is always only onestationary response. FSSMs of this group have also themaximum values of the random behaviour coefficient DR.The form of the state-transitions structure, for differentinitial state conditions, allows us to identify three differentsubgroups within the group A.

In Sections 5 and 6, we will show that FSSMs of thisgroup are those that provide the corresponding turbo codewith the best BER performance, among all the 63 casesunder study. This conclusion is confirmed by the EXITchart analysis.

FSSMs of this group have encryption polynomials g(x)that always have one of its three coefficients equal to zero.Table 1 describes the FSSMs of the different groups,detailing the number of closed cycles, cycles withtransients, responses without cycles and stationary responsesfor each FSSM, and the value of the correspondingrandom behaviour coefficient DR. FSSMs in group A withthe best performance are those for which all the possiblevalues of x, the set f0,1,a,a2

g, applied at the input of thepolynomial block, generates one of these same values at theoutput.

4.1.1 Subgroup A.1: FSSMs with 15 closed cycles oflength LC ¼ Lmax ¼ 15. All FSSMs of this group havecoefficient a2 ¼ 0. There are eight FSSMs in this group,which show sequences of states that are limit cycles of themaximum length. These limit cycles are also closed cycleswhich run in the same way from any initial state condition.This subgroup shows the highest value of the randombehaviour coefficient DR.

4.1.2 Subgroup A.2: FSSMs with 12 closed cycles oflength 6, three closed cycles of length 3 and one stationaryresponse.

There are 12 FSSMs in this subgroup. All these FSSMshave 15 closed cycles and one stationary response. In thissubgroup coefficient a1 ¼ 0. Closed cycles are shorter thanin the case of subgroup A.1, and values of the randombehaviour coefficient DR are smaller than those ofsubgroup A.1.

4.1.3 Subgroup A.3: FSSMs with 15 closed cycles oflength 3 and one stationary response. There are fourFSSMs in this subgroup. All these FSSMs have also onlyclosed cycles and one stationary response. All FSSMs ofthis subgroup have coefficients a1 ¼ 1 and a2 ¼ 0. Values


of the random behaviour coefficient DR are slightly smallerthan those of subgroup A.2.

4.2 Group B

In this group, some responses are of the form of cycles withtransients, but there are other responses without cycles and/or stationary responses. The random behaviour coefficient forFSSMs of this group is always DR , 0.3. In Sections 5 and 6,we will show that FSSMs of this group would provide thecorresponding turbo code with poorer BER performance incomparison with the schemes of group A. This conclusionis confirmed by the EXIT chart analysis.

4.2.1 Subgroup B.1: FSSMs with six closed cycles oflength 6, and six cycles with transients of length 6, twocycles of length 2 and two responses without cycles. Thereare 12 FSSMs in this subgroup and they do not showstationary responses. There are just six closed cycles that areshorter than those of subgroup A.1. The values of therandom behaviour coefficient DR are smaller than those ofsubgroups A.1–A.3.

4.2.2 Subgroup B.2: FSSMs with four closed cycles oflength 4, and 12 cycles with transients of length 4. Thereare six FSSMs in this subgroup. These FSSMs do notshow stationary responses. All FSSMs of this subgrouphave coefficient a1 ¼ 1. There are just four closed cycles oflength 4 that are shorter than the cases of subgroups A.1and A.2. Values of the random behaviour coefficient DR aresmaller than those of subgroups A.1–A.3, and comparableto FSSM’s behaviour of subgroup B.1.

4.2.3 Subgroup B.3: FSSMs with six closed cycles oflength 3, six cycles with transients of length 3, tworesponses without cycles and two stationary responses.There are 12 FSSMs in this subgroup. There are just sixclosed cycles of length 3 that are shorter than the case ofsubgroups A.1–A.3, B.1 and B.2. Values of the randombehaviour coefficient DR are smaller than those ofsubgroups A.1–A.3, and comparable and slightly higherthan FSSM’s behaviour of subgroups B.1 and B.2.

4.2.4 Subgroup B.4: FSSMs with two closed cycles oflength 2, 12 responses without cycles and two stationarypoint responses. There are six FSSMs in this subgroup. AllFSSMs of this subgroup have coefficient a1 ¼ 1. Values ofthe random behaviour coefficient DR are smaller than thoseof subgroups A.1–A.3, and comparable to FSSM’sbehaviour of subgroup B.3.

