Motivated by a recently published robust geometric programming approximation, a generalized approach for approximating efficiently the max* operator is presented. Using this approach, the max* operator is approximated by means of a generic and yet very simple max operator, instead of using additional correction term as previous approximation methods require. Following that, several turbo decoding algorithms are obtained with optimal and near-optimal bit error rate (BER) performance depending on a single parameter, namely the number of piecewise linear (PWL) approximation terms. It turns out that the known max-log-MAP algorithm can be viewed as special case of this new generalized approach. Furthermore, the decoding complexity of the most popular previously published methods is estimated, for the first time, in a unified way by hardware synthesis results, showing the practical implementation advantages of the proposed approximations against these method

Martina, Maurizio

Masera, Guido

Mathiopoulos P.T

Papaharalabos, S.

English

Motivated by a recently published robust geometric programming approximation, a generalized approach for approximating efficiently the max* operator is presented. Using this approach, the max* operator is approximated by means of a generic and yet very simple max operator, instead of using additional correction term as previous approximation methods require. Following that, several turbo decoding algorithms are obtained with optimal and near-optimal bit error rate (BER) performance depending on a single parameter, namely the number of piecewise linear (PWL) approximation terms. It turns out that the known max-log-MAP algorithm can be viewed as special case of this new generalized approach. Furthermore, the decoding complexity of the most popular previously published methods is estimated, for the first time, in a unified way by hardware synthesis results, showing the practical implementation advantages of the proposed approximations against these methods

