DEVICE AND METHOD FOR ENCODING OR DECODING MESSAGES BY MEANS OF A POLAR CODE

A device is configured for encoding an input sequence comprising message bits into a codeword using a polar code. The device is configured to sequentially encode each of a plurality of blocks of the input sequence by applying a sliding window to the input sequence, wherein each block of the input sequence is encoded based on an XOR operation of the block and a previous block of the input sequence to obtain a codeword block of the codeword, and sequentially output each obtained codeword block of the codeword.

FIELD

The present disclosure relates to encoding and decoding data. In particular, it relates to encoding and decoding data using a polar code. Thereby, embodiments of the present disclosure provide devices and methods for encoding and decoding data by applying a sliding window.

BACKGROUND

Channel codes are essential in all digital communications systems. A system for forward error correction (FEC) coding, also called generally a coding scheme, consists of an encoder at the transmitter side and a decoder at the receiver side. The encoder adds redundancy to the data to be transmitted, i.e. additional redundant data, and the decoder exploits this redundancy to correct transmission errors, such that the receiver can obtain the transmitted data free of errors despite a noisy communication channel between encoder and decoder.

In particular, polar codes are linear block codes that rely on the polarization effect (see, e.g., E. Arikan, “Channel polarization: A method for constructing capacity achieving codes for symmetric binary-input memoryless channels”, IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3051-3073, July 2009), which allows to sort bit positions of so-called bit-channels in order of reliability. As the code length goes toward infinity, the polarization phenomenon influences the reliability of bit-channels, which are either completely noisy or completely noiseless. Even more, the fraction of noiseless bit-channels equals the channel capacity.

The reliability of the channels can be determined according to the Bhattacharyya parameter:

where W is a binary memoryless symmetric channel, and W (y|0), W (y|1) are transition probabilities, Y is the output alphabet and Z is the Bhattacharyya parameter. The lower the Bhattacharyya parameter, the more reliable the channel. Other methods may be used to estimate bit-channel reliabilities. For example, a density evolution method (DE) may be used, and for additive white Gaussian noise (AWGN) the channels reliabilities may be determined according to a Gaussian approximation (GA). Other categories of noisy channels may be modelled such as a binary symmetric channel (BSC) or a binary erasure channel (BEC), for example, using Monte-Carlo statistical methods.

For finite practical code lengths, the polarization of bit-channels is incomplete. Therefore, there are bit-channels that are partially noisy. The polar encoding process consists in the classification of the bit-channels into two groups: the most reliable bit-channels that will carry the information bits and are indexed by the information set, and the less reliable bit-channels that are fixed to a predefined value (usually 0) and are indexed by the frozen set. In case of finite code lengths, the bit-channels with the highest reliability are selected to form the information set, while the remaining bit-channels are frozen.

Polar codes are based on the kernel matrix:

Encoding of such a polar code of length N=2″ and information length K is explained in the following. The frozen set F of size N-K is chosen, as described above. The bits uiof the input vector u are set to 0 for i∈F and to the information bits otherwise. The codeword x is computed as x=uT with the transformation matrix T=T2⊗n, ⊗n denoting the n-fold Kronecker product.

As a generalization, different kernels of different sizes can be introduced in the code design, obtaining a multi-kernel polar code (see, e.g., V. Bioglio et al., “Multi-kernel construction of polar codes”, 2017 IEEE International Conference on Communications Workshops (ICC Workshops), Paris, 2017, pp. 761-765.). As a result, the transformation matrix takes the form T=Ta⊗Tb⊗ . . . ⊗Tg, and the frozen set has to be calculated accordingly.

Polar code decoding can be based on a Successive Cancellation (SC) decoding algorithm (see, e.g., the aforementioned work of Arikan), which is inherently sequential. It can be viewed as a binary tree search, where bits are estimated at leaf nodes, and the tree is traversed depth-first, with priority given to the left branch. In SC decoding, the decoder starts with a decision for bit u1and feeds this decision back into the decoding process. Then, it makes a decision of bit u2and feeds this decision back into the decoding process. It proceeds in this fashion until it obtains the design for the last bit uN.

SC list decoding (SCL) is an enhanced version of SC, wherein the decision is postponed to the end of the decoding process, and is usually performed with the help of a cyclic redundancy check (CRC) (see, e.g., O. Afisiadis et al., “A low-complexity improved successive cancellation decoder for polar codes”, in Asilomar Conf. on Signals, Syst., and Comput., November 2014, pp. 2116-2120).

When too many information bits need to be transmitted, they are usually divided into blocks and transmitted separately on an independent codeword of length M. However, independent transmissions increase the block error rate of the system, since information is correctly recovered, only if all the S codewords are decoded correctly. Even a single error in one of the transmission results in an overall decoding failure.

