TECHNICAL FIELD The present disclosure relates to an apparatus for encoding an input sequence by means of polar coding. Moreover, the disclosure relates to a corresponding method for encoding the input sequence by means of polar coding.

BACKGROUND

Channel coding deals with error correction on noisy channels. A channel encoder maps a message into a codeword by adding redundancy, and this redundancy helps the codeword to be mapped back to the message at the receiving side, even if the channel disturbs the transmission (e.g. adds noise). In general, the design of the channel code depends on the transmission channel, i.e., the channel code should be designed depending on the characteristics of the transmission channel.

A common approach is to consider additive white Gaussian noise (AWGN) channels for designing codes, even if the actual transmission channel is not AWGN. This assumption is mostly a good assumption, because codes designed for AWGN channels mostly work also well on other practical channels. However, for certain cases, this assumption causes a performance degradation.

The existing solutions mainly consider designing codes for a fixed channel (e.g. an AWGN channel), and then use an interleaver after encoding to shuffle the codeword. In this way, an averaging effect takes place, i.e., the transmitted codeword experiences an averaged channel. This method is simple, but suboptimal.

Thus, there is a need for an improved apparatus for encoding input sequences. ...,ARY

In view of the above-mentioned problems and disadvantages, embodiments of the present invention aim to improve the conventional apparatuses and methods for encoding input sequences. An object is thereby to provide an apparatus for encoding sequences, wherein the apparatus enables improved performance when transmitting the sequences over a channel. The apparatus should also be able to perform flexible encoding. The obj ect is achieved by the embodiments provided in the enclosed independent claims.

Advantageous implementations of the embodiments are further defined in the dependent claims.

According to a first aspect, the invention relates to an apparatus for encoding an input sequence of N bits u = into a codeword c of length N, wherein the codeword c is for transmission over a communication channel comprising K parallel channels, wherein each parallel channel is characterized by a channel parameter p ₁ ,..., p _K , wherein the apparatus comprises a processor configured to divide the input sequence u into K subsequences u _i , i = 1, ..., K on the basis of the channel parameters p ₁ ,..., p _K , obtain K polarized subsequences c _i , wherein each polarized subsequence c _i is based on polar coding applied to each of the K subsequences u _i , and obtain the codeword c , wherein the codeword c is based on a polarizing transform F _K applied to the K polarized subsequences c _i. In particular, for example the processor may be configured to divide the input sequence u into K subsequences u _i , i = 1, ... , K on the basis of the channel parameters p ₁ ,..., p _K , apply a polar coding to each of the K subsequences u _i in order to obtain K polarized subsequences c _i , and apply a polarizing transform F _K to the K polarized subsequences c _i in order to obtain the codeword c .

The input sequence contains message bits (information bits) and frozen bits.

This provides the advantage that the encoding is flexible, and at the same time the performance is very high. In an embodiment, the polarizing transform F _K is a binary lower bidiagonal matrix of size K by K and is, for example, given by:

A lower bidiagonal matrix is defined as a matrix with non-zero entries along the main diagonal and the diagonal below. In binary case, all non-zero elements are ones.

In an embodiment, a polarized subsequence c _i is given by: c _i = u _i · G _i , wherein G _i is a generator matrix, and wherein the subsequence u _i comprises a set of information bits and a set of frozen bits.

In an embodiment, the generator matrix G _i is given by: wherein the symbol denotes a Kronecker product and wherein IF is given by:

In an embodiment, the processor is further configured to: map the K polarized subsequences c _i to amplitude-shift keying (ASK) or quadrature amplitude modulation (QAM) symbols, wherein the ASK and QAM symbols have different modulation orders. In an embodiment, the processor is further configured to divide the K polarized subsequences c _i into M groups of subsequences, each group comprising one or more subsequencesKi,..., K _m , wherein Kj is the number of subsequences in the j ^th group; obtain K, further polarized subsequences for group j (j=l, ... ,M), wherein the K, further polarized subsequences are based on a polarizing transform F _j applied to the K _j polarized subsequences c _i in group j, and map the K1+K2+ ... +K _M further polarized subsequences to ASK or QAM symbols, wherein each further polarized subsequence is mapped to a different ASK (or QAM) bit-level, wherein each ASK (or QAM) symbol is constructed by using at most one of the further polarized subsequences per group. In particular, the processor may be configured to apply a polarizing transform F _j to the K _j polarized subsequences c _i , wherein each of the polarizing transforms is applied to one of the groups, in order to obtain K, further polarized subsequences for each group with index i=l,„M

In an embodiment, the K channels of the communication channel are each considered as an additive white Gaussian noise (AWGN) channel.

In an embodiment, the channel parameters p ₁ ,..., p _K of the f parallel channels correspond to respective signal-to-noise ratios.

In an embodiment, a set of frozen bit indices of the polar coding is selected according to the 5G NR standard.

According to a second aspect, the invention relates to a method for encoding a input sequence of N bits u = ( u ₀ , ... , u _N-1 ) into a codeword c of length N, wherein the codeword c is for transmission over a communication channel comprising K parallel channels, wherein each parallel channel is characterized by a channel parameter p ₁ ,..., p _K , wherein the method comprises dividing the input sequence u into K subsequencesu _i , i = 1, K on the basis of the channel parameters p ₁ . p _K , applying a polar coding to each of the K subsequences u _i in order to obtain polarized subsequences c _i , and applying a polarizing transform F _K to the K polarized subsequences c _i in order to obtain the polar codeword c . The method according to the second aspect can be extended to implementation forms corresponding to the implementation forms of the apparatus according to the first aspect. In particular, the method according to the second aspect may have implementation forms comprising the feature(s) of the corresponding implementation form of the client device.

The method of the second aspect provides the same advantages and effects as described above for the apparatus of the first aspect and its respective embodiments.

According to a third aspect, a computer program product is provided, which includes program code for performing the method according to the second aspect and its implementation forms, when the program code is run by a processor.

According to still a further aspect, the invention also relates to a computer program product comprising a (non-transitory) computer readable medium and said mentioned computer program, wherein said computer program is included in the computer readable medium.

It has to be noted that all devices, elements, units and means described in the present application could be implemented in the software or hardware elements or any kind of combination thereof. All steps which are performed by the various entities described in the present application as well as the functionalities described to be performed by the various entities are intended to mean that the respective entity is adapted to or configured to perform the respective steps and functionalities. Even if, in the following description of specific embodiments, a specific functionality or step to be performed by external entities is not reflected in the description of a specific detailed element of that entity which performs that specific step or functionality, it should be clear for a skilled person that these methods and functionalities can be implemented in respective software or hardware elements, or any kind of combination thereof. BRIEF DESCRIPTION OF DRAWINGS

The above described aspects and implementation forms of the present invention will be explained in the following description of specific embodiments in relation to the enclosed drawings, in which:

FIG. 1 shows a schematic representation of a system comprising an exemplary apparatus for encoding an input sequence according to an embodiment; FIG. 2 shows a schematic representation of polar transforms according to an embodiment;

FIG. 3 shows a polarizing transform according to an embodiment; FIG. 4 shows a schematic representation of a generator matrix according to an embodiment;

FIG. 5 shows a schematic representation of a multi-level coding (MLC) scheme according to an embodiment; and

FIG. 6 shows a schematic representation of a method for encoding an input sequence according to an embodiment.

DETAILED DESCRIPTION OF EMBODIMENTS

The reason why common approaches are suboptimal is that the information about the channel is discarded and not exploited.

As an illustrative example let us assume that there are two parallel channels, where one of the channels is very good and the other one is very bad. With the conventional approach, the codeword is transmitted over both channels, and only half of the codeword is received, which results in a large performance degradation. If the transmitter has channel knowledge, then it can transmit only over the good channel and _l eave the bad channel idle. In such extreme cases, usage of the Channel State Information at the Transmitter (CSIT) leaving bad channels idle can lead to improvement.

FIG. 1 shows a schematic representation of a system 100 comprising an exemplary apparatus 101 for encoding an input sequence according to an embodiment.

The apparatus 101 is configured for encoding an input sequence of N bits u = (u ₀ , ...,u _N-1 ) into a codeword c of length N, wherein the codeword c is for transmission over a communication channel 102 comprising K parallel channels, wherein each parallel channel is characterized by a channel parameter p ₁ ,..., p _K .

The apparatus 101 comprises a processor 101a configured to divide the input sequence u into K subsequences u _i , i = 1, ... , K on the basis of the channel parameters ^p ₁ ,..., p _K , obtain K polarized subsequences c _i , wherein each polarized subsequence c _i is based on polar coding applied to each of the K subsequences u _i and obtain the codeword c , wherein the codeword c is based on a polarizing transform F _K applied to the K polarized subsequences c _i .

In an example this can be obtained by a processor configured to apply a polar coding to each of the K subsequences u _i in order to obtain K polarized subsequences c _i , and apply a polarizing transform F _K to the K polarized subsequences c _i in order to obtain the codeword c .

The encoded codeword c is then transmitted to an apparatus 103, for example a decoder, in order to be decoded.

The communication channel 102 can be or comprise parallel AWGN channels, which can be seen as a generalization of AWGN channels. In an embodiment, the parallel AWGN channels comprise channel state information at the transmitter (CSIT), and polar coding is considered as channel codes. OFDM sub-carriers or MIMO sub-channels can be seen as examples where parallel AWGN channels occur. In order to obtain the best performance, channel codes should be designed depending on the actual characteristics of the channel. For example, if a channel is considered that consists of k parallel AWGN channels, the channel code should be designed by considering the parameters pi, ..., pk of k channels.

As an example, the each parameter may correspond to the signal to noise ratio (SNR) of that parallel channel. In particular the parameter may be channel state information at the transmitter or the SNR.

In an embodiment, the message sequence is divided into multiple parts depending on the channel parameters, each of them is encoded (with different polar codes) to polarized sequences, and then the polarized sequences are combined using an additional polarization step (an additional polarization kernel) in order to obtain the final polarized sequence. In this way, the resulting final polarized sequence is a single codeword, which is built from polarized sequences that matches to the channel parameters.

This provides the advantage that the problem of code design based on k parameters is converted into a simpler problem of designing k different codes that are optimized for a single parameter.

In one embodiment, the scheme is further simplified by using the same polar code with different parameters to obtain each polarized sequence. As an example, 5G NR polar coding procedure can be used in order to obtain the polarized sequences (each of them may be obtained using a different rate depending on the channel parameter), and then combine the resulting polarized sequences by the additional kernel.

Numerical evaluations show that this kind of construction and encoding performs almost as good as designing a single code based on k parameters.

A channel, which consists of k parallel channels is considered, wherein each parallel channel has a different channel parameter p1, ..., pk. Then, the encoding of a message can be performed as follows: 1. Divide the message sequence into k subsequences depending on the channel parameters p ₁ ,..., p _K .

2. Perform polar transforms (possibly with different kernels) to each of the sub- sequences to obtain k polarized sequences. 3. Perform a polar transform with a further kernel to the concatenation of the k polarized sequences to obtain the final polarized sequence.

4. Map the resulting sequence to channel input symbols using a symbol mapper.

FIG. 2 shows a schematic representation of polar transforms according to an embodiment.

In general, a binary polar code of block length n and dimension k is defined by a set F of n-k frozen positions, and the polar transform , which denotes the log2(n)- fold Kronecker power of:

Polar encoding can be represented by: where the n - k frozen positions F in u = [u1,..., u _n ] are set to predetermined values. The vector c = [c1,..., c _n ] is the code word. Let I be the set of k unfrozen bit indices, i.e., the k information bits are collected in the vector at the indices given in I. The code rate is given as R = k/n. Polar codes can be decoded using simple successive cancellation (SC) decoder, or more complex SC list decoders.

If a transmission of n channel symbols over k parallel channels is considered, a simple approach would be to use K polar codes of length n _k =n/K. To avoid the loss due to the reduced block length, K polar codes can be combined using an additional kernel of size K, as depicted in Fig. 2 In this embodiment, k polar transforms (labeled as ‘Trafo’ in the figure) are used, where the 5G NR standard can be used in order to select the frozen bits for each block. The outputs of the blocks on the left are transformed by the additional kernel F _K of size K. The outputs corresponding to the i-th block are transmitted over the i-th channel.

