Belief-Propagation Decoding of LDPC Codes with Variable Node-Centric Dynamic Schedules
Tofar C.-Y. Chang, Pin-Han Wang, Jian-Jia Weng, I-Hsiang Lee, Yu T. Su
aa r X i v : . [ c s . I T ] F e b Belief-Propagation Decoding of LDPC Codeswith Variable Node–Centric DynamicSchedules
Tofar C.-Y. Chang,
Member, IEEE , Pin-Han Wang, Jian-JiaWeng,
Member, IEEE , I-Hsiang Lee, and Yu T. Su,
Life Senior Member, IEEE
Abstract
Belief propagation (BP) decoding of low-density parity-check (LDPC) codes with various dynamicdecoding schedules have been proposed to improve the efficiency of the conventional flooding schedule.As the ultimate goal of an ideal LDPC code decoder is to have correct bit decisions, a dynamic decodingschedule should be variable node (VN)-centric and be able to find the VNs with probable incorrectdecisions and having a good chance to be corrected if chosen for update. We propose a novel andeffective metric called conditional innovation (CI) which serves this design goal well. To make themost of dynamic scheduling which produces high-reliability bit decisions, we limit our search for thecandidate VNs to those related to the latest updated nodes only.Based on the CI metric and the new search guideline separately or in combination, we develop sev-eral highly efficient decoding schedules. To reduce decoding latency, we introduce multi-edge updatingversions which offer extra latency-performance tradeoffs. Numerical results show that both single-edgeand multi-edge algorithms provide better decoding performance against most dynamic schedules andthe CI-based algorithms are particularly impressive at the first few decoding iterations.
Index Terms
LDPC codes, belief propagation, informed dynamic scheduling, decoding schedule, 5G New Radio.
February 23, 2021 DRAFT
I. I
NTRODUCTION
Low-density parity-check (LDPC) codes are known to provide near-capacity performancewhen the belief propagation (BP) algorithm is utilized for decoding [1]. These codes have beenused in many applications such as deep-space network, disk storage, satellite communicationsand have adopted by several wireless communication standards, e.g., IEEE 802.11 (WiFi) [2]and 5G New Radio (NR) [3].The conventional BP algorithm performs message-passing on the code graph based on the flooding scheduling: the variable-to-check (V2C) messages sent from all variable nodes (VNs)to the linked check nodes (CNs) are updated and propagated simultaneously, so are the check-to-variable (C2V) messages. However, such a fully-parallel decoding schedule often requiresmany iterations to converge and necessitates complicated interconnections and large memoryfor hardware implementation. Therefore, sequential and semi-sequential decoding scheduleshave been proposed for improving the convergence speed and/or reducing the implementationcomplexity [5]-[16]; some even provide improved converged error rate performance. The non-flooding schedules are generally categorized into two classes–the ordered schedules and thedynamic schedules. The former class is also referred to as the standard sequential scheduling(SSS) strategies. The SSS-based BP decoders include the layered BP (LBP) [5], shuffled BP [6],and their variants [7], [8]. They converge at least twice faster than the conventional BP decoderand require less processing time and simpler hardware implementation [19], [20].The dynamic schedules modify the message-passing order based on newest available informa-tion. The informed dynamic scheduling (IDS) strategies form a popular subclass of the dynamicschedules. It makes an element-wise comparison of two sets we refer to as the current andprecomputed message sets. The former set can be the set of the C2V, V2C messages sent orVNs’ total log-likelihood ratios (LLRs) computed in the last update (messages or LLRs maybe updated in different time instants) and the elements of the latter set are the correspondingvalues if updated. These messages are functions of the channel values associated with eachcoded bits (or VNs) which vary from a codeword to another and the messages collected fromconnecting VNs or CNs which vary with each update. Hence, a proper dynamic schedule
DRAFT February 23, 2021 which adjusts the message-passing order according to these two message sets may yield fasterconvergence speed and lower error rate. The IDS strategies forward only the best precomputedmessage(s) according to a certain metric. In the residual BP (RBP) algorithm [9], the current andprecomputed C2V message sets are adopted, and element-wise differences between these twosets are called residuals . The RBP algorithm passes only the C2V message for the edge with themaximum residual among all code graph edges. It yields better convergence speed in comparisonwith the SSS and flooding scheduling based BP decoders but suffers from inferior convergederror rate performance. The degraded performance is due in part to the greedy behavior that thedecoder may keep updating only a small group of edges [9]. Hence, several algorithms wereproposed to prevent such a greedy event [9]-[12]. Other recent works have put more emphasison improving the RBP algorithm’s error rate performance. In [13]-[15], the message updatingpriority is mainly based on the VN decisions’ stability while the method proposed in [16] makesuse of the VNs’ total LLR difference and increases the chance of an unreliable VN to obtain theinformation originated from some reliable VNs. All these works have tried to improve the errorrate performance and/or convergence speed of an LDPC code decoding by selecting the optimaledge(s) for updating. Healy et al. [17] simplified the precomputing task by considering, for aCN, only two connecting edges with the smallest V2C magnitudes and selecting the one withthe larger C2V residuals as this CN’s candidate edge. Among all candidate edges, the one withthe largest C2V residual is chosen and the corresponding C2V message is propagated. Wang et al. [18] developed a fixed LBP decoding schedule which arranges the C2V message-passingorder according to the least-punctured and highest-degree principle. The authors also proposeda dynamic LBP schedule which slightly outperforms the fixed one.As the ultimate decoding goal is to have correct VN decisions, an effective schedule should beVN-centric and focus on accurately identifying the incorrect or unreliable VN decisions duringdecoding. It should give higher updating priority to those which are most likely to be corrected.The VN reliability measurements, e.g., decision reversion [13]-[15], the unsatisfied CN number[14], [16] and the change of VNs’ total LLRs [14]-[16] were used implicitly to identify incorrectbit decisions and the unreliable VNs were given higher priority for update. On the other hand, BP
February 23, 2021 DRAFT decoding is usually performed in LLR domain for computational simplicity and for the fact thatthe likelihood can be recovered from its LLR value. However, the likelihood is a true bit decisionreliability indicator and the change of a VN’s LLR is not linearly proportional to the likelihoodor conditional probability variation. Thus there is clearly a need to develop a new metric to suitthe purpose of correcting the most proper erroneous or most unreliable decisions. Moreover,many IDS decoders need to globally search for an edge/node for update and a reduction of thesearch range is necessary.In this paper, we propose an efficient metric, which we call conditional innovation (CI) , toestimate the potential likelihood improvement of a VN. CI is defined as the difference of a VN’scurrent and precomputed conditional posterior probabilities. We show that it also reflects thereliability or correctness of the corresponding VN decision. We further verify that a larger CInot only implies that the corresponding VN decision is more likely to be erroneous but alsohave a higher probability of being corrected if updated. The need of search range reductionand the intuition that the latest updated messages tend to be more trustworthy than the othersmotivate us to introduce an updating strategy that limits the next update candidates to thoseVNs which can be reached by the latest updated VNs in just two hops. We demonstrate thatthe proposed strategy does enhance the reliability of the propagated messages and narrow thecandidate selection range.Making use of these two concepts separately or in combination, we derive several efficientscheduling algorithms. In particular, by adopting the CI as the reliability measure in the schedul-ing strategies, we develop a CI based RBP (CIRBP) algorithm which is able to identify andcorrect most erroneous decisions in the first few iterations. Therefore, our CIRBP algorithmprovides excellent error rate performance at the early decoding stage and is shown to outperformthe existing IDS-based BP decoders. We also propose the latest-message-driven (LMD) strategywhich uses the latest updated C2V messages to determine the next updated VN. We call the BPdecoders which employ the LMD strategy as the LMD-based RBP (LMDRBP) algorithms. TheLMDRBP algorithms not only use the newest updated messages in selecting the next updatedVN but allow these newest messages to be passed with higher priority. Simulation results indicate
