Butt, Muhammad Fasih

Hanzo, Lajos

Ng, Soon Xin

English

Southampton (e-Prints Soton)

754 IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 9, SEPTEMBER 2009On the Union Bounds of Self-ConcatenatedConvolutional CodesSoon Xin Ng, Senior Member, IEEE, Muhammad Fasih Uddin Butt, Student Member, IEEE, andLajos Hanzo, Fellow, IEEEAbstract—In this contribution, the union bounds of self-concate-nated convolutional codes (SECCCs) are derived for communica-tions over both Additive White Gaussian Noise (AWGN) and un-correlated Rayleigh fading channels. The truncated union boundsofSECCCsareveryusefulforstudyingthecorrespondingbiterrorratio (BER) ﬂoors. Based on the truncated union bounds, variousSECCCs can be designed for a desired BER ﬂoor without the needof time-consuming Monte-Carlo simulations.Index Terms—Code design, self-concatenated convolutionalcodes, union bound.I. INTRODUCTIONTURBO codes based on parallel-concatenated convolu-tional codes (PCCCs) using two or more constituentConvolutional Codes (CCs) were proposed in [1]. The dis-covery of turbo codes was a breakthrough in coding theory,because they are capable of operating near the Shannon limit[2]. Serially-concatenated convolutional codes (SCCCs) [3]have been shown to yield a performance comparable, and insome cases superior, to turbo codes. Self-concatenated con-volutional codes (SECCCs) proposed by Benedetto et al. [4]and Loeliger [5] constitute another attractive iterative detectionaided code-family.SECCCs exhibit a low complexity, since they invoke only asingle encoder and a single decoder. Near-capacity SECCCshave been designed in [6] based on extrinsic informationtransfer (EXIT) charts [7]. All concatenated coding schemesincluding SECCCs tend to exhibit a Bit Error Ratio (BER) ﬂoorin the medium to high signal-to-noise ratio (SNR) region. How-ever, the BER ﬂoor of SECCCs has not been analyzed in theliterature. While the EXIT chart analysis is only accurate whena sufﬁciently long interleaver is used, the BER ﬂoor analysisusing truncated union bound is valid for arbitrary interleaverManuscript received May 01, 2009; revised May 01, 2009. First publishedJune 10, 2009; current version published July 01, 2009. This work was sup-ported by the European Union under the auspices of the OPTIMIX project aswell as that of the COMSATS Institute of Information Technology (CIIT), Is-lamabad under the auspices of Higher Education Commission. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Nikolaos V. Boulgaris.S. N. Ng and L. Hanzo are with the Communications Research Group,University of Southampton, Southampton SO17 1BJ, U.K. (e-mail:sxn@ecs.soton.ac.uk; lh@ecs.soton.ac.uk).M. F. U. Butt is with the Communications Research Group, University ofSouthampton, Southampton SO17 1BJ, U.K., on leave from the Departmentof Electrical Engineering, CIIT, Islamabad, Pakistan (e-mail: mfub06r@ecs.soton.ac.uk).Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/LSP.2009.2024784Fig. 1. Schematic of the SECCC encoder. The notations ￿ and ￿ denotethe information sequences of the hypothetical upper and lower component en-coder, respectively, while the puncturer output sequences of the hypotheticalupper and lower component encoder are denoted as ￿ and ￿ , respectively.lengths. Hence, the BER ﬂoor analysis is important for codedesign. More speciﬁcally, the union bound constitutes a usefulcode design technique [8]–[10], which was also employed forthe design of antenna selection schemes [11].In this contribution, we ﬁrst study the similarities and dif-ferences between PCCCs and SECCCs in Section II. Then, wehighlight the union bound derivation for conventional CCs inSection III, before we derive the union bound of SECCCs inSection IV. The union bound derived is then compared to oursimulation results in Section V and our conclusions are offeredin Section VI.II. SYSTEM MODELTheschematicoftheSECCCencoderemployingarecursive systematic convolutional (RSC) encoder and apuncturer is shown in Fig. 1. As seen from Fig. 1, thebit sequence is simply the interleavedversion of the original bit sequence .After the parallel-to-serial (P/S) conversion, we can computethe information sequence of the hypothetical upper SECCCcomponent code as . Interestingly,we can view the information sequence of the hypothetical lower1070-9908/$25.00 © 2009 IEEEAuthorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on July 1, 2009 at 10:16 from IEEE Xplore.  