Method of adapting a general quadrature amplitude modulation complex symbol constellation for quadrature spatial modulation and apparatus implementing the method

Abstract

A method of adapting a general quadrature amplitude modulation, M-QAM, symbol constellation for quadrature spatial modulation, QSM, comprises selecting, from a M-QAM complex symbol constellation, or a first subset thereof, a second subset of symbols whose projections onto axes of a corresponding I / Q-domain are distinguishably spaced apart on the I and Q axis, respectively. Alternatively, a transformation for the M-QAM complex symbol constellation, or the subset thereof, is determined targeting to yield, in the corresponding I / Q-domain, individual locations of projections of the constellation points that are distinguishably spaced apart on the I and Q axis, respectively, and applying the determined transformation to the M-QAM complex symbol constellation, or the subset thereof, yielding a correspondingly modified symbol constellation. The second subset or the modified symbol constellation, respectively, is provided to a transmitter and / or to a receiver.

Description

[0001]202402524 1 METHOD OF ADAPTING A GENERAL QUADRATURE AMPLITUDE MODULATION COMPLEX SYMBOL CONSTELLATION FOR QUADRATURE SPATIAL MODULATION AND APPARATUS IMPLEMENTING THE METHOD 5 FIELD OF THE INVENTION The present invention relates to wireless transmission of information via index modulation (IM), and targets to enable or facilitate integrated sensing and communications (ISAC), also referred to as joint communication and sensing (JCAS) or joint radar and communication (JRC), where wireless signals are 10 simultaneously used for communication and environment perception, i.e., generating information about the environment and objects therein. NOTATIONS Throughout this specification, bold symbols represent vectors or matrices. Scalar 15 values are denoted herein by lowercase letters in italics, as in x. SuperscriptsTand H, respectively denote the transpose and complex conjugate transpose of a vector or matrix. BACKGROUND 20 Integrated sensing and communications (ISAC) has recently been recognised as a new key technology for beyond fifth generation (B5G) and sixth generation (6G) wireless communications systems. While ISAC is expected to enable new applications, the fundamental trade-off between sensing and communication functionalities is also certain to put pressure on the equally important objective of 25 curbing the power consumption of B5G and 6G systems. In fact, B5G and 6G systems are required to significantly improve over existing systems not only in terms of energy and spectral efficiencies, but also on data throughput, reliability, latency, coverage, and user capacity, motivating the investigation of novel technologies such as millimetre-wave (mmWave) / Terahertz (THz) communications, massive 30 multiple-input multiple-output (mMIMO) and cell-free MIMO (CF-MIMO), reconfigurable intelligent surfaces (RISs), and index modulation (IM). 202402524 2 In IM a transmitter activates only a subset out of the available resources for each transmission instance for transmitting information symbols and other purposes. For example, in spatial modulation a transmitter with multiple transmit antennas activates only a subset out of the available antennas for each transmission instance 5 for transmitting information symbols and beamforming. To compensate for a possible loss in information due to a lower number of transmitted symbols, information is also encoded in the selection pattern of the respectively activated resources, e.g., transmit antennas. If the decoder can 10 determine which resources are activated, the corresponding information can be decoded. Among the alternatives mentioned above, IM is attractive for future wireless communication systems since the frugal utilization of resources at the transmitter 15 and the encoding of information in the respective activated subsets of the totally available resources enables high-efficiency operation while at the same time exploiting the information that is combinatorially encoded "energy-less" or “energy-free” in the resource activation patterns to achieve high spectral-efficiency. The sparse utilization of radio frequency (RF) chains enabled by IM is highly 20 appealing to millimetre-wave (mmWave) and Terahertz (THz) systems, in particular for reducing the required total transmit energy. The characteristic of IM also lends great flexibility since the symbols actually transmitted are not directly relevant to the resource subset activation scheme itself, making it transparent to other promising technologies, as demonstrated by a quickly expanding literature on the integration 25 of IM in connection with mMIMO, CFMIMO, RISs and ISAC. Generally, an IM scheme activating P selected patterns for transmitting out of a total of N resource allocation patterns can convey up to additional information bits on top of those conveyed over the selected P resources themselves. Therefore, 30 with sufficiently large N and an adequate P, very large amounts of information can be encoded in the pattern or index domain, such that IM has shown the potential to outperform fully dense, i.e., P = N, systems in certain scenarios, with the added bonus of improving the energy- and spectral efficiency, freeing the unused 202402524 3 resources to other functionalities, such as opportunistic communications and radar / sensing, etc. Figure 1 shows a simplified schematic representation of IM with multiple antennas 5 representing the resources that can be individually activated. Among the multiple antennas, at each transmit instance, respectively delimited by the dashed vertical lines distributed across the time domain, only a subset of antennas is used for transmitting, each of the subsets being unique and thus carrying information in itself. Within IM beamforming is still possible by appropriately controlling the phase and 10 amplitude of the analogue transmit signals transmitted by each antenna, and / or appropriate precoding. Note that resources that can be used in IM are not limited to antennas in a multi-antenna transmitter, but also comprise subcarriers, time slots, carrier frequencies, signal constellations, frequency-time matrices, space-time matrices, etc. 15 Motivated by the advantages of IM, a rapid development of enhanced spatial modulation (SM) designs aiming to maximise energy and spectral-efficiencies, and minimise error rates and decoding complexities can be found in recent literature. An excellent example is the quadrature spatial modulation (QSM) scheme found in in 20 "Quadrature spatial modulation," IEEE Trans. Veh. Technol., vol.64, no.6, pp. 2738-2742, Jun.2015, by R. Mesleh, S. S. Ikki, and H. M. Aggoune, in which the SM technique is applied independently to each in-phase and quadrature (IQ) component of the transmit symbols, such that P real-parts and P imaginary-parts of the symbols are independently transmitted from the corresponding selected P 25 transmit antennas from a total of NTtransmit antennas. The total number of antenna pattern combinations in QSM can be expressed by the combinatorics binomial coefficient. ^^^^ = ^^ ^^^ ^ . QSM results not only in a two-fold increase in the rate ofthe spatially-modulated information, but also in improved bit error rate (BER) performance compared to the original SM scheme discussed by R. Y. Mesleh, 30 H. Haas, S. Sinanovic, C. W. Ahn, and S. Yun, in "Spatial modulation," IEEE Trans. Veh. Technol., vol.57, no.4, pp.2228-2241, Jul.2008. In addition to virtually doubling the codebook size and hence the spectral efficiency in the spatial domain, 202402524 4 QSM is also well suited for use in ISAC applications, since the actual transmit symbols need not all be used for carrying information, leaving some or even all symbols for configuration exclusively for sensing purposes, e.g., using them as pilot signals whose properties are known at the receiver. 5 Further examples in this line of work are the improved generalised quadrature spatial modulation (GQSM) designs incorporating space-time coding (STC) and other techniques to optimise the antenna activation patterns, e.g., as discussed in “Space-time block coded spatial modulation," IEEE Transactions on 10 Communications, vol.59, no.3, pp.823-832, 2011, by . Başar, U. Aygölü, E. Panayirci, and H. V. Poor, and in "Scalable quadrature spatial modulation," IEEE Trans. on Wireless Commun., vol.21, no.11, pp.9293-9311, 2022, by H. S. Rou, G. T. F. Abreu, H. Iimori, D. González G., and O. Gonsa. 15 To summarise, (G)QSM schemes achieve high bit error rate (BER)-performance as well as high energy- and spectral- efficiencies by modulating information both in transmitted symbols and in coded combinatorial activations of subsets of multiple transmit antennas, exceeding the performance of conventional SM methods on all parameters. 20 The price of the aforementioned advantages of IM is, however, the burden cast onto the receiver in terms of the computational complexity required to detect the activated resources at each given transmission instance, given all possible resource activation patterns in the codebook of size . 