POLAR codes have been proven to achieve the symmetric. Reduce the Complexity of List Decoding of Polar Codes by Tree-Pruning

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1 Reduce the Complexity of List Decoding of Polar Cos by Tree-Pruning Kai Chen, Bin Li, Hui Shen, Jie Jin, and David Tse, Fellow, IEEE arxiv:58.228v [cs.it] 9 Aug 25 Abstract Polar cos unr cyclic redundancy chec aid successive cancellation list (CA-SCL coding can outperform the turbo cos and the LDPC cos when co lengths are configured to be several ilobits. In orr to reduce the coding complexity, a novel tree-pruning scheme for the SCL/CA-SCL coding algorithms is proposed in this paper. In each step of the coding procedure, the candidate paths with metrics less than a threshold are dropped directly to avoid the unnecessary computations for the path searching on the scendant branches of them. Given a candidate path, an upper bound of the path metric of its scendants is proposed to termined whether the pruning of this candidate path would affect frame error rate (FER performance. By utilizing this upper bounding technique and introducing a dynamic threshold, the proposed scheme letes the redundant candidate paths as many as possible while eeping the performance terioration in a tolerant region, thus it is much more efficient than the existing pruning scheme. With only a negligible loss of FER performance, the computational complexity of the proposed pruned coding scheme is only about 4% of the standard algorithm in the low signal-to-noise ratio (SR region (where the FER unr CA-SCL coding is about.., and it can be very close to that of the successive cancellation (SC cor in the morate and high SR regions. Inx Terms Polar cos, successive cancellation coding, tree-pruning. I. ITRODUCTIO POLAR cos have been proven to achieve the symmetric capacity on binary-input discrete memoryless channels unr a low-complexity successive cancellation (SC coding algorithm []. Although the polar cos asymptotically achieve the channel capacity, the performance unr SC coding is unsatisfying when the co length is of the orr of ilobits. Several alternative coding schemes have been proposed to improve the finite-length performance of polar cos, such as successive cancellation list (SCL [2], successive cancellation stac (SCS[3] and belief propagation (BP [4] coding algorithms. It is reported that polar cos unr the CRC-aid SCL/SCS (CA-SCL/SCS coding algorithms can achieve a better frame error rate (FER performance than the LDPC and turbo cos when the co lengths are configured to several ilobits [5][6][7]. Therefore, polar coding is believed to be a competitive candidate in future communication systems. Since the CA-SCS coding requires a large stac to store the candidate paths which leads to a high space complexity, CA-SCL coding algorithm is of more interest [8][9][]. This wor was supported by the Science and Technology Proect of Shenzhen (o. JSGG K. Chen, B. Li, J. Jin, and H. Shen are with the Communications Technology Research Lab., Huawei Technologies, Shenzhen, P. R. China ( aichen@ieee.org. D. Tse is with the Dept. of Electrical Engineering, Stanford University, CA , USA ( dntse@stanford.edu. evertheless, to achieve competitive performance against LDPC or turbo cos, a morate-sized list is required in CA- SCL coding. In that case, the computational complexity of the CA-SCL cor is still high. As stated in [], SCL coding can be regard as a path searching procedure on the co tree. In orr to reduce the complexity of SCL coding, tree-pruning technique is exploited by avoiding unnecessary path searching operations [2]. In orr to eep the FER loss in an acceptable region, [2] computes the pruning threshold in a very conservative way. Only the candidate paths with metrics much less than the maximum one are pruned. It wors well when the signalto-noise ratio (SR is high, where the metric of the correct path is usually much larger than the others. However, this existing pruning technique is no longer efficient when woring in the relative low SR region where the FER unr CA-SCL coding is about.., while it is exactly the operating regime for cellular networs. In this paper, we propose to compute the threshold using the sum of the survival path metrics. To evaluate how much a pruned candidate path would affect FER performance, we propose a metric upper bound of its scendants. Utilizing this upper bounding technique, a dynamic threshold is further proposed. The proposed scheme letes