Table Of ContentCan Punctured Rate-1/2 Turbo Codes Achieve a
Lower Error Floor than their Rate-1/3 Parent Codes?
Ioannis Chatzigeorgiou, Miguel R. D. Rodrigues, Ian J. Wassell Rolando Carrasco
Digital Technology Group, Computer Laboratory Department of EE&C Engineering
University of Cambridge, United Kingdom University of Newcastle, United Kingdom
Email: {ic231, mrdr3, ijw24}@cam.ac.uk Email: [email protected]
7 Abstract—Inthispaperweconcentrateonrate-1/3systematic II. PERFORMANCE EVALUATION OF
0 parallel concatenated convolutional codes and their rate-1/2 RATE-1/3TURBO CODES
0 punctured child codes. Assuming maximum-likelihood decoding
2 over an additive whiteGaussian channel,we demonstrate that a Turbo codes, in the form of symmetric rate-1/3 PCCCs,
n rate-1/2non-systematicchildcodecanexhibitalowererrorfloor consist of two identical rate-1/2 recursive systematic
a than that of its rate-1/3 parent code, if a particular condition convolutional encoders separated by an interleaver of size N
J is met. However, assuming iterative decoding, convergence
[10]. The information bits are input to the first constituent
of the non-systematic code towards low bit-error rates is
9 problematic. To alleviate this problem, we propose rate-1/2 convolutional encoder, while an interleaved version of the
partially-systematiccodesthatcanstillachievealowererrorfloor informationbitsareinputtothesecondconvolutionalencoder.
]
T than that of their rate-1/3 parent codes. Results obtained from Theoutputoftheturboencoderconsistsofthesystematicbits
I extrinsicinformationtransferchartsandsimulationssupportour ofthefirstencoder,whichareidenticaltotheinformationbits,
. conclusion.
s the parity check bits of the first encoder and the parity check
c I. INTRODUCTION bits of the second encoder.
[
A puncturedconvolutionalcode is obtainedby the periodic It was shown in [11] and [12] that the performance of
1 elimination of symbols from the output of a low-rate parent a PCCC can be obtained from the input-redundancy weight
v
convolutional code. Extensive analyses on the structure and enumeratingfunctions(IRWEFs)oftheterminatedconstituent
5
6 performanceof puncturedconvolutionalcodeshasshown that recursive convolutional codes. The IRWEF for the case of a
0 their performance is always inferior than the performance of convolutionalcode C assumes the form
1 their low-rate parent codes (e.g. see [1], [2]).
0 AC(W,Z)= AC WwZj, (1)
The performance of punctured parallel concatenated w,j
07 convolutionalcodes (PCCCs), also known as punctured turbo Xw Xj
/ codes, has also been investigated. Design considerationshave whereAC denotesthenumberofcodewordsequenceshaving
s w,j
beenderivedby analytical[3]–[5]as wellassimulation-based
c parity check weight j, which were generated by an input
: approaches [6]–[8], while upper bounds to the bit error sequence of weight w. The overall output weight of the
v
probability (BEP) were evaluated in [5], [9]. The most recent
i codewordsequence,forthecaseofasystematiccode,isw+j.
X papers [7]–[9] demonstrate that puncturing both systematic
The conditional weight enumerating function (CWEF),
r and parity outputs of a rate-1/3 turbo code results in better AC(w,Z), provides all codeword sequences generated by an
a
high-rate turbo codes, in terms of BEP performance, than
input sequence of weight w. Consequently, the relationship
puncturing only the parity outputs of the original turbo code.