DRAFT February 23, 2021 that the LMDRBP algorithms are able to surpass the existing RBP decoders for most cases whilerequiring less or the same computation efforts in selecting the updated node or edge. Combiningboth LMD strategy and CI metric, the resulting LMD-CIRBP algorithm obtains very impressivedecoding gain, especially at the early iterations, at the cost of moderate complexity increase.As the edge-wise updating strategies presented in [9]-[12] are performed in a fully-serialmanner, i.e., only one of the precomputed messages is propagated in each update, they entaillong decoding delays. As far as the decoding delay is concerned, those which adopt multi-edgeupdating ([9], [13]-[16]) and propagate more than one messages per update clock are morepractical. The increased parallelism reduces the decoding latency but may cost performanceloss. We develop multi-edge updating versions of our CIRBP and LMD-CIRBP algorithms withthe degree of parallelism as an adjustable parameter to provide performance-latency tradeoff.Experimental results show that, using a judicial chosen parallelism, our multi-edge LMD-CIRBPalgorithm achieves much reduced latency per iteration with little or no performance loss withrespect to its single-edge counterpart.The rest of this paper is organized as follows. In Sec. II, we give a brief review of knownRBP algorithms and the corresponding IDS strategies used. The properties of the proposed CImetric are analyzed and the CIRBP algorithm is presented in Sec. III. In Sec. IV, we discuss theLMD scheduling strategy and present the LMDRBP and LMD-CIRBP algorithms. The numericalresults and complexity analysis of our decoders are provided in Sec. V. In Sec. VI, we introducethe multi-edge updating versions of the CIRBP and LMD-CIRBP algorithms and give relatedsimulation results. Finally, we draw concluding remarks in Sec. VII.II. P
RELIMINARIES
A. Message Updating for RBP Decoding
A binary ( N , K ) LDPC code C of rate R = K/N is characterized by an M × N parity-checkmatrix H = [ h mn ] , where the entry h mn determines if the n th VN v n and the m th CN c m onthe associated bipartite code graph is connected. For a coded BPSK system, a binary codeword u = ( u , u , · · · , u N − ) , u n ∈ { , } is modulated to the sequence x = ( x , x , · · · , x N − ) , February 23, 2021 DRAFT where x n = 1 − u n for ≤ n < N , and then transmitted over an AWGN channel. Thecorresponding received noisy sequence and tentative decoded decision vector are respectivelydenoted by y = ( y , y , · · · , y N − ) and ˆ u = (ˆ u , ˆ u , · · · , ˆ u N − ) , where y n = x n + w n and w n , ≤ n < N , are i.i.d. zero-mean AWGN with variance σ .Let L C m → n be the C2V message from c m to v n in a BP-based decoder, L V n → m be the V2Cmessage from v n to c m , and L n be the total LLR of v n . For all m, n such that h mn = 1 , L C m → n and L V n → m are initialized as and y n /σ , respectively. We denote by M ( n ) = { m | h mn = 1 } the index set of CNs connected to v n and by N ( m ) = { n | h mn = 1 } the index set of VNs linkedto c m on the associated code graph. We further define M ( n ) \ m and N ( m ) \ n respectivelyas the set M ( n ) with m excluded and the set N ( m ) with n excluded. For the BP decodingalgorithm, the V2C messages sent from v n to c m , m ∈ M ( n ) , are calculated by L V n → m = 2 y n σ + X m ′ ∈M ( n ) \ m L C m ′ → n , (1)and the C2V message sent from c m to v n , n ∈ N ( m ) are updated by L C m → n = 2 tanh − Y n ′ ∈N ( m ) \ n tanh (cid:18) L V n ′ → m (cid:19) . (2)The total LLR of v n L n = 2 y n σ + X m ∈M ( n ) L C m → n , (3)is used to make tentative decoding decision ˆ u n = bsgn ( L n ) , where bsgn ( a ) = 0 if a ≥ andbsgn ( a ) = 1 otherwise.The original RBP algorithm [9] repeats the following four-step message updating procedure: Compute the C2V messages ˜ L C m → n by (2) for all ( m, n ) where h mn = 1 and the corre-sponding C2V message residuals (also referred to as C2V residuals for simplicity) by R C m → n = | ˜ L C m → n − L C m → n | . (4) Determine the C2V edge to be updated ( m ∗ , n ∗ ) = arg max ( m,n ) R C m → n . (5) DRAFT February 23, 2021 Perform the sole update L C m ∗ → n ∗ ← ˜ L C m ∗ → n ∗ . (6) After the updated C2V message is received by v n ∗ , the decoder updates and propagates theV2C messages L V n ∗ → i , i ∈ M ( n ∗ ) \ m ∗ based on (1).As only the C2V message L C m ∗ → n ∗ is updated and sent, we refer to { ˜ L C m → n } as the precomputedC2V messages. A decoding iteration is counted after E C2V messages are propagated, where E is the number of edges on the code graph. The decoder makes tentative codeword check atthe end of each iteration and stops when a valid codeword is found or the maximum iterationnumber has been reached. B. Other Scheduling Strategies
As mentioned before, several improved RBP algorithms have made an effort to avoid up-dating a small group of edges repeatedly. In particular, the node-wise RBP [9] decoder allowssimultaneously updates of more than one C2V message, the quota-based RBP [10] limits eachedge’s update times per iteration and the silent-variable-node-free RBP (SVNF-RBP) method[10] requires that every VN’s intrinsic message should be passed to a connecting CN with afixed updating order. The dynamic SVNF-RBP (DSVNF-RBP) algorithm [11] relaxes the fixedupdating order constraint. The residual-decaying-based RBP algorithm [12] scales the residualvalue of a message by a factor which decays with the number of times the same edge has beenupdated, thereby reducing the probability of its further update within an iteration. Among thesederivatives of the RBP algorithm, we found that, for many practical LDPC codes, the SVNF-RBPalgorithm not only provides improved decoding performance but is computational efficient.Besides preventing the greedy updating behavior, many schedules were designed to enhancethe decoding efficiency by detecting unreliable tentative VN decisions as soon as possible. Forexample, one can locate the VNs which are likely to have incorrect LLR signs and give themhigher updating priority [11], [13]-[16]. When a VN is updated, it automatically sends V2Cmessages to its connecting CNs like Step of the RBP algorithm. The DSVNF-RBP algorithm[11] first considers those C2V edges connecting to the unsatisfied CNs. As an unsatisfied CN February 23, 2021 DRAFT must link to at least one incorrect VN decision, updating the edges participating in unsatisfiedCNs may help reversing the erroneous decisions.In [13]-[15], the reliability of ˆ u n is judged by checking if it changes sign after an update.Let ˜ L n = 2 y n /σ + P m ∈M ( n ) ˜ L C m → n be the precomputed LLR of v n . In [13] and [14], a VN’stentative decision bsgn ( L n ) is regarded as unstable if bsgn ( L n ) = bsgn ( ˜ L n ) and the unstableVNs are given higher updating priority. In [15], a VN’s reliability is judged by checking ifthe associated tentative bit decisions remain unchanged in three consecutive updates. Among theunreliable VNs, the one with the largest total VN LLR difference | ˜ L n − L n | is chosen for update.In [16], the VNs are further classified into four types according to a certain decision reliabilitymetric so that the most unreliable VN can be updated by using the most reliable local messageson the code graph.As mentioned in the previous section, a decoding schedule should be VN-centric and focuson selecting an edge which can help its connected VN to make a better bit decision. We thusopt to have a schedule that prioritizes improving the most unreliable VN decisions. We adopt anVN-then-edge strategy which determines the targeted VN and from its connecting edges, selectone for C2V message update. How such an approach serves our design goal will become clearin the subsequent discourse.III. C ONDITIONAL I NNOVATION AND
