Semi-analytical Max-Log MAP Flexible Demodulation Method

Abstract

The invention relates to a flexible decoding method for a received digital signal, comprising, for a signal () in the complex plane received and to be interpreted into a vector of Q bits, a calculation of log-likelihood ratios for each bit of the vector.The digital signal is encoded into symbols in the complex plane according to a Gray-type or quasi-Gray labeling law, and a table associates each symbol in the constellation with other symbols in the complex plane. The process comprises, after receiving said signal, an identification step (100) of the symbol in the constellation closest to the signal in the complex plane, followed by an extraction step (101, 102) from memory of at least some of said other symbols associated by the lookup table with said closest identified symbol, and finally, by a linear combination (105) of products of the integer and imaginary values ​​of said extracted symbols and the received signal, the provision to the receiver controller of values ​​for the log-likelihood ratios of the Q bits. Abbreviated figure: Figure 5.

Description

Title of the invention: SEMI-ANALYTICAL MAX-LOG MAP FLEXIBLE DEMODULATION METHOD technical field

[0001] The invention falls within the general field of telecommunications, and more specifically within that of receivers for wireless communication systems, but also within that of receivers for wired communication systems. It focuses on flexible, or probabilistic, demodulation algorithms.

[0002] In the context of digital communications using linear modulations, the aim is to transmit information through the physical propagation channel by means of symbols derived from finite alphabets of complex numbers, also called constellations. These symbols are used to modulate in amplitude and phase one or more carrier frequencies used to generate the signal to be transmitted. This is quadrature amplitude modulation (QAM).

[0003] The implementation presented in this document provides an alternative method for implementing max-log-MAP soft demodulation (MAP meaning Maximum A Posteriori – this is a method that maximizes the a posteriori probability of having decided on the correct symbol or the correct bit sent), which allows for an improved performance-complexity trade-off compared to the prior art. The invention is proposed in particular for non-square constellations. Clarification of the technical context

[0004] Consider a single-user (SU) communication system, consisting of a transmitter and a receiver, where a bit block b of length Kh is processed by a classical encoder for Bit Interlevated Coded Modulation (BICM). This BICM encoder generates B data blocks, each composed of K QAM symbols belonging to a constellation X c C, where C is the field of complex numbers.

[0005] In detail, the BICM encoder is composed of the following operations. First, an error-correcting code C associates Kb bits of information with Kc coded bits, also called master code bits, because they are generated by a master code having a fixed coding rate. Next, the Kc coded bits are provided to a rate-matching / interleaver block, which can punch or repeat the input coded bits and interleave them to obtain a codeword d, consisting of Kd interleaved bits, this length being adapted to the resources available in the transmitter's frame for sending information. The interleaved bits are divided into B blocks of resource in such a way that Kd = BKQ, where BK are the available physical resources, M is the modulation order (i.e., the constellation size M =}X|) and Q — logQA / . Thus, if A j is the associated 2-bit vector

[0006] to the symbol -¾ such that «4 / Y(b) \ then d can be written

[0007] At the receiver, after time and frequency synchronization, and possibly equalization or other interference or jamming attenuation processes, the post-processed received symbols are designated by xf. Ideally, the xf are assumed to be observations generated by an equivalent channel modeled by complex circular additive white Gaussian noise (AWGN) added to the transmitted symbols, having a noise variance of 1. If a turbo receiver is used, then the upstream processing can exchange probabilistic (or soft) information about the bits with the decoder, in the form of logarithms of the bit likelihood ratios, called log-likelihood ratios (LLRs). The LLRs are denoted a a priori coming from the decoder to the demodulator and the upstream processing, and we note 'CS ^Rs extrinsic which are supplied to the decoder. Note that here the term "decoder" above includes the inverse operations corresponding to the error correction coding operations, and the bitrate adaptation and interleaving blocks.

[0008] The invention presented below relates to the soft demodulation operation (also called soft demapper or soft demapping), the purpose of which is to calculate extrinsic LLRs L^d]^ (LLR of the qth bit) on the coded bits, using the received and processed symbols (estimated equalized / received), the variance of the noise and residual interference 14 (post-equalized variance / received) and the a priori LLRs La(dkq) for k — 1, ..., K etq = 1, ■ Q.

[0009] This process is independent of the symbol index k, which is why this index is omitted in what follows.

[0010] In the exposition, X® and x| denote the subsets of the constellation with the qth bit set to 0 or 1.

[0011] On the receiver side, to approach optimal detection and decoding performance, it is necessary to identify the constellation symbols that were actually chosen by the transmitter, based on the received signal containing noise and possibly interference. The most robust methods for achieving this are the so-called "soft demodulation" techniques, which perform probabilistic estimations of these symbols, by calculating the log-likelihood ratios (LLRs, as mentioned above) of the bits used to label the transmitted symbol. These LLRs are subsequently referred to as bit LLRs (or bit LLRs) to distinguish them from the log-likelihood ratios calculated on the elements (symbols) of the signal constellation used by the system. These latter LLRs are called symbol LLRs.

[0012] The exact solution to the problem of computing extrinsic LLRs is known, but nevertheless, the complexity of implementing this solution is not always feasible for practical use, depending on the size of the alphabet (i.e., the constellation of signals) used. Thus, various variants of this solution with simplifying approximations have been proposed. In particular, excellent solutions exist for constellations with square geometric shapes (in English, "square QAM," Quadrature Amplitude Modulation: 4-QAM, 16-QAM, 64-QAM, 128-QAM, etc.), but the solutions in the literature for constellations that are not square or rectangular (examples of non-square constellations are shown in [Fig. 1], [Fig. 2], and [Fig. 3]) can still have excessively high complexity or detection performance losses that are too significant to neglect.

[0013] Soft demodulation in the general case, without approximation

[0014] The optimal soft demapping is that which implements the log-MAP algorithm [1]. In this case, the extrinsic LLRs are given by

[0015] - logL f

[0016] with the a priori likelihood on the symbols calculated as follows:

[0017] L a {a) = log[P (a) / P (a ref ) ], has ref , aeX

[0018] where, n / \ \ r / j \ . * * w _ By choosing the LJ logP (a) - -Lq^(p-^{a)La{dq) + constant, V«eX reference symbol aref equal to the one with the label of Q bits all zeros, [0, ...0] = Favorable cases of Gray and near-Gray labeling

[0019] The function that associates Q bits with a symbol of a given constellation (the so-called "labeling" function) can be chosen to facilitate calculations. The choice of labeling has a significant impact on how the a priori symbol LLRs { La{ a)} on x are related to the a priori bit LLRs j | G .

[0020] Reference [2] contains an approximation used for calculating LLR bits in the case commonly used in practice where the labeling between bits and symbol (i.e., the function —x which associates a bit vector d, called the label, with v GX, the symbol of the The constellation X) is created using Gray's method (indicated by "Gray mapping"). A labeling law is said to be "Gray's law" if it satisfies the following property: for each symbol in the constellation with a given label d, all symbols in the constellation that are neighbors of that symbol (i.e., having a minimal Euclidean distance from it) have a binary label that differs from d by at most one bit. A labeling is said to be quasi-Gray when this property is generally respected for all points in the constellation except for a few exceptions (e.g., symbols that have more than Q neighbors, where Q is the label length, cannot all satisfy this rule).

[0021] In the particular case of Gray labeling (or Gray mapping), where by construction the binary label between neighboring symbols changes only on a single bit, the a priori bit LLRs of a symbol each carry roughly an equal amount of mutual information on each bit [2]. Thus, the following approximation is accurate 100221

[0023] In other words, for constellations with Gray or near-Gray labeling (hereafter referred to as "quasi-Gray"—the Gray labeling property is satisfied for almost all symbols in the constellation), it is possible to perform the calculation (exact or approximate) of the extrinsic bit LLRs directly on symbol likelihoods. Then, if a posteriori bit LLRs are required, the a priori bit LLRs are added separately to the extrinsic bit LLRs thus calculated.

[0024] The presence of logarithmic and exponential operations in the preceding formula nevertheless makes this technique cumbersome in terms of computational complexity.

[0025] Hardware implementation - flexible max-log-MAP demapping

[0026] The preceding formula is often rewritten in terms of max-star, max* functions with

[0027] rti \ * i |.rf-«]2 \ * / ly-a'l” \

[0028] where rnax* ( a, b ) - max ( a, b) + f ( a - b ) and f _• x log( 1 + irM ) is a function called Jacobi function, and if there are more than two arguments, this function is associative and can be applied two by two sequentially on multiple arguments.

[0029] With this approach, the complexity of logarithms and exponentials is replaced by the complexity of computing maximum functions on two sets of Mil elements (essentially comparisons), and by computing M Jacobi functions. The Jacobi function has a regular behavior and decays rapidly; it can therefore be implemented by a one-dimensional look-up table (LUT) or approximated by an ap Piecewise linear proximation. In this case, the LUT, being one-dimensional, is a list, whose output element is chosen according to an input which is a range of values ​​from the real number line.

[0030] In order to further reduce complexity, in the prior art, the calculation of the Jacobi function is often completely omitted. In this case, we speak of flexible max-log-MAP demapping, and the extrinsic LLR bits are obtained with [° 031 ] Le ( d q ) «[ min a eX i ( | x e - a ' | 2 ) - min^ (J x e - a | 2 ) ].

[0032] This results in a significant reduction in computational complexity, in return of a loss of performance.

[0033] Furthermore, the computational complexity of the flexible max-log-MAP demapping may still remain prohibitive for higher order constellations (e.g. 16-QAM / APSK and higher numbers of different symbols, possibly with amplitude and phase modulation, APSK meaning amplitude and phase-shift keying, PSK meaning phase-shift keying), because this operation still requires calculating M distances, then performing for each of the log?M bits, a total of M comparisons, and this process is repeated for each symbol.

[0034] The flexible max-log-MAP demapping has nevertheless been widely used for the practical implementation of most modern communication systems, which operate with flexible input channel decoders (i.e. with LLRs on the coded bits).

[0035] Reference [8] provides an overview of the use of this technique for the material implementation of many constellations from various standards.

[0036] Various strategies exist for an efficient hardware implementation of the operations involved in the max-log-MAP formula, such as the calculation of the Euclidean distance, maxima, and comparisons, as can be seen in references [9],

[10] , and

[11] . In [9], the authors explore architectures with multipliers and multiplexers for the calculation of distances and maximum operations, and they are able to parallelize the calculation of LLRs over logyV stages, allowing a throughput equal to the symbol rate. In

[10] and

[11] , the authors explore the impact of hardware architecture choices related to the instruction set available on these targets on the efficiency of implementing the max-log-MAP algorithm on FPGA (Field-Programmable Gate Array) or ASIC (Application-Specific Integrated Circuit) hardware.

[0037] Although there is a wide range of architectures on max-log-MAP in general, when considering the simple case of a constellation with Gray labeling, hardware implementations based on real distance and the calculation of the function maximum throughput results are 3 to 6 times slower compared to architectures based on the analytical formulas of max-log-MAP LLRs. This result is shown in Table 8 of reference [8], where reference

[99] (L. Ali et al, 2015, here reference

[12] ) significantly outperforms the alternatives in references [9],

[10] and

[11] .

[0038] Moreover, the advantages in terms of complexity of a direct implementation of max-log-MAP are limited for high-order constellations, as mentioned above (the number of distances to be calculated and the size of the set grows linearly with the size of the constellation, or, equivalently, exponentially with the number of bits Q in the binary label).

[0039] To further reduce the computational burden of soft demapping, known efficient practical implementations (reference

[12] ) rely on analytical expressions for extrinsic LLRs that are computed using the geometric properties of each constellation. In particular, it is possible to directly translate the max-log-MAP approximation into accurate analytical equations for certain constellations with Gray labeling (see references [3] and [5]). This is notably the case for linear Puise Amplitude Modulation (PAM) and the corresponding square QAM constellations, which yield equations whose computational complexity is a function of Q = log^M.Furthermore, in the case of BPSK (binary phase modulation or 2-PSK) and QPSK (quadrature phase modulation) with Gray labeling, the max-log-MAP and the exact log-MAP coincide, and the analytical expressions of the max-log-MAP are therefore ideal.