the redundant candidate paths as many as possible while eeping the performance terioration in a tolerant region, thus it is much more efficient than the existing pruning scheme. The remainr of the paper is organized as follows. Section II reviews the basics of polar coding. Section III scribes the proposed tree-pruning scheme for SCL coding. A path metric upper bound of the scendants of some given candidate path and a dynamic threshold configuration method are proposed. Section IV provis the performance and complexity analysis based on the simulation results. Finally, Section V conclus the wor. A. otation Convention II. PRELIMIARIES In this paper, we use calligraphic characters, such as X and Y, to note sets, and X to note the number of elements in X. We write the Cartesian product of X and Y as X Y, and write the n-th Cartesian power of X as X n. Further, we write Y\X to note the sset of Y with elements in X exclud. We use notation v to note a -dimension vector (v, v 2,, v and v i to note a svector (v i, v i+,, v, v of v, i,. Particularly when i >, v i is a vector with no elements in it and the empty vector is noted by φ. We write v,o to note the

2 2 svector of v with odd indices (a : ; is odd. Similarly, we write v,e to note the svector of v with even indices (a : ; is even. For example, for v 4, v 3 2 = (v 2, v 3, v 4,o = (v, v 3 and v 4,e = (v 2, v 4. Further, given a inx set A, v A note the svector of v which consists of v i s with i A. B. Polar Coding and SC Decoding We are given a binary-input memoryless channel W : X Y with input alphabet X = {, } and output alphabet Y, the channel transition probabilities are W (y x, x X, y Y. For co length = 2 n, n =, 2,, and information length K, i.e. co rate R = K/, polar coding over W proposed by Arıan can be scribed as follows: After channel combining and splitting operations on inpennt uses of W, we get successive uses of synthesized binary input channels W (i, i =, 2,,, with transition probabilities where W (i (y, u i u i = W (y u = 2 W (y u ( u i+ X i W (y i x i (2 and the source bloc u are supposed to be uniformly distributed in {, }. { } The reliabilities of the polarized channels W (i can be evaluated by using nsity evolution [3], and is usually more evaluated efficiently by calculating Bhattacharyya parameters [] for binary erasure channels (BECs or by using Gaussian approximation [4] for binary-input AWG (BIAWG channels. To transmit a message { bloc } of K bits, the K most reliable polarized channels with indices i A are piced out for carrying these information bits; a fixed bit sequence called frozen bits are transmitted over the others. The inx set A {, 2,, } is called the information set and A = K, and its complement set which is noted by A c is called the frozen set. As mentioned in [], polar cos can be cod using successive cancellation (SC coding algorithm. In [], it is further scribed as a path searching procedure on a coding tree. The metric of a coding path u i can be measured using a posteriori probability P (i (ûi y W (i i= = if i A c and û i u i W (i (y,ui u i 2P(y otherwise (3 When û i is not a wrong frozen bit, the above path metric can be recursively computed as P (2i 2 = û 2i {,} P (i 2,o û 2i,e (i P,e 2 (4 + P (2i 2 = P (i 2,o û 2i,e (i P,e (5 2 + where n, = 2 n, i. Thus, SC coding can be scribed as a greedy search algorithm on the co tree. In each level, only the one of two scendants with larger path metric is selected for further expansion. { ( h i y û i =, û i i A u i i A c (6 where h i ( y, û i { if P (i (ûi = y otherwise C. Improved SC Decoding Algorithms (i (ûi P y The performance of SC is limited by the bit-by-bit coding strategy. Whenever a bit is wrongly termined, there is no chance to correct it in the rest of the coding procedure. Theoretically, the performance of the maximum a posteriori probability (MAP coding (or equivalently ML coding, since the inputs are assumed to be uniformly distributed can be achieved by traversing all the -length coding paths in the co tree. But this brute-force search taes exponential complexity and is impossible to be implemented for practical co lengths. Two improved coding algorithms, SCL coding and SCS coding, are proposed in [2] and [3]. Both of these two algorithms allow more than one edge to be explored in each level of the co tree. During the SCL(SCS coding, a set of candidate paths are obtained and stored in a list(stac. Combining the ias of SCL and SCS, a coding algorithm named successive