Quantization for Soft-Output Demodulators in Bit-Interleaved Coded Modulation Systems

Transcription

1 Quantization for Soft-Output Demodulators in Bit-Interleaved Coded Modulation Systems Clemens Novak, Peter Fertl, and Gerald Matz Institut für Nachrichtentechnik und Hochfrequenztechnik, Vienna University of Technology {cnovak, pfertl, Abstract We study quantization of log-likelihood ratios(llr) in bit-interleaved coded modulation (BICM) systems in terms of an equivalent discrete channel. We propose to design the quantizer such that the quantizer outputs become equiprobable. We investigate semi-analytically and numerically the ergodic and outage capacity over single- and multiple-antenna channels for different quantizers. Finally, we show bit error rate simulations forbicmsystemswithllrquantizationusingarate/lowdensity parity-check code. I. INTRODUCTION Bit-interleaved coded modulation(bicm) is an attractive scheme for wireless communications where a block of informationbitsismappedtotransmitsymbolsviaachannelencoder and a symbol mapper separated by a code bit interleaver []. At the receiver side, a demodulator(demapper, detector) calculates log-likelihood ratios(llr) for the code bits, which are de-interleaved and passed to the channel decoder. Theoretically, one real-valued LLR per code bit needs to be computed and stored by the receiver. Clearly, practical digital implementations can only use finite word-length approximations of real numbers, which motivates the study of LLR quantization. We note that LLR quantization is also relevant for wireless(relay) networks that perform distributed turbo and network coding by exchanging soft information between different nodes. Optimal LLR quantization maximizing informationrateforthespecialcaseofbpskmodulationoveran AWGN channel was considered in[]. However, an extension of this approach to other channels and modulations appears infeasible. Thus, we consider a different quantizer design in this paper which allows for simple implementation while only slightly degrading information rate. More specifically, our contributions are as follows: We propose to quantize the LLRs such that the quantizer outputs become equiprobable and provide appropriate reliability information to the channel decoder. We investigate the impact of the proposed LLR quantization on ergodic and outage rate, using a semi-analytical approach for single-input single-output(siso) systems with BPSK and Monte-Carlo simulations otherwise. We develop a method for designing and implementing the proposed quantizer during data transmission. We provide bit error rate() simulations for BICM systems with LLR quantization using low-density paritycheck(ldpc) codes. The paper is organized as follows. Section II presents the system model and Section III discusses the proposed LLR quantization based on an equivalent discrete channel. In SectionsIVandV,westudythesystemcapacityofSISOand MIMO-BICM systems, respectively. The estimation of the quantizer parameters is addressed in Section VI and results are provided in Section VII. II. SYSTEM MODEL We consider a MIMO-BICM system with M T transmit antennasand M R receiveantennas(siso-bicmcanbeviewed asspecialcasewith M T =M R =).Ablockdiagramisshown infig..asequenceofinformationbits b[n ]isencodedusing an error-correcting code, passed through a bitwise interleaver Πandthenscrambledbyapseudo-randomsequence p l [n]. The uniformly distributed interleaved and scrambled code bits aredemultiplexedinto M T antennastreams( layers ).Ineach layer,groupsof mcodebitsaremappedto(complex)data symbols x k [n] A, k =,...,M T ; here, A denotes the symbol alphabet of size A = m. The transmit vector at symboltime nisgivenby x[n] (x [n]... x MT [n]) T and carries R 0 =mm T interleavedcodebits c l [n], l=,...,r 0. Assumingflatfading,thelength-M R receivevectorequals y[n] = H[n]x[n] + w[n]. () Here, H[n]isthe M R M T MIMOchannelmatrixand w[n] CN(0,σ I)denotesthecomplexGaussiannoisevector.Inthe following, we will omit the time index n to simplify notation. At the receiver, the max-log demodulator calculates LLRs forthecodebits c l accordingto[3] Λ l = σ [ min x X 0 l y Hx min y Hx ]. () x Xl Here, X b l denotesthesetoftransmitvectorsforwhich c l = b. These LLRs (or approximate/quantized versions thereof) are de-scrambled by the sequence p l [n] = p l [n], deinterleaved and used by the channel decoder to obtain bit estimates ˆb[n].Thesymmetricnoisedistributionandtheuse ofthescrambleryieldthesymmetries f Λ (ξ) = f Λ ( ξ)and f Λ c (ξ c = ) = f Λ c ( ξ c = 0)forthe(un)conditionalLLR distribution.hence,knowledgeof f Λ c (ξ c = )issufficient for characterizing Λ.

