Fifty-second Annual Allerton Conference Allerton House, UIUC, Illinois, USA October 1 - 3, 2014
Concatenations of Polar Codes with Outer BCH Codes and Convolutional Codes Ying Wang and Krishna R. Narayanan Department of Electrical and Computer Engineering Texas A&M University {
[email protected] and
[email protected]}
Abstract—We analyze concatenation schemes of polar codes with outer binary BCH codes and convolutional codes. We show that both BCH-polar and Convolutional-polar (Conv-polar) codes can have frame error rate that decays exponentially with the frame length, which is a substantial improvement over standalone polar codes. With the increase in the cutoff rate of the channel after polarization, long constraint-length convolutional codes with sequential decoding suffice to achieve a frame error rate that decays exponentially with the frame length, whereas the average decoding complexity is low. Numerical results show that both BCH-polar codes and Conv-polar codes can outperform stand-alone polar codes for some lengths and choice of decoding algorithms used to decode the outer codes. For the additive white Gaussian noise channel, Conv-polar codes substantially outperform concatenated Reed Solomon-polar codes with a careful choice of the lengths of inner and outer codes. Index Terms—polar code, cutoff rate, concatenation, BCH, convolutional code, puncture, sequential decoding.
I. I NTRODUCTION Polar codes are known for their ability to achieve the capacity of many symmetric discrete memoryless channels with explicit construction and decoding methods [1]. However, the error performance of polar codes in the finite length regime is not good [2]. One reason is that the Successive Cancellation (SC) decoder is weak and is susceptible to error propagation. Another reason is that the codes themselves are weak, which can be seen from the low minimum distance of the code [3] and the poor error decay rate with length [4]. One way to improve the finite length performance of polar codes is to improve the decoding method. Belief propagation (BP) decoding for polar codes has been proposed in [3], but the improvement is not significant. SC list decoding can provide substantial improvement over SC decoding [5]. However, the scaling exponent cannot be improved with any finite list size [6], and both the memory space and the decoding complexity increase with the list size. Another way to improve the finite length performance is to change the structure of the code, such as concatenation with other codes or exploiting nonbinary construction scheme. Several such techniques have been proposed in the literature. For instance, the concatenation of polar codes with cyclic redundancy check can even outperform some Low-Density Parity-Check (LDPC) codes under list decoding. A new family of codes that interpolates between This work was supported by the Qatar National Research Foundation under grant NPRP 5-597-2-241.
978-1-4799-8009-3/14/$31.00 ©2014 IEEE
polar codes and Reed Muller codes is introduced in [7]. The concatenation of polar codes with RS codes proposed in [8] is shown to increase the error decay rate of the code to be almost exponential in the length but the field size of RS codes would increase exponentially with the length of the polar code. The concatenation of polar codes with short block codes is proposed in [9] by usage of the frozen bits which preserves the code rate. However the performance is not substantially improved, as repetition or Hamming codes are not strong enough to provide adequate protection. The concatenation of polar codes with LDPC codes is proposed in [10], showing that with a modified BP decoding, a better performance can be achieved. In [11] and [12] concatenation of polar codes with an interleaved Reed-Solomon code is proposed. The interleaver is designed in such a way that after interleaving, the equivalent channels experienced by signals with a same index in the inner polar codes are independent and identically distributed (i.i.d.). One can then aggregate those bit channels with a same index and employ an outer code on top of the inner polar code. In [11], Mahdavifar et al. show that the concatenation of polar codes with RS (referred to as RS-polar) codes can increase 1− the error decay rate to be O(2−N ), where N is the frame length of the code. In [12], Trifonov and Semenov view the concatenation schemes as multilevel coding and multistage decoding schemes and apply the design criteria in [13] to the problem considered. They then provide some examples where empirical results for designs with outer BCH codes (referred to as BCH-polar in this paper) are presented. In this paper, we extend the above results in the following ways, which we believe are new results. Firstly, we provide an analysis of BCH-polar codes and show that for these codes also, the error rate decays exponentially with the frame length. Secondly, we propose the concatenation of binary convolutional codes with polar codes (referred to as Conv-polar) and we show that this scheme can also achieve an exponential frame error decay rate, while the decoding complexity remains small. In fact, depending on the constraint length and decoding algorithm used for convolutional codes, one can trade-off performance for complexity smoothly. Finally, we provide a comparison between the performances of BCH-polar and Conv-polar codes with the stand-alone polar codes and RS-polar codes. We show that both codes can outperform stand-alone polar codes. For moderate lengths, when decoding with a maximum likelihood
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decoding algorithm, the BCH-polar code outperforms standalone polar codes for the binary erasure channel. For the additive white Gaussian noise (AWGN) channel, Conv-polar codes can substantially outperform both stand-alone polar codes and RS-polar codes. The proposed schemes are based on two important insights. Firstly, for RS-polar codes, the SC decoder works best at the bit level whereas the RS code is designed over a non-binary field resulting in a mismatch with the SC decoder. This results in an inefficiency in assigning rates of the outer codes. The use of binary codes provides a more natural way to assign different rates to different bit channels. Secondly, since the polarization process increases the cutoff rate of the channel [14], by using convolutional codes with rate smaller than the cutoff rate, one can obtain excellent error performance with a very low complexity sequential decoder. The rest of the paper is organized as follows. In Section II, we review background material on polar codes, the interleaved structure, and the cutoff rate. The general rate optimization scheme of the concatenated codes is given in section III. In Section IV, BCH-polar codes are discussed and detailed analysis of error probability is provided. The novel Conv-polar codes are proposed in Section V along with some discussion including analysis of the error probability. In Section VI, numerical results are given and Section VII concludes the paper. II. BACKGROUNDS In this section, we review the channel polarization process and the SC decoder. We then review the generalized concatenated framework which involves a Guess-Varanasi type interleaver. At the end of this section, we provide the definition of the channel cutoff rate which will be important for the discussion on the proposed Conv-polar codes. A. Polar codes Polar codes are generated by a channel combining and splitting process. Assume that the block length of the code is n, un1 are the bits to be encoded, xn1 are codedwords and y1n are the received bits. Let W (y|x) be the channel transition probability. First n copies of the channel are combined to create the channel Wn (y1n |un1 ), which is define as Wn (y1n |un1 ) = W n (y1n |un1 Gn ),Qwhere Gn is the generator n matrix, and W n (y1n |xn1 ) = i=1 W (yi |xi ). The channel splitting process splits Wn back into a set of n bit channels: P (i) 1 n n Wn (y1n , ui−1 1 |ui ) = 2n−1 un Wn (y1 |u1 ), i = 1, · · · , n. i+1
(i)
Let I(W ) be the capacity of the channel. The channels Wn polarize in the sense that the fraction of indices for which (i) Wn → 1 goes to I(W ) and the fraction of indices for which (i) Wn → 0 goes to 1 − I(W ) [1]. The construction process is to find a set of best channels based on the Bhattacharyya parameters and transmit information only on those channels. The SC decoder computes the Log-Likelihood Ratio (LLR) of each bit channel n L(i) ˆ1i−1 ) = log n (y1 , u
(i) Wn (y1n , u ˆi−1 1 |ui (i) n Wn (y1 , u ˆi−1 1 |ui
= 0) = 1)
(1)
1
2
k
n 1st polar code 2nd polar code
mth polar code 1st outer code
2nd outer code
kth outer code
Fig. 1. Concatenation of polar codes with binary codes
(i)
With recursive formula in computing Ln , the decoding complexity is O(n log n). The error decay rate of polar codes is β− O(2−n ) where β is the error exponent [4] and 0 < < 1/2. For Arikan’s original polar codes, β = 1/2. For nonbinary polar codes, β can be greater than 1/2. B. Interleaved structure The structure of the generalized concatenated scheme is shown in Fig. 1. In this scheme, k outer codes with a same block length m are concatenated and sent into a write columnwise transmit row-wise interleaver. Each length-k block of the output of the interleaver is further encoded by an (n, k) polar code. Hence, each row in Fig. 1 is a polar code, whereas each column is a codeword of an outer code or a frozen vector. One can easily see that the transmit signal consists of m polar codes and the frame length is N = mn. The receiver employs multistage decoding where at stage i, given the decoding output from stages 1, . . . , i − 1, all the bits with the same channel index i of different polar codes are decoded in parallel using SC decoding. Then all these bits are immediately decoded by an outer code. This structure ensures that the errors made in the early stages are already corrected before decoding the current bit and error propagation is thus substantially reduced. C. Cutoff rate The cutoff rate Rc of the channel is defined as [15] 2 XX p Rc = max − log2 Q(k) P (j|k) Q
j
(2)
k
where Q is the input probability distribution, and P (y|x) is the transition probability of the channel. We use (no , ko , ν) to represent a convolutoinal code with ko input bits and no output bits each time, and the constraint length of the code is ν. It is known that if the rate of the code is less than Rc , the average bit error rate of the convolutional code with sequential decoding can be bounded as [16] Pb ≤ Ke−no νρRc
(3)
where K is a constant independent of ν and Rc , and 0 ≤ ρ < 1. This is asymptotically as good as ML decoding [17].
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III. R ATE O PTIMIZATION SCHEME For concatenated polar codes in general, due to the channel polarization process, different bit channels would have different quality so one will need a flexible choice over the rates of different outer codes to satisfy the desired condition (for example, the target block error rate). In the following, we discuss a rate optimization scheme based on the equal block error rate design rule. Let pi be the error probability of the i-th bit channel of an (n, k) inner polar code, i = 1, ..., k. The block error rate PBi of each outer code depends on the specific codes used and the channel condition, i.e., PBi is a function of the outer code rate Roi and pi . If we use PE to represent the frame error rate of the concatenated code, then PE can be computed as PE = 1 −
k Y
(1 − PBi )
(4)
it’s implementable for the BEC since it’s equivalent to do the matrix inversion. In the following, we show how to choose the rates of outer codes with both BD and ML decoding, and then provide detailed analysis of the frame error rate for such construction where exponential decay rate is proved. We also analyze the decoding complexity of the above codes. A. BD decoding over a general channel For a BCH code of length m = 2l − 1 that can correct up to t errors, it has a rate [18] R0 ≥ 1 −
PBi =
If the target block error rate of each outer code is set to be the same PB∗ , PE can be bounded as PE ≤ 1 − (1 −
≈
kPB∗
(5)
To maximize the overall rate of the code, Roi can be chosen to be the maximum value that satisfies the condition that PBi ≤ PB∗ . Then the overall rate R of the concatenated code is Pk bRoi · mc (6) R = i=1 mn If we want to design codes with a fixed overall rate R∗ , then the rates of inner and outer codes should be optimized to achieve the minimum frame error rate. The optimization algorithm is as follows: 1) Compute pi for each bit channel of the inner polar code with density evolution. 2) Set P1 = 1 and P2 = 0. 3) If |P1 − P2 | > , set the target block error rate PB∗ for each outer code to (P1 + P2 )/2. Otherwise go to 6). 4) For each bit channel, find the maximum Roi that satisfies PBi ≤ PB∗ . 5) Compute k as the number of outer codes with Roi > 0. Compute R based on (6). If R ≥ R∗ , set P1 = PB∗ , compute PBi and then compute PE based on (4). If R ≤ R∗ , set P2 = PB∗ . Return to 3). 6) Find out the minimum value PEmin among all PE , which is the minimum achievable frame error rate of the code. Note that this algorithm is a modified version of the algorithm in [12]. IV. BCH- POLAR CODES In this section, we consider BCH codes as the outer codes. BCH codes are a large class of binary codes within which we can find a t-error correcting code of length m for 0 ≤ t ≤ b m−1 2 c. We first consider BCH codes with Bounded Distance (BD) decoding. To further improve the code performance, ML decoding is analyzed over the Binary Erasure Channel (BEC). Although ML decoding is too complex for general channels,
(7)
Assume that the i-th outer code can correct up to ti errors. Then PBi is given by
i=1
PB∗ )k
tl m
m X m j pi · (1 − pi )(m−j) , i = 1, · · · , k j j=t +1
(8)
i
The rate of the outer code Roi can be chosen based on the corresponding ti , i.e., roughly speaking Roi ≥ 1 − tmi l (in practice, one can find an exact relationship between Roi and ti [19]). B. ML decoding over a BEC channel Let u ∈ {0, 1}k denote the information sequence and x ∈ {0, 1}m denote the codeword. Let A be the set of unerased positions in the received vector. Then GA is denoted as the matrix containing columns only corresponding to the un-erased positions in the generator matrix, and z A is denoted as the received vector without the erased bits. u can be derived by solving uGA = xA
(9)
If GA is full rank, then u can be uniquely solved by the matrix inversion, thus the complexity is polynomial O(m3 ). We use BEC(α) to denote an erased channel with erasure probability α. For ML decoding, the erasure probability of the code can be estimated via the weight spectra of the code. If the number of the weight i codewords is P Ai , then the m block error rate PB can be computed as PB ≤ i=dm Ai αi . For an [m, k, dm ] BCH code where m is the length of the code, k is the number of information bits and dm is the minimum distance, the weight distribution of the code can be approximated as binomial, i.e., Ai = mi /2m−k [20]. The extended BCH code with length m0 = m + 1 can increase dm by 1, i.e., dm = 2t + 2. Then PBi is derived as 0
PBi ≤
m X j=2ti +2
m0 j αi j 2m0 −k
(10)
where αi is the erasure probability of the ith bit channel of the polar codes.
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C. Error probability analysis The following theorem shows that the frame error rate of BCH-polar codes decays exponentially with the frame length. Theorem 1. In the BCH-polar scheme, for any ∈ (0, 0.5), the frame error rate of the concatenated code is upper bounded 1− by N 2−N , where N is the frame length of the code. Proof. We know that the block error rate of the (n, k) inner 0.5− polar code is O(2−n ) asymptotically. The bit error rate pi ≤ PB for all i = 1, · · · , k. The error correction ability ti (1−Roi )m from (7). of the outer code can be derived as ti ≥ log 2 (m+1) Then the block error rate of the i-th outer code can be derived as follows: m X m j PBi = pi (1 − pi )m−j (11) j j=ti +1 m (12) ≤ pti +1 ti + 1 i (1−Roi )m m −n0.5− log (m+1) 2 (13) ≤ 2 ti + 1 1−Roi −1)m 2 (m+1)
−(n0.5− log
=2 Notice that t +1 mH( im )
m ti +1
(14)
can be bounded by Stirling’s approximation
2 . Here it’s simply bounded as 2m . We can pick n = N , m = N 1− and Roi = 1 − 2N −(0.5−) log2 (N 1− + 1). Roi → 1 as N → ∞. Plugging these into (14), we can get PBi ≤ 2−N
1−
(15)
The frame error rate PE is bounded as PE ≤ kPBi ≤ N 2−N
1−
(16)
Note that the concatenated code is still capacity achieving, as the rate of the inner code is asymptotically achieving the channel capacity and the rate of the outer code is approaching 1 for large enough n and m. D. Decoding complexity analysis In this section the decoding complexity of BCH-polar codes both with BD and ML decoding is analyzed. For BCH-polar codes with BD decoding, the complexity for decoding the inner polar codes is O(mn log n), Pnand the complexity for decoding the outer codes is O( i=1 ti 2 ), where ti is the error correction ability of each Pnouter code. Then the overall complexity is O(mn log n + i=1 ti 2 ). For length-N BCH codes, with ML decoding over the BEC channel, the decoding complexity is O(N 3 ). We can show that BCH-polar codes with ML decoding have much smaller decoding complexity than the same length BCH codes with ML decoding. The complexity for decoding the m polar codes is O(mn log n), and the complexity for decoding the outer codes is O(nm3 ). Thus the overall complexity is O(mn log n + nm3 ) = O(N log n + N 3 /n2 ) ≈ O(N 3 /n2 ), which is much smaller than O(N 3 ) if n is large. To have a lower decoding complexity, larger n is preferred.
V. P ROPOSED C ONV-P OLAR C ODES Convolutional codes have good error correction abilities with the advantage of soft decision decoding. A new class of concatenation schemes, Conv-polar codes, is proposed in this section. We use the interleaved structure mentioned above for concatenation. In the proposed scheme, we use convolutional codes as outer codes and polar codes as inner codes. Both Viterbi and sequential decoding are considered and the frame error rate is analyzed. It is shown that for the Conv-Polar codes, the frame error rate decays exponentially with the frame length. Note that the channel polarization process increases the cutoff rate. This makes convolutional codes particularly suitable for the concatenated framework as one can choose codes with large constraint lengths operating under the cutoff rates (increased by channel polarization) while keeping the complexity small by using sequential decoder. Similar to the BCH codes counterpart, for the proposed Conv-polar codes, one can individually allocate a suitable rate for each outer code. We then show that the Conv-polar codes have exponentially decaying frame error rate even with sequential decoding. This demonstrates the fact that the Convpolar scheme is both provably good and practically attractive. A. Viterbi decoding of convolutional codes For outer convolutional codes, optimal Viterbi decoding can be used when the constraint length and rate are not large. For the case when the constraint length is not large but the rate is high, the dual algorithm proposed in [21] can be applied. The decoding complexity is roughly the same as decoding the dual code. Another way to alleviate the high complexity problem for codes having high rates is to obtain a high-rate code from puncturing a low-rate mother code [22]. It has been shown that the decoding complexity remains almost the same as that of the mother code [21]. Also puncturing provides flexibility in designing a wide range of rates. Another advantage of puncturing is that one may use the same decoder for decoding various rates codes. The bit error rate Pb of the Viterbi decoder output for a convolutional code is bounded by the following: Pb ≤
∞ X
ad Pd
(17)
d=df ree
where df ree is the free distance of the code and ad is the number of bit errors associated with all the incorrect paths with Hamming distance d from the correct path. Pd is the probability that the decoder selects an incorrect path with distance d from the correct path. If the coded bits are modulated with q BPSK and sent through an AWGN channel, s then Pd = Q( 2dE N0 ), where Es is the symbol energy and N0 is the power spectrum the Gaussian noise. Q(x) is R ∞ density of 2 defined as Q(x) = x √12π e−t /2 dt. Then Pb can be bounded as r ∞ X 2dEs Pb ≤ ad Q( ) (18) N0
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d=df ree
average cutoff rate/capacity
0.98
and the rate for each outer code can be chosen very close to the cutoff rate of each bit channel.
0.96
C. Error probability analysis of Conv-polar codes In this section, theoretical analysis of error probability for the Conv-polar codes is provided. We show that even with sequential decoding, the Conv-polar codes can have exponentially decaying frame error rate.
0.94
0.92
0.9
Theorem 2. There exists a Conv-polar code with constraint length ν = αm, where α is a constant and m is the length of the convolutional code, such that for any discrete memoryless channel and for any overall rate R < C, the frame error rate 0 1− of the code PE is upper bounded by KN e−k ραN for any 0 < < 1, when sequential decoding is used. Here K and k 0 are constants, N is the frame length of the code, C is the capacity of the channel and 0 ≤ ρ < 1.
0.88
0.86
0.84
3
4
5
6
7 8 n=log2(N)
9
10
11
12
Fig. 2. The average cutoff rate of polar codes with length N = 2n
If the convolutional code is terminated to an (m, k) block code, then additional ν bits should be appended to drive the encoder to the zero state. This will result in rate loss. The rate k is actually m (1 − νk ). The block error rate of the ith outer code PBi is PBi ≤ kPbi (19) where Pbi is the bit error rate of the i-th outer code. A wide range of rates of convolutional codes are provided in [21] and [22]. Notice that there are several ways to avoid the rate loss. Tail biting is a good way to avoid the rate loss with a slight degradation in the performance [23]. B. Sequential decoding of convolutional codes If the constraint length ν is too large, the complexity of Viterbi decoder 2ν would become too high. In this case, sequential decoding can be applied to substantially lower the complexity. It is known that as long as the rate of the outer code is chosen to be less than the cutoff rate, the average computational effort of sequential decoders can be kept small and be independent of the constraint length [16]. It should be pointed out that the concatenation scheme benefits from the fact that the average cutoff rate of the channel will increase after channel polarization. The infinite-length polarization will eventually push the average cutoff rate all the way to the channel capacity. Fig. 2 plots the average cutoff rate of polar codes over AWGN channel (Eb/N 0 = 2dB) with length N = 2n . We can observe that the average cutoff rate increases towards the channel capacity as the length of the code increases. With the increased cutoff rate, better outer codes can be designed with larger constraint length, while decoding complexity is kept low. With sequential decoding, we can have a tradeoff between the performance and decoding complexity. Conv-polar codes are capacity achieving even with sequential decoding, since the average cutoff rate of the bit channels goes to channel capacity as the length of polar codes increases,
Proof. The average bit error rate of the ensemble of (no , ko , ν) convolutional code with sequential decoding can be bounded as follows when the rate Ro < Rc Pb ≤ Ke−no νρRc
(20)
where K is a constant, 0 ≤ ρ < 1 and Rc is the cutoff rate. The block error rate of the ensemble is PB ≤ mPb . This guarantees the existence of at least one (no , ko , ν) convolutional code whose block error PB ≤ Ke−no νρRc . For each equivalent bit channel, the cutoff rate Rci can be computed based on (2). Then, we choose an (noi , koi , ν) convolutional outer code for the i-th equivalent bit channel with rate Roi < Rci . Thus, the bit error rate for the i-th channel, Pbi ≤ Ke−noi νρRci < Ke−noi νρRoi . Then PE can be bounded as X X PE ≤ PBi < Kme−noi νρRoi (21) i
=
i
X
0
Kme−koi νρ ≤ Kmne−ko ραm
(22)
i
where k 0 = mini koi and n is the inner code length. If we pick n = N , m = N 1− , PE can be further bounded as 0
PE < KN e−k ραN
1−
(23)
As n increases, the inner codes will polarize and the cutoff rate of each bit channel will either go to 0 or 1. By choosing the Roi close to Rci , we can get the overall rate R → C. From the above theorem we can see that Conv-polar scheme also has an exponential decay of the frame error rate even with sequential decoding. When Viterbi decoding is used, one can trade off the decoding complexity for the exponent in the rate of decay of the error probability by choosing the constraint length to grow with the length at different rates. The following theorem makes this precise. Theorem 3. Consider a sequence of Conv-polar codes with inner polar code-length n equal to N for any 0 < < 1. Assume (noi , koi , ν) is the i-th outer code and Rci is the cutoff rate of the i-th channel. Let the constraint length
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1 ν = α ln m, where m is the outer code length, α > (1−)h and h = mini {noi Rci }. The frame error rate of the code is upper bounded by PE ≤ KN −(1−)(hα−1) . Meanwhile, the decoding complexity is proportional to N +(1−)α ln 2 if the Viterbi decoder is adopted.
0
10
−1
10
−2
Frame error rate
10
Proof. For an (no , ko , ν) convolutional code with Viterbi decoding, the bit error rate Pb can be upper bounded as [17] Pb ≤ Ke−no νRc
(24)
−3
10
−4
10
−5
10
The block error rate of the code PB is less than mPb . Then PE is bounded as X X PE ≤ PBi ≤ Kme−noi νRci ≤ Kmne−hν (25) i
−6
10
−7
10
i
where h = mini {noi Rci }. As ν = α ln m, we can get PE ≤ Kmne−hα ln m = Knm1−hα
0.38
0.4
0.42 0.44 0.46 0.48 Capacity of the BEC channel
0.5
0.52
0.54
Fig. 3. Frame error rate of BCH-polar codes for the BEC, R = 0.36, N = 214
(26)
If we pick n = N , m = N 1− , PE can be further derived as PE ≤ KN N (1−)(1−hα) = KN −(1−)(hα−1)
0.36
Polar code BCH−polar−BD,n=64,m=256 BCH−polar−BD,n=128,m=128 BCH−polar−BD,n=256,m=64 BCH−polar−ML,n=128,m=128 BCH−polar−ML,n=256,m=64
(27)
1 Since α > (1−)h , PE → 0 as N → ∞. The complexity of Viterbi decoding for each code is proportional to 2ν . Then we can get 2ν = 2α ln m = mα ln 2 . The overall decoding complexity of outer codes is proportional to nmα ln 2 = N +(1−)α ln 2 .
VI. N UMERICAL RESULTS We aim at designing concatenated codes with fixed overall rates. The design criterion is to minimize the frame error rate of the code by optimizing over the rates of inner and outer codes. In the first part BCH-polar codes are designed over the BEC. Both ML and BD decoding are considered for BCH codes. In the second part, we assume that the codewords are modulated with BPSK and sent through an AWGN channel. Conv-polar codes with Viterbi decoding are designed. All the results presented in this section are obtained by numerically evaluating bounds or approximations on the bit error rate and they are not obtained from montecarlo simulations. A. Estimation of bit channel error probabilities To design optimal rates of the outer codes, the equivalent bit channel as seen by the codewords in each column needs to be computed. The Bhattacharyya parameter is usually computed as an upper bound of the error probability of the channel. However it cannot be used in our design, as it doesn’t take into account that the previous bits are decoded correctly before decoding the current bit. Instead, we use numerical density (i) evolution to compute the density of the LLR Ln in (1) given i−1 i−1 that u ˆ 1 = u1 .
B. Frame error rate of BCH-polar codes BCH-polar codes are optimized for BEC(0.5). The overall rate is fixed to 0.36, and the frame length of the code is 214 . The extended BCH codes are used and the weight spectrum of the BCH code is obtained from [20]. Different lengths of inner and outer codes are considered. Fig. 3 plots the frame error rate of BCH-polar codes and the same length polar code over the BEC channel. It can be seen that BCHpolar codes with ML decoding significantly outperform the standard-alone polar code, whereas for this length, bounded distance decoding of BCH codes is unable to outperform the stand-alone polar code. It should be noted that even though ML decoding is used for the BCH code, the length of the BCH code is only 1/nth the overall length and hence the decoding complexity is substantially smaller than ML decoding of the whole concatenated code. C. Frame error rate of Conv-polar codes In this part we design Conv-polar codes over AWGN channel. The overall rate is fixed to 0.4 and the frame length is 214 . The codes are optimized at Eb /N0 = 2dB. The convolutional codes we use are from [22] with ν fixed to 8. Codes are punctured to get various rates from 1/8 to 7/8. All the convolutional codes are terminated with zero padding and the rate loss is taken into account. The frame error rate of Conv-polar codes computed by evaluating a truncated union bound on the bit error rate performance are shown in Fig. 4. We can observe that Conv-polar codes substantially outperform the same length polar codes with a good choice over m amd n. For finite length concatenated codes, there is a tradeoff between the lengths of inner and outer codes. Convpolar codes with n = 32, m = 512; n = 64, m = 256 and n = 128, m = 128 can all outperform the stand-alone polar code, while a concatenated code with n = 256, m = 64 is worse than the stand-alone polar code. Among the four
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can decay exponentially with the frame length. We have then proposed a new class of concatenation schemes, namely the Conv-polar codes. We have shown that such schemes can have exponentially decaying error rate with a low-complexity sequential decoder. Numerical results have indicated that both BCH-polar codes and Conv-polar codes can outperform the stand-alone polar code. Furthermore, the performance of Conv-polar codes is significantly better than RS-polar codes. Potential future work includes the concatenation of non-binary polar codes with non-binary outer codes. In this way, the error decay rate may be increased. Also joint decoding of inner and outer codes is currently under investigation.
0
10
polar code Conv−polar,n=32,m=512 Conv−polar,n=64,m=256 Conv−polar,n=128,m=128 Conv−polar,n=256,m=64
−1
Frame error rate
10
−2
10
−3
10
R EFERENCES −4
10
1
1.5
2 Eb/N0 in dB
2.5
3
Fig. 4. Frame error rate of Conv-polar codes for the AWGN channel, R = 0.4, N = 214
0
10
polar code RS−polar Conv−polar,n=16,m=512 Conv−polar,n=32,m=256 Conv−polar,n=64,m=128 Conv−polar,n=128,m=64
−1
Frame error rate
10
−2
10
−3
10
−4
10
1
1.5
2 Eb/N0 in dB
2.5
3
Fig. 5. Frame error rate of Conv-polar codes for the AWGN channel, R = 1/3, N = 213
schemes, codes with n = 64, m = 256 have the lowest frame error rate. To compare Conv-polar codes with RS-polar codes, we design codes with the same length and rate of RS-polar codes in [11]. The frame length is 213 and the rate is 1/3. The performance of both Conv-polar and RS-polar codes are shown in Fig. 5. The frame error rate of the RS-polar codes is reproduced from [11]. It can be observed that the RS-polar code is not better than the stand-alone polar code, while the Conv-polar code with a good choice of m and n can be significantly better than the stand-alone polar code. This shows the superiority of Conv-polar codes over RS-polar codes. VII. C ONCLUSION Concatenation schemes of polar codes with binary codes have been considered in this paper. We have reviewed BCHpolar codes and we have shown that the frame error rate
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