cancellation hybrid (SCH is proposed in [], which can achieve a better tra-off between computational complexity and space complexity. Moreover, with the help of CRC cos, polar cos cod by these improved SC coding algorithms are found to be capable of achieving the same or even better performance than turbo cos or LDPC cos [5] [7] [6]. Among these existing improved SC coding algorithms, benefitting from the limited requirement for the memory, (CA-SCL coding is the most interesting for hardware implementation [8] [9] [5] [6]. As shown in Fig., the processing loop of the standard SCL/CA-SCL coding is as follows: S For each candidate path, calculate the path metrics of its scendant paths; S2 Sort the metrics, and reserve at most L paths with the larger metrics and lete the others; S3 If any two of the survival paths share the same parent no, then a copy operation is performed to create separate woring spaces for these two paths; S4 For each survival path, update the partial-sum recursively; S5 The above loop is processed until the length of candidate paths reach. The candidate path with the largest (7

3 3 Initialize the list with a null path, and set i = p Fig.. i = i + yes no Calculate metrics of scendant paths Is frozen? no Sort the metrics, reserve the at most L largest path and lete the other Perform path copy operation if required Update the partial-sums of the survival paths i =? yes Select the path with largest path metric (which can pass CRC chec Pruning Delete the paths with metrics less than a threshold The flow chart of the pruned SCL/CA-SCL coding. path metric (when CRC embedd, the candidates which cannot pass CRC are dropped directly is piced out for the final cision. III. TREE-PRUIG SCHEME FOR SCL/CA-SCL DECODIG ALGORITHM A. The Proposed Pruning Scheme In orr to reduce the computational complexity of SCL coding, a pruning operation is add after the sorting operation (as shown in Fig.. If the metric of some candidate path is less than a threshold, it will be directly leted to avoid redundant path expansions and copy operations. In this paper, we propose to use the path metric sum of the (maximum L survival candidate paths after sorting: while coding { } the i-th bit, the metrics of the survival paths is P (i, where L i is inx set of the survival paths in the list after sorting operation, L i L; If the following inequality holds for some L i, the corresponding path is then leted, P (i L i < α i P (i (8 = where α i <. Particularly, if α i =, then no pruning is performed when coding this i-th bit. In the following part of this section, we ll discuss how to choose the value of {α i }. Performance Deterioration: Suppose that the correct path is still in the list after the sorting operation during coding the i-th bit. The probability of that the -th candidate is the correct path (i.e., the performance loss of leting this path is computed as Pr {-th path is the correct one} = P (i Li = P (i (9 Fig. 2. -vi The probability nsity function of LLR. +vi Pr ui log Pr ui 2 Statistical Threshold Configuration: Given a specific polar co, the channel property, and a tolerant FER performance loss P tol, the most direct way to configure α i is through Mote Carlo simulation. Initially, set α i = and simulate using standard (CA-SCL coding. During coding the i-th bit in each frame, the ratio of the metric of the correct path (until the i-th bit P c (i and the sum metric of the survival paths in the list is record; If the final coding result is correct and the ratio is less than α i, then update α i with this ratio, i.e., ( P c (i α i = min α i, L = P ( (i When the amount of simulated frame is large enough, the pruning operation based on (8, the FER performance loss can be very small. B. Dynamic Threshold Configuration The Monte Carlo configuration is pendant on the specific SR, co length, and co rate. Thus, it s quite difficult to use for practical application. For polar cos, the reliability of the polarized channels can be evaluated using Gaussian approximation [4] or some other techniques; in other words, the probability nsity functions (PDFs of the LLRs which corresponding to the receiving bits (conditioned on that the previous bits are correctly cod can be a priori information to the cor. In this ssection, we present a method to estimate the performance loss brought by pruning using these LLR distributions; and then, a dynamic threshold configuration method is proposed. Using the proposed thresholding method, the pruned (CA-SCL coding can fully utilize the tolerant performance terioration and thus lower the computational complexity. Path Metric Upper Bounds: The LLR PDFs can be obtained by using nsity evolution or Gaussian approximation [4]. Based on the PDF corresponding to a bit u i, we can fine a LLR region, such that the probability of the corresponding LLR taes values in [ l i, l i ] is larger than a pre-fined small probability P llr, { Pr l i log Pr{u } i = } Pr{u i = } l i P llr ( Therefore, when coding the i-th bit, if one candidate path has metric P (i, the metric of any its scendant path P ( at the -th level has an upper bound, P ( = P (i =i+ e li ( + e P ( (2 li