Moreover, many polar-code-based encoding and decoding devices and methods are too complex. Thus, the inventors have recognized that there is generally a need for improved devices for coding data using polar codes.

SUMMARY

In view of the above-mentioned disadvantages, the present disclosure provides improved devices and methods for encoding and decoding based on polar code. A polar-code based encoding device and method is provided, and a polar-code based decoding device and method is provided, these devices and methods can operate with reduced complexity.

According to a first aspect, a device for encoding an input sequence comprising message bits into a codeword using a polar code is provided. The device is configured to sequentially encode each of a plurality of blocks of the input sequence, by applying a sliding window to the input sequence, wherein each block of the input sequence is encoded based on an XOR (exclusive or) operation of the block and a previous block of the input sequence, to obtain a codeword block of the codeword, and sequentially output each obtained codeword block of the codeword.

The device of the first aspect, which may be implemented in an encoder, is designed for performing polar code based encoding with a sliding window, which reduces the encoding complexity. For instance, the device permits calculating the codewords as linear combinations of common codewords. This may reduce the cost of additional hardware to be installed in the device, since it can rely on standard polar code hardware. Moreover, the use of the device permits reducing the encoding latency, since message bits can be processed on-the-fly. Finally, the device has the same Block Error Rate (BLER) performance as a conventional device for encoding without sliding window based on a polar code.

In an embodiment of the device according to the first aspect, the previous block is a predetermined block or the previous block is equal to the last block of the input sequence.

This provides a simple but efficient implementation of the coding scheme.

In an embodiment of the device according to the first aspect, the device is further configured apply a transformation matrix, TM/2, of a polar code of length M/2, wherein TM/2=T2⊗n, with n=log2(M) and M being a size of the sliding window to the block and the previous block of the input sequence to obtain the codeword block of the codeword.

This provides an exemplary implementation of a sliding window for the polar code encoding.

In an embodiment of the device according to the first aspect, the device is further configured to calculate a first block on the basis of the predetermined block and the transformation matrix.

In an embodiment of the device according to the first aspect, the device is configured to determine a reliability of bit positions of each block of the input sequence determine an information set Iiof size Kifor each block of the input sequence ui, wherein Kicorresponds to a size of a corresponding block of the message bits miand wherein the set Iicorresponds to the Kibit positions of each block of the input sequence uihaving the highest reliability and generate each block of the input sequence uiby placing the message bits miin the positions defined by Ii.

According to a second aspect, a device for decoding a sequence of bits into an output sequence using a polar code is provided, wherein the sequence of bits represents a codeword after a transmission over a communication channel. The device is configured to sequentially decode each of a plurality of blocks of the sequence of bits by applying a sliding window, in order to obtain an auxiliary sequence, process the auxiliary sequence to obtain the output sequence, wherein each block of the output sequence is processed based on an XOR operation of a block of the auxiliary sequence and a previous block of the auxiliary sequence.

The device of the second aspect, which may be implemented in a decoder, is designed for performing polar code based decoding with a sliding window, which reduces the decoding complexity. This permits reducing the cost of additional hardware to be installed in the device, since the device can rely on standard polar code hardware.

In an embodiment of the device according to the second aspect, the device is further configured to obtain message bits from the output sequence.

According to a third aspect, a method for encoding an input sequence comprising message bits into a codeword using a polar code is provided. The method comprises sequentially encoding each of a plurality of blocks of the input sequence, by applying a sliding window to the input sequence, encoding each block of the input sequence based on an XOR operation of the block and a previous block of the input sequence, to obtain a codeword block of the codeword, and sequentially outputting each obtained codeword block of the codeword.

According to a fourth aspect, a method for decoding a sequence of bits into an output sequence using a polar code is provided, wherein the sequence of bits represents a codeword after a transmission over a communication channel. The method comprises sequentially decoding each of a plurality of blocks of the sequence of bits by applying a sliding window, in order to obtain an auxiliary sequence, processing the auxiliary sequence to obtain the output sequence, wherein each block of the output sequence is processed based on an XOR operation of a block of the auxiliary sequence and a previous block of the auxiliary sequence.

The methods of the third aspect and the fourth aspect achieve the same advantages as, respectively, the devices of the first aspect and the second aspect. The method of the third aspect may have implementation forms according to the implementation forms of the device of the first aspect. The method of the fourth aspect may have implementation forms according to the implementation forms of the device of the second aspect.

According to a fifth aspect, a computer program is provided, the computer program comprising a program code for performing the method according to the third or the fourth aspect, when executed on a computer.

