WO2022028672A1

WO2022028672A1 - Encoding and decoding scheme using symbol soft values

Info

Publication number: WO2022028672A1
Application number: PCT/EP2020/071854
Authority: WO
Inventors: Yi Qin; Branislav M POPOVIC
Original assignee: Huawei Technologies Co., Ltd.
Priority date: 2020-08-04
Filing date: 2020-08-04
Publication date: 2022-02-10
Also published as: CN116134735A

Abstract

The disclosure relates to an encoding and decoding scheme using symbol soft values. A first communication device (100) transmits a set of modulated symbols. The set of modulated symbols have been obtained based on set of coded bits by a linear transformation of a set of uncoded bits based on a coding matrix G which is a Kronecker product of two binary matrices G1 and G2. G1 is a n-th Kronecker power of matrix (I) and G2 is a K x K full rank binary matrix, where K is the modulation order of a modulation symbol constellation and K > 1, and where M ₀ is a multiple of K. A second communication device (300) receives the set of modulated symbols and obtains a set of decoded bits based on the set of received modulated symbols and a set of symbol soft values.

Description

ENCODING AND DECODING SCHEME USING SYMBOL SOFT VALUES

Technical Field

The disclosure relates to polar an encoding and decoding scheme using symbol soft values for improved coding performance.

Background

Polar code is a linear block error correcting code that is proven to achieve channel capacity for binary-input discrete memoryless channels (B-DMCs) with low encoding and decoding complexity. Due to these advantages, polar codes are utilized for transmission of uplink and downlink control signals for enhanced mobile broadband (eMBB) control channels in 3GPP New Radio (NR) system. However, if high order modulations, e.g., 16QAM, are used, the channels are no longer B-DMCs, and therefore polar code may not achieve the channel capacity.

Summary

An objective of examples of the disclosure is to provide a solution which mitigates or solves the drawbacks and problems of conventional solutions.

The above and further objectives are solved by the subject matter of the independent claims. Further advantageous examples of the disclosure can be found in the dependent claims.

According to a first aspect of the invention, the above mentioned and other objectives are achieved with a first communication device for a wireless communication system, the first communication device being configured to obtain a set of uncoded bits comprising M_o bits, wherein the set of uncoded bits comprises information bits; obtain a set of coded bits by a linear transformation of the set of uncoded bits based on a coding matrix G, wherein the coding matrix G is a Kronecker product of two binary matrices

G1 and G2, where G1 is a n-th Kronecker power of matrix and G2 is a K x K full rank

binary matrix, where K is the modulation order of a modulation symbol constellation and K > 1, and where M_o is a multiple of K obtain a set of modulated symbols by modulating the set of coded bits based on the modulation symbol constellation; and transmit the set of modulated symbols to a second communication device. A modulation order of a modulation symbol constellation can be understood as the number of bits carried in one modulation symbol. For example, if there are 2^K modulation symbols in the modulation symbol constellation, the modulation order is K.

To obtain a set of coded bits may be understood as to employ a polar code on a set of uncoded bits. Hence, polar codes are used for encoding.

An advantage of the first communication device according to the first aspect is that a polar-like code among every length-Zf bit segments is guaranteed. Thereby, the symbol soft values of every length-^ bit segments can be used for decoding at the receiver.

In an implementation form of a first communication device according to the first aspect,

An advantage of this implementation form is that it supports length M_o uncoded bits and guarantees that the number of coded bits is M_o.

In an implementation form of a first communication device according to the first aspect, K is a power of 2 when G2 is a log₂ K-th Kronecker power of matrix

In this case, G is a log₂ M₀ -th Kronecker power of matrix An advantage of this

implementation form is therefore that it guarantees a polar-like code among every length-^ bit segments for the special case that G is a log₂ M₀-th Kronecker power of matrix

In an implementation form of a first communication device according to the first aspect, obtaining the set of modulated symbols comprises obtain a subset of the set of coded bits, wherein the subset of the set of coded bits consists of entire segments of the set of coded bits; and obtain the set of modulated symbols by modulating the subset of the set of coded bits.

An advantage of this implementation form is that it guarantees that each entire bit segment is modulated or not, and therefore avoids modulate only a part of a bit segment.

In an implementation form of a first communication device according to the first aspect, obtaining the set of modulated symbols comprises obtain an extended set of coded bits, wherein the extended set of coded bits comprises the set of coded bits and one or more segments of the set of coded bits; and obtain the set of modulated symbols by modulating the extended set of coded bits.

An advantage of this implementation form is that it guarantees that each entire bit segment is modulated or not, and therefore avoids only modulate a part of a bit segment.

In an implementation form of a first communication device according to the first aspect, obtaining the set of modulated symbols comprises interleave the set of coded bits by interleaving entire segments of the set of coded bits; and obtain the set of modulated symbols by modulating the interleaved set of coded bits.

An advantage of this implementation form is that it guarantees that each entire bit segment is modulated to one modulation symbol, and therefore avoids modulate only a part of a bit segment.

In an implementation form of a first communication device according to the first aspect, a segment of coded bits is obtained according to

where k = 1, 2, 3 ... is an index of a bit in the set of coded bits and i_b = 1, 2, ... K.

An advantage of this implementation form is that a feasible bit segmentation solution is provided.

According to a second aspect of the invention, the above mentioned and other objectives are achieved with a second communication device for a wireless communication system, the second communication device being configured to receive a set of modulated symbols from a first communication device, wherein the set of received modulated symbols are associated with a set of coded bits and a modulation symbol constellation; obtain a set of symbol soft values based on the set of received modulated symbols and symbols of the modulation symbol constellation; and obtain a set of decoded bits associated with the set of coded bits based on the set of received modulated symbols and the set of symbol soft values. An advantage of the second communication device according to the second aspect is that the decoding is based on symbol soft values instead of bit LLRs. Thereby, improved performance is provided, e.g. in terms of lower error rate such as BLER.

In an implementation form of a second communication device according to the second aspect, the set of coded bits are obtained from a linear transformation of a set of uncoded bits based on a coding matrix G, wherein the coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th Kronecker power of matrix and G2 is a K x K

full rank binary matrix, where K is the modulation order of the modulation symbol constellation and K > 1, and where M_o is a multiple of K.

An advantage with this implementation form is that it guarantees a polar-like code among every length-^ bit segments. Thereby, the symbol soft values of every length-^ bit segments can be used for decoding at the receiver.

In an implementation form of a seco

nd communication device according to the second aspect,

An advantage with this implementation form is that it supports length-M₀ uncoded bit and guarantees that the number of coded bits is M_o.

In an implementation form of a second communication device according to the second aspect, each symbol soft value in the set of symbol soft values are obtained based on an inner product between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation when the properties of channel is unknown to the second communication device.

The channel here denotes the channel in which the set of modulated symbols are received. Hence, this is the case when the second communication device don't know the properties of the channel. For example, there is no reference or pilot signals for channel estimation or demodulation.

An advantage with this implementation form is to define the symbol soft value for the case channel is unknown, which is related to the correlation between received symbol and modulation symbols in the constellation. In an implementation form of a second communication device according to the second aspect, the inner product between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation is obtained according to

|YX^H | ² or YX^H where Y is the received modulated symbol in the set of received modulated symbols, X is the symbol of the modulation symbol constellation, and H is the conjugate transpose operator.

An advantage with this implementation form is that |YX^H |² is related to the probability that the transmitted signal is X if the received signal is Y, and YX^H is related to the probability that the transmitted signal is X if the received signal is Y and the angle between X and Y. YX^H can be used for the case that channels of different transmitted modulation symbols are similar. |YX^H |² can be used for all cases.

In an implementation form of a second communication device according to the second aspect, each symbol soft value in the set of symbol soft values is obtained based a difference between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation when the properties of a channel is known to the second communication device.

The channel here denotes the channel in which the set of modulated symbols are received. Hence, this is the case when the second communication device knows the properties of the channel.

An advantage with this implementation form is to define the symbol soft value for the case channel is known, which is related to the difference between received symbol and modulation symbols in the constellation.

In an implementation form of a second communication device according to the second aspect, the difference between the received modulated symbol in the set of received modulated symbols and the symbol of the modulation symbol constellation is obtained according to

where Y is the received modulated symbol in the set of received modulated symbols, X is the symbol of the modulation symbol constellation, and p is the signal-to-noise ratio of the received modulated symbol in the set of received modulated symbols. An advantage with this implementation form is to provide a symbol soft value for the case when the channel is known. p|Y - X|² is related to the difference between X and Y, where |Y — X|² is also called Euclidean distance between X and Y or Frobenius norm of the difference between X and Y.

In an implementation form of a second communication device according to the second aspect, obtaining the set of decoded bits comprises obtain the set of decoded bits based on soft values of decoded bit segments, wherein soft values of each decoded bit segment are determined based on soft values of at least one intermediate bit segment, and wherein soft values of each intermediate bit segment are determined based on soft values of two basic bit segments, wherein the sum of the two basic bit segments in Galois Field of two elements is equal to the intermediate bit segment, and wherein the soft values of the two bit segments are obtained based on the set of symbol soft values.

An advantage with this implementation form is that the decoder can calculate the soft values of decoded bit segments with low complexity.

In an implementation form of a second communication device according to the second aspect, obtaining the set of decoded bits comprises obtain the set of decoded bits based on soft values of decoded bit segments, wherein soft values of each decoded bit segment are determined based on soft values of at least one intermediate bit segment, and wherein soft values of each intermediate bit segment are determined based on initial soft values of the intermediate bit segment, soft values of one basic bit segment, and at least one decoded bit, and wherein the initial soft values of the intermediate bit segment are obtained based on the set of symbol soft values.

An advantage with this implementation form is that the decoder can calculate the soft values of decoded bit segments with low complexity by using this implementation.

According to a third aspect of the invention, the above mentioned and other objectives are achieved with a method for a first communication device, the method comprises obtaining a set of uncoded bits comprising M_o bits, wherein the set of uncoded bits comprises information bits; obtaining a set of coded bits by a linear transformation of the set of uncoded bits based on a coding matrix G, wherein the coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th Kronecker power of matrix and G2 is a K x K

full rank binary matrix, where K is the modulation order of a modulation symbol constellation and K > 1, and where M_o is a multiple of K\ obtaining a set of modulated symbols by modulating the set of coded bits based on the modulation symbol constellation; and transmitting the set of modulated symbols to a second communication device.

The method according to the third aspect can be extended into implementation forms corresponding to the implementation forms of the first communication device according to the first aspect. Hence, an implementation form of the method comprises the feature(s) of the corresponding implementation form of the first communication device.

The advantages of the methods according to the third aspect are the same as those for the corresponding implementation forms of the first communication device according to the first aspect.

According to a fourth aspect of the invention, the above mentioned and other objectives are achieved with a method for a second communication device, the method comprises receiving a set of modulated symbols from a first communication device, wherein the set of received modulated symbols are associated with a set of coded bits and a modulation symbol constellation; obtaining a set of symbol soft values based on the set of received modulated symbols and symbols of the modulation symbol constellation; and obtaining a set of decoded bits associated with the set of coded bits based on the set of received modulated symbols and the set of symbol soft values.

The method according to the fourth aspect can be extended into implementation forms corresponding to the implementation forms of the second communication device according to the second aspect. Hence, an implementation form of the method comprises the feature(s) of the corresponding implementation form of the second communication device.

The advantages of the methods according to the fourth aspect are the same as those for the corresponding implementation forms of the second communication device according to the second aspect.

The disclosure also relates to a computer program, characterized in program code, which when run by at least one processor causes said at least one processor to execute any method according to examples of the invention. Further, the disclosure also relates to a computer program product comprising a computer readable medium and said mentioned computer program, wherein said computer program is included in the computer readable medium, and comprises of one or more from the group: ROM (Read-Only Memory), PROM (Programmable ROM), EPROM (Erasable PROM), Flash memory, EEPROM (Electrically EPROM) and hard disk drive.

Further applications and advantages of the examples of the disclosure will be apparent from the following detailed description.

Brief Description of the Drawings

The appended drawings are intended to clarify and explain different examples of the invention, in which:

- Fig. 1 shows a first communication device according to an example of the disclosure;

- Fig. 2 shows a method for a first communication device according to an example of the disclosure;

- Fig. 3 shows a second communication device according to an example of the disclosure;

- Fig. 4 shows a method for a second communication device according to an example of the disclosure;

- Fig. 5 shows a wireless communication system according to an example of the disclosure;

- Fig. 6 shows more in detail the encoding and decoding schemes according to examples of the disclosure;

- Fig. 7 shows details of a polar decoder of a second communication device according to an example of the disclosure;

- Fig. 8 shows a butterfly-based decoder according to an example of the disclosure; and

- Fig. 9 shows performance results for an example of the disclosure.

Detailed Description

A transmitter employing polar code generally comprises at least three blocks: a polar encoder block, a bit segmentation block and a modulation block.

The polar encoder block includes a mapper which is used to map N information bits and M_o - N frozen bits and parity check bits (if any, generated based on information bits and predefined parity check functions) to a bit vector B of size 1 x M₀, where the vector length is M_o. After that, a polar code is applied as linear mapping from B to vector of coded bits C of size 1 x M₀, i.e. , C = BG in GF (2). G is a matrix with size M_o x M_o obtained by G = (G₀)®^{log2 M}° as the m₀-th Kronecker power of matrix

The bit segmentation block separates the coded bits into M/K bit segments and there are K bits in each bit segment. K is the number of bits conveyed by one modulation symbol.

The modulation block maps every K bits to a modulation symbol. If the modulation symbol contains more than one complex number, the modulation is called multi-dimensional modulation. Otherwise, the modulation is called single-dimensional modulation. The bit segmentation block and modulation block can also be considered as one combined block to generate modulation symbols based on coded bits.

A polar code receiver generally comprises two blocks: a demodulation block and a polar decoder block.

The demodulation block demodulates each received modulation symbol to estimate the Log- Likelihood Ratio (LLR) of each bit of the K modulated bits based on the received modulation symbol. If the channel is known at the receiver, the received modulation symbol is the symbol after equalization of the received signal. Otherwise, the received modulation symbol is the received signal. The first step of demodulation is to calculate the symbol soft-value of the demodulated symbol, which is defined as log probability of each constellation point, 2^K constellation points in total, being transmitted based on the received modulation symbol, i.e., as given in Eq. (1)

where X is a modulation symbol in the constellation, and Y is the received modulation symbol.

In the second step, the 2^K symbol soft values of each received modulation symbol are converted to bit soft value of each modulation bit, i.e., LLR value defined as in Eq. (2)

where b is a modulation bit. The method to obtain LLR(b) by LL(X) is introduced in Eq. (4). Alternatively, equivalent methods can be used to obtain LLR(b) based on X and Y.

The first step of the polar decoder is to estimate B, bit by bit, based on the M_o LLR values from demodulation block by using the successive cancellation (SC) polar decoding. The estimated bits of B are called decoded bits B. The principle of SC polar decoding to estimate the j-th bit

If is a frozen bit, = 0; If is a parity check bit, its value is obtained according to the previously decoded bits and parity check function;

11, otherwise where

•

= ) is the probability that Y was received and the previously decoded bits are [S⁽¹⁾,S⁽²⁾,

given the currently decoded bit is b.

Due to the recursive structure of the coding matrix, the SC polar decoding can be efficiently performed in a recursive manner by using a data flow graph with structure named a butterflybased decoder. In the recursive manner, the following f function and g function are used: f function: calculate the LLR of the sum of two bits by: with

• Input: LLR of bit a and b,

• Output: LLR of bit c = a ® b. g function: update the LLR of a bit based on its LLR and the LLR of another bit, and the sum of them is known:

with

• Input: estimated bit

• Output: updated LLR of bit α.

List decoding can be applied in this step to improve the decoding performance. By list decoding, the vector B is estimated bit by bit to obtain B. In particular, when the j-th bit (1 < i ≤ M_o) is estimated, the decoding algorithm determines a list of estimation of the first i bits. Each entry in the list contains a feasible estimate of the first i bits and the corresponding probability of this estimate. Comparing with unique decoding, which only outputs the most possible value of each bits, list decoding has a higher chance to achieve global optimal estimation. In addition, the maximum list size should be limited for complexity reason, e.g., no larger than L_max. The value of L_max is pre-defined, e.g., L_max = 8.

In the second step, the polar decoder gets the estimated information bits from B using a demapper which corresponds to the mapper at the transmitter. It can be found that the input of polar decoder are the M_o LLR values of bits, but the received signals are M_o/K modulation symbols and each corresponds to 2^K symbol soft values. How to fully utilize these symbol soft values to decode becomes an important issue.

The method in conventional solutions is to convert 2^K symbol soft values of each received modulation symbol to K LLR values by calculating the probability of each bit being a “0” or “1”.

The symbol soft value defined in Eq. (1) can be represented by ln?(x_/|Y), where Y is the received modulation, X_f is the f -th modulation symbol in the constellation and f e

where

• is the binary form of f, b^C^) is the j_b-th bit in b_f ,

• the numerator and denominator of Eq. (4) are probabilities that currently decoded bit is 1 and 0 given Y was received, respectively.

After this conversion, the LLR values can be directly used in a conventional polar decoder.

In the following disclosure it will however be demonstrated that the conventional calculation of LLR of each information bit, based on usage of coded-bits soft information under assumption of independent coded bits after demodulation, leads to erroneous LLR values of decoded bits if the codeword is generated by polar encoder for some modulation methods. It is also demonstrated that the same problem remains if the list decoding of a polar code is applied.