CIRBP D
ECODING
A. CI and VN Decisions
Updating the VNs with unreliable bit decisions to enhance the chance of reversing erroneousdecisions can significantly improve both the convergence speed and the converged error rate.This perhaps is the rationale behind some related works [13]-[15] that prioritize updating theunreliable VNs (i.e., the decision-changed VNs). The reliability metric used there was derivedfrom the stability of the VN decisions. Updating an unstable VN with the largest total LLR changemay help reversing the bit decision but not necessarily toward the correct one. Furthermore, thismetric tends to ignore unreliable VNs that have stable decisions but small LLR magnitudesand reduce their chances for improving reliability. Hence, we need a metric that avoids these
DRAFT February 23, 2021 shortcomings and, ideally, we would like this metric to be able to accurately predict the degreeof a VN decision’s correctness and its chance of being corrected if updated. In the followingparagraphs, we present a metric which possesses similar properties.Define O Z = { , , . . . , Z − } where Z ∈ Z + and p e ,n = Pr(ˆ u n = u n ) as the bit errorprobability of the current decision ˆ u n . The codeword error probability would be Pr( ˆ u = u ) = 1 − Y n ∈O N (1 − p e ,n ) . (7)Analogously, we denote by ˜ u n the decision after v n is updated (i.e., ˜ u n = bsgn ( ˜ L n ) ) and let ˜ p e ,n = Pr(˜ u n = u n ) . If only one VN is updated at one time and both p e ,n and ˜ p e ,n were available,to maximally lower the codeword error probability, it is reasonable to select a VN v n ∗ whichhas the best chance of improving its bit error probability for update. That is, n ∗ = arg max n ∈O N ( p e ,n − ˜ p e ,n ) . (8)Since p e ,n and ˜ p e ,n are not available, we seek for an alternate parameter which can help inferthe quantity ( p e ,n − ˜ p e ,n ) . We define the conditional posterior probabilities, Pr( u n = 0 | L n ) =exp( L n ) / (1 + exp( L n )) def = p ( L n ) and Pr( u n = 1 | L n ) = 1 − p ( L n ) def = p ( L n ) ; both aredeterministic function of L n and their values lie within [0 , . The proposed metric D n = | p ( L n ) − p ( ˜ L n ) | = | p ( L n ) − p ( ˜ L n ) | , (9)measures the new information about u n we may obtain if the update L n ← ˜ L n is carried out. ≤ D n < is thus called the conditional innovation (CI) henceforth.The usefulness of CI is derived from two interesting properties. For convenience, the messages { L n } , { ˜ L n } and { D n } are respectively modeled as random variables L , ˜ L and D . P =exp( L ) / (1 + exp( L )) , P = 1 − P , and ˜ P and ˜ P are similarly defined for ˜ L . The firstproperty has to do with the behavior of the function J ( γ ) , Pr ( the decision is correct | D ≥ γ )Pr ( the decision is incorrect | D ≥ γ )= Pr ( P ≥ . | D ≥ γ )Pr ( P < . | D ≥ γ ) , (10) February 23, 2021 DRAFT0 J ( ) J ( ): GA-DE Iter.=1 Iter.=2 Iter.=3 Simulated J ( ): Gallager (8000,4000) Iter.=1 Iter.=2 Iter.=3 SNR=1.5 dB
SNR=1.75 dB J ( ) (a) J ( γ ) obtained by GA-DE with rate=0.5, ( d v =4, d c =8) andsimulated J ( γ ) for Gallager (8000,4000) ( d v =4, d c =8) code. J ( ) Simulated J ( ): 802.11 (1944,972) Iter.=1 Iter.=2 Iter.=3 SNR=1.75 dB
SNR=2 dB J ( ) (b) Simulated J ( γ ) for the 802.11 (1944,972) code.Fig. 1: J ( γ ) obtained by GA-DE and simulation. where the second equality holds by assuming that the all-zero codeword is transmitted. Thisassumption is used throughout our analysis without explicitly mentioned or appeared in relatedconditional probability expressions.In Appendix A, we apply the Gaussian approximation (GA) based density evolution (DE)technique [21] to show that Property 1:
When the BP algorithm is applied to decode an LDPC code in AWGN channelsand the C2V messages can be modelled as i.i.d. Gaussian random variables, J ( γ ) , is a decreasingfunction of the threshold γ when the signal-to-noise ratio (SNR) is sufficient large.The i.i.d. C2V messages assumption is the same as that proposed in [21] and our proof issemi-analytic in the sense that some parts of the proof require computer based calculation. Theassumption of independent C2V messages [21] is valid if the LDPC code of concern is eithercycle-free or the iteration number of interest is smaller than half of the girth of the code so thatthe VNs do not receive correlated information. As an example, we consider a rate- . regularcode ensemble with CN degree d c = 8 and VN degree d v = 4 . We depict J ( γ ) of the firstthree iterations for the flooding schedule in Fig. 1(a). For comparison, we also present J ( γ ) forthe (8000 , Gallager code with ( d c , d v ) = (8 , [4] where the corresponding conditionalprobabilities are obtained by simulation. In Fig. 1(b), we show the simulated J ( γ ) for the 802.11 DRAFT February 23, 20211 (1944,972) code. For both cases, we find that J ( γ ) is a decreasing function of γ and, for a VNwhose D n is sufficiently large, the associated bit decision is likely to be incorrect. We furtherprove in Appendix B that Property 2:
Under the same assumptions of
Property 1 , if the current decision is incorrect( ˆ u n = u n ), i.e., for all P < / , the function F ( γ ) , Pr( ˜ P ≥ P | D ≥ γ )Pr( ˜ P < P | D ≥ γ ) (11)is always larger than , and it is a strictly increasing function of γ when γ ∈ [0 , P ) and goesto infinity when γ ∈ [ P , .This property implies that if the bit decision of a VN is incorrect, the larger the associated D n is, the greater the probability of making a correct decision after an update becomes, that is, Pr( p ( ˜ L n ) > p ( L n )) increases. These two properties indicate that we should give the VN withthe largest D n the highest updating priority. This VN has the highest probability of being bothincorrect (before update) and correctable (after update). B. The CIRBP Decoding Algorithm
Based on the above discussion, we propose the CI based RBP (CIRBP) algorithm as shown in
Algorithm 1 . The VN with the largest CI can be selected as the candidate VN for update as itis the VN which is most likely to yield an erroneous decision if D n ∗ ≥ γ and it is also the mostcorrectable if updated; otherwise, identifying the incorrect decision(s) becomes difficult and theC2V update would then follow the original RBP algorithm. Such threshold-based judgement isbased on our observation in Figs. 1(a) and 1(b) that the probability that a VN decision is wrongis a monotonic decreasing function of γ . For the selected VN, denoted by v n ∗ henceforth, theassociated incoming C2V message L C m ∗ → n ∗ , which has the maximum residual, is updated (lines8–9). Using this C2V message, v n ∗ then sends new V2C messages to c i , i ∈ M ( n ∗ ) \ m ∗ andthe associated messages ˜ L C i → j , R C i → j , ˜ L j and D j ∀ j ∈ N ( i ) \ n ∗ , will be calculated (lines 10–13).As J ( γ ) depends on the code structure, the iteration number, SNR and the decoding scheduleused and is not admitted in a closed-form expression. Its monotonicity property can only beproved semi-analytically. For practical concerns, we use a fixed γ and find that a properly February 23, 2021 DRAFT2
Algorithm 1
Conditional Innovation Based RBP (CIRBP) Algorithm Initialize all L C m → n = 0 and all L n = L V n → m = 2 y n /σ Generate all ˜ L C m → n by (2) and compute all R C m → n Compute all ˜ L n and D n Find n ∗ = arg max j { D j | j ∈ O N } if D n ∗ < γ then Find ( m ∗ , n ∗ ) = arg max ( i,j ) { R C i → j | h ij = 1 } and go to line 9 end if Find m ∗ = arg max i { R C i → n ∗ | i ∈ M ( n ∗ ) } Let L C m ∗ → n ∗ ← ˜ L C m ∗ → n ∗ . Propagate L C m ∗ → n ∗ , let R C m ∗ → n ∗ = 0 , and update L n ∗ for every i ∈ M ( n ∗ ) \ m ∗ do Generate and propagate L V n ∗ → i Compute ˜ L C i → j , R C i → j , ˜ L j and D j ∀ j ∈ N ( i ) \ n ∗ end for Go to line 4 if