PAPAHARALABOS S

MATHIOPOULOS P. T

MASERA, Guido

MARTINA, MAURIZIO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

09 November 2022POLITECNICO DI TORINORepository ISTITUZIONALEOn optimal and near-optimal turbo decoding using generalized max operator / Papaharalabos, S; MATHIOPOULOS P.,T; Masera, Guido; Martina, Maurizio. - In: IEEE COMMUNICATIONS LETTERS. - ISSN 1089-7798. - STAMPA. -13:7(2009), pp. 522-524. [10.1109/LCOMM.2009.090537]OriginalOn optimal and near-optimal turbo decoding using generalized max operatorPublisher:PublishedDOI:10.1109/LCOMM.2009.090537Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2265278 since:IEEEIEEE COMMUNICATIONS LETTERS, VOL. X, NO. XX, XXXXX 2009 1On Optimal and Near-Optimal Turbo DecodingUsing Generalizedmax∗ OperatorStylianos Papaharalabos,Member, IEEE, P. Takis Mathiopoulos,Senior Member, IEEE,Guido Masera,Senior Member, IEEE, and Maurizio Martina,Member, IEEE.Abstract—Motivated by a recently published robust geometricprogramming approximation, a generalized approach for approx-imating efficiently the max∗ operator is presented. Using this ap-proach, themax∗ operator is approximated by means of a genericand yet very simple max operator, instead of using additionalcorrection term as previous approximation methods require.Following that, several turbo decoding algorithms are obtainedwith optimal and near-optimal bit error rate (BER) performancedepending on a single parameter, namely the number of piecewiselinear (PWL) approximation terms. It turns out that the knownMax-Log-MAP algorithm can be viewed as special case of thisnew generalized approach. Furthermore, the decoding complexityof the most popular previously published methods is estimated,for the first time, in a unified way by hardware synthesis results,showing the practical implementation advantages of the proposedapproximations against these methods.Index Terms—Turbo codes, iterative decoding.I. I NTRODUCTIONOVER the past decade or so several algorithmic ap-proaches aiming to simplify the well-knownmax∗ oper-ator [1] for decoding turbo codes [2] have been proposed andanalyzed. These algorithms include: Improved Max-Log-MAP[3], Constant Log-MAP [4], Linear Log-MAP [5], AverageLog-MAP [6], and recently the algorithms in [7], [8]. Thepenalty paid for such approximations is a small bit errorrate (BER) performance degradation as compared with theperformance achieved by the optimal Log-MAP algorithm [9].However, although these sub-optimal algorithms are compu-tationally simpler as compared with the optimal solution, aadditional correction term is required to be added to themaxoperation in order to minimize performance degradation.Geometric programing is an optimization problem used invarious fields, such as information theory, analog/RF circuitdesign, power control of wireless communication networksand statistics [10]. However, its application to turbo decodinghas not been investigated so far. Recently, the authors in [11]have dealt with the convex log-sum-exp (lse) function andconsidered a robust geometric programming problem as robustS. Papaharalabos and P. T. Mathiopoulos are with the Institute for SpaceApplications and Remote Sensing (ISARS), National Observato y of Athens,Vas. Pavlou and Metaxa Str., P. Penteli, 15236, Athens, Greece ( -mail:spapaha@space.noa.gr).G. Masera and M. Martina are with the VLSI Lab, Dipartimento diElettronica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino,Italy.This research work was supported by EU IST NEWCOM++ (IST-216715)project.Manuscript received Month xx, 2009.linear programming problem. In particular, a constructiveal-gorithm was presented in order to find the best piecewise linear(PWL) approximation terms of the bivariatelse function. Theirresearch has shown that the exact number of PWL termsdepends on the approximation error resulting from the originalbivariatelse function.In this letter, we modify and apply for the first time theoptimal approximations presented in [11] to decode turbocodes noting that the bivariatelse function is equivalent tothemax∗ operator. Consequently, a generalized approximationfor the max∗ operator is obtained using: (i) a small numberof efficient PWL terms that ease a hardware implementation;and (ii) the very simplemax operator. The main feature of thisapproximation is that there is no need to use any correctionterm, in contrast to all previously published methods [4]–[8]. Furthermore, it turns out that the known Max-Log-MAPalgorithm can be viewed as special case of this new approach,when two PWL terms are considered. Our research has shownthat by considering four PWL approximation terms with theadditional use of scaling as in [3], the resulting turbo codeBER performance is essentially identical to the performanceof the optimal Log-MAP algorithm. Hardware synthesis resultshave shown that by considering three PWL approximationterms, the resulting algorithm outperforms most of the pre-viously published methods, i.e. [5]–[8], in terms of occupiedarea savings, and achieves near Log-MAP performance.II. REDUCED COMPLEXITY TURBO DECODINGIn this section the most important reduced complexity turbodecoding algorithms are reviewed. Consider an informationsequence ofN bits denoted with̄u = [u1, u2, . . . , uN ]. Thissequence is turbo encoded, then each coded bit is binaryphase-shift keying (BPSK) modulated, taking values from thealphabet{±1} with equal probabilities, and transmitted withbit energyEb over an additive white Gaussian noise (AWGN)channel with one-sided power spectral densityNo. At thereceiver the turbo decoder estimates the transmitted sequenceof bits.The max∗ operation, i.e. Jacobian logarithm, used in turbodecoding is defined as [1]max∗(x1, x2)4= log{exp(x1) + exp(x2)} = max(x1, x2)+ ln {1 + exp(− |x1 − x2|)} = max(x1, x2) + fc(|x1 − x2|)(1)where fc(|x1 − x2|) is a non-linear function referred to as‘correction term’ [9] and|.| denotes absolute value. For moreIEEE COMMUNICATIONS LETTERS, VOL. X, NO. XX, XXXXX 2009 2than two arguments, the Jacobian logarithm is applied recur-sively. For example, considering three arguments, it yieldsmax∗(x1, x2, x3) = max∗ {max∗(x1, x2), x3} . (2)For the Log-MAP algorithm, a look-up table (LUT) substi-tutesfc(|x1 − x2|), which is usually implemented with eightvalues [9]. If LUT is omitted, then the Log-MAP simplifiesto the Max-Log-MAP algorithm. In the past, several reducedcomplexity decoding algorithms have been obtained by ap-proximating fc(|x1 − x2|) with different methods, e.g. see[4]–[8], a summary of which is presented in Table I.Having evaluated the performance of reduced complexitydecoding algorithms given in Table I and as shown in Fig. 1(for the turbo code simulation parameters see the next section),it is concluded that at BER of10−4: (i) The Max-Log-MAPis the worst performing algorithm with approximately 0.4 dBdegradation as compared with Log-MAP; (ii) The maximumperformance degradation of reduced complexity algorithmsascompared with Log-MAP is approximately 0.1 dB; and (iii)the Linear Log-MAP and also the algorithm of [7] achieve thebest, i.e. near Log-MAP performance. A thorough complexityestimation of these algorithms is reported in the next section.III. O PTIMAL max∗ APPROXIMATIONS AND THEIRAPPLICATION TO TURBO DECODINGFrom pure mathematical curiosity, the authors in [11] have,instead of approximatingfc(|x1 − x2|), approximated (1) asa whole, i.e. themax∗ operator directly. Hence, (1) becomesmax∗(x1, x2)≈max(κ1∗x1+λ1∗x2+µ1, . . . , κi∗x1+λi∗x2+µi)(3)whereκi, λi, andµi are real positive values andi ≥ 2. Thebest PWL approximations of themax∗ operator with differentnumber of terms are shown in Table II. The approximationerror reduces in the order of√2/r2 and for practical applica-tions 5 ≤ r ≤ 10 has been considered [11]. It is underlinedthat in case of turbo decoding, ther = 2 approximation isidentical to the Max-Log-MAP algorithm.Performance evaluation results for various values ofr, withthe additional use of scaling as in [3], have shown that at BERof 10−5: (i) Both r = 5 and r = 4 approximations achieveessentially identical to the Log-MAP performance; and (ii)Ther = 3 approximation has performance degradation of less than0.03 dB against the Log-MAP algorithm. In order to ease ahardware implementation, ther = 3 andr = 4 approximationshave been modified, respectively asmax∗(x1, x2)≈max[x1, 0.5 ∗ (x1 + x2 + 1), x2] (4)max∗(x1, x2)≈max(x1, 0.25 ∗ x1 + 0.75 ∗ x2 + 0.5,0.75 ∗ x1 + 0.25 ∗ x2 + 0.5, x2).