Design Considerations for Iteratively-Decoded Source-Channel Coding Schemes

Transcription

1 Desig Cosideratios for Iteratively-Decoded Source-Chael Codig Schemes Ragar Thobabe * ad Jörg Kliewer * Uiversity of Kiel, Istitute for Circuits ad Systems Theory, Kiel, Germay Departmet of Electrical Egieerig, Uiversity of Notre Dame, Notre Dame, IN 46556, U.S.A. Abstract We address outer variable-legth ecodig of a first-order Markov source i a serially cocateated codig scheme with a ier recursive covolutioal code. Decodig is carried out iteratively betwee the costituet decoders for variable-legth ad covolutioal code. While variable-legth codes are commoly kow to be sesitive to trasmissio errors, we show that they ca lead to sigificat performace improvemets compared to fixed-legth source ecodig with optimized mappigs. Specifically, we propose a simple variablelegth code costructio with a free distace of two ad good compressio properties at the same time. Numerical results show that the performace gai of the proposed approach also holds for precoded ISI chaels where iterative joit source chael equalizatio ad decodig is employed at the receiver. I. INTRODUCTION The robust trasmissio of variable-legth ecoded source sigals over wireless chaels has become a active research area durig the last years. It is motivated by the icreasig demad o multimedia ad data services i third ad fourth geeratio wireless etworks. I such applicatios source compressio is usually carried out usig stadardized techiques which, i order to achieve high compressio gais, ofte employ variable-legth codes (VLCs). VLCs were origially desiged for error-free trasmissio scearios. I this form they are highly sesitive to trasmissio errors ad suffer from error propagatio; for example, sigle bit errors may lead to isertios ad deletios of symbols, causig a loss of sychroizatio. For this reaso, VLCs are usually avoided i classical wireless commuicatio applicatios. I order to make VLCs applicable to the peculiarities of wireless chaels, a variety of joit source-chael codig ad decodig schemes addressig the error-resiliet trasmissio of VLCs have bee proposed i the literature. This icludes the desig of error-resiliet VLCs, the joit optimizatio of VLCs ad chael codes, the robust decodig of VLCs, ad the joit decodig of VLCs ad chael codes. For delay- ad / or complexity-costraied trasmissio scearios those combied techiques are ofte more advatageous tha the classical separatio of source ad chael ecodig. As we will show i the followig, such schemes have the potetial to eve outperform schemes This work was partly supported by the Germa Research Foudatio (DFG) uder grat KL 1080/3-1 ad by the Uiversity of Notre Dame Faculty Research Program. based o fixed word legth source ecodig, which are usually ot affected by sychroizatio problems. A class of robust VLCs is give by reversible variablelegth codes (RVLCs) [1 4], which are a well established techique for robust video compressio. RVLCs exploit additioal redudacy i the VLC code words i order to guaratee bidirectioal decodig. If the umber of error evets is low, the impact of a sychroizatio loss ca be mitigated. I [5], Buttigieg itroduced variable-legth error-correctig codes (VLECs), which combie the variablelegth character of VLCs with distace costraits of chael codes at the cost of additioal redudacy. A key compoet for a reliable trasmissio of VLCs is give by the VLC decoder. The first maximum a posteriori probability (MAP) decoder was itroduced i [6], where a optimal VLC sequece estimatio was carried out by the Viterbi algorithm, applied to a appropriate VLC trellis. Aother trellis represetatio, which has become the stadard trellis for VLC decodig, was itroduced i [7]. It beefits from a efficiet state represetatio ad allows both a bitwise MAP ad a posteriori probability (APP) decodig with the BCJR algorithm [8] ad a optimal sequece estimatio with the Viterbi algorithm. Later o, both exact ad complexity-reduced approximate MAP decodig techiques for variable-legth source codes were proposed by differet authors, either by usig a modified or list Viterbi algorithm [9 14] or by usig sequetial decodig approaches [15], [16]. May approaches (e.g., [10 12]) exploit the redudacy due to the residual source correlatio after source ecodig for additioal error protectio. Geerally, this allows for a less powerful forward error correctio (FEC) code, resultig i a reductio of allocated badwidth or latecy for the overall trasmissio system. As reported i [17 21], a sigificat amout of residual source correlatio, modeled as first-order Markov process, ca be observed for several output parameters i state-of-the-art speech codecs ad the MPEG audio compressio scheme. For some parameters, eve a redudacy up to 50% ca be observed. Further improvemets ca be achieved by additioal forward error correctio combied with a joit decodig of source ad chael code. Depedig o the ecoder structure, this ca be obtaied by mergig the source ad the chael decoder, as proposed i [9], [22 24]. Sice the cocateatio of VLC ad chael code essetially represets a serial code cocateatio, aother optio is to apply iterative

