Method of receiving information transmitted via index modulation and apparatus implementing the method

Abstract

A method of decoding information coded in respective resource element activation patterns, REAP, of activated transmit resource elements, TREs, comprises calculating, based on the subset of REAPs known to be used at the transmitter for the present transmission instance, a probability for each TRE of having been activated, and determining, based on a receive signal, a channel matrix and the probability for each TRE of having been activated, estimates of at least the REAP for the present transmission instance. An interference cancellation is performed on the estimates, and the process is iteratively repeated. The REAP or the information represented by the REAP is then output.

Description

[0001] METHOD OF RECEIVING INFORMATION TRANSMITTED VIA INDEX MODULATION AND APPARATUS IMPLEMENTING THE METHOD

[0002] FIELD OF THE INVENTION

[0003] 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 simultaneously used for communication and environment perception, i.e., generating information about the environment and objects therein.

[0004] NOTATIONS

[0005] Throughout this specification, bold symbols represent vectors or matrices. Scalar values are denoted herein by lowercase letters in italics, as in x. SuperscriptsTandH, respectively denote the transpose and complex conjugate transpose of a vector or matrix.

[0006] BACKGROUND

[0007] 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 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 multipleinput multiple-output (mMIMO) and cell-free MIMO (CF-MIMO), reconfigurable intelligent surfaces (RISs), and index modulation (IM).

[0008] 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 for transmitting information symbols and beamforming.

[0009] 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 determine which resources are activated, the corresponding information can be decoded.

[0010] Among the alternatives mentioned above, IM is attractive for future wireless communication systems since the frugal utilization of resources at the transmitter 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 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 of IM in connection with mMIMO, CFMIMO, RISs and ISAC.

[0011] Generally, an IM scheme activating P selected patterns for transmitting out of a total of 2V resource allocation patterns can convey up to [log2additional information bits on top of those conveyed over the selected P resources themselves. Therefore, 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 resources to other functionalities, such as opportunistic communications and radar / sensing, etc.

[0012] Figure 1 shows a simplified schematic representation of IM with multiple antennas 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 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.

[0013] 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 "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 transmit antennas from a total of Nrtransmit 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 of the spatially- modulated information, but also in improved bit error rate (BER) performance compared to the original SM scheme discussed by R. Y. Mesleh, H. Haas, S. Sinanovic, C. W. Ahn, and S. Yun, in "Spatial modulation," IEEE Trans. Veh.

[0014] 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, 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. 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 “Spacetime block coded spatial modulation," IEEE Transactions on Communications, vol. 59, no. 3, pp. 823-832, 2011 , by . Ba§ar, U. Aygdlu, 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. limori, D. Gonzalez G., and 0. Gonsa.

[0015] 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.

[0016] 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 2LiOfl,2WJ.

[0017] Consider a system with / ?= 5 resource blocks out of which A = 3 are activated at any transmission instance. The index selection procedure selects a subset of A elements possible combinatorial patterns, which form a corresponding codebook I having a 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 I is likewise shown in the figure. In the index or activation vectors the non-zero positions indicate positions of activated resources.

[0018] It is obvious that such highly structured and deterministic codebook will inevitably 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 nonzero 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 configuration / pattern.

[0019] Returning to GQSM, such naive decoding requires searching over a combinatorial space of size • Mp, where NTis the number of transmit antennas, a is the codebook amplification factor (a = 1 for the classic GQSM), P is the number of symbols transmitted, and M 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 _ZVr>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 transmit antennas and with high data rate with large P.

[0020] 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 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.

[0021] 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 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.

[0022] 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, 0. A. Dobre, and S. Ikki, in "Optimum Low- Complexity Decoder for Spatial Modulation," IEEE Journal on Selected Areas 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

[0023] M. Simarro et al., in "Low Complexity Near-ML Sphere Decoding based on a MMSE ordering for Generalized Spatial Modulation," IEEE 31st Annual International 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.

[0024] More recently, methods aiming to directly reduce the order of the binomial 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,

[0025] Y. Liu, L. Gan, and L. Hanzo, achieve a reduction of the complexity order from to < a < 1, by leveraging the orthogonal matching pursuit (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 a.

[0026] In "An Efficient Vector-valued Belief Propagation Decoder for Quadrature Spatial Modulation," 2022 56th 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 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

[0027] 1 decoding cost to the order of the square root of the binomial coefficient, 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 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.

[0028] Still, the complexity of all known detectors is dependent on the parameters of the combinatorial space, i.e. , from the factorial relationship between the number of P activatable patterns out of a total of 2V resources.

[0029] It is, therefore, desirable to provide an improved method of decoding symbols in index modulated wireless transmissions that provides a low complexity while maximising the spectral and energy efficiency, and that permits optimising both the communication and sensing performance in ISAC applications.

[0030] SUMMARY OF THE INVENTION

[0031] This object is attained by the method of claim 1 , the apparatus of claim 8. A communication system, a vehicle equipped with an apparatus in accordance with the invention, a computer program product and a corresponding computer-readable medium are provided in claims 10 to 13, respectively. Advantageous embodiments and developments are provided in respective dependent claims.

[0032] 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 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.

[0033] In order to enable an accurate statistical inference the most important input is the 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.

[0034] The present invention builds 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 activation codebook shown in figure 2 into a corresponding index vector form that lends itself to low-complexity analytical treatment.

[0035] An important aspect of the proposed approach is reducing the decoding complexity at the receiver, which is prohibitive in massive set-ups under naive detection. A 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.

[0036] As mentioned above, many IM techniques can be found in the literature which 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 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.

[0037] This specification discloses, inter alia, a unit vector decomposition (UVD) formulation of GQSM signals that enables the concurrent estimation of the index positions as independent unit vectors. Further, several modifications to the transmitter constellation design and the detection algorithm are discussed, which enhance the performance of the detector.

[0038] Prior to describing the invention in greater detail the underlying GQSM system model and transmitter structure will be discussed. Consider a point-to-point (P2P) MIMO wireless communications system where the transmitter and the receiver are equipped with NTand NRantennas, respectively, such that the received signal vector y e (CWRX1corresponding to the transmission of the information vector x e CWTX1through the flat-fading MIMO channel H e £NRXNT jsdescribed by where w e cw«xlis a complex-valued additive white Gaussian noise (AWGN) vector with element-wise variance No, such that wn~ XN(0, No~) for n e {1, — ,NR}.

[0039] In a GSQM transmitter, the in-phase and quadrature (IQ) components of P transmit symbols s = {sf, •••>$£}, sp= sp+ jspe (C, with p e {1, •••, / ’}, taken from a complexvalued constellation Lcof size M |LC|, are sparsely mapped into the respective transmit vector components xRe CWTX1anc|xi e CNTXIoftheformfcl-th position fcp-th position fcp-th position ) ) yielding the transmit signal vector

[0040] 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 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 P = 1.

[0041] The real and imaginary parts of the transmit symbols are independent, i.e. , they can 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 kR= [fcR, — from the set of possible index vectors K of size 0 = |K|. Specifically, 0 = 21M7)] implies that only binary- encodable subsets of the possible activation pattern combinations are utilised, such that the size of the GQSM IQ activation vectors codebook A is given by

[0042] | A| = 0. Note also that the position of a symbol component in the vector corresponds directly to the activated antenna element at the transmitter.

[0043] By exploiting the possible combinatorial patterns of the symbol component positions, the GQSM scheme conveys information not only from the encoding of P symbols from S, but also from the selection of P positions out of NTpossible positions, respectively for both xRand x1.

[0044] In light of the above, the total information conveyed by the GQSM signal x is given by where BSpis the number of bits spatially encoded by the antenna position selection of P symbol parts, and BDgis the number of bits digitally encoded by the transmit symbols.

[0045] In the following a codebook example is given. For the sake of clarity, a small system with NT= 5, P = 3 is considered, such that a total of (^) = 10 activation vectors are possible, namely where the non-zero elements s*denote either spor sp, since the signal codebooks for both IQ parts are identical. A valid codebook for such a system is a subset A of the above, containing 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 a„, that is where the vector [1 , 2, 5]Tindicates that the resources with the indices 1 , 2 and 5 are activated. For the index vector set K the following distributions exist:

[0046] - the probability of selecting a given vector kqe K, which is uniformly distributed

[0047] - the probability of activating any given resource index re {1, 7?} 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

[0048] - the probability of each digit, i.e., the a-th element of the vector k being a specific resource index.

[0049] Out of these distributions, the last one is not uniform and exhibits a non-trivial distribution.

[0050] In the example provided in equations (4) and (5), the first digit of denoted by k±cannot take on the values of 4 and 5, and has a higher probability of 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.

[0051] To exploit this feature the present invention proposes a novel analytical modelling method of the index activation probability per digit of the IM-based codebooks.

[0052] 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 as where y e IR2WRX1, X e Kt2JVTX1, and w e IR2WRX1denote the IQ-decoupled counterparts of y,x, and w, respectively, and the IQ-decoupled channel matrix H e ]&2NRX2NT isfurther decomposed into two submatrices as

[0053] H = [HRHl], with HRe ]R2WRXWT and Hle ]R2WRXWT denoting the effective channel components for xRand x1, respectively.

[0054] The real domain IQ-decoupled form y = Hx + iv presented in equation (6) is the basis of most known QSM / GQSM detection algorithms, which generally seek to estimate the effective transmit signal x, in knowledge of y and H, i.e. , seek to solve the recovery problem

[0055] XML = argmin || y - Hx \\j, (8) where E is the discrete domain of the effective signal x, of cardinality While the linear recovery problem appears trivial, the challenge lies in the infeasible size of the discrete domain E of x e IR2W7’X1, with cardinality 0 ) • Mp, where [-J2= 2Llog2 ( )lis the flooring operation to the nearest power of 2.

[0056] As can been seen from the latter expression, the codebook size 0 scales at a geometric rate on P, and at a squared factorial rate on NT, such that complexity becomes prohibitive even for moderately large MIMO scenarios, which is why simulation results in the known literature exist only up to NT< 10, P < 3, even with lowest-complexity methods.

[0057] The problem described by equation (8) is NP-hard due to the discrete solution domain E, 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 various characteristics of the GQSM signal have been proposed. For example, the inherent sparsity of x 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 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 xRand x1are concurrently estimated by leveraging the independent and identically distributed (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 Decoder for Quadrature Spatial Modulation" (full citation further above) was shown to reduce the search space from

[0058] Building on the above, the present invention exploits a low-complexity decoder, 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 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 the combinatorial space, as seen in equation (2).

[0059] 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 combinatorics- free signal domain.

[0060] Exploiting the fact that each IQ symbol component occupies only a single position in the GQSM transmit vector x described in equation (2), i.e., is only transmitted from a single resource element, the transmit signal can be rewritten as a superposition of the symbol components multiplied by elementary activation vectors et, yielding where the activation vector ete {0,1}WT, with t e T {1, — ,NT}, is the t-th column of an NTx NTidentity matrix {0,1}WTas

[0061] In light of the reformulation in equation (9), the IQ-decoupled received signal vector in equation (6) becomes where the linear recovery of x has been transformed into the joint estimation problem of 2P activation vectors, i.e. , unit vectors. The reformulation in equation (11 ) better highlights the construct of the GQSM signals with inherently independent spatial and digital encoding, as illustrated in figure 3, which shows a corresponding multivariate bilinear factor graph with 2P symbol variables and 2P vector variables. The reformulation enables using independent estimation techniques when some of the decoupled variables are known, e.g., via pilot symbols.

[0062] The UVD model described above works for all spatial modulation (SM) schemes whose activation vectors are directly given by the unit vectors in S, e.g., generalized spatial modulation (GSM) and GQSM. There exist, however, enhanced SM-based schemes whose transmit symbols are not multiplied by elementary activation vectors in the space of {0,1}WT, but by optimized precoded dispersion vectors drawn from a complex space €Nt.

[0063] Let Z)R{1d be the sets from which the precoded dispersion vectors of a generic coded SM are drawn. Then, the corresponding transmit signal vector is given by

[0064] Next, the trivial identities are leveraged with eqe {0,1}<3X1, to rewrite equation (12) as where ERe GNTXQand E1e GNTXQare the dispersion vector codebook matrices constructed by concatenating the Q dispersion vectors in DRand D1respectively. It 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 ERand E1reduce to the identity matrix in the GQSM case.

[0065] Then, the IQ-decoupled form of the received signal corresponding to the transmit vector in equation (13) becomes where the effective channel H HE e ]R2WRX2<2 is constructed from the dispersion codebook E and the channel state information (CSI) H, both known at the receiver.

[0066] 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 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 Computational Advances in Multi-Sensor Adaptive Processing, 2023, pp. 301-305, in which dispersion vectors are generalized into dispersion matrices of size NTx T with T 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.

[0067] To elaborate, given the sets of dispersion matrices Z)Rand

[0068] D1{Dg}^ G (£NTXT, the transmit signal of a STC-based IM scheme becomes whose vectorized form is trivially given as vec

[0069] The UVD representation of equation (17) is similar to that of equation (14), except for the increased dimensionality to TNT, and the dictionary matrices of vectorized dispersion matrices ER[vec (DR), ■■■ , vec (DR)] G (C™TX<3ANC|

[0070] E1[vec (D[), , which yields the vectorized received signal model y where ITis the T x T identity matrix, the symbol ® denotes the Kronecker product, and H and w are the augmented effective channel matrix and AWGN vector, respectively.

[0071] 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, emphasizing however the generality of the methods.

[0072] Next, the prior distributions of the two sets of variables in the UVD model, namely the IQ symbol components and s?, and the elementary activation vectors ekR and e^, respectively, are analysed in order to enable their estimation via Bayesian inference methods.

[0073] Since QSM schemes typically employ complex symmetric constellations such as M-quadrature amplitude modulation (QAM), spand sf, and belong to the corresponding real-valued effective constellation S 9i{Lc} = 3{SC}, which is typically a pulse amplitude modulation (PAM) constellation of cardinality |S| VM. If that is not the case, and Zcis irregular, the formulation is trivially generalized by considering distinct PAM constellations for Sp and s^.

[0074] Succinctly, therefore

[0075] On the other hand, the probability mass function (PMF) of ekR is not trivial and is given by which can be alternatively described solely in terms of the corresponding activation indices kpand kpx, such that equation (20) can equivalently be written as

[0076] The challenge in equation (21 ) lies in the probabilities Pkp( / cp) which, far from being uniform and independent, is strongly correlated and differs significantly for each p, as a consequence of the combinatorial selection which yields finite and ordered index lists. To illustrate, consider the example shown in equation (5) for NT= 5 and P = 3, from which it can be seen that the first element k±has a higher probability of taking 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 k2and k3.

[0077] Fortunately, the analytical PMFs corresponding to the index vector elements kp,Yp, 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 t e T = {1, — ,NT} given by where

[0078] (p - 1) < t < (NT- P - p + 1). (23)

[0079] For each value p, the polynomial R^(t;NT~) gives the number of the mutually exclusive index t, out of a list of length P, confined to the interval T, such that the desired distributions are obtained by merely normalizing such polynomials, yielding Some examples of the discrete polynomial-based PMF given by equation (24) are compared to the true empirical distributions in figure 4, for various NTand P, which confirm that the expression is exact. The circles indicate the results of the proposed scheme, while the solid lines indicate the corresponding empirically-obtained true values of the activation indices of the index vectors in K.

[0080] For future convenience, the probability masses of the PMF is represented by the vector rpdefined as

[0081] In possession of the analytical prior PMFs of the indices kpgiven in equations (19) 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 are known Vp, 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 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.

[0082] First, the soft-replicas for the activation vector variables ekR and with p e {1, ••• , / ’} are defined for the n-th factor node, i.e., n-th pilot, with n e {1, — ,2NR}, as ekR.nand ekpi .n, respectively. The corresponding error covariance matrices of the activation vector soft-replicas are defined as where the expectation is taken over the corresponding unique PMFs of equation (25), for each p e {1, •••, ?}.

[0083] It can be seen from equation (26) and the system analysis provided further above 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 (-)Rreplaced by (-)1.

[0084] Computed naively, the covariances in equation (26) require a summation of NTouter products of dimensions NTx NT, with complexity O(N^). Thanks to the UVD and the sparsity of the unit vectors in S, however, equation (26) can be put into a closed-form of associated complexity O(N^), where the NTx NTsquare matrix ERnis given by with rp(t) denoting the t-th element of the vector of probability mass coefficients of rpfrom equation (26).

[0085] In hand of the soft-replica vectors and the error covariance matrices, the 2NRfactor nodes corresponding to the received symbols in y, each perform soft-interference cancellation (IC) on the received symbols ynfor the activation vectors as where (hR)Te IRlxW7,and (h^)Te IRlxW7,respectively denote the n-th rows of the channel components HRe ]R>2WTXWTE^2NTXNT whjCh are defined in equation (7).

[0086] As can be seen in equation (30) the soft-IC symbol is comprised of the true symbol part and a term related to the 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 where the conditional variance vRnis obtained via with the total variance vnprior to the soft-IC

[0087] The conditional PDFs of equation (31 ) are combined at the variable nodes to obtain the extrinsic belief bRn, i.e. , where self-interference cancellation (IC) is included for the computation of the extrinsic belief for the n-th factor node, as can be seen by the exclusion of the n-th conditional PDF from the variable node.

[0088] The resulting unsealed multivariate Gaussian PDF of the extrinsic beliefs are efficiently described in terms of the information vector and in terms of the precision matrix given by

[0089] In turn, the posterior Bayes-optimal soft-replicas are computed from the extrinsic beliefs via

[0090] Similar to equation (26), the expectation of ekpover its domain 8 can be efficiently described in a closed-form expression with reduced computational complexity, namely where the vector of unnormalized belief mass z£.nis given by with expG(•) denoting the element-wise exponentiation of a vector, Tjp.n^ denoting the elements at the t-th position of the vector i]p.n, and ARn(tdenoting the elements at the (t, t)-th postion of the matrix ARn.

[0091] 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 computational complexity, given by diag ( where |-|Gdenotes the element-wise absolute value operator.

[0092] Equations (29) to (40) describe the steps of one MP iteration to estimate the 2P activation vectors of the piloted GQSM with UVD, which yields the refined posterior soft-replica vectors and the corresponding error covariance matrices.

[0093] At the end of a r-th MP iteration, the soft-replica vectors and error covariance matrices are damped in order to prevent early convergence to a local optimum, thus where p e [0,1] is the damping factor, and the superscript (-)M denotes the r-th iterate of the variable. The last of the MP iterations, denoted by the index rconv, can be determined by well- known convergence criteria, such as the maximum number of iterations, i.e. , while given convergence threshold of the soft-replicas, i.e., while > £Tlh ■ After the last MP iteration, a belief consensus is taken over the n’

[0094] 2NRfactor nodes via which is equivalent to equation (34) without the self-IC.

[0095] The consensus information vectors and precision matrix diagonals are respectively given by

[0096] Finally, a hard-decision on the 2P activation vectors is made by evaluating the consensus PDFs for all NTvalid states of the activation vectors, i.e., which is equivalent to computing the consensus soft-replicas where and determining the index of the maximum value of the consensus soft-replica, i.e., fc? = argmax tET where ekR(t) denote the t-th element of the consensus softreplica vector ek«.

[0097] 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 steps in the decoder, with the operations on the factor nodes and the variable nodes shown in the left and right dashed box, respectively.

[0098] Inputs to the process 100 are the received signal y, the effective channel matrices HRand H1, pilot symbols spand sf, Yp, and the noise variance No. Outputs of the process are the estimated activation vectors ekR and Yp.

[0099] In step 105 the soft-replicas ekR.nand are initialised Yn and Vp, following the PMF determined via equation (25). In step 110 the error covariance matrices rp\ and of the activation vector soft-replicas are computed via equation (27). This marks the beginning of the iterative part of the process 100.

[0100] In step 115 a soft-IC is performed to obtain the real and imaginary components yp nand yplnvia equation (29). In step 120 the conditional variances v*nand v^.nare computed via equation (32), and in step 125 the information vectors y£:nand rfp.nrepresenting the unsealed multivariate Gaussian PDF of the extrinsic beliefs are computed via equation (35). Next, in step 130, the diagonal vectors of the precision matrices, diag (Ap.n) and diag (Ap.n), are computed via equation (40). In step 135 the respective expectations of the activation vectors, ekR.nand Ski n, are computed via equation (38), before the corresponding error covariance matrices of the activation vector soft-replicas, rp\ and Tp.n, are computed via equation (27) in step 140. In step 145 the expectations of the activation vectors, ekR.nand ekpi .n, and the corresponding error covariance matrices of the activation vector soft-replicas, rp\ and rp.n, are updated via damping using equation (41 ). Step 150 checks if a 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 ekR.nand ekpi .nare selected and provided to a belief consensus stage.

[0101] The termination criterion can include, for example, a predetermined number of iterations, or a convergence of the -optimal soft-replicas ekR.nand ekpi .n, 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 ekR.nand ekilies below the predetermined value.

[0102] Establishing the belief consensus comprises, Vp, in step 155, obtaining and if? via equation (43), and obtaining, in step 160, the precision matrix diagonals diag(AR) and diag via equation (44). The final activation vector estimates ekR are obtained via equation (45) in step 165.

[0103] In the following section the fundamental limitations and the consequent sources of errors of the decoder discussed above are identified, which mainly arise from the omission of the joint probability information between the unit vectors of the UVD. Subsequently, enhancements to the method are proposed to improve the detection 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 (-):nare omitted here, and only the real part variables (-)Rare described, as the same analysis applies to the imaginary part variables (-)1.

[0104] Consider two soft-replicas eRpand eR, of two unique unit vectors eRpand eR, with p p', and the respective transmit symbols and sR. Correct estimation results in the convergence of the effective symbol • eRpto the true value of • eRand illustrated in figure 7 a).

[0105] 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 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.

[0106] 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 proposed in the following.

[0107] Assume that multiple transmit symbols have high similarity, i.e. , |sR- sR| « 0. Such behavior is illustrated by observing Fig. 4 b and considering s? « sR, where the initialized effective symbols s? • eRpand s?, ■ eR, are consequently very similar in value. Therefore, the effective symbols appearing in the core MP steps of the UVD-GaBP decoder via equations (28)-(36), may converge to either of the true symbols • eRpor s?, • eR,, and potentially produce an errorneous duplicate estimate.

[0108] The first proposed approach to mitigate the duplicate convergence error is to modify the transmit symbol constellation, such that a sufficient difference in numerical value are ensured for the symbols in the respective real and imaginary domains.

[0109] To that end, consider the IQ-decoupled constellations ZR9i{Lc} and Z13{SC}, 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 Sc, the PAM constellations have a smaller cardinality than M, as multiple symbols in Zcmay convey the same amplitude in the IQ domain, leading to duplicate symbol values.

[0110] In order to avoid such duplicate symbols in the IQ domain, a simple rotation scheme can be applied to Lc, which leads to non-duplicate PAM symbols in both IQ domains. Furthermore, an optimal rotation angle 0* is obtained to maximize the minimum Euclidean distance between the symbols of the PAM constellation. There exists an optimal rotation angle 0* for each of the quadrants, but here only the rotation angle within the first quadrant is considered, i.e. , 0 e as shown in Table I for some sizes of the constellation.

[0111] The optimization problem is described by e' = argmax

[0112] »<».?) where Dmin(-) is a function on a set denoting the minimum Euclidean distance of its elements defined as

[0113] It can be observed that the optimal rotation illustrated in figure 8 b) is effectively equivalent to fitting a QAM constellation from M unique IQ-orthogonal symbols from the a larger scaled QAM constellation to maximize the minimum Euclidean distances between the symbols, as illustrated in figure 8 c).

[0114] 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. In addition to the optimal rotation of the symbol constellation which is pre-computed at the GQSM transmitter, the present invention as discussed hereafter proposes to enhance the detector introduced above. 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 PMFs Pefcp(ekp= et) of the Bayes-optimal soft-replica denoiser in equation (37) may be enhanced to first-order conditional PMFs Pekpi = t' . In other words, the first-order conditional PMFs yield the activation probability of the t-th index of the p-th unit vector, given the activation information of t '-th index of the p ' -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 that only the first-order conditional PMF is considered herein for practicality.

[0115] 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 greedy hard- determination of the most confident index pair (p, t) is made for each n-th factor node by evaluating equation (34), and finding the index pair yielding the maximal belief mass, i.e. , by solving argmax P(b£nI et) (51) p,t which is equivalent to obtaining the indices of the maximum values in the belief mass vector Zp.nin equation (39).

[0116] Under such greedy selection, the conditional PMF-based denoiser with the greedy indices Pn,tn, becomes to yield the soft-replica vectors incorporating the dependent information of the most confident unit vector variable.

[0117] 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. , system parameters NTand P, all values of the conditional PMF can be pre-computed.

[0118] Furthermore, a similar approach can also be performed after the convergence of the MP iterations to find the index of the strongest value within the converged beliefs in equation (42) pR, tR= argmax p,t which is equivalent to obtaining the indices of the maximum values in the belief masses zRin equation (47).

[0119] 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 as and the corresponding symbol and unit vector variables are nullified from the respective soft-replica, to be repeated for P successive IC iterations.

[0120] 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., kR= sort (p[R?-^[R]]) (55) denote the greedily selected activation index at the A-th iteration of the successive IC loop.

[0121] The steps of a method 200 incorporating the receiver-side enhancements proposed herein are presented below with reference to figure 9. Figure 10 shows a schematic block diagram of the method 200. Note that for P = 1, the enhancement is not necessary as there is only a single unit vector to evaluate, and the conditional 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 P = 1, as the duplicate convergence error cannot occur with a single unit vector variable.

[0122] Inputs to the process 200 are the received signal y, the effective channel matrices HRand H1, pilot symbols sRand sf, Yp, and the noise variance No. Outputs of the process are the estimated index vectors kRand k1.

[0123] In step 205 the soft-replicas ekR.nand are initialised Yn and Yp, following the PMF determined via equation (25). In step 210 the error covariance matrices rRnand Tp.nof the activation vector soft-replicas are computed via equation (27). This marks the beginning of the iterative part of the process 200.

[0124] In step 215 a soft-IC is performed to obtain the real and imaginary components yRnand ypl.nvia equation (29). In step 220 the conditional variances vRnand vp.nare computed via equation (32), and in step 225 the information vectors yRnand rfp.nrepresenting the unsealed multivariate Gaussian PDF of the extrinsic beliefs are computed via equation (35). Next, in step 230, the diagonal vectors of the precision matrices, diag and diag (Aj,.n), are computed via equation (40). In step 235 the conditional PMFs for pRand p^ are evaluated via equation (51 ) for finding the index pair yielding the maximal belief mass. In step 240 the expectations of the activation vectors, ekR.nand ekpi .n, are computed via equation (52), before the corresponding error covariance matrices of the activation vector soft-replicas, rp\ and rp.n, are computed via equation (27) in step 245. In step 250 the expectations of the activation vectors, ekR.nand and the corresponding error covariance matrices of the activation vector soft-replicas, rp\ and T^.n, 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 loop is repeated, from step 215, with the previously computed values as inputs. In the positive, “Yes”-branch of step 255, the optimal activation vector estimates ekR.nand ekpi .nare selected and provided to a belief consensus stage.

[0125] The termination criterion can include, for example, a predetermined number of iterations, or a convergence of the -optimal soft-replicas ekR.nand 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 ekR.nand ekpi .nlies below the predetermined value.

[0126] Establishing the belief consensus comprises, Vp, in step 260, obtaining and if? via equation (43), and obtaining, in step 265, the precision matrix diagonals diag(AR) and diag(Aj,) via equation (44). In step 270 the most confident index pair of unit vectors pRand p1and corresponding index activation information tRand Q within the converged beliefs are determined using equation (53), and stored, in step 275, for a X-th IC iteration. In step 280 IC is performed for obtaining y^ and yn' using 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 kRand k1are obtained via equation (55) in step 290 and output.

[0127] 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 NTtransmit antennas, NRreceive antennas, and P pilot symbols from the M-ary constellation, considering the three cases specified in

[0128] Table II below:

[0129] Table II

[0130] 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, however, in all cases considered.

[0131] 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, transmitted activation vectors.

[0132] 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 x 32 employing various numbers of symbols P. The BER plots, shown against the energy-per-bit-to-noise-power-spectral-density ratio (Eb / NQ~), indicate that the duplicate convergence of unit vectors, identified further above as a critical potential source of errors, does not occur for P = 1 but can be substantial for P > 1. The results also show, however, that the duplicate convergence behaviour is significantly mitigated by the further improvements to the underlying method 100 introduced above. The BER floor behaviour observed for P > 1 is, however, not only a consequence of duplicate convergence of unit vectors, but also a function of system size. To clarify, larger systems can sustain larger P values, without exhibiting error floors. In order to demonstrate that, consider the next set of results, given in figure 12, which show the performances achieved by method 200 in systems of size 64 x 64 and 96 x 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.

[0133] 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.

[0134] 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.

[0135] In order to enable a fair comparison between the proposed and conventional approaches, the total power and transmission rate BTrin 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 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 performance of the MUX-based system in figure 13.

[0136] 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. 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 X is evaluated in terms of number of floating point operations (FLOPs), which represents the cost of a single arithmetic operation of a real-valued scalar, and the complexity order O(-) representing the leading coefficient order of each system parameter variable.

[0137] 1) Brute-force ML Decoder:

[0138] In a brute-force ML detection, the likelihood metric for all possible codewords of the piloted GQSM codebook E is evaluated and the transmit signal with the minimum error is selected, i.e. ,

[0139] XML = argmin || y - Hx ||^. (56)

[0140] 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 codebook size (^T)2.

[0141] 2) IQ-decoupled Vector-valued GaBP Decoder:

[0142] 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 achieving a quadratic order complexity reduction by exploiting the half-sized 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 where T is the number of MP iterations.

[0143] 3) Proposed UVD-GaBP Decoder:

[0144] Following the proposed MP rules, the computational complexity of method 100 without enhancements, is obtained as

[0145] In contrast, the complexity of the enhanced UVD-GaBP decoder of method 200 is given by

[0146] 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 P,NT, and NR.

[0147] Figure 13 compares the full decoding complexity in FLOPs of the four analysed decoders, for increasing number of transmit antennas NTand varying values of symbols P. The number of receive antennas NRand the number of MP iterations T are fixed respectively at a reasonable value of NR= 64 and T = 100, as the two parameters only introduce a linear complexity order, as opposed to NTand P with more prominent effects, as can be seen in the summary of Table III below:

[0148] Table III - Computational complexity orders of the conventional GQSM decoders and the proposed UVD-GaBP decoders.

[0149] For P = 1, the combinatorial coefficient will reduce to NTand hence the ML decoder enjoys a significantly low complexity, however, for actual combinatorial IM codebooks even with P = 2 or P = 3, the complexity of the ML decoder and the IQ- VGaBP is shown to increase to an infeasible order with respect to NTand P, especially for mMIMO scenarios with NT> 16.

[0150] On the other hand, the proposed UVD-GaBP while requiring a higher complexity than that of the ML with small NTand P, a significant superiority can be observed for scenarios with larger system parameters of NTand P, where the rate of complexity gain with respect to increasing NTand P is shown to be very low, noting the logarithmic scale of the complexity plot.

[0151] 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 parametrized MIMO GQSM with NTand P, subject to the same decoding complexity cost.

[0152] The left subfigure first compares piloted GQSM systems requiring a decoding complexity in the order 2 x 109FLOPs, with the three selected scenarios: ML decoder with NT= 16, NR= 16, P = 3, the enhanced UVD-GaBP decoder of method 200 with NT= 24, NR= 24, and P = 3, and the UVD-GaBP decoder of method 100 with NT= 32, NR= 32, and P = 2. Similarly, the right subfigure compares piloted GQSM systems requiring a decoding complexity in the order 3 x IO10FLOPs, with: ML decoder with NT= 16, NR= 16, P = 4, the enhanced UVD-GaBP decoder (Alg. 2) with NT= 32, NR= 32, and P = 4, and also with NT= 48, NR= 48, and P = 3.

[0153] In light of the decreased complexity, the supported GQSM system of the UVD-GaBP decoder can be significantly scaled, especially larger-ordered, compared to the conventional ML decoder. The increased system parameters of NTand P inherently imply an increased spectral efficiency of the system, improving the BER performance with respect to the Eb / N0.

[0154] The Eb / N0gain is about 3 dB in the left figure under a decoding complexity of 2 x 109FLOPs, and the gain increases to 6 dB for higher complexity scales, shown in the right figure, under a decoding complexity of 3 x IO10FLOPs. This Eb / N0gain is expected 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.

[0155] 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 previously discussed.

[0156] In light of the foregoing description, in accordance with a first aspect of the present invention a method of decoding information represented by respective resource element activation patterns (REAP), formed, at a transmitter and for a present transmission instance, by activated subsets of transmit resource elements, TRE, out of a plurality of activatable TREs is presented. A TRE may be an antenna from a plurality of antennas, a subcarrier, a time slot, or the like, or combinations thereof. A subset of REAPs selected from all possible REAPs that can be formed from the plurality of activatable TREs is used at the transmitter for coding information to be transmitted. Each of the activated TREs transmits transmit symbols selected from a known set of symbols, with at least one transmit symbol transmitted by at least one TRE being a pilot symbol that is known at a receiver. The method comprises, at the receiver, receiving a receive signal comprising transmit symbols of all TREs activated for the present transmission instance, and calculating, based on the subset of REAPs known to be used at the transmitter for the present transmission instance, a probability for each TRE of having been activated. Based on at least one pilot symbol transmitted by at least one TRE, a channel matrix is estimated for at least the communication channel between the transmitter and the receiver comprising the at least one TRE, or such information is received. Based on the receive signal, the channel matrix and the probability for each TRE of having been activated, estimates for at least the REAP for the present transmission instance are calculated in a first iterative process while a termination criterion is not met. A belief consensus process is performed on the previously determined estimates and the corresponding error covariance matrices, targeted to yield the most confident REAP. The results are stored for a current repetition index of a second iterative process. An interference cancellation is performed at least on the received transmit symbols associated with the most confident TRE in the most confident REAP, yielding an estimated interference-free receive signal. In the second iterative process the first iterative determination process is repeated for all REAPs from the subset of REAPs, using the estimated interference-free receive signal as input. Finally, the REAP to be output is obtained from the all stored results of the second iterative process, and output. Alternatively, the information represented by the REAP may be output after corresponding decoding. The known subset of REAPs and / or the pilot symbols may be stored in a memory of the receiver.

[0157] In one or more embodiments of the method calculating the probability for each TRE of having been activated comprises applying a Bayesian inference method.

[0158] In one or more embodiments of the method determining comprises applying an iterative Gaussian belief propagation, GaBP, process.

[0159] In one or more embodiments of the method determining comprises computing denoised soft-replicas and covariance matrices based on a probability mass function, PMF, associated with the estimates and the corresponding error covariance matrices for at least the REAP for the present transmission instance. In one or more embodiments of the method determining comprises determining the most confident TRE within a REAP, at least for each REAP from the subset of REAPS.

[0160] In one or more embodiments of the method the TREs transmit quadrature-modulated signal components of the transmit symbols, and determining comprises representing the respective real and imaginary signal components of the received signal vector as respective products of a transmitted symbol component and an activation unit vector, applying an iterative GaBP-based MP process for jointly estimating the REAPs for the respective real and imaginary parts of the received signal vector, using the probability for each TRE of having been activated as initial activation unit vector, and decoding REAPs output by the iterative GaBP process for retrieving the transmitted information.

[0161] In accordance with a second aspect of the present invention a receiver configured for receiving index-modulated transmit symbols is presented. The receiver comprises one or multiple antennas, associated circuitry for processing radio frequency signals, one or more microprocessors, and associated volatile and non-volatile memory. The 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 receiver to implement or carry out one or more embodiments of the method in accordance with the first aspect of the present invention.

[0162] In one or more embodiments the circuitry for processing radio frequency signals comprises a low noise amplifier and / or a mixer configured for providing a representation of a received signal at an intermediate frequency. The mixer preferably uses a same oscillator signal as a transmitter co-located with the receiver in the wireless apparatus. The latter may enable using signals transmitted by the entity comprising the receiver, which are reflected off objects, for environment perception. Two or more receivers according to the second aspect of the invention may form a communication system.

[0163] The receiver in accordance with the second aspect of the invention may be arranged in a vehicle.

[0164] 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, microcode, etc., or an embodiment combining software and hardware aspects.

[0165] 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 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.

[0166] 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 a wireless apparatus in accordance with the second 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 wireless apparatus.

[0167] Computer program instructions, or code, for carrying out operations for 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 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 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).

[0168] 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 by the computer by means of a corresponding receiver, and that is transferred to and stored in a memory of the computer.

[0169] The described features, structures, or characteristics of the embodiments may be combined in any suitable manner. In this description, numerous specific details are 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 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 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. 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 more” unless expressly specified otherwise.

[0170] 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 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, 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.

[0171] It should be noted that, in some implementations or embodiments, the functions 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 to one or more blocks, or portions thereof, shown in the figures.

[0172] 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 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 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 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 via a "classic multiplex paradigm," in which symbols and pilots are transmitted simultaneously, for the purpose of communication and sensing, respectively.

[0173] Contrary to conventional decoders, which assume a uniform distribution of the activation vectors at the transmitter, the method proposed herein considers a scalar, 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.

[0174] 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 (loT) communications, e.g., in wireless factory and other industrial settings, extra-large scale MIMO and OFDM systems, and even in underwater acoustic communications.

[0175] 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 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 etc., and any IM-based modulation.

[0176] 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 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.

[0177] 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 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 implement or carry out embodiments of the method in accordance with the first aspect pf the invention as described herein.

Claims

CLAIMS1. A method (200) of decoding information represented by respective resource element activation patterns, REAP, formed, at a transmitter and for a present transmission instance, by activated subsets of transmit resource elements, TRE, out of a plurality of activatable TREs, a subset of REAPs selected from all possible REAPs that can be formed from the plurality of activatable TREs being used at the transmitter for coding information to be transmitted, each of the activated TREs transmitting transmit symbols (x) selected from a known set of symbols, at least one transmit symbol (x) transmitted by at least one TRE being a pilot symbol that is known at a receiver (400), the method comprising, at the receiver (400):- receiving a receive signal (y) comprising transmit symbols (x) of all TREs activated for the present transmission instance,- computing (210), based on the subset of REAPs known to be used at the transmitter for the present transmission instance, a probability for each TRE of having been activated,- estimating, based on at least one pilot symbol transmitted by at least one TRE, a channel matrix (H) for at least the communication channel between the transmitter and the receiver comprising the at least one TRE, or receiving such information,- determining (215-255), in a first iterative process while a first termination criterion is not met, and based on the receive signal (y), the channel matrix (H) and the probability for each TRE of having been activated, estimates (ekR.n, eki,.n) and the corresponding error covariance matrices (rp\, Tp.n) for at least the REAP for the present transmission instance,- performing (260-270) a belief consensus process on the previously determined estimates (ekR.n, eki,.n) and the corresponding error covariance matrices (r^n, rp.n), targeted to yield the most confident REAP, and storing (275) the results for a current repetition index of a second iterative process,- performing (280) an interference cancellation, IC, process at least on the received transmit symbols associated with the most confident TRE in the most confident REAP, yielding an estimated interference-free receive signal,- repeating (285) in the second iterative process, the first iterativedetermination process (215-255) for all REAPs from the subset of REAPs, using the estimated interference-free receive signal as input,- obtaining (290) the REAP from the all stored results of the second iterative process, and- outputting the REAP or the information represented by the REAP.

2. The method (200) of claim 1 , wherein computing (210) the probability for each TRE of having been activated comprises applying a Bayesian inference method.

3. The method (200) of claim 1 or 2, wherein determining (115-150) comprises applying an iterative Gaussian belief propagation, GaBP, process.

4. The method (200) of claim 3, wherein determining (215-255) comprises computing denoised soft-replicas and covariance matrices based on a probability mass function, PMF, associated with the estimates (ekR.n, eki n) and the corresponding error covariance matrices (r£n, r^.n) for at least the REAP for the present transmission instance.

5. The method (200) of any one or more of claims 1 to 4, wherein determining (215-255) comprises determining the most confident TRE within a REAP, at least for each REAP from the subset of REAPs.

6. The method (200) of one or more of the preceding claims, wherein the activatable TREs are antennas of a multi-antenna transmitter, subchannels, and / or time slots.

7. The method (200) of any one or more of claims 1 to 3, wherein the TREs transmit quadrature-modulated signal components of the transmit symbols, and wherein determining (215-255) comprises:- representing the respective real and imaginary signal components of the received signal vector (y) as respective products of a transmitted symbol component (sp) and an activation unit vector (et),- applying an iterative Gaussian belief propagation, GaBP, -based messagepassing, MP, process for jointly estimating the REAPs for the respective real and imaginary parts of the received signal vector (y), using the probability for each TRE of having been activated as initial activation unit vector, and- decoding REAPs output by the iterative GaBP process for retrieving the transmitted information.

8. Receiver (400) configured for receiving index-modulated transmit symbols, the receiver (400) comprising one or more antennas (402), circuitry (404) for processing radio frequency signals, 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 microprocessors (406), configure components of the receiver (400) to implement or carry out a method of any one of the preceding claims 1 to 7.

9. The receiver (400) of claim 8, wherein the circuitry (404) for processing radio frequency signals comprises a low noise amplifier and / or a mixer configured for providing a representation of a received signal at an intermediate frequency.

10. A communication system comprising two or more receivers (400) according to claim 8 or 9.

11. A vehicle comprising a receiver (400) according to claim 8 or 9.

12. Computer program product comprising computer program instructions which, when executed by a microprocessor of a receiver (400) according to claim 8 or 9, cause the receiver (400) and / or control hardware blocks, modules or components of the receiver (400), respectively, to execute the method of one or more of claims 1 to 7.

13. Computer readable medium or data carrier retrievably transmitting or storing the computer program product of claim 12.

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