4.2.5 Subgroup B.5: FSSMs with four closed cycles oflength 2, and 12 responses without cycles. There are threeFSSMs in this subgroup. All FSSMs of this subgroup havecoefficients a1 ¼ 0 and a2 ¼ 0. These FSSMs do not showstationary responses. Values of the random behaviourcoefficient DR are smaller than those of subgroups

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Table 1 Classification of FSSMs in terms of their state-transitions structures

Scheme dhfree DR LC ¼ 15 LC ¼ 6 LT ¼ 6 LC ¼ 4 LT ¼ 4 LC ¼ 3 LT ¼ 3 LC ¼ 2 Nocycles

Stationaryresponses

Group A

SubgroupA.1

(0 a 0) 4 0.6942 15 1

(0 a2 0) 4 0.6983 15 1

(1 a 0) 4 0.7186 15 1

(1 a2 0) 4 0.7289 15 1

(a a 0) 4 0.7128 15 1

(a a2 0) 4 0.7108 15 1

(a2 a 0) 4 0.7386 15 1

(a2 a2 0) 4 0.7260 15 1

SubgroupA.2

(0 0 1) 4 0.4478 12 3 1

(0 0 a) 4 0.4084 12 3 1

(0 0 a2) 4 0.3865 12 3 1

(1 0 1) 4 0.4397 12 3 1

(1 0 a) 4 0.5076 12 3 1

(1 0 a2) 4 0.4013 12 3 1

(a 0 1) 4 0.4482 12 3 1

(a 0 a) 4 0.5058 12 3 1

(a 0 a2) 4 0.3861 12 3 1

(a2 0 1) 4 0.4527 12 3 1

(a2 0 a) 4 0.4021 12 3 1

(a2 0 a2) 4 0.3999 12 3 1

SubgroupA.3

(0 1 0) 4 0.3789 15 1

(1 1 0) 4 0.4245 15 1

(a 1 0) 4 0.3853 15 1

(a2 1 0) 4 0.4074 15 1

Group B

SubgroupB.1

(1 a 1) 4 0.2494 6 6 2 2

(1 a a) 4 0.1468 6 6 2 2

Continued

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Table 1 Continued


Stationaryresponses

(1 a2 1) 4 0.1404 6 6 2 2

(1 a2 a2) 5 0.2482 6 6 2 2

(a a 1) 5 0.2324 6 6 2 2

(a a a2) 4 0.2254 6 6 2 2

(a a2 a) 4 0.2206 6 6 2 2

(a a2 a2) 5 0.2477 6 6 2 2

(a2 a a) 4 0.1423 6 6 2 2

(a2 a a2) 4 0.1917 6 6 2 2

(a2 a2 1) 4 0.1636 6 6 2 2

(a2 a2 a) 4 0.2378 6 6 2 2

SubgroupB.2

(1 1 a) 5 0.1762 4 12

(1 1 a2) 4 0.1335 4 12

(a 1 1) 4 0.2087 4 12

(a 1 a) 4 0.1763 4 12

(a2 1 1) 4 0.2148 4 12

(a2 1 a2) 4 0.1344 4 12

SubgroupB.3

(0 a 1) 5 0.2454 6 6 2 2

(0 a a) 4 0.2833 6 6 2 2

(0 a2 a) 4 0.2161 6 6 2 2

(0 a2 1) 4 0.2830 6 6 2 2

(0 a2 a) 4 0.2232 6 6 2 2

(0 a2 a2) 5 0.2411 6 6 2 2

(1 a a2) 4 0.2379 6 6 2 2

(1 a2 a) 4 0.2308 6 6 2 2

(a a a) 4 0.2977 6 6 2 2

(a a2 1) 4 0.2980 6 6 2 2

(a2 a 1) 5 0.2557 6 6 2 2

(a2 a2 a2) 5 0.2600 6 6 2 2

SubgroupB.4

(1 1 1) 4 0.2383 2 12 2

(a 1 a2) 4 0.2443 2 12 2

(a2 1 a) 5 0.2376 2 12 2

Continued

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Table 1 Continued


Stationaryresponses

(0 1 1) 4 0.2361 2 12 2

(0 1 a) 5 0.2688 2 12 2

(0 1 a2) 4 0.3325 2 12 2

SubgroupB.5

(1 0 0) 6 0.1466 4 12

(a 0 0) 6 0.2481 4 12

(a2 0 0) 6 0.1793 4 12

T

A.1–A.3, and comparable to FSSM’s behaviour of subgroupsB.1 and B.2.

