WO2022135664A1

WO2022135664A1 - A unipolar pulse amplitude modulation for peak power constrained channels

Info

Publication number: WO2022135664A1
Application number: PCT/EP2020/087457
Authority: WO
Inventors: Tobias PRINZ; Thomas WIEGART; Gerhard Kramer
Original assignee: Huawei Technologies Co., Ltd.; Technische Universität München
Priority date: 2020-12-21
Filing date: 2020-12-21
Publication date: 2022-06-30

Abstract

The present disclosure relates to a data transmission procedure, in particular, to PAM-6 data transmission. The disclosure notably proposes using a class of punctured 2-dimensional symbol constellations to generate 1-dimensional PAM symbols by projection. A transmitting device, a signal format, and a transmission method using the punctured 2-dimensional symbol constellations are proposed. The transmitting device generates a sequence of symbols by mapping a message onto the punctured 2-dimensional symbol constellation. The punctured constellation is a full 2-dimensional symbol constellation minus four constellation symbols. N constellation symbols in the punctured constellation have fewer neighboring constellation symbols than they have in the full 2-dimensional symbol constellation. Transmission efficiency can thus be improved.

Description

A UNIPOLAR PULSE AMPLITUDE MODULATION FOR PEAK POWER

CONSTRAINED CHANNELS

TECHNICAL FIELD

The present disclosure relates to data transmission and, in particular, to data modulation based on a class of punctured 2-dimensional symbol constellations comprising 32 constellation symbols. A punctured 2-dimensional symbol constellation is a full 2-dimensional symbol constellation minus a subset of symbols, wherein the subset is also referred to as an “exclusion set”. The disclosure provides a transmitting device, a signal format, and a transmitting method, which are each based on the punctured 2-dimensional symbol constellation. The punctured 2- dimensional symbol constellation allows transmitting the two coordinates of each 2- dimensional symbol via pulse amplitude modulation (PAM), in particular, PAM-6. The punctured 2-dimensional constellation may be contained in the first quadrant of a complex plane to enable unipolar PAM. With the punctured 2-dimensional symbol constellation an improved data transmission over a peak power constrained channel can be achieved.

BACKGROUND

In digital communication, data to be transmitted may be mapped onto a discrete alphabet, often referred to as a symbol constellation, and may be transmitted over a channel.

In case of intensity modulation (with or without peak power constrained channels), unipolar PAM may be used for modulating the data. In this case, specific PAM symbol constellations with M constellation symbols may be selected, wherein M is a power of two. For instance, as is relevant for this disclosure, such a symbol constellation with M = 6 may be selected for realizing PAM-6.

The use of PAM-6 may lead to potential gains compared to the use of PAM-8 at a rate of 2 bits per channel. Assuming symbol-metric decoding (SMD), 0.23 dB can be gained with a soft decoder (SD), which is a decoder that takes soft information from the channel and 1.35 dB can be gained with a hard-decision (HD) decoder. Assuming bit-metric decoding (BMD), 0.33 dB can be gained with a SD and 0.75 dB can be gained with a HD decoder. However, binary data cannot be mapped directly to 6 constellation symbols. In addition, the above-mentioned restriction to the power of two results in rate loss in some operating regions.

Notably, a 1 -dimensional symbol constellation can be obtained as projection of a two dimensional constellation on two orthogonal axes. Thus, 2-dimensional symbol constellations are a natural choice for the generation of 1 -dimensional PAM symbol constellations whose cardinality is not a power of two.

The size of a symbol constellation determines how many bits can be mapped onto each constellation symbol. A symbol constellation of 2ⁿ constellation symbols can carry n bits per constellation symbol. For example, a 2-dimensional symbol constellation with 32 constellation symbols may carry 5 bits per constellation symbol.