Each information bit can be expressed as a linear combination of coded bits in GF(2): at the polar encoder, the information bits are inserted to a length-M₀ binary vector B (size 1 x M_o) together with frozen bits and parity check bits, if any. Then the coded bits vector C (size 1 x M_o) of the same length-M₀ are obtained by a linear mapping as

C = BG (mod2) (5) where

C is the vector of coded bits,

B is the vector composed by information bits and frozen bits, (size M_o x M₀) is the log₂ M₀-th Kronecker power of the matrix

Due to the fact tha for any full rank

square matrices E and D, we have in GF(2). Hence, B = CG(mod2). So,

each information bit in B can be expressed as a linear combination of coded bits C in GF(2) with coefficients “0” or “1” in a column of G.

The problem of computing LLR of information bits in conventional polar decoder: in Eq. (5) we assume there are N information bits and M_f frozen bits in B, the length of B is M_o = N + M_f and also equals to the length of coded bit vector C. Consider an information bit b_L as the j-th bit in B, it is a linear combination of M_o coded bits in C with coefficients as the j-th column of G, which can be expressed as

in GF(2) where

We will compute the LLR of bt in Eq. (6), where the LLR is defined in Eq. (2). According to Eq.

(2) and (6), the LLR of information bit bt can be expressed as

(7)

In order to compute Eq. (7), a conventional polar decoder decomposes Eq. (7) into an expression consisting of LLR values of each coded bit. In particular, it is shown that the LLR of the sum of two statistically independent random binary variables U₁ and U₂ can be expressed as

If we assume

are statistically independent from each other, where

• z_u is an arbitrary index in Z_t,

• Zi\{z_u] is the set of all indexes in Zj except for

_u, t\( _u} { i \ _u) then LLR of b_L can be expressed as (for brevity, we use U₂ to represent ^cz ^as given

in Eq. (9))

In Eq. (10), we use

to represent the LLR of b_t under the assumption:

Assumption 1 ^are statistically independent.

The problem of using Eq. (10) in a conventional polar decoder is that Assumption 1 may not always hold In order to show what will happen if Assumption 1 does not hold, we will compare the

By inserting Eq. (2) into (10), we obtain

not statistically independent.

To demonstrate that U_r and U₂ might not be statistically independent, in the next section, we prove that it is the case for the 16QAM modulation. ^are not statistically independent at the receiver for 16QAM: in

this section, we first prove a general conclusion for 16QAM that two coded bits demodulated from the same 16QAM symbol are not statistically independent at the receiver Then based on this conclusion, we will give a corollary for 16QAM that ^are

not statistically independent at the receiver.

In 3GPP TS38.211, a 16QAM symbol can be generated by 4 bits. Without loss of generality, we consider a 16QAM symbol generated by four bits [a_lt a₂, a₃, a₄] as

Without loss of generality, we will prove that the demodulated a₄ and a₂ are not statistically independent by computing the conditional probability Pr(a₁ = a7|a₂ = a^, Y).

Assuming that the received symbol is Y, the probability that a₄ =

and a₂ = cT₂ can be expressed as

where

Substituting Eq. (15) and (16) into (14), we can obtain

For any 4 bits [a^, a^, a^, a^], Pr(X[aj,aj,aj,aj]|T) in Eq. (15) can be further expressed as

where σ² is the power of the noise of the channel. Substituting Eq. (13) and (18) into (17), after some mathematical derivations (see the Appendix 1), we arrive at

where Re{/} is the real part of Y. It is obvious from (19) that depends on a₂, implying the conclusion:

Conclusion 1: For 16QAM symbol generated by Eq. (13), <¾ is not independent from a₂. In Eq. (19), if <¾ is demodulated to be <¾, it means that

> 0.5, which is equivalent to exp and further equivalent to

Moreover, for the case (1 - 2%)Re{T) > 0, it can be observed from (19) that

. Therefore, we have the following conclusion:

Conclusion 2: for 16QAM, if <¾ is demodulated to be ¾, there must be

These conclusions can also be observed from the following numerical evaluation, in which we let the 16QAM symbol generated by Eq. (13) transmit through AWGN channel and compare the following bit error rates (BER):

• BER of a 1.

• BER of <¾ under condition: demodulated α₂ = 1,

• BER of <¾ under condition: demodulated α₂ = 0.

It can be found that the BER of α₁ is significantly related to whether a₂ is demodulated as “1” or “0” at the receiver. Thus, α₁ and a₂ are statistically dependent, i.e. , conclusion 1 is verified. We can find that BER curve of α₁ is lower and steeper on condition demodulated a₂ = 1 than on condition demodulated a₂ = 0, which implies that

0, Y) when <¾ is demodulated to be Therefore, conclusion 2 is verified.

We have proved for 16QAM that two bits, α₁ and a₂, demodulated from the same 16QAM symbol are not statistically independent. In order to prove that

are not statistically independent, we let α₁ = U₁ and a₂ be one bit in the summation the feasibility is proved in Appendix 2.

Considering the following two facts:

1) α₁ and a₂ are not statistically independent as proved above,

2) and α₂ are not statistically independent since α₂ is summed in U₂,

we can conclude that α₁ and U₂ are not statistically independent. Due to α₁ = U₁, we have the following corollary:

Corollary: are not statistically independent.

Evaluation for 16QAM using Eq. (12) to compute LLR results in decoding error: we have proved in section 2 and 3 that in Eq. (8) may not equal to LLR(bt) in Eq. (12) because the bits

demodulated from the same 16QAM symbol may be not statistically independent. In this section, we will provide example and numerical evaluation for 16QAM to show that using Eq. (10) in conventional decoder may further result in decoding error.

At the transmitter, we use Eq. (5) to generate coded bits. In particular, we consider length M_o = 8 polar code with M_inf0 = 4 information bits and M_f = 4 frozen bits, where the 4 information bits are assumed as [1,1, 1,1], The B and G in Eq. (5) are obtained as follows:

• In this example, we assume information bits are inserted to d₄, b₆, b₇, b₈ of the vector B as

B = [0,0,0, b₄, 0, b₆, b₇, b₈] = [0,0, 0,1, 0,1, 1,1] (20) with “0” in Eq. (20) being frozen bits, and b₄, b₆, b₇, b₈ being information bits.

Then, the coded bits C can be computed by Eq. (5) and we obtain

C = BG (mod2) = [0,1, 1,0, 1,0, 0,1]

After that, the first and second 4 bits in C are modulated into two 16QAM symbols, and

according to Eq. (13), respectively. The channel is AWGN channel. In this example,

we assume that SNR=0dB to simplify calculations. The received symbol vector Y is assumed as

We compare the following two decoders, i.e. bit-LLR based decoder and symbol soft value based decoder.

Bit-LLR based decoder is adopted at the receiver: when bit-LLR based decoder (i.e. the conventional polar decoder) is adopted, the LLR of each information bit is calculated by Eq. (13). By using the received signal in Eq. (21), the

computed by Eq. (11) are as follow

0

Then, we can obtain the decoded information bits

decoding error happens.

Symbol soft value based decoder is adopted at the receiver: the symbol soft value is defined in Eq. (1). In this example, the symbol soft value of modulation symbol α forr the t-th

received modulation symbol can be written as

where is one modulation symbol generated by 4 bits [α₁ α₂ , α₃, α₄] as in Eq.

(12),

• t Ε {1,2} is the index of received modulation symbol,

• Y_t is the t-th received modulation symbol.

Since there is one-to-one mapping between modulation symbol and 4 bits [α₁, α₂, α₃, α₄] as in (13), the LL(x_{[ai.a2.a3.a4]}, t) in Eq. (22) can be mapped to the probability of the values of the 4 bits [α₄, α₂, α₃, α₄], which can be used in Eq. (12) to compute LLR(bt).

In particular, let α₄ = c₄, α₂ = c₂, α₃ = c₃, α₄ = c₄ for generating the first modulation symbol by Eq. (13) and let α₁ = c₅, α₂ = c₆, α₃ = c₇, α₄ = c₈ for the second modulation symbol, where c₄ to c₈ are the 1^st to 8^th coded bits in C, respectively. Let c_Zu = c₁(i.e., z_u = 1) as an example, the Pr([c_Zu, U₂ ] = [0,0] | Y) in Eq. (12) can be expressed as:

SzeZi\{i] ^cz=° which is computed based on the symbol soft values. Similarly, we obtain

Finally, LLR(bJ can be calculated based on symbol soft values.

The way to compute Pr([c_Zu, U₂ ] = [0,0] |Y) in Eq. (23) based on symbol soft value is different from the conventional method given in Eq. (10) and (11). We compute LLR(bt) by Eq. (23) based on these symbol soft values, where { }

is calculated as in Eq. (18). The final results are as follow:

Then, we can obtain the decoded information bits i.e. , the decoding

is correct.

