Sliding Window Polar Codes

Sliding Window Polar Codes Valerio Bioglio, Carlo Condo arXiv:2004.07767v1 [cs.IT] 16 Apr 2020 Mathematical and Algorithmic Sciences Lab Huawei Technologies France SASU Email: tvalerio.bioglio,[email protected] Abstract—We propose a novel coupling technique for the design of polar codes of length N , making them decodable through a sliding window of size M ă N . This feature allows to reduce the computational complexity of the decoder, an important possibility in wireless communication downlink scenarios. Our approach is based on the design of an ad-hoc kernel to be inserted in a multi-kernel polar code framework; this structure enables the sliding window decoding of the code. Simulation results show that the proposed sliding window polar codes outperform the independent blocks transmission in the proposed scenario, at the cost of a negligible decoding overhead. Fig. 1: Wireless communication downlink scenario. Index Terms—Polar Codes, window decoding, information coupling I. I NTRODUCTION Polar codes [1] are linear block codes relying on the channel polarization effect, that allows to create virtual bit-channels of different reliability. In a polar code of length N and dimension K, the information is transmitted through the K most reliable bit-channels, while the remaining bit-channels are “frozen” to a predetermined value, usually zero. Originally, „ polar  codes 1 0 were based on the polarization kernel T2 “ , which 1 1 limited code length N to be a power of 2. This limitation has been overcome thanks to the discovery of polarization properties of larger kernels, for which kernels providing the best polarization properties have been identified [2]. In parallel, multi-kernel polar codes [3] have been proposed to create polar codes of virtually any length with efficient implementations and fast decoding [4], [5]. In this work, we focus on the communication between a transmitter and a receiver having different computational capabilities, namely when the receiver is less powerful than the transmitter, e.g. in the wireless communication downlink scenario depicted in Figure 1. In our model, the transmitter is able to create polar codewords of length N , while the receiver can handle only polar codewords of length M ă N . The straightforward solution for this problem is to divide the information in S “ N {M blocks and transmit separately each block on a different codeword of length M . However, independent transmissions increase the block error rate (BLER) of the system, since transmitted information is recovered only if all the S codewords are decoded correctly; a single error in one of the transmissions results in an overall decoding failure. In this paper, we propose a novel polar code design for codes of length N which can be decoded using a sliding window framework, i.e. considering M ă N received symbols per decoding step, and using the result of one step to facilitate the next. Recent works have already shown the benefits of inserting memory in the polar code encoding and decoding process [6], [7]. Moreover, windowed decoding has been proposed for the decoding of spatially coupled codes [8] for convolutional LDPC codes [9]; a similar approach has been proposed in [10] to construct partially information coupled polar codes. Our approach differs from the aforementioned techniques, as it is based on the definition of an ad-hoc kernel enabling the sliding window decoding of a particular multi-kernel polar code. Accordingly, the frozen set needs to be designed on the basis of the resulting transformation matrix. Finally, we show through both theoretical analysis and simulations that the proposed code design is able to outperform independent blocks transmission in the proposed scenario with negligible decoding overhead. II. BACKGROUND In this section, we review the basic concepts of polar codes design and decoding that will be useful for description and analysis of the proposed sliding window polar codes. A. Basic channel transform The basic combination of two independent binary discrete memoryless channels (IB-DMCs) W0 and W1 defined over the same alphabets Wi : X Ñ Y, with X “ t0, 1u, is depicted in Figure 2 and performed by the channel transformation matrix T2 . The transition probabilities of the split channels W ´ and W ` are given by 1 ÿ W0 py0 |u0 ‘ u1 qW1 py1 |u1 q , (1) W ´ py0 , y1 |u0 q “ 2 x PX 1 W ` py0 , y1 , u0 |u1 q “ W0 py0 |u0 ‘ u1 qW1 py1 |u1 q . (2) W´ t“3 X0 U0 W0 Y0 α β t“2 W` ℓ X1 U1 W1 α Y1 Fig. 2: Basic combination of two IB-DMC channels. αr t“0 û0 The one-step transformation defined by T2 can be rewritten using operators g and f