5 Ber performance of nonlinearturbo codes using FSSMS designedover GF(4)FSSMs, like those described in Section 4, are utilised asconstituent encoders of nonlinear systematic 1/2-rateconvolutional turbo codes over GF(4). The scheme underconsideration is described in Fig. 1, and makes use ofpuncturing, and a random interleaver of lengthLINT ¼ 10 000. The random interleaver is the same for allthese schemes. The LOG MAP BCJR algorithm [9], witheight iterations, is used in the decoding process. All otherparameters of the turbo code remain the same for all theschemes. This analysis is developed for the waterfall regionof the BER performance, as shown in Fig. 3.

Figure 3 BER performances of some turbo code schemesconstructed using FSSMs defined over GF(4)


It can be seen that schemes of subgroup A.1 perform verywell and close to an equivalent scheme operating over thebinary field GF(2), considered as a reference forcomparison purposes. Other schemes that belong to thesubgroup A.2 have also very good BER performances. Inparticular, the (1 0 a) turbo code, constructed with FSSMswith a random behaviour coefficient DR ¼ 0.5076, and the(a 0 a) turbo code, constructed with FSSMs with arandom behaviour coefficient DR ¼ 0.5058.

Turbo codes constructed using FSSMs of subgroup A.1,and some of subgroup A.2, can be considered as the bestamong those under study. All these schemes areconstructed using FSSMs with the highest value of therandom behaviour coefficient DR.

The (a2 0 1) turbo code, constructed with FSSMs with arandom behaviour coefficient DR ¼ 0.4527, has quite goodBER performance, but it is worse than that of subgroupA.1, and some other FSSMs of subgroup A.2.

The remaining BER performances are always worse thanthose of the above-described schemes, and correspond toschemes for which the random behaviour coefficient valueis smaller, and it is DR , 0.45. For example, the (0 1 0)turbo code has a BER performance slightly worse than the(a2 0 1) turbo code, and the corresponding randombehaviour coefficient DR reduces slightly from DR ¼

0.4527 to 0.3789.

In general terms, turbo codes constructed using FSSMs ofgroup B cannot achieve the best BER performance. All theseturbo codes are constructed using FSSMs whose randombehaviour coefficients are always DR , 0.3. These schemesutilise FSSMs with many cycles with transients, and quiteshort length. This seems to be an undesirable characteristicfor a given FSSM that is constituent encoder ofconvolutional turbo codes.

There are, however, some schemes that have quitegood BER performances in this group. They are always


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slightly worse than that of the (a2 0 1) turbo code of thesubgroup A.2.

On the other hand, group B presents those turbo codes withthe worst BER performance among all the codes under study:the (a2 a 1), (1 a 1) and (a a a) turbo codes, for which therandom behaviour coefficient is DR , 0.3.

Summarising, best nonlinear turbo codes designed overGF(4) use FSSMs of group A (especially subgroups A.1and A.2), for which the response to all initial stateconditions are always closed cycles, with only one stationaryresponse. This fact is taken into account in the calculationof the random behaviour coefficient DR, which is always DR

. 0.37 in group A. Nonlinear turbo codes designed overGF(4) that use FSSMs of group B have, in general, poorerBER performances than those that use FSSMs of group A,and they are never the best achievable.

6 Exit chart analysis of nonlinearturbo codes using FSSMS designedover GF(4)Ten Brink [10–16] introduced a very useful tool for theanalysis of iterative decoders, which is known as the EXITchart. The EXIT chart is an especially good tool for theanalysis of the waterfall region, and also illustrates thebehaviour of the code in the other two regions of the BERperformance curve of a turbo code [10–17]. For each valueof Eb/N0, it describes the relationship between the mutualinformation of the a priori information and the message bitinformation, IA, and the mutual information of theextrinsic information and the message bit information, IE.