The examples above show that when symbol soft value based decoder can perform correct decoding, the bit-LLR based one may lead to decoding error. Furthermore, BLER simulation shows that the symbol soft value based decoder is statistically less prone to decoding error than the bit-LLR based decoder, i.e., about 20% reduction of BLER at SNR=0dB.

In the examples above, list decoding is not discussed. If list decoding is applied, decoding processes based on each entry in the list are the same as in the examples above. Therefore, the same problem exists for list decoding and the same solution can be applied.

Polar codes can achieve channel capacity for binary-input discrete memoryless channels (B- DMCs). However, if high order modulations, e.g., 16QAM, are used, the channels are no longer B-DMCs, and therefore polar codes may not achieve the channel capacity. By using the proposed encoder and decoder scheme herein disclosed, higher throughput can be achieved compared to bit-based decoder because the channel is DMC with symbol-input.

The same problem can be found for any channel coding if an information bit can be expressed as the sum of at least two correlated bits. For example, in low-density parity-check (LDPC) code, if there is a parity-check function includes one information bits and two other coded bits, and the two coded bits are correlated to each other due to modulation, e.g., 16QAM in the example in IDF, the same problem can be found.

Non-binary decoder is also considered for turbo code when non-binary turbo encoder is used, i.e., duo-binary turbo convolution code. In duo-binary turbo convolution code, the input of encoder is quaternary or with higher order.

Comparing with a duo-binary turbo convolution encoder, a polar encoder is binary encoder and not based on convolution. At the decoder, SC polar decoding can be efficiently performed in a recursive manner by a butterfly-based decoder, which is different from the iteration based duo-binary turbo convolution decoder. Due to the different structures of encoder and decoder, the functions in duo-binary turbo convolution code cannot be used for polar code. Therefore, a symbol soft value based polar decoder and corresponding encoder are desired.

For the above reasons a first communication device 100 and a second communication device 300 are herein disclosed according to examples of the invention. The first communication device 100 act as a transmitter and the second communication device 300 act as a receiver in the herein given examples but are not limited thereto.

Fig. 1 shows a first communication device 100 according to an example of the invention. In the example shown in Fig. 1 , the first communication device 100 comprises a processor 102, a transceiver 104 and a memory 106. The processor 102 may be coupled to the transceiver 104 and the memory 106 by communication means 108 known in the art. The first communication device 100 may further comprise an antenna or antenna array 110 coupled to the transceiver 104, which means that the first communication device 100 may be configured for wireless communications in a wireless communication system. That the first communication device 100 may be configured to perform certain actions can in this disclosure be understood to mean that the first communication device 100 comprises suitable means, such as e.g. the processor 102 and the transceiver 104, configured to perform said actions.

The processor 102 of the first communication device 100 may be referred to as one or more general-purpose central processing units (CPUs), one or more digital signal processors (DSPs), one or more application-specific integrated circuits (ASICs), one or more field programmable gate arrays (FPGAs), one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, and one or more chipsets.

The memory 106 of the first communication device 100 may be a read-only memory, a random access memory, or a non-volatile random access memory (NVRAM).

The transceiver 104 of the first communication device 100 may be a transceiver circuit, a power controller, an antenna, or an interface which communicates with other modules or devices.

In examples, the transceiver 104 of the first communication device 100 may be a separate chipset or being integrated with the processor 102 in one chipset. While in some examples, the processor 102, the transceiver 104, and the memory 106 of the first communication device 100 are integrated in one chipset. According to examples of the disclosure the first communication device 100 is configured to obtain a set of uncoded bits comprising M_o bits, wherein the set of uncoded bits comprises information bits. The first communication device 100 is further configured to obtain a set of coded bits by a linear transformation of the set of uncoded bits based on a coding matrix G. The coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th

Kronecker power of matrix and G2 is a K x K full rank binary matrix, where K is the

modulation order of a modulation symbol constellation and K > 1, and where M_o is a multiple of K. The first communication device 100 is further configured to obtain a set of modulated symbols by modulating the set of coded bits based on the modulation symbol constellation. The first communication device 100 is further configured to transmit the set of modulated symbols to a second communication device 300.

Generally, a Kronecker product of a first matrix A and a second matrix B of size M x N is to generate a third matrix C, where the element in the (M(d₁ - 1) + d ₂)-th row and (/V(d₃ - 1) + d₄)-th column of C is equal to the product of the element in the c^-th row and d₂-th column of A and the element in the d₃-th row and d₄-th column of B. Further, a n-th Kronecker power of matrix is the result of Kronecker producted by itself for n - 1 times.

Fig. 2 shows a flow chart of a corresponding method 200 which may be executed in a first communication device 100, such as the one shown in Fig. 1. The method 200 comprises obtaining 202 a set of uncoded bits comprising M_o bits, wherein the set of uncoded bits comprises information bits. The method 200 further comprises obtain 204 a set of coded bits by a linear transformation of the set of uncoded bits based on a coding matrix G, wherein the coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th

modulation order of a modulation symbol constellation and K > 1, and where M_o is a multiple of K. The method 200 further comprises obtaining 206 a set of modulated symbols by modulating the set of coded bits based on the modulation symbol constellation. The method 200 further comprises transmitting 208 the set of modulated symbols to a second communication device 300.

In examples of the invention, the coding matrix G may equal to a Kronecker product of the two binary matrices G1 and G2.

In examples of the invention, integer n is given by the formula:

In further examples of the invention, K is a power of 2 when G2 is a log₂ -th Kronecker power of matrix

Fig. 3 shows a second communication device 300 according to an example of the invention. In the example shown in Fig. 3, the second communication device 300 comprises a processor 302, a transceiver 304 and a memory 306. The processor 302 is coupled to the transceiver 304 and the memory 306 by communication means 308 known in the art. The second communication device 300 may be configured for both wireless and wired communications in wireless and wired communication systems, respectively. The wireless communication capability is provided with an antenna or antenna array 310 coupled to the transceiver 304, while the wired communication capability is provided with a wired communication interface 312 coupled to the transceiver 304. That the second communication device 300 is configured to perform certain actions can in this disclosure be understood to mean that the second communication device 300 comprises suitable means, such as e.g. the processor 302 and the transceiver 304, configured to perform said actions.

The processor 302 of the second communication device 300 may be referred to as one or more general-purpose CPUs, one or more DSPs, one or more ASICs, one or more FPGAs, one or more programmable logic devices, one or more discrete gates, one or more transistor logic devices, one or more discrete hardware components, and one or more chipsets.

The memory 306 of the second communication device 300 may be a read-only memory, a random access memory, or a NVRAM.

The transceiver 304 of the second communication device 300 may be a transceiver circuit, a power controller, an antenna, or an interface which communicates with other modules or devices.

In examples, the transceiver 304 of the second communication device 300 may be a separate chipset or being integrated with the processor 302 in one chipset. While in some examples, the processor 302, the transceiver 304, and the memory 306 of the second communication device 300 are integrated in one chipset. According to examples of the disclosure the second communication device 300 is configured to receive a set of modulated symbols from a first communication device 100. The set of received modulated symbols are associated with a set of coded bits and a modulation symbol constellation. The second communication device 300 is further configured to obtain a set of symbol soft values based on the set of received modulated symbols and symbols of the modulation symbol constellation. The second communication device 300 is further configured to obtain a set of decoded bits associated with the set of coded bits based on the set of received modulated symbols and the set of symbol soft values.

Fig. 4 shows a flow chart of a corresponding method 400 which may be executed in a second communication device 300, such as the one shown in Fig. 3. The method 400 comprises receiving 402 a set of modulated symbols from a first communication device 100. The set of received modulated symbols are associated with a set of coded bits and a modulation symbol constellation. The method 400 further comprises obtaining 404 a set of symbol soft values based on the set of received modulated symbols and symbols of the modulation symbol constellation. The method 400 further comprises obtaining 406 a set of decoded bits associated with the set of coded bits based on the set of received modulated symbols and the set of symbol soft values.

In examples of the invention, the second communication device 300 obtains the set of coded bits from a linear transformation of a set of uncoded bits based on a coding matrix G. The coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th Kronecker power of matrix and G2 is a K x K full rank binary matrix. K is the modulation

order of the modulation symbol constellation and K > 1, and M_o is a multiple of K.

In further examples of the invention, integer n is given by the formula:

Fig. 5 shows a communication system 500 according to an example of the invention. The wireless communication system 500 comprises a first communication device 100 and a second communication device 300 configured to operate in the communication system 500. For simplicity, the communication system 500 shown in Fig. 5 only comprises one first communication device 100 and one second communication device 300. However, the communication system 500 may comprise any number of first communication devices 100 and any number of second communication devices 300 without deviating from the scope of the invention.

In the communication system 500, the first communication device 100 act as a transmitter and the second communication device 300 act as a receiver. In other examples, the reverse case is possible. It is illustrated in Fig. 5 that the first communication device 100 transmits a set of modulated symbols to the second communication device 300 over a radio channel 510. Upon reception of a set of modulated symbols from the first communication device 100, the second communication device 300 obtains a set of symbol soft values and further obtain a set of decoded bits associated based on the set of received modulated symbols and the set of symbol soft values. It is further noted from Fig. 5 that the first communication device 100 is illustrated as a network access node, such as a base station; and the second communication device 300 is illustrated as a client device, such as a User Equipment. However, examples of the disclosure are not limited thereto.

For the polar code encoding and decoding scheme according to examples of the disclosure at least two objectives are achieved. The first objective is to provide for polar code an efficient method to calculate in the decoder of the second communication device 300 correct LLR values for each information bit, using the output signal of the demodulator of the transmitted modulation symbols. The second objective is to guarantee that the proposed decoder can be efficiently performed in a recursive manner by using a data flow graph with butterfly-based decoder structure, and therefore low decoding complexity can be achieved.

The concept of “bit segment” used in this disclosure may be defined as a segment of continuous bits in a bit stream, e.g., coded bits. The definition of soft value of a bit segment may be: (24)

where Pr(a = a₀ |Y) is the probability that a = a₀ given that Y was received. If the bit segment is used to generate a modulation symbol X_a, we have

In examples of the disclosure the following points may be made for the second communication device 300 with general polar decoder based on symbol soft value.

R1 : The input of the decoder of the second communication device 300 is symbol soft value. For a modulation symbol X and the received symbol Y (if the channel is known at the receiver, Y the received symbol after equalization), the symbol soft value is related to |YX^H |² or YX^H for the case the channel is unknown at the receiver, or related to p|Y - X|² for the case the channel is known at the receiver, where p is the SNR.

R2: The LLR of an information bit in the decoder of a codeword generated by a polar encoder, is calculated by using a corresponding set of soft values of modulation symbols used to transmit coded bits whose linear combination produces in the first communication device 100 the observed information bit. In particular, assume the i -th information bit b_L is a linear combination of a set of coded bits c_z, z e Z_L as defined in (6), i

can be calculated as

where

• S_x, b₂, ... , bi-! are previously estimated bits,

• M_o is the length of coded bits,

• K is the modulation order,

• C is the vector of coded bits c_t,

• C^(t) is the t-th bit segment of C, which is defined as C^(t) =

,

• X_c(_t) is the modulation symbol generated based on the coded bit segment C^(t),

• LL(x_c(_t), t) = ln Pr{x_t = X_c(_t) |Y_t} is the symbol soft value, which is the log probability of modulation symbol X_c(_t) being transmitted if the t-th received symbol is Y_t,

• i^{s a} sub-matrix which include the first to the (j - l)-th column of G (includes all rows).

In orderto implement the decoder in Eq. (25) with low complexity, the following novel f function and g function may be applied in a recursive manner at the second communication device 300 which is different from the f and g functions used in conventional decoders as previously described.

R3: The LLR of b_t in Eq. (25) can be calculated based on the following novel f function and g function: Definition of f function: calculate the soft values of the sum of two independent bit segments in GF(2) by:

with

• Input: soft values of bit segments a and b,

• Output: soft value of bit segment c = a ® b,

Definition of g function: update the soft values of a bit segment a based on its soft values and the soft values of another bit segment b, and the sum of these two bit segments c in GF (2) is known:

• Input: estimated sum of bit segment c = affib = c₀ , soft values of bit segments a and b,

• Output: soft value of bit segment a,

In order to apply the recursive manner at the second communication device 300, the first communication device 100 needs to guarantee the recursive structure of coding matrix for coded bit segments correspond to each transmitted modulation symbol. In particular, the following points may be made for the second communication device 300. the number of coded bits

and K is the modulation order, G₂ is a full rank binary matrix of size K

T2: If rate matching is applied, entire bit segment (of length K) should be kept, removed or repeated in the rate matched bits. This is explained more in detail in the following disclosure.

T3: If interleaving is applied after encoding, the interleaving should be bit-segment-level instead of bit-level, i.e., only change the order of each entire bit segment (length K). This is also explained more in detail in the following disclosure.

In examples of the invention, interleaving may be considered as part of rate matching. In this case, the rate matching includes at least two steps: a first step which is to select or remove or add some bits for repetition; and a second step which is interleaving. In this case, point T2 above is designed for the first step, and point T3 is for the second step.

With reference to Fig. 6 which illustrates further examples of the disclosure will hereby be described and explained. The terminology, expressions, systems design, etc. according to 3GPP NR may be used but is not limited thereto.

The second communication device 300 includes a demodulation block 320 coupled to a polar decoder block 322. If there are interleaving and/or rate matching performed at the first communication device 100, conventional de-interleaving and/or inverse operation of rate matching may be applied correspondingly at the second communication device 300 but are not illustrated in Fig. 6.

Demodulation block 320: the input of demodulation block 320 is a set of received modulation symbols which has been transmitted by the first communication device 100 over a radio channel 510. In the demodulation block 320 the probability of each symbol is calculated, which is equivalent to a symbol soft value of the symbol. The output of the demodulation block 320 that is provided to the polar decoder block 322 are symbol soft values instead of bit LLRs as in conventional solutions. A symbol soft value corresponds to a modulation symbol X in constellation and the received modulation symbol Y. It represents or relates to the probability that modulation symbol X was transmitted from the first communication device 100.

If the channel 510 is unknown at the second communication device 300, the probability may be calculated based on XY^H or |XY^H |², and therefore the symbol soft value can be XY^H or |XY^H |². The reason is that the logarithmic value of probability that X was transmitted given received symbol Y is proportional to |XY^H |². lf XY^H is used as symbol soft value, the phase information of the channel 510 is also considered. When calculating the soft value of the sum of two bit segments (corresponding to two symbols), we assume the phase information of the channel 510 of the two symbols are the same, i.e. , calculated by vector addition. Otherwise, |XY^H | ² may be used and phase information of the channel 510 is not considered, and therefore there is no restriction on channel phase, i.e., the soft value of the sum is calculated by scalar addition. Alternatively, |XY^H|² can be replaced by |XY^H| . Moreover, if symbols X and Y are matrixes, which means that there are multiple transmission antennas at the first communication device 100, the |XY^H|² above can be replaced by trace(XY^HYX^H).

In other words, each symbol soft value in the set of symbol soft values are obtained based on an inner product between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation when the properties of the channel 510 is unknown to the second communication device 300.

The inner product between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation may be obtained according to |YX^H|² or YX^H where Y is the received modulated symbol in the set of received modulated symbols, X is the symbol of the modulation symbol constellation, and H is the conjugate transpose operator.

On the other hand, if the channel 510 is known at the second communication device 300 and equalization is applied before demodulation, the symbol soft value can be p|Y - X| ² , where p is the SNR of the channel 510. The reason is that the logarithmic value of probability that X was transmitted given received Y is proportional to p|Y - X|². Optionally, p|Y - X|² can be replaced by Jp | Y - X| .

In other words, each symbol soft value in the set of symbol soft values is obtained based a difference between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation when the properties of the channel 510 is known to the second communication device 300.

The difference between the received modulated symbol in the set of received modulated symbols and the symbol of the modulation symbol constellation may be obtained according to p | Y - X|² where Y is the received modulated symbol in the set of received modulated symbols, X is the symbol of the modulation symbol constellation, and p is the signal-to-noise ratio (SNR) of the received modulated symbol in the set of received modulated symbols.

Moreover, the relative value of the symbol soft value can also be used by the second communication device 300. For example, when Xy is taken as reference (X^ is one constellation point), e.g., f = 0 or other values within {0,1, ... 2^K - 1}, the relative value can be |XY^H|² - |X_ZY^H|² for unknown channel at the second communication device 300, and for known channel at the second communication device 300.

Assuming there are 2^K modulation symbols in the constellation, each received modulation symbol will generate 2^K corresponding symbol soft values which are provided to the polar decoder 322.

Polar decoder 322: the polar decoder block 322 of the second communication device 300 includes two subblocks, i.e. a decoding block 330 and a de-mapper block 332 which corresponds to two steps, i.e.:

1 . Produce a decoded bit vector B by estimating the probability of B segment by segment, i.e. the decoding block 330,

2. Select information bits from B, i.e. the de-mapper block 332.

The second step, which corresponds to the de-mapper 332 of the polar decoder 322 is the same as in a conventional polar decoder. Hence, we only focus on the first step and therefore the decoding block 330.

Fig. 7 illustrates an exemplary solution which may be performed in the decoding block 330 and comprises three steps l-lll.

In step I in Fig. 7, the second communication device 300 obtains symbol soft values of each modulation symbols, which are the soft values of corresponding coded bit segments.