In an embodiment, the computer program can be provided on a non-transitory computer-readable storage or recording medium. The computer program, when executed by a processor, causes the processor to perform the method according to the third or the fourth aspect.

In an embodiment, the computer program is implemented in programmable hardware. The programmable hardware may comprise components such as application-specific integrated circuits (ASICs), field-programmable arrays (FPGAs), digital signal processors (DSPs), or general-purpose processors.

DETAILED DESCRIPTION

FIG. 1shows a data communication system100comprising a device101(“encoder”) according to an embodiment of the present disclosure. The device101is configured to encode an input sequence u, which comprises the bits of a message m, into a codeword x using a polar code.

The device101is generally configured to obtain the input sequence u to be transmitted (the “input sequence” may also be termed “information word” or “input vector”), and to produce the codeword x, which may contain redundancy. This codeword x may, then, be transmitted over a noisy communication channel102to a device103(“decoder”) for decoding, e.g., according to another embodiment of the present disclosure. This transmission typically introduces errors. The noisy signal may be received by the decoder103as a sequence of bits y (the “sequence of bits” may also termed “output vector”). The decoder103may then use the received bit values to calculate estimates of the transmitted codeword x and the transmitted message m. The set of possible codewords is called the code, or the channel code. In this embodiment, a polar code is used at the encoder101to encode the input sequence u. Both the encoder101and decoder103may know the polar code and, thus, may be aware of positions of the frozen bits F or information set I, respectively. The information set I (sometimes also called “reliability sequence”) may be used by the decoder103in both determining the input sequence u (e.g. during successive decoding) and in extracting the message bits m from the input sequence u.

The encoder101, according to the embodiment of the present disclosure, is configured to sequentially encode each of a plurality of blocks uiof the input sequence u, by applying a sliding window to the input sequence u. Each block uiof the input sequence u is encoded based on an XOR operation of the block uiand a previous block ui-1of the input sequence u, to obtain a codeword block x′iof the codeword x′. Further, the encoder101is configured to sequentially output each obtained codeword block x′iof the codeword x′.

Moreover, the decoder103is configured to decode the sequence of bits y into an output sequence u using the polar code. The sequence of bits y represents the codeword x after transmission over the (noisy) communication channel102. The decoder103, according to an embodiment of the invention, is configured to sequentially decode each of a plurality of blocks of the sequence of bits y by applying a sliding window, in order to obtain an auxiliary sequence u′. Further, the decoder103is configured to process the auxiliary sequence u′ to obtain the output sequence u, wherein each block of the output sequence u is processed based on an XOR operation of a block of the auxiliary sequence and a previous block of the auxiliary sequence ui-1′. Furthermore, the device103can be configured to obtain message bits m from the output sequence u.

For the sake of the completeness, in the following, a summary of the theory behind the sliding window polar coding, as applied in embodiments of the present disclosure, is given.

An exemplary sliding window polar code of length N and dimension K, which it is decodable through a sliding window of size M, may be designed as follows. Given S=N/M, its transformation matrix can be defined as T=W2S⊗TM/2,where TM/2=T2⊗n, with m=log2(M), is the transformation matrix of a classical polar code of length M/2, while W2Sis a full binary lower triangular matrix of size 2S. This matrix is given by a square matrix of size 2S×2S having ones on and below the diagonal, and zeros above the diagonal. The resulting transformation matrix T can be described as a multi-kernel polar code (see the aforementioned work of Bioglio et al.). This also permits to calculate the frozen set F accordingly.

In the following, as also exemplarily illustrated inFIG. 2, an example of the proposed framework for a sliding window polar code of length N=16 and dimension K=8 with window size M=8 is given. It is assumed that a message m=00011110 is to be encoded. This message m can be encoded by means of the matrix T=W4⊗T4, wherein numerical representation of the matrices W4, T4, and T are shown as example inFIG. 3. In order to perform the encoding of the message m, the information set I or, equivalently, the frozen set of bits F is needed. For the generation matrix T, the information set I is shown inFIG. 3and may be given by I={4,8,11,12,13,14,15,16}. Moreover, as illustrated inFIG. 3, from the information set I, sub-information sets I1={4}, I2={4}, I3={3,4}, I4={1,2,3,4} may be obtained as well as the four corresponding sub-input vectors u1to u4populated accordingly (seeFIG. 2).