4 4 Fig. 3. Path Metric Level-i Fig (24,52 Eb/ =.5dB Upperbound of Path Metrics Max Metrics by Simulation Avg Metrics by Simulation Bit Inx Level-( i+ Level-( i+2 The upper bound of path metric. Survival Paths Pruned Paths Descendants of Pruned Paths which would survive Descendants of Pruned Paths which wouldn't survive A graphic monstration of the pruned paths which would survive. where i <. ote that for bit inx [i, ], every bit effects the value of P ( to some extent, no matter it s an information bit or a frozen bit. Specifically, for an information bit with relatively high reliability, i.e., with a large l i, its impact on P ( is consired negligible; for a frozen bit, since the value of l i is relatively smaller, its impact on P ( is more significant. Fig. 3 gives the simulation result of a (24, 52 polar co unr BIAWGC with SR.5dB. The coding algorithm is CA-SCL with L = 32. The maximum and average values of the path metric during coding each bit are recod. To guarantee the inequality (2 holds with probability larger than 9, we set P llr = 9. As shown in the figure, the simulation data is well bound by (2. 2 Threshold Computation: In this ssection, we propose a new threshold computation method which can fully utilize the pre-fined tolerant FER performance loss P tol. As previously stated, pruning operation during coding u i will cause some FER performance loss; When expansion at level-(i + on the co tree, the loss brought by the pruned path at level-i is accumulated, i.e., the paths which cause performance loss during coding u i+ inclu not only the newly pruned paths but also the scendants of the pruned paths at level-i. Thus, when coding at level-i on the co tree, the FER performance loss is computed based on both the newly pruned paths at level-i and the scendants of all the previously pruned paths which would be in the list. A graphic illustration is given in Fig. 4. To estimate the FER loss brought by the pruning operations, the pruned path should be record. Let S i be the active pruned path during coding the first i bits, i.e., u i. For each pruned path S i, the level inx t when it is pruned, along with the corresponding path metric p and the estimated performance loss q which is computed using (9, is record. Obviously, t i. Based on (t, p, q, the maximum metric value at the i-th level of the scendants of the pruned path can be computed using P (i in (2, Z (i = p P (t + P (i (3 The performance loss P (i which is brought by the pruning operations during coding the first i bits is evaluated as follows: S Find the survival paths in the list L i which are with metrics larger than the maximum Z (i {, } L i = L i, P (i max Z (i L i (4 S i the number of these found paths is L i ; S2 Find (L L i pruned records with indices S i S i which has the larger estimated performance losses, i.e., for any S i and S i \S i, we have q q, where S i = L L i. S3 The performance loss P (i is upper bound by P (i S i q (5 The threshold α i is termined by the tolerant performance loss P tol and the loss introduced in the previous coding process P (i, L α i = i\r i P (i (6 L i P (i where inx set R i indicates the candidates to be pruned and is the largest sset of L i which satisfies ( P tol P (i (7 P (i R i L i P (i After the pruning, the set of pruned records S i is updated as follows: S Combing S i and the newly pruned paths which are induced by R i, the obtained temporary inx set is noted by T i ; S2 Find the L pruned records with largest losses in T i, the result indices form the set T i, i.e., T i T i, for any T i and T i \T i, we have q q. S3 The minimum value of the metric upper bounds of the pruned records in T i is Z min = min T i Z (i (8 S4 S i is obtained by inactivating all the records in T i with estimated metric less than Z min { } S i = T i, Z (i Z min (9 ote that, initially, S =.