(5)In terms of implementation, synthesizable VHDL descrip-tions have been produced for themax∗ approximations of (4)and (5) as well as for Log-MAP with LUT, Max-Log-MAPand the algorithms shown in Table I. In order to derive faircomparisons, the same area optimization effort of the synthesistool must be guaranteed for all cases. To this purpose, althoughall considered implementations of themax∗ operation arepure combinational architectures, registers have been placedat the architecture inputs and output. This allows setting aunique clock frequency constraint for all considered cases,fCK = 200 MHz. Synthesis results obtained in terms of areaoccupied by the combinational part on a 130 nm standard cellCMOS technology are given in Table III for precision metricsrepresented with 8, 10 and 12 bits, respectively. These resultsshow, on the one hand, that the proposedr = 3 approximationoutperforms most of the previously published methods, i.e.[5]–[8]. In particular, its occupied area is 35% smaller than thatrequired by the Log-MAP algorithm and it is only inferior tothe Constant Log-MAP algorithm by 8%. On the other hand,the proposedr = 4 approximation has comparable complexitywith the method in [7] and outperforms [5], [6], [8].Performance evaluation results have been obtained for themost efficient method, i.e. Constant Log-MAP, ther = 3and r = 4 proposed approximations, and these are illustratedin Fig. 2. A 16-states turbo code is considered with codingrate equal to 1/2 and generator polynomials(1, 33/23)o inoctal form representing the feed-forward and backward poly-nomials, respectively. Furthermore, an information sequenceof N = 103 bits is assumed, whereas the total number oftransmitted frames is105. A pseudo-random turbo interleaveris considered and at the receiver a maximum of 10 decoding it-erations are performed. In order to reduce computer simulationtime, a genie stopping rule is assumed at the turbo decoder. Incomputer-based simulations the scaling factor, denoted with s,was constant when varying the number of decoding iterations,having the following values: (i) 0.65 for Max-Log-MAP; (ii)0.9 for Log-MAP; (iii) 0.85 for Constant Log-MAP; and (iv)0.75 for the rest of the algorithms.As shown in Fig. 2, the best performance is achieved bythe Constant Log-MAP algorithm and the proposedr = 4approximation followed by ther = 3 approximation. In moredetail, at BER of10−4 and with respect to the Log-MAPalgorithm: (i) The Constant Log-MAP algorithm has perfor-mance degradation of approximately 0.03 dB; (ii) Ther = 4approximation has comparable performance with the ConstantLog-MAP algorithm; and (iii) Ther = 3 approximation hasperformance degradation of 0.05 dB. However, bothr = 3and r = 4 approximations offer practical implementationadvantages with respect to other methods.IV. CONCLUSIONIt has been shown that themax∗ operator used in turbodecoding can be simplified into different number of PWLapproximation terms in an efficient way usingmax onlyoperation. The proposedr = 3 approximation is 0.05 dBinferior to the Log-MAP algorithm but 35% much simpler.Further, it outperforms, in terms of occupied area savings,most of the previously published methods, such as Linear Log-MAP [5], Average Log-MAP [6], and the algorithms from[7], [8]. The proposedr = 4 approximation is only 0.03 dBinferior to the Log-MAP algorithm and it it less complex thanthe methods presented in [5], [6], [8].IEEE COMMUNICATIONS LETTERS, VOL. X, NO. XX, XXXXX 2009 3TABLE ITHE MOST IMPORTANT REDUCED COMPLEXITY TURBO DECODING ALGORITHMS USING DIFFERENT APPROXIMATION INfc(|x1 − x2|) OF (1).Constant Log-MAP [4] Linear Log-MAP [5] Average Log-MAP [6]fc(|x1 − x2|){3/8, if |x1 − x2| < 20, otherwise max(ln 2 − 0.25 ∗ |x1 − x2| , 0){ln 2 + 0.5 ∗ (x1 + x2), if |x1 − x2| < 2 ∗ ln 20, otherwiseRef. [7] Ref. [8] Max-Log-MAPfc(|x1 − x2|) ln 2 ∗ 2−|x1−x2| max(0, ln 2 − 0.5 ∗ |x1 − x2|) 0TABLE IIBEST APPROXIMATIONS OF THEmax∗ OPERATION USING DIFFERENTNUMBER OF PWL TERMS AND DEPLOYINGmax ONLY OPERATION [11].No. of Resultingmax∗ Approximation No. of Approx.Terms (r) max Ops. Error2 max(x1, x2) 1 0.693max(x1,3 0.5 ∗ (x1 + x2) + 0.693, 2 0.223x2)max(x1,0.271 ∗ x1 + 0.729 ∗ x2 + 0.584,4 0.729 ∗ x1 + 0.271 ∗ x2 + 0.584, 3 0.109x2)max(x1,0.167 ∗ x1 + 0.833 ∗ x2 + 0.45,5 0.5 ∗ x1 + 0.5 ∗ x2 + 0.693, 4 0.0650.833 ∗ x1 + 0.167 ∗ x2 + 0.45,x2)TABLE IIIOCCUPIED AREA COMPARISON(SQUAREµm) OF DIFFERENTmax∗APPROXIMATIONS ON A 130 NM STANDARD CELL TECHNOLOGY ANDVARIOUS PRECISION METRICS REPRESENTATIONS.Algorithm 8 bits 10 bits 12 bitsMax-Log-MAP 250.133 312.666 373.182Log-MAP 1022.72 1276.888 1613.76r = 3 of (4) 653.573 831.086 1006.583r = 4 of (5) 883.534 1149.804 1426.160Constant Log-MAP [4] 599.108 764.519 931.946Linear Log-MAP [5] 978.342 1264.784 1559.296Average Log-MAP [6] 1069.116 1363.627 1555.261Ref. [7] 891.602 1135.684 1377.748Ref. [8] 1137.701 1456.418 1758.988REFERENCES[1] A. J. Viterbi, “An intuitive justification and a simplifiedimplementationof the MAP decoder for convolutional codes,”IEEE J. Sel. AreasCommun., vol. 16, no. 2, pp. 260–264, Feb. 1998.[2] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannolimiterror correcting coding and decoding: Turbo codes,” inIEEE Int. Conf.Commun. (ICC), Geneva, Switzerland, May 1993, pp. 1064–1070.[3] J. Vogt and A. Finger, “Improving the Max-Log-MAP turbo decoder,”IEE Electron. Lett., vol. 36, no. 23, pp. 1937–1939, Nov. 2000.[4] W. J. Gross and P. G. Gulak, “Simplified MAP algorithm suitable forimplementation of turbo decoders,”IEE Electron. Lett., vol. 34, no. 16,pp. 1577–1578, Aug. 1998.[5] J. F. Cheng and T. Ottosson, “Linearly approximated Log-MAP algo-rithms for turbo decoding,” inIEEE Vehic. Techn. Conf. (VTC), Spring,Tokyo, Japan, May 2000, pp. 2252–2256.[6] B. Classon, K. Blankenship, and V. Desai, “Channel coding for 4Gsystems with adaptive modulation and coding,”IEEE Wireless Commun.Mag., vol. 9, no. 2, pp. 8–13, Apr. 2002.[7] H. Wang, H. Yang, and D. Yang, “Improved Log-MAP decodingalgorithm for turbo-like codes,”IEEE Commun. Lett., vol. 10, no. 3,pp. 186–188, Mar. 2006.[8] S. Talakoub, L. Sabeti, B. Shahrrava, and M. Ahmadi, “An improvedMax-Log-MAP algorithm for turbo decoding and turbo equalizt on,”IEEE Trans. Instrum. Meas., vol. 56, no. 3, pp. 1058–1063, Jun. 2007.[9] P. Robertson, E. Villebrun, and P. Hoeher, “A comparison of optimal andsub-optimal MAP decoding algorithms operating in the log domain,” in0.75 1 1.25 1.510−410−310−210−1100Eb/No (dB)BERMax−Log−MAPAverage Log−MAPRef. [8]Constant Log−MAPLog−MAPLinear Log−MAPRef. [7]Fig. 1. Turbo code BER performance comparison with the most importantreduced complexity decoding algorithms.0.75 1 1.25 1.5 1.7510−610−510−410−310−210−1Eb/No (dB)BERr=2 terms (Max−Log−MAP), s=0.65r=3 terms, s=0.75r=4 terms, s=0.75Constant Log−MAP, s=0.85Log−MAP, s=0.9Fig. 2. Turbo code BER performance with efficient approximations f themax∗ operator and different number of PWL terms. The extrinsic informationis scaled by a factor ofs.IEEE Int. Conf. Commun. (ICC), Seattle, USA, Jun. 1995, pp. 1009–1013.[10] S. Boyd and L. Vandenberghe,Convex optimization. CambridgeUniversity Press, 2004.[11] K. L. Hsiung, S. J. Kim, and S. Boyd, “Tractable approximate robustgeometric programming,”Springer Optim. Eng. Journal, vol. 9, no. 2,pp. 95–118, Jun. 2008, available on-line.