2 source-chael decodig (ISCD), which was proposed by Bauer ad Hageauer [25] for VLCs ad by Goertz i [26] for fixed-legth mappigs. Further ISCD approaches for VLCs were preseted by several authors [27 33] ad differ i the realizatio of the employed soft-iput / soft-output VLC decoder. I the followig, we address the joit desig of source ad chael ecodig for ISCD. While earlier work i [34], [35] focuses o the trade-off betwee the explicit redudacy itroduced by the chael ad the source ecoder, we cosider correlated source sigals ad rate-1 chael codes. The resultig ecodig schemes are optimized both i terms of covergece properties ad error-floor performace based o the desig guidelies for serial cocateated codes give i [36], [37] ad a aalysis of the extrisic iformatio trasfer characteristics [38]. Specifically, we propose a ovel variable-legth code costructio with a miimum Hammig distace of two which allows the resultig source-chael ecodig ad decodig scheme to outperform the best kow fixed-legth mappigs for ISCD [39] without itroducig additioal redudacy. Simulatio results are preseted for a biary trasmissio over the AWGN chael, where we also address the case of additioal iter-symbol iterferece (ISI). This paper is orgaized as follows. Sectio II presets a overview of the trasmissio system ad iterative sourcechael decodig. I Sectio III we propose a ovel VLC code costructio which is specifically tailored to the system beig optimized i Sectio IV. Fially, simulatio results are give i Sectio V. II. TRANSMISSION SYSTEM We cosider the sceario depicted i Fig. 1, where the trasmissio of a source-chael ecoded real-valued source U 1 over a white Gaussia oise (AWGN) chael with additioal ISI is addressed. A. Source Model ad Source Ecoder I the followig, we address the case where redudacy due to both residual itraframe correlatio ad o-uiform source symbol distributio is iheret i the source sigal. Therefore, we assume that the source vector U cosists of K real-valued correlated source symbols U k Ê, which may be geerated by a first-order autoregressive process. Such a source process ca be obtaied by filterig ucorrelated Gaussia oise with a recursive filter H(z) = z/(z a) where a is the correlatio coefficiet. After W -bit quatizatio, we obtai the idex vector I with elemets I k II, II = {0,...,2 W 1}, draw from the distributio Pr(I k ). The residual idex correlatio is modeled by a first-order Markov process, characterized by the idex-trasitio probabilities Pr(I k I k 1 ). These probabilities are later exploited at the receiver side as additioal a priori kowledge, leadig to a ehaced error protectio. I this cotext, they may be iterpreted as a soft parity check. 1 Radom variables ad vectors of radom variables are represeted by upper-case letters, the correspodig realizatios are represeted by lowercase letters. I a fial ecodig step, the quatizatio idices I k are mapped symbol-wise to their biary represetatio, resultig i a legth-n bit vector with elemets B {0,1}. For compariso purposes we cosider mappigs realized by variable-legth codes (VLCs) with the codeword set VLC ad the mea word legth l ad also mappigs with a fixed word legth W. B. Chael Ecoder ad Precoder After source ecodig, the biary source vector B is permuted by a iterleaver π. The resultig bit vector B with elemets B is the iput to the chael ecoder, which cosists of a recursive systematic covolutioal (RSC) code with rate R RSC = 1, puctured from a rate-1/2 mother code with geerator polyomials (G r,g f ) 8. G r deotes the feedback polyomial ad G f the feedforward polyomial. I the absece of ISI, the RSC ecoder serves as a ier scrambler. Pucturig is the performed radomly such that a fractio of p sys [0,1] of the systematic bits ad a fractio of p par = 1 p sys of the parity bits are removed from the bit stream. This pucturig method is beeficial sice, for a give mother code, a optimizatio relies solely o the parameter p sys. If a additioal degradatio by ISI is cosidered, the RSC ecoder serves as a recursive rate- 1 precoder. I this case, all systematic bits are puctured (p sys = 1). Furthermore, we restrict ourselves to the case where the precoder solely cosists of a feedback polyomial. C. Chael Model I the followig, we cosider a biary trasmissio employig biary phase shift keyig (BPSK); i.e., the symbols X m at the output of the ecoder are take from the set { 1,1}. The ISI o the chael is modeled by a real-valued liear filter with the fiite impulse respose h 0 (m). If o ISI is cosidered, h 0 (m) = γ 0 (m) holds with γ 0 (m) = 1 for m = 0 ad γ 0 (m) = 0 for m 0. We apply a ormalizatio of the impulse respose h 0 (m) such that E s = E{ X 2 } = E{ Z 2 } = E z. A additioal white Gaussia oise process N with the oise variace σ 2 N = 1/(2E s /N 0 ) is the added to the output Z of the ISI filter. Correspodigly, the observatios Y m at the output of the chael are characterized by the likelihood p Y Z (y z). D. Iterative Source-Chael Decoder ad Equalizer The realized trasmitter structure is equivalet to a serial cocateatio of two compoet codes which are separated by a iterleaver [36]. Accordigly, both joit decodig of the source ad chael code ad joit source decodig ad equalizatio of the ISI chael ca be realized by the iterative decodig scheme depicted i Fig. 1. I the followig, we give a brief descriptio of the iterative source-chael decoder discussed i [33], which is here exteded to the case of trasmissio chaels sufferig from ISI. For further readig we refer to [33].