A PAM-6 data transmission may be generated using a 2-dimensional symbol constellation. For example, by projecting a 2-dimensional 32-QAM constellation 101 - as it is shown in FIG. 1 - to one dimension. Given a constellation symbol from the complex symbol constellation 101, the real part may be transmitted as one PAM-6 symbol, and the imaginary part may be transmitted as another PAM-6 symbol. Thus, one 32-QAM symbol corresponds to two PAM-6 symbols. Five bits can be directly mapped to one 32-QAM symbol. If the 32-QAM symbol constellation is the 32-QAM symbol constellation illustrated in Fig. 1, the left side, this method is often just referred to as PAM-6. In this disclosure, however, it will be referred to as “Cross 32-QAM/PAM-6” or only “Cross 32-QAM”.

However, the “Cross 32-QAM” symbol constellation is not optimal for PAM-6 under a peak power constraint, in particular, a 1 -dimensional peak-power constraint. FIG. 1 shows also another exemplary symbol constellation 102, which is referred to as “Donut 32-QAM” symbol constellation in this disclosure, but which is also not optimal for PAM-6 data transmission over a peak power constrained channel.

SUMMARY

In view of the above, an objective is thus to find a class of symbol constellations that improves PAM-6 data transmission. The disclosure desires, in particular, to achieve gains as mentioned- above, at least gains as close as possible to the theoretical ones. The class of symbol constellations should specifically be optimized for PAM-6 for data transmission over a peak power constrained channel, i.e., under a peak power constraint.

These and other objectives are achieved by the embodiments provided in the enclosed independent claims. Advantageous implementations of the embodiments of the invention are further defined in the dependent claims.

The embodiments of the invention base on the following considerations regarding the “Cross 32-QAM” symbol constellation 101 shown in FIG. 1.

All constellation symbols of the 2-dimensional symbol constellation 101 are in principle equally probable, however, the 1 -dimensional constellation symbols of the projected symbol constellation, are not equally probable. Thus, under a 1 -dimensional peak power constraint, the symbols 0 and ^P_max (i e., the outer constellation symbols of the symbol constellation 101) should be used more often than the inner constellation symbols. Alternatively, all constellation symbols should be used with the same probability at high signal-to-noise ratio (SNR). However, the “Cross 32-QAM/PAM-6” method mentioned above transmits the outer 1-dimensional constellation symbols corresponding to the symbol constellation 101 the least often (due to the missing corner symbols).

Thus, for embodiments of the invention, this disclosure suggests using a different class of 32- QAM symbol constellations. These new symbol constellations may be obtained by taking a regular square QAM symbol constellation with 36 constellation symbols (6 x 6) lying in the first quadrant of the complex plane, and dropping four constellation symbols other than the corner symbols to obtain a punctured symbol constellation with 32 constellation symbols. By further optimization, punctured 32-QAM symbol constellations that lead to a significant performance improvement when implementing PAM-6 under a peak power constraint can be obtained. Notably, the peak-power constraint applies to the PAM constellation and therefore is a 1 -dimensional constraint.

That is, an advantage of embodiments of the invention is a better performance of PAM-6 data transmission over a peak power constrained channel. At the same time, the complexity compared to the “Cross 32-QAM” symbol constellation 101 shown in FIG. 1 is not increased. A further advantage of some embodiments of the invention is that the proposed symbol constellations allow Gray labeling.

A first aspect of this disclosure provides a transmitting device configured to: generate a sequence of symbols by mapping a message onto a punctured 2-dimensional symbol constellation comprising 32 constellation symbols, wherein the punctured 2-dimensional symbol constellation is a full 2-dimensional symbol constellation comprising 36 constellation symbols minus an exclusion set, wherein the exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation, wherein the four constellation symbols are selected such that A constellation symbols in the punctured 2-dimensional symbol constellation have fewer neighboring constellation symbols than they have in the full 2-dimensional symbol constellation, and wherein A is in a range of 9 to 16; and transmit the sequence of symbols.

In particular, A constellation symbols in the punctured 2-dimensional symbol constellation have fewer neighbors at minimum distance than they have in the full 2-dimensional symbol constellation. For transmitting the sequence of 2-dimensional symbols, these symbols may be further mapped to 1 -dimensional PAM symbols (a pair of PAM symbols per 2-dimensional symbol).

In this disclosure, “neighbor” means “closest neighbor” (unless specified otherwise). In other words, the neighbors of a given symbol are the one or more symbols that are closer to the given symbol than any other symbols of the constellation.

