Method for shaping the probabilistic amplitude distribution of modulation symbols, producing computer program and corresponding device.

Abstract

Method for shaping the probabilistic amplitude distribution of modulation symbols, corresponding computer program and device. The invention relates to a method for shaping the probabilistic amplitude distribution of modulation symbols belonging to a constellation of q-QAM type symbols. Such a method comprises, for at least one given modulation symbol carrying m bits of information, including m-2 amplitude bits and 2 quadrant bits, a non-uniform (E420) non-binary convolutional encoding applied to an element of a finite field of order two to the power of na, the element being associated with na amplitude bits among the m-2 amplitude bits, the non-uniform encoding delivering an amplitude parity element associated with na updated amplitude bits.The occurrence rate of the amplitude parity element is linearly dependent on at least one probability associated with a constellation symbol having an amplitude identified by m-2 amplitude bits comprising the updated na amplitude bits, the probability following a Gaussian distribution of the constellation symbol amplitudes. ABBREVIATED FIGURE: [Fig.4].

Description

Title of the invention: Method for shaping the probabilistic amplitude distribution of modulation symbols, corresponding computer program product and device. Field of invention

[0001] The field of the invention is that of the transmission of digital data via a transmission channel, e.g. a radioelectric, optical, acoustic, etc. transmission channel.

[0002] The invention relates more particularly to a method for shaping the probabilistic amplitude distribution of a plurality of modulation symbols carrying digital data to be transmitted.

[0003] The invention has applications, in particular, but not exclusively, in the fields of digital data transmission, whether by radio frequency, optical, acoustic, etc. Prior art and its drawbacks

[0004] In order to achieve the capacity of a digital communications channel, two main optimization paths are classically followed:

[0005] - optimization of error-correcting coding; and

[0006] - optimization of the probabilistic distribution of data at the input of the channel of communication.

[0007] Historically, the first path has been the most followed. Several codes of the forward error correction (FEC) type have been discovered. In particular, the first families of error-correcting codes such as the Golay code and convolutional codes were first invented, then more elaborate codes, which proved to be close to the optimum, were invented: turbo codes, low-density parity-check (LDPC) codes and polar codes.

[0008] On the contrary, the second way has remained an open question for some time. In the article by G. Bôcherer, "Probabilistic signal shaping for bit-metric decoding,” IEEE Int. Symp. on Inform. Theory. Honolulu, HI, USA: IEEE, Jun 2014, pp. 431-435 and in the article by G. Bôcherer, F. Steiner, and P. Schulte, ^Bandwidth efficient and ratematched low-density parity-check coded modulation,” IEEE Trans. On Commun., vol. 63, no. 12, pp. 4651-4665, 2015, probabilistic amplitude shaping or PAS (for “Probabilistic Amplitude Shaping”) was developed and coupled with FEC codes to ensure good error correction and to optimize the amplitude distribution of the symbols of mo dulation at the channel input. Since then, hundreds of studies have attempted to build an optimal PAS system. In particular, it has been shown that emulating a Gaussian distribution of the amplitude of the modulation symbols at the channel input achieves the highest mutual information in the case of an additive white Gaussian noise (AWGN) transmission channel.

[0009] However, the design of features allowing the shaping of the probabilistic amplitude distribution remains an open question for today's researchers, targeting low-complexity but high-performance features. For example, the article by P. Schulte and G. Bôcherer, “Constant composition distribution matching” IEEE Trans. on Inf. Theory, vol. 62, no. 1, pp. 430-434, 2015, proposes different algorithms for such features. However, although allowing to obtain the optimal amplitude distribution, the techniques proposed in these works introduce a significant latency in the communication chain. Indeed, the proposed techniques are based on arithmetic codings requiring a large data depth. As a result, the target distribution cannot be achieved for short data frame sizes.

[0010] There is thus a need for a technique for shaping the probabilistic amplitude distribution of the modulation symbols, making it possible in particular to reduce the latency in the transmission chain, eg in order to be applicable to short data frame sizes. Statement of the invention

[0011] In one embodiment of the invention, a method is proposed for shaping the probabilistic amplitude distribution of modulation symbols belonging to a constellation of symbols of type <?-QAM, avec q égal à deux puissance m et m un entier naturel supérieur ou égal à quatre. Chaque symbole de la constellation véhicule m bits d’information, m-2 bits, dits bits d’amplitude, identifiant une amplitude du symbole dans le plan complexe, les 2 autres bits, dits bits de quadrant, parmi les m bits identifiant une appartenance du symbole à un quadrant du plan complexe. Selon un tel procédé, un dispositifélectronique effectue, pour au moins un symbole de modulation donné véhiculant m bits d’information dont m-2 bits d’amplitude et 2 bits de quadrant, un encodage de mise en forme comprenant :

[0012] - for at least one set of na amplitude bits among the m-2 amplitude bits, n has an even natural integer, an encoding, called non-uniform encoding, of non-binary convolutional type applied to an element of a finite field of order two to the power na, the element being defined among the elements of the finite field as being associated with the na bits of amplitude. The non-uniform encoding delivers an amplitude parity element defined among the elements of the finite field as being associated with updated na amplitude bits.