3 U Q I VLC Ecoder/ Bit Mappig B π B RSC Ecoder/ Precoder X Source Ecoder h 0 (m) Û, Î Log-APP Source Decoder/ Sequece Estimatio Pr(I k I k 1 ) A (s) L(B) E (s) π 1 π E (c) L(B ) A (c) Log-APP RSC Decoder/ Equalizatio p Y Z (y z) h 0 (m) Y Z N p Y Z (y z) Fig. 1. Model of the trasmissio system, cotaiig source ecodig, chael ecodig / precodig, ad iterative source chael decodig for the AWGN chael, ad iterative source decodig ad equalizatio i the case of iter-symbol iterferece. 1) Compoet Decoders: The iterative decoder cosist of two soft-iput / soft-output (SISO) decoders, correspodig to the outer source ecoder ad the ier chael ecoder / ISI chael. All decoders are realized by trellisbased a posteriori probability (APP) decoders. They are implemeted i the logarithmic domai (Log-APP decoder) ad provide log-likelihood ratios (LLRs) [40] for the correspodig symbols at the ecoder side. Based o the observatios Y = [Y 1,...,Y M ] for the trasmitted symbols X m at the output of the commuicatio chael ad o the a priori iformatio A (c) = [A (c) 1,...,A(c) N ] for the iterleaved symbols B, the ier decoder computes the vector L(B ) := [ ] L(B 1 Y,A (c) ),...,L(B N Y,A (c) ) of a posteriori LLRs for the iterleaved source bits B ( Pr(B L(B Y,A(c) ) := l =0 Y,A (c) ) ) Pr(B =1. Y,A(c) ) Hece, if o ISI is preset o the chael, we apply the BCJR algorithm [8] to the trellis represetatio of the uderlyig RSC code. I the case of ISI, the RSC trellis is replaced by a super trellis for the ISI chael ad the recursive precoder, which ca be see as a geeralizatio of the DPSK/ISI super trellis proposed i [41]. I order to costruct the super trellis, the precoder ad the filter associated with the ISI are combied to a sigle ecoder with a biary feedback polyomial ad a real-valued feedforward polyomial similar to the direct form II realizatio for recursive real-valued filters [42]. Correspodigly, for biary iput sequeces the trellis states of the super trellis ca be directly obtaied from the states of this filter. We also assume that perfect kowledge of the oise statistics p Y Z (y z) ad the chael impulse respose h 0 (m) is available at the decoder. For the outer source decoder we cosider the Log-APP VLC decoder proposed i [33]. By appropriate modificatios of the classical BCJR algorithm for the trellis proposed i [7] the residual source correlatio ca be exploited for additioal error protectio. Due to a low umber of trellis states this results i a efficiet decodig algorithm which is especially well-suited for the decodig of large block legths. Similar to the ier ecoder, the outer decoder computes the vector [ ] L(B) := L(B 1 A (s) ),...,L(B N A (s) ) of a posteriori LLRs o the source bits B ( L(B A (s) Pr(B =0 A (s) ) ) ) := l Pr(B =1 A (s) ) by cosiderig additioal a priori kowledge i form of the idex-trasitio probabilities Pr(I k I k 1 ). Sice the outer decoder decodes o the VLC codebits ad ot o the symbols of the desired source idex vector I, a subsequet sequece estimatio o the VLC trellis [7] must be employed i order to obtai the idex ad source vector estimates Î ad Û. 2) Iterative Decodig: The iterative decodig process is based o the exchage of extrisic iformatio [40] betwee the two compoet decoders. Therefore, the a posteriori LLRs L(B A(c),Y) ad L(B A (s) ) of the two decoders are separated ito two terms correspodig to the a priori iformatio A (c) ad A(s) ad the extrisic iformatio E (c) ad E(s). By employig the otatio X \ = [X 1,...,X 1,X +1,...,X N ] ad uder the assumptio of equiprobable bits B ad B we obtai 2 L(B A(c),Y) = L(B A(c) \,Y) +L(B }{{ } A(c) }{{ } ) =:E (c) =A (c) for the ier chael decoder / equalizer ad L(B A (s) ) = L(B A (s) \ ) }{{} =:E (s) for the outer source decoder. The a priori LLRs A (c) of the biary symbols B the extrisic LLRs E (c) ad A(s) + L(B A (s) ) }{{} =A (s) (1) (2) are direct observatios ad B, respectively, while carry the iformatio ad E(s) 2 Note that for the LLRs A (.) the idetity L(B A (.) ) = A (.) holds.