On optimal and near-optimal turbo decoding using generalized max operator

PORTO Publications Open Repository TOrino

IEEE COMMUNICATIONS LETTERS, VOL. X, NO. XX, XXXXX 2009 1
On Optimal and Near-Optimal Turbo Decoding
Using Generalized max∗ Operator
Stylianos Papaharalabos, Member, IEEE, P. Takis Mathiopoulos, Senior Member, IEEE,
Guido Masera, Senior Member, IEEE, and Maurizio Martina, Member, IEEE.
Abstract—Motivated by a recently published robust geometric
programming approximation, a generalized approach for approx-
imating efficiently the max∗ operator is presented. Using this ap-
proach, the max∗ operator is approximated by means of a generic
and yet very simple max operator, instead of using additional
correction term as previous approximation methods require.
Following that, several turbo decoding algorithms are obtained
with optimal and near-optimal bit error rate (BER) performance
depending on a single parameter, namely the number of piecewise
linear (PWL) approximation terms. It turns out that the known
Max-Log-MAP algorithm can be viewed as special case of this
new generalized approach. Furthermore, the decoding complexity
of the most popular previously published methods is estimated,
for the first time, in a unified way by hardware synthesis results,
showing the practical implementation advantages of the proposed
approximations against these methods.
Index Terms—Turbo codes, iterative decoding.
I. INTRODUCTION
OVER the past decade or so several algorithmic ap-proaches aiming to simplify the well-known max∗ oper-
ator [1] for decoding turbo codes [2] have been proposed and
analyzed. These algorithms include: Improved Max-Log-MAP
[3], Constant Log-MAP [4], Linear Log-MAP [5], Average
Log-MAP [6], and recently the algorithms in [7], [8]. The
penalty paid for such approximations is a small bit error
rate (BER) performance degradation as compared with the
performance achieved by the optimal Log-MAP algorithm [9].
However, although these sub-optimal algorithms are compu-
tationally simpler as compared with the optimal solution, an
additional correction term is required to be added to the max
operation in order to minimize performance degradation.
Geometric programing is an optimization problem used in
various fields, such as information theory, analog/RF circuit
design, power control of wireless communication networks
and statistics [10]. However, its application to turbo decoding
has not been investigated so far. Recently, the authors in [11]
have dealt with the convex log-sum-exp (lse) function and
considered a robust geometric programming problem as robust
S. Papaharalabos and P. T. Mathiopoulos are with the Institute for Space
Applications and Remote Sensing (ISARS), National Observatory of Athens,
Vas. Pavlou and Metaxa Str., P. Penteli, 15236, Athens, Greece (e-mail:
spapaha@space.noa.gr).
G. Masera and M. Martina are with the VLSI Lab, Dipartimento di
Elettronica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino,
Italy.
This research work was supported by EU IST NEWCOM++ (IST-216715)
project.
Manuscript received Month xx, 2009.
linear programming problem. In particular, a constructive al-
gorithm was presented in order to find the best piecewise linear
(PWL) approximation terms of the bivariate lse function. Their
research has shown that the exact number of PWL terms
depends on the approximation error resulting from the original
bivariate lse function.
In this letter, we modify and apply for the first time the
optimal approximations presented in [11] to decode turbo
codes noting that the bivariate lse function is equivalent to
the max∗ operator. Consequently, a generalized approximation
for the max∗ operator is obtained using: (i) a small number
of efficient PWL terms that ease a hardware implementation;
and (ii) the very simple max operator. The main feature of this
approximation is that there is no need to use any correction
term, in contrast to all previously published methods [4]–
[8]. Furthermore, it turns out that the known Max-Log-MAP
algorithm can be viewed as special case of this new approach,
when two PWL terms are considered. Our research has shown
that by considering four PWL approximation terms with the
additional use of scaling as in [3], the resulting turbo code
BER performance is essentially identical to the performance
of the optimal Log-MAP algorithm. Hardware synthesis results
have shown that by considering three PWL approximation
terms, the resulting algorithm outperforms most of the pre-
viously published methods, i.e. [5]–[8], in terms of occupied
area savings, and achieves near Log-MAP performance.
II. REDUCED COMPLEXITY TURBO DECODING
In this section the most important reduced complexity turbo