2 equivalent discrete channel b[n ] encoder Π cl[n] p l[n] DEMUX map. map. x[n] MIMO channel y[n] demodulator MUX p l[n] Λ l[n] Q( ) d l[n] Π decoder ˆb[n ] Fig.. Block diagram of a MIMO-BICM system. III. LLR QUANTIZATION TheLLRsin()canattainanyrealvalue.Wenextstudy how to quantize these LLRs. While in practice the demodulator will directly deliver quantized LLRs, the efficient calculation ofquantizedllrsisoutofthescopeofthispaper. Weconsideraq-bitquantizercharacterizedby K = q bins I k = [i k,i k ], k =,...,K.Weusetheconvention i 0 =, i K = andassumesymmetricbins(thisismotivated bythesymmetryofthellrdistributions),withboundaries i k sorted in ascending order. The quantizer Q( ) maps the LLR Λ l toadiscretellr d l accordingto d l = Q(Λ l ) = λ k if Λ l I k. Here, λ k I k isthe kthquantizationlevel. In the following, we consider the equivalent discrete channel with binary input c {0,} and K-ary output d {λ,...,λ K }.Here, cand dareobtainedbyrandomlypicking abitposition l =,...,R 0 accordingtoauniformdistribution. This models a situation where the outer channel code is blind to the bit positions within the symbol labels. The crossoverprobabilities p bk = Pr{d = λ k c = b} = Pr{Λ I k c = b}ofthischannelaregivenby p bk = f Λ c (ξ b)dξ, I k where f Λ c (ξ b)istheconditionalprobabilitydensityfunction (pdf)ofthellr Λgiventhat c = b(averagedwithrespect tobitposition l).notethat Pr{d = λ k } = Pr{Λ I k } = (p 0k + p k ).Themutualinformation(capacity) I = I(c;d) ofthisdiscretechannelisgivenby[4] I = b=0 k= K p bk p bk log. (3) p 0k + p k If the LLR distribution f Λ c (ξ b) and hence the transition probabilities p bk areaveragedwithrespecttothestatisticsof the physical channel H(reflecting fast fading), the quantity I describes the ergodic rate achievable over the equivalent channel(cf.[5]). Otherwise(quasi-static fading), the transition probabilities p bk, and thus the rate I, change with every realization of the channel matrix H. Here, the probability p out (r) = Pr{I R}, 0 R R 0 (4) characterizes the rate(denoted R) versus outage trade-off[5]. Designing the quantizer to maximize the mutual information I(c; d) appears analytically infeasible in general(for BPSK andnofadingthesolutionisgivenin[]).hence,wepropose adifferentapproach:since c Λ disamarkovchain,the dataprocessinginequalityimplies I(c;d) I(c;Λ).Inorder for I(c;d)tobeascloseaspossibleto I(c;Λ)(forfixed K), our proposed quantizer maximizes the mutual information I(Λ;d).With H( )denotingentropy,itfollowsthat I(Λ;d) = H(d) H(d Λ) and H(d Λ) = 0 because d is a deterministic function of Λ. H(d) is maximized by a uniform distribution of dandtherefore,thequantizerboundaries i k, k =,...,K, have to ensure that Pr{d=λ k } = p 0k + p k =, k =,...,K. (5) K UsingtheunconditionalcumulativeLLRdistribution F Λ (λ) = Pr{Λ λ} = λ [ fλ c (ξ c = 0) + f Λ c (ξ c = ) ] dξ,the optimal boundaries can be obtained by finding the arguments forwhich F Λ (λ) = k/k,i.e., ( k ) i k = F Λ K k =,...,K. (6) Wenotethatforthecapacityin(3)onlythebins(boundaries)arerelevant,i.e.,theactualquantizationlevels λ k do not influence the achievable rate. However, these values are important in order to provide the channel decoder(e.g., a belief propagation decoder) with correct reliability information[6]. In view of the equivalent discrete channel, we hence propose to choose the quantization levels as corresponding LLRs λ k = log Pr{c = d = λ k} Pr{c = 0 d = λ k } = log p k p 0k. (7) Wefinallynotethat λ k I k. IV. SISO-BICM SYSTEMS WITH BPSK MODULATION WenextstudyinmoredetailthecaseofaSISOsystem (M T = M R = )withbpskmodulation(r 0 = bpcu)in Rayleighfading.Here,thesystemmodel()becomesrealvaluedandsimplifiesto y = hx + w,with h N(0,), w N(0,σ /),and x = c {,}.Then,theLLR Λin()equals Λ = hy σ = h(hx + w). (8) σ Theresultsinthissectionalsoapplytotheinphaseandquadraturephase ofsisosystemswithgray-labeledqpskandtothetwolayersofbpskmodulated MIMO systems.

3 Max. Achievable Rate [bpcu] bit bit Fig.. Comparison of ergodic capacity for SISO-BICM with BPSK and different quantizer word-lengths. Outage Probability R=/4bpcu bit bit R=3/4bpcu Fig. 3. Outage probability for quasi-stationary fading for SISO-BICM with BPSKforrate R=/4bpcuand R=3/4bpcu. A. Ergodic Capacity Conditioned on c = x =, the LLR can be rewritten as Λ = σ z T Az,where z = ( h ) w T σ N(0,I)and ( ) σ/ A =. σ/ 0 Usingtheeigenvaluedecomposition A = UΣU T,with U orthogonaland Σ =diag{σ,σ },where σ, = ± +σ, we further obtain Λ = σ zt Σ z = ] [σ z σ + σ z. Here, z = U T z N(0,I)duetotheorthogonalityof U. Thus, Λ is a linear combination of two independent chi-square random variables with one degree of freedom. The distribution f Λ c (ξ c = )canthusbeshowntobegivenby(cf.[7]) f Λ c (ξ c=) = σ π exp( ξ + σ ) K 0 ( ξ ), (9) where K 0 ( ) denotes the modified Bessel function of the secondkindandorder0. Using(9), one can determine the LLR distribution, the LLR quantization(cf.(6)), and the ergodic capacity of the equivalent channel. Numerical results for the rate in bits per channel use(bpcu) versus SNR achievable with our proposed LLR quantizersofdifferentword-length qareshowninfig..asa reference, we also show the capacity of non-quantized max-log demodulation(labeled ). Hard-output demodulation (i.e., -bit quantization) incurs a significant performance loss compared to non-quantized demodulation (more than 5 db at rate / bpcu). With -bit and 3-bit LLR quantization, performance remains within db of the non-quantized case up to rates of approximately / bpcu and 3/4 bpcu, respectively. B. Outage Capacity Additionally conditioning on the channel coefficient h, it followsstraightforwardlythat Λ c N(xγ,γ)with γ = h /σ.thisallowstocalculatethetransitionprobabilitiesof the equivalent channel as ( ) ( ik (b )γ ik (b )γ p bk = Q Q ). γ γ The outage probability can thus be evaluated according to(4). Numericalresultsof p out (r)versussnrforquasi-stationary fadingwithrate R = /4bpcuandwith R = 3/4bpcuare showninfig.3.llrquantizationwithmorethan bitsis required to offer performance gains at medium-to-high outage probability. At high SNR, the gap between the non-quantized caseandallquantizeddemodulators(q =,,3)is.5dB and.5dbfor R = /4bpcuand R = 3/4bpcu,respectively. Here, q > 3isrequiredtoclosethisgapandtoreachoutage probabilities close to the non-quantized case. V. MIMO SYSTEMS AND HIGHER-ORDER MODULATION In the following, we investigate LLR quantization for MIMO systems and higher-order constellations. Since in this case analytical expressions for the LLR distribution are hard to obtain in general, the remaining discussion is based exclusively on numerical results. For the capacity results in this section, we used empirical LLR distributions obtained from Monte-Carlosimulationstodeterminethebins I k suchthat Pr{Λ I k } = /K(cf.(5)).Intheremainderofthepaper, we will consider a MIMO system with Gray-labeled 6- QAMmodulation(here, R 0 =8bpcu). A. Ergodic Capacity We evaluated the capacity in (3) under the assumption of ergodic spatio-temporally i.i.d. fast Rayleigh fading for various quantizer word-lengths q. To this end, we estimated the transition probabilities p bk by means of Monte-Carlo simulations after having determined the optimal bins based on 0 5 channelrealizations.fig.4showstheresultsobtained.it can be seen that the extreme case of -bit quantization yields a considerable performance loss in comparison to the nonquantizedcase(e.g., 3dBSNRlossat 4bpcu).Increasingthe number of quantization levels reduces this gap significantly(to 0.5 db and 0. db for -bit and 3-bit quantization, respectively, at 4 bpcu). Even though at higher rates slightly increasing SNR gaps are observed, these results suggest that 3-bit LLR quantization is sufficient for practical purposes. In order to illustrate the impact of the LLR quantizer design on ergodic capacity, we consider the same MIMO-BICM

4 Max. Achievable Rate [bpcu] bit bit Fig. 4. Ergodic capacity of a MIMO-BICM system with Gray-labeled 6-QAM for different LLR quantization word-lengths. Required rate=6 bpcu rate=4 bpcu 8 rate= bpcu 6 bit reference 4 bit non uniform bit proposed Interval Boundaries i 3 = i Fig.5. SNRrequiredforatargetrateof, 4, 6bpcuvs.quantizerboundary i 3 ofa-bitllrquantizer( MIMO-BICMsystemwithGray-labeled 6-QAM).Boundaries i 3 ofproposedquantizeraremarkedbygreendots. system with Gray-labeled 6-QAM modulation for various - bit(i.e., 4-level) LLR quantizers. Since symmetric quantization hereamountsto i = 0and i = i 3,theboundary i 3 is sufficient to index all quantizers in this case. Fig. 5 plots thesnrrequiredtoachievetargetratesofbpcu,4bpcu and 8bpcu versus i 3, respectively. As a reference we also show the required SNR using a -bit quantizer with uniform distribution and the required SNR for -bit quantization(hard demodulation).itcanbeseenthatforratesofbpcuand 4 bpcu the -bit quantizer with uniform distribution requires thesamesnrasthe-bitquantizerwithoptimalchoiceof i 3 (thisisthequantizerproposedin[]).forratesof6bpcu the SNR loss of the -bit quantizer with uniform distribution isaboutdbcomparedtotheoptimalquantizer. B. Outage Capacity We next provide numerical results for the outage probability in(4) for quasi-static fading. Fig. 6 shows the outage probability p out (r)versussnrfordifferentquantizerwordlengthsand R=bpcuand R=6bpcu.Fromtheasymptotic slopes of these curves it is seen that the diversity order equalsinallcases.for R = bpcuand R = 6bpcu,hard demodulation(q = )isrespectively 4.8dBand.8dBaway from the non-quantized case at high SNR. LLR quantization withand3bitsperformsonlyslightlybetteratverylow Outage Probability R=bpcu R=6bpcu bit bit Fig. 6. Outage probability for quasi-stationary fading for MIMO with 6-QAMusingGraylabelingforrate R=bpcuand R=6bpcu. outage probability, but offer significant gains at medium-tohighoutageprobability.for R = bpcu,thesnrlossof LLRquantizationwith,,and3bitsat p out =0 equals 4dB,.4dB,and0.4dB,respectively. VI. ESTIMATION OF QUANTIZATION PARAMETERS The computation of the quantization boundaries i k and quantizationlevels λ k accordingto(6)and(7),respectively, requiresthellrdistributions f Λ (ξ)and f Λ c (ξ c),whichin general are unknown. We thus address on-the-fly estimation of the quantization parameters. The boundaries can be estimated by using an empirical estimate of the unconditional LLR distribution F Λ (ξ),whichcanbeobtainedfromareasonable number of non-quantized LLRs. In contrast, determining the quantization levels λ k by estimating f Λ c (ξ c) is more difficult since the code bits areunknownatthereceiver.hence,weproposetousethe following simple parametric model, which is motivated by numerical results for the case with 6-QAM(other system parameters may require a different model): { αβ α+β exp(αξ) ξ < 0, f Λ c (ξ c=) = αβ α+β exp( βξ) ξ 0. (0) Toestimatethetwoparameters α > 0and β > 0,wechoose twobins Ī and Ī andusethenon-quantizedllrs Λto obtainempiricalestimates ˆP i, i =,,oftheprobabilities P i (α,β) = Pr{Λ Īi} = f Λ (ξ)dξ, Ī i with f Λ (ξ) = [ f Λ c (ξ c = 0) + f Λ c (ξ c = ) ] /.Thesystem ofequations P i (α,β) = ˆP i canthenbesolvednumerically to obtain estimates of α and β. The transition probabilities of the equivalent channel and the quantization levels are then computedbasedon(0)usingtheestimatesof αand β. VII. NUMERICAL RESULTS To verify the capacity results, we performed simulations for SISO- and MIMO-BICM systems in ergodic Rayleigh fastfading.thechannelcodewasaregularldpccode with rate / and block length The LDPC code was designed using the EPFL web-tool at

5 Limit=3.6dB Limit=3.9dB Limit=4.56dB Limit=8.88dB bit bit Fig. 7. performance for a rate-/ LDPC coded SISO-BICM system with BPSK and different LLR quantization word-lengths Limit=.79dB Limit=3.34dB Limit=4.85dB offl., onl., offl., bit onl., bit offl., bit onl., bit Fig.8. performanceforarate-/ldpccoded MIMOsystem with Gray-labeled 6-QAM and different LLR quantization word-lengths. A. SISO-BICM We first consider a BPSK-modulated SISO-BICM system with LLR quantizers designed using the analytical results from Section IV. Fig. 7 shows the for our proposed LLR quantizers with different word-length together with the theoretical SNR thresholds(obtained from Fig. ). All curves are reasonably close to the respective SNR thresholds (obtained from Fig. and indicated by vertical lines). The gaps of -bit, -bit, and 3-bit LLR quantization to the non-quantized caserespectivelyequal6.db,.db,and0.4db. B. MIMO-BICM Fig. 8 shows two(strongly overlapping) sets of curves for the MIMO-BICM system with Gray-labeled 6- QAM and different LLR quantization word-lengths. One set of curves(labeled offl. ) pertains to an offline design of the LLR quantizer, whereas the other set(labeled onl. ) estimates the quantization parameters on-the-fly according to Section VI. The gap to the theoretical SNR thresholds(obtained from Fig.4andindicatedbyverticallines)equals 0.6dBfor 3-bit and -bit quantization and db for -bit quantization(hard demodulation). Furthermore, the proposed on-the-fly estimator for the LLR quantizer parameters performs extremely well in this setup(virtually indistinguishable from the offline design). To illustrate the importance of the correct choice of the LLR quantization levels, Fig. 9 shows versus quantization optimal quantizer level Quantization Level λ = λ Fig. 9. versus quantization level for -bit quantization at an SNR of.8 db using a rate-/ LDPC code( MIMO, 6-QAM, Gray labeling). level λ = λ forthesamemimosystemasbeforewith- bitllrquantizationatansnrof.8db.here,theoptimal quantizerlevel λ =.6(indicatedbyadashedverticalline) achievesaof Itisseenthattheachieved by the belief propagation decoder is quite sensitive to the choiceof λ ;for λ.5or λ 4.3,hasdeteriorated toabout 0 (i.e.,bymorethanordersofmagnitude). VIII. CONCLUSION We considered bit-interleaved coded modulation systems with demodulators providing quantized log-likelihood ratios (LLR). We provided design rules which lead to easily implementable LLR quantizers and studied the information rates of the equivalent discrete channel in the ergodic and outage regime. Numerical results for capacity and bit error rate showed that LLR quantization using a small number of bits is often sufficient. We also proposed simple procedures to estimate the quantizer parameters during data transmission. ACKNOWLEDGEMENTS The authors are grateful to Jossy Sayir for drawing their attention to the problem of quantized soft information and to Joakim Jaldén for helpful comments. This work was supported by the STREP project MASCOT(IST-06905), the Network of Excellence NEWCOM++(IST-675), and by the FWF Grant S0606 Information Networks. REFERENCES [] A. Guillén i Fàbregas, A. Martinez, and G. Caire, Bit-interleaved coded modulation, Foundations and Trends in Communications and Information Theory, vol. 5, no. -, pp. 53, 008. [] W. Rave, Quantization of log-likelihood ratios to maximize mutual information, IEEE Signal Processing Letters, vol. 6, pp , Apr [3] S. H. Müller-Weinfurtner, Coding approaches for multiple antenna transmission in fast fading and OFDM, IEEE Trans. Signal Processing, vol. 50, pp , Oct. 00. [4] T.M.CoverandJ.A.Thomas,ElementsofInformationTheory. New York: Wiley, 99. [5] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Boston(MA): Cambridge University Press, 005. [6] S. Schwandter, P. Fertl, Clemens Novak, and G. Matz, Log-likelihood ratio clipping in MIMO-BICM systems: Information geometric analysis and impact on system capacity, in Proc. IEEE ICASSP-009, pp , 009. [7] M. K. Simon, Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists. New York: Springer, 00.