According to the first aspect, the punctured 2-dimensional symbol constellation is the set difference of the full 2-dimensional symbol constellation and the exclusion set (excluded subset of symbols). This means that the exclusion set is not included in the punctured 2-dimensional symbol constellation. For example, the punctured 2-dimensional symbol constellation may be obtained by omitting the constellation symbols of the exclusion set from the full 2-dimensional symbol constellation. The exclusion set accordingly comprises one or more constellation symbols from the full 2-dimensional symbol constellation. The full 2-dimensional symbol constellation is accordingly the punctured 2-dimensional symbol constellation plus the exclusion set. By using the punctured 2-dimensional symbol constellation, the transmitting device of the first aspect may perform PAM-6 data transmission with improved performance. In particular, the theoretical gains mentioned above can be at least nearly achieved. PAM-6 obtained from the punctured 2-dimensional symbol constellation is better under a peak power constraint than PAM-6 obtained from the “Cross 32-QAM” symbol constellation or from the “Donut 32- QAM” symbol constellation shown in FIG. 1.

The punctured 2-dimensional symbol constellation may be represented in a complex plane with two orthogonal dimensions, which may be denoted as in-phase (I) and quadrature (Q) components. All of the 32 constellation symbols of the punctured 2-dimensional symbol constellation may be arranged in the first quadrant of the complex plane. Neighboring constellation symbols may be constellation symbols that are located directly next to each other in the horizontal direction or the vertical direction of the (complex) plane of the punctured 2- dimensional symbol constellation. Notably, a distance between constellation symbols in the diagonal direction is larger, and thus such constellation symbols are not considered to be neighboring constellation symbols in this disclosure. Labelling may be applied directly to the punctured 2-dimensional symbol constellation in some cases.

In an implementation form of the first aspect, N equals 16.

These punctured 2-dimensional symbol constellations are the best for performing PAM-6 under a peak power constraint when starting from a regular square 36-QAM symbol constellation. Notably, seven different symbol constellations with N = 16 are possible.

In an implementation form of the first aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set does not comprise any corner constellation symbol of the 2-dimensional symbol constellation.

In an implementation form of the first aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set comprises four inner constellation symbols of the 2-dimensional symbol constellation.

In an implementation form of the first aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation with a 6x6 arrangement of the 32 constellation symbols arranged in 6 columns and 6 rows; and the exclusion set comprises the constellation symbol arranged in the second column and the second row, the constellation symbol arranged in the second column and the fifth row, the constellation symbol arranged in the fifth column and the second row, and the constellation symbol arranged in the fifth row and the fifth column.

This punctured 2-dimensional symbol constellation is also referred to as “Framed Cross 32- QAM” symbol constellation in this disclosure. Advantageously, this symbol constellation allows a Gray labeling of the constellation symbols, which is particularly beneficial if the transmitting device of the first aspect is to use the symbol constellation with BMD.

In an implementation form of the first aspect, each constellation symbol in the punctured 2- dimensional symbol constellation is associated with a bit label comprising 5 bits.

In an implementation form of the first aspect, the constellation symbols in the punctured 2- dimensional symbol constellation are associated with a Gray bit labeling or quasi-Gray bit labeling.

In an implementation form of the first aspect, the transmitting device is further configured to generate the punctured 2-dimensional symbol constellation by excluding the exclusion set from the full 2-dimensional symbol constellation.

Accordingly, the transmitting device itself may calculate the punctured 2-dimensional symbol constellation used for the mapping of the message to be transmitted, i.e., it may also determine itself the exclusion set. The transmitting device may, however, also obtain the exclusion set (e.g., a precomputed exclusion set), and may determine the punctured 2-dimensional symbol constellation based on the full 2-dimensional symbol constellation and the obtained exclusion set. Alternatively, the transmitting device may also obtain directly the constellation symbols of the punctured 2-dimensional symbol constellation, i.e., in this case the transmitting device does not generate the punctured 2-dimensional symbol constellation.

In an implementation form of the first aspect, the symbols are transmitted over a peak power constrained channel. For transmission over the channel, as mentioned above, the symbols may first be projected to 1 -dimensional PAM symbols.