between the CWEF and the IRWEF is
The aim of this paper is to explore whether rate-1/2
punctured turbo codes can eventually achieve better AC(W,Z)= AC(w,Z)Ww. (2)
performance than their parent rate-1/3 systematic turbo codes w
X
on additive white Gaussian (AWGN) channels. Assuming
A relationship between the CWEF of a PCCC, P, and the
maximum-likelihood (ML) decoding, we demonstrate that,
CWEF of C, which is one of the two identical constituent
contrary to punctured convolutional codes, punctured turbo
codes, can be easily derived only if we assume the use
codes yielding lower error floors than that of their parent
of a uniform interleaver, an abstract probabilistic concept
code can be constructed. Nevertheless, we cannot be
introducedin[12].Inparticular,ifN isthesizeoftheuniform
conclusive when suboptimal iterative decoding is used. For
interleaverandAC(w,Z)istheCWEFoftheconstituentcode,
thisreason,wealsostudytheconvergencebehaviorofiterative
the CWEF of the PCCC, AP(w,Z), is equal to
decoding and we investigate whether the performance of the
proposedrate-1/2puncturedturbocodesconvergestowardsthe AC(w,Z) 2
theoretical error floor, at low bit error probabilities. AP(w,Z)= . (3)
N
(cid:2) (cid:3)
ThisworkwassupportedbyEPSRCunderGrantGR/S46437/01. w
(cid:18) (cid:19)
TheIRWEF ofP,AP(W,Z), canbethencomputedfromthe A symmetric rate-1/2 NS-PCCC, P′, can be obtained by
CWEF, AP(w,Z), in a manner identical to (2). puncturing the systematic output of a rate-1/3 PCCC, P,
The input-output weight enumerating function (IOWEF) which consists of two identical rate-1/2 recursive systematic
provides the number of codeword sequences generated by convolutional codes. However, P′ can also be seen as a
an input sequence of weight w, whose overall output weight PCCC constructed using two identical rate-1 non-systematic
is d, in contrast with the IRWEF, which only considers the convolutional codes, each one of which has been obtained
output parity check weight j. If P is a systematic PCCC, the by puncturing the systematic bits of a rate-1/2 systematic
corresponding IOWEF assumes the form convolutional code, identical to the one used in P. If C′ is
the punctured rate-1 non-systematic convolutional code and
BP(W,D)= BP WwDd, (4)
w,d C is the parent rate-1/2 systematic convolutional code, their
Xw Xd IRWEFs,AC′(W,Z)andAC(W,Z)respectively,areidentical,
where the coefficients BP can be derived from the i.e.,
coefficients AP of the IRWwE,dF, based on the expressions AC′(W,Z)=AC(W,Z), (9)
w,z
BP =AP , and d=w+j. (5) since, by definition, the IRWEF does not provide information
w,d w,j
about the weight of the systematic bits of a codeword. Thus,
The IOWEF coefficients BP can be used to determine a puncturing of the systematic bits of C will not cause any
w,d
tight upper bound on the BEP for ML soft decoding for the change in its IRWEF.
case of an AWGN channel, as follows Eitherbyapplyingthesamereasoningorbyconsidering(2)
and (3), we find that the IRWEF of the rate-1/2 NS-PCCC,
P ≤ 1 wBP Q 2RPEb ·d , (6) AP′(W,Z), is identical to the IRWEF of its parent code,
B N d w w,d r N0 ! AP(W,Z), i.e.,
XX
where RP is the rate of the turbo code, which in our case is AP′(W,Z)=AP(W,Z). (10)
equal to 1/3. The upper bound can be rewritten as
Puncturinghas an effect only when calculating the IOWEF
P ≤ P(w), (7) ofP′,BP′(W,D).Weusethenotationd′todenotetheoverall
B
w weight of a codeword sequence after puncturing as opposed
X
whereP(w) is the contributionto theoverallBEPof allerror to d, which refersto the overallweightof the same codeword
events having information weight w, and is defined as sequence before puncturing. Therefore the IOWEF of P′ can
be expressed as
P(w)= NwBwP,dQ 2RNPEb ·d . (8) BP′(W,D)= BwP,′d′WwDd′. (11)
d r 0 ! w d′
X XX
Benedetto et al. showed in [12] that the upper bound on Since all systematic bits are punctured, the weight w of the
the BEP of a PCCC using a uniform interleaver of size N informationbitsdoesnotcontributetotheoverallweightd′ of
coincides with the average of the upper bounds obtainable the punctured codeword sequences, and hence it follows that
FfroormsmthaellwvhaoluleescloasfsNof, dtheeterumppineirstbicouinntdercleaanvebres voefrsyizleooNse. BwP,′d′ =APw,j, and d′ =j. (12)
compared with the actual performance of turbo codes using From (5) and (12) we find that the relationship between the
specific deterministic interleavers. However, for N ≥ 1000, IOWEF coefficients, BwP,′d′ and BwP,d, is as follows
igtenhearsallbyeepnerfoobrmserbveetdterththatanradnedteormmliynisgteicneinrateterldeaivnetrerdleeasivgenrss BwP,′d′ =BwP,(d′+w) or BwP,′(d−w) =BwP,d (13)
[13]. Consequently, the upper bound provides a good since
indication of the actual error rate performance of a PCCC, d=d′+w. (14)
when long interleavers are considered.
Thisisequivalenttosayingthatifallinformationsequencesof
III. PERFORMANCE EVALUATION OF weightw areinputtobothP andP′,theoverallweightofthe
RATE-1/2PUNCTURED NON-SYSTEMATICPCCCS generated codeword sequences follows the same distribution
in both cases, but is shifted by w in the case of P′.