Restrictions apply.NG et al.: ON THE UNION BOUNDS OF SELF-CONCATENATED CONVOLUTIONAL CODES 755Fig. 2. Schematic of the SECCC encoder and decoder when communicating over Rayleigh fading channels. The notation ￿￿￿￿ denotes the LLR of bit ￿ and thethe supercripts ￿, ￿ and ￿ denote a priori, a posteriori and extrinsic nature of the LLR, respectively.SECCC component code as the interleaved version ofusing an odd–even separation (OES) based interleaver .More explicitly, the OES interleaver consists of two componentinterleavers, where the odd position of the bit sequence ispermuted based on the mapping of , while the evenposition of the bit sequence is permuted based on the inverseof the mapping , namely on .We apply a puncturer that removes the interleaved bit se-quence as well as all parity bits corresponding to the bit se-quence in order to yield the output sequence , as shownin Fig. 1. The resultant puncturing rate is given byand the SECCC output sequence consists of only the inputbit sequence aswellas theparitybit sequence correspondingto , as shown in Fig. 1. The SECCC encoder consists of boththe rate- RSC encoder and the rate- puncturer. Hence, thecoding rate of the SECCC encoder, as shown in Fig. 1, is givenby .Although is punctured from , we can obtain the Log-Likelihood Ratio (LLR) of the bits in by interleaving theLLRs associated with the bits in obtained from the MAPdecoder, as shown in Fig. 2. Hence, the output sequenceseen in Fig. 1 is similar to that of the upper component encoderof a turbo code, where all parity bits corresponding to the odd-position information bits are punctured. Similarly, all parity bitscorresponding to the odd-position information bits at the outputsequenceofthehypotheticallowercomponentcode ,asseenin Fig. 1, are punctured.Based on these observations, we are able to compute theunion bound of SECCCs, as detailed in Section IV.III. UNION BOUNDS OF CONVOLUTIONAL CODESThe pair-wise error probability (PWEP) is deﬁned as theprobabilitythatthemodulatedsymbolsequenceis wrongly decoded as another modulated symbol sequence. The PWEP, which depends on both themodulation scheme as well as on the code structure and thecommunication channel, can be expressed as [8](1)where istheSNR,while isthesquaredEuclideandis-tance between the modulated symbol sequences and , whencommunicating over AWGN channel, which is given by(2)where represents the set of indices satisfying the condi-tion of . The number of elements in the set is givenby , which quantiﬁes the number of erro-neous modulated symbols in the sequence , when comparedto the correct sequence .When communicating over uncorre-lated Rayleigh fading channels, the PWEP can be shown to be[12](3)In this contribution, we will derive the union bound of SECCCsbased on BPSK modulation. Note that when BPSK modulationis employed, we have , whenever since. Based on this simpliﬁcation, the PWEP forthe AWGN channel can be expressed from (1) as(4)Similarly, the PWEP for the uncorrelated Rayleigh fadingchannel can be simpliﬁed from (3) as(5)where is also referred to as theeffective Hamming distance,which quantiﬁes the diversity order of the code.The union bound of the average BER of a coding scheme canbe expressed as [8](6)where is the number of information bits per -bit codedsymbol and is the distance spectrum of the code, given by(7)where is the information weight denoting the number of er-roneous information bits in an encoded sequence and is theparity weight quantifying the number of erroneous parity bitsAuthorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on July 1, 2009 at 10:16 from IEEE Xplore.  Restrictions apply.756 IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 9, SEPTEMBER 2009in an encoded sequence. More explicitly, is the two-di-mensional Weight Enumerating Function (WEF), quantifyingtheaveragenumberofsequenceerroreventshavinganinforma-tion weight of and a parity weight of . Hence, the Hammingdistance is given by .IV. UNION BOUNDS OF SECCCSThe WEF of SECCCs can be expressed as(8)where and are the WEFs of the hypotheticalupperandlowercomponentcodes,respectively,whiletheeffec-tive parity weight of an SECCC is given by(9)where and are the parity weights of the hypotheticalupper and lower component codes, respectively. As we can seefrom Fig. 1, the information sequence of the upper componentencoder consists of the original information sequenceand its interleaved version . Hence, if the original informa-tion sequence has an information weight of , then the in-formation sequence of the upper component encoder willhave an information weight of . The same also applies to thelower component code. Hence, we have andin (8).The term in (8) denotes the probability of occurrencefor all the associated error events having information bit er-rors, whenemploying aself-concatenatedbit-interleaverhavinga length of bits. The evaluationof is basedonthe noveluniform self-interleaver concept, which may be interpreted asthe extension of the uniform bit-interleaver concept proposed in[9]. More speciﬁcally, a uniform self-interleaver may be parti-tioned into two bit-interleavers, as deﬁned in Deﬁnition 1.Deﬁnition 1: A uniform self-interleaver of length bitsis a probabilistic device, which maps a given input sequenceof length bits having an information weight of bits intoall possible permutations in the odd and even partitions of anequivalent odd-even-separation based interleaver of lengthhaving an information weight of , with