The EXIT function is defined as [10, 17]

IE ¼ Tr

IA , Eb

N0

� �(4)

Similarly, an EXIT chart can also describe the mutualinformation IA as a function of the mutual information IE,which can be useful for understanding the process in whichthe extrinsic information of the current iteration becomesthe a priori information of the following iteration.

This analysis is applied to the turbo codes designed in thispaper using the proposed FSSMs under study. Somerelationship is found between the EXIT chart appearanceand the structure of the state transitions and randomness ofthe output for the all-zero input case, quantitativelyindicated by the value of the defined random behaviourcoefficient DR.

Fig. 4 shows two EXIT charts of quite similar aspect,corresponding to schemes of subgroups A.1 and A.2.Fig. 4a is the EXIT chart of the (a2 a 0) turbo code, andFig. 4b is the EXIT chart of the (a 0 a) turbo code. The‘bottleneck’ region of the EXIT chart is given at aroundEb/N0 ’ 0.7 dB for the (a2 a 0) turbo code with anFSSM of subgroup A.1, whereas this is given at aroundEb/N0 ’ 0.9 dB for the (a 0 a) turbo code with an FSSMof subgroup A.2.

As seen in Fig. 3, the BER performance of the (a2 a 0)turbo code is better than that of the (a 0 a) turbo code, inthe region of Eb/N0 ’ 0.7 dB to Eb/N0 ’ 1.2 dB, inagreement with the above EXIT chart analysis. However,the BER performance of the (a 0 a) turbo code is betterthan that of the (a2 a 0) turbo code, in the region of

Figure 4 EXIT chart for the turbo code using an FSSM with coefficients

a EXIT chart for the turbo code using an FSSM with coefficients (a2 a 0) (subgroup A.1)b EXIT chart for the turbo code using an FSSM with coefficients (a 0 a) (subgroup A.2)

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around Eb/N0 ’ 1.2 dB to Eb/N0 ’ 1.5 dB. This seems tobe related to the shape of the EXIT chart. Other turbo codes,designed using FSSMs of the same subgroups, A.1 and A.2,show EXIT charts like those seen in Figs 4a and b,respectively.

Fig. 3 shows the BER performances of reasonably goodturbo codes, like the (a2 0 1), the (0 1 0) and the (0 1 a)turbo codes. Random behaviour coefficients are smallerthan those of FSSMs of subgroup A.1, and thecorresponding BER performances are around 0.6 dB inaverage worse than for the above presented cases.

Fig. 5a shows the EXIT chart of the (a2 0 1) turbo code,subgroup A.2, where the case Eb/N0 ¼ 0.7 dB is notincluded for clarity, because curves IE ¼ Tr (IA, Eb/N0) andIA ¼ Tr (IE, Eb/N0) intersect before the point IE ¼ Tr (IA,Eb/N0) ¼ IA ¼ Tr (IE, Eb/N0) ¼ 1 for that value of Eb/

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N0. The EXIT chart is quite similar to that of the (a 0 a)turbo code of the same subgroup A.2. However, separationbetween curves IE ¼ Tr (IA, Eb/N0) and IA ¼ Tr (IE, Eb/N0) is very small for the EXIT chart of the (a2 0 1) turbocode. This means that the number of iterations required fora good BER performance at Eb/N0 ’ 0.9 dB, for instance,is quite large, and larger than the number of iterationsutilised for calculating the BER performances shown inFig. 3.

Fig. 5b shows the EXIT chart of the (0 1 0) turbo code ofsubgroup A.3. Curves IE ¼ Tr (IA, Eb/N0) and IA ¼ Tr (IE,Eb/N0) intersect before the point IE ¼ Tr (IA, Eb/N0) ¼

IA ¼ Tr (IE, Eb/N0) ¼ 1 for Eb/N0 ¼ 0.7 dB, Eb/N0 ¼

0.9 dB and Eb/N0 ¼ 1.1 dB. This type of intersectingpoint will be referred to as an anticipated intersectingpoint, and it indicates that iterative decoding is notefficient at those values of Eb/N0. In this case, the

Figure 5 EXIT chart of the turbo codes

a EXIT chart of the (a2 0 1) turbo codeb EXIT chart of the (0 1 0) turbo codec EXIT chart of the (0 1 a) turbo code


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separation of the curves IE ¼ Tr (IA, Eb/N0) and IA ¼ Tr (IE,Eb/N0) is very small, so that the required number of iterationsfor some values of Eb/N0 is quite large, and larger than thenumber utilised in this paper, for a suitable performance.The bottleneck shape of the EXIT chart starts at Eb/N0 ’1.3 dB, and is present at high values of the coordinate IA inthe EXIT chart representation. The presence of anticipatedintersecting points is an indication of a poorer BERperformance with respect to schemes for which theseanticipated points are not present in the EXIT chart.

