Fixed-Threshold Polar Codes

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1 Fixed-Threshold Polar Codes Jig Guo Uiversity of Cambridge Albert Guillé i Fàbregas ICREA & Uiversitat Pompeu Fabra Uiversity of Cambridge guille@ieee.org Jossy Sayir Uiversity of Cambridge jossy.sayir@eg.cam.ac.uk Abstract We study a family of polar codes whose froze set is such that it discards the bit chaels for which the mutual iformatio falls below a certai fixed threshold. We show that if the threshold, which might deped o the code legth, is bouded appropriately, a codig theorem ca be proved for the uderlyig polar code. We also give accurate closed-form upper ad lower bouds to the miimum distace of the resultig code whe the desig chael is the biary erasure chael. I. ITRODUCTIO Chael polarizatio, itroduced by Arıka [], is a pheomeo by which, give a biary-iput discrete memoryless chael, virtual chaels betwee the bits at the iput of a liear ecoder ad the chael output sequece are created, such that the mutual iformatio i each of these chaels coverges to either zero or oe as the code legth teds to ifiity; the proportio of chaels with mutual iformatio close to oe coverges to the origial chael s mutual iformatio. These virtual chaels are created by recursively applyig chael combiig ad splittig steps. Polar codes of rate R = K are liear codes whose geerator matrix is such that its rows iduce the K virtual chaels with highest mutual iformatio amog all possible chaels. The scheme behaves as if ucoded bits were set through these chaels. This costructio, together with polarizatio, explicitly gives a code of rate close to the mutual iformatio of the chael with vaishig error probability. We propose a differet costructio of polar codes. Istead of choosig the best K virtual chaels, we choose all chaels whose mutual iformatio is above a certai threshold which might deped o the code legth. This ew costructio is show to preserve the capacity-achievig property of Arıka s origial costructio as log as the threshold fuctio is bouded appropriately. This costructio iduces accurate closed-form upper ad lower bouds to the miimum distace of the resultig codes whe the desig chael is the biary erasure chael BEC. Our results sharpe existig bouds i the literature o the miimum distace of polar codes [2]. The paper is orgaized as follows. otatio ad preiaries are give i Sectio II. Fixed-threshold polar codes are discussed i Sectio III. Proofs of our mai results ca be foud i Sectio IV. This work has bee fuded i part by the Europea Research Coucil uder ERC grat agreemet , by the FP7 etwork of Excellece EWCOM# uder grat agreemet ad by the Spaish Miistry of Ecoomy ad Competitiveess uder grat TEC C II. OTATIO AD PRELIMIARIES I this sectio, we itroduce our otatio. We also state some relevat results i the literature that are eeded. Cosider a biary discrete memoryless chael B-DMC W : {0, } Y with trasitio probability W y x, where Y is the output alphabet. The chael iput-output mutual iformatio with equiprobable iputs is deoted by IW ad the correspodig Bhattacharyya parameter is deoted by ZW. Let be the chael block legth ad defie x =x,...,x ad y = y,...,y to be the legth- iput ad output sequeces, respectively. The vector chael from x to y is defied as W y x. Cosider the matrix [ ] 0 G 2 =, ad deote by G = G 2 the matrix, correspodig to the Kroecker product by itself = log 2 times. The iformatio bits are deoted by u =u,...,u, with u i {0, } for i =,...,. The, we defie W y u = W y ug as the vector chael iduced from the iformatio bits whe applyig the liear trasformatio G. This step is commoly termed chael combiig. Chael splittig is a procedure that geerates biary-iput chaels out of W y u assumig a successive decoder. More precisely, the i-th chael, for i =,...,, is geerated as follows W i y,u,...,u i u i = u i+,...,u 2 i W y u 2 ad assumes that the bits u,...,u i are kow. Similarly, we deote the Bhattacharyya parameter of the i-th chael, for i =,...,, as Z i = Z W i y,u,...,u i u i. We let Z deote the radom variable correspodig to Z i. For future use, we defie the chael iverse labelig fuctios b j : {,...,} {0, } for j =,...,, such that b j i returs the j-th bit i the biary label of chael i atural labelig is assumed, for j =,..., ad i =,...,. Lemma Chael Polarizatio []: For ay B-DMC W, the chaels {W i } for i =,..., have the property that i {,...,} : I W i δ, ] i {,...,} : I W i [0,δ = IW, = IW. 3

2 I words, Lemma implies that as the mutual iformatio of the chaels W i for i =,..., teds to either oe or zero, ad that the fractio of chaels with mutual iformatio close to oe is i the it IW. I terms of covergece rate, we have the followig. Lemma 2 [3]: For ay B-DMC W ad costat 0 <β< 2, we have Pr Z 2 β = IW. 4 θ The followig is a corollary of [4, Th. 3]. Lemma 3: For ay B-DMC W ad costat 0 <β< 2, we have Pr Z < 2 β = IW. 5 A. Polar Codes I order to costruct a polar code of rate R = K, exactly K rows of G must be selected. Differet methods for choosig these rows yield differet codes. I particular, Arıka s origial costructio, first fixes the rate R, ad the selects the rows of G whose idices give highest I W i over all i =,...,. Alteratively, polar codes ca be defied through the set of discarded rows. This is called the froze set F; its complemet is deoted by F c. If the rate is fixed, the cardiality of the froze set is obviously F = K ad the froze set is F = { i =,..., I W i <θ } where θ is the mutual iformatio threshold that makes F = K. Arıka s threshold fuctio is difficult to characterize aalytically ad eve to compute umerically. Fig. illustrates the evolutio of the threshold of Arıka s costructio for R =0.3, 0.4 i a BEC with IW = 2. As show i [2, Lemma 6.2], the miimum distace of polar codes ca be expressed as a fuctio of the froze set. Let w H b,...,b deote the fuctio that outputs the Hammig weight of bits b,...,b. Lemma 4 [2]: For ay choice of froze set F, the miimum distace d mi of polar codes is give by 6 d mi =mi i F c 2wHbi,...,bi. 7 III. FIXED-THRESHOLD COSTRUCTIO We propose a differet method for costructig polar codes. Arıka s origial codes, have a fixed rate, ad the froze set is determied by all sub-chaels with mutual iformatio smaller tha a certai threshold, such that F = K see 6. I the proposed costructio, a mutual iformatio threshold fuctio θ is fixed, ad the froze set is give as F = { i =,..., I W i <θ }. 8 The threshold fuctio θ cosidered here is a closed-form fuctio of, thus easily computable. As a result, the umber Arıka R =0.3 Arıka R =0.4 θ = 2 β, β = 2 5 θ = 2 β, β = 4 Fig.. Thresholds of Arıka s fixed-rate costructio for R = 0.3, 0.4 ad fixed-threshold polar codes with threshold θ over the BEC with IW = 2. of chaels i the froze set is i geeral ot fixed by the rate, ad hece, the rate itself depeds o. Fig. also shows the threshold fuctios θ = 2 β for β = 2 5, 4. A. Codig Theorem For fixed-threshold polar codes, the rate R of the code is i geeral a fuctio of. I order for these codes to have a valid codig theorem, the threshold fuctio θ must be such that, as grows, the rate R coverges to IW ad the error probability P e 0. We first show the coditios uder which R coverges to IW. Lemma 5 Rate covergece: Cosider a B-DMC W, ad a fixed-threshold polar code of legth ad threshold fuctio θ. Let R deote the rate of the code. If there exists a 0 such that for > 0 such that the threshold fuctio satisfies the bouds α 2 β θ α 2 2 β 2, 9 where α m > 0, 0 <β m < 2, m =, 2, the R = IW. 0 Lemma 5 shows that asymptotically, ay threshold meetig the bouds i 9 is such that R IW. I particular, ay costat threshold θ 0, will result i R IW. Fig. 2 illustrates the results of Lemma 5. We ca see that the speed of covergece depeds o the threshold θ. It is implicit i the figure that some threshold fuctios that fulfill the bouds i Lemma 5 approach the mutual iformatio from above, i.e., R = IW +δ for some δ > 0 such that δ 0. By the coverse to the chael codig theorem [5], such codes will have a error probability bouded away from zero. The followig result characterizes the set of threshold fuctios that yield vaishig error probability.