In an implementation form of the first aspect, the symbols are transmitted using pulse amplitude modulation, PAM, in particular PAM-6.

Thus, the transmitting device may employ PAM-6 for data transmission under a peak power constraint, and may reach an optimal performance due to the punctured 2-dimensional symbol constellation. Notably, the transmitting device may thereby map the real and imaginary part (or the two coordinates) of each constellation symbol to a pair of 1 -dimensional PAM-6 symbols (as also explained above)

In an implementation form of the first aspect, N equals 16 and the punctured 2-dimensional symbol constellation is represented by one of the following diagrams:

each dot in each of the above diagrams representing a constellation symbol, the constellation symbol comprising a first PAM-6 symbol and a second PAM-6 symbol, an x-coordinate of the dot indicating the first PAM-6 symbol, a y-coordinate of the dot indicating the second PAM-6 symbol.

In an implementation form of the first aspect, the transmitting of the sequence of symbols comprises: converting the sequence of symbols to a signal matched to a transmission channel between the transmitting device and a receiving device; and transmitting the signal to the receiving device.

A second aspect of this disclosure provides a signal format for a transmitting device, wherein the signal format is based on: a punctured 2-dimensional symbol constellation, wherein the punctured 2-dimensional symbol constellation is a full 2-dimensional symbol constellation comprising 36 constellation symbols minus an exclusion set, wherein the exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation, wherein the four constellation symbols are selected such that TV constellations symbols in the punctured 2-dimensional symbol constellation have less neighboring constellation symbols than they have in the full 2-dimensional symbol constellation, and wherein /Vis in a range of 9 to 16.

In an implementation form of the second aspect, N equals 16.

In an implementation form of the second aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set does not comprise any corner constellation symbol of the full 2-dimensional symbol constellation.

In an implementation form of the second aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set comprises four inner constellation symbols of the full 2-dimensional symbol constellation.

In an implementation form of the second aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation with a 6x6 arrangement of the 32 constellation symbols arranged in 6 columns and 6 rows; and the exclusion set comprises the constellation symbol arranged in the second column and the second row, the constellation symbol arranged in the second column and the fifth row, the constellation symbol arranged in the fifth column and the second row, and the constellation symbol arranged in the fifth row and the fifth column. In an implementation form of the second aspect, each constellation symbol in the punctured 2- dimensional symbol constellation is associated with a bit label comprising 5 bits.

In an implementation form of the second aspect, the constellation symbols in the punctured 2- dimensional symbol constellation are associated with a Gray bit labeling or quasi-Gray bit labeling.

In an implementation form of the second aspect, N equals 16 and the punctured 2-dimensional symbol constellation is represented by one of the following diagrams:

The signal format of the second aspect allows achieving the advantages and effects described above for the transmitting device of the first aspect.

A third aspect of this disclosure provides a method for transmitting a message, the method comprising: generating a full 2-dimensional symbol constellation, the full 2-dimensional symbol constellation comprising 36 constellation symbols; generating a punctured 2- dimensional symbol constellation by excluding an exclusion set from the full 2-dimensional symbol constellation, wherein the exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation, wherein the four constellation symbols are selected such that N constellations symbols in the punctured 2-dimensional symbol constellation have fewer neighboring constellation symbols than they have in the full 2-dimensional symbol constellation, and wherein /Vis in a range of 9 to 16; and generating a sequence of symbols by mapping the message onto the punctured 2-dimensional symbol constellation.

In an implementation form of the third aspect, N equals 16.

In an implementation form of the third aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set does not comprise any corner constellation symbol of the full 2-dimensional symbol constellation.

In an implementation form of the third aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set comprises four inner constellation symbols of the full 2-dimensional symbol constellation.

In an implementation form of the third aspect, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation with a 6x6 arrangement of the 32 constellation symbols arranged in 6 columns and 6 rows; and the exclusion set comprises the constellation symbol arranged in the second column and the second row, the constellation symbol arranged in the second column and the fifth row, the constellation symbol arranged in the fifth column and the second row, and the constellation symbol arranged in the fifth row and the fifth column.

In an implementation form of the third aspect, each constellation symbol in the punctured 2- dimensional symbol constellation is associated with a bit label comprising 5 bits.

In an implementation form of the third aspect, the constellation symbols in the punctured 2- dimensional symbol constellation are associated with a Gray bit labeling or quasi-Gray bit labeling.

In an implementation form of the third aspect, the method further comprises generating the punctured 2-dimensional symbol constellation by excluding the exclusion set from the full 2- dimensional symbol constellation.

In an implementation form of the third aspect, the symbols are transmitted over a peak power constrained channel.

In an implementation form of the third aspect, the symbols are transmitted using pulse amplitude modulation, PAM, in particular PAM-6.

In an implementation form of the third aspect, the punctured 2-dimensional symbol constellation is represented by one of the following diagrams:

A fourth aspect of this disclosure provides a computer program comprising a program code for performing, when running on a computer, the method according to the third aspect or any of its implementation forms.

A fifth aspect of the present disclosure provides a non-transitory storage medium storing executable program code which, when executed by a processor, causes the method according to the third aspect or any of its implementation forms to be performed. 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 (embodiments of the invention) will be explained in the following description of specific embodiments in relation to the enclosed drawings, in which

FIG. 1 shows examples of 2-dimensional symbol constellations, in particular, the “Cross 32-QAM” symbol constellation and the “Donut 32-QAM” symbol constellation.

FIG. 2 shows a transmitting device according to an embodiment of the invention, which uses a punctured 2-dimensional symbol constellation as proposed in this disclosure.

FIG. 3 shows examples of punctured 2-dimensional symbol constellations proposed in this disclosure, which may be used by a transmitting device according to an embodiment of the invention.

FIG. 4 shows an example of the punctured 2-dimensional symbol named “Framed Cross

32-QAM” with labeling, which may be used by a transmitting device according to an embodiment of the invention, and shows in comparison the “Cross 32- QAM” symbol constellation and the “Donut 32-QAM” symbol constellation with labeling, respectively. FIG. 5 compares an achievable rate and a frame error rate (FER) of the “Framed Cross 32-QAM” symbol constellation with the Cross 32-QAM” symbol constellation and the “Donut 32-QAM” symbol constellation, respectively, for a soft-decision (SD) decoder.

FIG. 6 compares an achievable rate and a frame error rate (FER) of the “Framed Cross 32-QAM” symbol constellation with the Cross 32-QAM” symbol constellation and the “Donut 32-QAM” symbol constellation, respectively, for a hard-decision (HD) decoder.

FIG. 7 shows method according to an embodiment of the invention.

DETAILED DESCRIPTION OF EMBODIMENTS

FIG. 2 shows a transmitting device 200 according to an embodiment of the invention. The transmitting device 200 may be, or may be included in, an optical transmitter, for example, a coherent optical transmitter. The transmitting device 200 may compute driving signals for the optical transmitter. The transmitting device 200, or the optical transmitter comprising the transmitting device 200, may transmit data (messages) over a peak power constrained channel, for instance, to a receiving device 210.

The transmitting device 200 is configured to generate a sequence of 2-dimensional symbols 203, and to transmit the sequence of symbols 203, e.g., to the receiving device 210. Thereby, the symbols 203 may be transmitted over the peak power constrained channel and/or may be transmitted using PAM, in particular, PAM-6. For transmitting the symbols 203, the transmitting device 200 may be configured to convert the sequence of symbols 203 to a signal matched to the peak power constrained channel between the transmitting device 200 and the receiving device 210, and may then transmit the signal over the channel to the receiving device 210. In particular, the sequence of symbols 203 may thereby be mapped to 1 -dimensional PAM symbols using projection (each symbol 203 to a pair of 1-dimensional PAM symbols), and the 1 -dimensional PAM symbols may be sent over the channel.

