Study of Design of Rate-Compatible Polar Codes Based on Non-Uniform Channel Polarization

Polar codes, introduced by Arikan [1], were proved to achieve the symmetric capacity of binary input symmetric discrete memoryless channels (B-DMCs) under a Successive Cancellation (SC) decoder as the codelength goes to infinity. The block error rate converges to zero as the code length $N$ goes to infinity. Polar codes are based on the phenomenon of channel polarization. The channel polarization theorem states that, as the codelength $N$ goes to infinity, the polarized bit-channels become either a noiseless channels or pure noise channels. The information bits are transmitted over the noiseless bit-channels and the pure noise bit-channels are set to zero (frozen bits). The construction of a polar code involves the identification of channel reliability values associated to each bit to be encoded. This identification can be performed for a code length and a specific signal-to-noise ratio. Therefore, polar code construction refers to selecting an appropriate frozen bit position pattern and several polar code construction algorithms have been proposed. Polar codes were recently selected by the 3GPP group as the channel codes for the upcoming 5th generation mobile communication standard (5G) uplink/downlink control channel [2].

In the construction of polar codes, we recall that the code length $N$ of standard polar codes is limited to powers of two, i.e., $N=2^{n}$, however, the code length flexibility is required for practical applications. There are several construction techniques applied in the standard polar code model proposed by Arikan [1] considering the SC decoder. Among the most well-known polar codes construction techniques are the Bhattacharyya-based approach, proposed by Arikan [1], the Density Evolution (DE) schemes of Mori [3][4] and Tal [5], the Gaussian Approximation (GA) technique of [6] and the Polarization Weight (PW) algorithm [7]. The Bhattacharyya parameter-based approach was proposed along with Monte-Carlo (MC) simulations to estimate bit channel reliabilities [1]. The MC method needs to estimate the average error probability of each bit-channel via simulations. Therefore, encoding and SC decoding are required in each simulation. In the DE method, we need to compute the probability density function (PDF) of the log-likelihood-ratio (LLR) of each channel first and then choose the channels that are most likely to be correct. Since DE includes function convolutions, its precision is limited by the complexity. Similar to the MC method, the DE method has high computational cost. Extending the ideas of [3] and [4], Tal [5] devised two approximation methods by which one can get upper and lower bounds on the error probability of each bit-channel using efficient schemes of degrading and upgrading quantization. A bit-channel reliability estimation method for AWGN channels based on DEGA has been proposed in [6], giving accurate results with a significant reduction in complexity if compared with DE. The PW algorithm [7] is a recent attempt to study the partial order (PO) [8] for designing universal polar codes, which performs as well as GA but with much lower complexity. An important characteristic of polar codes is that the channel orderings are channel-dependent. PO is based on the observation that the channel orderings are degraded according to the binary expansion of their indexes. Therefore, in this method the channel orderings are not channel-dependent. PO provides us with quality information for all channels. This can be used to simplify code construction of polar codes. In [9] and [10] a comparative study of the performance of polar codes constructed by various techniques using the AWGN channel and SC decoder has been carried out. Construction of polar codes based on the AWGN channel and GA have been reported in [11] and [12]. In [9] the design of good polar codes with any construction method has been verified with the SC decoding error rate for various scenarios and polar code construction algorithms. Polar codes can also be constructed and adapted to a specific decoder, for example, construction of polar code for SCL decoding [13] and BP decoding [14][15]. In [16] the authors propose a genetic algorithm framework that jointly optimizes the code construction and rate with a specific decoder.

To obtain code length flexibility, arbitrary kernels and multi-kernel (MK), puncturing, shortening and extension are typically performed. Polarization matrices of various sizes, for example 3x3, 5x5 and 7x7, were used to build a polar code of any length. BCH kernel matrices proposed in [17] and the code decompositions proposed in [18] both have restrictions on the size of the kernel matrices. Square polarizing kernels larger than two have been proposed in [19], [20] and [21], while a polar code construction with mixed kernel sizes has been proposed in [22][23]. By considering different polarizing kernels of alternate dimensions, MK improves block length flexibility. Although the general coding and decoding structure follows the same structure of Arikan’s standard polar codes, there is an increased complexity with the generalization. Polar codes construction using the Reed-Muller (RM) rule [24][25][26] can improve the performance of the error rate.