[0013] The non-uniform encoding is configured so that an occurrence rate of the amplitude parity element is linearly dependent on at least one probability associated with a constellation symbol having an amplitude identified by m-2 amplitude bits comprising the updated na amplitude bits. The probability follows a Gaussian distribution of the constellation symbol amplitudes in the complex plane. The shaping encoding outputs an updated modulation symbol carrying the m information bits of the given modulation symbol with the updated na amplitude bits in place of the na amplitude bits.

[0014] Thus, the invention proposes a new and inventive solution for shaping the probabilistic amplitude distribution of modulation symbols belonging to a constellation of q-QAM type symbols.

[0015] More particularly, the known techniques for such shaping are based on an arithmetic coding of the information bits, which requires a large data depth. In contrast, the proposed technique implements a non-binary convolutional type encoding applied to a set of bits representative of the modulation symbols. Being based on a convolutional coding, the performance of the proposed technique proves to be largely independent of the size of the information frame at its input. Such an approach thus presents a reduced latency of the transmission chain, thereby making it particularly suitable, e.g., for transmission systems implementing short data frames.

[0016] In certain embodiments, each symbol of the constellation has a position in the complex plane corresponding to a Gray coding of the m bits of information carried by the symbol.

[0017] In some embodiments, said at least one set of na amplitude bits comprises the most significant bits among the m-2 amplitude bits.

[0018] In some embodiments, the non-uniform encoding is implemented for a single set of na amplitude bits among the amplitude bits, with na equal to m-2, outputting an amplitude parity element associated with m-2 updated amplitude bits. The occurrence rate of the outputted amplitude parity element is proportional to a probability associated with an amplitude of constellation symbols identified by the m-2 updated amplitude bits associated with the outputted amplitude parity element. The updated modulation symbol carries the m information bits of the given modulation symbol with the m-2 updated amplitude bits in place of the m-2 amplitude bits.

[0019] Thus, encoding all of the amplitude bits in one step makes it possible to further reduce the latency of the transmission chain.

[0020] In some embodiments, non-uniform encoding is implemented to a plurality of different sets of na amplitude bits among the m-2 amplitude bits, na an even natural number, outputting a plurality of respective amplitude parity elements corresponding to different sets of updated na amplitude bits. The occurrence rate of the output amplitude parity element in a given implementation is proportional to a sum of probabilities associated with constellation symbol amplitudes identified by m-2 amplitude bits including the updated na amplitude bits associated with the output amplitude parity element in the given implementation. The updated modulation symbol carries the m information bits of the given modulation symbol with the different sets of updated na amplitude bits in place of the different sets of na amplitude bits.

[0021] Thus, the encoding of all or part of the amplitude bits in several steps makes it possible to keep a lower number of encoder states than in the case where a single step of encoding the amplitude bits is envisaged. This is obtained at the expense of an increase in latency compared to the latter case.

[0022] In some embodiments, the number na of bits is identical for all sets of na bits in the plurality.

[0023] Thus, the number of states of the encoder naturally remains regular at each step of encoding the amplitude bits, thereby simplifying its implementation. However, in other embodiments, the number na of bits is different for one (or more) sets of the plurality.

[0024] In some embodiments, the plurality of different sets of na amplitude bits comprises the m-2 amplitude bits.

[0025] In certain embodiments, the shaping encoding comprises:

[0026] - an encoding, called uniform encoding, of non-binary convolutional type applied to a element of a finite field of order four, the element being defined among the elements of the finite field as associated with the 2 quadrant bits, delivering a quadrant parity element defined among the elements of the finite field as associated with 2 updated quadrant bits.

[0027] The uniform encoding is configured so that an occurrence rate of the quadrant parity element is proportional to a probability associated with a constellation symbol belonging to a quadrant of the complex plane identified by the 2 updated quadrant bits. The probability follows a uniform distribution law of the constellation symbols in the quadrants of the complex plane.

[0028] Thus, encoding the quadrant bits in addition to the amplitude bits makes it possible to guarantee a shaping encoding corresponding to a code with a non-zero distance, thereby making the decoding more efficient.

[0029] In some embodiments, the uniform encoding implements elements memory, an integer greater than or equal to one.

[0030] Thus, the number of states of the uniform encoder can be obtained equal to the number of states of the non-uniform encoder even in the case where the non-uniform encoder is defined on a finite field of order greater than four. This makes it possible to simplify the decoding.

[0031] In some embodiments, the non-uniform encoding implements memory elements, na being equal to two.

[0032] Thus, the number of states of the uniform encoder and the non-uniform encoder is identical, both encoders being defined over a finite field of order four. The overall structure of the shaping encoder is further simplified.

[0033] In some embodiments, the non-uniform encoding implements a single memory element, na being expressed as 2 / / .

[0034] Thus, the number of states of the uniform encoder is equal to the number of states of the non-uniform encoder even though the latter is defined on a finite field of order greater than four. This makes it possible to simplify the decoding.

[0035] The invention also relates to a computer program comprising program code instructions for implementing a method as described above, according to any one of its different embodiments, when executed on a computer.