In step II in Fig. 7, the second communication device 300 calculates the probability of the q-th decoded bit segment according to the formula

In step III in Fig. 7, the second communication device 300 estimates for the q-th decoded bit segment as the bit segment with the largest probability. If the q-th decoded bit segment is the last bit segment to estimate, the decoding is finished and an estimation of B is outputted as shown in Fig. 7. Otherwise, the second communication device 300 will calculate the probability of the next decoded bit segment, i.e. for q = q + 1, as illustrated with the feedback line in Fig. 7 from step III to II.

An idea of the first step is to estimate each bit segment of B in turn based on a bit segment level successive cancellation (SC) polar decoding algorithm:

where P

| , , , is the probability that the currently decoded bit segment is given Y was received and the previously decoded bit segments were . The feasible values of bit segment should guarantee that the frozen bit is

0 and the parity check bit is correct. According to Eq. (29), the LLR of the i-th bits b_i in B can be calculated by Eq. (25). To further calculate Eq. (25), it can be expressed as in Eq. (11) as ^^^(^_^)

Similar to Eq. (23), the probability

_can be calculated as:

=

where: ● x_t is the t-th modulation symbol, ● x_C ^(t) is the modulation symbol generated based on the bit segment C^(t), ● LL(x_c ^(t), ) = ln Pr{x_t = x_c(t)|Y_t} is the symbol soft value defined in (22), where Y_t is the ^-th received symbol, ● G_(:,1:kq) is a sub-matrix which include the first to the ^^-th column of ^ including all rows, ●

(according to the proof in section 1.2.2.1) and

condition of the probability). The rest probabilities in Eq. (27) can be calculated in the same way as in Eq. (20), and finally we can obtain Eq. (25). Therefore, we can obtain the LLR of each bit in B based on the symbol soft values according to Eq. (27) and Eq. (28). This is important and different from conventional LLR-based polar decoders as in Eq. (11). The SC decoder based on Eq. (26) to (28) is novel because there is no LLR of coded bit needed/calculated at the second communication device 300. By using Eq. (26) and (27) problem mentioned previously can be avoided.

Moreover, since list decoding is applied, when making decision of the q-th decoded bit segment by Eq. (26), the decoder should output all feasible estimation of to

the list in descending order of P

If the list length exceeds the maximum length L_max after estimating the q-th bit segment, then keep L_max estimations of with the largest probability. Finally, after decoding all the bit segments of B,

output the most possible estimation.

The computation of Eq. (26) to (28) can be efficiently performed in a recursive manner by using a bit segment level data flow graph with structure named a butterfly-based decoder. In Appendix 3, we prove that the structure of butterfly-based decoder is the same as in conventional polar decoder with y- coded bits.

Two functions, i.e., a f function and a g function, may be used in the butterfly-based decoder which is shown Fig. 8. The f function is used to calculate the probabilities of the sum of two independent bit segments in GF (2), i.e. f function: for two independent bit segments a and b, if c = affib, then the probability of c satisfies

If we convert the probability in Eq. (29) to the symbol soft value, the following f function for symbol soft value can be obtained:

In other words, in examples of the disclosure the f function may be formulated as: obtaining the set of decoded bits comprises obtain the set of decoded bits based on soft values of decoded bit segments. The soft values of each decoded bit segment are determined based on soft values of at least one intermediate bit segment (e.g., c in Eq. (29) and (30)), and the soft values of each intermediate bit segment are determined based on soft values of two basic bit segments(e.g., bit segments a and b in Eq. (29) and (30)). The sum of the two basic bit segments in Galois Field (GF) of two elements is equal to the intermediate bit segment, and the soft values of the two bit segments are obtained based on the set of symbol soft values.

The g function is used to calculate the probabilities of the sum of two bit segments in GF(2) when one of them has already been estimated, i.e. g function: for two bit segments a and b, if c = affib and c is known as c = c₀, then the probability of a can be updated as

If we convert the probability in Eq. (31) to the symbol soft value, the following g function the symbol soft value can be obtained:

which can be used for calculating Eq. (28) in butterfly-based decoder. LL(b = a_offic_o) =

In other words, in examples of the disclosure the g function may be formulated as: obtaining the set of decoded bits comprises obtain the set of decoded bits based on soft values of decoded bit segments. The soft values of each decoded bit segment are determined based on soft values of at least one intermediate bit segment (e.g., a in Eq. (31) and (32)), and the soft values of each intermediate bit segment on the left side of Eq.

(32)) are determined based on initial soft values of the intermediate bit segment (e.g., soft value on the right side of Eq. (32)), soft values of one basic bit segment (e.g.,

basic bit segment is b in Eq. (31) and (32)), and at least one decoded bit (e.g., decoded bits in c₀ in Eq. (31) and (32)). The initial soft values of the intermediate bit segment are obtained based on the set of symbol soft values.

With reference back to Fig. 6, the first communication device 100 on the other hand comprises a polar encoder block 120 coupled to a bit segmentation block 122 which in turn is coupled to a modulation block 124.

Polar encoder block 120: the polar encoder block 120 includes two subblocks, i.e., a mapper block 130 and a linear transformation block 132. The first step and correspondingly the mapper block 130, which is inverse to the de-mapper block 332 in the polar decoder 322, is the same as in conventional polar encoders and will therefore not be described more in detail. The mapper block 130 obtains N bits and outputs a bit vector B with M_o uncoded bits which are provided to the linear transformation block 132.

For the second step and corresponding linear transformation block 132, which is inverse transform of decoding block 330 in the polar decoder 322, the bit vector B is linearly transformed using coding matrix G to obtain bit vector C with M_o coded bits i.e., C = BG. A conventional polar coding matrix G = can be used here, where M_o is the

number of coded bits. The bit vector C is provided to the bit segmentation block 122.

Alternatively, a set of coding matrices can also be used here as extension of the coding matrix used in linear transformation block 132. In particular, let

be the q-th bit segment of C, and define

where K is the modulation order.

Let C be the bit vector composed

is the q-th bit segment of C). According to Eq. (5) and (33), the coding matrix to generate C from B can be expressed as

which is a block coding matrix. The decoding is based on the structure of G_b(ocfe and rl oi® independent from transformation matrix j ^log2 between

Thus, any full

rank binary transformation matrix G₂ is feasible for the proposed decoder. Therefore, the coding matrix can be extended to

where G₂ can be any full rank binary matrix of size K x K.

From Eq. (35), the coded bits are generated based on a block coding matrix G_b(ocfe in Eq. (33) with recursive structure, and G₂ is a linear operation from bit segment

Thus, the recursive structure can be guaranteed if each modulation symbol is generated based on each bit segment which requires that the operations after encoding should not mix any two bit segments

Bit segmentation block 122: the soft values of each coded bit segment are needed at the polar decoder. In order to obtain the soft value for the bit segment at the second communication device 300, the same bit segmentation will be used at the first communication device 100, and each bit segment will be used to generate one modulation symbol. Therefore, the bit segmentation is in the bit segmentation block 122: the q-th coded bit segment includes the [K(q - 1) + l]-th to [Kq]-th bits in the vector C obtain from the linear transformation block 132.

The output of the bit segmentation block 122 is parallel bits segments with K number of bits each which are provided to the modulation block 124.

Modulation block 124: modulation is a mapping from the bit segments provided by the bit segmentation block 122 to modulation symbols in the modulation block 124. A difference from conventional modulation is that the modulation order K herein is power of 2 if

* . This is because that the transformation matrix between

and needs log₂ K to be an integer, i.e., K is power of 2.

In implementations of the invention, there may be one or more additional steps at the first communication device 100 and corresponding steps at the second communication device 300. Some examples are hereby described.

Interleaving: interleaving is generally to change order of coded bits in order to make the transmission more robust. However, the second communication device 300 needs to get the soft values of each entire original (before interleaving) coded bit segment. Hence, the interleaving should be bit-segment-level instead of bit-level, i.e., interleaving is to change order of entire bit segments.

Therefore, in examples of the disclosure the first communication device 100 obtains the set of modulated symbols based on interleave the set of coded bits by interleaving entire segments of the set of coded bits; and obtain the set of modulated symbols by modulating the interleaved set of coded bits.

Rate matching: rate matching is to change the length of coded bits to satisfy the scheduled resources. Assume the number of needed coded bits is M according to the scheduled resources and modulation order, it may not equal to M_o. In particular, if M < M_o, M bits can be selected from M_o coded bits at the transmitter. If M > M_o, repetition of coded bits can be used to generate M bits at the first communication device 100. The requirement of rate matching is that entire coded bit segment should be kept, removed or repeated in the rate matched bits. The reason is that selecting, removing or repeating half bit segment will make it impossible for the second communication device 300 to obtain the soft value of the whole bit segment. Therefore, in examples of the disclosure the first communication device 100 obtains the set of modulated symbols based on obtain a subset of the set of coded bits. The subset of the set of coded bits consists of entire segments of the set of coded bits. The first communication device 100 further obtains the set of modulated symbols by modulating the subset of the set of coded bits.