Various shortening and puncturing methods for polar codes have been reported in [27]-[41]. Generally, puncturing or shortening causes a loss of performance because when the number of bits punctured or shortened increases the code length decreases, degrading the performance. A review on puncturing and shortening techniques can be found in [27], which includes designs based on the column weights (CW) of the generator matrix and the reversal quasi-uniform puncturing scheme (RQUP) based on bit reversed permutation. In practice, the design of the code is made for $N=2^{n}$ and rate-compatible code design techniques use a specifically chosen criterion for shortening or puncturing and generate a codeword with length $M$ that is $2^{n-1}<M<2^{n}$. In [28]-[30] puncturing methods have been reported using belief propagation (BP) decoding based on optimization techniques employing retransmission schemes such as Hybrid Automatic Repeat reQuest (HARQ). Different properties of punctured codes have been explored: minimum distance, exponent binding, stop tree drilling, and the reduced generating matrix method [31]-[34]. Schemes that depend on the analysis of density evolution were proposed in [35] and [36]. On the other hand, shortening methods have been studied with SC decoding. In shortening techniques, we freeze a bit channel that receives a fixed zero value. The decoder, however, uses a plus infinity log-likelihood ratio (LLR) for that code bit as it is often assumed that this value is known. The study in [37] proposed a search algorithm to jointly optimize the shortening patterns and set of frozen bits. The work in [38] devised a simple shortening method, reducing the generator matrix based on the weight of the columns (CW). In [39] the reversal quasi-uniform puncturing scheme (RQUP) for reducing the generator matrix has been proposed. The authors in [40] presented a polarization-driven (PD) shortening technique for the design of rate-compatible polar codes, which consists of reducing the generator matrix by relating its row index with the channel polarization. Recently, the PW algorithm has been used in a puncturing and shortening technique reported in [41].

Extension of polar codes have been studied in [42], [43] and [44]. For HARQ schemes it was proposed that an arbitrary number of incremental coded bits can be generated by extending the polarization matrix such that multiple retransmissions are aggregated to produce a longer polar code with extra coding gain. The complexity of the encoder and decoder are similar to that of standard polar codes. Nevertheless, there is a significant increase in complexity when designing flexible-length polar codes by concatenated codes [45], [49] and by asymmetric kernel construction [47]. In both cases the code construction is specific to each kernel dimension without generalization gains. Chained polar subcodes [48] were shown to be an effective method for achieving length compatible polar codes.

In this paper, we present the concept of non-uniform polarization, which allows the construction of rate-compatible polar codes considering the polarization of channels with an arbitrary distribution of the Bhattacharyya parameter. Generalized channel polarization schemes have been proposed in [49] and [50]. In [49] the authors present a rate-matching scheme for multi-channel polar codes, which can be directly applied to bit-interleaved coded modulation schemes. Moreover, in [50] polar coding schemes are introduced for independent parallel channels in which the channel parameters are no longer fixed but exhibit some random behaviors.

The original construction of Arikan [1] considers a single Bhattacharyya parameter to all channels, which we call uniform polarization. We develop a Non-Uniform Polarization technique based on the Gaussian Approximation (NUPGA) for designing rate-compatible polar codes of arbitrary length [54]. We then develop NUPGA-based shortening and extension algorithms for rate-compatible polar code designs. In the proposed techniques the shortened or extended channels are re-polarized for a more efficient choice of noiseless channels. We also present a generalization of the GA algorithm, which is used for both polarization of the initial channel and re-polarization of the shortened channel, and we present a simplified construct technique for extended polar codes. Simulations compare the proposed NUPGA techniques with existing approaches. A key feature of the proposed designs in that, the encoder and decoder structures are the same as that of the original polar codes [1] and require the same complexity.

In summary, the main contributions of this work are:

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  a novel NUPGA approach for designing rate-compatible polar codes along with a proof that it achieves capacity;

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  NUPGA-based shortening and extension techniques for designing rate-compatible polar codes;

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  an extensive simulation study that compares the NUPGA and existing design techniques.

The rest of the paper is organized as follows. In Section II, we present a brief review of the fundamentals of polar codes required for our exposition, outline their notation, and describe their encoding and SC/SCL decoding. In Section III, we review the principles of polarization theory of polar codes. In Section IV, we discuss the non-uniform polarization of polar codes and show that similarly to uniform polarization, it achieves channel capacity. In Section V, we give a detailed description of the proposed NUPGA-based shortening and extension techniques along with the pseudo-codes for their implementation. We present our simulation results in Section VI and the conclusions in Section VII.

In this section we review some fundamental aspects of polar codes that will help in the exposition of the proposed techniques. Using the notation proposed by Arikan [1], we describe key principles of the encoder and the decoder of polar codes.

Given a B-DMC $W:\mathcal{X}\to\mathcal{Y}$, where $\mathcal{X}=$ $\{0,1\}$ and $\mathcal{Y}$ denote the input and output alphabets, respectively, we define the channel transition probabilities as $W(y|x)$, $x\in\mathcal{X}$, $y\in\mathcal{Y}$. As demonstrated in [1], after channel combining and splitting operations on $N=2^{n}$ independent uses of $W$, we get $N$ successive uses of synthesized binary input channels $W^{(i)}_{N}$ with $i=1,2,\ldots,N$. $K$ represents the most reliable sub-channels. The indices of $K$ form the set of information $\mathcal{A}$. They are the ones that carry the information bits. The indices of the remaining sub-channels are included in the complementary set $\mathcal{A}^{\text{c}}$, and can be set to fixed bit values, all zeros.

A polar codes scheme is uniquely defined by three parameters: code-length $N=2^{n}$, code-rate $R=K/N$ and an information set $\mathcal{A}\in$ $[N]$ with cardinality $K$. For encode we have $x^{N}_{1}=u^{N}_{1}\textbf{G}_{N}$, where $\textbf{G}_{N}$ is the transform matrix, $u^{N}_{1}\in\{0,1\}^{N}$ is the source block and $x^{N}_{1}\in\{0,1\}^{N}$ is the codeword. The source block $u^{N}_{1}$ is comprised of information bits $u_{\mathcal{A}}$ and frozen bits $u_{\mathcal{A}^{\text{c}}}$. The $N$-dimensional matrix can be recursively defined as $\textbf{G}_{N}=\textbf{B}_{N}\textbf{F}^{\otimes n}_{2}$, where $\otimes$ denotes the Kronecker product, $\textbf{F}_{2}=\footnotesize\left[\begin{array}[]{cc}1&0\\ 1&1\end{array}\right]$ and $\textbf{B}_{N}$ is the bit-reversal permutation matrix, which can be omitted without loss of generality. Then, the codeword is transmitted to the receiver through AWGN channel.

We may adopt the SC decoder so that the information bits are estimated as [1]:

Having known the decoded bits $\hat{u}^{N}_{1}=(\hat{u}_{1},\ldots,\hat{u}_{N})$ and received sequence $y^{N}_{1}=(y_{1},\ldots,y_{N})$, the likelihood ratio (LR) message of $u_{i}$, ${\rm LR}(u_{i})=\frac{W^{(i)}_{N}(y^{N}_{1},\hat{u}_{1}^{i-1}|0)}{W^{(i)}_{N}(y^{N}_{1},\hat{u}_{1}^{i-1}|1)}$, can be recursively calculated using the SC decoding algorithm [1]. Then, $\hat{u_{i}}$, an estimated value of $u_{i}$, can be computed by:

where $h_{i}:\mathcal{Y}^{\text{N}}\times\mathcal{X}$ ${}^{i-1}\to\mathcal{X}$, $i\in\mathcal{A}$, are decision functions defined as:

for $y_{1}^{N}\in\mathcal{Y}^{\text{N}}$, $\hat{u}_{1}$ ${}^{i-1}\in\mathcal{X}$ ^i-1.

The LR output from the node at row $i$ and stage $j$ is denoted as $L(i,j)$, according to the decoding tree in [1]. In order to calculate $L(i,j)$, as proposed in [1], we compute the following equation recursively:

where $f$ and $g$ functions were defined in [1] as:

Notice that in (6), $\hat{u}_{sum}$ is the modulo-2 sum of partial previous decoded bits. The term $\hat{u}_{sum}$ depicts the successive operation in the SC algorithm. The decision on the current bit strongly depends on the estimate of previously decoded bits. A log-likelihood ratio (LLR)-based SC algorithm has been developed in [51] to simplify the design. Accordingly, (5)-(6) in the natural domain are transformed to the following equations in the logarithm domain:

Similar to LDPC decoding, a min-sum approximation [51] can be further employed to reduce the complexity of (7):

where (8)-(9) describe the LLR version of the SC algorithm.

For the sucessive cancelation list (SCL) decoder [52], let $S^{(i)}$ denote the set of candidate sequences in the $i$th step of the decoding process, and $|S^{(i)}|$ is the size of $S^{(i)}$. Let $L$ be the maximum allowed size of the list and $T$ be a threshold for pruning with $T\leq 1$. The SCL algorithm can be described as follows:

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  bits are estimated successively with index $i=1,2,\ldots,n$; for each candidate in the list,

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  generate two $i$-length sequences for decoding $\hat{u}_{i}$ as bit 0 and bit 1 by SC decoding;

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  if the number of candidates $|S^{(i)}|$ is not larger than $L$, there is nothing to do;

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  otherwise, reserve $L$ candidates with the largest probabilities and drop the others from $S^{(i)}$;

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  check each candidate $\hat{u}_{i}\in S^{(i)}$, if $P(\hat{u}_{1}^{i})<T\max\limits_{\hat{u}_{1}^{i}\in S^{(i)}}P(\hat{u}_{1}^{i})$, eliminate $\hat{u}_{1}^{i}$ from $S^{(i)}$.

For sequence estimation, after all the bits are examined, re-encode every candidate in the list and calculate the corresponding likelihood probabilities. Select the one with the maximal probability as the sequence estimate:

In this section, we recall the theoretical aspects of the construction of polar codes and show that the channel polarization theory can be generalized to non-uniform channels. In this scenario, the main aspects of the channel polarization theory are maintained, namely, the conservation of the associated channel capacity and the induction to the polarized channel [1].

In this section, we show a generalization of the equations presented in Section III.B, and we verify that for non-identical channels, that is, with different Bhattacharyya parameters, they also converge for channel polarization.

If the channels are independent but not uniform, then $W_{(i)}:\mathcal{X}_{\text{(i)}}\to\mathcal{Y}_{\text{(i)}}$, as shown in Fig. 5, where we rewrite (17) as

We consider $N$ and $N^{\prime}$ with the same cardinality and in such a way that their transition probabilities $W_{(i)}(y|x)$ and $W_{(j)}(y|x)$ may differ if $i\neq j$. A non-identical channel corresponds to a channel with different Bhattacharyya parameters. As in [1], given any B-DMC $W_{(i)}$ the same definitions of the symmetric capacity (18) and the Bhattacharyya parameter (19) are adopted as performance measures, and rewritten as

We notice that the relation in (20) is equivalent and can be rewritten as

where (21) remains valid.

The $W_{2}$ channel with the transition probabilities of (22) and (23) can be written as

Using the BEC channel again as an example, for a comparison with Section III.B, for the case of the parameter $Z$ we have:

The reliability terms in (29) and in (30) are rewritten as

The cumulative rate in (31) and the reliability in (32) are then given by

We can show that (26) can be obtained by performing the following operations:

The non-uniform polarization channel for any set of B-DMCs $W_{N}^{(i)}$, $i\in[1,N]$, for arbitrary small $\delta\leq 0$ there exist polar codes that achieve the sum capacity $I_{s}=\sum_{1}^{N}I_{i}$, $I_{i}=I(X_{i},Y_{i})$ in the sense that as $N\rightarrow\infty$, which is a power of 2, the fraction of indices $i\in[1,N]$ satisfies

where the values of $I_{N}^{(i)}$ converge to $\{0,1\}$, which is a novel result related to the established result for uniform channel polarization in (33).

In this section, we describe the proposed NUPGA design algorithms which employ the non-uniform construction. In the proposed NUPGA design algorithms, the tree process for the recursive channel polarization can be redesigned according to Fig.6 without loss of generalization, with the nodes being calculated using the functions described in (29) and (30).