[0036] The invention also relates to a device for shaping the probabilistic amplitude distribution of modulation symbols belonging to a constellation of symbols of type <?-QAM, avec q égal à deux puissance m et m un entier naturel supérieur ou égal à quatre, chaque symbole de la constellation véhiculant m bits d’information, m-2 bits, dits bits d’amplitude, identifiant une amplitude du symbole dans le plan complexe, les 2 autres bits, dits bits de quadrant, parmi les m bits identifiant une appartenance du symbole à un quadrant du plan complexe. Un tel dispositif d’estimation comprend une machine de calcul reprogrammable ou une machine de calcul dédiée configurée pour mettre en œuvre les étapes du procédé de mise en forme de la distribution probabiliste en amplitude de symboles de modulation selon l’invention (selon l’un quelconque des différents modes de réalisation précités).Thus, the characteristics and advantages of this device are the same as those of the corresponding steps of the method of shaping the probabilistic amplitude distribution of modulation symbols described previously. Therefore, they are not detailed further. List of figures

[0037] Other aims, characteristics and advantages of the invention will appear more clearly on reading the following description, given as a simple illustrative, and non-limiting, example, in relation to the figures, among which:

[0038] [Fig. 1] represents a transmitter and a receiver according to an embodiment of the invention;

[0039] [Fig.2] represents, in the complex plane, a constellation of 16-QAM type symbols according to an embodiment of the invention;

[0040] [Fig.3] represents an example of a device structure that can be implemented in the transmitter of [Fig.1] and allowing the implementation of the steps of the shaping method of [Fig.4] according to an embodiment of the invention;

[0041] [Fig.4] represents the steps of a method of shaping the pro distribution amplitude babbler of modulation symbols according to an embodiment of the invention;

[0042] [Fig.5] represents an example of a non-binary convolutional type encoder structure according to an embodiment of the invention;

[0043] [Fig.6] represents an example of a trellis representative of the shaping encoding implemented in the shaping method of [Fig.4] according to an embodiment of the invention;

[0044] [Fig.7] represents the steps of a method for determining the transitions between states of a non-binary convolutional type encoder as implemented in the shaping method of [Fig.4] according to an embodiment of the invention;

[0045] [Fig.8] represents an example of transitions obtained by implementing the method of [Fig.7] when the symbol constellation is of type 16-QAM;

[0046] [Fig.9] represents the BER (for “Bit Error Rate” in English) obtained with a known PAS technique and by implementing the method of [Fig.4] for different lengths of data frames;

[0047] [Fig. 10] represents the FER (for “Frame Error Rate” in English) obtained with a known PAS technique and by implementing the method of [Fig.4] for different lengths of data frames.

[0048] Detailed description of embodiments of the invention

[0049] The general principle of the invention is based on the use of a non-binary convolutional type encoding applied to a set of bits representative of the modulation symbols whose probabilistic amplitude distribution is to be shaped. Indeed, the known techniques for such shaping are based on an arithmetic coding of the information bits, which requires a large data depth. Conversely, being based on a convolutional coding, the performance of the proposed technique proves to be largely independent of the size of the information frame at its input. The proposed technique thus has a reduced latency, which makes it particularly suitable e.g. for transmission systems implementing short data frames.

[0050] We now present, in relation to [Fig.l] a transmitter 100 and a receiver 120 according to one embodiment of the invention.

[0051] More particularly, the communication channel 110 is an optical channel. Thus, the transmitter 100 is an optical transmitter and the receiver 120 is an optical receiver. However, in other embodiments, the communication channel 110 is a radio frequency propagation channel. In such embodiments, the transmitter 100 and the receiver 120 are respectively a radio frequency transmitter and a radio frequency receiver. In other embodiments, the communication channel 110 is an acoustic propagation channel. In such embodiments, the transmitter 100 and the receiver 120 are respectively an acoustic transmitter and an acoustic receiver.

[0052] Returning to [Fig.l], the transmitter 100 comprises a device 111 for generating modulation symbols from information bits. According to the present technique, the modulation symbols belong to a constellation of symbols of the q-QAM type, with q equal to two to the power of m and m a natural integer greater than or equal to four. More particularly, each symbol of the constellation carries m bits of information. Furthermore, it is assumed that among the m bits, m-2 bits, called amplitude bits, identify an amplitude of the symbol in the complex plane, the other 2 bits, called quadrant bits, among the m bits identify a membership of the symbol in a quadrant of the complex plane.

[0053] This is illustrated in [Fig.2] in the case of a 16-QAM constellation, in which the position of each symbol in the complex plane codes 4 bits of information. More specifically, the 2 bits framed in dotted lines have identical values ​​for all the symbols belonging to the same quadrant of the complex plane. These are the quadrant bits defined above. The other 2 bits define the amplitude of the symbols in the constellation. These are the amplitude bits defined above. Indeed, the symbols represented by a circle have the same amplitude (or module) and amplitude bits which have the same value. The same is true for the symbols represented by crosses and for the symbols represented by dots.

[0054] For example, the generation of modulation symbols is based on Gray coding of the information bits in order to associate a set of bits with a given symbol of the constellation.

[0055] The same type of encoding of information bits can be implemented for all constellations of symbols of type <?-QAM, avec q égal à deux puissance m et m un entier naturel supérieur ou égal à quatre. Dans de telles constellations, chaque symbole de la constellation véhicule m bits d’information. On suppose ici que la correspondance entre les m bits d’information et les symboles de modulation est telle que les m bits d’information comprennent m-2 bits d’amplitude et 2 bits de quadrant.

[0056] Returning to [Fig.l], the transmitter 100 comprises a shaping device 112 of the probabilistic amplitude distribution of the modulation symbols generated by the module 111. More particularly, the shaping device 112 is configured to implement the method of shaping the probabilistic amplitude distribution of modulation symbols as described below in relation to [Fig.4]. The device 112 delivers updated modulation symbols having an amplitude distribution in the complex plane which follows a Gaussian distribution law.