Furthermore, in examples of the disclosure the first communication device 100 obtains the set of modulated symbols based on obtain an extended set of coded bits. The extended set of coded bits comprises the set of coded bits and one or more segments of the set of coded bits. The first communication device 100 further obtains the set of modulated symbols by modulating the extended set of coded bits.

In this disclosure a segment of coded bits may be defined and obtained according to the formula

Moreover, as aforementioned there are two main cases at the second communication device 300 which have implications for the decoding procedure, i.e. the case when the second communication device 300 knows the properties of the channel 510 and the case when the second communication device 300 does not know the properties of the channel 510 which has previously been discussed. Knowledge of the channel 510 may e.g. relate to knowledge of the SNR, SNIR, phase rotation, or other relevant channel properties. These channel properties may be estimated based on reception of reference or pilot signals. However, information about the channel properties may also be received from other communication devices, e.g. in control signaling. Firstly, the case when the second communication device 300 do not know the properties of the channel 510 will be described further and thereafter the case when the second communication device 300 knows the properties of the channel 510.

The following aspects are when the properties of the channel 510 are unknown to the second communication device 300.

Modulation: in this case, multi-dimensional modulation can be used due to unknown channel at the second communication device 300. Each multi-dimensional modulation symbol contains multiple elements, i.e., as a vector x. Demodulation: the received symbol vector on the time-frequency (T-F) resources for mapping the t-th modulation symbol is y_t . There are two options that may be employed by the demodulation block 320.

relative values. In this embodiment, we use |y_tx * |² as example.

Option 2: the output of demodulation where x^ is the f-th symbol vector in the

constellation. In order to reduce complexity, the demodulator can output some y_tx * for some x_f with large value. The second option can be applied only when the channel of some

modulation symbols can be considered to be the same or similar to each other. Hence, in NR a base station/network access node may need to send control signal to a UE to indicate the time and/or frequency resource size(s) that can be considered as using the same beam/precoder or considered as the same channel. Otherwise, the resource size can be preconfigured or decided by receiver.

Polar decoder: symbol soft value based list polar decode to estimate B: the LLR of each bit in B can be estimated by Eq. (25) or by a recursive polar decoder. For the recursive polar decoder, according to Appendix 3 a butterfly-based decoder can be used as shown in Fig. 8 to estimate B . The butterfly-based decoder may be part of the decoder block 330 of the second communication device 300 in examples of the invention.

The butterfly-based decoder obtains 2^K symbol soft values of each received modulation symbol from the demodulation block 320. The estimation of B in the butterfly-based decoder includes stages as shown in Fig. 8, and each stage includes operations. Denote

the u-th operation of the v-th stage as The input of operation

'^s the output of operation O^. There is no operation at stage 0.

For other stages:

• The operation

is a f function if

is even,

• The operation

is a g function if

is odd. The f and g functions have been discussed previously and simplified operations are as follow:

For the decision, due to list decoding, the decision blocks in Fig. 8 need to output probabilities of all feasible decoded bit segments, which are represented by path metric (PM) values.

After all of the decisions have been made, output B whose bit segments are generated by multiplying by bit segments with the largest PM values from the decision

operations.

Alternatively, G-L can be multiplied by the binary indices of the input soft values, and therefore the indices of the input soft values are changed. According to Appendix 3 below, the butterfly-based structure of proposed recursive decoder in Fig. 8 with symbol soft value input is the same as conventional SC polar decoder. Since the complexity of conventional SC polar decoder is 0(M₀ logM₀), the propose recursive decoder is also with the complexity of order of O(M₀ logM₀).

In this example of the invention, the channel 510 is known at the second communication device 300. The channel 510 may e.g. be estimated based on detection of pilot symbols or reference symbols transmitted together with data symbols from the first communication device 100 to the second communication device 300 which is illustrated in Fig. 5. In this case, equalization is necessary before demodulation. Therefore, the set of received modulation symbol used in the demodulation block is a set of modulation symbols after equalization. The following aspects are special for this case.

Modulation: at the first communication device 100, the only difference compared to the embodiment when the channel is unknown is that one modulation symbol may be one complex value or a vector of complex values.

Demodulation: at the second communication device 300, the demodulation block is different from the example when the channel is unknown. After equalization, the symbol on the T-F resources for mapping the t-th modulation symbol is y_t. The output of demodulation are is the /-th symbol in the constellation, p is the SNR at the receiver. In

order to reduce complexity, the demodulation may output p\y_t - Xy |² for some x^ with large

value. Alternatively, the output can be a value computed based on pIy_t - x_f I², for example, exp (p\y_t - Xy|²), or relative values, p can also be expressed where a² is the

noise power after equalization.

Polar Decoder: at the second communication device 300, the f function in polar decoder is different from the example when the channel is unknown since the output of demodulation is different. The f function is as follow:

Performance results

We use a link-level evaluation on block error rate (BLER) for comparing the proposed polar decoder according to example of the disclosure and a conventional polar decoder in the case that Multi-dimensional modulation is applied and channel is unknown at the receiver. In Table 1 the simulation parameters are presented.

Table 1 : simulation parameters

The evaluation results are shown in Fig. 9 in which the x-axis shows the SNR in dB and the y- axis error rate in BLER. It can be found from Fig. 9 that a 2.3dB SNR gain can be achieved by the proposed polar decoder (solid line in Fig. 9) compared to the conventional decoder (dashed line in Fig. 9).

Appendix 1

Proof of equation (19):

First, we define the normalization factor

Substituting (18), (A1) into (17), we can obtain

where Re{} and Im{} are the real part and imaginary part, respectively. According to (12), we have

It can be found that R is only related to α₁ and α₂, } is only related

to α ₃ and α ₄. Thus — for any α₁, and are the

same for any a₃ and a₄. Hence, (A2) can be simplified as

Substituting (A3) into (A5) and replacing {0,1} by {a_r ,1 - a^, we can obtain

Then, (19) is obtained.

Appendix 2

Proof of: a₂ is in the summation

Since U₁ = c_Zu = a_±, the index of a_r and a₂ within the coded bit vector C are z_u and z_u + 1, respectively. Moreover, due to z_u e Z_t, according to the definition of Zj in (6), we have g_{Zu i} = 1. It can be proved that g i = 1 as follows:

Consider as in (5), it is obvious that g_k+lii = 1 if k is odd and g_{k i} = 1.

Since every 4 coded bits are modulated into one 16QAM symbol and a_r is the first bit modulated in a 16QAM symbol, the index of a_r within C must be odd, i.e. , z_u is odd. Because g_Zu ,i = 1 and z_u is odd, we can obtain g_Zu+lft = 1.

According to the definition of Zj in (6), the index of α₂ (i-e-, ^zu + 1) belongs to Zj due to q_Zu+i,i = 1. Therefore, a₂ is involved in the summation

Appendix 3

Proof of: The structure of butterfly-based decoder for symbol soft value based decoder is the same as in conventional polar decoder with

coded bits.

Proof: Since and C = BG, we can obtain

where B^w is the q-th bit segment in B, I_KxK is identity matrix. It can be observed that C can

block matrix with block size K x K, and the value of each block is l_KxK multiplied by an factor -( Thus, for the bit segment in B^1£fi , G_b(ocfe have the same structure as the conventional polar encoder with coded bits. Therefore, at the receiver,

the same butterfly-based decoder structure for conventional polar decoder with

coded bits

can be used for the proposed symbol soft value based decoder.

A client device in this disclosure includes but is not limited to: a UE such as a smart phone, a cellular phone, a cordless phone, a session initiation protocol (SIP) phone, a wireless local loop (WLL) station, a personal digital assistant (PDA), a handheld device having a wireless communication function, a computing device or another processing device connected to a wireless modem, an in-vehicle device, a wearable device, an integrated access and backhaul node (IAB) such as mobile car or equipment installed in a car, a drone, a device-to-device (D2D) device, a wireless camera, a mobile station, an access terminal, an user unit, a wireless communication device, a station of wireless local access network (WLAN), a wireless enabled tablet computer, a laptop-embedded equipment, an universal serial bus (USB) dongle, a wireless customer-premises equipment (CPE), and/or a chipset. In an Internet of things (IOT) scenario, the client device may represent a machine or another device or chipset which performs communication with another wireless device and/or a network equipment.

The UE may further be referred to as a mobile telephone, a cellular telephone, a computer tablet or laptop with wireless capability. The UE in this context may e.g. be portable, pocket- storable, hand-held, computer-comprised, or vehicle-mounted mobile device, enabled to communicate voice and/or data, via the radio access network, with another entity, such as another receiver or a server. The UE can be a station (STA), which is any device that contains an IEEE 802.11 -conformant media access control (MAC) and physical layer (PHY) interface to the wireless medium (WM). The UE may also be configured for communication in 3GPP related LTE and LTE-Advanced, in WiMAX and its evolution, and in fifth generation wireless technologies, such as NR.

A network access node in this disclosure includes but is not limited to: a NodeB in wideband code division multiple access (WCDMA) system, an evolutional Node B (eNB) or an evolved NodeB (eNodeB) in LTE systems, or a relay node or an access point, or an in-vehicle device, a wearable device, or a gNB in the fifth generation (5G) networks. Further, the network access node herein may be denoted as a radio network access node, an access network access node, an access point, or a base station, e.g. a radio base station (RBS), which in some networks may be referred to as transmitter, “gNB”, “gNodeB”, “eNB”, “eNodeB”, “NodeB” or “B node”, depending on the technology and terminology used. The radio network access nodes may be of different classes such as e.g. macro eNodeB, home eNodeB or pico base station, based on transmission power and thereby also cell size. The radio network access node can be a station (STA), which is any device that contains an IEEE 802.11 -conformant MAC and PHY interface to the wireless medium. The radio network access node may also be a base station corresponding to the 5G wireless systems.

Furthermore, any method according to examples of the disclosure may be implemented in a computer program, having code means, which when run by processing means causes the processing means to execute the steps of the method. The computer program is included in a computer readable medium of a computer program product. The computer readable medium may comprise essentially any memory, such as a ROM (Read-Only Memory), a PROM (Programmable Read-Only Memory), an EPROM (Erasable PROM), a Flash memory, an EEPROM (Electrically Erasable PROM), or a hard disk drive.

Moreover, it is realized by the skilled person that examples of the first communication device 100 and the second communication device 300 comprises the necessary communication capabilities in the form of e.g., functions, means, units, elements, etc., for performing the solution. Examples of other such means, units, elements and functions are: processors, memory, buffers, control logic, encoders, decoders, rate matchers, de-rate matchers, mapping units, multipliers, decision units, selecting units, switches, interleavers, de-interleavers, modulators, demodulators, inputs, outputs, antennas, amplifiers, receiver units, transmitter units, DSPs, MSDs, TCM encoder, TCM decoder, power supply units, power feeders, communication interfaces, communication protocols, etc. which are suitably arranged together for performing the solution.

Especially, the processor(s) of the first communication device 100 and the second communication device 300 may comprise, e.g., one or more instances of a Central Processing Unit (CPU), a processing unit, a processing circuit, a processor, an Application Specific Integrated Circuit (ASIC), a microprocessor, or other processing logic that may interpret and execute instructions. The expression “processor” may thus represent a processing circuitry comprising a plurality of processing circuits, such as, e.g., any, some or all of the ones mentioned above. The processing circuitry may further perform data processing functions for inputting, outputting, and processing of data comprising data buffering and device control functions, such as call processing control, user interface control, or the like.

Finally, it should be understood that the disclosure is not limited to the examples described above, but also relates to and incorporates all examples within the scope of the appended independent claims.

Claims

1. A first communication device (100) for a communication system (500), the first communication device (100) being configured to obtain a set of uncoded bits comprising M_o bits, wherein the set of uncoded bits comprises information bits; obtain a set of coded bits by a linear transformation of the set of uncoded bits based on a coding matrix G, wherein the coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th Kronecker power of matrix and G2 is a K x K full rank

binary matrix, where K is the modulation order of a modulation symbol constellation and K >

1. and where M_o is a multiple of K obtain a set of modulated symbols by modulating the set of coded bits based on the modulation symbol constellation; and transmit the set of modulated symbols to a second communication device (300).

2. The first communication device (100) according to claim 1 , wherein

3. The first communication device (100) according to claim 1 or 2, wherein K is a power of 2 when G2 is a log₂ K'-th Kronecker power of matrix

4. The first communication device (100) according to any one of the preceding claims, wherein obtaining the set of modulated symbols comprises obtain a subset of the set of coded bits, wherein the subset of the set of coded bits consists of entire segments of the set of coded bits; and obtain the set of modulated symbols by modulating the subset of the set of coded bits.

5. The first communication device (100) according to any one of the preceding claims, wherein obtaining the set of modulated symbols comprises obtain an extended set of coded bits, wherein the extended set of coded bits comprises the set of coded bits and one or more segments of the set of coded bits; and obtain the set of modulated symbols by modulating the extended set of coded bits.

6. The first communication device (100) according to any one of the preceding claims, wherein obtaining the set of modulated symbols comprises interleave the set of coded bits by interleaving entire segments of the set of coded bits; and obtain the set of modulated symbols by modulating the interleaved set of coded bits.

7. The first communication device (100) according to any one of claims 4 to 6, wherein a segment of coded bits is obtained according to

8. A second communication device (300) for a communication system (500), the second communication device (300) being configured to receive a set of modulated symbols from a first communication device (100), wherein the set of received modulated symbols are associated with a set of coded bits and a modulation symbol constellation; obtain a set of symbol soft values based on the set of received modulated symbols and symbols of the modulation symbol constellation; and obtain a set of decoded bits associated with the set of coded bits based on the set of received modulated symbols and the set of symbol soft values.

9. The second communication device (300) according to claim 8, wherein the set of coded bits are obtained from a linear transformation of a set of uncoded bits based on a coding matrix G, wherein the coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th Kronecker power of matrix and G2 is a K x K full rank binary matrix, where

K is the modulation order of the modulation symbol constellation and K > 1, and where M_o is a multiple of K.

10. The second communication device (300) according to claim 9, wherein

11. The second communication device (300) according to any one of claims 8 to 10, wherein each symbol soft value in the set of symbol soft values are obtained based on an inner product between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation when the properties of the channel (510) is unknown to the second communication device (300).

12. The second communication device (300) according to claim 11 , wherein the inner product between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation is obtained according to

|YX^H | ² orYX^H where Y is the received modulated symbol in the set of received modulated symbols, X is the symbol of the modulation symbol constellation, and H is the conjugate transpose operator.

13. The second communication device (300) according to any one of claims 8 to 12, wherein each symbol soft value in the set of symbol soft values is obtained based a difference between a received modulated symbol in the set of received modulated symbols and a symbol of the modulation symbol constellation when the properties of the channel (510) is known to the second communication device (300).

14. The second communication device (300) according to claim 13, wherein the difference between the received modulated symbol in the set of received modulated symbols and the symbol of the modulation symbol constellation is obtained according to p|Y - X|² where Y is the received modulated symbol in the set of received modulated symbols, X is the symbol of the modulation symbol constellation, and p is the signal-to-noise ratio of the received modulated symbol in the set of received modulated symbols.

15. The second communication device (300) according to any one of claims 8 to 14, wherein obtaining the set of decoded bits comprises obtain the set of decoded bits based on soft values of decoded bit segments, wherein soft values of each decoded bit segment are determined based on soft values of at least one intermediate bit segment, and wherein soft values of each intermediate bit segment are determined based on soft values of two basic bit segments, wherein the sum of the two basic bit segments in Galois Field of two elements is equal to the intermediate bit segment, and wherein the soft values of the two bit segments are obtained based on the set of symbol soft values.

16. The second communication device (300) according to any one of claims 8 to 15, wherein obtaining the set of decoded bits comprises obtain the set of decoded bits based on soft values of decoded bit segments, wherein soft values of each decoded bit segment are determined based on soft values of at least one intermediate bit segment, and wherein soft values of each intermediate bit segment are determined based on initial soft values of the intermediate bit segment, soft values of one basic bit segment, and at least one decoded bit, and wherein the initial soft values of the intermediate bit segment are obtained based on the set of symbol soft values.

17. A method (200) for a first communication device (100), the method (200) comprising: obtaining (202) a set of uncoded bits comprising M_o bits, wherein the set of uncoded bits comprises information bits; obtaining (204) a set of coded bits by a linear transformation of the set of uncoded bits based on a coding matrix G, wherein the coding matrix G is a Kronecker product of two binary matrices G1 and G2, where G1 is a n-th Kronecker power of matrix and G2 is a K x K

full rank binary matrix, where K is the modulation order of a modulation symbol constellation and K > 1, and where M_o is a multiple of K\ obtaining (206) a set of modulated symbols by modulating the set of coded bits based on the modulation symbol constellation; and transmitting (208) the set of modulated symbols to a second communication device (300).

18. A method (400) for a second communication device (300), the method (400) comprising: receiving (402) a set of modulated symbols from a first communication device (100), wherein the set of received modulated symbols are associated with a set of coded bits and a modulation symbol constellation; obtaining (404) a set of symbol soft values based on the set of received modulated symbols and symbols of the modulation symbol constellation; and obtaining (406) a set of decoded bits associated with the set of coded bits based on the set of received modulated symbols and the set of symbol soft values.

19. A computer program with a program code for performing a method according to claim 17 or 18 when the computer program runs on a computer.