The transmitting device 200 is further configured to generate the sequence of symbols 203 by mapping a message 201 (i.e., data to be transmitted, e.g. obtained or generated by the device 200) onto a punctured 2-dimensional symbol constellation 202 comprising 32 constellation symbols. As mentioned, the 2-dimensional constellation symbols may be further mapped to 1- dimensional PAM symbols via projection. The punctured 2-dimensional symbol constellation 202 may be one of a class of punctured 2-dimensional symbol constellations. Examples of the punctured 2-dimensional symbol constellations 202 are shown in FIG. 3 (e.g., symbol constellations explained below).

The punctured 2-dimensional symbol constellation 202 is a full 2-dimensional symbol constellation minus an exclusion set, wherein the exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation. That is, the constellation symbols included in the exclusion set are included in the full 2-dimensional symbol constellation, but not (anymore) in the punctured 2-dimensional symbol constellation. The four constellation symbols are selected such that A constellation symbols in the punctured 2-dimensional symbol constellation - i.e., the remaining constellation symbols after removing the four constellation symbols of the exclusion set - have fewer neighboring constellation symbols than they have in the full 2-dimensional symbol constellation - i.e., before removing the exclusion set. In embodiments of the invention A is in a range of 9 to 16, i.e., Amay be one of: 9, 10, 11, 12, 13, 14, 15, and 16.

The punctured 2-dimensional symbol constellation 202 with the 32 constellation symbols allows performing an improved 32-QAM/PAM-6 data transmission under a peak power constraint, i.e., PAM-6 data transmission can be improved at least compared to using the “Cross 32-QAM” symbol constellation 101 and “Donut 32-QAM” symbol constellation 102 shown in FIG. 1.

The design (or generation, e.g., by the transmitting device 200) of the punctured 2-dimensional symbol constellation 202 may start with the full 2-dimensional symbol constellation comprising 36 constellation symbols. For instance, the latter may be a uniformly spaced, unipolar 36-QAM symbol constellation (with 6 x 6 constellations symbols). Then, four constellation symbols are removed from the full 2-dimensional symbols constellation, such that only 32 constellation symbols remain. These 32 constellation symbols can be labeled by five bits. The four removed constellation symbols form the exclusion set.

The four constellation symbols are, in particular, removed such that the number of constellation symbols, which lose at least one closest neighbor, is in a range of A = 9 to 16, and is ideally maximized to N= 16. For example, in any constellation in which the symbols are arranged in an orthogonal lattice, the closest neighbors of a given constellation symbol are those constellation symbols that are located directly next to the given constellation symbol in the horizontal or vertical direction.

Assuming that the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation, the condition 9 < N < 16 can be satisfied if the exclusion set does not comprise any corner constellation symbol of the full 2-dimensional symbol constellation, and/or if the exclusion set comprises four inner constellation symbols of the full 2-dimensional symbol constellation.

The maximum number of the constellation symbols, which may lose at least one direct neighbor - after the exclusion set has been removed - is 16. This leads to seven possible symbol constellations, which may be considered “optimal” for the purpose of this disclosure. These seven symbol constellations are shown in FIG. 3. In each symbol constellation, each dot represents a constellation symbol. Notably, since the transmitting device may use PAM-6 for data transmission, each constellation symbol (each dot) may comprise a first PAM-6 symbol and a second PAM-6 symbol, wherein an x-coordinate of the dot indicates the first PAM-6 symbol, and a y-coordinate of the dot indicates the second PAM-6 symbol.

In FIG. 3, the punctured 2-dimensional symbol constellation 202a named “Framed Cross 32- QAM” allows a Gray labeling, which is beneficial when the symbol constellation is to be used with BMD. For this symbol constellation 202a, the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation with a 6 x 6 arrangement of the 32 constellation symbols arranged in 6 columns and 6 rows. The exclusion set comprises the constellation symbol arranged in the second column and the second row, the constellation symbol arranged in the second column and the fifth row, the constellation symbol arranged in the fifth column and the second row, and the constellation symbol arranged in the fifth row and the fifth column. After removing this exclusion set, from the full 2-dimensional symbol constellation, the shown symbol constellation 202a is obtained.

