Methods for optimal and scalable quadrature space-time modulation

Abstract

A computer-implemented method of scalable space-time quadrature modulation for multidimensional wireless systems, comprising configuring multiple transmit antennas to respectively represent in-phase spatial constellation symbols in an in-phase spatial constellation and quadrature-phase spatial constellation symbols in a quadrature-phase spatial constellation, and mapping source data to the in-phase and quadrature-phase spatial constellation symbols represented by the multiple transmit antennas, the method constructing sets with equal multiplicity of transmit antenna activation, thereby ensuring maximum possible transmit diversity and reducing the required complexity, making this feasible for a larger number of antennas.

Description

[Technical Field]

[0001] The present invention relates to the field of decoding digital communications in overloaded channels. [Background technology]

[0002] In advanced spatial modulation (SM) schemes, only a subset of the transmit antennas are activated per symbol slot via a covariance matrix that determines the activation pattern. This selection is determined by a so-called index vector, which holds integer values ​​corresponding to the indices of the covariance matrices that are simultaneously activated. Until state-of-the-art (SotA) techniques, the construction of the aggregate index vector was simply performed by the combination order, which leads to unequal allocation of transmit antennas and thus to a loss of transmit diversity.

[0003] Spatial modulation (SM) technique is a well-known technique for MIMO systems, to which the present invention introduces a novel way to transmit additional information bits via transmit antenna selection (spatial dimension) in addition to the traditional information bits via symbol selection (constellation dimension). From traditional SM, which only allows selection of one antenna and therefore limits the total number of antennas to a power of two, generalized SM has been developed to allow any total number of antennas by selecting a combination of multiple antennas from the available transmit antennas.

[0004] And by combining the concept of space-time code (STC) with SM-MIMO, transmit diversity is introduced into SM, adding the time dimension by allowing multiple symbol periods in the code design. Important works include the space-time shift keying (STSK) scheme based on linear dispersion (LD) codes by Sugiura et al., and space-time block coding (STBC)-based schemes such as STBC-SM and STBC-CSM.

[0005] In parallel, a contrasting technique for improving transmit diversity has been introduced by using quadrature spatial modulation (QSM) schemes, in which the ideas of conventional SM are applied separately to the real and complex components of the transmitted signal.

[0006] This idea is complemented by increasing transmit diversity in diversity-achieving quadrature spatial modulation (DA-QSM). Finally, more recently, Enhanced Diversity-Achieving Quadrature Spatial Modulation (EDA-QSM) incorporates ideas from STBC-SM into DA-QSM and has been shown to outperform all its predecessors.

[0007] The EDA-QSM method for SotA has the following limitations: - Coding gain is not optimal The maximum allowed symbol period is limited to two -The number of transmitting antennas is limited to an even number Suboptimal spectral efficiency

[0008] This approach allows for efficient utilization of the large stationary sources associated with massive MIMO systems, but comes with the drawback of requiring a similarly large number of radio frequency (RF) chain components. This efficient utilization of the RF chain makes SM attractive for future wireless systems, such as Beyond 5G (B5G), which will continue to make extensive use of millimeter wave (mmWave) bands, and sixth-generation (6G) networks, which are expected to also incorporate terahertz and visible light communication (VLC) [6].

[0009] However, a major drawback of early SM schemes was that only one antenna was selected per transmission, severely limiting the achievable SE. To circumvent this limitation, generalized spatial modulation (GSM) schemes were later developed, in which multiple antennas were selected for each transmission, leading to significant improvements in SE. However, another drawback of early SM methods, including GSM, was that they focused on increasing SE without a corresponding effort to reduce BER, for example, by utilizing transmit diversity. This limitation motivated the idea of ​​combining SM with space-time coding (STC). Examples include space-time shift keying (STSK) based on linear dispersion (LD) coding, methods incorporating space-time block coding (STBC), and spatial modulation with cyclic structure (CSM).

[0010] Based on this knowledge, further optimization of SM transmitter design led to the discovery of the quadrature spatial modulation (QSM) technique, in which the SM concept is applied independently to the real and imaginary components of the modulated signal via dedicated space-time dispersion matrices. This idea has been further developed in a series of QSM techniques using progressively improved dispersion matrix designs, including the diversity-enabled quadrature spatial modulation (DA-QSM) scheme incorporating Alamouti codes, and the more recent extended diversity-enabled quadrature spatial modulation (EDA-QSM) method, in which the dispersion matrix is ​​constructed using full-diversity full-rate (FDFR) codes with block-wise sphere decodability

[24] . Among all the above schemes, EDA-QSM is the currently known best QSM scheme in terms of both BER and SE performance.

[0011] Despite these advantages, the EDA-QSM scheme, and consequently, the prior QSM schemes, still have two main drawbacks. First, the covariance matrix used in the QSM schemes proposed so far is based on a 2x2 STBC, which limits both the diversity and coding gain achieved by the method. Regarding this first limitation, TWe demonstrate herein that QSM designs based on STBCs with size T not proportional to are fundamentally suboptimal in the SE sense. A second drawback is that current QSM detection schemes are based either on exhaustive maximum likelihood (ML) detectors or, at best, on sphere detectors. It is noteworthy here that, contrary to previous assertions, sphere decoding in fact still has an average complexity that grows exponentially with the number of simultaneously decoded symbol periods. This result is supported by several findings, where a cubic closed-form expression for the expected complexity of a sphere detector is derived, and where it is shown that lattice reduction does not improve the tail exponents of the sphere detector's complexity distribution. Regarding this second limitation, we demonstrate in practice that the complexity of both ML- and sphere detection (SD)-based QSM receivers scales exponentially with n, exponent P, such that these techniques are fundamentally not scalable with respect to QSM systems. T and T. In other words, the current QSM scheme has a serious and twofold scalability problem in that there are no scalable transmitter and receiver designs. Summary of the Invention [Means for solving the problem]

[0012] Motivated by this problem, in this application, we propose a scheme that is scalable to any block size, i.e., n T , T, and P are unconstrained, and the receiver can decode it in polynomial time, i.e., for medium P and large n TWe propose a novel QSM solution that is practical for and T. Furthermore, we find that the proposed QSM scheme has every possibility to optimize SE, diversity, and coding gain. To this end, we first introduce an optimal FDFR golden STBC code for the design of the QSM covariance matrix. The golden code is a fast-decodable STBC known to be optimal, i.e., FDFR with the highest coding gain for Gaussian constellations, and has been shown to be generally constructible for any block size. The resulting optimized scalable QSM (OS-QSM) scheme is the first proposed method to date with this feature.

[0013] The new OS-QSM design is further enhanced by a novel algorithm for selecting the indices of the covariance matrix employed in this scheme, which ensures optimal and diverse utilization of all space-time resources, as all transmit antennas are utilized with equal frequency and likelihood across multiple block transmissions.Finally, to ensure feasible decodability for a scalable transmitter design, a novel Greedy Boxed Iterative Shrink-Threshold Algorithm (GB-ISTA) QSM detector based on a sparse recovery method is proposed.

[0014] Thanks to its sparse signal processing approach, the proposed decoding scheme does not require any restrictions on the core code design, unlike prior sphere detection methods that require block-diagonal fast decodability. However, in addition, and most importantly, a major advantage of the new proposed GB-ISTA QSM receiver is that it does not require a large codebook space search, unlike ML and state codeword block-matched sphere decoding (SCMB-SD). In fact, the complexity order of the proposed receiver is cubic in T, quadratic in P, and n T It has been shown to be first order only at

[0015] In general, the proposals herein can be summarized as follows: · Spectral efficiency-optimality: A closed-form expression for the optimal number of coded symbols P required for QSM to achieve SE-optimality is given, which, combined with the rate-optimality condition for STBC, highlights the importance of systematic scalability of STBC size T in the design of SE-optimal QSM schemes. · Optimal diversity and coding gain: A new Golden Code-based Quadrature Phase Spatial Modulation (GQSM) transmission scheme is obtained through the design of a covariance matrix based on a 2x2 Golden Code, which is known to achieve optimal coding gain over integer symbol constellations. · Transmitter scalability: The new GQSM design is generalized by extending the 2x2 Golden Code to a TxT FDFR STBC variant, enabling it to scale for any n T , T, and P are applicable to the OS-QSM method. · Optimal resource utilization: Method 1 provides a new mechanism for selecting an optimal set of distribution matrix indices, which ensures that all Q space-time resources are utilized evenly over time, as required for optimal diversity gain. Scalability at the receiver: A new low-complexity greedy iterative reduced threshold algorithm (ISTA)-based demodulation algorithm for GSM systems is proposed, which is not only feasible on a larger scale due to its linear complexity, but also applicable to other STBC-QSM systems. Receiver complexity: T and n with P as the exponent T In contrast to the ML and SD receivers, which are exponential in T, P, and n, a new complexity expression for the proposed receiver is derived, which is cubic in T, quadratic in P, and n T It has been shown to be first order with respect to

[0016] In future wireless communication systems beyond 5G (B5G) and sixth generation (6G), one of the major expectations is that the number of transmit and receive antennas in a multiple-input multiple-output (MIMO) configuration will increase significantly to support the requirements.

[0017] The main problem or challenge with existing solutions arises when the system is extended to associated massive MIMO, which may make the SCMB-SD approach impractical due to the nature of the tree search algorithm.

[0018] In other words, as the number of antennas increases, i.e., in larger MIMO systems, spatial modulation becomes problematic / infeasible.

[0019] The proposed (Greedy ISTA decoding) method is designed to decode (Golden) Enhanced Diversity-Enabled Quadrature Phase Spatial Modulation (EDAQSM) signals for high-order MIMO systems with reasonable complexity.

[0020] Thus, spatial modulation is practically unfeasible when the number of antennas increases, i.e., in the case of larger MIMO systems. Therefore, its benefits may not be fully realized in practice. The proposed method significantly reduces the required complexity, making spatial modulation practically feasible with larger numbers of antennas (16x16 systems).

[0021] This invention will have a significant impact on mmW-based communications, a growing area of ​​advanced connectivity technology.

[0022] 1) Multiple Access Transmitter Design: In particular, thanks to the generalization of the Golden Code introduced here, it is possible to operate with highly sparse received signals in combination with spatial modulation, which allows overlapping of multiple users without causing significant multi-user interference.

[0023] 2) Multidimensional Modulation: Currently, the design is "only" in space and time, but it is intended to extend it to include the frequency domain as well. The result would be "tensor-based" modulation in frequency, space, and time. In architectures similar to those we have developed so far, the extra dimensions should only tend to make the per-user signal sparser, again allowing more users to overlap without increasing the effective multi-user interference.

[0024] 3) Establishing the connection by coding (transmitter side aspect). Coding is an aspect that we have not focused enough on so far in the context of this project. The main contributions have been on the performance and robustness of detection in overloaded systems (for practical problems). This work makes it possible to address detection and decoding simultaneously (joint optimization).

[0025] This, in simpler terms, enables communication systems with higher capacity and more users per unit of bandwidth, enabling ultra-high density communication scenarios.

[0026] These and other objects, features, and advantages of the present invention will become more apparent from a consideration of the drawings and detailed description.

[0027] This problem has not been addressed in any previous spatial modulation scheme, even in SotA, Enhanced Diversity-Enabled Quadrature Phase Spatial Modulation (EDA-QSM), and no solution has been found so far.

[0028] To address the above-mentioned issues and remove the limitations of the SotA scheme (EDA-QSM), an optimal and scalable quadrature spatial modulation scheme (OS-QSM) is proposed. Specifically, the design considers the highlighted sections of the QSM signal generation flowchart.

[0029] OS-QSM utilizes the Full Diversity Full Rate (FDFR) Golden STC and its generalization, the FDFR Full STC, to achieve optimal coding gain up to T = 6. These STCs are used instead of the Sezginer-Sari-Biglier (SSB) STBC used in EDA-QSM, and the necessary power adjustments and extensions are applied to construct the final dispersion matrix for QSM.

[0030] Specifically, the portion of the QSM signal generation flowchart / block diagram in Figure 1 highlighted by the dotted lines represents a solution to the above problem.

[0031] OS-QSM can be used in any system and situation, replacing the previous SM approach. Systems that can only support a small number of RF chains compared to the number of antennas should particularly benefit from SM's characteristics. Cellular networks and V2X communications can run with better efficiency and data rates (eMBB).

[0032] Complex matrices and vectors are denoted by bold uppercase and lowercase letters, and their elements are denoted by regular lowercase letters with subscripts, denoted X, x, and x, respectively. i The real and imaginary parts of a complex number x are expressed as x R and x I For future convenience, we will use the complex vector X=[x1,x2,...,x n ] T For each, the associated decomposed vector

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[0033] All previous state-of-the-art spatial modulation (SM) schemes have decoded SM signals using ML, as in most spatial modulation (SM) schemes, or using modified tree search algorithms, as in the EDA-QSM state-of-the-art SM scheme, without any notable attempt at sparse detection or greedy techniques.

[0034] Tree search algorithms are highly impractical due to the highly combinatorial nature of SM and the resulting search space size. Furthermore, SM imposes various restrictions on the structure of the signal to be decoded, such that simple low-complexity decoding is not possible.

[0035] A problem associated with the prior art is that there is no sparse detection solution to this problem, and solutions using other techniques are characterized by prohibitive complexity.

[0036] Massive multiple-input multiple-output (MIMO) systems in Beyond 5G (B5G) and sixth generation (6G) wireless communications are expected to incorporate a large number of transmit and receive antennas.

[0037] The use of spatial modulation (SM) and its variants, such as quadrature spatial modulation (QSM), is one of the promising candidates for massive MIMO systems. However, as the system size increases, the classical decoding complexity of SM schemes becomes infeasible (i.e., the complexity becomes intractable) because SotA uses a maximum likelihood (ML) decoding algorithm or an ML-based tree search algorithm.

[0038] The method constructs a set of equal multiplicity transmit antenna activations, as shown in Figure 5, which ensures the maximum possible transmit diversity. This was not the case in previous SotA approaches, where there was no equal multiplicity and therefore non-optimal transmit diversity was utilized from the number of available antennas, i.e., all antennas were used equally).

[0039] All existing spatial modulation schemes benefit, as they all already utilize index vector selection (or can be reformulated to do so), which can be used to improve data rates and increase efficiency in cellular networks and V2X communications (eMBB).

[0040] These and other objects, features, and advantages of the present invention will become more apparent from a consideration of the drawings and detailed description.

[0041] The proposed decoder has significantly lower complexity than existing ML and tree search algorithms. The decoder has quadratic complexity with respect to the number of transmit antennas.

[0042] Furthermore, the proposed decoder also provides greater flexibility in transmitter development, since it does not require design constraints such as block diagonal or orthogonal properties in the transmission scheme, but only the sparsity inherent in the spatial modulation scheme. In other words, the proposed decoder can accommodate many different coding structures (flexibility).

[0043] The decoder can be used in any MIMO system where the number of antennas is expected to be so large that ML techniques are infeasible.

[0044] This scheme can be used in cellular networks and V2X communications to increase efficiency (eMBB).

[0045] These and other objects, features, and advantages of the present invention will become more apparent from a consideration of the drawings and detailed description.

[0046] 1. An embodiment of an optimal and scalable quadrature space-time modulation (OS-QSM) method that configures multiple transmit antennas to respectively represent in-phase and quadrature space constellation symbols in an in-phase and quadrature space constellation, and maps source data to the in-phase and quadrature space constellation symbols represented by the multiple transmit antennas, wherein the method applies an optimal and scalable quadrature space-time modulation scheme (OS-QSM) that provides the maximum possible coding gain for the resulting quadrature space-time modulation.

[0047] Another embodiment of a computer-implemented decoding method features a modification to the Iterative Reduced Threshold Algorithm (ISTA) via boxing with range restriction and hard thresholding.

[0048] Another embodiment of the computer-implemented decoding method features an Iterative Shrinking Threshold Algorithm (ISTA) via Boxing Hardening, greedy selection of antenna position indices and symbol estimates, and their independent decoding of corresponding antenna modulation bits and symbol modulation bits.

[0049] Another embodiment of the computer-implemented decoding method includes a process that runs in parallel with the greedy selection to ensure that a valid estimate of an index vector from a given finite set of index vectors is produced as output, and to apply interference cancellation using the identified values: Check before every iteration whether a final confirmation can be computed from the currently decoded index, while keeping track of which index was obtained from the greedy selection; If final confirmation is not possible, the interference from the preceding greedy selection is removed and the next iteration is performed.

[0050] Another embodiment features a generalization of the Golden Code and its combined spatial modulation and operational computation on highly sparse received signals without large scale multi-user interference at multiple user overlaps.

[0051] Another embodiment features a sparser signal per user and still allows more users to overlap without increasing the effective multi-user interference.

[0052] Another embodiment features a receiver (R) for a communication system having a processor, a volatile and / or non-volatile memory, and at least one interface adapted to receive signals on a communication channel, wherein the non-volatile memory stores computer program instructions that, when executed by the microprocessor, configure the receiver to implement the decoding method of one or more of the above embodiments.

[0053] Another embodiment features a computer program product including computer-executable instructions that, when executed on a computer, cause the computer to perform the decoding method of one or more of the above embodiments.

[0054] Another embodiment features a computer-readable medium that stores and / or transmits the above computer program product.

[0055] Another embodiment features a vehicle unit comprising a communication system having a receiver (R) in the vehicle, the system being adapted to perform the decoding method of one or more of the above embodiments.

[0056] Another embodiment features a vehicle having one or more of the vehicle units described above.

[0057] All aspects of this application may be incorporated into mobile devices, base stations, and components of wireless systems. All components described may be incorporated into vehicles.

[0058] BRIEF DESCRIPTION OF THE DRAWINGS For a fuller understanding of the nature of the present invention, reference should be made to the following detailed description which should be read in conjunction with the accompanying drawings.

[0059] A quadrature spatial modulation (QSM) scheme is considered, which distributes n T A large number of bits can be carried while combining transmitting a relatively small number of all P M-ary modulation symbols from the dynamic selection of bulk transmissions. [Brief explanation of the drawings]

[0060] [Figure 1] 1 is a schematic diagram showing the general structure of a QSM transmission scheme; [Figure 2] 1. The spectral efficiency of the OS-QSM scheme for T=2, 4, and 8 in a given system with nT=8 and M=4. [Figure 3] The influence of T and M on the optimal ratio P* / T between the number of transmitted symbols and the number of epochs. [Figure 4] 10 shows the behavior of fractional peak spectral efficiency as a function of nT for different sizes of T and M. [Figure 5] 1 is a bipartite graph representing the spatiotemporal resource usage associated with each index vector kn in a QSM system with P=3 and Q=8. Specific examples of k1, k37, and k54 are explicitly shown. [Figure 6] This is a comparison between the ISTA threshold function Λ(s;τ) and the BH-ISTA threshold function Π(s;τ) according to Eq. (29). [Figure 7]Figure 7 shows the convergence of ûΠ(η) and ûΛ(η) from (28) and (30), respectively, as a function of the number of iterations η. Figure 7a shows the sparsity convergence with various thresholds. Figure 7b shows the MSE convergence with the optimal threshold. [Figure 8] FIG. 1 is a schematic diagram showing the proposed GB-ISTA receiver architecture for QSM demodulation. [Figure 9] Figure 9 shows the effect of scalable parameters on the complexity of a QSM receiver. Figure 9a shows fixed P and various T as a function of nT. Figure 9b shows fixed T and various P as a function of nT. Figure 9c shows fixed nT and various T as a function of P. [Figure 10] This is the impact of scalability on the BER performance of the GB-ISTA detection OS-QSM method with a fixed SE. [Figure 11] The effect of scaling P on the BER performance of the GB-ISTA detection OS-QSM scheme. [Figure 12] This is a comparison of previous spatial modulation methods. [Figure 13] This is a comparison of the golden EDA-QSM and the original EDA-QSM with ML decoding. [Figure 14] Comparison of Golden EDA-QSM vs. Original EDA-QSM by theoretical ABEP. [Figure 15] The effect of nT on the spectral efficiency of EDA-QSM. [Figure 16] The effect of nT of P on the spectral efficiency of EDA-QSM. [Figure 17] This is the impact on the spectral efficiency of EDA-QSM in the GISTA feasible region. [Figure 18] This is the average number of GISTA iterations by GISTA decoding. [Figure 19] FIG. 10 is a diagram of the decoding complexity between the ML decoder and the GISTA decoder. [Figure 20] Performance comparison between GISTA decoder and full ML decoder (nT 4). [Figure 21] This is a comparison of GISTA decoding performance. DETAILED DESCRIPTION OF THE INVENTION

[0061] The following is a detailed description of the concepts, system / network architecture, and detailed design of many aspects of a wireless communication network intended to address 5G requirements and use cases. It should be understood that the terms "requirement," "need," or similar language describe desirable features or functionality of the system in the context of advantageous designs for particular embodiments, and do not indicate essential or critical elements for all embodiments. Therefore, each requirement and each feature, although described below as essential, critical, needed, or similar language, should be understood to be optional.

[0062] In the following discussion, this wireless communication network, including radio devices, radio access networks, and core networks, will be referred to as "NX." It should be understood that the term "NX" is used herein merely as a label for convenience. Implementations of wireless devices, radio network equipment, network nodes, and networks that include some or all of the features detailed herein may, of course, be referred to by any of a variety of names. For example, future 5G specification developments may use the terms "New Radio," or "NR," or "NR multiplex mode," and it will be understood that some or all of the features described herein with respect to NX may be directly applicable to these NR specifications. Similarly, while various technologies and features described herein are directed to "5G" wireless communication networks, specific implementations of wireless devices, radio network equipment, network nodes, and networks that include some or all of the features detailed herein may or may not be referred to by the term "5G." The present invention relates not only to all individual aspects of NX, but also to developments in other technologies, such as LTE, in interworking and interacting with NX. Moreover, such individual aspects and such individual developments constitute separable embodiments of the present invention.

[0063] Figure 1 shows a schematic diagram illustrating the general structure of the QSM transmission scheme.

[0064] A. System Model n T A transmitter with n transmit antennas employs SM. R Consider a point-to-point (P2P) MIMO communication system exchanging information with a receiver equipped with receive antennas. The received signals corresponding to T consecutive time slots, where the channel is assumed to be constant, can be compactly written as

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[0065] In the above equation,

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[0066] In what follows, we assume that the quasi-static Rayleigh fading channel matrix H is known at the receiver but not at the transmitter. Also, note that the channel power per matrix element is unitary, so the basic signal-to-noise ratio (SNR) is given by

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[0067] Regarding equation (2) and again referring to FIG. 1, it is clear that in the QSM scheme, the bit string b is of length

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[0068] From the above, the design of a particular QSM scheme essentially involves the selection of Q covariance matrices A in sets A and B. q and B q The index vector k indicates the choice of the method adopted in the construction of each of the R and k I It can be said that the selection of a set K containing

[0069] To illustrate how a prior art (SotA) QSM scheme can be transformed into the general framework described in equation (2), we first consider a QSM scheme. In this case, the variance matrix (i.e., T=1, Q=n T ) is converted into a variance vector, which is given by A q =e q and B q =je q (3) In the above equation, e q I Q is the qth column of the index vector k R and k I No specific design criteria are given for the selection of the index.

[0070] Next, in the DA-QSM system, in order to utilize transmit diversity, a two-column covariance matrix (i.e., T=2, Q=n T ) is adopted. In particular, in this method,

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[0071] From the above, we can see that the DA-QSM method essentially improves over the QSM method by increasing diversity, i.e., by extending the transmission instance from T = 1 to T = 2. However, the covariance matrix of the DA-QSM method is still real, just like the QSM method, which suggests that the additional multiplexing function is not aggregated and the coding gain is not optimized.

[0072] In contrast, the EDA-QSM method improves on the latter in both aspects. In particular, in this scheme, the dispersion matrix is ​​more elaborately designed as follows:

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[0073] Through the above brief description, it can be easily understood that the fundamental difference between the DA-QSM method and the EDA-QSM method is that the dispersion matrix of EDA-QSM is complex-valued, and the orthogonality between the real and imaginary dimensions is better utilized to obtain multiplexing and coding gain.

[0074] However, two fair criticisms can be made about the above scheme, and indeed, to the best of our knowledge, about all existing QSM methods for SotA proposed so far: a) the scheme does not scale systematically across space and time simultaneously for any T > 2, and b) the achieved coding gain is not optimal. Alleviating these two limitations is the aim of our first contribution, described in the next section.

[0075] Optimized and scalable quadrature spatial modulation transmitter design A. Optimal spectral efficiency of QSM systems Given the number of bits carried by the transmission of each QSM transmission symbol X, as in equation (2), and the fact that such a transmission requires the use of T consecutive channels, SEζ for any QSM scheme is given by:

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[0076] Figure 2 shows the T 1 shows the spectral efficiency of the OS-QSM scheme for T=2, 4, and 8 in a given system with T=8 and M=4.

[0077] The binomial coefficient in equation (8)

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[0078] Motivated by the above discussion, we seek an analytical expression for the optimal ratio P / T that maximizes the SE when n T and M are given. This can then be used to determine the relative SE reduction that occurs when setting T < nT in a large-scale system with n T → ∞. To this end, we consider the upper and lower bounds of the binomial coefficient. That is,

Equation

[0079] By using Equation (9) in Equation (8), the following boundary can be obtained.

Equation

[0080] By differentiating the latter equation with respect to P, the following equation is obtained.

Equation

[0081] Assuming the expression in Equation (11) is equal to zero, Q = Tn TThe optimal number of symbols P that maximizes the SE of a QSM system employing an M-ary constellation with space-time resources is * We obtain the following analytically implicit expression that determines

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[0082] However, the desired P * Recalling that is also as large as possible, equation (12) implies that

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[0083] We emphasize that the elegant result presented in (14) is general to any QSM system. It follows that the optimal ratio P / T that maximizes the SE of a QSM system is T In other words, for any given M and n T For this, the optimal QSM for the SE is one where P / T is n, as shown and confirmed by the simulation results shown in Figure 3. T It should correspond linearly to

[0084] That is, Fig. 3 shows the optimal ratio P between the number of transmitted symbols and the number of epochs. * The effects of T and M on / T are shown.

[0085] It should also be noted that the QSM covariance matrix is ​​generally constructed on the basis of an STBC characterized by a T × T square encoding matrix. Therefore, for QSM to be SE-optimal, P must be n T If the underlying STBC itself must scale up accordingly, then the code size T must also scale up accordingly to maintain SE-optimality. In other words, equation (14) also implies that, to achieve SE-optimization, a QSM scheme carrying M-ary symbols must scale up accordingly for n transmit antennas. T This suggests that we must employ an underlying full-rate STBC whose size scales linearly with the number of

[0086] T=n T Note that setting n is not a scalable proposition, not only because it implies that having the same number of RF chains in the transmitter would be prohibitively expensive, but also because it would require a fully dense signal and, consequently, a prohibitively complex ML receiver. This observation motivates the comparison shown in Figure 4, which shows the n obtained by QSM schemes employing STBCs of different sizes. T As a function of and for different M, P * The maximum achievable spectral efficiency ζ occurs at * This shows the part where T is large enough but still n T The significantly smaller QSM method also T It can be seen that as long as is sufficiently large, asymptotically near-optimal SE is achieved.

[0087] Figure 4 shows the relationship between n and T for different sizes of T and M. T 1 shows the behavior of the fractional peak spectral efficiency as a function of .

[0088] Given these results, in the next section we introduce a new QSM transmitter design, including both a description of how to construct a QSM covariance matrix based on an optimal STBC of any size, and a new systematic mechanism for obtaining a set of related index vectors used to select an index vector during transmission. For clarity, we first take a simple example for the 2x2 case to introduce the construction of the covariance index set for optimal diversity gain. An extension of the scheme to generalized T follows.

[0089] B. Golden (2x2) covariance matrix and optimal index set Before describing the construction of the proposed covariance matrix, without loss of generality and for ease of comparison with existing methods, the transmit signal matrix X according to equation (2) satisfies the unit average transmit power constraint for each active transmit antenna and is expressed as the constellation

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[0090] The construction of the QSM dispersion matrix based on the latter golden code is S GLet C be the auxiliary matrix i and D i These are the real part s of each encoded i-th symbol, respectively. i R and the imaginary part s i I is used to modulate the

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[0091]

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[0092] Regarding the scaling coefficients in equation (18), the denominator 1 / sqrt(5) inherits the coefficients of the golden code as in equations (15) and (16), and the numerator sqrt(2) is the transmission power constraint

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[0093] The Golden Code is known to perform better than the SSB code employed in the EDA-QSM scheme, but has a very similar structure to the latter, and as will be demonstrated later through simulated comparisons, the utilization of the SSB code in the construction of the above dispersion matrix improves the performance of the QSM scheme beyond that briefly described by itself.

[0094] However, to improve the performance of the QSM scheme by adopting STBC, i.e.

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[0095]

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[0096] Figure 5 shows a bipartite graph representing the spatiotemporal resource usage associated with each index vector kn in a QSM system with P=3 and Q=8. 37 , and k 54 Specific examples of

[0097] As shown in the graph, K * Given index set k from n The inclusion of a given index in a set k is associated with a particular resource usage (and possibly multiplicity) and is identified by the graph edges intercepted by the enclosures enclosing the corresponding index. n is the set of resources r n The notation k is used to indicate that n ⇒r n Using the set k n μ to indicate the multiplicity of resource rq in kn Use (rq).

[0098] For example, the use of resource r1 = {2 × (1,1), (1,2), (2,1), 2 × (2,2)} results in having k1 = [1,2,3] in K, and μ k1 (1,1)=μ k1 Let (2,2) = 2, and we can write k1 ⇒ r1 succinctly. Similarly, μ r37 (3,2)=μ r37 Let (4,1)=2, and k 37 =[3,4,5]⇒r 37 ={(1,1),(1,2),(2,1),(2,2),(3,1),(4,2)}, and k 54 =[5,6,8]⇒r 54 ={(3,1),2×(3,2),2×(4,1),(4,2)}.

[0099] From all the above, it is clear that to optimize the performance of the QSM scheme, in order to avoid redundancy and uneven utilization of spatiotemporal resources, the set K of indices of the distribution matrix (including the corresponding resource set R) must satisfy the following condition: a) Any two index vectors k in the set n and k m cannot be equal (i.e., k n ≠k m ,∀n≠m) b) No two elements in each index vector can be equal (i.e., [k n ] i ≠[k n ]j,∀k n and i≠j) c) Utilization of all available resources must be ensured (i.e., μK(r q )>0 ∀r q ∈R) d) All resources are used every time (i.e., μK(r1) = μK(r Q )) e) To allow encoding of the codeword, the cardinality of the set must be a power of two (i.e.,

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[0100] For example, in Table I, the index vector K={k1,k2,k3,k5,···,k8,k 10 ,k 11 ,k 19 ,···,k 23 ,k 26 ,k 27 ,k 28 ,k 35 ,···,k 38 ,k 41 ,k 42 ,k 47 ,k 48 ,k 50 ,···,k 56}. The reader can verify that this choice of K ensures that all resources in the associated set R have multiplicity 24. In contrast, the first 32 index vectors in Table I, i.e., K = {k1,...,k 32 Simple truncation of} results in a non-uniform usage pattern, where μ K (1,1)=μ K (2,2)=32,μ K (1,2)=μ K (2,1)=28,μ K (3,1)=μ K (4,2)=19, and μ K (3,2)=μ K This is clearly not optimal, since (4,1)=17, which means that antennas 1 and 2 will be used much more frequently than antennas 3 and 4. The problem of choosing the optimal set K, as described and illustrated above, is related to a classic problem in combinatorial graph theory known as the vertex cover problem. However, in this context, the problem has the following additional difficulties: a) the graph in question is divided into two groups, b) equal multiplicity coverage is required, and c) nodes must be selected in three subsets at a time.

[0101] [Table 1]

[0102] Method 1: Greedy construction of a set K of optimal index vectors

[0103] [Table 2]

[0104] These particularities make the problem unique, to the best of our knowledge, and unsolvable by known variations of the vertex cover algorithm. Fortunately, we can exploit the highly symmetric structure of the associated bipartite graph to design an efficient way to solve the selection problem at hand. Therefore, with a slight abuse of notation, we define the multiplicity of the distribution matrix index q in a set K as μ K (Note that there is no ambiguity in the definition of resource multiplicity, since the distribution matrix index is a single number, whereas the spatiotemporal resources are pairs.) Then, as can be seen from Figure 5, due to the symmetry of the graph, μ K (1)==μ K A solution K for (Q) means a solution R for which each of the spatiotemporal resources {(1,1),(1,2),(2,1),(2,2),(3,1),(4,2)} has the same multiplicity. Therefore, the problem can be solved efficiently by greedy index selection, as described in Method 1.

[0105] C. Optimal generalized design (T × T) P, T, and n T With the above greedy optimal index vector selection algorithm being general to any T, the final limiting factor preventing the generalization of QSM to any T is the construction of the covariance matrix based on an STBC of any size. This obstacle is overcome by considering the design of a QSM covariance matrix based on a full FDFR STBC.

[0106] A T × T FDFR STBC is a T-by-T FDFR STBC where the average transmitted energy per antenna is normalized to 1, an energy efficiency shaping constraint is applied, and an SE-preserving lower bound on the coding gain (a.k.a. non-erasing determinant) is maximized. 2Finally, for a given T∈N + For , the design can be written as:

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[0107] Note that the full FDFR STBC is a perfect generalization of the 2x2 Golden Code. To see this, consider the case where T=2 and the corresponding lattice generator matrix

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[0108] As a result, to adopt a full FDFR STBC in the design of a QSM, the corresponding auxiliary dispersion matrix Ci and D i It follows that it is sufficient to decompose the core code structure of equation (19) in terms of

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[0109] Following this, the complete set of covariance matrices A and B can be constructed, i.e.

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[0110] Next, we focus on constructing the optimal set of index vectors, K, by directly generalizing the method described in Subsection III-B. Indeed, as can be seen by examining Eq. (21), each auxiliary matrix, C i and D i is a T × T sparse matrix obtained from a circular rotation of a diagonal matrix containing only the T nonzero elements of R.

[0111] Method 2 QSM signal generation

[0112] [Table 3]

[0113] Therefore, all the associated covariance matrices obtained from Equation (22) are sparse matrices with only T non-zero entries, corresponding to the T spatio-temporal resources utilized. In other words, while in the golden QSM scheme in Subsection III-B, each covariance matrix index q is associated with two resources, in the full STBC-based construction described here, each index q is associated with T resources, and the corresponding bipartite graph shown in Figure 5 is T·n T Index (circular) nodes and T n T It is simply extended to a similar graph with resource (rectangle) nodes and each resource node is connected to T index nodes, and vice versa. As a result, the greedy strategy described earlier remains valid, as evidenced by the fact that Method 1 applies to general T. We next summarize the structure of the proposed scalable QSM scheme in Method 2.

[0114] IV. PROPOSED RECEIVER DESIGN A. Sparse Formulation of QSM Receivers Methods 1 and 2 introduced above together demonstrate that OS-QSM transmitter design is feasible and tractable. However, there is no true scalability without feasibility, and to complete the task, we also need to show that the proposed OS-QSM design can be effectively decoded with reasonable complexity.

[0115] To put the problem in context, given P, M, T, and n T For Q=T·nT, the ML receiver is

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[0116] We emphasize that this issue applies not only to the OS-QSM scheme described in Subsection III-B, but also to the current SotA QSM method, since the above example is for T = 2, which is the size of the core code used in the current SotA QSM method. Furthermore, we emphasize that the use of an SD receiver is also infeasible in large-scale cases, since the nature of the tree search algorithm still requires excessive computational complexity in large systems. Finally, we note that useful properties such as fast decodability and block diagonality are known to be infeasible without sacrificing optimality for STBCs of any size, so a scalable detector for the QSM method cannot rely on such features.

[0117] Based on the above, we present a combinatorial factor that is infeasible without relying on tree search or specific properties of STBC.

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[0118] In addition, given the prior information about the coding structure, the proposed decoder is able to detect any QSM signal.

[0119] The core idea of ​​our approach is to make full use of a sparse representation of the QSM signal across the entire channel (i.e., all available spatio-temporal resources), which is assumed to be known at the receiver. The proposed decoding method then leverages an iterative shrunken threshold algorithm (ISTA) to greedily extract estimates of the symbol and dispersion indices, resulting in a significant reduction in complexity compared to ML- and SD-based methods. To this end, we first combine equations (1) and (2) to consider a vectorized form of the QSM received signal, given by

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[0120] Equation (23) is the combined real and imaginary separated information and noise vectors

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[0121] To elaborate on equation (27) with an example, let P = 3, T = 2, and n T Consider a system with 4 bits, and a particular bit sequence b=[b R ,b I ,b S ], the selected index vector is k R =k 10 = [1,3,7] and k I =k 47 = [4,5,7], then the corresponding combined information vector is

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[0122] In principle, this latter feature could be used in the design of the SD receiver for the OS-QSM system proposed above, similar to how block separability was used in the design for the EDA-QSM system. Naturally, the problem with this approach is that the separated symbol vectors

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[0123] Considering the focus on scalability leading to controlled complexity at the receiver, two suitable candidate methods to be applied to demodulation of OS-QSM are the Generalized Approximate Message Passing (GAMP) algorithm and the Iterative Reduced Threshold Algorithm (ISTA), both of which are based on a signal vector u of size 2Tn T However, it is well known that the GAMP algorithm strongly depends on the specific structure of the measurement matrix and the independence of the received signals, which in the case of QSM cannot be assumed as a direct consequence of the use of STBC in the covariance matrix in general. If the necessary conditions are not met, the performance of the GAMP receiver will be poor, characterized by an error floor at high SNR.

[0124] Therefore, motivated by this fact, we choose to follow an ISTA-based approach in the design of a low-complexity demodulator for QSM systems, as described below. In particular, a method for detecting QSM signals is introduced that is based on a dedicated ISTA variation that incorporates modifications to both the threshold function and the index vector estimation process, especially QSM detection.

[0125] B. Greedy-boxed ISTA-based QSM decoder Considering the standard ISTA iteration,

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[0126] Figure 6 compares the ISTA threshold function Λ(s;τ) with the BH-ISTA threshold function Π(s;τ) (29).

[0127] By incorporating this modification, we obtain the boxed hard ISTA (BH-ISTA) receiver, which can be written as follows:

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[0128] The computational cost of repeatedly evaluating Eq. (30) is

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[0129] Figure 7 shows the results of Eqs. (28) and (30), respectively, as a function of the number of iterations η.

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[0130] Figure 7(a) shows the sparsity convergence with different thresholds.

[0131] Figure 7(b) shows the MSE convergence with the optimal threshold.

[0132] In particular, Fig. 7(a) shows the,equivalent of the,η,as a function of,η,for various values ​​of the threshold parameter,τ,.

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[0133] In fact, as a result of boxing and hard thresholding,

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[0134] Further details are given in Section VA. Next, Figure 7(b) shows that the mean squared error (MSE) obtained with the proposed BH-ISTA method is superior to that obtained with the conventional ISTA, which demonstrates the effectiveness of the boxed hard threshold modification proposed here for demodulating QSM signals. However, the index {k R ,k I The question remains how to efficiently detect the bits associated with the choice of}∈K. To do this, another addition is introduced to the ISTA-based sparse detector: a greedy hard detection procedure for each recovered symbol, together with a simultaneous update of equation (30). This can be explained as follows:

[0135] Consider multiple runs of the BH-ISTA iteration described in equation (30). Before the mth run, modifications are made to y, G, and u, which can be expressed by rewriting equation (30) as follows:

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[0136] η * Let be the last iteration of the m-th run of the latter estimator, and let its corresponding result be

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[0137] First, the hard-detection version

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[0138] The remaining amount is then updated as follows:

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[0139] If the detection process is error-free, then after exactly m=2P runs, the sequence

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[0140] However, more generally, even at a concentration P,

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[0141] P K Let (q) denote the projection of sequence q onto set K, where either a sequence k∈K or an empty set φ is returned by the projection, depending on whether q contains a sequence from K or not. The notion of greedy selection is consistent if there are multiple valid k∈K in the combination of elements in the feasible elements, with lower indices of q (rather than the element values ​​themselves) being preferred. Then, equation (34) can be expanded as follows:

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[0142] In simple terms, Equation (37) is the estimation vector for the next run after the mth run of the BH-ISTA detector.

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[0143] Obviously, the only alternative is to

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[0144] As above, updates to ym and Gm are also

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[0145] FIG. 8 is a schematic diagram showing the proposed GB-ISTA receiver architecture for QSM demodulation.

[0146] The procedure described in equations (31)-(33) and (35)-(39) represents a greedy (i.e., symbol-by-symbol and index-set-by-index-set) modification of the GB-ISTA detector introduced earlier, which is called the greedy boxing iterative reduced-threshold algorithm for QSM demodulation.

[0147] At the end of the process, the estimate of the selected variance matrix index vector

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[0148] Method 3: Greedy-boxed (hard) ISTA receiver for QSM systems

[0149] [Table 4]

[0150] Method 4a: Greedy ISTA Decoder

[0151] [Table 5]

[0152] Method 4b: Distributed index determiner: index_est(·)

[0153] [Table 6]

[0154] The following table shows the spectral efficiency η of the n×n generalized EDA-QSM.

[0155] [Table 7]

[0156] Given n T Compare pairs when n and P increase by the same factor while maintaining the same sparsity for . (4,n R ,1),n=2,η=4,s=1 / 8 (4,n R ,2),n=4,η=4,s=1 / 8

[0157] When this effect is compared with a larger increase in P, striking and novel results are observed. (4,n R ,2),n=2,η=6,s=1 / 4 (4,n R ,4),n=4,η=7,s=1 / 4

[0158] For the same "average sparsity" in time, the spectral efficiency increases with n, and T increased as the value increased. (16,n R,2),n=2,η=10,s=1 / 16 (16,n R ,4),n=4,η=11.5,s=1 / 16

[0159] V. Complexity and Performance Analysis In this section, we analyze the performance of the proposed OS-QSM through computer simulations. To focus on the scalability of the system, all the simulation results shown are for a relatively large number of transmit antennas (i.e., n T ≥ 6) and for increasing numbers of transmission slots (i.e., T ≥ 2), where the number of digitally modulated transmitted symbols P and the cardinality M of the corresponding constellations are adjusted on a case-by-case basis to highlight the key findings of each simulated experiment. To our knowledge, no simulation results for QSM schemes using such parameters have appeared in the literature to date, due to the prohibitive computational complexity of existing receivers.

[0160] A. Complexity: GB-ISTA vs. ML and SCMB-SD receivers With the latter caveat in mind, we begin by evaluating the decoding complexity of the scaled QSM system, in particular by deriving the complexity orders of the conventional ML and SCMB-SD approaches, as well as the proposed GB-ISTA algorithm described in Section IV. For any given n T , T, and P, the brute force ML decoder generates P digitally modulated symbols s∈S P To transmit the real and imaginary parts of R ,k I}∈K, all possible

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[0161] The practical infeasibility of ML-based detection for QSM systems is due to the fact that the number of transmitted symbols P is larger than all theoretically scalable quantities n TThis is clearly highlighted by equation (40) to highlight the fact that , is an exponential in the complexity order of , T, and M. We next show that this challenge cannot be fully alleviated by SD approaches. To that end, we consider, again ideally and for simplicity, that SD can reduce the search radius to a single symbol, so that the factor MP in equation (40) can be neglected. In other words, we can see that the complexity order associated with an SD-based QSM receiver can, at best, be reduced to

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[0162] From the latter, we can conclude that the only advantage of SD in relation to scalable QSM schemes is that it allows scaling of the digital constellation density M, which does not only have a negative impact on the corresponding BER, but is also not a significant factor in increasing the system SE. This means that the total number of bits carried by a QSM scheme is

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[0163] Finally, we address the computational complexity of the proposed GB-ISTA. First, from equations (31) to (36), we see that the GB-ISTA receiver obtains the spatially coded bits b from the sparse recovery process, not from the search. R and b I , i.e., we directly obtain the values ​​and locations of the non-zero elements of û. As a result of eliminating such combinatorial searches, we can obtain a scalable parameter n as shown in the following complexity analysis of step 3 of Method 3. T, T, and P have significantly smaller effects on GB-ISTA.

[0164] 1) Method 3 takes as input the effective matrix G given by equation (27). Its construction involves the sparse block-diagonal matrix

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[0165] 2) Next, the GB-ISTA receiver executes the evaluation formula (31) multiple times. The first step is

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[0166] The cost of implementing this operation is (2Tn T )(2Tn R )+(2Tn R )+(2Tn R )(2Tn T )+(2Tn R ) flops, but as shown in Figure 7(a), the sparsity of u^(η) quickly reduces to its actual value of 2P, so the complexity of that step is 8PTn R +4Tn R =4Tn R(2P+1). Then, the 2Tn required for the boxed hard threshold function Π T Including flops, η * Considering that iterations are required, the total cost associated with each evaluation run of Eq. (31) is η * (4Tn R (2P+1)+(2Tn T ) flops.

[0167] 3) After the convergence of Eq. (31), the receiver obtains the sparse estimate vector

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[44] at negligible cost.

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[0168] 4) Considering that the cost of removing the elements, interference, and sequences represented by equations (37) to (39) is negligible, the next significant cost for the receiver is the verification of the obtained index. In particular, after at least P runs, the location index

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[0169] 5) Finally, as explained in line 11 of Algorithm 3, GB-ISTA calculates the index vector

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[0170] From the above, the total complexity order of GB-ISTA can be estimated as follows:

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[0171] The per-bit complexity orders of the ML and proposed GB-ISTA decoders are obtained by dividing the expressions in (40) and (42) by the number of bits detected per transmission.

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[0172] Figure 9 shows the impact of the scalable parameters on the complexity of the QSM receiver.

[0173] Figure 9(a) shows the n T Figure 1 shows the relationship between the temperature and the temperature of the object as a function of T for a fixed P.

[0174] Figure 9(b) shows the n T Figure 1 shows the relationship between P and T for various P values ​​as a function of T.

[0175] Figure 9(c) shows the relationship between P and fixed n T , showing various T.

[0176] B. BER performance of the proposed OS-QSM scheme As shown above, we take advantage of the significant complexity reduction achieved by GB-ISTA over ML detection to evaluate the BER performance of the proposed OS-QSM scheme decoded via GB-ISTA. In general, our simulated experiments show that the proposed OS-QSM can be realized with a relatively large number of transmit antennas, using relatively few space-time resources per transmission, while achieving fairly high spectral efficiency and very low E b The aim is to further demonstrate that very low BER can be achieved at / N0.

[0177] Figure 10 shows the impact of scalability on the BER performance of the GB-ISTA detection OS-QSM scheme with fixed SE.

[0178] To that end, the first set of results shown in Figure 10 shows the P / T ratio kept constant and M adjusted so that all curves correspond to systems with the same spectral efficiency, n TWe compare the BER performance of the proposed methods for,various values ​​of,T,and,P,.,Note that the curve for,T,=2,can be considered as a reference corresponding to the,EDA-QSM scheme for SotA, although the results shown,actually incorporate the enhancements described in Subsection III-B,,namely,a) the use of an optimal golden code as opposed to a blockwise,spheredecodable FDFR-STBC, and b) the improvement due to the,utilization of the optimal construction of the index vector set,K,as given in Method 1.

[0179] Two important facts can be seen from the results in Figure 10. First, the significant improvement in BER achieved by scaling is only seen when the number of receive antennas is significantly larger (n R = 12). This shows that the gain is not only due to increased diversity (since receive diversity is already large), but also due to the coding gain obtained from the use of the optimal FDFR-STBC adopted in the design of OS-QSM. Second, the results shown were actually simulated down to fairly low BERs using a conventional computer (i.e., not a particularly powerful machine) and in settings that are virtually impossible to simulate with ML or SD-based receivers. The latter point is supported by the T This is reinforced by the results on the right side of Fig. 10, which includes the curves for the =12 system, and serves to further emphasize the true feasibility of the proposed GB-ISTA receiver.

[0180] However, one criticism that may be raised regarding the results of Figure 10 is that the value of P adopted here is too large for the corresponding T and n T The parameterization used in Figure 10 is such that all systems have the same SE to allow direct comparison under comparable conditions, and again, all curves are in terms of E rather than SNR. b I want to clarify again that this is also why it is plotted against / N0.

[0181] Figure 11 shows the impact of scaling P on the BER performance of the GB-ISTA detected OS-QSM scheme.

[0182] In any case, to dispel any doubts about the ability of the proposed OS-QSM design and the corresponding GB-ISTA receiver to actually achieve a feasible and optimized spectral efficiency combined with a low BER, additional results obtained by varying P up to the optimal value given in equation (14) are shown in Figure 11. Note that due to the truncation operation in the expression of the achievable SE given in equation (8), values ​​of P adjacent to the value given in equation (14), i.e., P*, are also optimal (since they result in exactly the same SE). For example, as in Figure 10, n T When = 6, T = 2, and M = 4, P * = 8, but all values ​​P = {7, 8, 9} result in ζ = 16 in equation (8). Similarly, as in Figure 11, all values ​​P = {10, 11, 12, 13} result in n T = 6, T = 3, and M = 4 yield the largest SE, ζ = 17.

[0183] With these caveats in mind, turning to the results obtained in Figure 11, we see that when upscaling P, only a very modest degradation in BER is observed, and in fact it becomes smaller as T gets larger, as shown in Figure 11. This is a small and fair price to pay for nearly doubling the spectral efficiency of the system. We conclude that the slight BER degradation observed when upscaling the ratio P / T towards optimizing SE is due to the fact that n T We find that upscaling T, T, or both results in a corresponding reduction in the sparsity of the vectorized received signal, which tends to be less significant in systems with higher diversity and coding gain. This trend is indeed observable in Figure 11, where increasing T=2 to T=3 narrows the gap between the BER curves.

[0184] Figure 12 compares the spatial modulation methods available to date.

[0185] Furthermore, the proposed decoder also provides greater flexibility in transmitter development, since it does not require design constraints such as block diagonal or orthogonal properties in the transmission scheme, but only the sparsity inherent in the spatial modulation scheme. In other words, the proposed decoder can accommodate many different coding structures (flexibility).

[0186] The decoder can be used in any MIMO system where the number of antennas is expected to be so large that ML techniques are infeasible. This scheme can be used to increase efficiency in cellular networks and V2X communications (eMBB).

[0187] FIG. 13 shows a comparison of golden EDA-QSM and original EDA-QSM with ML decoding, showing a comparison of golden EDA-QSM versus original EDA-QSM.

[0188] Figure 14 shows the comparison of the golden EDA-QSM versus the original EDA-QSM by theoretical ABEP.

[0189] Figure 15 shows the spectral efficiency of the EDA-QSM versus n T The figure shows a comparison of the impact of

[0190] Figure 16 shows the spectral efficiency of the EDA-QSM versus the n T This shows the impact of

[0191] Figure 17 shows the impact of EDA-QSM on the spectral efficiency in the GISTA feasible region.

[0192] FIG. 18 shows the average number of GISTA iterations by GISTA decoding.

[0193] FIG. 19 shows the decoding complexity between the ML decoder and the GISTA decoder.

[0194] Figure 20 shows the performance comparison between the GISTA decoder and the full ML decoder (n T 4) is shown.

[0195] Figure 21 shows the performance comparison of GISTA decoding.

[0196] This application describes a new transmitter and receiver design for the QSM system, which uses n transmit antennas. T A design is proposed that focuses on scalability in terms of the number of , the number of transmission instances T, and the number of coded M-ary symbols P, as well as its performance optimization in terms of SE, diversity, and coding gain. This contribution shows that to achieve optimality in SE, the QSM scheme T This is motivated by the demonstrated fact that the need to scale up T, T, and P is necessary, which is not possible with the SotA approach. On the transmit side, the newly proposed OS-QSM scheme differs from SotA alternatives in that its covariance matrix is ​​designed based on the FDFR STBC and the covariance matrix index selection is performed via a novel greedy algorithm, ensuring that all of the transmitter's space-time resources are evenly utilized across multiple transmissions. Second, at the receiver, the proposed technique, thanks to its reliance on the sparsity structure of QSM signaling, eliminates the combinatorial nature of existing ML- or SD-based approaches and contributes to a novel ISTA-based receiver that further enables system scaling in terms of feasibility. Indeed, a complexity analysis is provided, which shows that, in contrast to ML and SD detectors, which have exponential complexity in T and nT with P as the exponent, making them infeasible in scaled scenarios, the proposed GB-ISTA receiver is third-order on T, second-order on P, and nT. T It is shown that the proposed method enjoys a complexity order that is only first order. Simulation results for a scaled configuration not previously shown in the relevant literature confirm both the high performance and feasibility of the proposed OS-QSM scheme and GB-ISTA receiver.

Claims

1. 1. An optimal and scalable quadrature space-time modulation (OS-QSM) computer-implemented method for configuring multiple transmit antennas, comprising: configuring a plurality of transmit antennas to respectively represent in-phase spatial constellation symbols in an in-phase spatial constellation and quadrature-phase spatial constellation symbols in a quadrature-phase spatial constellation; mapping source data to the in-phase and quadrature-phase spatial constellation symbols represented by the plurality of transmit antennas; the method applies an optimal and scalable quadrature spatial modulation scheme (OS-QSM) that provides the maximum possible coding gain for the resulting quadrature spatial modulation; Modifications to the Iterative Shrinkage Threshold Algorithm (ISTA) are made through range restriction, hard thresholding, and boxing. A method characterized by:

2. 2. The method of claim 1, characterized by performing an Iterative Shrinking Threshold Algorithm (ISTA) via Boxing Hard, greedy selection of antenna position indices and symbol estimates, and their independent decoding of corresponding antenna modulation bits and symbol modulation bits.

3. A process running in parallel with the greedy selection is performed to ensure that a valid estimate of an index vector from a given finite set of index vectors is produced as output, and to apply interference cancellation using the identified values: - checking before every iteration whether a final confirmation can be computed from the currently decoded index, while keeping track of which index has been obtained from the greedy selection; If the final check is not possible, remove the interference from the previous greedy selection and perform the next iteration. The method of claim 2.

4. 2. The method of claim 1, characterized by a generalization of the Golden Code and the combined operation of spatial modulation on a highly sparse received signal without generating large-scale multi-user interference in the overlap of multiple users.

5. A receiver (R) of a communication system having a processor, a volatile and / or non-volatile memory and at least one interface adapted to receive signals on a communication channel, wherein the non-volatile memory stores computer program instructions that, when executed by a microprocessor, configure the receiver to perform the method according to any one of claims 1 to 4.

6. A computer program product comprising computer executable instructions which, when executed on a computer, cause said computer to perform the method of any one of claims 1 to 4.

7. A computer-readable recording medium for storing and / or transmitting the computer program product of claim 6.

8. A vehicle unit comprising a communication system with a receiver (R) in the vehicle, said system being adapted to carry out the method according to any one of claims 1 to 4.

9. A vehicle comprising one or more vehicle units according to claim 8.

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