Fig. 5c shows the EXIT chart of the (0 1 a) turbo code ofsubgroup B.4. Curves IE ¼ Tr (IA, Eb/N0) and IA ¼ Tr (IE,Eb/N0) have anticipated intersecting points for Eb/N0 ¼

0.7 dB, Eb/N0 ¼ 0.9 dB, Eb/N0 ¼ 1.1 dB and Eb/N0 ¼

1.3 dB. These curves join in the vicinity of IE ¼ Tr (IA,Eb/N0) ¼ IA ¼ Tr (IE, Eb/N0) ¼ 1 at Eb/N0 ¼ 1.5 dBand higher values of Eb/N0, defining a region which is notexactly of a bottleneck shape, but it is rather similar to twonon-parallel straight lines that approximate their anticipatedintersecting point.

By inspection of the EXIT charts of the (a2 0 1), the(0 0 1) and the (0 1 a) turbo codes, we can say that thecorresponding EXIT chart shows only one anticipatedintersecting point, and a suitable bottleneck shape regionfor the former code, and displays many anticipatedintersecting points, and a rather missing bottleneck shaperegion for the latter. Accordingly, the correspondingrandom behaviour coefficient DR diminishes.

There is a strong relationship between the EXIT chart of aturbo code constructed using FSSMs of group A, and thecorresponding state-transitions structure and value of therandom behaviour coefficient of these FSSMs. Thus, allthe turbo codes constructed using FSSMs of subgroup A.1


have EXIT charts which do not show anticipatedintersecting points in the range of Eb/N0 ¼ 0.7 dB to Eb/N0 ¼ 1.9 dB, like that seen in Fig. 4a.

On the other hand, all the turbo codes constructed usingFSSMs of subgroup A.2 have EXIT charts of the form ofFig. 4b, which shows an anticipated intersecting point forEb/N0 ¼ 0.7 dB.

In the same way, all the turbo codes constructed usingFSSMs of subgroup A.3 have EXIT charts of the form ofFig. 5c, which has anticipated intersecting points for Eb/N0

¼ 0.7 dB, Eb/N0 ¼ 0.9 dB and Eb/N0 ¼ 1.1 dB.Therefore the best BER performance, equivalentlydetermined by the inspection of the corresponding EXITchart, is given for turbo codes constructed using FSSMs ofsubgroup A.1, which show the highest values of therandom behaviour coefficient and also responses to theall-zero input that are always closed cycles of maximumlength, and also for turbo codes constructed using FSSMsof the subgroup A.2, which show relatively high values ofthe random behaviour coefficient and also responses to theall-zero input that are always closed cycles of lengths 6 and 3.

Then, turbo codes constructed using FSSMs of thesubgroup A.3 are those with reasonable high values ofthe random behaviour coefficient and also responses to theall-zero input that are always closed cycles of length 3. Thecorresponding EXIT charts predict a BER performance ofturbo codes constructed using FSSMs of group A, whichdegrades from those using FSSMs of subgroups A.1 andA.2 to those using FSSMs of subgroup A.3.

Fig. 3 shows the BER performances of poor turbo codeslike the (a2 a 1) and the (a a a) turbo codes.

Figure 6 EXIT chart of the turbo codes of subgroub B.3

a EXIT chart of the (a2 a 1) turbo code (FSSM of subgroup B.3)b EXIT chart of the (a a a) turbo code (FSSM of subgroup B.3)

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Fig. 6a shows the EXIT chart of the (a2 a 1) turbo code. Itpresents curves IE ¼ Tr (IA, Eb/N0) and IA ¼ Tr (IE, Eb/N0)with many anticipated intersecting points in the EXIT chartfor Eb/N0 ¼ 0.7, 0.9, 1.1, 1.3, 1.5 and 1.7 dB.