4 provided by the remaiig observatios A (c) \,Y, ad A (s) respectively. Accordig to (1) ad (2), with the otatio E (s) = [E (s) 1,...,E(s) N ] ad E(c) = [E (c) 1,...,E(c) N ], \, the extrisic LLRs E (c) ad E(s) ca be obtaied by subtractig the a priori iput A (c) ad A(s) from the a posteriori LLRs L(B A(c),Y) ad L(B A (s) ) as show i Fig. 1. After (de-)iterleavig, the extrisic LLRs provided by the oe decoder become the a priori iput of the other decoder. Sice o extrisic iformatio is available at the output of the outer decoder durig the iitial decodig step, the a priori LLRs A (c) of the ier decoder are iitialized with zeros. The iterative process cotiues util a error-free trasmissio is achieved or a appropriate stoppig criterio is satisfied. Note that after the first half iteratio the quatities derived by the ier Log-APP decoder are o loger exact a posteriori LLRs: BJCR-based stadard decoders cosider the iput vectors Y ad A (c) to be mutually idepedet. But i fact, the a priori iput A (c) is a fuctio of Y. Thus, exact a posteriori LLRs may oly be obtaied if the Log- APP decoder addresses these depedecies i the derivatio of the LLRs. E. Code Rate of the Trasmissio System Sice the source correlatio ameliorates the achievable error protectio, the overall code rate R of the trasmissio system relies o both the explicit redudacy of the chael code ad the amout of residual source redudacy. Therefore, we associate the source redudacy with a code rate R S. Due to the serial cocateatio of source ad chael ecodig, the overall code rate R of the trasmissio system is the give by the product of the code rate R S of the source ecoder ad the code rate R RSC of the RSC code: R = R S R RSC. With the etropy H(B) of the vector B, we ca defie the code rate R S of the source sigal as R S := H(B) N = H(B) K l, where l = W for a fixed word legth with W -bit quatizatio. The code rate R S ca be separated ito the product of two code rates R corr ad R Map. While the code rate R corr correspods to the redudacy due to the itraframe correlatio, the code rate R Map is associated with the redudacy itroduced by the bit mappig or the VLC. With H(B) = H(I) we obtai R S = H(I) K H(I k ) } {{ } =:R corr H(I k) l }{{} =:R Map. (3) By applyig the chai rule of etropy, the code rate of the itraframe redudacy may be simplified further. Thus, for a first-order Markov source, the code rate R corr for the itraframe redudacy may be expressed as R corr = H(I k) + (K 1) H(I k I k 1 ) K H(I k ) H(I k I k 1 ). H(I k ) The last approximatio holds for a log block legth K 1. III. EVEN-WEIGHT VARIABLE-LENGTH CODES I the followig, we propose a simple VLC costructio which provides good compressio properties ad at the same time guaratees a miimum Hammig distace of d mi = 2 betwee code sequeces of equal legth. These properties are especially useful for the desig of efficiet ecodig schemes for iterative source-chael decodig, which will be discussed i Sectio IV. I order to satisfy the distace costrait d mi = 2, we oly cosider codewords with eve Hammig weight i the codeword set VLC of the VLC; further costraits as symmetry or reversibility are eglected. Accordigly, we refer to the resultig VLC as eve-weight variable-legth code (EW VLC). It ca be see as a simple variable-legth error-correctig code [5]. The costructio of the EW VLC for a set II of source symbols I k is parametrized by the miimum code word legth l mi. I the followig, l mi is set to the miimum word legth of the Huffma code, derived for the uderlyig source distributio Pr(I k ). The derivatio of the codeword set VLC for the EW VLC ca the be summarized as follows: 1. Iitializatio of - the curret code word legth l=l mi, - the umber of assiged code words A=0, ad - the set of available legth-l codewords V l = {b b l }. 2. The code word cadidates b available i the set V l are separated ito the set of code words with eve Hammig weight E l := {b b V l, w H (b) is eve} ad ito the set of code words with odd Hammig weight O l := {b b V l, w H (b) is odd}. 3. We assig a umber of mi{ E l, II A} code words b E l with eve Hammig weight to the code word set VLC of the EW VLC. 4. The umber of assiged code words, the set of available code words, ad the curret code word legth are icremeted accordig to A = A + mi{ E l, II A} V l+1 = {b=[b 1,...,b l+1 ] [b 1,...,b l ] O l } l = l If all code words are assiged (A = II ), the code costructio is completed; otherwise retur to step 2. After costructig the code word set VLC, the codewords are assiged to the symbols I k i II such that highly probable symbols are represeted by short codewords. Eve though the distributio of the source symbols I k is ot cosidered for the code costructio, Table I verifies the good compressio properties of the EW VLC. For the distributio of the Eglish alphabet ad for a Laplacia, a Gaussia, ad a uiform distributio with II = 32, it compares the code rate R Map of the EW VLC to the code

5 rates achieved by the Huffma code ad the asymmetric ad symmetric reversible variable-legth codes (RVLCs) from [3], [4]. It becomes obvious that the EW VLC outperforms the RVLCs with respect to their compressio properties. Furthermore, for a Laplacia ad a Gaussia distributio, the performace of the EW VLC is close to that of the Huffma code. Code rate R Map Egl. alphabet Laplace Gaussia uiform Huffma EW-VLC asym. RVLC [3] sym. RVLC [3] asym. RVLC [4] sym. RVLC [4] TABLE I COMPARISON OF THE CODE RATE R Map FOR THE EVEN-WEIGHT VARIABLE-LENGTH CODE (EW VLC) (d mi = 2), THE HUFFMAN CODE (d mi = 1), AND THE REVERSIBLE VARIABLE-LENGTH CODES (RVLCS) PROPOSED IN [3] (d mi = 1) AND [4] (d mi = 2). IV. ENCODER DESIGN FOR ITERATIVE SOURCE-CHANNEL DECODING AND EQUALIZATION We ow address the optimizatio of the source ad chael ecoder i Fig. 1 for iterative source-chael decodig. The properties of