Rates higher than 1/3 can be achieved by puncturing
The upper bound on the BEP for ML soft decoding for the
the output of a rate-1/3 turbo encoder. Punctured codes
case of an AWGN channel takes the form
are classified as systematic (S), partially systematic (PS) or
non-systematic (NS) dependingon whether all, some or none P ≤ P′(w), (15)
B
of their systematic bits are transmitted [7]. In this section w
X
we concentrate specifically on rate-1/2 NS-PCCCs, because where P′(w) is given by
their weight enumerating functions can be easily related to
wthiellwneoiwghbteenduemmeornasttirnagtefdu.nctions of their parent codes, as it P′(w)= d′ NwBwP,′d′Q r2RNP0′Eb ·d′! (16)
X
and RP′ = 1/2. Taking into account (13) and (14), we can 10−3
rewrite (16) as a function of d and BP , as follows Upper bound, Rate−1/3 PCCC(1,5/7,5/7)
w,d P(2), Rate−1/3 PCCC(1,5/7,5/7)
10−4 P(3), Rate−1/3 PCCC(1,5/7,5/7)
P′(w)= NwBwP,dQ 2RNP′Eb ·(d−w) (17) 10−5 UPP′′p((23p))e,,r RR baaottueen−−d11,// 22R NNatSSe−−−PP1/CC2CC NCCS((−11P,,55C//77C,,55C//(771)),5/7,5/7)
XIdV. PERFORM ArNCE A0NALYSIS ! bability10−6
o
Pr
neeBdesfotroe bceonmtineutinsog ttohatthea draetreiv-1a/t2ionNoSf-PaCCcoCndciatinonacwhhieicvhe Bit Error 10−7
a better ML bound than its parent rate-1/3 PCCC, we first 10−8
enumerate a number of results, derived and justified in [14]:
1) The minimum information weight w for recursive 10−9
min
convolutional encoders is w =2.
min 10−10
2) For recursive constituentcodesand long interleavers, the 0 1 2 3 4 5 6 7
E/N (dB)
b 0
contribution to the overall BEP of all error events with
oddinformationweightisnegligible.Furthermore,asthe Fig. 1. Upper bounds and contributions to the BEP of all error events
withinformationweightof2and3,fortherate-1/2NS-PCCC(1,5/7,5/7)
interleaver size increases, the contribution to the overall
anditsparentrate-1/3 PCCC.Theinterleaver sizeis1,000.
BEP of all error events with information weight w is
min 10−3
dominant.
Upper bound for parent rate−1/3 PCCC
3) The upper bound of a PCCC which uses recursive 10−4 Upper bound for rate−1/2 NS−PCCC
constituent codes, depends on its free effective distance
dfree.eff,whichcorrespondstotheminimumoveralloutput 10−5
Itisawlseoigshtrtawighhetnfotrhwearidnftoormveartiifoynthwaeti,gahltthiosuwghmipn.uncturingof bability10−6 PCCC(1,2/3,2/3)
o
tohnetshyestBemEPa,titchbeitpsreovfiaourasltye-h1i/g3hPliCgChtCedaftfreecntdssthsetilulpappeprlyb.ound Error Pr10−7 PCCC(1,7/5,7/5)
Based on (7) and (15), the rate-1/2 NS-PCCC, P′, will Bit 10−8 PCCC(1,5/7,5/7)
achieve a better bound than its parent, if
10−9
P′(w)< P(w). (18)
PCCC(1,17/15,17/15)
w≥2 w≥2 10−10
X X 0 1 2 3 4 5 6 7 8 9 10
For large interleaver sizes, the dominant terms will be P′(2) Eb/N0 (dB)
and P(2), thus (18) reducesto P′(2)<P(2), or equivalently
Fig.2. Comparissonofupperboundsforvariousrate-1/2NS-PCCCsand
Q(f′(d))< Q(f(d)), (19) theirparentrate-1/3 PCCCs.Theinterleaver sizeis10,000.