introduced in [11] to degrade and enhance channel parameters as W ´ “ W0 g W1 , W ` “ W0 f W1 . ℓ β t“1 βr (3) (4) The Bhattacharyya parameter of channel Wi is defined as ÿa ZpWi q “ Wi py|0qWi py|1q (5) yPY and can be used to evaluate the channel reliability, since it provides an upper bound on the probability that an error occurs under ML decoding [1]. The Bhattacharyya parameter of the transformed channels can be calculated on the basis of that of the original channels as ZpW0 g W1 q ď 1 ´ p1 ´ ZpW0 qq ¨ p1 ´ ZpW1 qq , (6) ZpW0 f W1 q “ ZpW0 qZpW1 q . (7) B. Polar codes Polar codes rely on the polarization acceleration enabled by the n-fold Kronecker product of the basic channel transformation kernel T2 ; for a polar code length N “ 2n , its channel transformation matrix is defined as TN “ T2bn . As the code length goes toward infinity, bit-channels become completely noisy or completely noiseless, and the fraction of noiseless bit-channels approaches the channel capacity. In case of finite code lengths, however, the polarization of bit-channels is incomplete, generating bit-channels that are partially noisy. The Bhattacharyya parameter of these intermediary bit-channels can be tracked throughout the polarization stages to estimate their reliabilities. For an pN, Kq polar code, the indices of the K most reliable bit-channels are selected to form the information set I. The input vector u “ ru0 , u1 , . . . , uN ´1 s is then created by assigning the K message bits to the entries of u whose indices are listed in I; the remaining entries of u, forming the frozen set F , are set to zero. The codeword x is finally calculated as x “ u ¨ TN . C. Successive-Cancellation Decoding In [1], the native decoding algorithm for polar codes, called successive cancellation (SC), was proposed as well. The decoding process is portrayed in Figure 3 as a depth-fist binary tree search, with priority given to the left branch. Given a node at stage t, let us define as α the set of 2t likelihoods received from its parent at stage t`1, initialized as the channel û1 û2 û3 û4 û5 û6 û7 Fig. 3: SC decoding tree of an p8, 4q polar code. White and black nodes are frozen and information bits respectively. information at the root node. The node computes 2t´1 soft values composing αl , that is then sent to the left child at stage 2t´1 . From the left child, 2t´1 partial sums composing β l are received, allowing for the calculation of likelihood vector αr to be sent to the right child, that returns the β r partial sum vector. The 2t -element partial sum vector β can be finally computed and sent to parent node at stage t ` 1. At t “ 0, the decoded bit ûi are estimated by taking hard decisions on the basis of received likelihoods. To improve the error-correction performance of SC at moderate code lengths, SC list (SCL) decoding has been proposed in [12]. It relies on L parallel SC decoders working on different paths. Every time an information bit is estimated, the paths are doubled, with L decoders considering it a 0, and L a 1. To each path is associated a metric, such that the L paths with the largest path metric are discarded, and the decoding proceeds until last information bit has been decoded. III. S LIDING W INDOW P OLAR C ODES Here we describe how to design, encode and decode a polar code of length N “ 2n and dimension K such that it is decodable through a sliding window of size M “ 2m . A. Sliding window kernel The sliding window kernel WS is the cornerstone of the proposed construction. It is defined by the full binary lower triangular matrix of size S, i.e. the square matrix of size S ˆS having ones on and below the diagonal, and zeros above the diagonal, as depicted in Figure 4a. The matrix WS imposes a channel transformation that can be described as a cascade of basic channel transformations, as represented in Figure 4c. This structure imposes to decode input bit ui using only two channel likelihood out of S, namely x̃i and x̃i`1 , the former being modified by the hard decisions taken on bits 0 ď j ă i. We will see how to leverage on this property to construct multi-kernel polar code mixing WS with classical polar kernel T2bn to enable for a sliding window decoding mechanism. If the encoded bits are transmitted over channel W, the virtual channel Wi experienced by input bit ui transmitted through channel transformation matrix WS undergoes the transformation Wi “ W g pW f W f . . . f Wq, loooooooooooomoooooooooooon i`1 times (8) S » — — — — WS “— — — – 0 1 1 .. . 1 1 1 (a) Matrix representation. fi ZpW0 q ZpW1 q ffi ffi ffi ffi ffiS ffi ffi fl ZpWS´1 q û0 û1 .. . WS .. ZpWq . (b) Code design. ûS´2 ûS´1 .. . .. . x0 x1 xS´2 xS´1 (c) Channel combination. Fig. 4: Kernel WS description. for 0 ď i ă S ´ 1, while for the last channel WS´1 “ loooooooooomoooooooooon W f W f ... f W . (9) S times The Bhattacharyya parameter of these virtual channels, depicted in Figure 4b, can hence be calculated as # 1 ´ p1 ´ ZpWqq ¨ p1 ´ ZpWqi q if i ă S, ZpWi´1 q ď ZpWqS if i “ S. (10) The Bhattacharyya parameter can be used to evaluate the bit-channel reliability for various channel models. For binary erasure channels (BECs), the bit error probabilities can be directly calculated from them, while under additive white Gaussian noise (AWGN) channels, density evolution under Gaussian approximation (DE/GA) technique can be implemented on their basis [13]; the latter algorithm estimates the likelihood distribution of the polarized channels by tracking their mean at each stage of the SC decoding tree. Given the block representation of kernel WS depicted in Figure 4c, the bit error probability δi of bit ui under BEC transmission can be calculated as # 1 ´ p1 ´ δq ¨ p1 ´ δ i q if i ă S , δi´1 “ (11) δS if i “ S , where δ is the bit erasure probability of the original BEC. For the AWGN channel, the equations tracking the likelihood mean µi of input bit ui for WS are given by # φ´1 p1 ´ p1 ´ φpµqq ¨ p1 ´ φpiµqqq if i ă S, µi´1 “ Sµ if i “ S, (12) where φp¨q is defined as φpxq “ $ &1 ´ % ?1 2 πx 1 ş8 ´8 tanh ˆ z e 2 ˙ 2 pz´xq 4x dz if x ą 0, (13) if x “ 0, and can be approximated as described in [13]. B. Code design As for classical polar codes, the design of a sliding window polar code entails the selection of its transformation matrix T and frozen set F . Given S “ N {M , the transformation matrix of the code is defined as T “ WS b TM , where TM “ T2bm is the transformation matrix of a polar code of length M . A sliding window polar code can hence be described as a particular multi-kernel polar code [3] constructed by placing kernel WS before the transformation matrix of a classical polar code of length M . Note that the position of WS with respect to the other kernels is not interchangeable. The Tanner graph of the resulting transformation matrix T is depicted in Figure 5. The frozen set can be designed according to the general multi-kernel polar code approach [3] using equations described in Section III-A. Given the structure of T , however, the calculation of channel bit reliabilities can be simplified as follows. The Tanner graph depicted in Figure 5 shows that the channel transformation imposed by T can be seen as the juxtaposition of S polar code transformations of length M , each one altering a different channel whose reliability depends on WS . Given the transmission channel W, it is thus possible to initially calculate the Bhattacharyya parameter of the virtual channel Ws experienced by the s-th polar code Ps defined by TM using (10); then, classical polar codes design equations can be used to evaluate the bit-channel reliabilities of input bits ups´1qM , ups´1qM`1 , . . . , usM´1 . With reference to Figure 5, the Battacharyya parameters at the left of each TM block would be independently calculated using the inputs at their right, that have already undergone one further polarization step through WS . Finally, all the bitchannels of input vector u are sorted in order of reliability, where the indices of the N ´ K least reliable bit-channels form the frozen set F , and the indices of the remaining K bit-channels form the information set I of the code. C. Encoding The straightforward encoding process for the proposed sliding window polar codes follows the standard polar code technique. The K message bits are inserted in the input vector u according to the information set previously calculated, namely storing their values in the indices listed in I, while the remaining bits of u are set to zero. Codeword x can be calculated as x “ u ¨ T , where T is the transformation matrix of the code calculated as previously described. Codeword x is then transmitted through the channel. However, the particular structure of WS allows for an alternative encoding algorithm based on previously described set of S polar codes P1 , . . . , PS of size M . The information up1q up2q u0 u1 uM´1 uM uM`1 u2M´1 tp1q .. . .. . TM TM .. . .. . .. . .. . .. . t WS WS .. . .. . .. . .. . uN ´M`1 uN ´1 .. . TM W W y0 y1 xM´1 xM xM`1 W W W yM´1 yM yM`1 x2M´1 W y2M´1 y p1q y p2q p2q .. . .. . uN ´M upSq x0 x1 .. . .. . t WS xN ´M W xN ´M`1 W .. . xN ´1 W .. . pSq yN ´M yN ´M`1 y pSq yN ´1 Fig. 5: Tanner graph of transformation matrix T of sliding window polar code. set Is of polar code Ps can be extracted from the global information set I as the set of entries of I comprised between ps ´ 1q ¨ M and s ¨ M ´ 1, for s “ 1, . . . , S. Similarly, partial input vectors us for 1 ď s ď S are extracted from u or can be created on the basis of the message bits. Each partial input vector can be encoded independently through polar encoding of length M , obtaining S partial codewords tp1q , . . . , tpSq . Finally, codeword x is obtained by backward accumulation of the partial codewords, starting from the last one, i.e. x “ rtp1q ‘ . . .‘ tpSq , tp2q ‘ . . .‘ tpSq , . . . , tpS´1q ‘ tpSq , tpSq s. This encoding strategy, depicted in Figure 6, follows the structure of sliding window kernel WS and permits to maintain the classical polar encoding structure, however slightly increasing the encoding latency. In fact, each parallel encoder requires log2 M steps to encode the M -length polar codes, then S steps to obtain x, for a total of log2 M ` S steps, i.e. slightly larger than the log2 N “ log2 M ` log2 S steps required for the encoding of a standard polar code of length N . D. Decoding The decoding of the proposed sliding window polar code design is performed in S SC decoding steps, each one employing M likelihoods. Each SC decoding step outputs M bits, representing the estimated input vector û; each SC decoding step takes M received symbols as input, conveniently modified by the M estimated input bits decoded at the previous step. In the following, logarithmic likelihood ratios (LLRs) are used in SC decoding, as proposed in [14]. The decoding procedure is detailed in Algorithm 1. Let us suppose the N channel LLRs to be stored in vector y. This vector is initially split into S sub-vectors of size M as y “ ry p1q |y p2q | . . . |y pSq s. The LLR buffer l is initialized with Algorithm 1 Sliding Window Successive Cancellation 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Load information set I Load channel LLRs y Initialize buffer l “ y p1q for s “ 1 . . . S ´ 1 do Is “ ti ´ ps ´ 1q ¨ M | i P I, ps ´ 1q ¨ M ď i ă s ¨ M u ûpsq “ Decodepl ‘ y ps`1q , Is q xpsq “ ûpsq ¨ TM psq l “ p´1qx ¨ l ` y ps`1q IS “ ti ´ pS ´ 1q ¨ M | i P I, pS ´ 1q ¨ M ď i ă S ¨ M u State ûpSq “ Decodepl, IS q return û “ rûp1q , . . . , ûpSq s the first M entries of y, namely fixing l “ y p1q . At decoding step s, the decoder performs the SC decoding of the pM, Ks q polar code Ps defined as follows. The information set Is is calculated as the set of entries of I comprised between ps´1q¨ M and s ¨ M ´ 1, while the transformation matrix is given by TM . The estimated input vector ûpsq is obtained through SC decoding of code Ps , using vector v “ l ‘ y ps`1q as channel LLRs, where y psq is the sub-vector of y containing its entries from s ¨ M to ps ` 1q ¨ M ´ 1 as y psq piq “ ypps ´ 1q ¨ M ` iq and A‘B “ 2 tanh´1 ptanhpA{2q ¨ tanhpB{2qq » sgnpAq ¨ sgnpBq ¨ minpA, Bq. (14) (15) Vector xpsq “ ûpsq ¨ TM is then calculated to be used as partial sum, namely to update the LLR buffer as l “ psq p´1qx ¨ l ` y ps`1q . The decoding of last sub input vector ûpSq constitutes an exception since it is obtained using directly up1q up2q m pM, K1 q encoder pM, K2 q encoder tp2q ûp1q pM, KS q encoder y ps`1q pM, Ks q SC decoder buffer l ûpsq ûp2q .. . .. . upSq tp1q Fig. 7: Decoding scheme of sliding window polar codes. tpSq ûpSq 11 FULL SW-M = 128 IND-M = 128 SW - M = 256 IND - M = 256 10 Fig. 6: Encoding scheme of sliding window polar codes. 9 Eb /N0 8 the content of the LLR buffer l, i.e. imposing v “ l. Finally, the input vector estimation û is obtained juxtaposing the sub input vectors as û “ rûp1q , . . . , ûpSq s. This decoding strategy can be run on-the-fly during the reception of channel signals. This procedure is depicted in Figure 7. An SC list decoder [12] for the proposed sliding window polar codes can be easily implemented on the basis of the described decoding algorithm. 7 6 5 4 3 2 0 100 200 300 400 500 600 700 800 900 1000 K E. Performance analysis In this section, we analyze the performance of the proposed scheme both in terms of latency and BLER. Under AWGN channels it is possible to exploit the results of the DE/GA algorithm to approximate the BLER of SC decoding of polar codes. Considering binary phaseshift keying (BPSK) modulation, the mean values of the γ{10 , where γ represents channel LLRs are given by 4 K N 10 the channel Eb {N0 in dB. Under the Gaussian assumption, the bit error probability Pea pui q is related to the LLR mean value µi as Pe pui q “ Qp µi {2q, where Qp.q denotes the tail probability of the standard Gaussian distribution. The block error probability under SC decoding can hence be approximated by ÿ ˆc µi ˙ PeSC „ . (16) Q 2 iPI Equation (16) can be used directly for sliding window polar codes designed under DE/GA. An estimate of the BLER under SC in case of the independent transmission of S uncorrelated polar codewords with the same length M is given by PeSC „ S SC 1 ´ p1 ´ pSC e q , where pe is the expected BLER of a single codeword transmission, calculated according to (16). Figure 8 shows the theoretical EB {N0 required by each code construction to reach the target BLER of 10´3 as a function of dimension K for different window sizes M . Independent transmission (IND) can be seen as the state-ofthe-art in the envisaged scenario: the transmitter divides the K message bits into S “ N {M messages of K 1 “ K{S bits, that are encoded and transmitted independently using S polar codes of length M and dimension K 1 . The transmission is successful if all S blocks are decoded correctly. In case of full polar code (FULL) transmission, the transmitter ignores the limitations at the receiver and transmits a codeword obtained using the full pN, Kq polar code designed according to the Fig. 8: Minimum SNR to reach target BLER 10´3 for codes at N “ 1024. best frozen set, which is decoded smoothly at the receiver. Finally, in the proposed sliding window decoding (SW) the transmitter designs and encodes a polar codeword using the described technique. The receiver uses the proposed sliding window decoder to decode the received signal. We can see that the proposed SW technique outperforms IND for all rates at equal window dimension, with a substantial gain of 1 dB for some of the rates. Moreover, for M “ 256 sliding window polar codes show a gap of less than 0.5dB from the optimal FULL. Unfortunately, theoretical bounds on the BLER performance of polar codes under SCL are unknown, so it is not possible to perform a similar analysis for list decoders. Regarding the SC decoding latency, for the first S´1 phases the sliding window decoder requires 2M time steps to decode a length-M polar code and update the buffer l, while 2M ´ 2 time steps are sufficient for the last decoding, leading to a total latency of 2M S ´ 2 “ 2N ´ 2 time steps. This is equal to the latency of an SC decoder for a non-systematic code of length N under the same assumption. However, the implementation complexity of the proposed sliding window decoder is that of a polar decoder of length M , plus M memory elements to store buffer l and the logic to update it. We can thus conservatively identify as a complexity upper bound that of a decoder for a code of length 2M : consequently, if N ě 2M , the proposed decoder always yields a lower implementation cost than a length-N decoder. A similar analysis can be done for SCL decoding, obtaining the same outcome. IV. S IMULATION RESULTS In this section we present the BLER performance of the proposed sliding window design and decoding of polar codes, 100 100 10-2 10-2 BLER 10-1 BLER 10-1 FULL - SC SW - SC IND - SC FULL - L = 8 SW - L = 8 IND - L = 8 10-3 10-3 FULL - SC SW - SC IND - SC FULL - L = 8 SW - L = 8 IND - L = 8 10-4 1 1.5 2 10-4 2.5 3 3.5 4 4.5 Eb /N0 Fig. 9: N “ 1024, M “ 128. 1 1.5 2 2.5 3 3.5 4 4.5 Eb /N0 Fig. 10: N “ 8192, M “ 1024. Fig. 11: BLER comparison of different polar codes for rate K{N “ 1{4. compared to state-of-the-art independent block transmissions and optimal full polar code transmission. We study a scenario where the transmitter sends K bits to the receiver at a code rate R “ K{N , but the receiver has limited decoding capabilities and can handle only M ă N bit. Figure 11 shows the performance of the IND, FULL and SW strategies for codes of rate R “ K{N “ 1{4. The FULL case is used as a benchmark of the best possible achievable BLER. The codes are decoded using either the SC or SCL algorithms with list size L “ 8. Black curves correspond to the SC bound calculated according to (16), while black markers depict simulation results obtained through SC decoding. We can see that the bounds perfectly match the simulations, hence they can be considered as reliable approximations of the SC decoding curves for the proposed code lengths. Red curves with red markers correspond to SCL decoding simulations. The proposed solution outperforms IND under both SC and SCL decoding, while approaching the optimal performance represented by FULL. The entity of the gain provided by SW with respect to IND is significant, corresponding to 1.5dB. V. C ONCLUSION In this work, we presented a novel multi-kernel construction that allows a sliding window decoding approach, thanks to which only a fraction of the codeword bits need to be received before bits can be decoded. 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