5 5 x FER Overlapped: L32, L32 Ptol=e-5 Existing, L32 Ptol=.*FER Existing, L32 Ptol=e-5 Proposed, L32 Ptol=.*FER Proposed, L8 wo pruning L6 wo pruning L32 wo pruning L32 Ptol=e-5 Existing L32 Ptol=.*FER Existing L32 Ptol=e-5 Proposed L32 Ptol=FER Proposed L32 Ptol=.*FER Proposed Average Complexity L32 Ptol=e-5 Existing L32 Ptol=.*FER Existing L32 Ptol=e-5 Proposed L32 Ptol=FER Proposed L32 Ptol=.*FER Proposed Eb/(dB Eb/(dB Fig. 5. FER performances unr different coding schemes. Fig. 6. Average computational complexity. C. Complexity The complexity of (CA-SCL coding consists of three parts: the path extension (inclus the updating of path metrics (4 (5 and the partial-sums, path metric sorting, path copy, and partial-sum updating. Applying pruning, many redundant path extensions along with path copies are avoid. Since the computational complexity to obtain a length- path is O( log [], and in the best case only one path is preserved in the list, thus the computational complexity is reduced by O(L log. However, calculating the threshold itself introduces additional compactions. For each information bit, the metrics of the survival paths and the pruned paths are add up to compute the threshold, thus the complexity increases with O(L. Thus, the computational complexity can be reduced by orr of O(L log if the P tol is set to a proper value. Moreover, when one of the two scendants of a single parent path is pruned, there is no longer need for the path copy operation. In fact, it is the usual case especially when the corresponding polarized channel is with high reliability. Therefore, the number of required path copies is also reduced. As to the path metric sorting, the least reliable paths are required to be piced out when computing the threshold, so the pruning does not reduce the sorting complexity. IV. SIMULATIO RESULTS In this section, we analyze the performance of the proposed pruned (CA-SCL coding algorithm via simulation. The simulated polar co has co length = 24 and the co rate R = /2, which is constructed unr E b / =.5dB using Gaussian Approximation [4]. The information bloc is assumed to have 6 embedd CRC bits, and CA-SCL coding is applied. Fig. 5 shows the FER performances unr different L values and pruning techniques. Fig. 6 and Fig. 7 show the corresponding average computational complexity and average number of path copies, respectively. The average computational complexity is evaluated in terms of the number of metric recursive operations, which are fined in (4 and (5. Here, Fig. 7. Average Times of Path Copies L32 Ptol=e-5 Existing L32 Ptol=.*FER Existing L32 Ptol=e-5 Proposed L32 Ptol=FER Proposed L32 Ptol=.*FER Proposed Eb/(dB umber of path copy operations. we pay more attention to the SR region where the FER is around.., which is the interesting Particularly, the thresholds of sum statistical is obtained by Monte Carlo simulation. As shown in the figures, when P tol taes a relative conservative value (compared with the FER, that is 5, all the pruning technique do not introduce noticeable loss in FER; while the proposed scheme has much lower complexity than the existing scheme in [2]. When coding with CA-SCL with L = 32 and P tol = FER, the performance is teriorated and very close to standard CA-SCL with L = 6, while the average complexity is even lower than the standard one with L = 8. Further, when P tol =. FER, the FER performance loss is less than.db, but the complexity is reduced by 5% 75%. Fig. 8 compares the FER and FER loss of the proposed pruning scheme and [2] unr different target losses P tol. The E b / is fixed to.5db. As shown in the figure, when P tol FER, the actual FER loss is very close to the target P tol ; while the actual loss of [2] is far less than the target. That means, compared with [2], the proposed pruning scheme utilizes the tolerant FER loss much more efficiently, thus it is