[0057] In order to be able to implement the steps of the shaping method of [Fig.4], according to one embodiment of the invention the shaping device 112 is an electronic device comprising ([Fig.3]) a random access memory 303 (for example a RAM memory), a processing unit 302 equipped for example with a processor, and controlled by a computer program stored in a read-only memory 301 (for example a ROM memory or a hard disk). At initialization, the code instructions of the computer program are for example loaded into the random access memory 303 before being executed by the processor of the processing unit 302.

[0058] This [Fig. 3] illustrates only one particular way, among several possible ones, of producing the shaping device 112 so that it performs certain steps of the shaping method of [Fig. 4] (according to any one of the embodiments described above in relation to [Fig. 4]). Indeed, these steps can be carried out indifferently on a reprogrammable computing machine (a PC computer, a DSP processor or a microcontroller) executing a program comprising a sequence of instructions, or on a dedicated computing machine (for example a set of logic gates such as an FPGA or an ASIC, or any other hardware module).

[0059] In the case where the formatting device 112 is produced with a reprogrammable computing machine, the corresponding program (i.e. the sequence of instructions) may be stored in a removable storage medium (such as for example a CD-ROM, a DVD-ROM, a USB key) or not, this storage medium being partially or totally readable by a computer or a processor.

[0060] Returning to [Fig.l], the transmitter 100 comprises an FEC coding device 113 of a known type. The device 113 delivers modulation symbols encoded from the updated modulation symbols delivered by the shaping device 112 and, where appropriate, the modulation symbols delivered by the generation device 111.

[0061] The transmitter 100 also comprises a modulation device 114 of a known type. The device 114 delivers the optical signal to be transmitted from the encoded modulation symbols.

[0062] The receiver 120 includes devices implementing the sy- functionalities metrics of those described above for transmitter 100.

[0063] More particularly, the receiver 120 comprises a demodulation device 124 of a known type. The demodulation device 124 delivers encoded modulation symbols estimated from the optical signal received via the channel 110.

[0064] The receiver 120 comprises an FEC decoding device 123 implementing the symmetrical functionalities of the device 113. The FEC decoding device 123 delivers updated modulation symbols estimated from the estimated encoded modulation symbols delivered by the demodulation device 124.

[0065] The receiver 120 comprises a device 122 for compensating for the shaping performed by the shaping device 112. The compensation device 122 delivers estimated modulation symbols from the estimated updated modulation symbols delivered by the FEC decoding device 123.

[0066] The receiver 120 comprises a device 121 for estimating the information bits conveyed by the estimated modulation symbols delivered by the compensation device 122.

[0067] We now present, in relation to [Fig.4], the steps of a method for shaping the probabilistic amplitude distribution of modulation symbols according to an embodiment of the invention.

[0068] Such a method comprises, for at least one given modulation symbol belonging to a constellation of symbols of type <?-QAM, avec q égal à deux puissance m et m un entier naturel supérieur ou égal à quatre, une étape E400 d’encodage de mise en forme de la distribution probabiliste en amplitude du symbole de modulation donné délivrant un symbole de modulation mis à jour.

[0069] More particularly, step E400 comprises a step E410 of encoding, called uniform encoding, of non-binary convolutional type applied to an element of a finite field, or Galois field, of order four GF(4). The element is defined among the elements of the finite field GF(4) as associated with the 2 quadrant bits of the given modulation symbol. Step E410 delivers a quadrant parity element defined among the elements of the finite field GF(4) as associated with 2 updated quadrant bits. The uniform encoding is configured so that an occurrence rate of the quadrant parity element is proportional to a probability associated with a constellation symbol belonging to a quadrant of the complex plane identified by the 2 updated quadrant bits, the probability following a uniform distribution law of the symbols of the constellation in the quadrants of the complex plane.

[0070] Such non-binary convolutional encoding is described for example in the article by R. Klaimi, C. Abdel Nour, C. Douillard and J. Farah, “Design of Low-Complexity Convo-lutional Codes over GF(q),” in IEEE Global Commun. Conf. (GLOBECOM), Abu Dhabi, UAE, Dec 2018. [Fig.5] illustrates the structure of a non-binary convolutional encoder binary in a case with two memory elements. More specifically, when considered with a given finite field of order q, the input elements I and the output elements (s, p) are elements of the same finite field, while the lattice of the code shows q2 states. The recursion is ensured by the constants {r 1, r2} e GF(q)2, and the parity elements p are calculated using the constants {al, a2, a3, a4} e GF(q)4. Furthermore, in order to obtain an occurrence rate of the parity element according to a given probability, the parameters of the code, e.g. the constants {al, a2, a3, a4} according to the embodiment of [Fig.5], are determined e.g. as described further below in relation to [Fig.7].

