Relaxed Channel Polarization for Reduced Complexity Polar Coding

Relaxed Channel Polarization for Reduced Complexity Polar Coding Mostafa El-Khamy, Hessam Mahdavifar, Gennady Feygin, Jungwon Lee, and Inyup Kang Modem R&D, Samsung Electronics, San Diego, CA 92121. {mostafa.e, h.mahdavifar, gennady.f, jungwon2.lee, inyup.kang}@samsung.com Abstract—Arıkan’s polar codes are proven to be capacity-achieving error correcting codes while having explicit constructions. They are characterized to have encoding and decoding complexities of l log l, for code length l. In this work, we construct another family of capacityachieving codes that have even lower encoding and decoding complexities, by relaxing the channel polarizations for certain bit-channels. We consider schemes for relaxing the polarization of both sufficiently good and sufficiently bad bit-channels, in the process of channel polarization. We prove that, similar to conventional polar codes, relaxed polar codes also achieve the capacity of binary memoryless symmetric channels. We analyze the complexity reductions achievable by relaxed polarization for asymptotic and finite-length codes, both numerically and analytically. We show that relaxed polar codes can have better bit error probabilities than conventional polar codes, while having reduced encoding and decoding complexities.

I.

I NTRODUCTION

Polar codes, introduced by Arikan [1], [2], are the first family of codes with explicit construction that asymptotically achieve the capacity of binary input symmetric discrete memoryless channels, as the block length goes to infinity. Polar codes can be encoded and decoded with relatively low complexity. Both the encoding complexity and the successive cancellation (SC) decoding complexity of polar codes are O(l log l), for code length l [1]. Reduction in decoding latency of SC decoding can be observed through interleaved concatenation of shorter polar codes [3]. The decoding latency and memory requirements of polar decoders can be reduced to O(l) with a semi-parallel architecture for SC decoding [4], where efficiency is achieved without a significant throughput penalty by sharing processing resources and taking advantage of the regular structure of polar codes. Latency and algorithmic reductions for polar decoding of Arikan’s polar codes were observed by simplifying the decoder to decode all bits in a rate-one constituent code simultaneously [5], [6]. The encoding and decoding latencies of polar codes can also be reduced to O(l) through multi-dimensional polar transformations [7]. In this paper, we propose methods to reduce both the encoding and decoding computational complexities of polar codes, by means of relaxing the channel polarization. The resultant codes are called relaxed polar

codes. In practical scenarios, codes have finite block lengths and are designed with a specific information block lengths and rate in order to satisfy a certain error rate. Due to the nature of channel polarization, the error probability of certain bit-channels decrease (or increase) exponentially at each polarization step. Hence, the encoding and decoding complexities may be reduced, without performance degradation, by stopping the further polarization of certain bit-channels if their polarization degrees hit suitable thresholds. For Arıkan’s polar code with length l, each bit-channel is polarized log l times. However, for the proposed relaxed polar codes, some bit channels will be fully polarized log l times, and the polarization of the remaining bit-channels will be relaxed, where their polarization is aborted if they become sufficiently good or sufficiently bad with less than log l polarization steps. The obvious advantage of relaxed polarization is savings in both encoding and decoding computational complexities, since there will be no channel transformations done at relaxed nodes while encoding, and there will be no need to calculate new likelihood ratios (LRs) at relaxed nodes while decoding. Another advantage of relaxed polarization is the reduction in space (area and memory) complexity in practical implementations. That is because decoding of relaxed polar codes will result in smaller likelihood ratios, or smaller log-likelihood ratios (LLRs) in case the computation is done in the log domain, than that for fully polarized polar codes. Hence, smaller bit-width will be required for LR calculation and storage. Relaxed polarization has the effect of reducing the number of processing nodes required at lower levels of the polarization tree. Also, by appropriate permutation of the bit channels, one expects to be able to eliminate the wiring for the wider butterflies in FFT-like SC decoders for relaxed polar codes. The reduction in bit-width and number of processing nodes required for relaxed polar code decoders has the compound effect of reduction in power consumption. Hence, with relaxed polarization, one can expect even more efficient implementations (or less throughput penalty) with semi-parallel hardware architectures [4], as well as reduced latency with multi-dimensional transformations [7]. Moreover, it is found that with careful construction of relaxed polar codes, there is no error performance degradation and that relaxed polar codes can have a

lower bit-error rate than conventional polar codes with the same rate. II.

R ELAXED C HANNEL P OLARIZATION

In this section, we define relaxed polarization. We prove that, similar to conventional polar codes, relaxed polar codes can asymptotically achieve the capacity of a binary memoryless symmetric (BMS) channel. We also prove that the bit-channel error probability of relaxed polar codes is at most that of conventional polar codes, and without rate-loss. For a conventional polar code of length l, a channel f (i) W goes through n = log l polarization levels. Let W 2t denote the ith bit-channel of a relaxed polar code at (i) polarization level t. Similarly, W2t is the corresponding bit-channel for a conventional polar code. Define the set of good bit-channels of a relaxed polar code according to the channel W and a positive constant β < 1/2 as n o f , β) def f (i) ) < 2−lβ/l , (1) Gel (W = i ∈ [l] : Z(W l

where [l] , {1, 2, . . . , l}. Similarly, let Gl (W, β) be the set of good bit-channels of conventional polar codes. For a relaxed polar code, consider two independent f (i) copies of a parent bit-channel W 2t to be polarized into two descendent bit-channels at level t+1, corresponding to a code of length 2t+1 , via the following channel transformation     (2i−1) f (2i) (i) f f (i) f (2) → , W W W , W 2t 2t 2t+1 . 2t+1 (i)

f t at level t < n be Let the polarized channel W 2 sufficiently good, such that it already satisfies the definition of a good channel at the target length 2l , i.e. its −lβ f (i) /l for Bhattacharyya parameter (BP) Z(W 2t ) < 2 1 a positive constant β < /2. Then, the idea of relaxed polarization is to stop further polarization of this good channel, and the corresponding node in the polarization tree is called a relaxed node. The channels of all the descendents of a relaxed node are the same as that of the relaxed parent node and will also be relaxed. The ith bit channel is the channel taking the ith bit at the input, while assuming perfect knowledge of previous bits indexed by 1, 2, . . . , i − 1. The bit-channel transformation at the relaxed node is given by 1 t+1

f (2i−1) W (y12 2t+1

, u2i−2 |u2i−1 ) = 1

t+1

f (2i) (y12 W 2t+1

|u2i ) = , u2i−1 1

t

f (i) W (y12 , u2i−2 1,o |u2i−1 ) 2t t+1

f (i) (y22t +1 , u2i−2 W 1,e |u2i ). 2t

Relaxing the further polarization of sufficiently good channels is called good-channel relaxed polarization 1 Let un n 1,e and u1,o denote the even-indexed and odd-indexed elements of the vector un 1 = (u1 , u2 , . . . , un ), respectively. For any BMS channel W : X → Y , let W (y|x) denote the probability of receiving y ∈ Y given that x ∈ X = {0, 1} was sent, for any x ∈ X and y ∈ Y .

(GC-RP). The next theorem shows that GC-RP codes asymptotically achieve the capacity of BMS channels. Theorem 1: For any BMS channel W , with capacity f ,β)| |Gel (W C(W ), and any constant β < 1/2, liml→∞ = l C(W ). Proof: Consider a relaxed channel at level t < n, −lβ f (i) where n = log l. Then Z(W /l. Then the 2t ) < 2 BPs of all its 2n−t descendents at level n are equal to f (i) e f Z(W 2t ) and are in Gl (W , β). If a channel belongs to Gl (W, β) in the conventional code, then it must have polarized to a good channel at level n or earlier. If it polarized at level n, then by definition it also belongs f , β). Otherwise, its parent has polarized to a to Gel (W good channel at level t < n. With relaxed polarization, this channel and all its 2n−t − 1 siblings will also be in f , β). Therefore, Gl (W, β) ⊂ Gel (W f , β) and hence Gel (W e f |Gl (W , β)| ≥ |Gl (W, β)|. Since, for conventional polar codes liml→∞ |Gl (W, β)|/l = C(W ) [2], the theorem follows. Hence, similar to conventional polar codes, the frame error rate (FER) of relaxed polar codes can be β shown to be upper bounded by 2−l . Next, we compare the bit-error rate of a relaxed polar code with that of Arıkan’s polar code. The error probability of a BMS channel W with uniform input is defined by 1 X E(W ) = min{W (y|0), W (y|1)}. (3) 2 y∈Y