established bit mappigs ad variablelegth codes with respect to the desig guidelies for serial cocateated chael codes will be aalyzed. We will show i the followig that both covetioal bit mappigs ad VLCs lead to several drawbacks, which ca be avoided by employig EW VLCs proposed i the previous sectio. For both EW VLCs ad a scheme with fixed-legth source ecodig by employig the optimized bit mappigs from [39], we fially preset the results of a optimizatio of the ier chael ecoder ad precoder for the AWGN chael ad the ISI chael, respectively. A. Aalysis of Serial Cocateated Iterleaved Codes ad Desig Rules Geerally, the optimizatio of cocateated ecoders i cojuctio with iterative decodig ca be approached from two differet poits of view: o the oe had, the compoet codes have to be chose i such a way that good codes with respect to the distace properties are geerated. O the other had, we have to esure i the code desig that covergece of the iterative decoder ca be achieved. Both aspects are importat for the reliability of the resultig trasmissio system: a system sufferig from a high error floor is as much useless as a strog code that ca ot be decoded. 1) Distace Properties: A upper boud o the bit error probability P err for serial cocateated chael codes was derived i [36]. The aalysis of the upper boud leads to importat isights ito the desig of powerful serial cocateated codes. It shows that serial cocateated codes if properly desiged ca beefit from a iterleaver gai. It is defied as the factor by which the P err is decreased for a icreasig iterleaver legth N. Geerally, a iterleaver gai ca be obtaied [36] 1) if the miimum Hammig distace d (o) mi of the outer code satisfies d (o) mi 2 ad 2) if the ier ecoder is recursive. If these coditios are satisfied, the iterleaver gai is give by [36] P err N d (o) mi 2. Obviously, a large miimum Hammig distace d (o) mi of the outer ecoder leads to a large iterleaver gai. Furthermore, it is beeficial to maximize the effective free distace of the ier code ad to choose the outer ecoder as o-recursive [36]. 2) Covergece Aalysis: A powerful framework for the aalysis of the covergece behavior of the iterative decoder is give by extrisic iformatio trasfer characteristics (EXIT charts) [38]. Based o iformatio theoretic measures, it allows for precise predictios of the performace of serial cocateated codes uder iterative decodig, without ruig the iterative decoder. For a EXIT charts aalysis, SISO decoders are treated as iformatio filters. They are characterized by the amout of mutual iformatio betwee their extrisic output ad the correspodig quatity at the ecoder, give the mutual iformatio betwee the a priori iput ad the uderlyig symbols at the ecoder. I the followig, we refer to the mutual iformatio at the a priori iput of the source decoder ad the chael decoder / equalizer as I (s) apri := I(A(s) ;B ) ad I (c) apri := I(A(c) ;B ), respectively, ad to the mutual iformatio at the extrisic output of the source decoder ad the chael decoder / equalizer as I (s) extr := I(E (s) ;B ) ad I (c) extr := I(E (c) ;B ), respectively. Furthermore, we deote the mutual iformatio I(Y m ;X m ) betwee the symbols Y m ad X m at the output ad iput of the commuicatio chael as I c := I(Y m ;X m ). Correspodigly, the EXIT characteristics specifyig the source decoder ad the chael decoder / equalizer are give by the mappigs X (s) : I (s) apri I (s) extr = X (s) ( I (s) apri ) X (c) : I (c) apri I (c) extr = X (c) ( I (s) apri I c ). ad Note that, sice the ier decoder also utilizes the a priori iput for the chael observatios, the EXIT characteristic of the ier chael decoder / equalizer is coditioed o I c. Geerally, EXIT characteristics represet bijective mootoically icreasig fuctios from the iterval [0, 1] to [0, 1], ad thus, the iverse mappigs X (s) 1 ad X (c) 1 are well defied. I order to aalyze the covergece behavior of the iterative decoder, the EXIT characteristic X (c) of the ier (chael) decoder ad the iverse EXIT characteristic X (s) 1 of

6 the outer (source) decoder are plotted ito the same diagram. The iverse has to be applied for the outer ecoder sice the extrisic output of the oe decoder becomes the a priori iput of the other decoder. Predictios of the covergece behavior ca be obtaied by drawig a decodig trajectory (cf. [38]) betwee the two characteristics. Covergece is possible if the ier EXIT characteristic X (c) lies above X (s) 1. A itersectio of the EXIT characteristics specifies the fixpoit of the iterative decodig procedure. Correspodigly, i order to avoid a covergece towards high bit error rates, the compoets of the ecoder should be chose i such a way that the EXIT charts itersect at (I apri,i extr ) = (1,1) bit / chael use. As a cosequece, the ier ecoder must be recursive [37], [38] ad the miimum Hammig distace d (o) mi of the outer ecoder must satisfy d(o) mi 2. 3 Both recommedatios cofirm the desig guidelies derived from the aalysis of the upper boud of the bit error probability i [36]. Aother desig criterio follows from the area property of the EXIT charts, proved i [43] for the biary erasure chael. It states that the area A(X (s) 1 ) uder the iverse EXIT characteristic of the outer decoder correspods to its code rate R (o), while the area A(X (c) ) uder the ier EXIT characteristic X (c) is lower or equal to the ratio of the mutual iformatio I c of the commuicatio chael ad the ier code rate R (i) : A(X (s) 1 ) = R (o) ad A(X (c) ) I c /R (i). Sice equality holds for R (i) = 1, it is thus reasoable to choose ier rate-1 codes. Otherwise, the trasmissio system may suffer from a capacity loss [43], which may icrease the decodig threshold towards higher chael sigalto-oise ratios (SNR). B. Iverse EXIT Characteristics for the Ier Source Decoder For both stadard ad optimized bit mappigs ad the EW VLC itroduced i Sectio III the iverse EXIT characteristics for the ier source decoder [33] are depicted i Fig. 2. The EXIT characteristics are obtaied for source vectors of K = symbols, a correlatio coefficiet a = 0.9, ad a 4-bit scalar quatizatio with the Lloyd-Max quatizer. As show i Fig. 2, for stadard bit mappigs as the Gray mappig or the atural biary mappig, the outer source decoder provides oly a low amout of iformatio at its extrisic output, give perfect a priori iformatio (I (s) apri = 1 bit / chael use). Early itersectios with the EXIT fuctio of the ier decoder are the cosequece, leadig to a poor performace of the trasmissio system. To overcome this problem, optimized mappigs have bee proposed i [44], [39], which are based o a maximizatio of the distortio for the case of oe-bit errors ad o a maximizatio of the extrisic iformatio for optimal a priori iformatio, respectively. The iverse EXIT characteristic for 3 Eve though this coditio is satisfied for chael codes, it will become importat for the desig of the source ecoder i the ext sectio. I (s) apri Gray Mappig Natural Biary Adrat et al RVLC symm. RVLC symm. RVLC, d f =2 EW VLC I (s) extr Fig. 2. Iverse EXIT characteristics X (s) 1 of the (outer) source ecoder / decoder pair: compariso betwee stadard ad optimized bit mappigs, reversible VLCs, ad the EW VLC for a first-order autoregressive process with correlatio coefficiet a = 0.9, quatized by a 4-bit Lloyd- Max quatizer. the mappig from [39] is icluded i Fig. 2, verifyig the superiority of the optimized mappigs to the covetioal oes. However, we are still ot able to obtai iterative decoder covergece at (I apri,i extr ) = (1,1) bit / chael use i the EXIT chart. A similar behavior of the iterative decoder ca be expected for the asymmetrical RVLC proposed i [3]. By cosiderig a additioal symmetry costrait i the RVLC desig [3], the robustess ca be further improved, however, at the cost of a decreased code rate R Map. I this case, the mea word legth l is larger tha the quatizatio word legth W, i.e., the symmetrical RVLC itroduces some redudacy. Neither the aalyzed mappigs or the RVLCs from [3] fulfill the coditio d (o) mi 2. As a cosequece, trasmissio systems which employ oe of these codes ca geerally ot beefit from a iterleaver gai. RVLCs which satisfy the distace costrait d (o) mi 2 are proposed i [4]. The iverse EXIT characteristic for the symmetrical RVLC from [4] is icluded i Fig. 2. Assumig a appropriate (recursive) ier ecoder, covergece at (I apri,i extr ) = (1,1) bit / chael use becomes possible, ad the iterleaver gai ca be exploited. Ufortuately, as we ca see i Table I ad coclude from the area uder the iverse EXIT characteristic, the code rate is further decreased, i.e., additioal redudacy is itroduced due to the distace costrait. This drawback ca be avoided with the EW VLC from Sectio III. As Fig. 2 shows, (I apri,i extr ) = (1,1) bit / chael use is achieved i the EXIT chart whereas a slight compressio ca be obtaied compared to the bit mappigs with costat word legth. Thus, the EW VLC allows for a good covergece behavior of the iterative decoder while maitaiig good compressio properties at the same time.

7 (a) 1 (b) I extr, (c) I (s) apri 0.5 EW VLC 0.25 AWGN, E b /N 0 =2 db, RSC (13, 14) 8 ISI, E b /N 0 =5.6 db, ucoded ISI, E b /N 0 =5.6 db, DPC ISI, E b /N 0 =5.6 db, GPC (13, 1) I (c) apri, I (s) extr I extr, (c) I (s) apri 0.5 Adrat et al AWGN, E b /N 0 =2.5 db, RSC (75, 1) 8 ISI, E b /N 0 =5.6 db, ucoded ISI, E b /N 0 =5.6 db, DPC ISI, E b /N 0 =5.6 db, GPC (23, 1) I (c) apri, I (s) extr Fig. 3. Optimizatio of the ier chael ecoder / precoder for the EW VLC (a) ad the optimized bit mappig from [39] (b): AWGN chael without ISI, ucoded AWGN chael with ISI chael, ad precoded AWGN chael with ISI. C. Code Matchig for the Ier ad Outer Ecoder I the followig, we focus o the desig of the ier RSC code / precoder. The goal is to fid a good-matchig ier code whose EXIT characteristic lies above the iverse EXIT characteristic of the outer source decoder. Therefore, we restrict ourselves to the EW VLC, itroduced i the previous sectio, sice it offers the best trade-off betwee compressio capabilities ad distace properties. As a bechmarkig scheme for our EW VLC-based approach we cosider fixedlegth quatizer idices with a optimized bit mappig [39] i the source ecodig stage. As Fig. 2 shows, the outer source decoder provides oly a small amout of extrisic iformatio I (s) extr for I (s) apri 0.25 bit / chael use for both EW VLC ad optimized bit mappig. Therefore, i order to avoid a early itersectio of the EXIT characteristics, the ier chael ecoder / precoder has to be chose such that i absece of a priori kowledge (I (c) apri = 0 bit / chael use) a miimum amout of extrisic iformatio I (c) extr(0) 0.1 bit / chael use is guarateed. 1) AWGN Chael without ISI: For rate-1 RSC codes, which are obtaied by pucturig the systematic bits, it was observed i [38] that oly codes whose feedforward polyomials cosist of a sigle coefficiet are able to geerate extrisic iformatio I (c) extr(0) > 0 bit / chael use without additioal a priori kowledge. However, especially for a short code memory, these codes have the drawback that oly few cadidates are available such that the code desig becomes quite costraied. I [38], this weakess was compesated by systematic dopig, i.e., for RSC codes whose feed-forward polyomials cosist of more tha oe coefficiet, a low umber of systematic bits was trasmitted i additio to the parity bits. However, this approach poses a problem i our sourcechael ecodig setup. I order to achieve I (c) extr(0) 0.1 bit / chael use the umber of additioally trasmitted systematic bits would become too large such that the decrease of the coderate R RSC caot be eglected. As a remedy, i [33] we have applied puctured RSC codes where both systematic ad parity bits are prued from the bit stream i order to fid a good-matchig rate-1 RSC code. However, due to the high degree of freedom, a optimizatio becomes fairly complex. The desig of appropriate pucturig patters ca be simplified by cosiderig radomly puctured RSC codes as itroduced i Sectio II-B. The result of the optimizatio for the EW VLC from Sectio III is show i Fig. 3(a). A good-matchig code is foud by radomly pucturig a fractio p sys = 92% of the systematic bits ad a fractio of p par = 8% of the parity bits from the memory-3 RSC code with geerator polyomials (13,14) 8. Fially, for the optimized bit mappig from [39] we apply the memory-5 RSC code [45], where the correspodig EXIT characteristic is show i Fig. 3(b). It is obtaied by pucturig all systematic bits from the rate-1/2 mother code with the geerator polyomials (75,1) 8. Sice i this case the EXIT characteristic for the source decoder does ot reach (I apri,i extr ) = (1,1) bit / chael use i the EXIT chart, a RSC code with larger memory must be applied. 2) AWGN Chael with ISI: For the aalysis of the ier equalizer EXIT characteristic, we employ the memory-4 ISI chael from [46] give by the impulse respose h 0 (m) = γ 0 (m)+0.46 γ 0 (m 1) γ 0 (m 2) γ 0 (m 3) γ 0 (m 4). It itroduces severe ISI to the trasmitted data ad is thus well suited as test chael for turbo equalizatio [47]. I the followig, we assume perfect kowledge of the chael coefficiets h 0 (m) at the receiver. Without additioal precodig, the ISI chael ca be iterpreted as a o-recursive o-biary ier ecoder, leadig to a violatio of the desig rules, summarized i

8 (a) 10 0 (b) SERL EW VLC, K =20000 EW VLC, K =2000 EW VLC, K =200 Adrat et al., K =20000 Adrat et al., K =2000 Adrat et al., K = E b /N 0 i db R SNR i db 12 7 EW VLC, K =20000 EW VLC, K =2000 EW VLC, K =200 2 Adrat et al., K =20000 Adrat et al., K =2000 Adrat et al., K = E b /N 0 i db Fig. 4. Compariso betwee optimized mappigs [39] ad EW VLCs for the AWGN chael with iterative source-chael decodig: symbol error rate (SER) versus E b /N 0 (a) ad sigal-to-oise ratio (SNR) versus E b /N 0 after source recostructio (b), for varyig block legths K {200, 2000, 20000}. Sectio IV-A. Accordigly, we ca observe i Fig. 3 that the EXIT characteristic for the APP equalizer applied to the ucoded chael does ot reach I (c) extr = 1 bit / chael use for I (c) apri 1 bit / chael use. Early itersectios with the outer source decoder characteristics result for both the EW VLC i Fig. 3(a) ad the optimized bit mappigs i Fig. 3(b). A simple approach to make the ISI chael recursive is to apply a differetial precoder (DPC) i form of a memory-1 accumulator. However, eve though the EXIT characteristic for the equalizer ad the differetial precoded ISI chael approaches (I apri,i extr ) = (1,1) bit / chael use i the EXIT chart, early itersectios still remai. To overcome this problem, we cosider a geeralized precoder (GPC). As described i Sectio II-B, it is realized by a rate-1 RSC code with feedforward polyomial G f = 1 which is obtaied by pucturig all systematic bits from the bit stream. The results of a code search are icluded i Fig. 3: for the EW VLC (a), the memory-3 RSC code (13,1) 8 offers good properties, allowig for a robust covergece behavior of the iterative decoder / equalizer. As already observed i the AWGN case, the code search for the optimized bit mappig leads to a code with higher memory. We foud the memory-4 RSC code (23,1) 8 as a well-suited cadidate. The correspodig EXIT characteristic is show i Fig. 3(b). Note that, sice the memory of all cosidered precoders is lower or equal to the memory of the ISI chael, the decoder complexity stays the same as for the equalizatio of the ucoded chael. V. SIMULATION RESULTS I the followig, we verify the good performace of the eve-weight variable-legth code (EW VLC) from Sectio III i the cotext of iterative source-chael decodig. Mote Carlo simulatios are performed for both the BPSKmodulated AWGN chael with ad without ISI by usig the code costructios from Sectio IV-C. The results are preseted i Figs. 4 ad 5 for the symbol error rate (SER L ) based o the Leveshtei metric 4 ad for the distortio i terms of recostructio sigal-to-oise ratio (R SNR). I order to allow a fair compariso betwee differet approaches the chaels are parametrized by the chael SNR E b /N 0 related to the trasmit eergy per iformatio bit E b =E s /R. The simulatios are carried out for source vectors of K = symbols, a correlatio coefficiet a = 0.9, ad a 4-bit scalar quatizatio with the Lloyd-Max quatizer. For both the EW VLC ad the optimized mappig we achieve approximately the same overall code rate of R All iterleavers are realized by s- radom iterleavers. A. AWGN Chael without ISI Fig. 4 shows the good performace of the EW VLC combied with the radomly puctured RSC code (13,14) 8. For a block legth of K = symbols ad E b /N db, a error-free trasmissio is achieved after 40 iteratios, withi 0.6 db of the capacity limit for R = Compared to the fixed-legth approach with a optimized mappig from [39] ad a memory-5 RSC code, a gai of 0.5 db i chael SNR with respect to clear chael quality ca be observed i Fig. 4(b). For the optimized mappig covergece is obtaied after 10 iteratios, eve though the trasmissio suffers from a residual SER of approximately For a itermediate block legth of K = 2000 symbols the performaces of both systems deteriorate. Despite the VLC approach ow also shows a error-floor, it still outperforms the optimized mappig: for E b /N db ad K = 2000 symbols, the achieved distortio is comparable to the oe obtaied by the fixed-legth-based scheme for a block legth 4 The Leveshtei metric of two sequeces gives the umber of isertios, deletios, ad / or substitutios which are eeded to trasform oe sequece ito the other. A efficiet approximatio is proposed i [5].