decoding algorithms are reviewed. Consider an information
sequence of N bits denoted with u¯ = [u1, u2, . . . , uN ]. This
sequence is turbo encoded, then each coded bit is binary
phase-shift keying (BPSK) modulated, taking values from the
alphabet {±1} with equal probabilities, and transmitted with
bit energy Eb over an additive white Gaussian noise (AWGN)
channel with one-sided power spectral density No. At the
receiver the turbo decoder estimates the transmitted sequence
of bits.
The max∗ operation, i.e. Jacobian logarithm, used in turbo
decoding is defined as [1]
max∗(x1, x2)
4
= log{exp(x1) + exp(x2)} = max(x1, x2)
+ ln {1 + exp(− |x1 − x2|)} = max(x1, x2) + fc(|x1 − x2|)
(1)
where fc(|x1 − x2|) is a non-linear function referred to as
‘correction term’ [9] and |.| denotes absolute value. For more
IEEE COMMUNICATIONS LETTERS, VOL. X, NO. XX, XXXXX 2009 2
than two arguments, the Jacobian logarithm is applied recur-
sively. For example, considering three arguments, it yields
max∗(x1, x2, x3) = max
∗ {max∗(x1, x2), x3} . (2)
For the Log-MAP algorithm, a look-up table (LUT) substi-
tutes fc(|x1 − x2|), which is usually implemented with eight
values [9]. If LUT is omitted, then the Log-MAP simplifies
to the Max-Log-MAP algorithm. In the past, several reduced
complexity decoding algorithms have been obtained by ap-
proximating fc(|x1 − x2|) with different methods, e.g. see
[4]–[8], a summary of which is presented in Table I.
Having evaluated the performance of reduced complexity
decoding algorithms given in Table I and as shown in Fig. 1
(for the turbo code simulation parameters see the next section),
it is concluded that at BER of 10−4: (i) The Max-Log-MAP
is the worst performing algorithm with approximately 0.4 dB
degradation as compared with Log-MAP; (ii) The maximum
performance degradation of reduced complexity algorithms as
compared with Log-MAP is approximately 0.1 dB; and (iii)
the Linear Log-MAP and also the algorithm of [7] achieve the
best, i.e. near Log-MAP performance. A thorough complexity
estimation of these algorithms is reported in the next section.
III. OPTIMAL max∗ APPROXIMATIONS AND THEIR
APPLICATION TO TURBO DECODING
From pure mathematical curiosity, the authors in [11] have,
instead of approximating fc(|x1 − x2|), approximated (1) as
a whole, i.e. the max∗ operator directly. Hence, (1) becomes
max∗(x1, x2)≈max(κ1∗x1+λ1∗x2+µ1, . . . , κi∗x1+λi∗x2+µi)
(3)
where κi, λi, and µi are real positive values and i ≥ 2. The
best PWL approximations of the max∗ operator with different
number of terms are shown in Table II. The approximation
error reduces in the order of
√
2/r2 and for practical applica-
tions 5 ≤ r ≤ 10 has been considered [11]. It is underlined
that in case of turbo decoding, the r = 2 approximation is
identical to the Max-Log-MAP algorithm.
Performance evaluation results for various values of r, with
the additional use of scaling as in [3], have shown that at BER
of 10−5: (i) Both r = 5 and r = 4 approximations achieve
essentially identical to the Log-MAP performance; and (ii) The
r = 3 approximation has performance degradation of less than
0.03 dB against the Log-MAP algorithm. In order to ease a
hardware implementation, the r = 3 and r = 4 approximations
have been modified, respectively as
max∗(x1, x2)≈max[x1, 0.5 ∗ (x1 + x2 + 1), x2] (4)
max∗(x1, x2)≈max(x1, 0.25 ∗ x1 + 0.75 ∗ x2 + 0.5,
0.75 ∗ x1 + 0.25 ∗ x2 + 0.5, x2).
(5)
In terms of implementation, synthesizable VHDL descrip-
tions have been produced for the max∗ approximations of (4)
and (5) as well as for Log-MAP with LUT, Max-Log-MAP
and the algorithms shown in Table I. In order to derive fair
comparisons, the same area optimization effort of the synthesis
tool must be guaranteed for all cases. To this purpose, although
all considered implementations of the max∗ operation are
pure combinational architectures, registers have been placed
at the architecture inputs and output. This allows setting a
unique clock frequency constraint for all considered cases,
fCK = 200 MHz. Synthesis results obtained in terms of area
occupied by the combinational part on a 130 nm standard cell
CMOS technology are given in Table III for precision metrics
represented with 8, 10 and 12 bits, respectively. These results
show, on the one hand, that the proposed r = 3 approximation
outperforms most of the previously published methods, i.e.
[5]–[8]. In particular, its occupied area is 35% smaller than that
required by the Log-MAP algorithm and it is only inferior to
the Constant Log-MAP algorithm by 8%. On the other hand,
the proposed r = 4 approximation has comparable complexity
with the method in [7] and outperforms [5], [6], [8].
