A CHANNEL ENCODER AND A METHOD FOR ENCODING AN INFORMATION WORD

TECHNICAL FIELD

Generally, the present invention relates to the field of channel coding. More specifically, the present invention relates to a channel encoder and a corresponding method for encoding an information word.

BACKGROUND

Channel codes are essential in all digital communications systems. A system for forward error correction (FEC) coding, also called 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 obtains the transmitted data free of errors despite the noisy communication channel. Figure 1 shows such a communication system 100, wherein the data u to be transmitted, termed information word, is given to an encoder 101 , which produces the codeword x containing redundancy. This is then transmitted over a noisy communication channel 103 which typically introduces errors. The output vector y is provided to a decoder 105, which produces estimates of the transmitted codeword and the transmitted data. The set C of possible codewords is called the code, or channel code, and the following is particularly concerned with such a code.

For complexity reasons at the encoder and the decoder side, linear codes over finite fields are typically employed. The following description is provided for the finite field F ₂ = {0,1} of size 2 for the sake of simplicity; however, everything holds similarly for other fields or rings. A code C of length N and dimension K, for short, an ( N, K ) code, may be defined by a generator matrix G of size K x N\

C = {x uG: u e F ₂ }. In that case, the encoder that maps the information word u of length K to the code word x of length N is given by x = uG where addition and multiplication are over the binary field {0,1}. Alternatively, the code C may be defined by the parity check matrix H of size (N— K) x N\

C = {x E F ₂ ^w : XH ^T = 0}

where H ^T denotes the transpose of H.

For notational convenience, M = N - K will also be used in the following. It is to be noted that by the definition of a linear code via the parity check matrix H, a vector x is a code word if and only if xH ^T = 0.

For a given generator matrix, check matrices can be determined and vice versa (see F.J. MacWilliams & N.J.A. Sloane: The Theory of Error-Correcting Codes. North-Holland Publishing, 1977).

For an efficient encoding and improved error performance, check matrices are typically decomposed into two parts: H = [H _S H _P ] , where H _p is a full rank QV - K) x (N - K ) matrix.

The expression [H _S , H _p \ denotes the concatenation of H _s and H _p .The part H _p is called the parity forming part and H _s the syndrome forming part. Systematic encoding works as follows: to a message u of length K, redundancy bits p of length N - K are appended. The resulting vector x = [ u, ] is a code word of length N if xH ^T = 0 <= p = uHj{H _p ) ¹ . The parity bits p can thus be calculated from the message u in two steps: a first step of calculating the syndrome s = uH and a second step of calculating the parity bits p = Such encoding is called systematic, because a code word consists of the unaltered message u with the parity bits p appended.

Frequency bands for data transmission are a very expensive and restricted resource in wireless, fibre-optic and copper-cable communication. To mitigate this bandwidth limitation, higher-order modulation is required, where more than 1 bit is mapped to each real-dimensional time-frequency slot. Common higher-order modulation formats are quadrature amplitude modulation (QAM) and amplitude phase-shift keying (APSK).

In coherent transmission, the electric energy required to transmit one signal point is proportional to the square of its absolute value. Consequently, the outer points of a modulation format require significantly more energy than the inner points. The energy- efficiency of a transmission scheme can therefore be improved by transmitting the inner points more frequently than the outer points. Techniques for implementing such power- efficient non-uniform signalling are called probabilistic shaping.

Probabilistic shaping includes mapping to a shaping set S Q F ^w containing sequences with desired properties. For example, the set S Q {0,1} ³ with the four sequences of lowest weight is

5 = {000,001,010,100}.

Arbitrary bit strings can be mapped into sequences in S by successively applying the map:

00 ^| ® 000

01 ·® 001

10 ·® 010

11 ^| ® 100.

It is to be noted that the shaping set S is not linear, e.g., adding 001 e S and 010 e S results in Oil, which is not in the set S. Mapping into shaping sets can be realized by distribution matching (DM), which refers to algorithms dedicated for this task (P. Schulte & G. Bocherer,“Constant composition distribution matching,” IEEE Trans. Inf. Theory, vol. 62, pp. 430, 2016). Figure 2 shows an example of a conventional communication system 200 comprising on the side of the channel encoder a distribution matcher 201 and a FEC encoder 203.

Probabilistic amplitude shaping (PAS) is an efficient way to combine probabilistic shaping with forward error correcting (see G. Bocherer et. al.,“Bandwidth efficient and rate- matched low-density parity-check coded modulation”, IEEE Trans. Commun., vol. 63, pp. 4651 , 2015). As displayed in figure 2, a distribution matcher (DM) 201 serves as the shaping device, followed by a linear FEC encoder 203 (see G. Bocherer et. al., “Bandwidth efficient and rate-matched low-density parity-check coded modulation”, IEEE Trans. Commun., vol. 63, pp. 4651 , 2015).

In a probabilistic amplitude shaping (PAS) scheme, only the message is shaped and the parity is appended un-shaped. In the modulator, shaped bits select the signal point amplitudes and the un-shaped bits select the signal point signs, which is the reason for the name“probabilistic amplitude shaping”. Formally, as also illustrated in figure 3, a distribution matcher g 301 maps a message t e F ^L to a shaped message u = g(t ) e S c F ^K and then u is encoded systematically. The resulting vector [u, p] has the following properties: [u, p]H ^T = 0, i.e., [ u, p ] e C; the message part is shaped, i.e., u e 5; and the parity part is un-shaped, i.e., p may take any value in F ^N~K without restriction.

Conventional systematic encoding and probabilistic amplitude shaping split the parity check matrix in a syndrome former H _s 303 and a parity former H _p 305 which is square and full rank, as shown in figure 3. The parity p = S(H _p ) is fully determined by the sydrome s and no shaping constraints can be applied in the calculation of the parity. Consequently, probabilistic amplitude shaping has an un-shaped parity part p, which is severely sub- optimal in some important cases.

For instance, in a hybrid automatic repeat request (HARQ) with incremental redundancy protocol as employed in 5G network systems, the additional parity bits are transmitted when the receiver emits a NACK message. Since the parity is unshaped in probabilistic amplitude shaping, the power efficiency of the incremental redundancy is sub-optimal.

Furthermore, when data is modulated onto the amplitude of the signal for intensity modulation, all bits must be shaped for power-efficient transmission. Consequently, un- shaped parity bits are sub-optimal. The un-shaped parity part also impacts on any scenario where optimal signalling requires to shape some or all of the parity bits.

In light of the above, there is a need for an improved channel encoder and a

corresponding method allowing, in particular, for encoding an information word into a shaped codeword with shaped parity bits more efficiently. SUMMARY

It is an object of the invention to provide an improved channel encoder and a

corresponding method allowing, in particular, for encoding an information word into a shaped codeword with shaped parity bits more efficiently.

The foregoing and other objects are achieved by the subject matter of the independent claims. Further implementation forms are apparent from the dependent claims, the description and the figures.

Power efficient signalling requires to transmit signal points with low power more often than signal points with large power. This translates for the forward error correction (FEC) system into the requirement that certain bit patterns should occur more often than others: for instance, the bits output by the FEC encoder can be required to follow a non-uniform distribution with zeros being output more often than ones.

Embodiments of the invention enable optimal transmission by providing an efficient encoder and a corresponding method to unambiguously map a message to a vector for a linear code and a non-linear shaping set, i.e., to encode an information word into a shaped codeword with shaped parity bits in an efficient manner.

More specifically, according to a first aspect the invention relates to a channel encoder configured to encode an information word u of length K - J bits with K > J > 0 into a code word x of length N bits with N - K = M > 0, wherein the code word x contains the information word u and M + J parity bits p and wherein the channel encoder is configured to implement a probabilistic shaping scheme for the M + J parity bits p such that the code word x = [ u, p ] is a code word of a linear code C with dimension K and length N and the M + J parity bits p fulfil a first shaping constraint, i.e. associated with the probabilistic shaping scheme (as already described above, [u, p] denotes the concatenation of u and p).

Thus, an improved channel encoder is provided, allowing for encoding an information word into a shaped codeword with shaped parity bits in an efficient manner. In a further possible implementation form of the first aspect, the channel encoder is configured to implement the probabilistic shaping scheme such that further the information word u fulfils a second shaping constraint.

In a further possible implementation form of the first aspect, the first shaping constraint is equal to the second shaping constraint.

In a further possible implementation form of the first aspect, the first shaping constraint requires that the number of 0 bits is substantially different from the number of 1 bits of the M + J parity bits p of the code word x.

In a further possible implementation form of the first aspect, the number of 0 bits is substantially larger than the number of 1 bits of the M + J parity bits p of the code word x, wherein the channel encoder is further configured to map the M + J parity bits p to a plurality of signal points having at least one low energy level and one high energy level by mapping the 0 bits of the M + J parity bits p to the low energy level signal points.

In a further possible implementation form of the first aspect, the number of 1 bits is substantially larger than the number of 0 bits of the M + J parity bits p of the code word x, wherein the channel encoder is further configured to map the M + J parity bits p to a plurality of signal points having at least one low energy level and one high energy level by mapping the 1 bits of the M + J parity bits p to the low energy level signal points.

In a further possible implementation form of the first aspect, the channel encoder comprises a distribution matcher, wherein the distribution matcher is configured to generate the information word u of length K - J bits on the basis of L data bits with K - ] > L.

In a further possible implementation form of the first aspect, the channel encoder is configured to encode the information word u of length K - J bits with K > J > 0 to a code word x of length N bits on the basis of a parity check matrix H, wherein the channel encoder is configured to use the parity check matrix H in the following form:

H = [H _S H _P ] , wherein H _s denotes a syndrome former of size (N - K) x ( K -/) and H _p denotes a non square full row rank parity former of size (N - K) x (N - K + /).

In a further possible implementation form of the first aspect, the channel encoder is configured to generate a syndrome s of length M bits using the syndrome former H _s of size (N— K) x ( K -/) and the information word u of length K - ] on the basis of the following equation: s = uH , wherein H denotes the transpose of the syndrome former H _s .

In a further possible implementation form of the first aspect, the channel encoder is configured to generate the M + J parity bits p on the basis of the following equation: p// = s, wherein H _p denotes the transpose of the parity former H _p .

In a further possible implementation form of the first aspect, the parity check matrix H defines a linear forward error correction, FEC, code such that the FEC code has good error correcting capabilities.

In a further possible implementation form of the first aspect, the parity check matrix H defines a low-density parity check, LDPC, code.

In a further possible implementation form of the first aspect, the channel encoder is configured to consider the parity former H _p to be the parity check matrix of a linear code and to determine the M + J parity bits p on the basis of the syndrome s using the following equation: p// = s . In a further possible implementation form of the first aspect, the channel encoder is configured to generate a Trellis representation of the parity former H _p and to determine the parity bits p from the syndrome s by applying a Viterbi algorithm on the Trellis representation of the parity former H _p .

In a further possible implementation form of the first aspect, the channel encoder is configured to use a cost function for the Viterbi algorithm, wherein the cost function is defined by, in particular identical to the first shaping constraint.

In a further possible implementation form of the first aspect, the parity former H _p has a diagonal structure optimized for efficient parity forming, in particular a Trellis

representation of the parity former H _p with a reduced number of states.

According to a second aspect the invention relates to a method of encoding an information word u of length K - J bits with K > J > 0 into a code word x of length N bits with N - K = M > 0, wherein the code word x contains the information word u and M + J parity bits p, by implementing a probabilistic shaping scheme for the K - J bits of the information word u and the M + J parity bits p such that the code word x = [u, p] is a code word of a linear code C and the M + J parity bits p fulfil a first shaping constraint, i.e. associated with the probabilistic shaping scheme.

Thus, an improved method is provided, allowing for encoding an information word into a shaped codeword in an efficient manner.

According to a third aspect the invention relates to a computer program product with program code for performing the method according to the second aspect when executed on a computer.

The invention can be implemented in hardware and/or software.

BRIEF DESCRIPTION OF THE DRAWINGS

Further embodiments of the invention will be described with respect to the following figures, wherein: Fig. 1 shows a schematic diagram illustrating a communication system comprising an encoder, a communication channel and a decoder;

Fig. 2 shows a schematic diagram illustrating a communication system implementing a probabilistic amplitude shaping scheme;

Fig. 3 shows a schematic diagram illustrating an encoder for probabilistic amplitude shaping;

Fig. 4 shows a schematic diagram illustrating a channel encoder according to an embodiment; and

Fig. 5 shows a schematic diagram illustrating a method of encoding an information word according to an embodiment.

In the various figures, identical reference signs will be used for identical or at least functionally equivalent features.

DETAILED DESCRIPTION OF EMBODIMENTS

In the following description, reference is made to the accompanying drawings, which form part of the disclosure, and in which are shown, by way of illustration, specific aspects in which the present invention may be placed. It is understood that other aspects may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense, as the scope of the present invention is defined by the appended claims.

For instance, it is understood that a disclosure in connection with a described method may also hold true for a corresponding device or system configured to perform the method and vice versa. For example, if a specific method step is described, a corresponding device may include a unit to perform the described method step, even if such unit is not explicitly described or illustrated in the figures. Further, it is understood that the features of the various exemplary aspects described herein may be combined with each other, unless specifically noted otherwise. As will be described in more detail in the following, embodiments of the invention allow providing shaped parity bits, whereas the conventional probabilistic amplitude shaping (PAS) scheme has un-shaped parity bits. Embodiments of the invention offer in particular the advantage of optimal shaping with integration of forward error correction (FEC) for any input distribution.

Figure 4 shows a schematic diagram of a channel encoder 400 according to an embodiment. The channel encoder 400 is configured to encode an information word u of length K -] bits with K > J > 0 into a code word x of length N bits with N - K = M > 0. The code word x contains the information word u and M + J parity bits p.

The channel encoder 400 is configured to implement a probabilistic shaping for the M + J parity bits p such that the code word x = [u, p] is a code word of a linear code C and the M + J parity bits p fulfil a first shaping constraint, and the information word u fulfils a second shaping constraint (as already described above, [u, p] denotes the concatenation of u and p).

In an embodiment, the first shaping constraint can be equal to the second shaping constraint.

In an embodiment, the first shaping constraint requires that the number of 0 bits is substantially different from the number of 1 bits of the M + J parity bits p of the code word x.

In an embodiment, the number of 0 bits can be substantially larger than the number of 1 bits of the M + J parity bits p of the code word x, wherein the channel encoder is further configured to map the M + J parity bits p to a plurality of signal points having at least one low energy level and one high energy level by mapping the 0 bits of the M + J parity bits p to the low energy level signal points.

Alternatively, the number of 1 bits can be substantially larger than the number of 0 bits of the M + J parity bits p of the code word x, wherein the channel encoder is further configured to map the M + J parity bits p to a plurality of signal points having at least one low energy level and one high energy level by mapping the 1 bits of the M + J parity bits p to the low energy level signal points. In an embodiment, the channel encoder 400 comprises a distribution matcher 401 that is configured to generate the information word u of length K - J bits on the basis of L data bits with K - J > L.

In a further embodiment, the channel encoder 400 is configured to encode the information word u of length K - J bits with K > J > 0 to a code word x of length N bits on the basis of a parity check matrix H, wherein the channel encoder 400 is configured to use the parity check matrix H in the following form:

H = [H _s , H _p ] , wherein H _s denotes a syndrome former 403 of size N - K) x ( K -/) and H _p denotes a non-square full rank parity former 405 of size (N - K) x (N - K + /).

In an embodiment, the parity check matrix H can define a linear forward error correction, FEC, code such that the FEC code has good error correcting capabilities. Also, the parity check matrix H can define a low-density parity check, LDPC, code.

Moreover, the channel encoder 400 can be configured to generate a syndrome s of length M bits using the syndrome former H _s 403 of size (N - K) x (K—J) and the information word u of length K - J on the basis of the following equation: s = uH , wherein H denotes the transpose of the syndrome former H _s 403.

The channel encoder 400 is configured to generate the M + J parity bits p, or to consider the parity former H _p to be the parity check matrix of a linear code and to determine the M + J parity bits p on the basis of the syndrome s and the following equation: p// = s, wherein // denotes the transpose of the parity former H _p 405. In a further embodiment, the channel encoder 400 is configured to generate a Trellis representation of the parity former H _p 405 and to determine the syndrome s by applying a Viterbi algorithm to the Trellis representation of the parity former H _p 405. Thus, the parity former H _p 405 can have a diagonal structure optimized for efficient parity forming, in particular a Trellis representation of the parity former H _p 405 with a reduced number of states.

In a further embodiment, the channel encoder 400 is configured to use a cost function for the Viterbi algorithm, wherein the cost function is defined by, in particular identical to the first shaping constraint.

The following shows in detail how the channel encoder 400 according to the embodiment enables efficient encoding of messages t into shaped code words x e S n C for linear codes C C F" and in general non-linear shaping sets S e F ⁿ by using the shaping FEC scheme.

As shown in figure 4, the channel encoder 400 can divide the parity check matrix into the syndrome former H _s 403 and the non-square M x (M + /) parity former H _p 405 that has full row rank. As already mentioned above, for the parity bits p the encoder 400 can use the solution of the underdetermined linear equation:

p = sH that fulfills the first shaping criteria, i.e., the resulting code word [ u, ] is in the shaping set and it is also a code word.

The parity check matrix of a linear code can be represented by H = [ H _S , H _P ] with the following properties: syndrome former H _s 403 is (N - K) x (K - /); and parity former H _p 405 is (N - K) x (N - K + /) = M x (M + /) with full row rank M. In particular, H _p is non square.

The encoding process includes the following steps: a first step of calculating shaped message u = g(t ) of length K - ] bits by using as in probabilistic amplitude shaping the distribution matcher 401 , such as a constant composition distribution matcher (CCDM), (PAS); a second step of calculating the syndrome s = uH of length M = N - K bits; a third step of calculating the shaped length M + J parity bits p such that pH _p = s and

[u, p] e 5.

Since H _p has more columns than rows, the linear equation pH _p = s is underdetermined and has more than one solution. The parity former 405 can then choose the solution that is optimal according to the first shaping criteria.

The parity former 405 can be implemented using existing algorithms, depending on the shaping criteria. For instance, when the shaping criteria require low weight, parity forming is equivalent to compressed sensing and appropriate algorithms can be used. When the shaping criteria require low weight, parity forming is equivalent to syndrome decoding and appropriate algorithms can be used. A general parity forming matrix can be represented by a Trellis and syndrome decoding can be implemented by running the Viterbi algorithm on the trellis. Moreover, a general shaping criterion can be implemented by running the Viterbi algorithm on the Trellis.

As illustrated in figure 4, the channel encoder 400 can generate a final output, i.e. the code word x by combining the output of the distribution matcher 401 and the output of the parity former 405 via a multiplexer 407.

Figure 5 shows a schematic diagram illustrating a method 500 of encoding an information word u of length K - J bits with K > J > 0 into a code word x of length N bits with N - K = M > 0, wherein the code word x contains the information word u and M + J parity bits p.

The method 500 comprises the following step 501 : implementing a probabilistic shaping scheme for the K - J bits of the information word u and the M + J parity bits p such that the code word x = up is a code word of a linear code C and the M + J parity bits p fulfil a first shaping constraint, i.e. associated with the probabilistic shaping scheme.

While a particular feature or aspect of the disclosure may have been disclosed with respect to only one of several implementations or embodiments, such feature or aspect may be combined with one or more other features or aspects of the other implementations or embodiments as may be desired and advantageous for any given or particular application. Furthermore, to the extent that the terms "include", "have", "with", or other variants thereof are used in either the detailed description or the claims, such terms are intended to be inclusive in a manner similar to the term "comprise". Also, the terms "exemplary", "for example" and "e.g." are merely meant as an example, rather than the best or optimal. The terms "coupled" and "connected", along with derivatives may have been used. It should be understood that these terms may have been used to indicate that two elements cooperate or interact with each other regardless whether they are in direct physical or electrical contact, or they are not in direct contact with each other.

Although specific aspects have been illustrated and described herein, it will be

appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent implementations may be substituted for the specific aspects shown and described without departing from the scope of the present disclosure. This application is intended to cover any adaptations or variations of the specific aspects discussed herein.

Although the elements in the following claims are recited in a particular sequence with corresponding labeling, unless the claim recitations otherwise imply a particular sequence for implementing some or all of those elements, those elements are not necessarily intended to be limited to being implemented in that particular sequence.

Many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the above teachings. Of course, those skilled in the art readily recognize that there are numerous applications of the invention beyond those described herein. While the present invention has been described with reference to one or more particular embodiments, those skilled in the art recognize that many changes may be made thereto without departing from the scope of the present invention. It is therefore to be understood that within the scope of the appended claims and their equivalents, the invention may be practiced otherwise than as specifically described herein.