d≥Xdfree.eff d≥Xdfree.eff
where f′(d) and f(d) are defined as AftersubstitutingRP′ andRP with1/2and1/3respectively,
we find that a rate-1/2 NS-PCCC can achieve a lower upper
f′(d)= 2RP′Eb ·(d−2), bound on the BEP, over an AWGN channel, than its rate-1/3
N
r 0 (20) parent PCCC only if the effective free distance of the parent
2R E
P b PCCC, d , meets the condition
f(d)= ·d, free.eff
N
r 0
d >6. (24)
according to (8) and (17). free.eff
FunctionQ(ξ) is a monotonicallydecreasingfunctionof ξ, The contributions of the error events with weight 2 and 3,
where ξ is a real number. Consequently, if ξ1 and ξ2 are real for the previouscase, are examined in Fig.1. We observe that
numbers with ξ1 > ξ2, it follows that Q(ξ1) < Q(ξ2), and up to a certain value of Eb/N0, the rate-1/2 PCCC(1,5/7,5/7)
vice versa, i.e, exhibits a lower upper bound than its parent code. Note that
for this range of E /N values, the inequality P′(2)< P(2)
Q(ξ )<Q(ξ )⇔ξ >ξ . (21) b 0
1 2 1 2 is satisfied. Although P(3) and P′(3) do not significantly
Therefore, inequality (19) is satisfied if affecttheboundsatlowE /N values,theyplayanimportant
b 0
f′(d)>f(d), for everyd≥d . (22) roleat higherEb/N0 values.However,for a largerinterleaver
free.eff
size (e.g., N = 10,000), P′(2) is dominant, as explained
Both f′(d) and f(d) are monotonically increasing functions, previously. More specifically, P′(2) determines the upper
therefore (22) holds true if only f′(dfree.eff)>f(dfree.eff), or bound of the rate-1/2 NS-PCCC(1,5/7,5/7), which is always
2RP′ lower than the upper bound of its parent rate-1/3 PCCC for
d > . (23)
free.eff RP′ −RP therangeofEb/N0valuesinvestigated,asweobserveinFig.2.
1 1 1
E/N=4.5 dB
b 0
0.8 E/N=1.5 dB 0.8 0.8
b 0
E/N=1.5 dB
b 0
0.6 0.6 0.6
E/N=1.0 dB
I,IE1A2 Eb/N0=0.2 dB I,IE1A2 b 0 I,IE1A2 Eb/N0=0.8 dB
0.4 0.4 0.4
0.2 0.2 0.2
Eb/N0=0.2 dB, N=106 Eb/N0=1.0 dB, N=106 Eb/N0=0.8 dB, N=106
Eb/N0=1.5 dB, N=103 Eb/N0=4.5 dB, N=103 Eb/N0=1.5 dB, N=103
0 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
I ,I I ,I I ,I
A1E2 A1E2 A1E2
(a) Rate-1/3 PCCC(1,5/7,5/7) (b) Rate-1/2 NS-PCCC(1,5/7,5/7) (c) Rate-1/2 PS-PCCC(1,5/7,5/7)
Fig.3. Extrinsicinformationtransfercharacteristicsofiterativedecodingforvariousturbocodesusinganinterleaversizeof106bits.Twodecodingaveraged
trajectories foreachcasearealsoplotted; oneforaninterleaver sizeof103 bitsandtheotherforaninterleaver sizeof106 bits.
In Fig.2, we also observe that in all cases except uses the extrinsic output I = T (0,E /N ) of the first
E1,1 b 0
for PCCC(1,2/3,2/3), the rate-1/2 NS-PCCCs achieve better decoderasa-prioriknowledge,i.e.,I =I .Theextrinsic
A2,1 E1,1
bounds than their parent PCCCs. The reason is that the free output of the second decoder, I = T (I ,E /N ), is
E2,1 A2,1 b 0
effective distance of the parent rate-1/3 PCCC(1,2/3,2/3) is forwarded to the first decoder to become a-priori knowledge
d =4, thuscondition(24) is notmet. For all other turbo during the next iteration, i.e., I =I , and so on. Note
free.eff A1,2 E2,1
codes investigated in Fig.2, d >6 holds true. that convergence to the (I ,I )=(1,1) point, i.e., towards
free.eff A E
low BEPs, occurs if the transfer characteristics do not cross.