9 (a) 10 0 (b) SERL EW VLC, GPC EW VLC, DPC EW VLC, ucoded Adrat et al., GPC Adrat et al., DPC Adrat et al., ucoded E b /N 0 i db R SNR i db 12 7 EW VLC, GPC EW VLC, DPC EW VLC, ucoded 2 Adrat et al., GPC Adrat et al., DPC Adrat et al., ucoded E b /N 0 i db Fig. 5. Compariso betwee optimized mappigs ad EW VLCs for the ISI chael (ucoded, differetial precodig (DPC), ad geeralized precodig (GPC)) ad iterative source-chael decodig ad equalizatio: symbol error rate (SER) versus E b /N 0 (a) ad sigal-to-oise ratio (SNR) versus E b /N 0 after source recostructio (b). of K = symbols. If the block legth is further reduced to K = 200 symbols, the optimized mappig becomes superior for the chael SNR regime show i Fig. 4. The SER compariso i Fig. 4(a) for differet block / iterleaver legths shows the behavior we expect accordig to the discussio from Sectio IV-B: due to a miimum distace of d (o) mi = 1, the system based o the optimized mappig suffers from a residual error floor for E b /N db. I cotrast, the EW VLC with d (o) mi = 2 takes advatage of the available iterleaver gai: while the SER oly slowly decreases for E b /N db ad a block legth of K = 2000 symbols, o error evet could be observed for E b /N db ad K = symbols. B. AWGN Chael with ISI The results for ucoded ad precoded trasmissio i the presece of ISI are show i Fig. 5 for a blocklegth of K = symbols. The results cofirm the poor performace of iterative source decodig ad equalizatio for the ucoded ad differetially precoded (DPC) ISI chael, predicted i Sectio IV-C. For both the EW VLC-based ad the mappig-based scheme early itersectios betwee the EXIT characteristics lead to high symbol error rates ad high distortios. Compared to the EW VLC, which suffers from sychroizatio losses, the optimized mappig turs out to be more robust i this eviromet. However, sigificat improvemets ca be obtaied by cosiderig the geeralized precoders (GPC) proposed i Sectio IV-C. For the EW VLC a error-free trasmissio is ow achieved for E b /N db after 50 iteratios betwee the SISO source decoder ad the equalizer. I accordace to the results for the AWGN chael without ISI we agai observe a gai of 0.5 db for the EW VLC-based techique compared to the approach based o a optimized mappig, which ow shows a slightly decreasig error floor for E s /N db. VI. CONCLUSIONS We have preseted a simple variable-legth code costructio which is well suited for trasmissio systems employig iterative source-chael decodig. While providig good compressio properties compared to, e.g., RVLCs the proposed eve-weight variable-legth code (EW VLC) guaratees a miimum Hammig distace of two betwee equal-legth code sequeces. Both features are the key to trasmissio systems which are extremely efficiet i terms of required overhead for error protectio ad robust at the same time. As the simulatio results for correlated Gauss- Markov sources show, the proposed EW VLC facilitates reliable commuicatio close to the AWGN chael capacity solely by exploitig residual source redudacy. This also holds true if the chael is additioally affected by ISI. Fially, a compariso shows that the proposed EW VLCbased scheme outperforms the best fixed-legth bit mappig optimized for iterative source-chael decodig. These results suggest to employ well-desiged variable-legth codes istead of fixed-legth mappigs i the cotext of iterative source-chael decodig. REFERENCES [1] Y. Takishima, M. Wada, ad H. Murakami, Reversible variable legth codes, IEEE Tras. Commu., vol. 43, o. 2/3/4, pp , Feb./Mar./April [2] J. We ad J. D. Villaseor, Reversible variable legth codes for efficiet ad robust image ad video codig, i Proc. IEEE Data Compressio Coferece, Sowbird, UT, Mar. 1998, pp [3] C. W. Tsai ad J. L. Wu, O costructig the Huffma-code based reversible variable legth codes, IEEE Tras. Commu., vol. 40, o. 9, Sept [4] K. Laković ad J. Villaseor, O desig of error-correctig reversible variable legth codes, IEEE Commuicatios Letters, vol. 6, o. 8, pp , Aug [5] V. Buttigieg, Variable-Legth Error-Correctig Codes, Ph.D. thesis, Departmet of Electrical Egieerig, Uiversity of Machester, Machester, Eglad, 1995.

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