equal probability ofgiven by(10)where , which characterizes the traditional-bit uniform interleaver having an information weight ofbits. If there are bit errors in the information sequence, thenthere will be bit errors in the ‘odd’ sequence as well asanother bit errors in the ‘even’ sequence , since issimply the interleaved version of the sequence.The WEF for an SECCC having a block length ofencoded symbols and a total of number of trellis states canbe calculated as follows. We can deﬁne the State Input-Redun-dancy WEF (SIRWEF) for a block of SECCC-encoded sym-bols as(11)where is the number of paths in the trellis enteringstate at symbol index , which have an information weightof and a parity weight of . The notations and representdummy variables. For each -bit coded symbol at index , theterm can be calculated recursively as follows:(12)where represents the speciﬁc -bit input symbol that triggersthe transition from state at index to state at index ,while the terms and can be formulated as(13)where and are the information weight and the parityweight,respectively,ofthetrellispathsenteringstate atindex. Furthermore, is the informa-tion weight of the -bit information symbol that triggers thetransition from state to andis the parity weight between and , where is the encoded-bitsymbolcorrespondingtothetrellisbranchinthetransitionfrom state to and is the actual encoded -bit symbol atindex .Again,alltheparitybitsin (or )correspondingto the odd-position information bits are punctured. Note thatthe parity weight contribution corresponding to a puncturedparity bit equals to zero.Let the encoding process commence from state 0 at index 0and terminate at any of the possible states at index . Thenthe WEF used in (7) is given by(14)Note that for linear codes [13] the distance proﬁle of the codeis independent of which particular encoded symbol sequence isconsidered to be the correct one. Hence, for the sake of sim-plicity, we can assume that the all-zero encoded symbol se-quence is transmitted.Based on all the above equations, the union bound of anSECCC employing BPSK modulation can be shown to be(15)when communicating over AWGN channels and(16)when communicating over uncorrelated Rayleigh fading chan-nels, where .V. RESULTS AND DISCUSSIONSLet us now compare the BER performance of CCs andSECCCs to their union bounds truncated at a maximum Ham-ming distance of , wherethe maximum information and parity weights considered areand , respectively. Figs. 3 and 4 showsthe BERs of our simulations and bounds of the CCs andAuthorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on July 1, 2009 at 10:16 from IEEE Xplore.  Restrictions apply.NG et al.: ON THE UNION BOUNDS OF SELF-CONCATENATED CONVOLUTIONAL CODES 757Fig. 3. Simulations and truncated union bounds of BPSK-assisted CC andSECCC, when communicating over AWGN channels. The union bounds aretruncated at a maximum Hamming distance of ￿ ￿￿ ￿ . The SECCCemploys an interleaver of length 12000 bits and 8 decoding iterations.Fig. 4. Simulations and truncated union bounds of BPSK-assisted CC andSECCC, when communicating over uncorrelated Rayleigh fading chan-nels. The union bounds are truncated at a maximum Hamming distance of￿ ￿￿ ￿ . The SECCC employs an interleaver of length 12000 bits andeight decoding iterations.SECCCs employing BPSK modulation, when communicatingover both AWGN and uncorrelated Rayleigh fading channels.Both the CC and SECCC employ an RSC code based on agenerator polynomial of expressed in octal format.As shown in Figs. 3 and 4, the truncated union bound quan-tiﬁes the BER ﬂoor of SECCCs quite accurately. Hence, wecan design various SECCCs having various desired BER ﬂoorsusing the proposed truncated union bound.VI. CONCLUSIONSA useful union bound has been derived for BPSK-basedSECCCs, when communicating over both AWGN and un-correlated Rayleigh fading channels. The union bound canbe truncated in order to conveniently analyze the BER ﬂoorof SECCCs. This union bound can be used together withEXIT charts in order to design near-capacity SECCCs oper-ating at a given desired BER ﬂoor. It can also be extendedto high-order modulation schemes for designing bandwidthefﬁcient SECCCs.REFERENCES[1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near shannon limiterror-correctingcoding anddecoding: Turbocodes,”in Proc.IEEEInt.Conf. Communications, 1993, pp. 1064–1070.[2] C. E. Shannon, “A mathematical theory of communication,” Bell Syst.Tech. J., vol. 27, pp. 379–656, Jul. 1948.[3] S.Benedetto,D.Divsalar,G. Montorsi,andF.Pollara,“Serialconcate-nationofinterleaved codes:Performanceanalysis, design,anditerativedecoding,” IEEE Trans. Inform. Theory, vol. 44, pp. 909–926, 1998.[4] S. Benedetto, D. Divsalar, G. 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On the Union Bounds of Self-Concatenated Convolutional Codes

On the Union Bounds of Self-Concatenated Convolutional Codes

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