Fig. 6b shows the EXIT chart of the (a a a) turbo code,where curves IE and IA have anticipated intersecting pointsfor all the shown values of Eb/N0. This indicates thatiterative decoding does not make sense at those values ofEb/N0, confirming that the corresponding BERperformances are the poorest among all the evaluated casesin Fig. 3. In spite of slight differences between the twoEXIT charts seen in Figs. 6a and 6b, they both correspondto poor BER performance turbo codes.

Many of the turbo codes constructed using FSSMs ofsubgroups B.1–B.5 are characterised by EXIT charts of theform of Figs. 6a and 6b.

However, some schemes of group B are exceptions to thisrule, and present EXIT charts of the form of Fig. 5b. Theseexceptions are only present in subgroups B.1 and B.3. Thus,the (1 a2 1), the (a a a2) and the (a2 a2 1) turbo codes(FSSMs of subgroup B.1), and the (0 a a2), the (0 a2 1)and the (a a2 1) turbo codes (FSSMs of subgroup B.3),have EXIT charts of the form of Fig. 5b. The highestquality EXIT chart present in group B is of the form ofFig. 5b, and there are no turbo codes in this group withEXIT charts of the form of Figs. 4a and 4b.

In this sense, we can say that, in the case of turbo codesconstructed using FSSMs of group B, there is no completecorrespondence between the state-transitions structure of anFSSM designed over GF(4), and the form of the EXITchart of the turbo code constructed using this FSSM.

The best EXIT charts verified for turbo codes constructedusing FSSMs of group B is like that shown in Fig. 5b, wherethere are three anticipated intersecting points for Eb/N0 ¼

0.7, 0.9 and 1.1 dB.

In general, the form of the EXIT chart has a strongrelationship with two of the coefficients of the polynomialg(x), a1 and a2. Most of the FSSMs of the same subgrouphave very similar trellis structures, but slight differencesbetween these trellis structures define some differences inthe EXIT chart of the corresponding turbo code.

7 ConclusionsIn this paper, we have analysed the performance of turbocodes designed with FSSMs that operate over the finitefield GF(4). A polynomial included in the structure ofthese FSSMs and defined also over GF(4) characterises thebehaviour of these FSSMs. The selected topology presents63 different turbo coding schemes that were studied interms of their BER performances, and equivalently of theirEXIT chart representations.


It is found that some characteristics of the constituentFSSMs are relevant to the BER performance of thecorresponding turbo code. On one hand, FSSMs areclassified by the structure of the sequence of states that themachine has in response to the all-zero input, for differentinitial state conditions. On the other hand, these FSSMsare also characterised by the level of randomness that thecorresponding output shows for the all-zero input.

A coefficient called the random behaviour coefficient isdefined, which takes into account not only the randomnessof the output, but also the state-transitions structure,measured over the responses for all the different initial stateconditions, verified for the all-zero input case.

As a result, FSSMs can be classified into two main groups,called groups A and B. These two groups present someinternal differences, and they are organised in the form ofsubgroups.

The main characteristic of FSSMs of group A is that theresponses for different initial conditions of the all-zero inputcase are always closed cycles, and there is only one stationaryresponse. The length of the closed cycles goes from themaximum length LC ¼ 15, to the minimum LC ¼ 2.FSSMs of group A show the highest value of the randombehaviour coefficient, which is always DR . 0.5. It is alsofound that turbo codes constructed using FSSMs thatbelong to group A show the best BER performances amongall the turbo codes under study. This is confirmed by theEXIT chart analysis. This analysis shows a strongrelationship between the form of the EXIT chart (classifiedby the number of anticipated intersecting points that thischart presents), and the state-transitions structure of theconstituent FSSM, in the case of group A. Slight differencesin the shape of the EXIT chart can however appear.

On the contrary, FSSMs of groupB have responses thatmixclosed cycles with cycles with transients, and also responseswithout cycles. There are also stationary responses in somecases. The main difference with respect to group A is thatnot all the non-stationary responses of these FSSMs areclosed cycles. Since the best turbo codes of group B showEXIT charts of the form of Fig. 5b, similar to the form ofthe EXIT chart of the worst subgroup of group A, thesubgroup A.3, it is concluded that the best FSSMs that areused as constituent encoders of a turbo code are those thatbelong to subgroups A.1 and A.2.