3 R Arıka R = 2 θ = 3 4 θ = 2 β, β = 5 θ = 2 β, β = 2 5 θ =2 β, β = 5 θ =2 β, β = Fig. 2. Rate covergece for differet threshold fuctios θ over a BEC with IW = 2. Theorem Codig Theorem: Cosider a B-DMC W, ad a fixed-threshold polar code of legth ad threshold fuctio θ. Let P e,θ deote the error probability of this code. Let γ,γ 2 > 0, β > 2, 0 <β 2 < 2 be fixed. If there exists a 0 such that for > 0, the threshold fuctio is such that the, γ β θ γ 2 2 β 2. P e,θ =0. 2 Theorem shows that as, thresholds close to will yield polar codes for which both the rate coverges to IW ad the probability of error vaishes; this rules out costatthreshold codes. B. Miimum Distace I this sectio, we give closed-form upper ad lower bouds to the miimum distace of fixed-threshold polar codes whe the desig chael is the BEC. As the results will show, the bouds are very easy to compute ad accurately characterize the miimum distace. BEC desigs perform well i multiple chaels, icludig the biary-symmetric chael ad the additive white Gaussia oise chael AWG [6]; the performace of BEC codes over the latter chael is show to be very close to that of AWG-tailored polar codes. Our bouds, stated i Theorem 2, follow after the followig results, the first of which beig a extesio of [3, Lemma ]. Lemma 6: Let A : R R : x 2x x 2 ad B : R R : x x 2. Suppose a sequece of umbers x 0,x,...,x is defied by specifyig x 0 :0 x 0 ad the recursio x i+ = f i x i with f i {A, B}. Suppose {0 i : f i = A} = k ad {0 i :f i = B} = k. The, x B k A k x 0, 3 where A k deotes the applicatio of fuctio Aktimes. Proof: Observe that both Ax ad Bx are mootoically icreasig fuctios over x [0, ]. We first have that A Bx B Ax. 4 The upper boud i 3 correspods to choosig f 0 =...= f k = A, f k =...= f = B. Suppose {f i } is ot chose as above. The, there exists j {,..., } for which f j = B ad f j = A. Accordig to Eq. 4, we ca always achieve a larger value by swappig f j ad f j. Lemma 7: Cosider the chaels W i for i =,..., resultig from = log steps of chael combiig ad splittig with matrix G 2 over a BEC with mutual iformatio IW. Cosider the biary label of each chael b i,...,b i, its correspodig Hammig weight w H b i,...,b i, ad defie S k = { i =,..., w H b i,...,b i = k } 5 to be the set of chael idices such that the Hammig weight of its biary label is equal to k for i =,..., ad k =,...,. Defie The, i k arg max i S k I W i. 6 b i k,...,b i k =,...,, 0,...,0. 7 }{{}}{{} k k I words, the above result states that for a fixed Hammig weight k of the biary label of the chael idexi =,...,, placig the k oes i the most sigificat bit positios yields the maximum mutual iformatio over all such chaels. Proof: We otice that Ax =2x x 2 ad Bx =x 2 are the recursive formulas to calculate mutual iformatio of the bit chaels for a BEC of mutual iformatio IW. The, for i S k Lemma 6 states that I W i B k A k IW. 8 ad that chael i k achieves the boud 8 with equality. ow, assume the miimum distace d mi is such that d mi =2 k0. The, from the defiitio of the froze set i 8 ad Lemma 4 k 0 must be such that I <θ I. 9 W i k 0 W i k 0 ote that, for a fixed threshold θ, k 0 is the miimum resp. maximum iteger solutio of the upper boud resp. lower boud i 9. Theorem 2: Cosider a BEC with mutual iformatio IW = ɛ. The miimum distace d mi =2 k0 of fixedthreshold polar codes of legth ad threshold θ is such

4 θ = 2 β, β = 2 5 Fixed rate R =0.35 Fixed rate R =0.49 d mi dmi I W = 8 I W = 2 I W = 7 8 Fig. 3. Miimum distace of fixed-threshold polar codes with θ = 2 β with β = 2 over the BEC. 5 that k 0 ca be bouded as k 0 log 2 log2 c c 2 c 0, 20 k 0 log 2 log2 c c 0, 2 where c 0 = log 2 { log 2 ɛ},c = l θ,c 2 = l ɛ. As opposed to the origial polar codes [], where the miimum distace ca oly be computed oce the froze set is determied, the bouds give i Theorem 2 are give as closed-form expressios ad are very simple to calculate. I Fig. 3 we show the miimum distace bouds of fixedthreshold polar codes with threshold fuctio θ = 2 β with β = 5 i a BEC chael with IW = 7 8, 2 ad 8. The dotted lies correspod to the actual miimum distace. Crosses resp. circles correspod to poits where the upper resp. lower boud gives the actual d mi of the code. As we ca see from the figure, the bouds are tight i the sese that either the upper or lower boud gives the actual d mi of the code. Fig. 4 compares the actual miimum distace of fixedthreshold polar codes with that of Arıka s fixed-rate polar codes. By checkig the rate-covergece i Fig. 2, we observe that for fixed-threshold polar-codes, the slower the covergece to IW, the higher the miimum distace. We also ote that the regios where the miimum distaces of fixedthreshold ad fixed-rate polar codes coicide, is the regio where their correspodig rates are close. I this case, both costructios would give approximately the same code. A. Proof of Lemma 5 IV. PROOFS Cosider first the upper boud i 9 ad let θ = α 2 2 β 2 with 0 <β 2 < 2. For I W i θ, usig [, Eq. 2] ad x 2 2 x gives Z W i 2α2 2 2β ɛ Fig. 4. Miimum distace d mi for Arıka s fixed-rate polar codes with R =0.35, 0.49 ad for fixed-threshold polar code with threshold fuctio θ = 2 β with β = 2 5 for a BEC with IW = 2. There exists a such that, for >, we ca fid 0 < β < 2, that 2α 2 2 2β 2 2 β. 23 The above implies that Pr I W i θ Pr Z W i 2 β. 24 Similarly, for I W i <θ, we use l + x x ad [, Eq. ] to show that Z W i α 2 2 β 2 > log 2 e. 25 There exists a 2 such that for > 2, we ca fid 0 < β < 2, that ad hece Pr I W i <θ α 2 2 β 2 log 2 e Pr 2 β, 26 Z W i > 2 β. 27 Accordig to the fixed-threshold costructio, i {,...,} : I W i [θ, ] R =. 28 We also have that i {,...,} : I W i [0,θ R + =. 29

5 Takig its i {,...,} : I W i R [θ, ] = 30 i {,...,} : Z W i 2 β 3 = IW 32 where 30 follows from 24 ad the last step follows from Lemma 2. Similarly, i {,...,} : I W i [0,θ 33 i {,...,} : Z W i > 2 β 34 = IW. 35 Therefore, we have that R = IW as desired. Covergece of the lower boud is proved similarly ivokig Lemma 3. Give that both upper ad lower bouds o θ are such R = IW, ay threshold fuctio that for large eough ca be bouded as 9 will have that the correspodig R will coverge to IW. B. Proof of Theorem First, we otice that for a give chael W ad block legth, the smaller θ, the higher the rate, ad the larger the error probability P e,θ. Therefore, we cocetrate o the lower boud ad prove that for θ = γ β,β > 2 that P e,θ =0. 36 Usig [, Eq. 2] ad x 2 2 x get that Z Thus, W i P e,θ Sice β > 2, we have that 2γ β i F c Z W i max Z i F c, 38 W i, 39 2γ β 2, γ β 2 =0, 42 ad hece P e,θ =0. 43 C. Proof for Theorem 2 We prove the upper boud. Cosider b i k,...,b i k =,...,, 0,...,0. 44 }{{}}{{} k k To fid d mi =2 k0 we eed to fid the smallest k such that I W i k θ. For the BEC, we have that I W i k = ɛ 2k 2 k 45 where we have defied ɛ = IW. If we take logarithms we have that k must be such that 2 k l ɛ 2k l θ. 46 It ca be show that l ɛ 2k cɛ 2k 47 where c = l ɛ ɛ, ad that I W i k give i 45 oegative ad icreasig i k. This implies that the miimum k that solves 46 is upper bouded by the miimum k that solves which is i tur the same that solves where ξ 2 k cɛ 2k l θ, 48 2 k k log 2 ɛ + log 2 ξ log 2 ɛ l θ c. Further takig logarithms we have that 49 k log 2 k log 2 ξ log 2 log 2 ɛ. 50 Observe that upper boudig the R.H.S. of 50 by log 2 log 2 ξ log 2 log 2 ɛ implies a upper boud to the solutio for k. Hece, k = log 2 log 2 ξ log 2 log 2 ɛ 5 is a iteger solutio of 50, provig the upper boud. The lower boud is proved similarly, usig that l x x for x 0 to boud the L.H.S. of 47. ACKOWLEDGMET The authors thak Erdal Arıka for helpful discussios ad meetigs facilitated by the FP7 etwork of Excellece EWCOM# ad its predecessor EWCOM++. REFERECES [] E. Arıka, Chael polarizatio: A method for costructig capacityachievig codes for symmetric biary-iput memoryless chaels, IEEE Tras. If. Theory, vol. 55, o. 7, pp , July [2] S. B. Korada, Polar codes for chael ad source codig, Ph.D. thesis, Ecole Polytechique Fédérale de Lausae, Switzerlad, [3] E. Arıka ad E. Telatar, O the rate of chael polarizatio, i IEEE It. Symp. If. Theory, Jue 2009, pp [4] S. H. Hassai, R. Mori, T. Taaka, ad R. Urbake, Rate-depedet aalysis of the asymptotic behavior of chael polarizatio, IEEE Tras. If. Theory, ov [5] R. G. Gallager, Iformatio theory ad reliable commuicatio, Wiley, 968. [6] R. Mori ad T. Taaka, Performace of polar codes with the costructio usig desity evolutio, IEEE Commu. Letters, vol. 3, o. 7, pp , Jul

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