Stopping Condition is not satisfiedchosen γ suffices to give outstanding performance. The chosen γ cannot be too small for thenCI is no longer a reliable indicator in identifying the incorrect yet correctable bit decision. Butif γ is too large, the probability Pr( D n ∗ ≥ γ ) becomes very small and our CIRBP decoder willrely on the conventional LLR residual most of the time and gives diminishing gain against theoriginal RBP decoder.IV. L ATEST -M ESSAGE -D RIVEN S CHEDULE AND
LMDRBP A
LGORITHMS
A. LMD Scheduling Strategy
Most IDS strategies focus on using some message reliability metric to select the C2V messagesto be propagated. On the other hand, the update criteria presented in the previous section and in[11], [16], are implicitly designed to select a VN such that the selected one can make a betterbit decision. Both approaches eventually improve the reliability of the V2C messages which thetarget VN is going to deliver and the resulting decoders do yield performance better than thatof the standard BP decoder with the same iteration or edge update number. It is reasonable toconjecture that not only the V2C messages emitted from the latest updated VN ( v n ∗ ) but also DRAFT February 23, 20213 the subsequent C2V messages forwarded by the connecting CNs become more trustworthy. Thisconjecture suggests that the decoding schedule prioritize using the messages originated fromthose nodes which are just updated and possess the newest information. An extra benefit ofconsidering only newly updated nodes and messages is the reduction of the search range forfinding a suitable C2V message or VN for the next update.Based on this idea and following the VN-centric guideline, we propose the latest-message-driven (LMD) RBP (LMDRBP) algorithm as described in
Algorithm
2. In this algorithm wecompare the C2V residuals of the latest renewed C2V messages, i.e., the messages forwardedby those CNs which just received new V2C messages from the latest-updated VN, and select theVN v n ∗ associated with the maximum C2V residual as the next update target. For the selectedVN, we compare all its connected C2V messages—both new and old—and accept only theone with the largest residual (lines 9–11). By doing so, we reduce the VN search range to thenearest neighboring VNs of the latest updated VN but not the C2V message search range ofthe targeted VN and avoid favoring a certain group of edges. The total LLR of v n ∗ and theassociated V2C messages are updated, and then the CNs linking to v n ∗ precompute their C2Vmessages and residuals to complete an update procedure (lines 4–8). This procedure repeats untilthe stopping condition is satisfied. The numerical results presented in the next section show that Algorithm of Algorithm 2 and send the C2V message corresponding to the maximum residual found in line . This modified version is referred to as the simplified LMDRBP (sLMDRBP) algorithm. B. LMD-based CIRBP Algorithm
If the information carried by the latest updated C2V messages is more reliable, the relatedprecomputed VN total LLRs ( ˜ L n ’s) and the CI values can also be more trustworthy after February 23, 2021 DRAFT4
Algorithm 2
Latest-Message-Driven RBP (LMDRBP) Algorithm Initialize all L C m → n = 0 and all L V n → m = 2 y n /σ Generate all ˜ L C m → n by (2) and compute all R C m → n Find ( m ∗ , n ∗ ) = arg max ( i,j ) { R C i → j | h mn = 1 } Let L C m ∗ → n ∗ ← ˜ L C m ∗ → n ∗ . Propagate L C m ∗ → n ∗ , let R C m ∗ → n ∗ = 0 , and update L n ∗ for every i ∈ M ( n ∗ ) \ m ∗ do Generate and propagate L V n ∗ → i Compute ˜ L C i → j and update R C i → j ∀ j ∈ N ( i ) \ n ∗ end for Find ( m ′ , n ′ ) = arg max ( i,j ) { R C i → j | i ∈ M ( n ∗ ) \ m ∗ , j ∈ N ( i ) \ n ∗ } Find ˆ m = arg max i { R C i → n ′ | i ∈ M ( n ′ ) } and let m ′ ← ˆ m Let ( m ∗ , n ∗ ) ← ( m ′ , n ′ ) Go to line 4 if
Stopping Condition is not satisfiedincorporating these newest messages. Combining the concepts of the LMD schedule and theCIRBP decoder can then improve the accuracy of the VN reliability judgement. Since theCIRBP decoder selects the target VN by comparing VNs’ CI values, we modify the LMDbased schedule by letting the updated VN be decided by the last-updated CI values instead ofthe C2V residuals. With the modified schedule, we have LMD-based CIRBP (LMD-CIRBP)decoding algorithm described in
Algorithm 3 .To determine the initial updated VN and edge, we simply select the VN with the globalmaximum CI be the initial targeted VN (line 4). The initial chosen edge will be the one whichhas the maximum residual among all candidate C2V messages to be sent to the targeted VN (line5). Let L C m ∗ → n ∗ be the selected C2V message and c m ∗ and v n ∗ respectively be the correspondingCN and VN. The V2C messages from v n ∗ (i.e., L V n ∗ → i ) would be updated, and then all associatedprecomputed messages, C2V residuals, and CIs will also be renewed (lines 7–10). For all VNsin the set U ( m ∗ , n ∗ ) , { ˜ n | ˜ n ∈ N ( ˜ m ) \ n ∗ , ˜ m ∈ M ( n ∗ ) \ m ∗ } , the one with the maximum CIis chosen as the next update target which accepts the C2V message from one of its connectingedges with the maximum C2V residual (lines 11–12). The above procedure will be repeateduntil the stopping condition is met. The LMD-CIRBP algorithm enjoys the advantages of bothCIRBP and LMDRBP decoders–it not only has better chance to locate the VNs which indeed DRAFT February 23, 20215 need to be updated but requires much less search complexity since only those CI values for theVNs in U ( m ∗ , n ∗ ) need to be compared. Algorithm 3
LMD-Based CIRBP (LMD-CIRBP) Algorithm Initialize all L C m → n = 0 and all L V n → m = 2 y n /σ Generate all ˜ L C m → n by (2) and compute all R C m → n Compute all ˜ L n and D n Find n ∗ = arg max j { D j | j ∈ O N } Find m ∗ = arg max i { R C i → n ∗ | i ∈ M ( n ∗ ) } Let L C m ∗ → n ∗ ← ˜ L C m ∗ → n ∗ , propagate L C m ∗ → n ∗ , let R C m ∗ → n ∗ = 0 , and update L n ∗ for every i ∈ M ( n ∗ ) \ m ∗ do Generate and propagate L V n ∗ → i Compute ˜ L C i → j , R C i → j , ˜ L j and D j ∀ j ∈ N ( i ) \ n ∗ end for Find n ′ = arg max j { D j | j ∈ U ( m ∗ , n ∗ ) } . Let n ∗ ← n ′ and go to line 6 if Stopping Condition is not satisfiedV. N
UMERICAL R ESULTS AND C OMPLEXITY A NALYSIS
In this section, we compare the frame error rate (FER) performance and computationalcomplexity of the proposed and some known RBP decoders. The simulation setup is the sameas what was described in Sec. II-A, i.e., an LDPC coded data stream is BPSK-modulated andtransmitted over an AWGN channel with two-sided power spectral density N / σ . ThreeLDPC codes are considered: the (1944 , rate- / LDPC code of the IEEE 802.11 standard(WiFi) [2], and the (1848 , rate- / and (500 , rate- / LDPC codes used in the 5G NRspecification [3]. We denote these codes by W- , N- and N- , respectively. Accordingto 5G NR specification, N- is obtained by puncturing the first 56 VNs of a length- mother code generated based on Base Graph 1 (BG1) with lifting size 28 while N- is obtainedby puncturing the first 20 VNs of a length- mother code derived from Base Graph 2 (BG2)with lifting size 10. As mentioned in Section II, an iteration is defined as E C2V messagepropagations, and I max denotes the maximum allowed iteration number. February 23, 2021 DRAFT6
SNR=1.75 dBSNR=1.625 dB F E R Iteration F E R (a) W- code. SNR=2.2 dB F E R Iteration
5G NR (500,100) CIRBP, =0 CIRBP, =0.05 CIRBP, =0.1 CIRBP, =0.15 CIRBP, =0.2 CIRBP, =0.3 CIRBP, =0.5 CIRBP, =0.7 RBP
SNR=2 dB F E R (b) N- code.Fig. 2: FER convergence behaviors of CIRBP algorithms with different γ in decoding W- and N- codes Both the precomputed and actual propagated C2V messages are calculated by (2). If we usethe min-sum approximation [22] instead of (2) for the C2V message precomputations [9], [10],the computation load decreases possibly at the expense of performance degradation.
A. FER Performance
In Figs. 2(a) and 2(b), we show the effect of the CI thresholds ( γ ) on the CIRBP algorithm’sperformance in decoding the W- and N- codes at different SNRs ( E b /N ). Those curvesindicate that when γ ≤ . , the threshold provides an early-stage and converged performancetradeoff: γ = 0 or . gives the best st-iteration FER performance but γ = 0 . , . , or . results in better converged FER performance. Although not shown here, our simulationsconfirm that the N-1848 code renders similar behaviors. To avoid presenting too many curvesin one figure, we only present the CIRBP decoder performance using γ = 0 . and . for theremaining figures.Fig. 3(a) plots the error-rate performance of various IDS algorithms in decoding W- code at I max = 3 . At FER ≈ − , the CIRBP and LMD-CIRBP algorithms have about . dB gain with respect to the SVNF-RBP and DSVNF-RBP algorithms. The LMDRBP algorithmalso outperforms the SVNF-RBP and DSVNF-RBP algorithms at the same FER. In Fig. 3(b) weshow the FER and BER convergence behaviors of these algorithms in decoding the same code at DRAFT February 23, 20217 F E R E b / N (dB) B E R E b / N (dB) (a) FER and BER performance, I max = 3 Iteration F E R B E R (b) FER and BER convergence behaviors, SNR = . dBFig. 3: FER and BER performance and convergence behaviors for various IDS-based decoding algorithms; W- code. SNR = 1 . dB. These figures indicate that our algorithms outperform the SVNF-RBP/DSVNF-RBP (RDRBP) algorithms for I max < ( I max < ). In addition, the LMDRBP, sLMDRBPand LMD-CIRBP decoders yield better converged ( I max = 50 ) FER performance than that of theSVNF-RBP and RDRBP decoders. Among these decoders, the LMD-CIRBP algorithm gives byfar the best performance at the first iteration.For the LDPC codes used in IEEE 802.11 systems, the degrees of all VNs are at least twoand messages can be exchanged through every VN. However, there are several degree-1 VNsin the 5G NR codes (N- and N- ) and to decode these codes with the LMDRBP andLMD-CIRBP algorithms we have to make some modifications. More specifically, after the C2Vmessage L C m ∗ → n ∗ was sent, the next updated VN is selected from the VNs which link to c i for all i ∈ M ( n ∗ ) \ m ∗ . If v n ∗ is a degree-1 node, we allow the decoder to search for thenext updated VN from the set N ( m ∗ ) \ n ∗ and line 9 of Algorithm 2 is replaced by “Find ( m ′ , n ′ ) = arg max ( i,j ) { R C i → j | i ∈ M ( n ∗ ) , j ∈ N ( i ) \ n ∗ } ” while line 11 of Algorithm 3 is tobe modified as “Find n ′ = arg max j { D j | j ∈ N ( m ∗ ) \ n ∗ } ”.Shown in Fig. 4(a) is the error-rate performance of various IDS algorithms in decoding theN- code with I max = 3 . We find that the LMD-CIRBP algorithm outperforms the SVNF-RBPone by . and . dB at FER ≈ − and BER ≈ − , respectively. The CIRBP algorithmprovides . – . dB gain in comparison with the SVNF-RBP one. The LMDRBP decoders also February 23, 2021 DRAFT8
5G NR (500,100) 3 Iter. RBP [9] SVNF [10] DSVNF [11] RDRBP, =0.8 [12] CIRBP, =0.10 CIRBP, =0.15 sLMDRBP LMDRBP LMD-CIRBP F E R E b / N (dB) B E R E b / N (dB) (a) FER and BER performance, I max = 3 F E R
5G NR (500,100) 2.2 dB LBP [5] CIRBP, =0.10 RBP [9] CIRBP, =0.15 SVNF-RBP [10] sLMDRBP DSVNF-RBP [11] LMDRBP RDRBP, =0.8 [12] LMD-CIRBP
Iteration B E R (b) FER and BER convergence behaviors, SNR = . dBFig. 4: FER and BER performance and convergence behaviors for various IDS-based decoding algorithms; N- code.
5G NR (1848, 616) 3 Iter. RBP [9] SVNF [10] DSVNF [11] RDRBP, =0.7 [12] CIRBP, =0.10 CIRBP, =0.15 sLMDRBP LMDRBP LMD-CIRBP F E R E b / N (dB) B E R E b / N (dB) (a) FER and BER performance, I max = 3 F E R Iteration B E R
5G NR (1848, 616) 1.3 dB LBP [5] CIRBP, =0.10 RBP [9] CIRBP, =0.15 SVNF-RBP [10] sLMDRBP DSVNF-RBP [11] LMDRBP RDRBP, =0.7 [12] LMD-CIRBP (b) FER and BER convergence behaviors, SNR = . dBFig. 5: FER and BER performance and convergence behaviors for various IDS-based decoding algorithms; N- code. give performance better than that of the SVNF-RBP and DSVNF-RBP decoders. These decoders’corresponding convergence trends at SNR = 2 . dB are shown in Fig. 4(b), which confirm thatour CIRBP and LMD-CIRBP algorithms outperform existing algorithms for all iterations, and theLMDRBP and sLMDRBP decoders also outperform existing decoders except for the RBP andRDRBP decoders at the 1st iteration. It is wroth mentioning that the LMD-CIRBP algorithm’sfirst-iteration performance, FER ≈ × − , is quite impressive.In Figs. 5(a) and 5(b), we depict the performance of various IDS algorithms with I max = 3 and their convergence behaviors at SNR = 1 . dB in decoding the N- code. We see that, for DRAFT February 23, 20219 I max = 3 and at FER ≈ − or BER ≈ − , the CIRBP algorithm yields . dB gain against theSVNF-RBP algorithm, and the LMD-CIRBP decoder achieves the same FER gain but has lessthan . dB gain at the same BER. The LMDRBP decoders still yield performance better thanthat of the SVNF-RBP algorithm. Fig. 5(b) shows that the CIRBP and LMD-CIRBP algorithmsoutperform the SVNF-RBP and RDRBP algorithms when I max ≤ . With reduced search range,the LMD-CIRBP algorithm still give outstanding first-iteration and converged FER performance.Although the FER/BER vs. SNR curves are presented for I max = 3 only, Figs. 3(b), 4(b) and 5(b)indicate that, at selected SNRs, the CI-based decoders are better than other IDS-based algorithmsfor almost all I max of interest. B. Complexity Summary
We summarize the decoding complexity of the proposed algorithms and the original RBP,SVNF-RBP, and DSVNF-RBP algorithms in Table I in terms of the numbers of required C2Vprecomputations, CI computations, and real-number comparisons per update. In Table I, a “C2Vpre-update” includes precomputations of C2V messages and residuals, and a “CI update” includescomputing ˜ L n , table look-up of p ( · ) , and the evaluation of (9) with a total of three real-number subtractions/additions involved: two for updating ˜ L n and one for computing the CI. The“C2V residual and CI comparisons” counts the real-number comparisons needed for finding themaximum residual and CI value. ¯ d v and ¯ d c in Table I respectively denote average VN and CNdegrees, and ( ¯ d v , ¯ d c ) of the W- , N- , and N- codes are respectively (3 . , . , (4 . , . , and (3 . , . . For the sLMDRBP algorithm, a C2V message propagation isfollowed by ( ¯ d v − d c − − comparisons for deciding the next updated VN and the C2Vmessage to be forwarded. For the LMDRBP algorithm, ( ¯ d v −
1) ¯ d c − comparisons are requiredafter delivering a C2V message, where ( ¯ d v − d c − − of them are used for locating thetarget VN and the rest of them are for deciding the next updated C2V message. In LMD-CIRBPdecoding, passing a C2V message is followed by ( ¯ d v − d c − CI updates and ( ¯ d v −
1) ¯ d c − comparisons for choosing the ensuing targeted VN and the associated C2V message to be sent.For the CIRBP algorithm, there are ( ¯ d v − d c − CI updates after the C2V pre-updates.
February 23, 2021 DRAFT0
TABLE I: Complexity Summary
C2VPropagation V2CUpdate C2VPre-Update CI Update C2V Residual and CIComparisonsRBP 1 ¯ d v − d v − × ( ¯ d c − E − RDRBP E − SVNF-RBP ¯ d v ( ¯ d c − − DSVNF-RBP ≤ ¯ d v ( ¯ d c − − sLMDRBP ( ¯ d v − d c − − LMDRBP ( ¯ d v −
1) ¯ d c − LMD-CIRBP ( ¯ d v − d c −
1) ( ¯ d v −
1) ¯ d c − CIRBP ( ¯ d v − d c − N + (1 − κ )( ¯ d v −
1) + κ ( E − N : total VN number E : total edge number ¯ d v : averaged VN degree ¯ d c : averaged CN degree κ : Pr( D n ∗ < γ ) Then, N − and one comparisons are respectively used to search for the largest CI ( D n ∗ ) andcheck if D n ∗ ≥ γ . If D n ∗ ≥ γ , additional ¯ d v − comparisons are needed for selecting thecandidate CN; otherwise, we follow the original RBP schedule and perform E − comparisonsto find the C2V message conveying the maximum residual. Let κ = Pr( D n ∗ < γ ) , then κ is an increasing function of γ and on the average we need N + (1 − κ )( ¯ d v −
1) + κ ( E − comparisons to select the updated C2V message. Our simulation results indicate that κ varieswith the iteration number and is a function of SNR and the code used. For W- code, (theaveraged) κ ≈ . for SNR = 1 . – . dB; for N- code, κ = 0 . and . for SNR = 2 and . dB; for N- code, κ = 0 . and . for SNR = 1 . and . dB.Table I shows that compared with the RBP, RDRBP, and SVNF-RBP decoders, the proposedLMDRBP and sLMDRBP decoders are more computationally efficient for all codes used. TheLMD-CIRBP decoder is the most complicated except for the CIRBP one since it requires extracomplexity for CI update. The later decoder needs to perform global residual comparison withprobability κ . The numerical results discussed so far indicate that the proposed decoders providevarious tradeoffs between complexity and decoding performance, and the LMD-CIRBP decoderhas the best performance-complexity balance, offering improved performance at the cost oflimited complexity increase. DRAFT February 23, 20211
As mentioned in the last section, the CIRBP and LMD-CIRBP algorithms give impressivefirst-iteration FER performance and a valid codeword is likely to be obtained within one iteration(i.e., before E C2V message updates), significantly reducing the average decoding complexity.VI. M
ULTI -E DGE U PDATING S TRATEGIES
The decoding schedules discussed so far all adopt a single-edge updating strategy that passingone C2V message per update. To reduce the decoding latency, we propose multi-edge CIRBP(ME-CIRBP) and multi-edge LMD-CIRBP (ME-LMD-CIRBP) algorithms in this section whichallow N P C2V messages to be propagated in parallel per update. Specifically, our multi-edgestrategy determines N P VNs to be updated and applies the single-edge strategies to each VN.For implementation efficiency, the number N p is fixed in each update.For a CIRBP based decoding, a simple and intuitive method for simultaneously updating N P VNs is to choose the nodes with the largest N P CI values which requires (at most) (2 N − P − N P / real-number comparisons. To further lower the complexity, we introduce a VN selectionmethod which selects N p indices from a candidate VN index set S for simultaneous updates.The set of the N p VN indices selected is denoted by P . Algorithm 4
A VN Selection Scheme
Input: N G : Group Number, N P : Selected VN Number, S : Input Search Set Output: P : Selected VN Index Set Initialize G i = ∅ for i = 0 , , . . . , N G − , P = ∅ G i ← G i ∪ { n } , where i = ⌊ D n × N G ⌋ , for every n ∈ S Find k ∗ = max { k : |Q ( k ) | ≤ N p } and let P = Q ( k ∗ ) if |P| < N P then Randomly choose N P − |P| elements from G N G − k ∗ − to form set G ′ N G − k ∗ − Let
P ← P ∪ G ′ N G − k ∗ − end if return P We first partition S into N G groups ( G , G , · · · , G N G − ) according to their CI values: for all n ∈ S , we let G i ← G i ∪ { n } if D n ∈ [ i/N G , ( i + 1) /N G ) , where N G is a predetermined designed February 23, 2021 DRAFT2 group number. We then find k ∗ = max { k : |Q ( k ) | ≤ N p } , where Q ( k ) def = S kj =1 G N G − j . If |Q ( k ∗ ) | = N p , we let P = Q ( k ∗ ) . Otherwise, we randomly select N P − |Q ( k ∗ ) | elements from G N G − k ∗ − to form G ′ N G − k ∗ − and set P = Q ( k ∗ ) ∪G ′ N G − k ∗ − . The procedure is formally describedin Algorithm 4 .Incorporating the above VN selection method into the CIRBP algorithm, we have the ME-CIRBP algorithm which we refer to as
Algorithm 5 . In this multi-edge updating schedule,the VNs whose indices belong to P are simultaneously updated. For each selected VN, thecorresponding incoming C2V message selection and the subsequent message renewal proceduresare the same as those of the CIRBP algorithm. Algorithm 5
Multi-Edge CIRBP (ME-CIRBP) Algorithm Initialize all L C m → n = 0 and all L n = L V n → m = 2 y n /σ Generate all ˜ L C m → n by (2) and compute all R C m → n Compute all ˜ L n and D n Find P by Algorithm 4 ( input: N G , N P , O N ) For all p ∈ P , perform lines - (by letting n ∗ ← p ) in Algorithm 1 in parallel Go to line 4 if
Stopping Condition is not satisfiedThe multi-edge version of the LMD-CIRBP algorithm (
Algorithm 6 ) is similarly structured:by combining the LMD-CIRBP decoder with
Algorithm 4 . In this algorithm, the first N P targeted VNs are found from O N . For each v p , p ∈ P , we simultaneously carry out the keymessage updating procedure of the LMD-CIRBP algorithm (i.e., lines – of Algorithm 3 ).For every p ∈ P , we update its associated C2V residuals and CI values, and then we find a VN v p ′ according to line 8 of Algorithm 6 as the next target VN and add p ′ to the temporary set P ′ .In case different v p ’s may suggest the same VN v p ′ so that |P ′ | < N P , we execute Algorithm4 to find the remaining N P − |P| VNs from those VNs which do not belong to P ′ .We plot the performance and convergence behaviors of the ME-CIRBP and ME-LMD-CIRBPalgorithms and their single-edge versions in decoding the W- code in Figs. 6 and 7. Thechannel and modulation scheme are the same as those specified in Sec. V. Fig. 6 shows that theME-CIRBP algorithm suffers from performance loss at early decoding iterations (but requires DRAFT February 23, 20213