The analysis shows that it is important for these FSSMs tohave all their non-stationary responses for the all-zero inputof the form of closed cycles, and also that the length ofthese closed cycles has to be long enough. Finally, thestrong correspondence between the state-transitionsstructure and value of the random behaviour coefficient ofthe FSSM, and the form of the EXIT chart of thecorresponding turbo code, verified in group A, is notcompletely given in group B.


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8 References

[1] BERROU C., GLAVIEUX A., THITIMAJSHIMA P.: ‘Near Shannonlimit error-correcting coding and decoding: turbo codes’.Proc. 1993 IEEE Int. Conf. Communications, Geneva,Switzerland, May 1993, pp. 1064–1070

[2] HANZO L., LIEW T.H., YEAP B.L.: ‘Turbo coding, turboequalisation and space-time coding, for transmission overfading channels’ (IEEE Press, Wiley, 2001)

[3] HEEGARD C., WICKER S.: ‘Turbo coding’ (Kluwer AcademicPublishers, 1999)

[4] PETRUZZI D.M., CASTINEIRA MOREIRA J., LEVIN D.G.: ‘Quasi-chaotic coding over GF(q)’, IEEE Trans. Commun., 2006,54, (3), pp. 462–468

[5] CASTINEIRA MOREIRA J., PETRUZZI D.M., LIBERATORI M.C., HONARY B.:‘Trellis hopping turbo coding’, IEE Proc., Commun., 2006,153, (6), pp. 966–975

[6] FREY D.R.: ‘Chaotic digital encoding: an approach tosecure communication’, IEEE Trans. Circuits Syst. II,Analog Digit. Signal Process., 1993, 40, pp. 660–666

[7] FREY D.R.: ‘Author’s reply to comments on chaotic digitalencoding: an approach to secure communication’, IEEETrans. Circuits Syst. II, Analog Digit. Signal Process., 1999,46, (11), pp. 1445–1447

[8] CHAMBERS W.G.: ‘Comments on chaotic digital encoding:an approach to secure communication’, IEEE Trans. CircuitsSyst. II, Analog Digit. Signal Process., 1999, 46, (11),pp. 1445–1447

Commun., 2008, Vol. 2, No. 5, pp. 630–641: 10.1049/iet-com:20070178

[9] BAHL L., COCKE J., JELINEK F., RAVIV J.: ‘Optimal decoding oflinear codes for minimising symbol error rate’, IEEE Trans.Inf. Theory, 1974, 20, pp. 284–287

[10] TEN BRINK S.: ‘Convergence behaviour of iterativelydecoded parallel concatenated codes’, IEEE Trans.Commun., 2001, 49, pp. 1727–1737

[11] TEN BRINK S.: ‘Convergence of iterative decoding’,Electron. Lett., 1999, 35, (10), pp. 806–808

[12] TEN BRINK S., SPEIDEL J., YAN R.: ‘Iterative demapping anddecoding for multilevel modulation’. Proc. IEEE GlobecomConf. 98, Sydney, NSW, Australia, November 1998,pp. 579–584

[13] TEN BRINK S.: ‘Convergence of iterative decoding’,Electron. Lett., 1999, 35, (13), pp. 1117–1119

[14] TEN BRINK S., KRAMER G., ASHIKHMIN A.: ‘Design of low-density parity-check codes for modulation and detection’,IEEE Trans. Commun., 2004, 52, (4), pp. 670–678

[15] ASHIKHMIN A., KRAMER G., TEN BRINK S.: ‘Extrinsic informationtransfer functions: a model and two properties’. Proc. Conf.Information Sciences and Systems, Princeton, NJ, 20–22March 2002, pp. 742–747

[16] SHARON E., ASHIKHMIM A., LITSYN S.: ‘EXIT functions for theGaussian channel’. Prov. 40th Annual Allerton Conf.Communication, Control, Computers, Allerton, IL, October2003, pp 972–981

[17] CASTINEIRA MOREIRA J., FARRELL P.G.: ‘Essentials of error-control coding’ (Wiley, 2006)

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EXIT chart analysis of nonlinear turbo coding over GF(4

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