Performance evaluation results have been obtained for the
most efficient method, i.e. Constant Log-MAP, the r = 3
and r = 4 proposed approximations, and these are illustrated
in Fig. 2. A 16-states turbo code is considered with coding
rate equal to 1/2 and generator polynomials (1, 33/23)o in
octal form representing the feed-forward and backward poly-
nomials, respectively. Furthermore, an information sequence
of N = 103 bits is assumed, whereas the total number of
transmitted frames is 105. A pseudo-random turbo interleaver
is considered and at the receiver a maximum of 10 decoding it-
erations are performed. In order to reduce computer simulation
time, a genie stopping rule is assumed at the turbo decoder. In
computer-based simulations the scaling factor, denoted with s,
was constant when varying the number of decoding iterations,
having the following values: (i) 0.65 for Max-Log-MAP; (ii)
0.9 for Log-MAP; (iii) 0.85 for Constant Log-MAP; and (iv)
0.75 for the rest of the algorithms.
As shown in Fig. 2, the best performance is achieved by
the Constant Log-MAP algorithm and the proposed r = 4
approximation followed by the r = 3 approximation. In more
detail, at BER of 10−4 and with respect to the Log-MAP
algorithm: (i) The Constant Log-MAP algorithm has perfor-
mance degradation of approximately 0.03 dB; (ii) The r = 4
approximation has comparable performance with the Constant
Log-MAP algorithm; and (iii) The r = 3 approximation has
performance degradation of 0.05 dB. However, both r = 3
and r = 4 approximations offer practical implementation
advantages with respect to other methods.
IV. CONCLUSION
It has been shown that the max∗ operator used in turbo
decoding can be simplified into different number of PWL
approximation terms in an efficient way using max only
operation. The proposed r = 3 approximation is 0.05 dB
inferior to the Log-MAP algorithm but 35% much simpler.
Further, it outperforms, in terms of occupied area savings,
most of the previously published methods, such as Linear Log-
MAP [5], Average Log-MAP [6], and the algorithms from
[7], [8]. The proposed r = 4 approximation is only 0.03 dB
inferior to the Log-MAP algorithm and it it less complex than
the methods presented in [5], [6], [8].
IEEE COMMUNICATIONS LETTERS, VOL. X, NO. XX, XXXXX 2009 3
TABLE I
THE MOST IMPORTANT REDUCED COMPLEXITY TURBO DECODING ALGORITHMS USING DIFFERENT APPROXIMATION IN fc(|x1 − x2|) OF (1).
Constant Log-MAP [4] Linear Log-MAP [5] Average Log-MAP [6]
fc(|x1 − x2|)
{
3/8, if |x1 − x2| < 2
0, otherwise max(ln 2− 0.25 ∗ |x1 − x2| , 0)
{
ln 2 + 0.5 ∗ (x1 + x2), if |x1 − x2| < 2 ∗ ln 2
0, otherwise
Ref. [7] Ref. [8] Max-Log-MAP
fc(|x1 − x2|) ln 2 ∗ 2−|x1−x2| max(0, ln 2− 0.5 ∗ |x1 − x2|) 0
TABLE II
BEST APPROXIMATIONS OF THE max∗ OPERATION USING DIFFERENT
NUMBER OF PWL TERMS AND DEPLOYING max ONLY OPERATION [11].
No. of Resulting max∗ Approximation No. of Approx.
Terms (r) max Ops. Error
2 max(x1, x2) 1 0.693
max(x1,
3 0.5 ∗ (x1 + x2) + 0.693, 2 0.223
x2)
max(x1,
0.271 ∗ x1 + 0.729 ∗ x2 + 0.584,
4 0.729 ∗ x1 + 0.271 ∗ x2 + 0.584, 3 0.109
x2)
max(x1,
0.167 ∗ x1 + 0.833 ∗ x2 + 0.45,
5 0.5 ∗ x1 + 0.5 ∗ x2 + 0.693, 4 0.065
0.833 ∗ x1 + 0.167 ∗ x2 + 0.45,
x2)
TABLE III
OCCUPIED AREA COMPARISON (SQUARE µm) OF DIFFERENT max∗
APPROXIMATIONS ON A 130 NM STANDARD CELL TECHNOLOGY AND
VARIOUS PRECISION METRICS REPRESENTATIONS.
Algorithm 8 bits 10 bits 12 bits
Max-Log-MAP 250.133 312.666 373.182
Log-MAP 1022.72 1276.888 1613.76
r = 3 of (4) 653.573 831.086 1006.583
r = 4 of (5) 883.534 1149.804 1426.160
Constant Log-MAP [4] 599.108 764.519 931.946
Linear Log-MAP [5] 978.342 1264.784 1559.296
Average Log-MAP [6] 1069.116 1363.627 1555.261
Ref. [7] 891.602 1135.684 1377.748
Ref. [8] 1137.701 1456.418 1758.988
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0.75 1 1.25 1.5
10−4
10−3
10−2
10−1
100
Eb/No (dB)
BE
R
Max−Log−MAP
Average Log−MAP
Ref. [8]
Constant Log−MAP
Log−MAP
Linear Log−MAP
Ref. [7]
Fig. 1. Turbo code BER performance comparison with the most important
reduced complexity decoding algorithms.
0.75 1 1.25 1.5 1.75
10−6
10−5
10−4
10−3
10−2
10−1
Eb/No (dB)
BE
R
r=2 terms (Max−Log−MAP), s=0.65
r=3 terms, s=0.75
r=4 terms, s=0.75
Constant Log−MAP, s=0.85
Log−MAP, s=0.9
Fig. 2. Turbo code BER performance with efficient approximations of the
max∗ operator and different number of PWL terms. The extrinsic information
is scaled by a factor of s.
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geometric programming,” Springer Optim. Eng. Journal, vol. 9, no. 2,
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On optimal and near-optimal turbo decoding using generalized max operator

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