[0071] Returning to [Fig.4], step E400 also comprises, for at least one set of na amplitude bits among the m-2 amplitude bits of the given modulation symbol, na an even natural integer, a step E420 of encoding, called non-uniform encoding, of non-binary convolutional type applied to an element of a finite field of order two to the power na, GF(2”fl). The element is defined among the elements of the finite field GF( 2“") as being associated with the na amplitude bits. Step E420 delivers an amplitude parity element defined among the elements of the finite field GF(2”n) as being associated with na updated amplitude bits.The non-uniform encoding is configured so that an occurrence rate of the amplitude parity element is linearly dependent on at least one probability associated with a constellation symbol having an amplitude identified by m-2 amplitude bits comprising the updated na amplitude bits, the probability following a Gaussian distribution law of the amplitudes of symbols of the constellation in the complex plane. The non-uniform encoding implemented in step E420 is thus of the same type as the uniform encoding implemented in step E410. However, the parameters of the code are different in order to obtain an occurrence rate of the amplitude parity elements which follows a Gaussian distribution of the amplitude of the updated symbols in the complex plane. An example of a method for determining the parameters of the code in order to obtain this result is described further below in relation to [Fig.7].

[0072] Returning to [Fig.4], step E400 delivers the updated modulation symbol carrying the m information bits of the given modulation symbol, but with the 2 quadrant bits updated in place of the 2 quadrant bits of the given modulation symbol and with the na amplitude bits updated in place of the na amplitude bits of the given modulation symbol.

[0073] In some embodiments, step E400 is implemented for a plurality of modulation symbols belonging to a constellation of symbols of type q-QAM, with q equal to two to the power of m and m a natural integer greater than or equal to four. In these embodiments, step E400 delivers a corresponding plurality of updated modulation symbols.

[0074] More particularly, in view of the definition of uniform and non-uniform encodings given above, it is observed that the constellation symbols updated on the basis of the 2 updated quadrant bits are uniformly distributed in the four quadrants of the complex plane.

[0075] Furthermore, the amplitude of the updated constellation symbols based on the updated amplitude na bits follows a Gaussian distribution in the complex plane.

[0076] In this perspective, according to certain embodiments, the na amplitude bits of the processed given symbol are the most significant na bits among the m-2 amplitude bits. Thus, the distribution of the updated constellation symbols is closer to a Gaussian distribution than when the na amplitude bits are chosen in any way.

[0077] In some embodiments, the non-uniform encoding is implemented for a plurality of different sets of na amplitude bits among the m-2 amplitude bits of the given modulation symbol being processed, na an even natural number, outputting a plurality of respective amplitude parity elements corresponding to different sets of updated na amplitude bits. As detailed further below in connection with [Fig.7] and [Fig.8], in such embodiments the occurrence rate of the output amplitude parity element during a given implementation is proportional to a sum of the probabilities associated with constellation symbol amplitudes identified by m-2 amplitude bits comprising the updated na amplitude bits associated with the output amplitude parity element during the given implementation.The updated modulation symbol carries the m information bits of the given modulation symbol with the updated different sets of na amplitude bits in place of the different sets of na amplitude bits.

[0078] Thus, the encoding of all or part of the amplitude bits in several steps makes it possible to keep a lower number of encoder states than in the case where a single step of encoding the amplitude bits is envisaged. This is obtained at the expense of an increase in latency compared to the latter case.

[0079] An example of a shaping encoding trellis obtained in such embodiments is illustrated in [Fig.6]. According to this example, N1 non-uniform encoding steps E420, C2 ... CN, are implemented on N different sets of na amplitude bits among the m-2 amplitude bits. Furthermore, the uniform encoding Ci implemented during step E410 works on elements of the finite field GF(4) and comprises memory elements, V an integer greater than or equal to one (eg is equal to one, two, three, four, etc.).

[0080] It is observed here that the number of states of the uniform encoder is obtained equal to the number of states of each implementation of the non-uniform encoder. This in fact makes it possible to simplify the decoding at reception. This equality in the number of states is obtained eg by playing on the number of memory elements and / or on the order of the finite fields on which the non-uniform codings implemented during the Nl encoding steps E420 are defined (i.e. the value of na considered at each implementation). The global shaping encoding thus appears as a non-binary convolutional type encoding varying in time via the alternation of the trellises associated with the uniform coding Ci and the non-uniform codings C2 ... CN.

[0081] In some embodiments, the number na of bits is identical for all sets of na bits of the plurality of different sets of na amplitude bits among the m-2 amplitude bits of the given modulation symbol being processed. Thus, the number of states of the encoder naturally remains regular at each step of encoding the amplitude bits, thereby simplifying its implementation. However, in other embodiments, the number na of bits is different for one (or more) sets of the plurality.

[0082] In some embodiments, the non-uniform encoding implemented during step E420 implements memory elements, na being equal to two. Thus, the uniform encoding implemented during step E410 and the non-uniform encoding implemented during step E420 work on elements of the finite field GF(4) and comprise memory elements. Thus, the overall structure of the shaping encoder is further simplified.

[0083] In certain embodiments, the uniform encoding implemented during step E410 comprises memory elements and the non-uniform encoding implemented during step E420 implements a single memory element, na being expressed as 2 / / . Thus, the number of states of the uniform encoder is equal to the number of states of the non-uniform encoder even though the latter is defined on a finite field of order greater than four. This makes it possible to simplify the decoding.

[0084] In some embodiments, the plurality of different sets of na amplitude bits comprises the m-2 amplitude bits.

[0085] In some embodiments, the non-uniform encoding is implemented for a single set of na amplitude bits among the amplitude bits of the processed given modulation symbol, with na equal to m-2, delivering an amplitude parity element associated with m-2 updated amplitude bits. The occurrence rate of the delivered amplitude parity element is proportional to a probability associated with an amplitude of constellation symbols identified by the m-2 updated amplitude bits associated with the delivered amplitude parity element. The updated modulation symbol carries the m information bits of the modulation symbol with the m-2 updated amplitude bits in place of the m-2 amplitude bits.

[0086] Thus, encoding all of the amplitude bits in one step makes it possible to further reduce the latency of the transmission chain.

[0087] In the aforementioned embodiments in which the set of m-2 amplitude bits of the processed given symbol is encoded, m is even. Conversely, in the embodiments in which m is odd, only a set comprising an even number of amplitude bits of the processed given symbol is encoded via the implementation of one or more non-uniform encoding steps E420. For example, the amplitude bits encoded via the implementation of one or more non-uniform encoding steps E420 are the most significant bits among the m-2 amplitude bits and the least significant bits are not encoded.

[0088] In certain embodiments, the uniform encoding step E410 is not implemented. Indeed, the modulation symbols obtained via the correspondence performed between the m information bits and the position of the symbols in the complex plane (e.g. via Gray coding) are often naturally equally distributed in the different quadrants of the complex plane. In such cases, the execution of step E410 is not necessary. However, the implementation of the uniform encoding step E410 in addition to step E420 makes it possible to guarantee a shaping encoding corresponding to a code with a non-zero distance, thereby simplifying the decoding.

[0089] We now present, in relation to [Fig.7], the steps of a method for determining the transitions between states of a non-binary convolutional type encoder as implemented in the shaping method of [Fig.4] according to an embodiment of the invention.

[0090] More particularly, in the case of uniform encoding (step E410 described above), the non-binary convolutional encoder responsible for generating the updated quadrant bits outputs parity elements according to a uniform distribution law. In other words, the possible parity elements at the output are equiprobable. If we refer e.g. to the encoder structure illustrated in [Fig.5], the parameters {rl, r2] and {al, a2, a3, a4] can thus be chosen in GF(4) to maximize the distance of the code obtained as proposed in the aforementioned article by R. Klaimi, C. Abdel Nour, C. Douillard and J. Farah.

[0091] Conversely, in the case of non-uniform encoding (step E420 described above), the non-binary convolutional encoder responsible for generating the updated amplitude bits does not comply with the same criteria for generating parity elements. More particularly, the transitions between states of the encoder are labeled with parity elements in order to ensure compliance with the target distribution. Secondarily, the distance of the code obtained is maximized in order to make the decoding more efficient. This is obtained e.g. via the choice of the recurrence polynomial (parameters {rl, r2] in the encoder structure illustrated in [Fig.5]) to ensure the bijective nature of the code.

[0092] Thus, if we call T(eb ..., eq) the table comprising the desired occurrences for each of the q parity elements, eb ..., eq, in the lattice of the non-uniform code:

[0093] - during a step E700, the parity element having the largest number of occurrences in the table T(eb ..., eq) is selected;

[0094] - during a step E710, a trellis transition is selected;

[0095] - during a step E720, it is tested whether the transition selected in step E710 has already has been assigned to a parity element or not. If the selected transition has already been assigned, then step E710 is implemented again. If the selected transition has not yet been assigned, step E730 is executed;

[0096] - during step E730, the minimum distance of the code is calculated under the assumption that the parity element selected in step E700 is assigned to the transition selected in step E710;

[0097] - during a step E740, it is tested whether the minimum distance of the code calculated in step E730 is zero or not. If the minimum code distance is zero, then step E710 is implemented again. If the minimum code distance is non-zero, step E750 is executed;

[0098] - during step E750, the parity element selected in step E700 is assigned to the transition selected in step E710. The number of occurrences associated with the parity element selected in step E700 is reduced by one unit in the table T(eb ..., eq);

[0099] - during a step E760, it is tested whether all of the occurrences of the elements of parities has been assigned or not. If all occurrences of the parity elements have been assigned, then the process stops. If all occurrences of the parity elements have not been assigned, step E700 is executed again.

[0100] In some embodiments, steps E730 and E740 are not implemented and step E750 is implemented directly after step E720.

[0101] The implementation of the method for determining [Fig.7] is more particularly illustrated via the example of [Fig.8]. In the present case, we reconsider modulation symbols belonging to a constellation of 16-QAM type symbols, as illustrated in [Fig.2]. The symbols belonging to this constellation carry four bits, including 2 quadrant bits and 2 amplitude bits. The uniform (step E410) and non-uniform (step E420) encoders thus work on finite fields of order 4, GF(4). Thus, the non-uniform encoding C2 is carried out via a single implementation of step E420 since a single set of 2 amplitude bits can be created in the present case for a given symbol. Furthermore, for simplicity, it is assumed that the encoders in question have a single memory element. Thus, the lattice associated with the shaping encoding has four states as illustrated in [Fig.2].

[0102] Furthermore, for the uniform encoding Ci a transition associated with each element, among the elements of GF(4), associated with a possible combination of quadrant bits part of each state. This leads to the uniform distribution of updated quadrant bits.

[0103] Conversely, suppose that the target distribution probabilities for the four sets of 2 bits of amplitude of the updated symbols are given by the vector pr = [0.125 0.1875 0.1875 0.5]. The trellis of the code considered consists of four states, with four transitions that diverge from each state, resulting in a total of sixteen transitions. Taking into account the distribution pr, the occurrence rate of the four parity elements associated with the four sets of 2 bits of amplitude of the updated symbols is given by the vector T(eb e2, e3, e4) = 16 * pr = [2 3 3 8]. Applying the method for determining transitions between states of [Fig.7] to the present case leads to the transitions illustrated in [Fig.8].