[0040] The analytical expressions for the QAM constellations (reference [3]) are given in Table 1, and the computational complexity of the soft demapping of a data symbol is provided.

[0041] The complexity estimate is given in terms of real additions c, real multiplications Mr, and comparisons Cr. Some addition or comparison operations sometimes appear, instead of multiplications, in the last column. This is to better account for multiplications with powers of 2, which can be performed with a bit shift in a fixed-point implementation.

[0042] [Tab. 1] Table 1: Analytical approximations of the flexible max-log-MAP demapper for QAM constellations (Mr, Mr and Cr; numbers of multiplications, additions and comparisons, respectively). Constellation Equations Complexity BPSK (2-PAM) L(^) =4R(xe)MM r+Ar QPSK (4-QAM) L(d4 = 2fiR(x-') / vi UdJ =272 I(.r) / vS 3Mr 16-QAM L(d2) = 4d(2d-\l^)\)l^ L(d2) = 4d\2d-\R(x')\)h' 4Mr+ 4 / 4,+4¾ L(d2) = L(d4) - d= V^ / ÏÔ 4di f xe ) / 1¾ J1 ( Xe ) 1 < 2d Sa^1(^) - d) / v%, I( xe ) > 2J I(xe) <-2d 4dR ( ) / 1¾ | R ( Xe ) | < d 8J(R(rj -d)lx^ R(xe)>2d %d(R(jd)+d) / ve, R(xe) <-2d 64-QAM L(d}) ~ ^(^2) = L(d3) = L(44) = ^ L^s) = 4 84(|I( W(|I( . 164( M4) =: 4 8d(|R 12J(|R , 16d( d= 1 / ^42 44((1(,¾) )-24) / 1¾ |I(^)| <4d 4d(6d- jl(» | ) / 1¾ 11(| >44 4d( |R(r) |-24) / 1¾ |R(» | < 4d 4d ( 6d -1R ( xe ) | ) / 1¾ | R ( xe ) | > 4d Sd(3d-\I(xe)\) / ^ <2d 4d(4d-|I(r') J) / 1¾ 2d< |I(x") | <6d 8d(5d-\ï(x)\yv» |I«H %d ( 3d -1R ( Xe ) | ) / 1¾ | R ( xe ) | < 2d M(4d-|R(.rcj|)M 2d< |R(r)| <6d 8d(5d-|R(xe) | ) / v$, |R«) | > 64 sign(I(.re) ) x d11(xe) 1 / 1¾ |I(^) | < 2d Xe) | -d) / 1¾ 2d < |I(X) | < 4d xe)\-2d) / v^, 4d< |I(xe)| <6d |I(^|-3<W |I(^)|> 64 sign(R(xe) ) x rf|R(r')| / ^ IR^)!^ (.V;) | -d)N^ 2d < iRCvQ | <4d (Xe) | -217) / 1¾ 4d < |R(xe) | < 6d |R(X')|-3i / )M |R(^)|>6J 5Mr + 3A,. +6Cr。

[0043] With regard to non-square constellations, the geometric characterization of max-log-MAP LLRs does not systematically provide a closed-form analytical expression. There are some well-known approximate expressions in the literature for 8-PSK ([4]) (that used in the efficient architecture of

[12] ) and for 16-APSK conforming to the DVB, Digital Video Broadcasting standard ([5]). These expressions and their computational costs are given in Table 2. For such a constellation, there are also alternative techniques that attempt to minimize the impact of these approximations, such as those in references [6] and [7], which seek to exploit polar coordinates to perform an exact max-log-MAP on an 8-PSK constellation. However, the computational cost of the polar coordinates is higher than the solution revealed below.

[0044] [Tab. 2] Table 2: Analytical approximations of the flexible max-log-MAP demapper for PSK / APSK constellations with published methods (Mr, Mr and Cr: numbers of multiplications, additions and comparisons, respectively). Constellation Equations Complexity 8-PSK [4] . / 2 3M f- + Ar ^(^3) 16-APSK «1, |R(.V) I < i4Mr + 8Ar + 4Cr (DVB) «xO - otherwise [5] «r,l = °Co - «v,l - a2, |l(.r)| >Tm otherwise t «3, otherwise ra2, |R(V')|>Tm otherwise L(dJ = ( IV - «x. 112 - ! - «rn 12 ) UdJ / . |2 1 |2\ , = 1 p - ayj | - | xe - aI ) / vJ UdA = 1.2703xR(» / ^. L(d4) = 1.2703 x 1(^)^ Tm = 0.5 ( Æ[COS( 5 ) + 7?2COS distance to the median between the two circles on the axes,

[0045] Use of binary constellation labeling

[0046] Furthermore, in order to get as close as possible to the max-log-MAP calculation, some recent work analyzes in more detail the properties of PSK constellations and certain APSK constellations (called "product APSK" in English), and it is found that the binary labeling of the constellation can be used for an efficient implementation.

[0047] In

[13] , the authors propose a new method for handling max-log-MAP demapping of PAM / QAM / PSK constellations (and certain symmetric APSKs) with Gray labeling using the formula:

[0048] T \ K _a* \ In the above expression: L e { d q ) « - I 1-2^ I----- • a* = ud ( Xe ) is the point (or symbol) in the constellation closest to f hardv Xe. It is chosen once uniquely, with ( ■ ) being the hard decision function of the constellation.

[0049] We hereafter denote V the qth bit (q between 1 and log2(M)) of its label a ct binary (this quantity takes the value 0 or 1). • is the point (or symbol) of the constellation closest to a* (at the place of Xe, as we would have in the ideal max-log-MAP) and which has the value । (this quantity takes the value 0 or 1) on the qth bit.

[0050] The authors derive a binary arithmetic for the PAM, QAM, and PSK constellations that allows the binary label of <4 to be calculated using the binary label of a*. The computational complexity of this method is therefore equivalent to the simplified hard detection complexity, i.e., QCr, followed by the calculation of the opposite label, which requires approximately Q arithmetic operations on the bits (comparable to comparisons), and finally Cr + 7Ar + 3Mr) operations to complete the calculation of the LLR using the formula above. Thus, a total of 3Cr + 7Ar + 3Mr) operations is required.

[0051] This method allows the exact calculation of max-log-MAP to be carried out by exploiting the properties of Gray labeling to find the nearest symbol <4 belonging to the set or X^ opposite to the symbol determined by hard detection (which is the nearest point of the received signal).

[0052] However, the arithmetic calculation of the opposite symbol and, more importantly, the calculation of the formula with Euclidean distances, still have a significant weight in the overall computational complexity.

[0053] Furthermore, document

[14] deals with the calculation of LLRs for QAMs with a generalization to non-uniform symbol spacings. This disclosure concerns more specifically on QAMs with a Gaussian geometric shape. It is significantly different from the invention proposed in this document, which is also not limited to a specific constellation shape. Summary of the invention

[0054] The problem addressed is soft demapping for estimating the log-likelihood ratios (LLRs) of the bits corresponding to the modulated symbols. The exact demapper is computationally expensive due to the exponentials and logarithms involved. The max-log demapper is also expensive because there are many distances to calculate. Analytical demappers approximate the max-log with geometric properties of each constellation, but they are less accurate for constellations that are not square QAMs.

[0055] We are looking for a compromise between performance and complexity.

[0056] The solution presented here is to use analytical expressions in conjunction with LUT lookup tables to approximate the max-log demapper. This is applicable to constellations with Gray or quasi-Gray labeling. It is ideal for non-square constellations such as PSK, APSK (e.g., DVB or MIL-STD-110C standards) of order 8 or higher. The idea is to pre-calculate, for a given constellation, the distance components that are linked by Gray or quasi-Gray labeling and store them in LUTs.

[0057] We thus propose a strategy derived from the max-log MAP, using the properties of Gray labeling, and noting the constellation point which is closest (identified during a so-called hard decision) to the signal to be demodulated Xe, we can write the equation of the log-likelihood ratio of each of the bits of the vector to be written to demodulate Xe.

[0058] To solve the aforementioned problems, a flexible decoding method for a digital signal received by a receiver performing quadrature demodulation is proposed. For a signal received in the complex plane and to be interpreted into a Q-bit vector—the signal in question being a noisy observation of the Q-bit vector encoded by a symbol in the complex plane—the method comprises calculating log-likelihood ratios for each bit of the vector. The method uses a pre-established lookup table stored in the receiver's memory. These principles are known, but the lookup table used in the invention is highly original and allows for increased performance or a reduction in the complexity of the method, which is advantageous for its practical implementation.

[0059] Remarkably, the digital signal is, according to the invention, encoded into symbols in the complex plane according to a Gray-type, or quasi-Gray-type, labeling law, and the table associates each symbol in the constellation with the other symbols in the complex plan.

[0060] The symbols associated with a symbol by the table are one per bit coded in a symbol, unless these symbols can be deduced from each other by application of a logical rule, in which case they may be fewer in the table, due to the presence of known redundancies.

[0061] The method further comprises, after receiving said signal, a step of identifying the symbol of the constellation closest to the signal in the complex plane, followed by a step of retrieving from memory at least some of said other values ​​in the complex plane associated by the lookup table with said closest identified symbol, and finally, by a set of arithmetic operations based on the real and imaginary values ​​of said extracted complex values ​​and of the received signal, providing the receiver controller with values ​​for the log-likelihood ratios of the Q bits for said received signal. The operations may be a combination of the real and imaginary values ​​extracted from the table, and the real and imaginary values ​​of the received signal, possibly a linear combination of products of these values ​​(in particular, monomials of degree 2).

[0062] This method is remarkable because it considerably reduces the calculations required, since most of the data to be used is available in LUTs and therefore easily accessible without computational effort. It is based on the mathematical and logical analysis presented below and represents an innovation compared to previous methods, since usable values ​​as log-likelihood ratios are obtained essentially by directly extracting pre-established values ​​from tables of quite modest size. The calculations performed on the extracted values ​​are simple multiplications and additions, excluding logarithms or exponentials, which does not consume computational resources.

[0063] According to optional and advantageous features: The table, or another pre-established table also stored in the receiver's memory, can associate real values, called distance values, with each symbol in the constellation. The extraction step then includes extracting positive real values ​​associated with the nearest identified symbol. The set of arithmetic and logical operations is then also based on these extracted positive real values, which are used to add a value independent of the received signal to the linear combination. The added element has the dimensions of a product of two signal coordinates in the complex plane, and is therefore typically a distance square. It is generally non-zero for constellations that are not simply polygonal. - the elements stored in memory are in the form of proportional values tionals to the difference between the input symbol of the table and the Q complementary symbols with respect to the bits of the label of said input symbol, or values ​​proportional to the difference between the square of the modulus of the input symbol of the table and the square of the modulus of the Q complementary symbols with respect to the bits of the label of said input symbol. - the constellation can be a PSK or APSK constellation, possibly symmetrical, but not necessarily. - the constellation is for example an 8-PSK constellation, or the constellation is for example a 16-APSK constellation. - during the extraction stage, consideration may be given, in addition to the symbol of the constellation closest to the signal in the complex plane, also to the result of additional geometric tests based on the received signal, to distinguish between neighboring symbols of the constellation which have the same given value of a certain bit. - the table associates each symbol of the constellation with other symbols in the complex plane, taking advantage of the constellation's symmetries to limit the size of the memory used. - the variance of noise and residual interference is taken into account in the calculation of the log-likelihood ratios of each bit of the vector. - the receiver is a turbo receiver and the prior likelihood log ratios of the turbo receiver are taken into account in the calculation of the likelihood log ratios of each bit of the vector.

[0064] Generally, at the receiver, synchronization begins by filtering and then performing time / frequency synchronization. Equalization then occurs by extracting the frame and conducting channel estimation for equalization. Next, detection occurs by performing demodulation, which is the object of the invention. Finally, errors are corrected by performing deinterlacing and decoding.

[0065] The method of the invention involves a limited number of multiplications and additions in addition to the hard decision (which involves comparisons and one or two possible additions and multiplications). In the case of 8-PSK, Q = 3, this results in 8 multiplications, 3 additions, and 3 comparisons, and the storage of a 3x8 LUT comprising two real values ​​forming a complex number. List of figures

[0066] The invention will now be described with reference to the accompanying figures, among which:

[0067] [Fig-1] The [Fig.1] presents an 8-PSK constellation with Gray labeling.

[0068] [Fig.2] Fig.2 shows a 16-APSK constellation conforming to the DVB standard with quasi-Gray labeling.

[0069] [Fig.3]

[0070] [Fig. 4] Figures 3 and 4 show 16-APSK and 64-APSK constellations compliant with MIL-STD-110C (Military Standard Interoperability and Performance Standards for Data Modems) with quasi-Gray labeling.

[0071] [Fig. 5] Fig. 5 shows the architecture of the semi-analytical demapping solution max-log-MAP for constellations with (quasi-)Gray labeling. Detailed description

[0072] The solution proposed here considerably reduces the computational burden of state-of-the-art proposals.

[0073] Instead of numerically calculating <4 and then calculating distances, it is proposed to exploit LUTs in which the symbols and corresponding distances of the pairs ( a*, which have been pre-calculated, are stored.

[0074] Thus, a memory usage on the receiver, which is very limited, is substituted for an effort in terms of calculations. The last formula in the previous section can be rewritten as follows:

[0075] \ L e (d q )~ -1 1-2^1---—— = I 1-2^1------- defining

[0076] 1-2^) et A^Ca*) = -(1-2^ ( j 4=1 ... Q

[0077] as quantities that depend on M possible values ​​of the point obtained with hard detection and* = ( ,xe; vJ ) (or its label it is possible to estimate the max-log-MAP with T / 7 \ )^(-^)1(^^,-( a*)) (1)

[0078] The A^^ are real numbers and the A^ are complex. In particular, for a PSK constellation, notably 8-PSK, A|ap$( a* ) = 0, for all a* ≥ 6, which further reduces the complexity of this method (this point is not obtained, however, for an APSK constellation), and the formula becomes T / jx R(;v9R(AM(a9 l+iGUKAa.A®'') ). (dq) ~ (2)

[0079] The preceding formula applies to all PSK constellations. A very interesting case is that of the 8-PSK constellation.

[0080] [Table 3] Table 3: Direct analytical approximations of the flexible max-log-MAP demapper according to the invention with the new LUT-based method for PSK / APSK constellations (Mr, Mr and Cr; multiplication, addition and comparison numbers, respectively). Constellation Equations Complexity 8-PSK Initial decision fi = |R(a“') | < |I(a"') 1 / 2 = R(r)<0 4=i(.xo <o Extraction de la mémoire ( 4,1 A„,. A^) = LUT_8PSK( 4) opérations arithmétiques et logiques L(di) = (R(x-)R(AcJ) +I(»l(AaJ) )A1 L(d2) = (R(^)R(Aj +1(.^)1( A^))A5 L(d3) = (R(a-)R(A^) +I(xOI( A^) ) / v< 8Mr+3Cr + 3A, + Stockage de deux LUTs 3x8 à sortie réelle (équivalent au stockage d’une LUT 3x8 à sortie complexe) 16-APSK (DVB) Décision initiale f= ( |I(.Vj | > |R(xe) 0 v ( ( |R(xO | < Tm) a ( |l(xO | < Tb- |R(^) | ) ) f2= ( |R(r) | > ^ / 3|I(» |) v (( |l(xA | < T,„) a ( |I(x'Q | < Th-|R(^) | ) ) / 3=r(x«)<0 4=1(-^) <0 T„ = 0.5( 4cos( 5 ) + Æ2COS (¾ ) ) : distance to the median between the two circles on the axes, Tb= (R + R2 ) / ^2* : the y-intercept of the perpendicular bisector between the radii of the two circles. Memory extraction ( A„.„ A^, A<f3, A„4) = LUT_16APSK( / r 4) (A|,|b.4,.,¾ âw> 4) = LUT_16APSK_P(44 4 4) Arithmetic and logical operations Udt) = (r(.v-)R(4ç|)-1(1^)1( ah1) +A^pj / v; L{<1^ = (R(r)R( Aff2)-I(v)l( Ao4 + Awù) / 4 ^(^3) = -1(^)1(^.3) Ud^ = (R(V)R( 4,.4) -1( A-) i( a„,4) + 4,4-4) / 4 14A4+ 10Ar + 8Cf + storage of three 4x16 LUTs with real output (equivalent to the storage of one 4x16 LUT with complex output and one 4x16 LUT with real output) .

[0081]

[0082]

[0083]

[0084]

[0085] The first way to implement formulas (1) and (2) with LUTs is to generate a LUT for the differences in squared distances (powers 2 of the distances), and especially a LUT for the difference in symbols, both indexed on a*, which requires that one dimension of the table is equal to the cardinality of the constellation (8 for 8-PSK, for example). Thus, the elements stored in memory (in the LUTs) are, as has been said, in the form of values ​​(A„j... g) proportional to the difference between the symbol at input of the table and, in the case presented here, each of the Q symbols complementary with respect to the bits of the label of said symbol at input, or of values ​​(A Ai >2 i) proportional to the difference between the square of the modulus of the symbol at input of the table and the square of the modulus of, again, each of the Q symbols complementary with respect to the bits of the label of said symbol at input. For example, as mentioned in Table 3 above, for the 8-PSK constellation, since Q = 3 bits, for each of the 8 symbols, we generate 3 LUT values ​​Aa,2, ^a3 with a complex output, indexed on 8 possible values ​​of a*, for a total of 3 x 2 x 8 = 48 real values, in a 3 x 8 = 24 complex output LUT (a LUT with 24 complex outputs). For the 16-APSK constellation, as mentioned in Table 3 above, for each of the 16 symbols, we generate 4 LUT values ​​with real output and 4 LUT values ​​with complex output for a total of 4x(l +2) x 16 = 192 real values ​​to be saved in memory, for example in two LUTs with respectively 4 x 16 = 64 real outputs and 4 x 16 = 64 complex outputs. Compared to this initial method of implementation, two variants are proposed, constituting the following improvements: - First, it was observed that choosing the opposite symbol solely based on a* does not always correspond to the max-log-MAP algorithm, which indicates that the symbol should be the one from the constellation closest to Xe and that also has the bit with the opposite sign, with respect to "*". Finding such a symbol is not a problem for Grays constellations like the 8-PSK, because the pair (a*, ) is always uniquely defined for any The received symbol X^. However, this is no longer true for a complex quasi-Gray constellation such as 16-APSK (defined in the DVB standard). In this case, for the same a*, there can be several choices depending on where the symbol X% is located within the Voronoi region of a*. It is therefore proposed to identify the "best" (i.e., closest to the received symbol) "4". xf) based on additional geometric tests on xf. The LUTs therefore incorporate a larger choice of pairs (a* j) indexed not only on o* but also on the results of these geometric tests on xf.

[0086] Thus, by noting Q = Q + Q, where Q is the number of bits per symbol, Q = log^M, M being the order of the constellation, and is the number of additional geometric tests, the LUTs s(q) and |2 are indexed on at most q = (Q - 1) bits, corresponding to _ 9C? values.

[0087] For example, in the case of a 16-APSK defined in DVB, after making a hard decision on the received symbol xf (which determines the choice of the Voronoi region, see Figure 2), there are 6 tests in total, leading to 64 possible configurations of ( a*, ) instead of the 16 possibilities determined by the hard decision a* alone. Using all these tests, the proposed solution is described in Table 4.

[0088] [Tab. 4] Table 4: Analytical approximations of the flexible max-log-MAP demapper for 16-APSK constellations with the new method based on LUTs and with increased accuracy through geometric tests (Mr, Mr and Cr: numbers of multiplications, additions and comparisons, respectively). Constellation Equations Complexity 16-APSK (DVB) Initial decision f, = |R(à-) | <73 |I(X)l / 2=|lCv)|<73|RCv)| ;3=|R(.r)|>r„ / 4=|i(r)|>r„ A= (v ((A ( WU H lRV) 1 )) L= (~6)v ( ( ~L) a < WO1 < Tb-|rm | ) ) Tm = ().5(^^05( J ) + ) ) : the distance to the median between the two circles on the axes, Tb = ( R^ + R^ ) / ^2* : the y-intercept of the perpendicular bisector between the radii of the two circles. Memory extraction ( A„a, A^ A^ Am) = LUT_16APSK( / f f2, f? f* / j A|„|> A|«^ =LUT_16APSK_P( / r 4 4 4 4 4) Arithmetic and logical operations 14M, + 10Ar + 8Cr + storage of three 4x64 LUTs with real output (equivalent to storing one 4x64 LUT with complex output and one 4x64 LUT with real output) LU = (rLIRULILAiH^H LÇdJ = (R(V)R(AU U(UI(AU +AkpjM L(d;) = 1 Ri r HU A 1 -1(^)1(44 + LW = (R(v)R(A.4) - I(v)i( a„,4) + A^pjA*;

[0089] Thus, the dimension M' of the LUTs can be multiplied by the number of LLRs bits per symbol Q and give the total space occupied QM' (recall that M is the order, size or cardinality - these words being synonyms here, of the constellation, and that Q is the base-2 logarithm of M - it is also the number of bits encoding a symbol, M'>M takes into account the binary geometric comparisons Q' which complement the hard decision).

[0090] For 8-PSK M = 8, Q = 3 and QM = 24.

[0091] For 16-APSK, for which this variant is usefully applicable, M' - 64, Q = 4, and QM' = 256. - Secondly, memory usage can be reduced by exploiting existing symmetries in the constellation and indexing the LUT on the output of relevant geometric tests on the received symbol xf. By considering the 16-APSK solution discussed previously (Table 4), eliminating the dependence on irrelevant tests for each LUT output, and exploiting the symmetry in the quadrants of the complex plane, the size of the LUTs is minimized for each LLR. This results in a total, as shown in Table 5, of two LUTs with a 2-bit indexed input (one dimension of 4) and two LUTs with a 3-bit indexed input (one dimension of 8), instead of four LUTs with a 6-bit indexed input (one dimension of 64 in Table 4). This 16-APSK demapping depends not only on the Voronoi regions of the constellation (see the outlines of the Voronoi regions in Figure 3) but also on the values ​​of Rl^) and I(Xe) relative to Tm. Similarly, for 8-PSK, the size of the LUTs can be reduced to a simple comparison, even though the number of operations required by the algorithm remains the same. This is shown in Table 5 where irrelevant entries are removed and the signs of the LLRs are adjusted according to the symmetry of the input.

[0092] [Tab. 5] Table 5: Analytical approximations of the flexible max-log-MAP demapper for PSK / APSK constellations with the new method based on LUTs. Constellation Equations Complexity 8-PSK Initial Decision |R(^)I < [1(^)1 Memory Extraction ( A„.,, Ao2, A(;2) =LUT_8PSK( / 1) Arithmetic and Logical Operations L(^) = ( |R(^) |R(AaJ) + |I(^ ) |I(Aæ1) ) / o\ L(d2) =sign(R(xf;))(|R(.r)|R(AfA2) + |I(xe)|I(AQ:2)) / v$ £(d3) = sign(I(r) ) ( |R(.u) |R( A^) + |I(.V) lUA^) ) / v$ W, + 3Cr + 3Ar + storage of two 3x2 LUTs with real output (equivalent to the storage of one 3x2 LUT with complex output) 16-APSK (DVB) Initial decision / ,= |R(.r')| < ^|I(^)| f2= |ICV) | < ^|R(x') | / ,= |R(.t')| > T,„ f\ = IHa*)! > Tm ( -a) v ( ( ~D a ( wu i< Tb-IRCO i ) ) / 6= ( ~L)v ( (~A)A < 1^)1 < T»- lRM 1 ) ) ZLUT1 ~ ^ + / 44'2 / 1 ZLUT2 ~ + / 3 + 2 / 2 ZLUT3 “ 1 + / 5 + 2 / 6 + 4 / 4 ZLUT4 ~ 1+ / 5 + 2 / 6 + 4 / 3 7^, = 0.5( / ^08(^ ) +Æ2cos(-g ) ) : the distance to the median between the two circles on the axes, Tb = : the y-intercept of the perpendicular bisector between the radii of the two circles Memory extraction (a„j, Awù ) =LUT_16APSK_L1(ilut1) (a„.2, Awù) =LUT_16APSK_L2(iLuT2) ( A^|44 = LmJ 6 APSKJ.3(>U!T3 ) 12Mr + 9Ar + 8C, + Storage of three 2x4 real output LUTs and three 2x8 real output LUTs again the groups of three LUTs equivalent to one complex output LUT and one real output LUT. ( A„.4, A|h|z4 ) = LUT_16APSK_L4( iLUT4) Arithmetic and logical operations LW = (|R(x') |R( A„j) - |I(a-)|I(A41) + Aw31) / vJ Ud,) = ( |Rœ)|R(Aa2) - |I(r) |I( A^) + AWÙ)AJ L(d3) =sign(R(^) ) ( |R(a-) ^R(A^) - |I(^) |l( A^) + A|(2p3) / v$ T(d4) = sign(I(.V) ) ( |R(a*) |R(A^) - |I(.r) |I(Aa4) + AW^)A4

[0093] In conclusion, this semi-analytical approach allows the development of specific solutions for 8-PSK, 16-APSK DVB-type constellations and other non-square QAM constellations, which can achieve performance close to (16-APSK) or even equal to (8-PSK) max-log-MAP. The general idea is to combine the use of low-complexity hard decisions (or their extensions to decisions on the pair (a\dg)) with LUTs on the differences of the symbols and their powers, as shown in [Fig. 5].

[0094] Specific proposals for 8-PSK and 16-APSK are given in Table 5. The invention for 8-PSK thus provides an alternative and advantageous way to calculate the exact max-log-MAP. The invention for DVB-compatible 16-APSK does not allow for the calculation of the exact max-log-MAP, since this constellation does not have ideal Gray labeling. Note that, for 16-APSK, the invention provides size-optimized LUTs, in particular two LUTs of size 8 for the first two LLRs and of size 16 for the last two LLRs. The total complexity and memory footprint of the LUTs of the invention disclosed herein are greater than in prior art solutions [4][5], but the performance loss is less than for these less complex prior art alternatives.

[0095] The implementation of the invention will depend on the software or hardware target in which this soft demapping operation is intended to be implemented. It will also depend on the types of quantities manipulated: depending on whether the numerical values ​​are implemented as fixed-point or floating-point numbers, the specific implementation of these operations will not be identical.

[0096] In a purely digital environment such as an FPGA or ASIC, with a fixed-point implementation, since manipulating the sign bits is very easy, it will be preferable to reduce the redundant quantities Aa# stored in LUTs, and to use more bit-by-bit comparisons to adjust the sign of these quantities as in Table 5

[0097] Conversely, on a DSP or processor, where a comparison is equivalent to an addition, Accessing LUTs can become much more efficient than performing operations like addition or multiplication when the memory where these LUTs reside is easily accessible. Therefore, an implementation with multiple LUTs containing redundant data, such as A and -A, would be preferable for latency reasons.

[0098] It is therefore important to consider the target when constructing LUTs and comparison rules for a given constellation.

[0099] The invention, without limiting the foregoing, relies on hard decisions and other relatively simple geometric comparisons to index lookup tables (LUTs) which contain the exact intermediate quantities needed to calculate the bit LLRs according to the max-log-MAP algorithm. This allows for an implementation of max-log-MAP that avoids calculating as many distances as there are points in the constellation, and subsequently comparing these distances. The method is therefore defined as semi-analytical because it combines analytical formulas with LUTs, unlike the prior art.

[0100] The revealed solution reduces the complexity of certain state-of-the-art solutions for computing extrinsic bit LLRs with the exact max-log-MAP method, while maintaining the same performance for non-square constellations (such as 8-PSK and Af-PSK M > 8, and Af-APSK constellations).

[0101] Some solutions in the state have very low complexity, but they closely approximate the calculation of the max-log-MAP method with analytical formulas, leading to a loss of performance. Compared to these methods, the solution revealed here is slightly more complex and requires storing pre-computed value tables (LUTs) in the receiver's memory, but it maintains the performance of the max-log-MAP method.

[0102] The invention is of interest for military communications. It could be of interest, for example, for HF communications. It applies in particular to 16-APSK and 64-APSK constellations conforming to the MIL-STD-110C standard (Military Standard Interoperability and Performance Standards for Data Modems) with quasi-Gray labeling.

[0103] Detection is possible by performing performance tests in a controlled environment, in particular by accessing PHY layer metrics.

[0104] We gain in performance for a given amount of computing power, therefore we gain in terms of energy consumption. This is very useful, particularly for military communications.

[0105] The architecture of the solution, represented in Figure 5, therefore consists of a hard 100 decision, typically based on Q' comparisons (plus some additions or multiplications for APSK constellations), followed by the use of a Symbol gap LUT 101: M' possible input values ​​for Q complex outputs, possibly with the parallel use of a distance square gap LUT 102: M' possible input values ​​for Q real outputs. The dashed elements are not required for PSK constellations, as the distance squares are all equal. Arithmetic operations 105 are then performed based on the values ​​extracted from the LUTs and the value of the signal to be decoded.

[0106] In terms of performance, 8-PSK offers gains of approximately 0.6–0.9 dB in a simple channel with additive white Gaussian noise (AWGN) for a target packet error rate below 10² and a coding rate of Uz. The gains are greater when the code efficiency is low. The gains are even greater if the channel is frequency-selective.

[0107] Ultimately, the proposed invention exploits the finite combinatorics of Gray labeling to create small LUTs, enabling more accurate LLR calculations with moderate complexity. Max-log MAP LLR quality is achieved with significantly less complexity, while simultaneously improving the quality of LLRs compared to analytical methods, all within a relatively controlled complexity.

[0108] Thanks to the invention, the quality of max-log MAP LLRs is achieved with less complexity, the quality of LLRs is improved compared to analytical methods, and the complexity is controlled. List of cited references

[0109] [1] S. ten Brink, J. Speidel and Ran-Hong Yan, “Iterative demapping and decoding for multilevel modulation," Proceedings of the IEEE GLOBECOM 1998, pp. 579-584 vol.l, Sydney, NSW, Australia, 8-12 Nov. 1998. [2] S. ten Brink, “Designing iterative decoding schemes with the extrinsic information transfer chart,” AEUInt. J. Electron / Commun., pp. 389-398 vol.54, no.6, 2000. [3] F. Tosato and P. Bisaglia, “Simplified Soft-Output Demapper for Binary In-terleaved COFDM with Application to HIPERLAN / 2”, Proceedings of IEEE ICC 2002, New York City, NY, USA, April 28. - May 2, 2002. [4] Michael K. Cheng, Dariush Divsalar, and Stéphanie Duy. "Structured low-density parity-check codes with bandwidth efficient modulation." Wireless Sensing and Processing IV. Vol. 7349. SPIE, 2009. [5] Chatzikontantinou Christos, and Spafaridis Xenofon. "A novel soft-demapping algorithm for 16-APSK, for Systems employing uncoded modulation.", Proceedings of the Pan-Hellenic Electrical and Computer Engineering Student Conférence, Vol. 7, Thessaloniki, Greece, Apr. 2014. [6] Li, Jianping and Yameng Shi. “Simplified Soft-output Demapper Based on a Linear Transformation Technique for M-ary PSK”, Sensors & Transducers, vol. 181, n° 10, Oct. 2014.

[0110] [7] A. Barre, E. Boutillon, N. Bias and D. Diaz, "A polar-based demapper of 8PSK démodulation for DVB-S2 Systems," SiPS 2013 Proceedings, Taipei, Taiwan, 16-18 Oct. 2013.

[0111] [8] Mostafa Rizk, Amer Baghdadi, Michel Jézéquel, "A Literature Survey on Al- gorithms and Hardware Architectures of Max-Log-MAP Demapping", Journal of Circuits, Systems and Computers, vol.31, no.03, 2022. [9] J. W. Park, M. H. Sunwoo, P. S. Kim and D.-I. Chang, “Low complexity soft-decision demapper for high order modulation of DVB-S2 System”, in Proc, ofthe IEEE International SoC Design Conférence (ISOCC), 02 Nov. 2008.

[10] A. Baghdadi and M. Jezequel, “ASIP-based universal demapper for multi-wireless standards”, IEEE Embedded Systems Letters, no. 1, pp. 9-13, May 2009.

[0112]

[11] M. Rizk, A. Baghdadi, M. Jezequel, Y. Mohanna and Y. Atat, “NISC-based soft-input soft-output demapper”, IEEE Transactions on Circuits and Systems, vol. 62, Nov. 2015.

[0113]

[12] I. Ali, U. Wasenmüller and N. Wehn, “A high throughput architecture for a low complexity soft-output demapping algorithm”, Advances in Radio Science, vol. 13, 2015.

[0114]

[13] Q. Wang, Q. Xie, Z. Wang, S. Chen and L. Hanzo, “A Universal Low- Complexity Symbol-to-Bit Soft Demapper”, IEEE Transactions on Vehicular Technology, vol. 63, no. 1, Jan. 2014.

[0115]

[14] US10003436B2, “Low-complexity LLR computation for nonuniform QAM constellations”.

Claims

Demands

1. A flexible decoding method for a digital signal received by a receiver performing quadrature demodulation, comprising, for a received signal (X^) in the complex plane and to be interpreted into a vector of Q bits, a calculation of log-likelihood ratios for each bit of the vector, the method using a pre-established lookup table stored in a memory of the receiver, and the method being characterized in that the digital signal is encoded into symbols in the complex plane according to a Gray-type or quasi-Gray-type labeling law, and the table associates to each symbol of the constellation other values ​​(A, ..., Aa, g) in the complex plane, the method comprising, after receiving said signal (X^), an identification step (100) of the symbol (a*) of the constellation closest to the signal in the complex plane, followed by an extraction step (101, 102) from memory of at least some of said other values ​​(A^j ...A^ g) in the complex plane associated by the lookup table with said nearest identified symbol, and finally, by a combination (105) of the real and imaginary values ​​of said values ​​in the complex plane extracted and of the received signal, the provision to the receiver controller of values ​​for the log-likelihood ratios ( Le {dkA} L ( dk_Q ) ) of the Q bits for said received signal.

2. Flexible decoding method according to claim 1, characterized in that the table or another pre-established table also stored in a memory of the receiver also associates with each symbol of the constellation of positive real values ​​(A^pj ... A|apq), the extraction step (101, 102) comprising the extraction from the memory of at least some of said positive real values ​​associated by the correspondence table with said nearest identified symbol, the combination being a linear combination (105) including a term independent of the received signal and based on said extracted positive real values.

3. A flexible decoding method according to claim 1 or claim 2, characterized in that the elements stored in memory are in the form of values ​​(A^j... A <z q) proportionnelles à la différence entre le symbole en entrée de la table et des Q symboles complémentaires par rapport aux bits de l’étiquette dudit symbole en entrée, ou de valeurs (a 2 • • • Ai (2 1 proportionnelles à la différence entre le ' a | ,1 4-1 a ,O l square of the modulus of the symbol input to the table and the square of the modulus of the Q complementary symbols with respect to the bits of the label of said input symbol.

4. A flexible decoding method according to any one of claims 1 to 3, characterized in that the constellation is a PSK or APSK constellation.

5. Flexible decoding method according to claim 4, characterized in that the constellation is an 8-PSK constellation.

6. Flexible decoding method according to claim 4, characterized in that the constellation is a 16-APSK constellation.

7. Flexible decoding method according to any one of claims 1 to 6, characterized in that during the extraction step (101, 102), consideration is given, in addition to the symbol (a*) of the constellation closest to the signal in the complex plane, also to the result of additional geometric tests based on the received signal (Xf), to distinguish between symbols of the constellation close to (fl*) which have the same given value of a certain bit.

8. A flexible decoding method according to any one of claims 1 to 7, characterized in that the table associates each symbol of the constellation with other symbols in the complex plane by taking advantage of the symmetries of the constellation to limit the size of the memory used.

9. Flexible decoding method according to any one of claims 1 to 8, characterized in that the variance (yj) of noise and residual interference is taken into account in the calculation of the log-likelihood ratios of each bit of the vector.

10. Flexible decoding method according to any one of claims 1 to 9, characterized in that the receiver is a turbo receiver and the prior log likelihood ratios of the turbo receiver are taken into account in the calculation of the log likelihood ratios of each bit of the vector.

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