V. CONVERGENCE CONSIDERATIONS
As an example, we consider the rate-1/3 systematic
TheupperboundontheBEPforMLsoftdecodingprovides
PCCC(1,5/7,5/7) to be the parent code. Fig.3(a) shows the
an accurate estimate of the suboptimal iterative decoder
transfer characteristics of the constituent decoders for the
performance at high E /N values, for an increasing number
b 0 parent PCCC using an interleaver of size N = 106. We
of iterations [12]. Since the performance of rate-1/3 PCCCs
see that for E /N = 0.2 dB, the decoder characteristics
graduallyconvergesto the ML bound,this boundcan be used b 0
do not intersect and the averaged decoding trajectory [15]
to predict the BEP error floor region of the corresponding
manages to go through a narrow tunnel. Fig.3(b) depicts the
code. However, when puncturing occurs, we need to explore
decoder characteristics for the rate-1/2 NS-PCCC employing
whether the performance of the iterative decoder eventually
an interleaver of the same size. For E /N = 1.0 dB,
convergesto the ML bound. For this reason, we use extrinsic b 0
the averaged trajectory just manages to pass through a
information transfer (EXIT) chart analysis [15], which can
narrow opening, which appears close to the starting point
accurately predict the convergence behavior of the iterative
(0,0). Therefore, for long interleavers, the performance of
decoder for very large interleaver sizes (e.g., N=106 bits).
the suboptimal iterative decoder for the rate-1/2 NS-PCCC
An iterative decoder consists of two soft-input/soft-output
convergestowardsthe errorfloor region,defined by the upper
decoders.Eachdecoderusesthereceivedsystematicandparity
bound, and eventually outperforms its rate-1/3 parent PCCC.
bits as well as a-priori knowledge from the previous decoder
However, convergence begins at a higher E /N value and a
to produce extrinsic information on the systematic bits. Ten b 0
larger number of iterations is required.
Brink described the decoding algorithm process using EXIT
chartanalysis[15].Tothisend,theinformationcontentofthe In Fig.3(a) and Fig.3(b) the averaged trajectories for the
a-priori knowledge is measured using the mutual information more practical interleaver size of 1,000bits are also depicted.
I between the information bits at the transmitter and the For E /N =1.5 dB the trajectory for rate-1/3 PCCC quickly
A b 0
a-priori input to the constituent decoder. Mutual information converges towards low BEPs. However, the trajectory for
I is also used to quantify the extrinsic output. The extrinsic rate-1/2 NS-PCCC dies away after 2 iterations, even for an
E
information transfer characteristics are then defined as a E /N value of 4.5 dB, due to the increasing correlations
b 0
function of I and E /N , i.e., I = T (I ,E /N ). By of extrinsic information. We attribute this problem to the
A b 0 E A b 0
plottingthemutualinformationtransfercharacteristicsofboth absence of received systematic bits, which causes erroneous
constituent decoders in a single EXIT chart, evolution of the decisions.As a result,errorpropagationprohibitsthe iterative
iterative decoding process can be visualised. decoder from converging. Thus, for small and more practical
Duringthefirstiteration,thefirstdecoderdoesnothaveany interleaver sizes, the performance of rate-1/2 NS-PCCC does
a-priori knowledge, thus I =0, while the second decoder not gradually approach the upper bound for ML decoding.
A1,1
To alleviate this problem, the turbo encoder could send 100 100
some systematic bits, while keeping the rate equal to 1/2
by puncturing some parity bits. In [9] we have presented
a technique for deriving good punctured codes and we bility10−2 bility10−2
have identified a rate-1/2 PS-PCCC that achieves the second ba ba
o o
best performance bound for ML decoding, after the rate-1/2 Pr Pr
or or
NcoSn-sPtiCtuCeCnt. Fdeigc.o3d(ce)rsshaoswwsetlhleastratnhsefearvecrhaagreadctetrriasjteiccstoroyf tfhoer Bit Err10−4 Bit Err10−4
N=106. A comparison with Fig.3(b) for the NS-PCCC case,
reveals that a lower Eb/N0 is required and less iterations 10−6 PCCC(1,7/5,7/5) 10−6 PCCC(1,5/7,5/7)
are needed, in order for the rate-1/2 PS-PCCC to converge. 0 1 2 3 0 1 2 3
E/N (dB) E/N (dB)
b 0 b 0
Furthermore,thetrajectoryforaninterleaversizeof1,000bits
Upper bound for rate−1/3 PCCC
reaches the top cornerof the EXIT chart for the same Eb/N0 Upper bound for rate−1/2 NS−PCCC
as the parent rate-1/3 PCCC, guaranteeing that the iterative Simulation for rate−1/3 PCCC (10 iterations)
decoder will converge towards low BEP values. Simulation for rate−1/2 NS−PCCC (10 iterations)
Simulation for rate−1/2 PS−PCCC (10 iterations)
VI. SIMULATION RESULTS
Fig. 4. Comparison between the upper bounds and simulation results for
The process for selecting rate-1/2 PS-PCCCs that lead to PCCCsusinganinterleaver sizeof1,000bits.
superiorperformancethantheir parentrate-1/3PCCCs can be
summarized in two steps. We first implement the technique
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