Depending on the sub-information set I1, . . . I4, the bits of the message m may be divided in four blocks m0=0001, m1=1, m2=1 and m3=10 (seeFIG. 2). The input vector blocks uiare, then, given by u0=0001, u1=0001, u2=0001 and u3=0010 (seeFIG. 2). The sliding window encoder101may first calculate x′1=u0·T4=1111 and transmits it, then other codeword block may be calculated sequentially obtaining x′2=(u1⊗u0)·T4=0000, x′3=(u2⊗u0)·T4=0000 and x′4=(u3⊗u0)·T4=0101 (seeFIG. 2).

At the decoder side, received blocks may be processed in couples to recover the auxiliary input vector u′. The first input vector is retrieved as u1=u′1=0001. After the second auxiliary input vector u′2=0000 has been decoded, the second input vector block can be calculated as u2=u′1⊕u′2=0001 and the message block m1=1 can be extracted. The other message blocks may be calculated accordingly with the subsequent auxiliary input vectors. InFIG. 4, the exemplary input vector u, its shifted vector ū, the vector u′ obtained on the basis of ū, and the corresponding codeword x′ are shown. The transmitted codeword x′ is received after transmission of the communication channel as vector y, as shown inFIG. 1.

The proposed polar code design and decoding procedure improves BLER performance in the Device-to-Device (D2D) scenario (seeFIG. 5) without increasing the decoding computational complexity, and in some cases permitting to reach the performance of the full length polar code.

In the following, some methods for determining the information set I are described.

Under AWGN channel, the DE/GA method (see, e.g., H. Vangala, E. Viterbo, and Y. Hong, “A comparative study of polar code constructions for the AWGN channel”, in arXiv preprint arXiv:1501.02473, 2015) can be used to evaluate bit-channels polarization of kernel W2stracking the LLR mean as:

where μ is the imput LLR mean and function ϕ can be calculated as described in J. Ha, J. Kim, and S. W. McLaughlin, “Rate-compatible puncturing of low-density parity-check codes”, IEEE Transactions on Information Theory, vol. 50, no. 11, pp. 2824-2836, 2004. Using the above metrics, the reliability of each bit of the input vector u can be calculated. The K bits having the highest reliability may form the information set I, while the indices of the remaining N−K bit-channels may form the frozen set F of the code. The K message bits may then be inserted in the input vector u according to the information I set previously calculated, namely storing their values in the indices listed in I, while the remaining bits of u are set to zero. Codeword x′ is then calculated as x′=u·T and transmitted through the channel102(seeFIG. 7).

The N channel LLRs may be stored in the vector y (seeFIG. 7). Sliding window decoder103may perform 2S polar decoding of (M/2, Kt) polar codes. Upper LLRs L0may be initialized to zero (seeFIG. 7). At step t, a sub-information set Itmay be calculated from the information set I as the set of entries of I comprised between

and t·M/2 reduced by (t−1)·M/2. This sub-information set may be used as the information set of a polar code having transformation matrix TM/2. The M/2 LLRs for this decoder103may be calculated as follows on the basis of y: the vector

is extracted from y, while a second vector L2of length M/2 is calculated as:

The LLRs to be given to the current decoder are calculated on the basis of these two vectors as =(L0+L1)L2, where

Next, the (M/2, Kt) polar code defined by Itmay be decoded via SC using L as channel LLRs. Decoding can be performed with any other polar decoder (device103), e.g. SCL. The resulting input vector utcan then used to calculate the partial sums used in the SC decoding as xt=ut·TM/2; these partial sums are then used to update the upper LLRs L0as L0=(L0+L1)·(1−2xt). When t=2S, decoding is concluded, and input vector u is calculated appending all the sub input vectors as u=[u1u2. . . u2S].

In the following, a mathematical demonstration of the robustness of the encoding performed by the device101and the decoding performed by the device103, according to embodiments of the present disclosure, is given.

As first, the device101may be configured to encode portions of the input vector u independently using a classical polar code. Moreover, the sliding window encoder mechanism may be implemented by the device101to perform on-the-fly encoding of incoming bits of message m.

The device101may divide the input vector u into 2S blocks of length M/2 as u=[u1u2. . . u2S] and polar encode each block independently, i.e. by matrix multiplication with channel transformation matrix of size M/2 as ci=ui·TM/2. These intermediary polar codewords cican be used to create a sliding window polar codeword x′, having a peculiar input vector u′, by transmitting first the last one and performing a XOR operation with all the subsequent polar codewords before transmission. As a consequence, the final transmitted codeword x′ is obtained as x′=[c2S|c1⊕c2S|c2⊕c2S| . . . |c2S-1⊕c2S].