Further in FIG. 3, the punctured 2-dimensional symbol constellations 202b and 202c, which are named “Grid 32-QAM” and “Grid 32-QAM (rotated)”, respectively, are the best symbol constellations under SMD, and in terms of a maximum achievable information rate. More 32-QAM punctured 2-dimensional constellations may be generated for being used by the transmitting device 200 according to embodiments of the invention, namely symbol constellations with TV = 15, 14, 13, 12, 11, 10, 9. However, these symbol constellations are not illustrated.

FIG. 4 shows an example of a Gray labeling of the “Framed Cross 32-QAM” symbol constellation 202a shown in FIG. 3. Further, FIG. 5(a) and FIG. 5(b) show numerical results for this symbol constellation 202 under soft- and hard BMD, respectively, and compare these results to similarly obtained results for the exemplary “Cross 32-QAM” symbol constellation 101 and “Donut 32-QAM” symbol constellation 102, which were introduced in FIG. 1. Notably, the “Donut 32-QAM” symbol constellation may be considered being a punctured 2-dimensional symbol constellation in the sense of this disclosure with N = 8, wherein the four central constellation symbols are removed from the full 2-dimensional symbol constellation.

In particular, FIG. 4 shows the labelled “Framed Cross 32-QAM” symbol constellation 202a, the labelled “Cross 32-QAM” symbol constellation 101, and the labelled “Donut 32-QAM” symbol constellation 102, i.e., with a bit labelling applied.

The respective performances were assessed with the following system model. The channel output was assumed to be given by Y = X + Z, wherein Z is Gaussian distributed with zero mean and noise variance <J². The peak signal-to-noise ratio (PSNR) is given by P_max/^ff2-

To get coded numerical results, LDPC codes from the 5GNew Radio standard were used. The system was simulated with a block/frame length of 3000 PAM-6 symbols (1500 32-QAM symbols) per frame, at a transmission rate of 2 bits per channel use.

FIG. 5 shows the achievable information rates, in (a), and the coded performance at a rate of 2 bits per channel use for soft-decoding. From the achievable rates, a gain of around 0.74 dB for the “Framed Cross 32-QAM” symbol constellation 202a compared to the other symbol constellations 101 and 102 can be expected. This gain is verified by the frame error rate simulations, shown in (b). FIG. 6 shows the same for hard-decoding. Here, from the achievable rates, shown in (b), a gain of around 0.5 dB of the “Framed Cross 32-QAM” symbol constellation 202a compared to the other symbol constellations 101 and 102 can be expected. This gain is verified by the frame error rate simulations, shown in (b).

FIG. 7 shows a method 700 for transmitting a message 201 according to an embodiment of the invention. The method 700 may be performed by the transmitting device 200 shown in FIG. 2, and the message 201 may be transmitted to a receiving device 210. The method 700 may also be performed only in parts by the transmitting device 200.

The method 700 comprises a step 701 of generating a full 2-dimensional symbol constellation, the full 2-dimensional symbol constellation comprising 36 constellation symbols. Step 701 may be a precomputation step, and is in this case not performed by the transmitting device 200. Further, the method 700 comprises a step 702 of generating a punctured 2-dimensional symbol constellation 202 by excluding an exclusion set 300 from the full 2-dimensional symbol constellation. The exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation. The four constellation symbols are selected such that N constellations symbols in the punctured 2-dimensional symbol constellation have less neighboring constellation symbols than they have in the full 2-dimensional symbol constellation, wherein N is in a range of 9-16. Also the step 702 may be a precomputation step, and is in this case not performed by the transmitting device 200. Further, the method 700 comprises a step 703 of generating a sequence of symbols 203 by mapping the message 201 onto the punctured 2- dimensional symbol constellation 202. The method 700 may further comprise a step of projecting the 2-dimensional symbols 203 to pairs of 1 -dimensional PAM symbols, and transmitting these PAM symbols over a channel.

The present disclosure has been described in conjunction with various embodiments as examples as well as implementations of the invention. However, other variations can be understood and effected by those persons skilled in the art and practicing the claimed embodiments of the 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.

Claims

1. A transmitting device (200) configured to: generate a sequence of symbols (203) by mapping a message (201) onto a punctured 2- dimensional symbol constellation (202) comprising 32 constellation symbols, wherein the punctured 2-dimensional symbol constellation (202) is a full 2-dimensional symbol constellation comprising 36 constellation symbols minus an exclusion set, wherein the exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation, wherein the four constellation symbols are selected such that N constellation symbols in the punctured 2-dimensional symbol constellation (202) have fewer neighboring constellation symbols than they have in the full 2-dimensional symbol constellation, and wherein A is in a range of 9 to 16; and transmit the sequence of symbols (203).

2. The transmitting device (200) according to claim 1, wherein:

N equals 16.

3. The transmitting device (200) according to claim 1 or 2, wherein: the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set does not comprises any comer constellation symbol of the full 2- dimensional symbol constellation.

4. The transmitting device (200) according to one of the claims 1 to 3, wherein: the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation; and the exclusion set comprises four inner constellation symbols of the full 2-dimensional symbol constellation.

5. The transmitting device (200) according to one of the claims 1 to 4, wherein: the full 2-dimensional symbol constellation is a square-shaped 2-dimensional symbol constellation with a 6x6 arrangement of the 32 constellation symbols arranged in 6 columns and 6 rows; and the exclusion set comprises the constellation symbol arranged in the second column and the second row, the constellation symbol arranged in the second column and the fifth row, the constellation symbol arranged in the fifth column and the second row, and the constellation symbol arranged in the fifth row and the fifth column.

6. The transmitting device (200) according to one of the claims 1 to 5, wherein: each constellation symbol in the punctured 2-dimensional symbol constellation (202) is associated with a bit label comprising 5 bits.

7. The transmitting device (200) according to one of the claims 1 to 6, wherein: the constellation symbols in the punctured 2-dimensional symbol constellation (202) are associated with a Gray bit labeling or quasi-Gray bit labeling.

8. The transmitting device (200) according to one of the claims 1 to 7, configured to: generate the punctured 2-dimensional symbol constellation (202) by excluding the exclusion set from the full 2-dimensional symbol constellation.

9. The transmitting device (200) according to one of the claims 1 to 8, wherein: the symbols are transmitted over a peak power constrained channel.

10. The transmitting device (200) according to one of the claims 1 to 9, wherein: the symbols are transmitted using pulse amplitude modulation, PAM, in particular

PAM-6.

11. The transmitting device (200) according to any one of the preceding claims, wherein N equals 16 and wherein the punctured 2-dimensional symbol constellation (202) is represented by one of the following diagrams:

12. The transmitting device (200) according to one of the claims 1 to 11, wherein transmitting the sequence of symbols (203) comprises: converting the sequence of symbols (203) to a signal matched to a transmission channel between the transmitting device (200) and a receiving device (210); and transmitting the signal to the receiving device (210).

13. A signal format for a transmitting device (200), wherein the signal format is based on: a punctured 2-dimensional symbol constellation (202), wherein the punctured 2-dimensional symbol constellation (202) is a full 2-dimensional symbol constellation comprising 36 constellation symbols minus an exclusion set, wherein the exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation, wherein the four constellation symbols are selected such that N constellations symbols in the punctured 2-dimensional symbol constellation have fewer neighboring constellation symbols than they have in the full 2-dimensional symbol constellation, and wherein /Vis in a range of 9 to 16.

14. A method (700) for transmitting a message (201), the method (700) comprising: generating a full 2-dimensional symbol constellation, the full 2-dimensional symbol constellation comprising 36 constellation symbols; generating a punctured 2-dimensional symbol constellation (202) by excluding an exclusion set from the full 2-dimensional symbol constellation, wherein the exclusion set comprises four constellation symbols of the full 2-dimensional symbol constellation, wherein the four constellation symbols are selected such that N constellations symbols in the punctured 2-dimensional symbol constellation have less neighboring constellation symbols than they have in the full 2-dimensional symbol constellation, and wherein A is in a range of 9 to 16; and generating a sequence of symbols (203) by mapping the message (201) onto the punctured 2-dimensional symbol constellation (202).

15. A computer program comprising a program code for performing, when running on a computer, the method (700) according to claim 14.