We can prove the following inequalities for a polarizing (i) bit-channel Wl/2 , 2 2      (i) (2i−1) (i) (4) E Wl = 2E Wl/2 − 2E Wl/2    2 (2i) (i) E Wl ≥ 2E Wl/2 . (5) Theorem 2: Consider a GC-RP code with goodf , β), and let Gl (W, β) be the good channel set Gel (W channel set of the fully polarized polar code. If f , β) = Gl (W, β), then the bit error probability Gel (W of the relaxed polar code is at most equal to that of P f (i) the fully polarized code, i.e. f ,β) E(Wl ) ≤ i∈Gel (W P (i) i∈Gl (W,β) E(Wl ) (i)

Proof: Consider one channel Wl/2 that is relaxed at f , β) = Gl (W, β), the last polarization level. Since Gel (W the indices 2i − 1 and 2i of its children are contained in the good channel sets of both codes. For the relaxed code, it follows that probability  sum error    of f (2i−1) + E W f (2i) = these two channels is E W l  l    (i) (i) f 2E Wl/2 = 2E Wl/2 . Together with summing (4) 2 In the rest of this paper, some proofs will be eliminated or abbreviated for clarity of presentation and page limitation.

    f (i) ≤ E W (2i−1) + and (5), it follows that 2E W l l/2   (2i) E Wl . Hence, the sum error probability of both channels in the relaxed code is at most their sum in the fully polarized code.

III.

R ELAXED P OLAR C ODE C ONSTRUCTIONS

In general, finite block length polar codes are constructed by fixing either a target FER E or target code rate R. We consider constructions of polar codes with code length l = 2n , at a target FER of E. A. Constructions for Erasure Channels f (i) To simplify notation, let the BP Zi,t , Z(W 2t ). At finite block lengths, we need to specify certain thresholds for the BPs in order to establish criteria for good-channel and bad-channel relaxed polarization. We proposed the following relaxed polarization schemes: 1) GC-RP: Node i at polarization level t is not further polarized if Zi,t < Tg 2) BC-RP: Node i at polarization level t is not further polarized if Zi,t > Tb 3) All-Channel Relaxed Polarization (AC-RP): Node i at polarization level t is not further polarized if Zi,t < Tg or Zi,t > Tb where Tg and Tb are the good and bad relaxation thresholds, respectively. Let Γ be the set of good bit-channels which are used to transmit the information bits. Then GC-RP codes satisfy the target FER E, if Tg = 2E/l. This is explained P by the fact that |Γ| ≤ l, which satisfies that FER ≤ i∈Γ Zi,n /2 ≤ |Γ| E/l ≤ E. In case of BC-RP, the bad channels are not further polarized if they become sufficiently bad, where the bad-channel relaxation threshold can be set to be Tb = 1 − Tg . To guarantee no rate loss from BC-RP, it should only log T be performed if n − t ≤ ⌈log2 log2 Tgb ⌉. The proposed 2 AC-RP relaxes the polarization of a bit-channel if it becomes either sufficiently good or bad. Since the bad bit-channels do not contribute to the FER, the target FER is still maintained with AC-RP. The complexity reduction R(n) is defined to be the ratio of the number of polarization operations that are

AC−RP Construction AC−RP Upper bound AC−RP Lower bound GC−RP Construction GC−RP Upper bound GC−RP Lower bound

0.9

0.8

0.7 Complexity Reduction

The concept of good-channel relaxed polarization, discussed so far, can be extended to bad-channel relaxed polarization (BC-RP) as follows. Consider the bit-channels in the polarization tree, at a level t < n, that are very bad. A bit-channel can be considered very bad if its Bhattacharyya parameter is very close to 1, or if none of its descendents will fall into the set of good bit-channels at the last level of polarization n. Hence, the polarization of very bad bit-channels can be stopped without affecting the final set of good bitchannels. Thus, with careful BC-RP, more complexity reductions can be possible, without degrading the code rate and error performance.

All−Channel Relaxed Polarization Bounds, Target FER =0.0001 1

0.6

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4 0.5 0.6 Channel Parameter p

0.7

0.8

0.9

1

Fig. 1. Complexity reduction by relaxed polarization on erasure channels with target FER of 10−4 , and code length 220 .

skipped due to relaxed polarization to the total number of polarization operations (A(n) = l log2 l) required for full polarization of a code of length l. The complexity reduction (CR) can be directly translated into encoding and decoding complexity ratios of (1 − R(n))−1 . In Fig. 1, the achieved complexity reduction ratio for a binary erasure channel with erasure probability p, BEC(p), is shown. It is observed that up to 85% CR is achievable, i.e. fully polarized (FP) code requires 6.6x more complexity than RP code. AC-RP will result in more complexity reduction than GC-RP as the channel becomes worse. The rate-loss is calculated as RLoss = RF P − RRP , where RF P and RRP are the rates of the codes which are constructed by full and relaxed polarization, respectively. The rate at a target FER E is calculated by aggregating the maximum number of bit-channels such that their accumulated BPs does not exceed 2E. The rate loss, for the constructions shown in Fig. 1, is less than 10−4 . Another important observation, from Fig. 1, is the symmetry of the AC-RP CR curve around p = 0.5. This is explained by the following Theorem 3, which can be proved by showing that the the polarization tree for BEC(p) and the polarization tree for BEC(1 − p) are isomorphic, where a node V with BP Z in the first tree is isomorphic to a node with BP 1 − Z in the second tree with one-to-one mapping by mirroring. Theorem 3: The complexity reduction with badchannel relaxed polarization of BEC(p) with threshold 1 − T is the same as that of good-channel relaxed polarization of BEC(1 − p) with threshold T . B. Constructions for General BMS Channels For general BMS channels, the Bhattacharyaa parameters are hard to compute at large block lengths. This is due to the exponential output alphabet size of the polarized bit-channels. Instead of exact calculation of BPs, they can be well approximated by bounding the output alphabet size of bit-channels via channel degrading and channel upgrading transformations [8]. The

channel degrading and upgrading operations provide lower bounds and upper bounds on the corresponding BPs. For AWGN channels, the BC-EP can also be reasonably approximated using density evolution and a Gaussian approximation [9]. Alternatively, for short codes, the BC-EP can be numerically evaluated using Monte-Carlo simulations, assuming a genie-aided SC decoder. For generality of description, let the EP of the j-th bit-channel at the t-th polarization level be bounded by P t,j ≤ Pt,j ≤ P t,j , (6) where P t,j and P t,j are the error probabilities of the corresponding upgraded and degraded channels respectively. In case the bit-channel error probabilty can be evaluated exactly or approximated by P˜t,j , then let P t,j = P t,j = P˜t,j . The choice of information set for fully polarized codes (FP) can be done according to an estimated BC-EP. For instance, the set Γ can be found by P accumulating as many good bit-channels, such that j∈Γ P n,j ≤ E, where E is the target FER. Then, the FP code is defined by Γ and has rate R = |Γ|/l.

In proposed GC-RP codes, a node will not be further polarized if the upper bound on its bit-channel error probability is lower than a certain threshold Eg . For BC-RP codes, a bit-channel will not be further polarized if the lower bound on its BC-EP exceeds an upper threshold Eb . The procedure for designing relaxed polar codes of length l at a target FER E on general BMS channels is specified in Algorithm 1. Each node j at level t in the polarization tree is associated with a label Relaxed(t, j) which is initialized to 0, and will be set to 1 only if this node will not be polarized. The BC-EP of each node in the relaxed-polarization tree is initialized to that of the full-polarization tree R P t,j = P t,j . With target FER E, the good-channel relaxation threshold is chosen P to be Eg = E/(Rl) to satisfy the target FER, i.e. j∈ΓR P n,j ≤ E. Let H(EW ) be the entropy of the channel W with fidelity EW , such that H(EW ) = 1−I(W ), where I(W ) is capacity of channel W . Then, the bad-channel relaxation threshold is chosen such that H(Eb ) = 1 − H(Eg ). For general BMS channels W with error probability EW , the approximation H(W ) ≈ h2 (EW ) can be used, based on the inequality H(W ) ≤ h2 (EW ) [1], [10], where h2 (ǫ) = −ǫ log2 (ǫ) − (1 − ǫ) log2 (1 − ǫ) is the binary entropy function. To guarantee that there is no rate loss from bad-channel relaxation if P t,j > Eb is satisfied at Step 8, then relaxation may only be done after verifying that the lower bound on the error probability of the best upgraded descendent channel of that node is still higher than Eg , which will be satisfied for practical frame errors E. The same procedure above can be used to construct GC-RP codes, by neglecting the  bad-channel relaxation condition P t,j > Eb in step 8.

Algorithm 1 Relaxed Construction for General BMS Channels 1: Stage 1: Calculate AC-RP bit-channel EP for target FER E and rate R 2: Set the GC relaxation threshold as Eg = E/(Rl) 3: Set the BC relaxation threshold as Eb = H −1 (1 − H(Eg )) 4: for t = 1 : n do 5: for j = 1 : 2t do 6: if Relaxed(t − 1, ⌈j/2⌉) = 1 then R 7: Relaxed(t, j) = 1, P t,j = R P  t,⌈j/2⌉  8: else if P t,j < Eg or P t,j > Eb then 9: Relaxed(t, j) = 1 10: end if 11: end for 12: end for 13: Stage 2: Construct AC-RP code with rate R R 14: Sort bit-channel EPs P t,j in ascending order 15: Select ΓR to have the Rl bit channels with smallest EP The complexity reduction achievable is analyzed by actual construction of the relaxed polar codes on AWGN channels. Up to 0.37 complexity reduction was observed on an AWGN channel with binary-input capacity C = 0.7. IV.

C OMPLEXITY R EDUCTION A NALYSIS

First, we consider the asymptotic analysis, where a family of capacity-achieving polar codes is assumed which is constructed with respect to a fixed parameter β < 1/2 and the set of good bit-channels Gl (W, β), for any block length l = 2n . Theorem 4: Let C be the capacity of the channel W . Then, for any ǫ > 0, small enough δ > 0, and large enough l, the complexity reduction ratio using the f , β), is at least GC-RP code, constructed with Gel (W
(C − ǫ) 1 − (2 + δ)β . In case of bad-channel relaxed polarization, we assume if none of the descendants of a certain node will belong to Gl (W, β), then the polarization at that node, and consequently all of its descendants, will be relaxed. Hence, we have the following upper bound. Theorem 5: For any ǫ, δ > 0 and large enough l = 2n , the complexity reduction ratio using the bad-channel relaxed polarization is at least (1 − C − ǫ)(1 −

(2 + δ) log n ). n

For asymptotically large n, and neglecting ǫ and δ, the complexity reduction ratio from good-channel and

bad-channel relaxed polarization is C(1−2β) and 1−C, respectively, which are both positive constant factors. The ratio of saved operations with AC-RP approaches 1−2βC. Hence, relaxed polarization can provide a nonvanishing scalar reduction in complexity, even as the code length grows infinitely. Next, we analyze the CR at finite block lengths. For simplicity, the analysis is done for an erasure channel BEC(p). The next two theorems, show a lower bound and an upper bound for GC-RP, respectively. Theorem 6: The good-channel relaxed polarization complexity reduction with erasure channel probability p is upper bounded by Rg (n) ≤ (n − tg )/n, m log T where tg = log2 log2 pg . l

(7)

γ ′ 2−tr ((n − tr )(n − tr − 2t + 1) + t(t − 1)) . 2n

The parameter γ ′ in Theorem 7 approaches C/2. In the above lower bound, a necessary condition is that t + 2tr > n to guarantee that no double counting occurs. Bounds for BC-RP can be proved independently or established by combining bounds on the GC-RP results with the symmetry result of Theorem 3, by replacing p with 1 − p in the bounds, and modifying other parameters accordingly. For a function F (p), let F t (p) denote the output from applying the function F recursively t times, with initial input p. CR from AC-RP can be analyzed lby combining m results on GC-RP and BC-RP. log T Let tg = log2 log2 pg and tb = mint {F t (p) > 1−Tg } 2 m l log Tg . for F (p) = 2p − p2 . Then, tb = log2 log 21−p 2 Observe that tb < tg if p > 0.5, else tb > tg if p < 0.5, and tb = tg if p = 0.5. Then, we have the following upper bound on AC-RP complexity reduction. Corollary 8: The all-channel relaxed polarization complexity reduction is upper bounded by Ra (n) ≤ (n − t′ )/n, ′

Rb (n) ≥

β ′ 2−tl ((n − tl )(n − tl − 2t + 1) + t(t − 1)) . 2n

Similar to Theorem 7, the above theorem holds under the condition that t + 2tl > n. We have observed that with the thresholds B = 2p − p2 and 1 − G = p2 , γ ′ of Theorem 7 and β ′ of Theorem 9 can be well approximated by (1 − p)/2 and p/2, respectively. To make notations consistent, let t′γ denote the level t in Theorem 7 and t′β denote the level t in Theorem 9.

2

For any polarization level t and some threshold B, let GB,t denote the set of bit channels at polarization level t with BP at most B i.e. GB,t = {i : Zi,t ≤ B}. l m 2 B−1 Theorem 7: Let tb = log2 loglog + 1, for 2p m l log T an arbitrary threshold B, and tr = log2 log2 Bg . 2 For a polarization level t ≥ tb , let γ ′ = mint≥tb | {i odd : i ∈ GB,t } |/2t . Then the GC-RP complexity reduction is lower bounded by Rg (n) ≥

l m log2 G−1 Theorem 9: Let tc = log2 log + 1, for an (1−p) 2 arbitrary threshold G. For a polarization level t ≥ t tc , letl β ′ = mint≥t m c | {i odd : Zi,t ≥ 1 − G} |/2 . Let log2 Tg tl = log2 log G , then the CR from BC-RP is lower 2 bounded by

(8)

′

where t = tg for p ≤ 0.5 and t = tb for p > 0.5, Lower bounds on BC-RP and AC-RP are shown below, where the good-channel and bad-channel relaxation thresholds should be chosen carefully, as described in Section III.

Corollary 10: The all-channel relaxed polarization is lower bounded by  β ′ 2−tl  2 (n − tl )(n − tl − 2t′β + 1) + t′β − t′β 2n  γ ′ 2−tr  2 + (n − tr )(n − tr − 2t′γ + 1) + t′γ − t′γ . 2n

Ra (n) ≥

The bounds of this section were evaluated and compared in Fig. 1 to the actual complexity reduction achievable by construction of relaxed polar codes with length l = 220 . The bounds well approximate the actual CR. They are minimized at p = 0.5, and symmetric around p = 0.5, which can be explained by the symmetry property of Theorem 3.

V.

D ECODING

OF RELAXED POLAR CODES

Decoding of relaxed polar codes can be done by a modified successive cancellation (SC) decoder. For a polar code of length l and BMS channel W , suppose that ul1 is the input vector and y1l is the received word. Consider a relaxed polar code constructed by Algorithm 1. At each level t = log l, for 1 ≤ i ≤ l, f (i) is not polarized and Relaxed(t, i) = 1 means that W l f (i) is fully polarized. Relaxed(t, i) = 0 means that W l In practical application of relaxed polar codes, the decoder will have prior knowledge of the polarization map Relaxed(t, i), which requires at most storage of 2l bits. For communication systems, the polarization map can be specified by the communication standard, similar to the specification of the parity-check matrices of block codes. For i = 1, 2, . . . , l, the SC decoder can computes (i) the likelihood ratio (LR) Ll of ui , given the channel outputs y1l and previously decoded bits u ˆi−1 recursively 1 (i) [1]. Hence, if Relaxed(t, i) = 0, the LR Ll can be

computed recursively as follows. (2i−1)

Ll

(9)

−1

10

(i) l/2 (i) 2i−2 l ,u ˆ2i−2 )Ll/2 (yl/2+1 ˆ1,o ˆ2i−2 1 + Ll/2 (y1 , u 1,e ) 1,e ⊕ u , (i) l/2 (i) 2i−2 2i−2 2i−2 l ˆ1,e ) ˆ1,o ) + Ll/2 (yl/2+1 , u ˆ1,e ⊕ u Ll/2 (y1 , u

(2i)

Ll

(y1l , u ˆ2i−2 ) 1

h

(y1l , u ˆ2i−1 ) 1 (i)

l/2

ˆ2i−2 ˆ2i−2 = Ll/2 (y1 , u 1,e ⊕ u 1,o )

i1−2ˆu2i−1

−2

10 Error Rate

=

(10)

(2i−1)

(i)

l/2

ˆ2i−2 ) = Ll/2 (y1 , u (y1l , u ˆ2i−2 1,o ) 1

(2i) Ll (y1l , u ˆ2i−1 ) 1

=

−4

10

−5

10

FER FP FER GC−RP FER AC−RP BER FP BER GC−RP BER AC−RP

−6

(11)

10

1

1.5

2

2.5

3

3.5

SNR (dB)

(i) l Ll/2 (yl/2+1 ,u ˆ2i−2 1,e ).

(12)

(1)

At the last stage when l = 1, the LRs are L1 (yi ) = W (yi |0)/W (yi |1). At the end, hard decisions are made (i) (i) on Ll , except for frozen bit-channels Wl where uˆi = ui = 0. Hence, l calculations for 1 + log l levels are required for decoding conventional FP polar codes. However, the decoding complexity of relaxed polar codes is linearly reduced by the ratio of relaxed nodes in the polarization tree. The error-rate performance by actual SC decoding of the RP codes on AWGN channels, constructed with Algorithm 1 is shown in Fig. 2. The FP good-channel set is optimized for the FP polar code. The GC-RP and ACRP codes have similar performance, with complexity reductions of 19.4% and 23.7%, respectively. For this example, RP codes have a better bit error rate (BER) than the FP code, which is justified by Theorem 2. Another important observation is that the proposed construction of relaxed polar codes is robust enough, such that it performs well over the whole range of simulated signal to noise ratios (SNRs), although the codes are constructed at one SNR. VI.

−3

10

(i)

l ,u ˆ2i−2 Ll/2 (yl/2+1 1,e ).

Otherwise, for a relaxed noded with Relaxed(t, i) = 1, the decoding is less complex as follows. Ll

n=12, R=0.554, CR(GC−RP)=0.194, CR(AC−RP)=0.237

0

10

C ONCLUDING R EMARKS

A new paradigm for polar codes, called relaxed polarization (RP), is introduced, where a bit-channel will not be further polarized if it has been already polarized to be sufficiently good or sufficiently bad. Hence, encoding and decoding of RP codes have lower computational and time complexities than those of conventional polar codes. We prove that, similar to conventional polar codes, RP codes are capacity achieving on BMS channels. Bounds on complexity reduction by RP are verified against actual constructions. Our asymptotic analysis shows a complexity reduction of 1 − 2βC for infinite block lengths on a channel with capacity C, and a positive constant β < 1/2. Compared to those of polarized polar codes, relaxed polar encoders and decoders will have less area, less power consumption, and less latency, with reductions proportional to the complexity reductions. The exact reductions in chip area, power consumption, and latency will depend on the specific hardware architecture. We also prove that

Fig. 2. RP code has length 212 and rate 0.55, and designed at C = 0.7 and E = 0.01. Error performance, for BPSK on AWGN channel with variance σ2 , versus SNR= 10 log10 (1/σ2 ).

RP codes can have lower information bit error rates than conventional polar codes of the same rate, which is also verified by numerical simulations of SC decoding of RP codes on AWGN channels. RP can also be applied to reduce complexity of compound polar codes [11]. R EFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

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