In this embodiment, a double-diagonal matrix is chosen as the additional kernel of size K, as shown in Fig. 3.

The overall transform matrix including the component codes and the additional kernel written as: wherein denotes the Kronecker product.

The proposed scheme has the advantage that one code can be used for all K channels simultaneously. For the K component codes of length n _k = n/K, polar codes as defined in the 5G standard can be used. Using the kernel FK given above, the matrix G _K can be represented as in FIG. 4, where each triangle corresponds to the lower triangular structure of a polar transform.

Summarizing the following steps can be performed in order to encode an input sequence:

1. divide the input message or sequence into K parts depending on the channel parameters;

2. choose the indices of the frozen and unfrozen bits according to a fixed rule (e.g. based on the 5G NR standard);

3. encode the chosen bits separately using K polar transforms; and

4. use an additional kernel of size K to further polarize K sequences to obtain final polarized sequence. FIG. 5 shows a schematic representation of a multi-level coding (MLC) scheme according to an embodiment.

The above exposed implementation forms can be further extended to support higher order modulation and adaptive modulation and coding. Multi-level coding (MLC) is known to perform well with polar coded modulation. In MLC, each ASK (or QAM) bit- level is encoded with a separate channel code.

The MLC scheme can be used in order to support adaptive modulation. Accordingly, each ASK bit-level is constructed according to the above exposed encoding scheme, and a different modulation order can be used for each parallel channel.

FIG. 5 shows an embodiment, wherein each rectangle corresponds to another multi kernel polar code that is constructed according to the above illustrated encoding scheme. The outputs are then combined in a way, such that ASK symbols are generated, where each ASK bit-level is constructed using a part of another polarized sequence. In this way, a single polar codeword is mapped to ASK symbols of different orders (in this case 8-ASK, 16-ASK and 32-ASK), which can be transmitted over three parallel channels.

In this scheme, ordering of the code-blocks is more flexible than for conventional schemes. The blocks in FIG. 5 could be reordered in size as e.g. 3, 3, 2, 3,1 without much loss.

Summarizing, the following steps can be performed in order to encode an input sequence:

1. divide the input sequence into K parts;

2. encode each of them using different polar codes to obtain K polarized sequences;

3. group the polarized sequences into m parts, containing at least one polarized sequence; 4. use a different polarizing kernel to combine the polarized sequences in each part into m further polarized sequences; and

5. use a symbol mapper to map the m further polarized sequences to ASK (or QAM) symbols, where we use as most one sequence from each of the further polarized sequences.

FIG 6 shows a schematic representation of a method 600 for encoding an input sequence according to an embodiment.

The method 600 for encoding a input sequence of N bits u = (u ₀ , ... , u _N-1 ) c of length N, wherein the codeword c is for transmission over a communication channel comprising K parallel channels, wherein each parallel channel is characterized by a channel parameter p ₁ ,..., p _K , wherein the method comprises the following steps: dividing 601 the input sequence u into K subsequences u _i ,i=1, ... , K on the basis of the channel parameters p ₁ ,..., p _K ; obtain polarized subsequences c _i , wherein each polarized subsequence c _i is based on polar coding applied to each of the K subsequences u _i. In particular, this may be achieved by applying 602 a polar coding to each of the K subsequences u _i in order to obtain polarized subsequences c _i ; and obtain the polar codeword c , wherein the codeword c is based on a polarizing transform F _K applied to the K polarized subsequences. In particular, this may be achieved by applying 603 a polarizing transform F _K to the K polarized subsequences c _i in order to obtain the polar codeword c .

The present invention has been described in conjunction with various embodiments as examples as well as implementations. However, other variations can be understood and effected by those persons skilled in the art and practicing the claimed invention, from the studies of the drawings, this disclosure and the independent claims. In the claims as well as in the description the word “comprising” does not exclude other elements or steps and the indefinite article “a” or “an” does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation.