Flexible IRHARQ Scheme for PolarCoded Modulation
Abstract
A flexible incremental redundancy hybrid automated repeat request (IRHARQ) scheme for polar codes is proposed based on dynamically frozen bits and the quasiuniform puncturing (QUP) algorithm. The length of each transmission is not restricted to a power of two. It is applicable for the binary input additive white Gaussian noise (biAWGN) channel as well as higherorder modulation. Simulation results show that this scheme has similar performance as directly designed polar codes with QUP and outperforms LTEturbo and 5GLDPC codes with IRHARQ.
I Introduction
Many communication channels are timevarying and unknown to the transmitter. Incremental redundancy hybrid automated repeat request (IRHARQ) as shown in Fig. 1 is a scheme that transmits additional redundancy bits until the data bits can be reconstructed. For turbo codes (such as those used in LTE), a low rate mother code is punctured with different patterns for several transmissions. The coding scheme for enhanced mobile broadband (eMBB) in 5G uses protographbased, Raptorlike LDPC codes [1] that allow for both flexible block length and code rate adaptation. The standard defines two base matrices that offer optimized performance for different operating regimes.
With cyclic redundancy check (CRC) outer codes and successive cancellation list (SCL) decoding [2], polar codes [3, 4] outperform stateoftheart turbo and LDPC codes in the short to medium length regime. Polarcoded modulation (PCM) is discussed in [5, 6, 7]. The performance comparison and efficient code design methods of three polarcoded modulation schemes are presented in [8, Fig. 11]. Multilevel polar coding (MLPC) with set partitioning (SP) labeling in [5] performs best and is around 1 dB more power efficient than an AR4JA [9] LDPC code decoded with 200 iterations.
A quasiuniform puncturing (QUP) algorithm was proposed in [10] to efficiently design lengthflexible polar codes, i.e., polar codes where the number of coded bits is not limited to be a power of two.
In [11], a scheme the authors called ?polar codes with incremental freezing? and in [12] ?parallelconcatenated polar codes? is proposed. The capacityachievability of this scheme is proved in [11] by using the nesting property. In [13], a polar code extension method is presented which outperforms the scheme in [11, 12] for finite block length.
In this work, we prove that the scheme in [13] can achieve capacity asymptotically in the block length under some design constraints. In addition, a lengthflexible IRHARQ scheme based on dynamically frozen bits and QUP is proposed. This scheme is extended to polarcoded modulation with amplitude shift keying (ASK) and quadrature amplitude modulation (QAM) constellations. Simulation results show that the polar codes designed by our algorithm have similar error correction performance as directly designed polar codes.
This work is organized as follows. In Sec. II, we review polar codes, PCM and QUP. We discuss existing and proposed IRHARQ schemes in Sec. III. Sec. IV provides design examples and numerical results. The performance of the proposed scheme is compared with directly designed polar codes and 4G/5G codes [14, 15] in additive white Gaussian noise (AWGN) channels. We conclude in Sec. V.
Ii Preliminaries
Iia Polar Coding
In this paper, uppercase letters denote the random variables (RV) while the corresponding lowercase letters are their realizations. The notation is short for . For an arbitrary subset of , is the complement of and denotes the vector of formed by the elements with indices in .
A binary polar code of block length and dimension is defined by the polar transform with matrix and frozen positions, where denotes the fold Kronecker power of the kernel
(1) 
Polar encoding can be represented by
(2) 
The vector denotes the codeword. The vector includes information bits and predefined frozen bits . and are called the information set and frozen set defined in [4]. SC decoding uses the channel observation and previous estimates to decode . Both encoding and SC decoding have complexity [4].
The polar code construction finds the most reliable bits in under SC decoding. The Monte Carlo (MC) construction was introduced in [3, 4], and needs extensive simulations. An information theoretical construction was introduced in [4]. The reliability of the th bit can be quantified by the mutual information (MI) . We can calculate these MIs for all by recursively calculating the MIs of the basic polar transform displayed in Fig. 2.
For the binary input additive white Gaussian noise (biAWGN) channel, density evolution [16] with Gaussian approximation (GA) [17] has much lower complexity and performs very close to the MC construction. The update rule for the basic polar transform is given by
(3)  
(4) 
The numerical approximations in [18] can be used for and . Additionally, the frame error rate (FER) under SC decoding can be estimated by
(5) 
(6) 
where
(7) 
The nesting property is introduced in [11]. The authors show that the reliability order of the polarized bits is independent on the channel quality for infinite length. Note that the nesting property generally does not hold for finite length.
IiB PolarCoded Modulation
We consider the following memoryless AWGN channel model with ASK constellations .
(8) 
where is zero mean Gaussian noise with variance one. Note that ASK constellations can be orthogonally extended to QAM constellations.
Polarcoded modulation (PCM) schemes are discussed in [5, 6, 7]. In [8], the performance of three schemes is compared and their efficient design algorithms are presented. MLPC with SP labeling in [5] provides the best performance. MLPC with code length works as following:

Encoding

Put information bits in vector and define
(9) 
Encode polar codes
(10) 
Map the code words to symbols for
(11)


Decoding

Demap and decode level 1
(12) 
Demap and decode level for
(13)

where denotes the input loglikelihood ratio (LLR).
MI demapper GA (MIDGA) construction calculates for and uses them to find the most reliable bits in with GA. Therefore, the required code rates of the polar codes are given by MIDGA. Note that the overall code rate of the system is and the transmission rate is .
IiC RateMatched Polar Codes
Because of the recursive structure of , polar codes usually have a block length that is a power of two. Punctured polar codes are introduced in [19, 20]. The punctured polar codes can be decoded with standard polar decoders for length . For punctured polar codes, bits in are not transmitted and the corresponding LLRs are set to zero. For GA construction, the initial MIs of the punctured bits are zero.
With the QUP algorithm, the first bits in are not transmitted, i.e.,
GA construction:  
after encoding:  
before decoding: 
This algorithm is called QUP because the punctured position in bitreversal representation [4] looks like uniformly distributed in . Let denote a QUP polar code with dimension , block length and punctured from bits mother codes with QUP algorithm, where has to the power of two.
Theorem 1.
For an QUPpolar code, the first bits in are frozen.
Proof.
For the MI update of the basic polar transform in Fig. 2, it is easy to show that
(14) 
As MI is always nonnegative, we have
(15)  
(16) 
Thus the number of channels with zero capacity is invariant. With QUP we set to zero. Because of the recursive structure of the zeros will propagate through the transform which causes for . ∎
Theorem 2.
All QUPpolar codes have the same encoding and decoding complexity as QUPpolar codes, where is a natural number and .
Proof.
The decoder for QUPpolar codes is shown in Fig. 3. Obviously, more than bits are punctured. Thus, are all frozen because of Theorem 1. According to the SC decoding, we first decode (upper decoder) and then decode (lower decoder) based on . The input of the lower decoder is . Thus for this QUPpolar code, we just need to run the lower decoder (dashed box in Fig. 3) which is the decoder for QUPpolar codes. With the same idea we can extend the theorem to mother code length .
∎
Iii IRHARQ with Polar Codes
The basic idea of IRHARQ is displayed in Fig. 1. The decoder receives bits from the th transmission and then decodes code. Using the CRC, the receiver may detect a decoding failure, in which case it requests the th transmission from the sender.
Iiia Existing Schemes: Polar Codes with Incremental Freezing
In [11, 12], polar codes with incremental freezing are proposed. The main idea is as follows. An polar code is transmitted first. When retransmission occurs, we additionally freeze ?most unreliable? information bits in the code and send them with an code in next transmission. After the nd transmission we decode the code and the first code successively. Note that the first code becomes a code with the estimation of the ?most unreliable? information bits. For the rd transmission (if needed), we freeze ?most unreliable? information bits in both the code and the code. The retransmissions are continued with the same manner until the decoding is successful. For infinite code length, this scheme achieves capacity [11] with the nesting property. We note that this scheme is equivalent to dividing the code into separated polar codes and decoding them successively, which causes a huge performance loss in the finite length regime.
IiiB Existing Schemes: Polar Extension
A polar code extension method is presented in [13]. Consider the st transmission with polar code , where denotes the codeword after the th transmission. When retransmission occurs, we design a polar code , let the information set in be a subset of the information set in and set . Then copy the bits, which are reliable in but unreliable in to the most reliable positions in . The code word can be presented by
(17)  
(18) 
Thus, only the first bits need to be transmitted, because the receiver already knows the remaining bits of the polar code from the previous transmission. The receiver first decodes the information bits in , copies them to as frozen bits and then decodes the remaining bits. The only difference between a directly generated polar code and the code after the nd transmission is the information set in which has to be a subset of the information set in . In this scheme, we decode an code after the nd transmission (). This extension is repeated until the decoding is successful. This extension method is not flexible because the transmission length has to be the same as the sum of all previous transmissions, i.e.,
(19) 
This scheme is capacityachieving under some constraints. According to the information theoretical construction in [4], designing a polar code for a channel with mutual information is equivalent to designing a polar code for and a polar code for , where
(20)  
(21) 
Let be the design MI for the th transmission and the code length after the th transmission (). Note that only bits are sent in the th transmission. Under the constraints
(22) 
the information set in is always a subset of the information set in because of the nesting property, i.e., every polar code after th transmission is an optimal polar code.
IiiC Proposed Scheme
Polar codes with dynamically frozen bits are proposed in [21] to improve the distance properties of polar codes. The idea is to predetermine some frozen bits as linear combinations of previous information bits. The corresponding ?dynamically frozen constraint? is needed to encode and decode. Our scheme is based on this technique and QUP. There are no constraints regarding the block length of any transmission.
We assume the system is designed for a maximum of transmissions. Let be the length of a mother polar code
(23) 
where denotes the ceiling function and denotes the length of the th transmission. In the proposed scheme, after the th transmission, a QUPpolar code is decoded. The main structure and design algorithm are displayed in Fig. 4 and Algorithm 1 respectively. The set is
(24) 
and and denote the information and frozen set of the QUPpolar code after the th transmission. The output dynamically frozen constraint is used for encoding and decoding. Note that . First bits are frozen to zero (Theorem 1), but their indices are neither in nor .
Obviously, for , the code is a normal QUPpolar code. Other codes are extended from previous codes with dynamically frozen bits. The bits which are already frozen have to be frozen for all extensions. This scheme is equivalent to the polar code extension method in [13] if is a power of two and .
For example, we consider and . The information bits are .

For the first transmission, for the QUPpolar code. Thus, and . We encode the vector by and transmit the last 7 bits .

For the second transmission, for the QUPpolar code. We precode with , which is in this case. We encode and send .
Because is a lower triangular matrix, does not change the first transmitted bits, i.e., . At receiver, we decode the QUPpolar code from the noisy version of . Note that is now a dynamically frozen bit with constraint .
In this example, the QUPpolar codes with optimal information set are decoded after every single transmission by using all received information.
IiiD Extension to MLC
With ASK, symbols ( bits) are transmitted in the th transmission. We replace line 1,2 in Algorithm 1 with an MLPC construction algorithm (MIDGA in this work).
Iv Design Examples and Simulation Results
direct polar  proposed polar  5GLDPC  LTEturbo  
st  7  7  7  7 
nd  7  7  7  7 
rd  7  7  7  7 
th  7  7  7  7 
In this section, three design examples for are shown in Fig. 5, Fig. 6, Fig. 7. We use 16 bits CRC with generator polynomial ?? for error detection. The polar codes are decoded by minsum approximated SCL decoding with list size 32. LogMAP decoding with 10 iterations and belief propagation (BP) with 50 iterations are used for LTEturbo and 5GLDPC codes, respectively. In the 8ASK example, BitInterleaved Coded Modulation (BICM) [22] is used for turbo and LDPC codes. Note that the directly designed polar codes (dashed curves) are QUPpolar codes only serve as a reference and can not work for an IRHARQ scheme.
The simulation results show that the polar codes generated by the proposed algorithm perform very similar to directly designed polar codes. In the 8ASK example, the proposed scheme performs approximately dB better than 5GLDPC codes after two to four transmissions.
Because of the extension constraint, it is hard to extend a heavily punctured polar codes. Consider an QUPpolar code, where is a positive integer and . Normally, the first bits are unreliable, while these bits are almost perfect in the extended code. This effect degrades the performance for all further extensions. For example, for , the rd and th polar codes perform much worse than directly designed code because and . Therefore, we should avoid using heavily punctured polar code for the th transmissions in biAWGN channel. However, this effect disappears for MLPC. We can design very good codes for . The reason should be the automatically controlled code rate for polar codes.
V Conclusion
In this paper, an IRHARQ scheme based on QUP and dynamically frozen bits for biAWGN channel and PCM is proposed. Simulation results show that the ratematched polar codes generated by the proposed algorithm perform very similar to directly designed QUPpolar codes.
For future work, this scheme can be applied for a fading channel, i.e. the channel information estimated by training symbols of the previous transmissions could be used to design polar codes for next transmission.
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