6 6 FER / FER Deterioration FER Target FER Target FER wo pruning FER Existing FER Existing FER Proposed FER Proposed Ptol Fig. 8. FER performance terioration comparison of the pruned coding scheme. with lower complexity. V. COCLUSIO In this paper, a tree-pruning technique to reduce the complexity of (CA-SCL is proposed. During the coding process, the candidate paths with metric less than a threshold are directly leted to avoid redundant path extensions. Based on the reliabilities of the information/frozen bits, an upper bound of the path metric is rived to estimate the terioration brought by the pruning operation. Utilizing this bound, a dynamic thresholding technique is presented. Compared with a similar existing scheme [2], the new proposed scheme can mae full use of the given tolerant performance terioration, and is much more efficient. [9] A. Balatsouas-Stimming, M. Bastani Parizi, A. Burg, LLR-based successive cencellation list coding of polar cos, IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, Florence, Italy, May 24. [] M. Monlli, S. H. Hassani, R. Urbane, Scaling Exponent of List Decors with Applications to Polar Cos, IEEE Information Theory Worshop (ITW, Sevilla, Spain, Sept. 23. [] K. Chen, K. iu, J. Lin, Improved successive cancellation coding of polar cos, IEEE Trans. on Commun., vol. 6, no. 8, pp. 3-37, 23. [2] K. Chen, K. iu, and J. Lin, A reduce-complexity successive cancellation list coding of polar cos, in IEEE 77th Vehicular Technology Conference (VTC Spring, Dresn, Germany, June 23. [3] R. Mori and T. Tanaa, Performance of polar cos with the construction using nsity evolution, IEEE Commun. Lett., vol. 3, no. 7, pp , Jul. 29 [4] P. Trifonov, Efficient sign and coding of polar cos, IEEE Trans. Commun., vol. 6, no., pp , ov. 22. [5] B. Yuan, K. K. Parhi, Successive Cancellation List Polar Decor using Log-Lielihood Ratios, 24 48th Asilomar Conference on Signals, Systems and Computers (ACSSC, pp , ov. 24. [6] G. Saris, P. Giard, A. Vardy, C. Thibeault, and W. J. Gross, Unrolled Polar Decors, Part II: Fast List Decors, arxiv:55.466v, May 25. REFERECES [] E. Arıan, Channel polarization: A method for constructing capacity achieving cos for symmetric binary-input memoryless channels, IEEE Trans. Inf. Theory, vol. 55, no. 7, pp , 29. [2] I. Tal and A. Vardy, List coding of polar cos, IEEE Int. Symp. Inform. Theory (ISIT, pp. -5, 2. [3] K. iu and K. Chen, Stac coding of polar cos, Electronics Letters, vol. 48, no. 2, pp , 22. [4]. Hussami, S. B. Korada, and R. Urbane, Performance of polar cos for channel and source coding, in IEEE Int. Symp. Inform. Theory, pp , 29. [5] I. Tal and A. Vardy, List coding of polar cos, arxiv:26.5v, May 22. [6] B. Li, H. Shen, and D. Tse, An adaptive successive cancellation list cor for polar cos with cyclic redundancy chec, IEEE Commun. Lett., vol. 6, no. 2, pp , 22. [7] K. iu and K. Chen, CRC-aid coding of polar cos, IEEE Commun. Lett., vol. 6, no., pp , Oct. 22. [8] G. Saris, P. Giard, A. Vardy, C. Thibeault, W. J. Gross, Increasing the Speed of Polar List Decors, arxiv:47.292, July 24.