25 Consider a system with R = 5 resource blocks out of which A = 3 are activated at any transmission instance. The index selection procedure selects a subset of A possible combinatorial patterns, which form a corresponding codebook ^ having a 30 total of 10 index or activation vectors, as exemplarily shown in figure 2. The activated resources are shown with the dot-pattern, while the unused resources are left blank. The resulting index-level codebook ^ is likewise shown in the figure. In the 202402524 5 index or activation vectors the non-zero positions indicate positions of activated resources. It is obvious that such highly structured and deterministic codebook will inevitably 5 have an effect on the decoding algorithm. The decoders must consider the various restrictions arising from the combinatorial selection, such as the finite number of non-zero elements, the limited positions of the non-zero elements, etc. Due to this finite discrete signal space, most conventional algorithms consider the entire codebook structure and “search” the index-level codebook for the most likely 10 configuration / pattern. Returning to GQSM, such naïve decoding requires searching over a combinatorial space of size ^ ⋅ # , where ^^ is the number of transmit antennas, ! is thecodebook amplification factor (! = 1 for the classic GQSM), ^ is the number of15 symbols transmitted, and # is the size of the complex symbol constellation. Due to the nature of the binomial coefficient scaling factorially, as NTand P grows in the regime of B5G mMIMO systems with NT>16, the number of total combinations grows super-exponentially in QSM systems. In other words, the codebook size becomes extremely large for massive MIMO scenarios with a large number NTof 20 transmit antennas and with high data rate with large P. This challenge has therefore motivated work on low-complexity decoders for IM, in its many variations. On the other hand, as the number of resources increases, the amount of information that can be encoded in the “spatial domain”, i.e., in the 25 resource activation patterns, significantly increases in comparison to the information that can be encoded in the digital symbol constellation domain. This suggests that the transmit information may ultimately solely be encoded in the resource domain, while the transmit symbols themselves can be fixed pilot symbols or specific waveforms for other functionalities, such as radar and sensing. 30 This means that the detector needs only to detect the resource patterns, in hand of the known transmitted symbols. However, this still poses a hard problem for the 202402524 6 decoder, since the knowledge of symbols only reduces the search space in the relatively smaller digital domain, but not on the true problematic resource combinatoric domain. For larger “resource spaces” this turns out to be infeasible using conventional maximum likelihood (ML) decoding methods. 5 To cite some examples, improvements over the brute-force ML approach detection methods based on reduced-search methods such as lattice and sphere decoders, as discussed by I. Al-Nahhal, E. Basar, O. A. Dobre, and S. Ikki, in "Optimum Low-Complexity Decoder for Spatial Modulation," IEEE Journal on Selected Areas 10 in Communications, vol.37, no.9, pp.2001-2013, 2019, by Z. Hu, F. Chen, Y. Liu, S. Liu, H. Yu, and F. Ji, in "Low-Complexity Detection for Multiple-Mode OFDM with Index Modulation," Physical Communication, vol.34, pp.38-47, 2019 or by M. Simarro et al., in "Low Complexity Near-ML Sphere Decoding based on a MMSE ordering for Generalized Spatial Modulation," IEEE 31st Annual International 15 Symposium on Personal, Indoor and Mobile Radio Communications, 2020, pp.1-6, are able to decrease the total decoding complexity in proportion to the total codebook size, but their cost are still dependent on the combinatorial factor. More recently, methods aiming to directly reduce the order of the binomial 20 coefficient have also been proposed. For instance, in "The achievable rate analysis of generalized quadrature spatial modulation and a pair of low-complexity detectors," IEEE Trans. on Veh. Technol., vol.71, no.5, pp.5203-5215, 2022, J. An, C. Xu, Y. Liu, L. Gan, and L. Hanzo, achieve a reduction of the complexity order from with ^^ ≤ ! ≤ 1, by leveraging the orthogonal matching pursuit25 (OMP) algorithm to operate over the probable non-zero indices of the sparse signal of the generalised quadrature spatial modulation (GQSM), of which IM is a specific type. It was shown thereby that a trade-off between computational complexity and optimal performance can be achieved by adjusting the value of α. 30 In "An Efficient Vector-valued Belief Propagation Decoder for Quadrature Spatial Modulation," 202256th Asilomar Conference on Signals, Systems, and Computers, 2022, pp.27-31, H. S. Rou, G. T. F. de Abreu, and T. Takahashi proposed a GQSM detector leveraging the message passing (MP) algorithm with vector-valued 202402524 7 variables under the Gaussian belief propagation (GaBP) framework, which was shown to eliminate the quadratic exponent in the combinatorial factor by employing a novel decomposition of the GQSM signal into two independent vectors. This reduces the decoding cost to the order of the square root of the binomial coefficient, % 5 while approaching the performance of a ML solution with fixed complexity gain of a quadratic order. Finally, non-classical approaches also exist that are independent of the binomial coefficient, including techniques based on machine learning and quantum 10 computing. Such alternatives have, nevertheless, their own limitations, such as the need for exhaustive offline training or expensive and not widely-available quantum computers, respectively. Still, the complexity of all known detectors is dependent on the parameters of the 15 combinatorial space, i.e., from the factorial relationship between the number of P activatable patterns out of a total of N resources. Further, conventional symbol constellations may not be optimally adapted for low-complexity decoding in ISAC applications. 20 It is, therefore, desirable to provide a method of adapting symbol constellations for improved low-complexity decoding in index modulated wireless transmissions that maximises the spectral and energy efficiency, and that permits optimising both the communication and sensing performance in ISAC applications. 25 SUMMARY OF THE INVENTION This object is attained by the method of claim 1, the apparatus of claim 6. A communication system, a computer program product and a corresponding computer-readable medium are provided in claims 7 to 9, respectively. 30 Advantageous embodiments and developments are provided in respective dependent claims. 202402524 8 A particular object of the present invention lies in addressing ISAC through IM in large scale systems, e.g., B5G massive MIMO. Embodiments of the present invention exploit piloted IM signals which, at the targeted large-scale systems, cannot be handled by known receivers due to the enormous computational 5 complexity. A novel decoding method presented herein that exploits the Gaussian belief propagation (GaBP) framework enjoys a complexity order that is independent of the combinatorial factor. In order to enable an accurate statistical inference the most important input is the 10 knowledge of the statistical distributions of the variables, i.e., the prior distribution. The prior probability of selecting a given activation pattern from the codebook with different activation patterns, is trivially uniform, i.e., all activation patterns have an equal probability of being selected. 15 The present invention builds, inter alia, on finding that the likelihood of activation of each individual resource in a given codebook is not uniform. Thus, the problem that is addressed by the present invention for enabling low complexity decoding of IM schemes is the prior distribution of each index, i.e., by transforming the index 20 activation codebook shown in figure 2 into a corresponding index vector form that lends itself to low-complexity analytical treatment. An important aspect of the proposed approach is reducing the decoding complexity at the receiver, which is prohibitive in massive set-ups under naïve detection. A 25 message passing (MP) decoder is discussed herein which is of polynomial order on other system parameters and entirely independent of the combinatorial coefficient of the codebook. As mentioned above, many IM techniques can be found in the literature which 30 address various issues such as the integration with other functionalities, e.g., radar and / or sensing, achievable data rates, decoding complexity and, of course, which resource domains to exploit and under which architecture. While the proposed method is applicable to a variety of IM schemes of general formulation, for the sake 202402524 9 of clarity of exposition this specification exemplarily focuses, without loss of generality, on the GQSM variation of IM, upon whose system and signal models the following derivation and analysis will be based. 5 This specification discloses, inter alia, modifications to the transmitter constellation design, that is particularly well adapted for a unit vector decomposition (UVD) formulation of GQSM signals, also discussed herein, which enables the concurrent estimation of the index positions as independent unit vectors. Further, modifications to the detection algorithm are discussed, which enhance the performance of the 10 detector. Prior to describing the invention in greater detail the underlying GQSM system model and transmitter structure will be discussed. 15 Consider a point-to-point (P2P) MIMO wireless communications system where the transmitter and the receiver are equipped with ^^and ^&antennas, respectively, such that the received signal vector ' ∈ ℂ^*×, corresponding to the transmission ofthe information vector - ∈ ℂ^.×, through the flat-fading MIMO channel / ∈ ℂ^*×^.is described by20 ' = / - + w ∈ ℂ^*×, 213where w ∈ ℂ^*×, is a complex-valued additive white Gaussian noise (AWGN)vector with element-wise variance ^4, such that w5 ∼ ^^20, ^43 for 8 ∈ 91, ⋯ , ^&;.25 In a GSQM transmitter, the in-phase and quadrature (IQ) components of ^ transmit symbols taken from acomplex-valued constellation ^ > of size # ≜ |^ >|, are sparsely mapped into therespective transmit vector components -A ∈ ℂ^.×, and -C ∈ ℂ^.×, of the formH I I%I-th position HJ-th position HJ-th position30 202402524 10 H%O -th position HJO -th position HJO -th positionC ^C, ^?C ,0, =^C L 22M3- =[0, ⋯ ,0, = , 0, ⋯ ,0, = , 0, ⋯ ^ , 0] ,yielding the transmit signal vector 5 Note that the superscripts R and I, respectively, denote the real and imaginary counterparts of the corresponding parameters in a general spatial modulation (GSM) signal model, and the superscript c in equations represents either of the real 10 and imaginary part whenever the two are treated identically in the equation. Further note that the classic quadrature spatial modulation (QSM) is a particular case of GQSM with ^ = 1.The real and imaginary parts of the transmit symbols are independent, i.e., they can 15 be transmitted from different antennas in (G)QSM. Consequently, the positions of the IQ vector components in equation (2) are described by two independent index vectors Q& ≜ RSA,, ⋯ , SA?, ⋯ , SA^TLand QC ≜ ⋯ , SC? , ⋯ , SC^ TLfrom the set oftors ^ of size ^ ≜ |^|. Specifically, ^ ≜ 2UVW Y.possible index vec X^ ^ Z ^[ implies thatonly binary-encodable subsets of the possible activation pattern combinations 20 are utilised, such that the size of the GQSM IQ activation vectors codebook ^ is given by|^| = ^. Note also that the position of a symbol component in the vectorcorresponds directly to the activated antenna element at the transmitter. 25 By exploiting the possible combinatorial patterns of the symbol component positions, the GQSM scheme conveys information not only from the encoding of ^ symbols from ^, but also from the selection of ^ positions out of ^^possible positions, respectively for both -Aand -C. 30 In light of the above, the total information conveyed by the GQSM signal - is given by 202402524 11 where ^acis the number of bits spatially encoded by the antenna position selection 5 of ^ symbol parts, and ^dXis the number of bits digitally encoded by the transmit symbols. In the following a codebook example is given. For the sake of clarity, a small system with ^ = 5 h^ , ^ = 3 is considered, such that a total of \i] = 10 activation vectors are10 possible, namely where the non-zero elements =?xdenote either =?Aor =?C, since the signal codebooks 15 for both IQ parts are identical. A valid codebook for such a system is a subset ^ of the above, containing 20 which can be also represented by a corresponding set of index vectors, each of which contains the indices of the non-zero elements in the corresponding activation vector }~, that is25 202402524 12 T where the vector [1, 2, 5] indicates that the resources with the indices 1, 2 and 5 are activated. For the index vector set ^ the following distributions exist: -the probability of selecting a given vector Q^ ∈ ^, which is uniformly5 distributed - the probability of activating any given resource index r ∈ 91, …, R ; under optimally generated codebooks, which is also uniform, i.e., the number of occurrences of each resource index over the entire codebook is the same - the probability of each digit, i.e., the a-th element of the vector k being a 10 specific resource index. Out of these distributions, the last one is not uniform and exhibits a non-trivial distribution. 15 In the example provided in equations (4) and (5), the first digit of denoted by S,cannot take on the values of 4 and 5, and has a higher probability of 20 taking a value of 1 than 2 or 3. Similarly, the other digits are non-uniformly nor trivially distributed, such that an accurate model or a derivation of the distribution is required. To exploit this feature the present invention proposes a novel analytical modelling 25 method of the index activation probability per digit of the IM-based codebooks. In view of the formulation of GQSM transmit signals described in equation (2), the received signal model of equation (1) can be rewritten in the IQ-decoupled form as30 202402524 13 denote the IQ-decoupledcounterparts of ', -, and w, respectively, and the IQ-decoupled channel matrix^ ∈ ℝ^^*×^^. is further decomposed into two submatrices as5 with ^A ∈ ℝ^^*×^. and ^C ∈ ℝ^^*×^. denoting the effective channel componentsfor xAand xC, respectively. 10 The real domain IQ-decoupled form y = ^^ + w presented in equation (6) is thebasis of most known QSM / GQSM detection algorithms, which generally seek to estimate the effective transmit signal ^, in knowledge of y and ^, i.e., seek to solve the recovery problem15 ^̂b^ = argmin ∥ ^ − ^^ ∥^^, 283^∈^where ^ is the discrete domain of the effective signal ^, of cardinality 20 While the linear recovery problem appears trivial, the challenge lies in the infeasible ^size of the discrete domain ^ of ¥ ∈ ℝ^^.×,, with cardinality ^ ≜ ¤ ⋅ #^, where ⌊⋅⌋^ ≜ 2⌊VWX^ 2⋅3⌋ is the flooring operation to the nearest power of 2.As can been seen from the latter expression, the codebook size ^ scales at a 25 geometric rate on ^, and at a squared factorial rate on ^^, such that complexity becomes prohibitive even for moderately large MIMO scenarios, which is why simulation results in the known literature exist only up to ^^ ≤ 10, ^ ≤ 3, even withlowest-complexity methods. 202402524 14 The problem described by equation (8) is NP-hard due to the discrete solution domain ^, which prevents classical convex optimization approaches, requiring instead a combinatorial search over a high-dimensional space for optimal solution. To circumvent this challenge, conventional low-complexity solutions leveraging 5 various characteristics of the GQSM signal have been proposed. For example, the inherent sparsity of ^ can be exploited by compressive sensing (CS)-based methods, and the discrete codebook structure can be utilised to design search-based methods such as sphere decoders and pruned-search decoders. Further, a vector-valued message passing (MP) method was recently proposed by 10 H. S. Rou et al., in "An Efficient Vector-valued Belief Propagation Decoder for Quadrature Spatial Modulation" (full citation further above), where the two variables-A and -C are concurrently estimated by leveraging the independent and identicallydistributed (i.i.d.) bivariate property of the decoupled vectors in equation (7). The bivariate vector-valued MP method in "An Efficient Vector-valued Belief Propagation 15 Decoder for Quadrature Spatial Modulation" (full citation further above) was shown to reduce the search space from ^ = ⋅2√#3^. Building on the above, the present invention exploits a low-complexity decoder, 20 which leverages a unit vector decomposition (UVD)-based GQSM system model integrated into the GaBP framework with purpose-derived index activation probability distributions, which achieves a remarkably low decoding complexity order that is entirely independent of the combinatorial factor 25 In the following description a piloted GQSM scenario is considered, where some or all symbol component values are known at the receiver. The pilot symbols are assumed to be arbitrary and can be utilised for other functionalities such as authentication, radar, channel estimation, etc. However, even with the known pilot symbols, the challenge remains in estimating the unknown indices of the symbols in 30 the combinatorial space, as seen in equation (2). 202402524 15 First, the unit vector decomposition (UVD) of the GQSM system model and the derivation of the novel index activation probability distributions arising from this decomposition will be discussed. As shall be clarified, the resultant index-wise distributions enables the design of an MP algorithm operating on a 5 combinatorics-free signal domain. Exploiting the fact that each IQ symbol component = occupies only a single position in the GQSM transmit vector - described in equation (2), i.e., is only transmitted from a single resource element, the transmit signal can be rewritten as a10 superposition of the symbol components multiplied by elementary activation vectors§¨, yielding where the activation vector §¨ ∈ 90,1;^., with ® ∈ ^ ≜ 91, ⋯ , ^^;, is the ®-th column15 of an ^^ × ^^ identity matrix ¯ which defines the set ℰ of orthonormal unit vectorson 90,1;^.as In light of the reformulation in equation (9), the IQ-decoupled received signal vector 20 in equation (6) becomes where the linear recovery of ^ has been transformed into the joint estimation problem of 2^ activation vectors, i.e., unit vectors. The reformulation in equation 25 (11) better highlights the construct of the GQSM signals with inherently independent 202402524 16 spatial and digital encoding, as illustrated in figure 3, which shows a corresponding multivariate bilinear factor graph with 2^ symbol variables and 2^ vector variables. The reformulation enables using independent estimation techniques when some of the decoupled variables are known, e.g., via pilot symbols. 5 The UVD model described above works for all spatial modulation (SM) schemes whose activation vectors are directly given by the unit vectors in ℰ, e.g., generalized spatial modulation (GSM) and GQSM. There exist, however, enhanced SM-based schemes whose transmit symbols are not multiplied by elementary activation 10 vectors in the space of 90,1;^., but by optimized precoded dispersion vectors drawn from a complex space ℂ^.. Let ·A ≜ be the sets from which theprecoded dispersion vectors of a generic coded SM are drawn. Then, the 15 corresponding transmit signal vector is given by Next, the trivial identities are leveraged 20 ≜½I∈ℂYY×¾ ≜½O∈ℂY×¾¸A~ ≜ ¹ R¸ºA,º, ⋯»º , ¸ºA^¼T ⋅ §~ and ¸C~ ≜ ¹ R¸ºC,º , ⋯»º , ¸ºC^¼ T ⋅ §~ , 2133with § ^×,~ ∈ 90,1; , to rewrite equation (12) as25 where ½A ∈ ℂ^.×^ and ½C ∈ ℂ^.×^ are the dispersion vector codebook matricesconstructed by concatenating the ¿ dispersion vectors in ·Aand ·Crespectively. It 202402524 17 is apparent that the formulation for the GQSM system described by equations (9) and (11) is the simplified case of this generalized UVD form, as the matrices ½A and½C reduce to the identity matrix in the GQSM case.5 Then, the IQ-decoupled form of the received signal corresponding to the transmit vector in equation (13) becomes 10 where the effective channel ^̃ ≜ ^ ∈ ℝ^^*×^^ is constructed from the dispersioncodebook  and the channel state information (CSI) ^, both known at the receiver. The structural equivalence between equations (15) and (11) is obvious, and it is evident that the decoding method to be described further below can be applied to 15 both cases. Following similar steps, it can be also shown that IM schemes incorporating space-time coding (STC), such as those proposed in “Scalable quadrature spatial modulation” (full citation further above) and H. S. Rou, et al., in “Enabling energy-efficiency in massive-MIMO: A scalable low-complexity decoder for generalized quadrature spatial modulation,” IEEE 9th International Workshop on 20 Computational Advances in Multi-Sensor Adaptive Processing, 2023, pp.301–305, in which dispersion vectors are generalized into dispersion matrices of size ^^ × Äwith Ä consecutive symbol periods, can also be represented, via a vectorization of the system model, by the same UVD formulation shown above, once again enabling the proposed technique presented further below to be applied. 25 202402524 18 To elaborate, given the sets of dispersion matrices ·A≜ and·C ≜ |ÅC ^ ^.×^~ ^~^, ∈ ℂ , the transmit signal of a STC-based IM scheme becomes 5 whose vectorized form is trivially given as vec 10 The UVD representation of equation (17) is similar to that of equation (14), except for the increased dimensionality to Ä^^, and the dictionary matrices of vectorized dispersion matrices ½A ≜ Rvec 2ÅA,3, ⋯ , vec \ÅA ^^^]T ∈ ℂ .×^ and½C ≜ Rvec 2ÅC C, 3, ⋯ , vec \Å^ ]T ∈ ℂ^^.×^, which yields the vectorized received signalmodel y15 where ¯^ is the Ä × Ä identity matrix, the symbol ⊗ denotes the Kronecker product,and ^̃ and Î̃ are the augmented effective channel matrix and AWGN vector, 20 respectively. 202402524 19 From the equivalence among equations (18), (15) and (11), it follows once again that the UVD model introduced further above for the GQSM scheme generalizes to other IM methods in other domains and higher dimensions. As indicated before, for the sake of simplicity, in this specification the GQSM scheme is exemplarily used, 5 emphasizing however the generality of the methods. Next, the prior distributions of the two sets of variables in the UVD model, namely the IQ symbol components =?Aand =?C, and the elementary activation vectors §HJI and §HJO , respectively, are analysed in order to enable their estimation via Bayesian 10 inference methods. Since QSM schemes typically employ complex symmetric constellations such as#-quadrature amplitude modulation (QAM), and =C? and belong to thecorresponding real-valued effective constellation ^ ≜ ℜ9^>; = ℑ9^>;, which is15 typically a pulse amplitude modulation (PAM) constellation of cardinality |^| ≜ √#. Ifthat is not the case, and ^>is irregular, the formulation is trivially generalized by considering distinct PAM constellations for =C and =?. Succinctly, therefore20 On the other hand, the probability mass function (PMF) of §HJI and is not trivial and is given by25 which can be alternatively described solely in terms of the corresponding activation indices S?Aand S?C, such that equation (20) can equivalently be written as 202402524 20 ¨∈^The challenge in equation (21) lies in the probabilities ℙÖ× which, far from being uniform and independent, is strongly correlated and differs significantly for each D, 5 as a consequence of the combinatorial selection which yields finite and ordered index lists. To illustrate, consider the example shown in equation (5) for ^^ = 5 and^ = 3, from which it can be seen that the first element S, has a higher probability oftaking on the value of 1, then 2 and 3, and can never take on the values 4 or 5. Similar restrictions apply for the other index vector elements S^and Si. 10 Fortunately, the analytical PMFs corresponding to the index vector elements S?, ∀D,can be derived using common tools of numerical and combinatorial analysis. In particular, consider the following bounded variations of Bernstein polynomials on the discrete variable ® ∈ ^ ≜ 91, ⋯ , ^^; given by15 where2D − 13 < ® < 2^^ − ^ − D + 13. 223320 For each value D, the polynomial ^^?2®; ^^3 gives the number of the mutuallyexclusive index ®, out of a list of length ^, confined to the interval ^, such that the desired distributions are obtained by merely normalizing such polynomials, yielding 25 Some examples of the discrete polynomial-based PMF given by equation (24) are compared to the true empirical distributions in figure 4, for various ^^and ^, which confirm that the expression is exact. The circles indicate the results of the proposed 202402524 21 scheme, while the solid lines indicate the corresponding empirically-obtained true values of the activation indices of the index vectors in ^. For future convenience, the probability masses of the PMF is 5 represented by the vector à?defined as In possession of the analytical prior PMFs of the indices S?given in equations (19) 10 and (24), respectively, next a GaBP-based MP algorithm for their estimation will be discussed. In particular, the MP rules for a piloted GQSM will be derived, i.e., where=A?, =C? are known ∀D, so as to enable sensing functionalities exploiting the symbols,as well as communication via the detection of activation patterns. It is noted that a bilinear MP decoder for this system could also be used, such that some of the pilots 15 could be replaced by information-carrying symbols to increase the rate of communications, albeit at the expense of some performance degradation in the ISAC function. First, the soft-replicas for the activation vector variables §HJI with D ∈20 91, ⋯ , ^; are defined for the 8-th factor node, i.e., 8-th pilot, with 8 ∈ 91, ⋯ ,2^&;, as§̂HJI:5 and respectively. The corresponding error covariance matrices of theactivation vector soft-replicas are defined as25 202402524 22 where the expectation is taken over the corresponding unique PMFs of equation (25), for each It can be seen from equation (26) and the system analysis provided further above 5 that the derivations corresponding to the two IQ components are identical. Therefore, for the sake of brevity, hereafter expressions are derived only for the real components, with the understanding that the same results apply to the imaginary components, only with the terms 2⋅3Areplaced by 2⋅3C. 10 Computed naïvely, the covariances in equation (26) require a summation of ^^outer products of dimensions ^^ × ^^, with complexity è2^i^3. Thanks to the UVDand the sparsity of the unit vectors in ℰ, however, equation (26) can be put into a closed-form15 of associated complexity è2^^^3, where the ^^ × ^^ square matrix êA?:5 is given by 20 with ë?2¨3 denoting the ®-th element of the vector of probability mass coefficients ofà? from equation (26).In hand of the soft-replica vectors and the error covariance matrices, the 2^&factor 25 nodes corresponding to the received symbols in y, each perform soft-interference cancellation (IC) on the received symbols y5for the activation vectors as 202402524 23 where 2îA53ñ ∈ ℝ,×^. and 2îC53ñ ∈ ℝ,×^. respectively denote the 8-th rows of thechannel components ^A ∈ ℝ^^.×^. and ∈ ℝ^^.×^. which are defined inequation (7). 5 As can be seen in equation (30) ^ the soft-IC symbol is comprised of the true symbol part and a term related to the 10 soft-replica error plus AWGN, such that the latter term can be approximated by a Gaussian scalar via the central limit theorem (CLT), which yields the conditional probability density functions (PDFs) of the soft-IC symbols conditional to a given activation vector as 15 where the conditional variance ^?A:5is obtained via 20 with the total variance ^5prior to the soft-IC 25 The conditional PDFs of equation (31) are combined at the variable nodes to obtain the extrinsic belief ç?A:5, i.e., 202402524 24 where self-interference cancellation (IC) is included for the computation of the 5 extrinsic belief for the 8-th factor node, as can be seen by the exclusion of the 8-th conditional PDF from the variable node. The resulting unscaled multivariate Gaussian PDF of the extrinsic beliefs are efficiently described in terms of the information vector ^ ^.×, ∈ ℝ , which is given10 by and in terms of the precision matrix given by15 In turn, the posterior Bayes-optimal soft-replicas are computed from the extrinsic beliefs via 20 Similar to equation (26), the expectation of §HJover its domain ℰ can be efficiently described in a closed-form expression with reduced computational complexity, namely 202402524 25 where the vector of unnormalized belief mass ^?A:5is given by 5 with exp⊙2⋅3 denoting the element-wise exponentiation of a vector, ^?A:52¨3denoting the elements at the ®-th position of the vector , and ΛA?:52¨,¨3denoting the elements at the 2®, ®3-th postion of the matrix ^A?:5.10 Note that equation (39) clarifies that only the diagonal elements of the precision matrix are required for the computation of the soft-replica vectors. Therefore, instead of computing the entire precision matrix as in equation (36), the vector of diagonal elements can be directly obtained to significantly improve the 15 computational complexity, given by diag \ where ∣⋅∣⊙denotes the element-wise absolute value operator. 20 Equations (29) to (40) describe the steps of one MP iteration to estimate the 2^ activation vectors of the piloted GQSM with UVD, which yields the refined posterior soft-replica vectors and the corresponding error covariance matrices. At the end of a ^-th MP iteration, the soft-replica vectors and error covariance 25 matrices are damped in order to prevent early convergence to a local optimum, thus 202402524 26 where ^ ∈ [0,1] is the damping factor, and the superscript 2⋅3[^] denotes the ^-thiterate of the variable. 5 The last of the MP iterations, denoted by the index ^conv, can be determined by well-known convergence criteria, such as the maximum number of iterations, i.e., while ^÷ùü, or a given convergence threshold of the soft-replicas, i.e., while ^> ^Th . After the last MP iteration, a belief consensus is taken over the 2^&10 factor nodes via which is equivalent to equation (34) without the self-IC. 15 The consensus information vectors and precision matrix diagonals are respectively given by ,20 Finally, a hard-decision on the 2^ activation vectors is made by evaluating the consensus PDFs for all ^^valid states of the activation vectors, i.e., §̃HJI = argmax ℙ\çA? ∣ §¨] , 2453§Õ∈ℰ25 which is equivalent to computing the consensus soft-replicas 202402524 27 where 5 and determining the index of the maximum value of the consensus soft-replica, i.e., SA? = argmax¨∈^ where ^̂HJI2®3 denote the ®-th element of the consensus softreplica vector §̂HJ*. 10 An example of the low-complexity decoding process 100 discussed above, dubbed "UVD-GaBP" for piloted GQSM schemes and described by equation (26)-(48), is summarized by the method steps provided below and depicted in the flow diagram in figure 5. Figure 6 shows a corresponding schematic block diagram of the process 15 steps in the decoder, with the operations on the factor nodes and the variable nodes shown in the left and right dashed box, respectively. Inputs to the process 100 are the received signal y, the effective channel matrices HR and HI , pilot symbols = and =?C∀D, and the noise variance N0. Outputs of the 20 process are the estimated activation vectors §HJI and § O J∀D. In step 105 the soft-replicas §̂HJI:5and §̂HJO:5are initialised ∀8 and ∀D, following the PMF determined via equation (25). In step 110 the error covariance matrices ä?A:5and of the activation vector soft-replicas are computed via equation (27). This 25 marks the beginning of the iterative part of the process 100. In step 115 a soft-IC is performed to obtain the real and imaginary components y‾A?:5and y‾C?:5 via equation (29). In step 120 the conditional variances ^A C?:5 and ^?:5 arecomputed via equation (32), and in step 125 the information vectors ^A?:5and ^C?:5 202402524 28 representing the unscaled multivariate Gaussian PDF of the extrinsic beliefs are computed via equation (35). Next, in step 130, the diagonal vectors of the precision matrices, diag and diag \C ^?:5], are computed via equation (40). In step 135 the respective expectations of the activation vectors, §̂HJI:5and are computed 5 via equation (38), before the corresponding error covariance matrices of the activation vector soft-replicas, ä?A:5and ä?C:5, are computed via equation (27) in step 140. In step 145 the expectations of the activation vectors, §̂HJI:5and and the corresponding error covariance matrices of the activation vector soft-replicas, ä?A:5and ä?C:5, are updated via damping using equation (41). Step 150 checks if a 10 termination criterion is met and, in the negative, “No”-branch of step 150, the iteration loop is repeated, from step 115, with the previously computed values as inputs. In the positive, “Yes”-branch of step 150, the optimal activation vector estimates §̂HJI:5and §̂HJO:5are selected and provided to a belief consensus stage. 15 The termination criterion can include, for example, a predetermined number of iterations, or a convergence of the -optimal soft-replicas §̂HJI:5and §̂HJO:5, within a predetermined range or below a predetermined value. Such convergence criterion can be fulfilled, e.g., when the average change between consecutive post-iteration soft-replicas §̂HJI:5and §̂HJO:5lies below the predetermined value. 20 Establishing the belief consensus comprises, ∀D, in step 155, obtaining ^C and ^?via equation (43), and obtaining, in step 160, the precision matrix diagonalsdiag\^A?] and diag\^C? ] via equation (44). The final activation vector estimates §̃HJIand §̃HJO are obtained via equation (45) in step 165. 25 In the following section the fundamental limitations and the consequent sources of errors of the decoder proposed above are identified, which mainly arise from the omission of the joint probability information between the unit vectors of the UVD. In other words, projecting multiple complex symbols to the same I / Q-domain 30 equivalent PAM constellation can result in different symbols being projected onto 202402524 29 the same location on the I or Q axis, making the projected symbols indistinguishable to the UVD detector. Thus, enhancements to the method are proposed herein to improve the detection 5 performance against such errors, at the cost of an additional computational complexity yet minimal structural change. Without loss of generality the dependencies on the observation node denoted by the subscript 2⋅3:5are omitted here, and only the real part variables 2⋅3Aare described, as the same analysis applies to the imaginary part variables 2⋅3C. 10 Consider two soft-replicas §̂A A A AHJ and §̂HJï of two unique unit vectors §HJ and §HJï withD≠ D^, and the respective transmit symbols =A A? and =?ï . Correct estimation results inthe convergence of the effective symbol =A ⋅ §̂A to the true value of = ⋅ §A HJ HJ , and=A?ï ⋅ §̂AHJï to =A?ï ⋅ §AHJï , as illustrated in figure 7 a).15 On the other hand, an erroneous estimation may arise when a soft-replica converges to the wrong unit vector, as illustrated in figure 7 b), leading to a duplicate estimate. Due to the combinatoric selection of the indices in a GQSM index vector and of general IM schemes as described further above, no duplicate indices may 20 exist within a single index vector. In the figure, the bold solid arrows indicate the correct convergence, while the thin solid arrows indicate further possible convergences. The dashed bold arrow indicates a convergence to the wrong unit vector. 25 In light of the above, two sources of such erroneous duplicate convergence are identified: a) high transmit symbol similarity, and b) lack of joint / dependent distribution information between the unit vector variables. In order to mitigate the error from the above-identified sources, three modifications to the GQSM transmitter and the UVD-GaBP decoder are discussed in the following. 30 202402524 30 Assume that multiple transmit symbols have high similarity, i.e., ^=A A ^? − =?ï ^ ≈ 0.Such behavior is illustrated by observing Fig. 7 b) and considering = A ≈ =?ï , wherethe initialized effective symbols ⋅ §̂AHJ and =A?ï ⋅ §̂AHJï are consequently very similarin value. Therefore, the effective symbols appearing in the core MP steps of the 5 UVD-GaBP decoder via equations (28)-(36), may converge to either of the true symbols =A ⋅ §AHJ or =A?ï ⋅ §AHJï , and potentially produce an errorneous duplicateestimate. The proposed approach to mitigate the duplicate convergence error comprises 10 selecting a subset of the transmit symbol constellation that ensures that a sufficient difference in numerical value is present for the symbols of the subset in the respective real and imaginary domains. An alternative approach to achieve the same goal comprises modifying the transmit symbol constellation, or a subset thereof, such that a sufficient difference in numerical value is ensured for the 15 symbols in the respective real and imaginary domains. To that end, consider the IQ-decoupled constellations ^A ≜ ℜ9^>; and ^C ≜ ℑ9^>;,which is equivalent to two PAM constellations respectively on the real and imaginary axes, as illustrated in figure 8 a). For a symmetric M-QAM constellation ^>, the PAM 20 constellations have a smaller cardinality than #, as multiple symbols in ^>may convey the same amplitude in the IQ domain, leading to duplicate symbol values. In order to avoid such duplicate symbols in the IQ domain, the invention propose a simple rotation scheme to ^>, which leads to non-duplicate PAM symbols in both IQ 25 domains. Furthermore, an optimal rotation angle∗is obtained to maximize the minimum Euclidean distance between the symbols of the PAM constellation. There exists an optimal rotation angle∗for each of the quadrants, but here only the rotation angle within the first quadrant is considered, i.e., ∈ , as shown inTable I for some sizes of the constellation. 30 202402524 31 The optimization problem is described by ∗= argmax # \ℜ|^$" ⋅ ^%^] $" %÷úû + #÷úû\ℑ|^ ⋅ ^ ^] 2493 5 where #÷úû2⋅3 is a function on a set denoting the minimum Euclidean distance of its elements defined as 10 It can be observed that the optimal rotation illustrated in figure 8 b) is effectively equivalent to fitting a QAM constellation from # unique IQ-orthogonal symbols from a larger scaled QAM constellation to maximize the minimum Euclidean distances between the symbols, as illustrated in figure 8 c). 15 In light of the above, given that the pilot symbols are selected uniquely, the rotated constellation ensures no duplicate or similar symbols in each of the decoupled IQ domain. 20 In addition to the optimal rotation of the symbol constellation which is a pre-computed at the GQSM transmitter, another approach is discussed below to enhance the detector also presented herein. In order to avoid the duplicate estimation error of the unit vectors the dependent information of the unit vectors can be incorporated in the form of conditional PMFs. Specifically, the prior PMFs25 of the Bayes-optimal soft-replica denoiser in equation (37) may beenhanced to first-order conditional PMFs ℙÓß×∣ ≡ ®^. In other words, the first-orderconditional PMFs yield the activation probability of the ®-th index of the D-th unit 202402524 32 vector, given the activation information of ®^-th index of the D^-th unit vector. While it is also possible to incorporate higher-order conditional PMFs by incorporating more than one conditional variable, the derivation and application of such higher-order PMFs require a significantly increased computational and space complexity, such 5 that only the first-order conditional PMF is considered herein for practicality. The evaluation of a conditional PMF requires hard-decided activation indices for a given unit vector, which is not available within the MP iterations fundamentally based on soft-replicas. Therefore within each iteration of the MP algorithm, a greedy10 hard-determination of the most confident index pair 2D, ®3 is made for each 8-thfactor node by evaluating equation (34), and finding the index pair yielding the maximal belief mass, i.e., by solvingĎA, ®̌A = a A5 5 rgmax ℙ\ç?:5 ∣ §¨] 2513?,¨15 which is equivalent to obtaining the indices of the maximum values in the belief mass vector ^?A:5in equation (39). Under such greedy selection, the conditional PMF-based denoiser with the greedy indices DA A5̌, ®̂5, becomes20 to yield the soft-replica vectors incorporating the dependent information of the most confident unit vector variable. 25 The conditional PMFs does not exhibit a closed-form expression unlike equation (24) for the independent PMFs, and therefore must be obtained via numerical evaluation. However, as the conditional PMFs are fixed for a given codebook, i.e., 202402524 33 system parameters ^^and ^, all values of the conditional PMF can be pre-computed. Furthermore, a similar approach can also be performed after the convergence of the 5 MP iterations to find the index of the strongest value within the converged beliefs in equation (42)ĎA, ®̌A = argmax?,¨ which is equivalent to obtaining the indices of the maximum values in the belief 10 masses ^?Ain equation (47). In hand of a converged index of the most confident unit vector variable, IC can performed on the received signals corresponding to the selected indices as15 and the corresponding symbol and unit vector variables are nullified from the respective soft-replica, to be repeated for ^ successive IC iterations. 20 Finally, the estimated index vectors can be obtained by concatenating and sorting the inherent activation indices from each greedy selection step without having to evaluate equations (45)-(48), i.e., Q̃A = sort \R®̌A , ⋯ , ®A[,] ̌[^] T] 255325 denote the greedily selected activation index at the / -th iteration of the successive IC loop. The steps of a method 200 incorporating the receiver-side enhancements 30 discussed above are presented below with reference to figure 9. Figure 10 shows a schematic block diagram of the method 200. Note that for ^ = 1, the enhancementis not necessary as there is only a single unit vector to evaluate, and the conditional 202402524 34 PMF-based denoiser is not applicable as there is no other unit vector to condition the extrinsic beliefs. In addition, the motivation of such algorithmic enhancements is not necessary for ^ = 1, as the duplicate convergence error cannot occur with asingle unit vector variable. 5 Inputs to the process 200 are the received signal y, the effective channel matrices HR and HI , pilot symbols =?Aand =?C∀D, and the noise variance N0. Outputs of the process are the estimated index vectors Q̃Aand Q̃C. 10 In step 205 the soft-replicas §̂HJI:5and §̂HJO:5are initialised ∀8 and ∀D, following the PMF determined via equation (25). In step 210 the error covariance matrices ä?A:5and of the activation vector soft-replicas are computed via equation (27). This marks the beginning of the iterative part of the process 200.15 In step 215 a soft-IC is performed to obtain the real and imaginary components y‾A?:5and y‾C via equation (29). In step 220 A C?:5 the conditional variances ^?:5 and ^?:5 arecomputed via equation (32), and in step 225 the information vectorsC and ^?:5representing the unscaled multivariate Gaussian PDF of the extrinsic beliefs are computed via equation (35). Next, in step 230, the diagonal vectors of the precision 20 matrices, diag \^A?:5] and diag \^C?:5], are computed via equation (40). In step 235 the conditional PMFs for D5̌Aand D5̌Care evaluated via equation (51) for finding the index pair yielding the maximal belief mass. In step 240 the expectations of the activation vectors, §̂HJI:5and are computed via equation (52), before the corresponding error covariance matrices of the activation vector soft-replicas, ä?A:525 and ä?C:5, are computed via equation (27) in step 245. In step 250 the expectations of the activation vectors, §̂HJI:5and §̂HJO:5, and the corresponding error covariance matrices of the activation vector soft-replicas, ä?A:5and ä?C:5, are updated via damping using equation (41). Step 255 checks if a termination criterion for the MP iteration loop is met and, in the negative, “No”-branch of step 255, the MP iteration 30 loop is repeated, from step 215, with the previously computed values as inputs. In 202402524 35 the positive, “Yes”-branch of step 255, the optimal activation vector estimates §̂HJI:5and §̂HJO:5are selected and provided to a belief consensus stage. The termination criterion can include, for example, a predetermined number of 5 iterations, or a convergence of the -optimal soft-replicas §̂HJI:5and §̂HJO:5, within a predetermined range or below a predetermined value. Such convergence criterion can be fulfilled, e.g., when the average change between consecutive post-iteration soft-replicas §̂HJI:5and §̂HJO:5lies below the predetermined value. 10 Establishing the belief consensus comprises, ∀D, in step 260, obtaining ^C and ^?via equation (43), and obtaining, in step 265, the precision matrix diagonalsdiag\^A?] and diag\^C? ] via equation (44). In step 270 the most confident index pairof unit vectors ĎAand ĎCand corresponding index activation information ®̌5Aand ® within the converged beliefs are determined using equation (53), and stored, in step 15 275, for a λ-th IC iteration. In step 280 IC is performed for obtaining y̌A C5 and y̌5using equation (54). Step 285 checks if a termination criterion for the IC iteration loop is met. in the negative, “No”-branch of step 285, the MP iteration loop is repeated, from step 215, with the previously computed values as inputs. In the positive, “Yes”-branch of step 285, the index vector estimates Q̃Aand Q̃Care obtained via 20 equation (55) in step 290 and output. The following discussion analyses the performance of the UVD-GaBP method and the enhancement modules discussed above based on simulations for different system sizes characterized by ^^transmit antennas, ^&receive antennas, and ^ 25 pilot symbols from the #-ary constellation, considering the three cases specified in Table II below: 202402524 36 Table II 5 The exemplary cases are selected to capture and highlight the improvements introduced by each of the three enhancement modules discussed herein, namely: method 100 is the baseline UVD-GaBP decoder; method 200 with conditional denoiser but without successive IC; and finally the full method 200 with both the conditional denoiser and successive IC. Optimized constellations are employed, 10 however, in all cases considered. To serve as a reference, the results are compared against the matched filter bound (MFB), which corresponds to the performance of a Genie-aided system in which the soft-replicas of the GaBP iterations are initialized with the actual, i.e., true, 15 transmitted activation vectors. The first set of results, depicted in figure 11, compares the BER performances of the proposed UVD-GaBP decoder for all cases in Table II and the corresponding MFB, in MIMO systems of size 32 × 32 employing various numbers of symbols ^. The20 BER plots, shown against the energy-per-bit-to-noise-power-spectral-density ratio201 / ^43, indicate that the duplicate convergence of unit vectors, identified furtherabove as a critical potential source of errors, does not occur for ^ = 1 but can besubstantial for ^ > 1. The results also show, however, that the duplicateconvergence behaviour is significantly mitigated by the further improvements to the 25 underlying method 100 introduced above. 202402524 37 The BER floor behaviour observed for ^ > 1 is, however, not only a consequence ofduplicate convergence of unit vectors, but also a function of system size. To clarify, larger systems can sustain larger ^ values, without exhibiting error floors. In order to demonstrate that, consider the next set of results, given in figure 12, which show the 5performances achieved by method 200 in systems of size 64 × 64 and 96 × 96.Note that simulation results for GQSM systems of such sizes have never been shown before, precisely due to the prohibitive complexity of conventional GQSM detection methods. 10 All in all, given that error floors do not affect large systems, and since the decoding complexity of such large systems is tamed by the proposed method it is fair to say that the proposed UVD-GaBP offers a systematic mechanism to employ massive MIMO in order to achieve high-performance ISAC functionalities with controlled complexity. 15 Note that in figure 11 the performance of the proposed piloted GQSM scheme is compared against a conventional multiplexed (MUX) MIMO-ISAC scheme, in which all antennas are active, but used to transmit symbols and pilots simultaneously, for the proposes of communications and sensing, respectively. 20 In order to enable a fair comparison between the proposed and conventional approaches, the total power and transmission rate ^ñóin bits / s / Hz of both systems is set to identical values, and the well-known linear GaBP is employed in the detection of the MUX scheme. Note, furthermore, that the proposed 25 UVD-GaBP-based MIMO-ISAC scheme is interference-free, unlike the conventional MUX-based approach, in which decoding errors made on communication symbol estimates may deteriorate sensing performance, while channel estimation errors made over pilots may impact BER performance, respectively. To the advantage of the conventional scheme, however, no interference is considered in evaluating the 30 performance of the MUX-based system in figure 13. 202402524 38 It can be seen that piloted GQSM in large systems, enabled by the proposed UVD-GaBP detector, outperforms a conventional MIMO-MUX system in which resources are split between communications and sensing functionalities. 5 In the following section, the decoding complexities of the conventional methods, the proposed UVD-GaBP decoder and the enhanced variants discussed above are derived, which highlight the significantly reduced complexity that is completely independent of the prohibitive combinatorial coefficient. Specifically, the computational complexity ^ is evaluated in terms of number of floating point 10 operations (FLOPs), which represents the cost of a single arithmetic operation of a real-valued scalar, and the complexity order ^2⋅3 representing the leading coefficient order of each system parameter variable. 1) Brute-force ML Decoder: 15 In a brute-force ML detection, the likelihood metric for all possible codewords of the piloted GQSM codebook ^ is evaluated and the transmit signal with the minimum error is selected, i.e.,^̂ ^b^ = argmin ∥ y − ^^ ∥^. 2563^∈^20 Following equation (56), the computational complexity of a brute-force decoder can be obtained as which reveals that the complexity of the ML decoder is trivially dependent on the full 25 codebook size \^^.]^. 2) IQ-decoupled Vector-valued GaBP Decoder: Next, the complexity of the IQ-decoupled vector-valued GaBP (VGaBP) scheme is analysed, which was shown to approach the optimal performance of the ML, while 30 achieving a quadratic order complexity reduction by exploiting the half-sized 202402524 39 decoupled i.i.d. variables of the GQSM in the IQ domain of equation (9). The decoding complexity of the IQ-decoupled VGaBP is given by 5 where ^ is the number of MP iterations. 3) Proposed UVD-GaBP Decoder: 10 Following the proposed MP rules, the computational complexity of method 100 without enhancements, is obtained as 15 In contrast, the complexity of the enhanced UVD-GaBP decoder of method 200 is given by 20 It can be seen that the intractable combinatorial coefficient of the conventional GQSM is not inherited by the proposed UVD-GaBP decoder, and instead replaced with a significantly lower polynomial-order complexity order term on the system size parameters ^, ^^, and ^&.25 Figure 13 compares the full decoding complexity in FLOPs of the four analysed decoders, for increasing number of transmit antennas ^^and varying values of symbols ^. The number of receive antennas ^&and the number of MP iterations ^ 202402524 40 are fixed respectively at a reasonable value of ^& = 64 and ^ = 100, as the twoparameters only introduce a linear complexity order, as opposed to ^^and ^ with more prominent effects, as can be seen in the summary of Table III below: 5 Table III - Computational complexity orders of the conventional GQSM decoders and the proposed UVD-GaBP decoders. For ^ = 1, the combinatorial coefficient will reduce to ^^ and hence the ML10 decoder enjoys a significantly low complexity, however, for actual combinatorial IM codebooks even with ^ = 2 or ^ = 3, the complexity of the ML decoder and theIQ-VGaBP is shown to increase to an infeasible order with respect to ^^and ^, especially for mMIMO scenarios with ^^ 8 16.15 On the other hand, the proposed UVD-GaBP while requiring a higher complexity than that of the ML with small ^^and ^, a significant superiority can be observed for scenarios with larger system parameters of ^^and ^, where the rate of complexity gain with respect to increasing ^^and ^ is shown to be very low, noting the logarithmic scale of the complexity plot. 20 Finally, figure 14 highlights the superiority of the low-complexity UVD-GaBP enabled piloted GQSM system with respect to the effective gain of the BER performance per decoded bit, in other words, illustrates the comparison of differently 202402524 41 parametrized MIMO GQSM with ^^and ^, subject to the same decoding complexity cost. The left subfigure first compares piloted GQSM systems requiring a decoding 5complexity in the order 2 × 109 FLOPs, with the three selected scenarios: MLdecoder with ^^ = 16, ^& = 16, ^ = 3, the enhanced UVD-GaBP decoder ofmethod 200 with ^^ = 24, ^& = 24, and ^ = 3, and the UVD-GaBP decoder ofmethod 100 with ^^ = 32, ^& = 32, and ^ = 2. Similarly, the right subfigurecompares piloted GQSM systems requiring a decoding complexity in the order10 3 × 10,4 FLOPs, with: ML decoder with ^^ = 16, ^& = 16, ^ = 4, the enhancedUVD-GaBP decoder (Alg. 2) with ^^ = 32, ^& = 32, and ^ = 4, and also with ^^ =48, ^& = 48, and ^ = 3.In light of the decreased complexity, the supported GQSM system of the UVD-GaBP 15 decoder can be significantly scaled, especially larger-ordered, compared to the conventional ML decoder. The increased system parameters of ^^and ^ inherently imply an increased spectral efficiency of the system, improving the BER performance with respect to the 01 / ^4.20 The 01 / ^4gain is about 3 dB in the left figure under a decoding complexity of2× 109FLOPs, and the gain increases to 6 dB for higher complexity scales, shown inthe right figure, under a decoding complexity of 3 × 10,4 FLOPs. This 01 / ^4 gain isexpected to increase with increasing complexity order, due to the combinatorial growth of the ML decoding complexity, as can be seen in the behaviour of Figure 13. 25 While the error floor behaviour is also still present in the proposed decoders as opposed to the ML decoders, the performance up to the error floor is superior, and the error floor is also expected to be mitigated in large systems as shown in the performance analysis further above, or via post-processing techniques as 30 previously discussed. In light of the foregoing description, in accordance with a first aspect of the present invention a method of adapting a general quadrature amplitude modulation 202402524 42 (M-QAM) complex symbol constellation ^>for quadrature spatial modulation (QSM) is presented. The method comprises receiving a M-QAM complex symbol constellation ^>, or a subset thereof, and selecting, from the M-QAM complex symbol constellation ^>, or the first subset thereof, a second subset of symbols 5 whose projections onto axes of a corresponding I / Q-domain are distinguishably spaced apart on the I and Q axis, respectively. Alternatively, the method comprises receiving a M-QAM complex symbol constellation ^>, or a subset thereof, and determining a transformation for the received M-QAM complex symbol constellation ^>, or the subset thereof, targeting to yield, in the corresponding I / Q-domain, 10 individual locations of projections of the constellation points that are distinguishably spaced apart on the I and Q axis, respectively. Preferably, selecting the second subset and determining the transformation, respectively, is targeted to yield individual locations of projections of the constellation points that are maximally spaced apart on the I and Q axis, respectively. In other words, the inter-symbol 15 distances of between the projected symbols in the I / Q domain are distinguishable and, preferably, maximal. The determined transformation is then applied to the received M-QAM complex symbol constellation ^>, or the subset thereof, yielding a correspondingly modified symbol constellation that is then provided to a transmitter and / or to a receiver. 20 In embodiments of the method determining the transformation comprises determining a rotation angle that rotates the constellation points of the received M-QAM complex symbol constellation ^>around the intersection of the I and Q axes which yields the desired distinguishably spacing of the projections. The rotation may 25 further comprise changing an amplitude of a respective complex symbol between the original and transformed constellation points. In embodiments of the method determining the transformation comprises determining individual rotation angles for each quadrant of the I / Q-domain. 30 The rotation angle may be determined in dependence of an order of the received M-QAM complex symbol constellation ^>. 202402524 43 In embodiments of the method the transformation maps constellation points of the received M-QAM complex symbol constellation ^>to different constellation points of a constellation of the same order or to constellation points of a constellation of a different order. This embodiment ensures compatibility with conventional M-QAM 5 standards. In embodiments of the invention at least the positions of pilot signals are transformed, while other symbols may keep their original locations. The transformation of the pilot symbols may comprise swapping locations with symbols 10 not subjected to the transformation. The computations required for adapting the symbol constellation require optimisation only over one variable, which is the rotation angle, and are low-complexity and can be pre-computed once. The proposed adaptation of the 15 symbol constellation is applicable to all types of constellations, rendering it suitable for any kind of SM-based modulation schemes, including SM, QSM, GSM, GQSM, STC-QSM, OS-QSM, etc. in a MIMO system, especially for massive MIMO P2P scenarios, leveraging UVD-GaBP based methods for detection. 20 In accordance with a second aspect of the present invention an apparatus configured for adapting a general quadrature amplitude modulation, M-QAM, complex symbol constellation ^>for quadrature spatial modulation, QSM is presented. The apparatus comprises one or multiple communication interfaces, one or more microprocessors, and associated volatile and non-volatile memory. The 25 elements or components are connected via one or more data and / or signal lines or buses. The non-volatile memory stores computer program instructions which, when executed by the one or more microprocessors, configure elements or components of the apparatus to implement or carry out one or more embodiments of the method in accordance with the first aspect of the present invention. 30 A communication system in accordance with a further aspect of the invention comprises one or more transmitters adapted to transmit symbols from a complex symbol constellation ^>as adapted for quadrature spatial modulation, QSM, in 202402524 44 accordance with the method of one or more of claims 1 to 5, one or more corresponding receivers, and one or more apparatus according to the second aspect of the invention. 5 As will be appreciated by one skilled in the art, aspects of the embodiments may be embodied as a system, apparatus, method, or program product. Accordingly, embodiments may take the form of an entirely hardware embodiment, an entirely software-implemented embodiment, including firmware, resident software, micro-code, etc., or an embodiment combining software and hardware aspects. 10 For example, the disclosed embodiments may be implemented as a hardware circuit comprising custom very-large-scale integration (VLSI) circuits or gate arrays, off-the-shelf semiconductors such as logic chips, transistors, or other discrete components. The disclosed embodiments may also be implemented in 15 programmable hardware devices such as field programmable gate arrays, programmable array logic, programmable logic devices, or the like. As another example, the disclosed embodiments may include one or more physical or logical blocks of executable code which may, for instance, be organised as an object, procedure, or function. 20 The method presented hereinbefore may be represented by computer program instructions. Accordingly, in accordance with a further aspect of the invention, a computer program product comprises computer program instructions which, when executed by a microprocessor of an apparatus in accordance with the second 25 aspect of the invention, cause the microprocessor to execute methods in accordance with the first aspect of the present invention, and to accordingly control hardware and / or software blocks or modules of the apparatus. Computer program instructions, or code, for carrying out operations for 30 embodiments may be any number of lines and may be written in any combination of one or more programming languages including an object- oriented programming language such as Python, Ruby, Java, Smalltalk, C++, or the like, and conventional procedural programming languages, such as the “C” programming language, or the 202402524 45 like, and / or machine languages such as assembly languages. The code may execute entirely on the user’s computer, partly on the user’s computer, as a stand-alone software package, partly on the user’s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the 5 remote computer may be connected to the user’s computer through any type of network, including a local area network (LAN), wireless LAN (WLAN), or a wide area network (WAN), or the connection may be made to an external computer, for example, through the Internet using an Internet Service Provider (ISP). 10 The computer program instructions may be retrievably stored or transmitted on a computer-readable medium or data carrier. The medium or the data carrier may by tangibly or physically embodied, e.g., in the form of a hard disk, solid state disk, flash memory device or the like. However, the medium or the data carrier may also comprise a modulated electro-magnetic, electrical, or optical signal that is received 15 by the computer by means of a corresponding receiver, and that is transferred to and stored in a memory of the computer. The described features, structures, or characteristics of the embodiments may be combined in any suitable manner. In this description, numerous specific details are 20 provided, such as examples of programming, software modules, user selections, network transactions, database queries, database structures, hardware modules, hardware circuits, hardware chips, etc., to provide a thorough understanding of embodiments. One skilled in the relevant art will recognize, however, that embodiments may be practiced without one or more of the specific details, or with 25 other methods, components, materials, and so forth. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of an embodiment. Reference throughout this specification to “one embodiment,” “an embodiment,” or similar language means that a particular feature, structure, or characteristic described in connection with the embodiment is 30 included in at least one embodiment. Thus, appearances of the phrases “in one embodiment,” “in an embodiment,” and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment, but mean “one or more but not all embodiments” unless expressly specified otherwise. 202402524 46 The terms “including,” “comprising,” “having,” and variations thereof mean “including but not limited to,” unless expressly specified otherwise. An enumerated listing of items does not imply that any or all of the items are mutually exclusive, unless expressly specified otherwise. The terms “a,” “an,” and “the” also refer to “one or 5 more” unless expressly specified otherwise. Where aspects of the embodiments are described in this specification with reference to schematic flowchart diagrams and / or schematic block diagrams of methods, apparatuses, systems, and program products according to embodiments 10 it will be understood that each block of the schematic flowchart diagrams and / or schematic block diagrams, and combinations of blocks in the schematic flowchart diagrams and / or schematic block diagrams, can be implemented by code. This code may be provided to a processor of a general-purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, 15 such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart diagrams and / or block diagrams. It should be noted that, in some implementations or embodiments, the functions 20 noted in the exemplary embodiments shown in the figures may occur out of the order shown in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. Other steps and methods may be conceived that are equivalent in function, logic, or effect 25 to one or more blocks, or portions thereof, shown in the figures. The invention presented hereinbefore permits combining the advantages of massive IM and ISAC techniques. Embodiments of the invention exploit the finding that in a massive multiple-input multiple-output (MIMO)-ISAC system, one can 30 convey information by relying solely on the massively-scaled IM scheme, reducing some or all transmit symbols to pilots, which can then be fully exploited for transmitting sensing or radar optimized symbols, or for other functionalities. The 202402524 47 method presented above is particularly advantageous in large antenna arrays, where the binomial coefficient quickly grows with the number of antennas. The proposed approach implies that such IM-enabled ISAC systems suffer less or 5 not at all from interference between the pilots, which are used as sensing signals or the like and the data, or information signal, thus naturally mitigating the potential performance degradation typical of conventional ISAC schemes. In fact, as has been shown herein, for sufficiently large sizes the proposed method outperforms conventional ISAC systems under the same rate and power constraints designed 10 via a "classic multiplex paradigm," in which symbols and pilots are transmitted simultaneously, for the purpose of communication and sensing, respectively. Contrary to conventional decoders, which assume a uniform distribution of the activation vectors at the transmitter, the method proposed herein considers a scalar, 15 digit-level distribution. Modelling or deriving the digit-level indices avoids the limitations associated with conventional methods using statistical inference, that an accurate prior distribution is not available, which renders an accurate estimation of the indices impossible. 20 The method and apparatus proposed herein can be used with great advantage in mobile communications such as vehicle-to-everything (V2X), high-speed railway communication systems, low earth orbit (LEO) satellite communications, communications with unmanned aerial vehicles (UAV), in particular in swarm settings, massive machine-type Internet-of-Things (IoT) communications, e.g., in 25 wireless factory and other industrial settings, extra-large scale MIMO and OFDM systems, and even in underwater acoustic communications. While the invention has been described herein using an exemplary mMIMO IM QSM system it is obvious to the skilled person having read and understood this 30 specification that the principles and methods developed and presented above also apply to general SM-based systems, including QSM, GSM, GQSM, STC-QSM, OS-QSM. 202402524 48 Detailed descriptions provided herein and discussed with reference to annexed drawings are intended as a description of exemplary configurations and are not intended to represent the only configurations in which the concepts described herein may be practiced. The detailed descriptions may include specific details 5 intended exclusively for the purpose of providing a thorough understanding of the various concepts. However, it will be apparent to those skilled in the art that these concepts may be practiced without these specific details. In particular, although terminology from 3GPP 5G NR may be used in this disclosure to exemplify embodiments herein, this should not be seen as limiting the scope of the invention. 10 Figure 15 shows a schematic block diagram of an exemplary receiver 400 in accordance with the invention. The receiver 400 is configured for receiving index-modulated transmit symbols and comprises two or more antennas 402, circuitry 404 for processing radio frequency signals, one or more microprocessors 15 406, and associated volatile 408 and non-volatile memory 410. The various components and elements of the apparatus 400 are communicatively connected via one or more data and / or signal lines or buses 412. The non-volatile memory 410 stores computer program instructions which, when executed by the one or more microprocessors 406, configure components of the wireless apparatus 400 to 20 implement or carry out embodiments of the method in accordance with the first aspect pf the invention as described herein.

Claims

202402524 49 CLAIMS 1. A method of adapting a general quadrature amplitude modulation, M-QAM, complex symbol constellation (^>) for quadrature spatial modulation, QSM, 5 comprising: - receiving a M-QAM complex symbol constellation (^>) or a first subset thereof, - selecting, from the M-QAM complex symbol constellation (^>), or the first subset thereof, a second subset of symbols whose projections onto axes of a 10 corresponding I / Q-domain are distinguishably spaced apart on the I and Q axis, respectively, or comprising - receiving a M-QAM complex symbol constellation (^>) or a first subset thereof, 15 - determining a transformation for the received M-QAM complex symbol constellation (^>), or the subset thereof, targeting to yield, in the corresponding I / Q-domain, individual locations of projections of the constellation points that are distinguishably spaced apart on the I and Q axis, respectively, 20 - applying the determined transformation to the received M-QAM complex symbol constellation (^>), or the subset thereof, yielding a correspondingly modified symbol constellation, and - providing the second subset or the modified symbol constellation, 25 respectively, to a transmitter and / or to a receiver.

2. The method of claim 1, wherein determining the transformation comprises determining a rotation angle that rotates the constellation points received M-QAM complex symbol constellation (^>), or the subset thereof, around the 30 intersection of the I and Q axes which rotation angle yields distinguishably spacing of the projections.202402524 50 3. The method of claim 2, wherein determining the transformation comprises determining individual rotation angles for each quadrant of the I / Q-domain.

4. The method of claim 2 or 3, wherein the rotation angle is determined in 5 dependence of an order of the received M-QAM complex symbol constellation (^>).

5. The method of one of claims 1 to 4, wherein the transformation maps constellation points of the received M-QAM complex symbol constellation 10 (^>), or the subset thereof, to different constellation points of a constellation of the same order or to constellation points of a constellation of a different order.

6. Apparatus (400) configured for adapting a general quadrature amplitude modulation, M-QAM, complex symbol constellation (^>) for quadrature spatial 15 modulation, QSM, the apparatus (400) comprising one or more communication interfaces (402), one or more microprocessors (406), volatile (408) and non-volatile memory (410), connected via one or more data and / or signal lines or buses (412), wherein the non-volatile memory (410) stores computer program instructions which, when executed by the one or more 20 microprocessors (406), configure components of the apparatus (400) to implement or carry out a method of any one of the preceding claims 1 to 5.

7. A communication system comprising one or more transmitters adapted to transmit symbols from a complex symbol constellation (^>) adapted for 25 quadrature spatial modulation, QSM, in accordance with the method of one or more of claims 1 to 5, one or more corresponding receivers, and at least one apparatus in accordance with claim 6.

8. Computer program product comprising computer program instructions which, 30 when executed by a microprocessor of an apparatus (400) according to claim 6, cause the apparatus (400) and / or control hardware blocks, modules or components of the apparatus (400), respectively, to execute the method of one or more of claims 1 to 5.202402524 51 9. Computer readable medium or data carrier retrievably transmitting or storing the computer program product of claim 8.

Images