[0104] In the present example, all the amplitude bits are encoded together in a single implementation of step E420 (case na equal to m-2). There is thus a direct correspondence between the desired probability for the amplitude of the updated symbol and the occurrence rate that must be considered to define the non-uniform encoding. This result remains true regardless of the number of amplitude bits. Thus, generally speaking, when all the amplitude bits are encoded together in a single implementation of step E420, the occurrence rate of the delivered amplitude parity element is proportional to the probability associated with the amplitude of constellation symbols identified by the m-2 updated amplitude bits.

[0105] Conversely, when only a portion of the amplitude bits is encoded during a given implementation of step E420 (case na strictly less than m-2 with na an even natural integer), the same amplitude parity element delivered during the implementation considered of step E420 corresponds to na updated amplitude bits associated with modulation symbols which may have different amplitudes, and therefore different associated probabilities. In this case, a marginalization technique may be implemented in order to determine the occurrence rate to be used for the amplitude parity element considered.

[0106] To illustrate this case, consider symbols belonging to a 64-QAM constellation. A given symbol carries six bits of information, including 2 quadrant bits and four amplitude bits, which are referred to as b3b2bib0 in the following. In order to obtain a given distribution of the amplitudes of the updated symbols, suppose for example that the updated amplitude bits b3b2bib0 must follow the probabilities given in the table [Table 1] below.

[0107] [Tableauxl] Bits b3b2bib0 Probabilities 0000 0.01 0001 0.07 0010 0.1 0011 0.01 0100 0.07 0101 0.07 0110 0.01 YES 0.07 1000 0.01 1001 0.07 1010 0.1 1011 0.07 1100 0.1 1101 0.07 1110 0.07 1111 0.1

[0108] It is also assumed that the four amplitude bits are encoded via 2 implementations of step E420, each implementation applying to a given set of 2 amplitude bits. More precisely, a first implementation of step E420 is applied to bits b3b2 and a second implementation of step E420 is applied to bits bib0. In order to implement steps E420, it is necessary to determine the occurrence rate of the amplitude elements associated with the set of bits b3b2 on the one hand, and with the set of bits bib0 on the other hand. To do this, the marginalization technique applied to the present configuration gives the occurrence rate to be used for the amplitude parity element considered as being the sum of the probabilities of the updated symbols whose amplitude bits include the bits corresponding to the amplitude parity element considered.Taking the case of table [Table 1] above, we obtain for bits b3b2 the associated probabilities given in table [Table 2] below.

[0109] [Tables2] Bits b3b2 Probabilities 00 = P(0000) + P(0001) + P(0010) + P(0011) = 0.19 01 = P(0100) + P(0101) + P(0110) + P(0111) = 0.22 10 = P(1000) + P(1001) + P(1010) + P(1011) = 0.25 11 = P(1100) + P(1101) + P(1110) + P(1111) = 0.34

[0110] The same calculation can be done for the bib0 bits.

[0111] Thus, when the non-uniform encoding (step E420) is implemented for a plurality of different sets of na amplitude bits among the m-2 amplitude bits, na an even natural integer, delivering a plurality of respective amplitude parity elements corresponding to different sets of updated na amplitude bits, the occurrence rate of the delivered amplitude parity element during a given implementation is proportional to a sum of the probabilities associated with amplitudes of constellation symbols identified by m-2 amplitude bits comprising the updated na amplitude bits associated with the delivered amplitude parity element during the given implementation.

[0112] We now present, in relation to [Fig.9] and [Fig. 10] the BER and the FER respectively obtained with a known PAS technique and by implementing the method of [Fig.4] for different lengths of data frames.

[0113] More particularly, the known PAS technique, designated by SoA below, is that described in the aforementioned article by P. Schulte and G. Bôcherer, “Constant composition distribution matching” IEEE Trans. on Inf. Theory, vol. 62, no. 1, pp. 430-434, 2015. The frame lengths, in number of symbols transmitted, are here:

[0114] - 144 for curves C900 and C1000 (SoA) and curves C930 and C1030 (process of [Fig.4]);

[0115] - 192 for curves C910 and C1010 (SoA) and C940 and C1040 (process of [Fig.4] ) ; And

[0116] - 1152 for curves C920 and C1020 (SoA) and C950 and C1050 (process of the [Fig.4]).

[0117] The curves in question show a better performance obtained for short frames by implementing the present technique compared to the PAS technique. known.

Claims

Claims

1. Method for shaping the probabilistic amplitude distribution of modulation symbols belonging to a constellation of g-QAM type symbols, with q equal to two to the power of m and m a natural integer greater than or equal to four, each symbol of the constellation carrying m bits of information, m-2 bits, called amplitude bits, identifying an amplitude of the symbol in the complex plane, the other 2 bits, called quadrant bits, among the m bits identifying a membership of the symbol in a quadrant of the complex plane, characterized in that an electronic device (112) performs, for at least one given modulation symbol carrying m bits of information including m-2 amplitude bits and 2 quadrant bits, a shaping encoding (E400) comprising: - an encoding (E410), called uniform encoding, of non-binary convolutional type applied to an element of a finite field of order four, the element being defined among the elements of the finite field as associated with the 2 quadrant bits, delivering a quadrant parity element defined among the elements of the finite field as associated with 2 updated quadrant bits, the uniform encoding being configured so that an occurrence rate of the quadrant parity element is proportional to a probability associated with a constellation symbol belonging to a quadrant of the complex plane identified by the 2 updated quadrant bits, the probability following a uniform distribution law of the constellation symbols in the quadrants of the complex plane; and - for at least one set of na amplitude bits among the m-2 amplitude bits, na an even natural integer, an encoding (E420), called non-uniform encoding, of non-binary convolutional type applied to an element of a finite field of order two to the power na, the element being defined among the elements of the finite field as being associated with the na amplitude bits, the non-uniform encoding delivering an amplitude parity element defined among the elements of the finite field as being associated with na updated amplitude bits, the non-uniform encoding being configured so that an occurrence rate of the amplitude parity element is linearly dependent on at least one probability associated with a constellation symbol having an amplitude identified by m-2 amplitude bits comprising the na updated amplitude bits, the probability following a distribution law

2.

3.

4.

5.

6. Gaussian of the constellation symbol amplitudes in the complex plane, the shaping encoding delivering an updated modulation symbol carrying the m information bits of the given modulation symbol with the updated na amplitude bits in place of the na amplitude bits and the updated 2 quadrant bits in place of the 2 quadrant bits. The method of claim 1, wherein the non-uniform encoding is implemented for a single set of na amplitude bits among the amplitude bits, with na equal to m-2, delivering an amplitude parity element associated with m-2 updated amplitude bits, the occurrence rate of the delivered amplitude parity element being proportional to a probability associated with an amplitude of constellation symbols identified by the m-2 updated amplitude bits associated with the delivered amplitude parity element, the updated modulation symbol carrying the m information bits of the given modulation symbol with the m-2 updated amplitude bits in place of the m-2 amplitude bits. The method of claim 1, wherein the non-uniform encoding is implemented for a plurality of different sets of na amplitude bits among the m-2 amplitude bits, na an even natural number, delivering a plurality of respective amplitude parity elements corresponding to different sets of updated na amplitude bits, the occurrence rate of the delivered amplitude parity element during a given implementation being proportional to a sum of the probabilities associated with constellation symbol amplitudes identified by m-2 amplitude bits comprising the updated na amplitude bits associated with the delivered amplitude parity element during the given implementation, the updated modulation symbol carrying the m information bits of the given modulation symbol with the different sets of updated na amplitude bits in place of the different sets of na amplitude bits. The method of claim 3, wherein the number na of bits is identical for all sets of na bits in the plurality. The method of claim 1, wherein the uniform encoding implements memory elements, an integer greater than or equal to one. The method of claim 5, wherein the non-uniform encoding implements memory elements, na being equal to two.

7. The method of claim 5, wherein the non-uniform encoding implements a single memory element, na being expressed as 2 / / .

8. A computer program product comprising program code instructions for implementing a method according to any one of claims 1 to 7, when said program is executed on a computer.

9. Electronic device (112) for shaping the probabilistic amplitude distribution of modulation symbols belonging to a constellation of symbols of the type <?-QAM, avec q égal à deux puissance m et m un entier naturel supérieur ou égal à quatre, chaque symbole de la constellation véhiculant m bits d’information, m-2 bits, dits bits d’amplitude, identifiant une amplitude du symbole dans le plan complexe, les 2 autres bits, dits bits de quadrant, parmi les m bits identifiant une appartenance du symbole à un quadrant du plan complexe, caractérisé en ce qu’il comprend une machine de calcul reprogrammable (302) ou une machine de calcul dédiée configurée pour effectuer, pour au moins un symbole de modulation donné véhiculant m bits d’information dont m-2 bits d’amplitude et 2 bits de quadrant, un encodage (E400) de mise en forme comprenant : - un encodage (E410), dit encodage uniforme,of non-binary convolutional type applied to an element of a finite field of order four, the element being defined among the elements of the finite field as associated with the 2 quadrant bits, delivering a quadrant parity element defined among the elements of the finite field as associated with 2 updated quadrant bits, the uniform encoding being configured so that an occurrence rate of the quadrant parity element is proportional to a probability associated with a constellation symbol belonging to a quadrant of the complex plane identified by the 2 updated quadrant bits, the probability following a uniform distribution law of the constellation symbols in the quadrants of the complex plane; and - for at least one set of na amplitude bits among the m-2 amplitude bits, na an even natural integer, an encoding (E420), called non-uniform encoding, of non-binary convolutional type applied to an element of a finite field of order two raised to the power of na,the element being defined among the elements of the finite field as being associated with the na amplitude bits, the non-uniform encoding delivering an amplitude parity element, defined among the elements of the finite field as being associated with na updated amplitude bits, the non-uniform encoding being configured so that an occurrence rate of the amplitude parity element is linearly dependent on at least one probability associated with a constellation symbol having an amplitude identified by m-2 amplitude bits comprising the updated na amplitude bits, the probability following a Gaussian distribution law of the amplitudes of symbols of the constellation in the complex plane, the shaping encoding delivering an updated modulation symbol carrying the m information bits of the given modulation symbol with the updated na amplitude bits in place of the na amplitude bits and the updated 2 quadrant bits in place of the 2 quadrant bits.

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