Algorithm 6
Multi-Edge LMD-CIRBP (ME-LMD-CIRBP) Algorithm Initialize all L C m → n = 0 and all L n = L V n → m = 2 y n /σ Generate all ˜ L C m → n by (2) and compute all R C m → n Compute all ˜ L n and D n Find P by Algorithm 4 ( input: N G , N P , O N ) For all p ∈ P , perform lines - in Algorithm 3 (by letting n ∗ ← p ) in parallel Set P ′ = ∅ for every p ∈ P do Find p ′ = arg max j { D j | j ∈ U ( m ∗ , p ) } where m ∗ = arg max i { R Ci → n ∗ | i ∈ M ( p ) } Let P ′ ← P ′ ∪ p ′ end for if |P ′ | < N P then Find P by Algorithm 4 ( input: N G , ( N P − |P ′ | ) , O N \ P ′ ) end if Let
P ← P ′ ∪ P Go to line 5 if
Stopping Condition is not satisfied F E R G =4, N P =27 ME-CIRBP, N G =4, N P =81 ME-CIRBP, N G =8, N P =27 ME-CIRBP, N G =8, N P =81 E b / N (dB) E b / N (dB) B E R (a) FER and BER performance, I max = 3 G =4, N P =27 ME-CIRBP N G =4, N P =81 ME-CIRBP N G =8, N P =27 ME-CIRBP N G =8, N P =81 F E R Iteration B E R (b) FER and BER convergence behaviors, SNR = . dBFig. 6: FER and BER performance of CIRBP and ME-CIRBP algorithms with different N P and N G in decoding W- code. February 23, 2021 DRAFT4 G =4, N P =27 ME-LMD-CIRBP N G =4, N P =81 ME-LMD-CIRBP N G =8, N P =27 ME-LMD-CIRBP N G =8, N P =81 F E R E b / N (dB) E b / N (dB) B E R (a) FER and BER performance, I max = 3 F E R G =4, N P =27 ME-LMD-CIRBP N G =4, N P =81 ME-LMD-CIRBP N G =8, N P =27 ME-LMD-CIRBP N G =8, N P =81 Iteration B E R (b) FER and BER convergence behaviors, SNR = . dBFig. 7: FER and BER performance of LMD-CIRBP and ME-LMD-CIRBP algorithms with different N P and N G in decodingW- code.TABLE II: Per-Iteration Complexity of CIRBP, ME-CIRBP, LMD-CIRBP, and ME-LMD-CIRBP Decoders C2VPropagation V2CUpdate C2VPre-Update CI Update C2V Residual and CIComparisons Comparisons forMulti-VN Selection(Algorithm 4)CIRBP
E E × ( ¯ d v − E × [( ¯ d v − d c − E × [( ¯ d v − d c − E × [ N + (1 − κ )( ¯ d v −
1) + κ ( E − E × ( d v −
1) (
E/N P ) × N G LMD-CIRBP E × [( ¯ d v −
1) ¯ d c − E × [( ¯ d v −
1) ¯ d c − ≤ ( E/N P ) × N G N : total VN number E : total edge number ¯ d v ( ¯ d c ) : averaged VN (CN) degree N G : group number N P : selected VN number only /N P decoding latency). As expected, the error-rate performance of both ME decodersimproves with a larger N G or a smaller N P . Fig. 7(b) demonstrates that, except for the case ( N G , N p ) = (4 , and at the very first iteration, the ME-LMD-CIRBP algorithm provides BERand FER performance comparable to that of its single-edge version. Both figures show that with ajudicial choice of ( N G , N P ) , the proposed ME algorithms yield similar or even better convergedperformance and, under a low latency constraint, they give far better FER performance.In Table II, we compare the per iteration complexities of the CIRBP, ME-CIRBP, LMD-CIRBP,and ME-LMD-CIRBP decoders. For ME-CIRBP decoder ( Algorithm 5 ), N P VNs are selectedby
Algorithm 4 and then updated. This select-VN-then-update procedure repeats
E/N P timesin one iteration (and propagate E C2V messages in total). We assume that the VN grouping in
DRAFT February 23, 20215
Algorithm 4 (line 2) can be simply performed by assigning n to G ⌊ D n × N G ⌋ or equivalently bypassing D n through an N G -level uniform quantizer. Hence, executing Algorithm 4 once requiresat most N G integer comparisons where (at most) N G − of them are for finding k ∗ (line 3)and the remaining ones are for checking if |P| < N P (line 4). The ME-CIRBP thus requires ( E/N P ) × N G integer comparisons for the VN selection in each iteration. As the ME-CIRBPdecoder need not compare CI after VN selection, it consumes only E × ( ¯ d v − real-valuecomparisons for comparing the C2V residuals of the selected VNs per iteration. The remainingoperations are the same as the CIRBP decoder. As summarized in Table II, when N × N P > N G ,the ME-CIRBP decoder requires less computational efforts compared with the CIRBP decoder.The complexity associated with the ME-LMD-CIRBP decoder can be similarly evaluated. As E C2V messages will be propagated in one iteration, the per-iteration complexity required forupdating messages/CIs and residual comparisons in the ME-LMD-CIRBP decoding (lines 5-10of
Algorithm 6 ) is the same as that needed by the LMD-CIRBP decoder. However, because
Algorithm 4 is executed at most
E/N P times in an iteration for the case |P ′ | < N P occurs in ME-LMD-CIRBP decoding (lines 11-13 of Algorithm 6 ), compared with the LMD-CIRBP decoder,the ME-LMD-CIRBP decoder may consume at most extra ( E/N P ) × N G integer comparisonsper iteration. To summarize, the ME-CIRBP algorithm generally consumes less computationaleffort compared with the CIRBP decoder but suffers from greater performance loss; the ME-LMD-CIRBP decoder may offer quite-nice performance-latency tradeoffs at the cost of slightlyincreased complexity. VII. C ONCLUSION
In this paper, we have presented novel IDS LDPC decoding schedules which apply a VNselecting metric called conditional innovation and a search complexity reduction criterion thatlimits our target VN/CN search range to those newly updated CNs and their connected VNs.The proposed schedules are VN-centric in the sense that the metrics used are aimed to improvethe reliability of the target VNs’ bit decisions by predicting the probability of reversing potentialincorrect decisions. Computer simulation results indicate that our schedules outperform knownschedules and achieve most impressive error rate performance gain in the first few iterations.
February 23, 2021 DRAFT6
Therefore, as far as the average computing complexity is concerned, the proposed schedulesdo not incur more computing burden. The converged FER performance of the LMD-basedalgorithms against their counterparts indicates that the search range reduction will eventuallyinclude those VNs that should be updated. The outstanding first-iteration performance of theLMD-CIRBP algorithm may be attributed to the decreasing probability of improper updateselections by considering only the shortlist candidates. To shorten the decoding delay, we developmulti-edge versions of the CIRBP and LMD-CIRBP algorithms by increasing the degrees ofparallelism in updating. The multi-edge versions are of low latency and are proved to be efficientin performance. A
PPENDIX
AA S
EMI - ANALYTIC P ROOF OF
Property 1
We verify Property 1 by evaluating (10) using the GA-DE technique [21]. Recall that D = | ˜ P − P | , where ≤ ˜ P , P < . Conditioning on D ≥ γ , the numerator of (10) is equal to Pr ( P ≥ . | D ≥ γ ) = Pr ( P ≥ max( γ, . | D ≥ γ ) = Z γ, . f P | D ( τ | D ≥ γ ) d τ = Z γ, . Pr( D ≥ γ | P = τ ) f P ( τ )Pr( D ≥ γ ) d τ, since P − γ ≥ ˜ P ≥ , where f ( · ) stands for probability density function (PDF); similarly, as P + γ ≤ ˜ P ≤ , the denominator of (10) is equal to Pr ( P < . | D ≥ γ ) = Z min(1 − γ, . Pr( D ≥ γ | P = τ ) f P ( τ )Pr( D ≥ γ ) d τ. Combining the above expressions then yields that J ( γ ) = R γ, . Pr( D ≥ γ | P = τ ) f P ( τ ) d τ R min(1 − γ, . Pr( D ≥ γ | P = τ ) f P ( τ ) d τ . (A.1)We now apply the GA-DE to obtain f P ( τ ) and Pr( D ≥ γ | P = τ ) . Note that in the GA-DE,all messages are modeled as i.i.d. consistent Gaussian random variables; specifically, the C2V(resp. V2C) messages are distributed according to N ( µ C , µ C ) (resp. N ( µ V , µ V ) ), where µ C (resp. µ V ) denotes the mean of the C2V (resp. V2C) messages. Due to the all-zero codewordassumption, the mean of the LLR of the received signal is µ = 2 /σ and hence we initialize DRAFT February 23, 20217 µ V = µ . For ( d v , d c ) regular LDPC codes, the µ C and µ V are recursively calculated by (wehave dropped the iteration index for notational simplicity): µ C = Φ − (cid:16) − [1 − Φ ( µ V )] d c − (cid:17) , (A.2) µ V = µ + ( d v − µ C (A.3)where Φ( µ ) is given in [21, Definition 1]. Similar recursions for irregular LDPC codes can befound in [21].Following the idea of the GA-DE, we approximate the total LLR L and the precomputed totalLLR ˜ L as consistent Gaussian random variables, i.e., L ∼ N ( µ L , µ L ) and ˜ L ∼ N ( µ ˜ L , µ ˜ L ) ,where µ L = µ + d v µ C . Moreover, their difference ∆ L , ˜ L − L is also approximated in the sameway with mean µ ∆ L = µ ˜ L − µ L , i.e., ∆ L ∼ N ( µ ∆ L , µ ∆ L ) . Using the above approximationsand the definitions L = ln( P /P ) and Q ( α ) = √ π R ∞ α e − β d β , we obtain f P ( τ ) = f L (cid:18) ln (cid:18) τ − τ (cid:19)(cid:19) , (A.4)and Pr( D ≥ γ | P = τ ) = Pr (cid:16) ˜ P ≥ min( τ + γ,
1) or ˜ P ≤ max( τ − γ, | P = τ (cid:17) = 1 − Z min( τ + γ, τ − γ, f ˜ L | L (cid:18) ln (cid:18) ˜ τ − ˜ τ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ln (cid:18) τ − τ (cid:19)(cid:19) d˜ τ = 1 − Z min( τ + γ, τ − γ, f ∆ L (cid:18) ln (cid:18) ˜ τ − ˜ τ (cid:19) − ln (cid:18) τ − τ (cid:19)(cid:19) d˜ τ = 1 − Q ln (cid:16) max( τ − γ, − max( τ − γ, (cid:17) − ln (cid:0) τ − τ (cid:1) − µ ∆ L √ µ ∆ L + Q ln (cid:16) min( τ + γ, − min( τ + γ, (cid:17) − ln (cid:0) τ − τ (cid:1) − µ ∆ L √ µ ∆ L . (A.5)Given µ L and µ ˜ L obtained from (A.2) for any fixed iteration, we can calculate J ( γ ) as afunction of γ using (A.5), (A.4), and (A.1). The GA-DE curves in Fig. 1(a) are the J ( γ ) ’s forthe first three iterations with the flooding schedule. The curves almost coincide with the simulatedones, and the decreasing property of J ( γ ) as claimed in Property 1 is also revealed. We remark February 23, 2021 DRAFT8 that similar behavior is observed for other LDPC codes of different rates and degree distributions.Moreover, our proof relies only on the assumption that µ ∆ L > whence is independent of theBP-based schedule used. A PPENDIX BP ROOF OF
Property 2
We prove that F ( γ ) > by considering two cases: γ ≥ P and γ < P . Note that the event { D ≥ γ } implies that ˜ P can lie in [0 , P − γ ] or [ P + γ, . When P < . and γ ≥ P , wemust have that ˜ P ∈ [ P + γ, and hence ˜ P ≥ P + γ with probability , resulting in that F ( γ ) = ∞ . For the case γ < P , we first rewrite (11) as F ( γ ) = Pr( { ˜ P ≥ P } ∩ { D ≥ γ } )Pr( { ˜ P < P } ∩ { D ≥ γ } ) = Pr( ˜ P ≥ P + γ )Pr( ˜ P ≤ P − γ ) = R P + γ f ˜ P (˜ τ ) d˜ τ R P − γ f ˜ P (˜ τ ) d˜ τ . (B.1)Since f ˜ P (˜ τ ) = f ˜ L (ln(˜ τ / (1 − ˜ τ ))) and ˜ L ∼ N ( µ ˜ L , µ ˜ L ) , we have the following expressions Z P + γ f ˜ P (˜ τ ) d˜ τ = Q ( g ( P , γ )) and Z P − γ f ˜ P (˜ τ ) d˜ τ = Q ( g ( P , γ )) for the terms in (B.1), where Q ( · ) is defined in Appendix A and g ( P , γ ) = ln (cid:16) P + γ − ( P + γ ) (cid:17) − µ ˜ L p µ ˜ L , g ( P , γ ) = µ ˜ L − ln (cid:16) P − γ − ( P − γ ) (cid:17)p µ ˜ L . With the above quantites, the expression in (B.1) is simplified as F ( γ ) = Q ( g ( P , γ )) Q ( g ( P , γ )) . (B.2)Since P < . , we obtain that (cid:20) ln (cid:18) P + γ − ( P + γ ) (cid:19) − µ ˜ L (cid:21) < (cid:20) µ ˜ L − ln (cid:18) P − γ − ( P − γ ) (cid:19)(cid:21) , which implies that Q ( g ( P , γ )) > Q ( g ( P , γ )) and hence F ( γ ) > .Based on the above derivation, it is clear that F ( γ ) = ∞ for γ ≥ P . We next show that F ( γ ) is strictly increasing for γ ∈ [0 , P ) . Specifically, we prove the following derivative is positive. d F ( γ )d γ = 1 p µ ˜ L ( Q ( g ( P , γ ))) × (cid:20) Q ′ ( g ( P , γ )) Q ( g ( P , γ ))( P + γ )(1 − ( P + γ )) − Q ′ ( g ( P , γ )) Q ( g ( P , γ ))( P − γ )(1 − ( P − γ )) (cid:21) (B.3) DRAFT February 23, 20219 where Q ′ ( α ) , d Q ( α )d α = − exp( − α / √ π . Recall the facts that Q ( α ) > , Q ′ ( α ) < ∀ α ∈ R , and d Q ′ ( α ) / d α = − αQ ′ ( α ) . Defining P ( α ) , Q ( α ) /Q ′ ( α ) , one can show that P ′ ( α ) , d P ( α ) / d α = [( Q ′ ( α )) + αQ ( α ) Q ′ ( α )] / ( Q ′ ( α )) > for α ≤ . For α > , we apply the inequality αQ ( α ) < − Q ′ ( α ) [23] to concludethat Q ′ ( α )( Q ′ ( α ) + αQ ( α )) > . Since P ′ ( α ) > for all α , i.e., P ( α ) is increasing, and g ( P , γ ) > g ( P , γ ) , we have that P ( g ( P , γ )) > P ( g ( P , γ )) . (B.4)Using (B.4) and the fact that ( P + γ )(1 − ( P + γ )) > ( P − γ )(1 − ( P − γ )) , we further obtain Q ′ ( g ( P , γ )) Q ( g ( P , γ ))( P + γ )(1 − ( P + γ )) > Q ′ ( g ( P , γ )) Q ( g ( P , γ ))( P − γ )(1 − ( P − γ )) . (B.5)Substituting (B.5) into (B.3) then shows that d F ( γ ) / d γ > for γ ∈ [0 , P ) .R EFERENCES [1] R. G. Gallager,
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