The rationale to obtain the described framework can be found in the structure of the channel transformation matrix of a sliding window polar code. The codeword may be created on-the-fly without waiting to collect all the bits of the message m. Starting from the original input vector u=[u1u2. . . u2S], this result can be obtained creating the auxiliary input vector u′=[u′1u′2. . . u′2S] imposing u′i=ui-1⊕uiwith u0=0. This auxiliary input vector u′ may then have the structure u′=[u1|u1⊕u2|u2⊕u3| . . . |u2S-1⊕u2S]. Given the nested property of frozen sets, it is known that Ii-1⊇Ii, where Iiis the information set of input vector block ui, so that u and u′ share the same frozen set and u′ is then well defined. This intermediary vector is in fact the input vector of codeword x′ calculated with the proposed framework, x′=u′·T, since u′·T=[u2S·TM/2|(u1⊕u2S)·TM/2|u2⊕u2S)·TM/2| . . . |(u2S-1⊕u2S)·TM/2]. As a consequence, the result of sliding window decoding of transmitted codeword x′ will be the intermediary input vector u′. However, input vector u is needed to extract message bits, hence a further post processing step is needed at the receiver to retrieve message bits.

At the side of the decoder103, channel outcomes y corresponding to codeword x′ can be decoded by applying the sliding window. As discussed previously, the result of the decoding process is the auxiliary input vector u′. This vector has to be post-processed to retrieve the original input vector u. This process consists of performing a XOR operation of each input vector block with the previously decoded one such that ui=u′i-1⊕u′iwith u′0=0. As a consequence, original input vector can be calculated as u=[u′1|u′1⊕u′2|u′2⊕u′3| . . . |u′2S-1⊕u′2S].

Therefore, the proposed framework permits to create codes based on polarization that have on-the-fly encoding and decoding. The last input vector block u2Smay be moved to the beginning of the message m. It can be supposed that the K bits of message m are going to be transmitted through a sliding window polar code of length N and dimension K with window size M and information set I. For each block uiforming the input vector u it is in this case possible to extract a sub-information set Iiof size Ki. The K message bits stored in vector m are then divided into 2S blocks as m=[m0m1m2. . . m2S-1] of size K0, K1, K2, . . . , K2S-1respectively, where |m0|=K2Sand I0=I2S. Each block x′iof encoded vector x′ is then calculated as x′i=(ui-1⊕u0)·TM/2for i=1, . . . , 2S, with the exception of first block that is calculated as x′1=u0·TM/2. Each block of codeword x′ is then calculated using only two sub-input vectors, namely the very first and the current one.

At the decoder103side, the sliding window decoding may output blocks of auxiliary input vector u′ sequentially. On-the fly decoding may calculate original input vector blocks as ui=u′i-1⊕u′iand extract message bits mi. Last input vector block u2Sis used to calculate first message block m0.

FIG. 6shows a communication system100comprising the device101for encoding the input sequence u according to an embodiment.

In particular,FIG. 6shows the wireless communication system100including a base station101and user equipment (UE)103where the UE103may be a portable device such as a smart phone or tablet. The base station101includes a transmitter and the UE103a receiver, whereby the base station101is able to transmit data to the UE103, for example, in a downlink or uplink connection102made according to a telecommunications protocol. Embodiments of the present disclosure may be applied in various communications systems. For example, it could be applied to any of a Global System for Mobile Communications (GSM), code division multiple access (CDMA), wideband code division multiple access (WCDMA), general packet radio service (GPRS), long term evolution (LTE), LTE frequency division duplex (FDD), LTE Time Division Duplex (TDD), a universal mobile telecommunications system (UMTS), enhanced mobile broadband (eMBB), ultra-reliable low-latency communications (URLLC) and massive machine-type communications (mMTC), or any 5th generation (5G) wireless communication system.

FIG. 8shows a method800for encoding an input sequence comprising message bits into a codeword using a polar code according to an embodiment.

The method800comprises the steps of:sequentially encoding801each of a plurality of blocks of the input sequence, by applying a sliding window to the input sequence,encoding802each block of the input sequence based on an XOR operation of the block and a previous block of the input sequence, to obtain a codeword block of the codeword; andsequentially outputting803each obtained codeword block of the codeword.

FIG. 9shows a method900for decoding a sequence of bits into an output sequence using a polar code, wherein the sequence of bits represents a codeword after a transmission over a communication channel102according to an embodiment.

The method900comprises: sequentiallydecoding901each of a plurality of blocks of the sequence of bits by applying a sliding window, in order to obtain an auxiliary sequence,processing902the auxiliary sequence to obtain the output sequence, wherein each block of the output sequence is processed based on an XOR operation of a block of the auxiliary sequence and a previous block of the auxiliary sequence.

In the claims as well as in the description, the word “comprising” does not exclude other elements or steps and the indefinite article “a” or “an” does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation.