How to select the optimal index vector for spatial modulation

Abstract

A computer-implemented decoding method for configuring multiple transmit antennas, each of which represents an inphase spatial constellation symbol in an inphase spatial constellation and an orthogonal spatial constellation symbol in an orthogonal spatial constellation, and mapping source data to the inphase spatial constellation symbols and the orthogonal spatial constellation symbols represented by the multiple transmit antennas, to construct an even multiplicity set of transmit antenna activations while ensuring maximum possible transmit diversity.

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 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. The construction of the index vector is naively performed in the combination order up until the state-of-the-art (SotA), but this leads to uneven allocation of transmit antennas and a degradation of transmit diversity.

[0003] Spatial modulation (SM) is a promising technique that can reduce the hardware complexity and cost of massively multiple-input multiple-output (MIMO) wireless communication systems without sacrificing bit error rate (BIR) and spectral efficiency (SE) performance. In particular, in SM schemes, information bits are embedded not only in the choice of transmitted symbols (also known as the consolation dimension) but also in the choice of transmit antennas utilized during transmission (also known as the spatial dimension).

[0004] Through this approach, the large number of stationary sources associated with massive MIMO systems can be efficiently utilized without requiring a similarly large number of radio frequency (RF) chain components. This efficient utilization of RF chains makes SM attractive for future wireless systems, such as fifth-generation and beyond (B5G), which will continue to use a large proportion of millimeter-wave (mm-wave) bands, and sixth-generation (6G) networks, which are expected to also incorporate terahertz and visible light communication (VLC) bands [6]. Summary of the Invention [Problem to be solved by the invention]

[0005] However, a major drawback of early SM schemes is that only one antenna is selected per transmission, severely limiting the achievable SE. To circumvent this limitation, the Generalized Spatial Modulation (GSM) scheme was later developed, in which multiple antennas are selected per transmission, leading to a significant increase in SE. However, another drawback of early SM schemes, including GSM, is that they focused exclusively on improving SE and did not equally focus on reducing BER, for example, through exploiting transmit diversity. This limitation motivated the idea of ​​combining SM with space-time coding (STC), such as the space-time shift keying (STSK) scheme based on linear dispersion (LD) coding, which incorporates space-time block coding (STBC) and spatial modulation with periodic structure (CSM).

[0006] This insight led to further optimization of SM transmitter designs, leading to the discovery of the quadrature spatial modulation (QSM) approach, in which SM concepts are applied independently to the real and imaginary components of the modulated signal via dedicated space-time coherence matrices. This idea has been further developed into a series of QSM techniques with progressively enhanced coherence matrix designs, including the diversity-achieved orthogonal spatial modulation (DA-QSM) scheme incorporating Alamouti codes, and the more recent scaled diversity-achieved orthogonal spatial modulation (EDA-QSM) scheme, in which the coherence matrix is ​​constructed using full-diversity full-rate (FDFR) codes with block-wise sphere decodability

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

[0007] Despite these advantages, the EDA-QSM scheme, and consequently the preceding QSM schemes, still suffer from two main drawbacks. The first drawback is that the covariance matrix used in the currently proposed QSM scheme is based on a 2x2 STBC, which limits both the diversity and coding gain achieved by the method. Regarding the first drawback, the inventors herein propose that n TWe show that QSM designs based on STBCs with size T not proportional to P are fundamentally suboptimal in the sense of SE. The second drawback is that current QSM detection schemes are based on exhaustive maximum likelihood (ML) or, at best, spherical detectors. It should be noted here that, contrary to the above assertion, the average complexity of spherical decoding grows exponentially with the number of jointly decoded symbol periods. This result is supported by several findings, including the derivation of a third-order closed-form expression for the expected complexity of a spherical detector and the demonstration that lattice reduction does not improve the tail exponent of the spherical detector's complexity distribution. Regarding the second constraint, we propose in this application that the complexity of both ML-based and spherical detection (SD)-based QSM receivers actually grows exponentially with P, i.e., n. T This shows that these techniques are fundamentally non-scalable in the realm of QSM systems due to the geometrical nature of τ and T. In other words, current QSM schemes face significant bidirectional scalability challenges, specifically the lack of scalable transmitter and receiver designs.

[0008] Inspired by the above problem, this application proposes a scheme that is scalable to any block size at the sender side, i.e., n T , T and P are unconstrained, and the receiver can decode it in polynomial time, i.e., for large n T We provide a practical new QSM solution with , T, and moderate P. Additionally, we find that the proposed QSM scheme offers a wide range of possibilities for optimizing SE, diversity, and coding gain. To this end, we first introduce an optimal FDFR golden STBC code in the design of the QSM covariance matrix. The golden code is a fast-decodable STBC known to be optimal, i.e., FDFR with maximum coding gain across Gaussian constellations, and has been shown to be generally configurable for any block size. The resulting optimized scalable QSM (OS-QSM) scheme is the first scheme proposed to date with this feature.

[0009] The novel OS-QSM design is further enhanced by a novel algorithm for selecting the indices of the covariance matrix employed in the scheme, ensuring that all transmit antennas are utilized with equal frequency and likelihood across multiple block transmissions, thus ensuring optimally diverse utilization of all spatio-temporal resources.Finally, to ensure feasible decodability of the scalable transmitter design, we propose a novel Greedy Boxed Iterative Shrinkage Binarization Algorithm (GB-ISTA) QSM detector based on sparse recovery.

[0010] Thanks to the sparse signal processing approach, the proposed decoding scheme does not require any constraints on the core code design, unlike conventional sphere detection methods that require block-diagonal fast decodability. However, additionally and most importantly, a major advantage of the proposed new GB-ISTAQSM receiver is that it does not require searching in a large codebook space, unlike block-wise sphere decoding matched to ML and state codewords (SCMB-SD). In fact, the complexity of the proposed receiver is order cubic in T, quadratic in P, and n T It has been shown that it is only linear for

[0011] Overall, the contributions of this paper can be summarized as follows: Spectral efficiency optimality: A closed-form expression for the optimized number P of coded symbols 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 novel Golden Code-based Orthogonal Spatial Modulation (GQSM) transmission scheme is obtained via the design of a coherence matrix based on a 2 × 2 Golden Code, which is known to achieve optimal coding gain over integer symbol constellations. Transmitter scalability: The novel GQSM design is generalized via extending the 2 × 2 Golden Code to its own T × T FDFR STBC variant, resulting in an OS-QSM scheme applicable to any nT, T, and P. · Optimal resource utilization: Method 1 provides a novel mechanism for selecting an optimal set of dispersion matrix indices, ensuring that all Q space-time resources are utilized equally over time, as required for optimizing diversity gains. Scalability at the receiver: A new low-complexity demodulation algorithm based on the Greedy Iterative Shrinkage Binarization Algorithm (ISTA) is proposed for the GSM system, which is not only feasible on a huge scale due to its high linear complexity, but also applicable to other STBC-QSM systems. Receiver complexity: A new formula for the proposed receiver complexity is derived, which is expressed as a function of T and n, with P as the exponent. T In contrast to the ML and SD receivers, which are geometrically T We show that it is linear for

[0012] Therefore, in this application, we present a method for constructing index vectors that respects configuration constraints while preserving uniformity of activation of multiple antennas.

[0013] The above problem has not been solved by any conventional method of spatial modulation, up to Enhanced Diversity-Achieving Quadrature Spatial Modulation (EDA-QSM) [1], which is a SotA and has not previously found a solution.

[0014] The method of the present invention iteratively constructs an optimal set of index vectors given system parameters such as the number of transmit antennas, the number of consecutive symbol periods, and the total number of transmitted symbols.

[0015] Specifically, this is represented by the dashed highlighted portion of the QSM signal generation flow / block diagram in Figure 1.

[0016] The objective function is the uniform allocation of transmit antenna indices in the final optimal set. The algorithm starts with an empty set and adds a vector to the set at each iteration. The vector added at each iteration contains the indices with the least multiplicity from the set being constructed.

[0017] The pseudocode for this method is listed in Method 1 on page 19, and a visual flow diagram of this method is shown in FIG.

[0018] Complex matrices and complex vectors are written in bold uppercase and lowercase letters, and their elements are denoted as X, x, and x, respectively. i The real and imaginary parts of a complex number x are written as x R and x I and for future convenience, we will denote it as a complex vector x=[x1,x2,…,x_ n ] T Accompanying isolated vector

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[0019] All previous state-of-the-art spatial modulation (SM) schemes use ML, like the majority of SM, or modified tree search algorithms, like the state-of-the-art SM scheme EDA-QSM, to decode SM signals, without making any notable attempts at sparse detection or greedy approaches.

[0020] Tree search algorithms are highly impractical due to the highly combinatorial nature of SM and the resulting large search space. Furthermore, SM imposes various constraints on the structure of the decoded signal, making naive low-complexity decoding impossible.

[0021] A problem with the prior art is that there is no sparse detection solution to this problem: solutions using other approaches are characterized by infeasible complexity. [Means for solving the problem]

[0022] Massive multiple-input multiple-output (MIMO) systems in fifth-generation (B5G) and sixth-generation (6G) wireless communications are expected to incorporate a large number of transmit and receive antennas.

[0023] 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 scale increases, the classical decoding complexity of SM schemes becomes infeasible (i.e., the complexity becomes intractable) because SotA uses maximum likelihood (ML) decoding or a tree search algorithm based on ML.

[0024] The proposed method is shown in the flow diagram of Figure 12.

[0025] The method builds a set of transmit antennas with equal activation multiplicities, resulting in maximum transmit diversity, as shown in Figure 13. This is not the case in conventional SotA, where the multiplicities are unequal, and therefore suboptimal transmit diversity is utilized from the number of available antennas, i.e., all antennas are used equally.

[0026] Advantageously, all existing spatial modulation schemes already utilize (or can be reformulated to utilize) index vector selection, which can be used to increase data rates and improve efficiency in cellular networks and V2X communications (eMBB).

[0027] These and other objects, features and advantages of the present invention will become more apparent from a consideration of the drawings and detailed description.

[0028] 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.

[0029] Furthermore, the proposed decoder also provides great flexibility in transmitter development, since it does not require design constraints such as block diagonal or orthogonal transmission schemes, but only the inherent sparsity of spatial modulation schemes. In other words, the proposed decoder can accommodate many different coding structures (flexibility).

[0030] The decoder can be used in any MIMO system where the number of antennas is large and therefore ML approaches would be infeasible. The scheme can be used to improve the efficiency of cellular networks and V2X communications (eMBB).

[0031] The above and other objects, features and advantages of the present invention will become more apparent from a consideration of the detailed description taken in conjunction with the drawings.

[0032] An embodiment of a computer-implemented method for selecting optimal index vectors for configuring multiple transmit antennas is to configure the multiple transmit antennas such that each antenna represents an in-phase spatial constellation symbol in an in-phase spatial constellation and an orthogonal spatial constellation symbol in an orthogonal spatial constellation, and to map source data to the in-phase spatial constellation symbols and the orthogonal spatial constellation symbols represented by the multiple transmit antennas, wherein the method constructs an even multiplicity set of transmit antenna activations that ensures the greatest possible transmit diversity.

[0033] Another embodiment of the computer-implemented decoding method is characterized by a modification of the Iterative Shrinkage Binarization Algorithm (ISTA) via boxing, range restriction and hard binarization.

[0034] Another embodiment of the computer-implemented decoding method is characterized by proceeding with a boxed-hard (ISTA) iterative reduced binarization algorithm, greedy selection of antenna indices and symbol estimate locations, and their independent decoding of corresponding antenna modulation and symbol modulation bits.

[0035] Another embodiment of the computer-implemented decoding method is characterized by ensuring that valid estimates of index vectors are produced as output from a given finite set of index vectors, where the process operates in parallel with greedy detection, and performing interference cancellation on the confirmed values; Check before every iteration whether a final confirmation is possible from the currently decoded index, keeping track of the index obtained from the greedy selection; If not possible, remove the interference from the previous greedy selection and perform the next iteration.

[0036] Another embodiment of the method is a method for transmitting a signal having a number of symbols P, a number of symbol slots T, and a number of transmit antennas n. T is input and proceeds.

[0037] The first step is to generate an empty set of valid vectors.

[0038] In the second step, a random seed vector is added to the set.

[0039] It then runs a routine to find the least frequently used index in the set.

[0040] In the next step, a vector with index a is selected and added to the set, checking whether the set has reached its maximum size; if not, the method proceeds to find the least frequently used index in the set; if not, the method ends.

[0041] Another embodiment is characterized by 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 the microprocessor, configure the receiver to implement the decoding method of one or more of the above-mentioned embodiments.

[0042] Another embodiment is characterized by a receiver with 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-described embodiments.

[0043] Another embodiment is characterized by a computer-readable medium that stores and / or transmits the computer program product described above.

[0044] Another embodiment is characterized by a vehicle unit including a communication system with a receiver (R) mounted on the vehicle, the system being adapted to perform the method according to one or more of the decoding methods of one or more of the above-described embodiments.

[0045] Another embodiment features a vehicle having one or more of the vehicle units described above.

[0046] All aspects of this application can be integrated with elements within a mobile device, a base station, or a wireless system. All of the above elements can be incorporated into a vehicle.

[0047] For a better understanding of the nature of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings. [Brief explanation of the drawings]

[0048] [Figure 1] 1 is a schematic diagram depicting the general structure of a QSM transmission system. [Figure 2]n is the spectral efficiency of the OS-QSM schemes with T=2, 4, and 8 in a given system where T=8 and M=4. [Figure 3] The influence of T and M on the optimal ratio P* / T of the number of transmitted symbols to the epoch. [Figure 4] The behavior of fractional peak spectral efficiency as a function of nT is shown for different magnitudes of T and M. [Figure 5] Fig. 1 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, explicitly showing the specific examples of k1, k37, and k54. [Figure 6] This is a comparison of the ISTA binarization and BH-ISTA binarization functions Λ(s;τ) from

[29] and Π(s;τ) from equation (29). [Figure 7] Figure 7 shows the convergence of ûΠ(η) and ûΛ(η) according to equations (28) and (30) as a function of the iteration number η. Figure 7a shows the convergence of sparsity at various thresholds. Figure 7b shows the convergence of MSE at the optimal threshold. [Figure 8] This is a schematic diagram depicting the configuration of the GB-ISTA receiver proposed for QSM demodulation. [Figure 9] Figure 9 shows the effect of scalability parameters on the complexity of a QSM receiver. Figure 9a shows fixed P and variable T as a function of nT. Figure 9b shows fixed T and variable P as a function of nT. Figure 9c shows fixed nT and variable T as a function of P. [Figure 10] The effect of scalability on the BER performance of the OS-QSM method detected by GB-ISTA with a fixed SE is shown. [Figure 11] The effect of scaling P on the BER performance of the GB-ISTA detection OS-QSM system. [Figure 12] FIG. 1 is a flow diagram of the proposed computer-implemented method for selecting optimal index vectors for spatial modulation. [Figure 13]Fig. 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, explicitly showing the specific examples of k1, k37, and k54. DETAILED DESCRIPTION OF THE INVENTION

[0049] n into the material according to the designed dispersion pattern T We consider a quadrature spatial modulation (QSM) scheme that can carry a huge number of bits while combining a relatively small number P of dynamically selected symbol transmissions, all M-ary modulated.

[0050] 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 phrases describe desirable features or functionality of the system in the context of advantageous designs of particular embodiments, and do not indicate necessary or mandatory elements of all embodiments. Accordingly, each requirement and capability described below as needed, important, desired, or similar phrases should be understood to be optional.

[0051] In the following description, the wireless communication network, including radio equipment, 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 radio equipment, radio network equipment, network nodes, and networks incorporating some or all of the features detailed herein may, of course, be referred to by any of a variety of names. It will be understood that future developments of 5G specifications may use terms such as "New Radio," "NR," or "NR multi-node," and that some or all of the features described herein in the context of 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 radio equipment, radio network equipment, network nodes, and networks incorporating some or all of the features detailed herein may or may not be referred to by the term "5G." The present invention relates to all individual aspects of NX, but also to the development of other technologies, such as LTE, in interaction with and interworking with NX. Furthermore, each such individual aspect and each such individual development constitutes a separable embodiment of the present invention.

[0052] Figure 1 shows a schematic diagram depicting the general structure of a QSM transmission system.

[0053] 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, assuming the channel is constant, can be written succinctly as

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[0054] In what follows, we assume that the quasi-static Rayleigh attenuation channel matrix H is known at the receiver but not at the transmitter, and state that the channel power per matrix entry is unitary, so the fundamental signal-to-noise ratio (SNR) is given by

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[0055] Regarding equation (2), referring again to Figure 1, in the QSM scheme, the bit sequence b is

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[0056] In view of the above, the design of a specific QSM scheme essentially involves the Q-variance matrix A q and B q and an index vector k that informs the selection of the covariance matrix used for each transmission. R and kI This can be said to result from the method adopted to select the set K containing

[0057] To show how state-of-the-art (SotA) QSM schemes can be incorporated into the general framework described in equation (2), we first consider the QSM scheme proposed in 119. In this case, we convert the covariance matrix into the covariance vector (i.e., T=1 and Q=n T ) to A q =e q And B q =je q; (3) where 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.

[0058] Next, in the DA-QSM system, a two-column dispersion matrix (i.e., T=2 and Q=n) is used to utilize transmission diversity. T ) is adopted. In particular, in this method,

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[0059] From the above, the DA-QSM method essentially improves over the QSM method by adding 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 covariance matrix of the QSM method, which means that no additional multiplexing capability is aggregated and the coding gain is not optimized.

[0060] In contrast, the EDA-QSM method improves on the latter in both respects. In particular, this method has a dispersion matrix of

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[0061] Through the above brief description, it is easy to see that the fundamental difference between the DA-QSM method and the EDA-QSM method is that the dispersion matrix of the EDA-QSM method is complex-valued, and therefore the orthogonality of the real and imaginary dimensions is better utilized to enjoy the multiplexing and coding gain.

[0062] However, two fair criticisms that can be made of all existing SotA QSM methods proposed to date, and indeed of our knowledge, are that a) they do not scale systematically across space and time simultaneously for any T > 2, and b) the achieved coding gain is not optimal. Relaxing these two constraints is the goal of our first contribution, described in the next section.

[0063] III. Optimized Scalable Orthogonal Spatial Modulation Transmitter Design A. Spectral efficiency optimization of QSM system Given the number of bits carried by the transmission of each QSM transmission symbol X according to equation (2), and the fact that such transmissions require T consecutive channel utilizations, SEζ for any QSM scheme is given by

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[0064] Figure 2 shows the T For a given system with T = 8 and M = 4, we show the spectral efficiency of the OS-QSM scheme with T = 2, 4, and 8.

[0065] The binomial coefficient in equation (8)

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[0066] The inventors consider the given n T For and M, we obtain an analytical expression for the optimal ratio P / T that maximizes SE, and use this to calculate T in a large-scale system. <n T Set n T →∞, we can determine the relative SE reduction that occurs. For this purpose, we can determine the upper and lower bounds of the binomial coefficient, i.e.

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[0067] Substituting equation (9) into equation (8) gives the following bound:

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[0068] By setting the expression in equation (11) equal to zero, Q = Tn T Optimal number of symbols P that maximizes SE in QSM systems with spatiotemporal resources * The following analytical implicit form is obtained that allows us to determine the M-ary constellation

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[0069] However, the desired P * Recalling that is also the maximum possible value, equation (12) suggests that

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[0070] We emphasize that the elegant result shown in (14) is general for any QSM system. This result shows that the optimal ratio P / T that maximizes the SE of a QSM system is found for n transmit antennas. T In other words, for any given M and n T On the other hand, the SE-optimized QSM is the one with P / T of n, as illustrated and confirmed by the simulation results shown in Figure 3. T must be linearly proportional to

[0071] This is because Figure 3 shows the optimal ratio P * This means that the influence of T and M on / T is shown.

[0072] Recall also that the QSM covariance matrix is ​​generally constructed based on an STBC characterized by a T × T square encoding matrix. As a result, for QSM to be SE-optimal, P must be n T If the underlying STBC itself needs to be proportional to the size of the code T in order to maintain SE-optimality, then the size of the code T must also be proportional to the number of transmit antennas n. In other words, equation (14) also implies that to achieve SE-optimality, a QSM scheme carrying M-ary symbols must be TThis suggests that we should adopt a full-rate STBC with a base rate proportional to

[0073] T=n T Note that setting P implies augmenting the transmitter with the same number of RF chains, which may be prohibitively expensive, and is not a scalable proposition, as it would result in a fully dense signal, which would also require an unpromisingly complex ML receiver. This observation is supported by the P obtained by QSM schemes employing STBCs of different sizes. * The maximum achievable spectral efficiency ζ occurs at * For different M, n T This motivates the comparison given in Figure 4, which shows the T QSM schemes with T significantly smaller than n T It can be seen that if is sufficiently large, it asymptotically achieves a near-optimal SE.

[0074] Figure 4 shows the behavior of fractional peak spectral efficiency for different sizes of T and M. T is shown as a function of .

[0075] Given these results, in the next section we present a new QSM transmitter design that includes both a description of how to construct a QSM dispersion matrix based on an optimal STBC of any size, as well as a new systematic mechanism for obtaining the associated set of index vectors used for selection during transmission. For illustrative purposes, we first present the construction of a set of dispersion indices for optimal diversity gain using a simple 2x2 example. An extension of the scheme to generalized T follows.

[0076] B. Golden (2x2) covariance matrix and set of optimal indices Before explaining the construction of the proposed covariance matrix, without loss of generality and to facilitate comparison with existing methods, let us consider the case where the transmit signal matrix X in Equation (2) is

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[0077] The construction of the QSM dispersion matrix based on the latter golden code is S G Let C be the auxiliary matrix i and D i and using these, the real part S of each coded i-th symbol is decomposed into i R and the imaginary part S i I Modulate the

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[0078] Regarding the scaling coefficients in equation (18), the denominator 1 / Sqrt(5) is inherited from the coefficients of the golden code, as in equations (15) and (16), while the numerator Sqrt(2) is the transmission power constraint.

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[0079] While golden codes are known to outperform the SSB codes used in the EDA-QSM system, their structure is very similar to that of the latter. Therefore, their use in constructing the dispersion matrix as described above inevitably leads to improved performance over the QSM system briefly discussed in Subsection II-B, as will be shown later through simulation comparisons.

[0080] However, it is possible to improve the performance of the QSM scheme employing STBC, i.e., to determine which covariance matrix to assign to the real and imaginary parts of each coded symbol.

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[0081] Also,

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[0082] Figure 5 shows the relationship between the index vector k in the QSM system with P=3 and Q=8. n is a bipartite graph representing the use of spatiotemporal resources associated with k1, k 37 , k 54 Specific examples of

[0083] As the graph shows, K * A given index set k from n The inclusion of a in a set K is associated with the use, sometimes multiplicity, of a particular resource identified by the edges of the graph enclosed by the boxes surrounding the corresponding indices. Hence the notation k n ⇒r n Using the index set k n is the resource r n suggests the use of μ kn (r q ) to find resource r in set k. n Indicates the multiplicity of

[0084] For example, the use of resource r1 = {2 × (1,1), (1,2), (2,1), 2 × (2,2)} is a consequence of having k1 = [1,2,3] in K, which can be succinctly written as k1 ⇒ r1, μk1(1,1) = μk1(2,2) = 2. Similarly, 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)}, μ r37 (3,2)=μ r37 (4,1)=2.

[0085] From all the above, it is clear that to avoid redundancy and uneven utilization of spatiotemporal resources and optimize the performance of the QSM scheme, the dispersion matrix index K (together with the corresponding resource set R) must satisfy the following condition: a) What two index vectors k and k in the set m cannot be equal (i.e., k n ≠k m ,∀n≠m); b) No two elements of each index vector can be equal (i.e., [k n ] i ≠[k n ] j ,∀k n, and i ≠ j); c) It must guarantee the utilization of all available resources (i.e. μ K (r q )>0∀r q ∈R); d) All resources are used equally frequently (i.e., μ K (r1)==μ K (r Q )), and finally e) codewords (i.e.,

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[0086] As an example, Table I shows a set of index vectors, K = {k1, k2, k3, k5, · · · , k8, k 10 ,k 11 ,k 19 ,···,k 23 ,k26,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 with this choice of K, the multiplicity of the associated set R is 24. In contrast, if we naively truncate the first 32 index vectors in Table I, i.e., K={k1, ,k 32} is 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 KThis leads to an uneven utilization pattern of (4,1)=17, which is clearly suboptimal since antennas 1 and 2 are used much more frequently than antennas 3 and 4. As discussed and illustrated above, the problem of selecting the optimal set K is related to a classic problem in combinatorial graph theory known as the vertex cover problem. However, in the context of this document, this problem suffers from the following issues: a) the graph in question is a bipartite graph, b) a cover with equal multiplicity is required, and c) nodes in the subset must be selected three at a time.

[0087] [Table 1]

[0088] [Table 2]

[0089] Due to these particularities, the problem itself is, to the best of our knowledge, unique and cannot be solved 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 algorithm to solve the selection problem at hand. To do so, we slightly borrow notation and define the multiplicity q of the distribution matrix index in a set K as μ K (Note that there is no ambiguity in the definition of resource multiplicity because the distribution matrix index is simply a numerical value, while the space-time resource is a pair of numbers.) Therefore, due to the symmetry of the graph, μ K (1)==μ K A solution K for (Q) implies a solution R for which each space-time resource {(1,1),(1,2),(2,1),(2,2),(3,1),(4,2)} has the same multiplicity. As a result, this problem can be solved efficiently by greedy index selection, as described in Method 1.

[0090] C. Optimal generalized design (T × T) P, T and n TThe final obstacle that prevents us from generalizing QSM to any T using the above-mentioned greedy optimal index vector selection algorithm, which is common in

[10] , 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.

[0091] T×T FDFR STBC is T 2 We encode the symbols in such a way that the average energy transmitted per antenna is normalized to unity, energy efficiency constraints are enforced, and an SE-preserving lower bound (a.k.a., a non-zeroing determinant) on the coding gain is maximized. Finally, for a given T∈N+, the design is described by

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[0092] Note that the full FDFR STBC is a perfect generalization of the 2 × 2 golden code of

[28] . To see this, it is sufficient to consider the case T = 2, and the corresponding lattice generating matrix

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[0093] From the above, to adopt the full FDFR STBC in the design of QSM, the core code structure in (19) can be symmetrically transformed into the corresponding auxiliary covariance matrix C i and D i , i.e.

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[0094] Following this, the full set of covariance matrices A and B can be constructed as follows:

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[0095] Next, we turn to constructing an 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 periodic rotation of a diagonal matrix containing only the T non-zero elements of R.

[0096] [Table 3]

[0097] As a result, all the associated covariance matrices obtained from (22) are sparse matrices with only non-zero elements corresponding to the T spatiotemporal resources used. 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, so the corresponding bipartite graph shown in Figure 5 is simply T·n T index (circular) nodes and T·n T The algorithm is expanded to index (rectangle) nodes, where each resource node is connected to an index node and vice versa. As a result, the greedy strategy described above is still valid, as evidenced by the fact that Algorithm 1 can be applied to general T. For the reader's convenience, we summarize the structure of the proposed scalable QSM scheme in Method 2.

[0098] IV. PROPOSED RECEIVER DESIGN A. Sparse formulation of QSM receiver Both methods 1 and 2 introduced above demonstrate that designing an OS-QSM transmitter is both feasible and tractable. However, there is no true scalability without feasibility, and therefore to complete the task, it is also necessary to demonstrate that the proposed OS-QSM design can be effectively decoded with reasonable complexity.

[0099] Specifically, the problem is to T , Q=T·n T On the other hand, the ML receiver is

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[0100] We emphasize that this issue applies not only to the OS-QSM scheme described in Subsection III-B, but also to current SotA QSM schemes such as

[20] –

[23] , where the core code size used in the latter scheme is T=2 as an example given above. We further emphasize that the use of SD receivers is not possible in the scaled case, since the nature of tree search algorithms still requires excessive computational complexity for 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, and therefore scalable detectors for QSM schemes cannot rely on such features.

[0101] Based on the above, in the following, we will consider the infeasible combination elements without relying on tree search or specific properties of STBC.

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[0102] The core idea of ​​our approach is to make the most 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 therefore utilizes an Iterative Shrinkage Binarization Algorithm (ISTA) to greedily extract symbol and dispersion indicator estimates, resulting in significant complexity reduction compared to ML and SD-based methods. To do so, we first combine equations (1) and (2) to obtain a vectorized form of the QSM received signal:

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[0103] Equation (23) can be further simplified by defining the combined real and imaginary part separation information and noise vector as:

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[0104] To specifically explain equation (27), let P=3, T=2, and n T Consider a system where b = 4, 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]. Therefore, the corresponding joint information vector is

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[0105] In principle, this latter feature can be used to design an SD receiver for the OS-QSM method proposed above, in the same way that block separability was used for the EDA-QSM method. The problem with this approach is, of course, that the separated symbol vectors

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[0106] Considering the focus on scalability leading to reduced complexity at the receiver side, two candidate methods suitable for application to OS-QSM demodulation are the Generalized Approximate Message Propagation (GAMP) algorithm and the Iterative Shrinkage Binarization Algorithm (ISTA), both of which are suitable for the 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, in the case of QSM, on the independence of the received signals, which cannot generally be assumed as a direct consequence of the use of STBC on the covariance matrix. If the necessary conditions are missing, the performance of the GAMP receiver will degrade, as characterized by an error floor at high SNRS.

[0107] Motivated by this fact, an ISTA-based approach was therefore chosen in the design of a low-complexity demodulator for QSM systems, which will be described in the sequel. In particular, we present a method specific to QSM detection that detects QSM signals based on ISTA-specific variations that make changes to both the binarization function and the index vector estimation process.

[0108] B. QSM decoder based on greedy boxed ISTA Consider the following standard ISTA recursion:

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[0109] Figure 6 shows a comparison of the ISTA binarization and BH-ISTA binarization functions Λ(s;τ) and Π(s;τ) (29).

[0110] By incorporating this change, the boxed hard ISTA (BH-ISTA) receiver is defined as follows:

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[0111] The computational cost of repeatedly evaluating Equation 30 is

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[0112] Figure 7 shows

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[0113] Figure 7(a) shows the convergence of sparsity for various thresholds.

[0114] Figure 7(b) shows the convergence of MSE to the optimal threshold.

[0115] In particular, Fig. 7(a)

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[0116] In fact, as a result of boxing and hard binarization,

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[0117] Further details are provided in Sections V-A. Figure 7(b) shows that the mean squared error (MSE) obtained with the proposed BH-ISTA approach is better than that obtained with the conventional ISTA, demonstrating the effectiveness of the box-shaped and hard binarization modifications proposed herein for demodulating QSM signals. However, the covariance matrix index {k R ,k I It remains to determine how to efficiently detect the bits associated with the choice of}∈K. To this end, we introduce another addition to the ISTA-based sparse detector: a greedy hard detection procedure for each recovered symbol and the accompanying update of equation (30), which can be written as follows:

[0118] We consider that the BH-ISTA iteration expressed by equation (30) can be performed multiple times, and before the mth iteration, changes are made to y, G, and u, so that equation (30) can be rewritten as follows:

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[0119] η * Let be the last iteration of the mth run of the latter estimator, and let the corresponding result be

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[0120] First,

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[0121] Then, the remaining amount

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[0122] If there are no errors during the detection process, after exactly m=2P runs, the sequence

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[0123] But more generally, errors can occur, e.g.

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[0124] P K Let (q) be the projection of sequence q onto set K. The projection returns either a sequence k∈K or an empty set φ, depending on whether q contains a sequence from K. If there are multiple feasible k∈K combinations of elements in q, then feasible elements with lower indices in q (rather than the element values ​​themselves) are preferred, as intended by greedy selection. Thus, equation (34) can be expanded to:

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[0125] In simple terms, equation (37) is that after the mth run of the BH-ISTA detector, the estimated vector for the next run is

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[0126] Obviously, the only other alternative is

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[0127] As above, y m and G m Updates are also hard decisions.

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[0128] FIG. 8 is a schematic diagram showing the structure of the proposed GB-ISTA receiver for QSM demodulation.

[0129] The procedure described in equations (31)-(33) and (35)-(39) is a greedy modification of the GB-ISTA detector introduced earlier, i.e., one symbol at a time and one index set at a time, and is called the greedy boxed iterative reduced binarization algorithm for QSM demodulation.

[0130] At the end of the process, an estimate of the index vector of the selected covariance matrix

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[0131] [Table 4]

[0132] V. Complexity and Performance Analysis In this section, we analyze the performance of the proposed OS-QSM through computer simulations. Since we focus on the scalability of the system, all simulation results presented are for a relatively large number of transmit antennas (i.e., n T≥ 6), an increasing number of transmission slots (i.e., T ≥ 2), the number of digitally modulated transmitted symbols P and the corresponding constellations M adjusted on a case-by-case basis. To the inventors' knowledge, simulation results for QSM schemes using such parameters have not appeared in the literature so far due to the high computational complexity of existing receivers.

[0133] A. Complexity: GB-ISTA vs. ML and SCMB-SD receivers From the latter perspective, we begin by evaluating the decoding complexity of the scaled QSM system by deriving the order of complexity, in particular, of the conventional ML and SCMB-SD approaches, as well as the proposed GB-ISTA algorithm described in Section IV.

[0134] Any n T , T and P, the brute-force ML decoder uses {k R ,k I}∈K, where K is the number of possible

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[0135] For ideality and simplicity, let us assume that each search takes one floating-point operation (flop):

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[0136] The practical impossibility of ML-based detection in QSM systems is clearly highlighted by equation (40), which shows that the number of transmitted symbols P is proportional to the number of all theoretically scalable quantities n T , T, and M. We then show that this challenge cannot be satisfactorily alleviated by the SD approach. To that end, again for the sake of ideality and simplicity, we consider the factor MP in equation (40) to be negligible, since SD allows the search radius to be reduced to a single symbol. In other words, we have shown that the order of complexity associated with a QSM receiver based on SD can be reduced to at most

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[0137] From the latter, with respect to the scalable QSM scheme, the only advantage of SD is that it allows scaling of the digital constellation cardinality M, which not only negatively impacts the corresponding BER but is also not the most important factor in increasing the system SE. The reason is that the total number of bits carried by the QSM scheme is

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[0138] Finally, we discuss the computational complexity of the proposed GB-ISTA. First, we consider the GB-ISTA receiver based on the spatially encoded bit b R and b I is not obtained from a search, but from the sparse restoration process, i.e., directly from the values ​​and positions of the non-zero elements of û. As a result of eliminating such combinatorial searches, we obtain a scalable parameter n T The impact of ,T, and P on GB-ISTA is significantly smaller as shown in the following complexity analysis of steps in Method 3.

[0139] 1) Method 3 takes the effective matrix G given in equation (27) as input, and its construction is T sparse block diagonal matrix with nonzero elements

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[0140] 2) The GB-ISTA receiver then performs multiple evaluations of equation (31), the first of which is

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[0141] This operation takes (2Tn T )(2Tn R )+(2Tn R )+(2Tn R )(2Tn T )+(2Tn R )=8T 2 n T n R +4Tn R Although it requires flops, the sparsity of u(η) quickly decreases to its actual value of 2P as shown in Figure 7(a), so the complexity of this step is more precisely 8PTn. R +4Tn R =4Tn R It is estimated that it takes (2P+1) flops. Then, the 2Tn required for the boxed hard binarization function Π T Steps including flops and η * The total cost of each evaluation of (31) is η * (4Tn R (2P+1)+2Tn T ) is estimated to be a flop.

[0142] 3) After the convergence of Eq. (31), the receiver obtains the sparse estimate vector

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[0143] 4) Considering that the cost of filtering elements, interference, and sequences represented by Eqs. (37)-(39) is negligible, the next significant cost for the receiver is the verification of the obtained indices. In particular, after at least P runs, a sufficient number of location indices is obtained.

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[0144] 5) Finally, as described in line 11 of Algorithm 3, GB-ISTA estimates the index vector

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[0145] From the above, the total complexity order of GB-ISTA is estimated as follows:

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[0146] The number of bits detected for each transmission of equations (40) and (42)

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[0147] Figure 9 shows the effect of the scalable parameters on the complexity of the QSM receiver. In Fig. 9(a), fixed P and variable T are T is shown as a function of . In Fig. 9(b), fixed T and variable P are T is shown as a function of . In Fig. 9(c), fixed n T , showing the variable T as a function of P.

[0148] B. BER performance of the proposed OS-QSM method Encouraged by the significant reduction in complexity achieved by GB-ISTA over ML detection as shown above, we evaluate the BER performance of the proposed OS-QSM scheme decoded via GB-ISTA. In general, our simulation experiments aim to further demonstrate 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 rather high spectral efficiency and achieving an extremely low BER at an extremely low E / Nb0.

[0149] Figure 10 shows the impact of scalability on the BER performance of the GB-ISTA detected OS-QSM scheme with fixed SE.

[0150] Therefore, the first set of results shown in Figure 10 shows the results for various n T We compare the BER performance of the proposed methods for T, P, and T. The curve for T=2 can be considered as a baseline corresponding to the SotA EDA-QSM scheme of

[22] ,

[23] , but we note that the results shown actually incorporate the improvements resulting from the enhancements described in Subsection III-B, namely: a) using the optimal golden code

[28] as opposed to the block-wise sphere-decodable FDFR-STBC of

[24] , and b) the optimal construction of the index vector set K given in Method 1.

[0151] Two important facts can be learned from the results in Figure 10. First, the number of receiving antennas is rather large (n RThe significant improvement in BER due to scaling is evident, despite the fact that the BER is 12. This indicates that the gain is not only due to the increased diversity (since the receiver diversity is already large), but also due to the coding gain resulting from the use of the optimal FDFR-STBC adopted in the OS-QSM design. Second, the results shown were actually simulated using a conventional computer (i.e., without the need for any particularly powerful machine), down to rather low BER, and for settings that are virtually impossible to simulate with ML- or SD-based receivers. The latter point further emphasizes the true feasibility of the proposed GB-ISTA receiver. T This is highlighted by the results on the right side of Figure 10, which includes the curve for the =12 system.

[0152] However, one criticism that can be made of the results of Figure 10 is that it means that the value of P adopted is not consistent with the corresponding T and n values ​​according to equation (14). T The parameterization used in Figure 10 makes it clear again that all systems have the same SE to allow direct comparison under identical conditions, which is incidentally also why all curves are plotted versus Eb / N0 rather than SNR.

[0153] Figure 11 shows the effect of P scaling on the BER performance of the GB-ISTA detected OS-QSM scheme.

[0154] 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, Fig. 11 shows additional results obtained by varying P up to the optimum value given in Eq. (14). Due to the floor function operation in the expression for the achievable SE given in Eq. (8), values ​​of P that are adjacent to the value given in Eq. (14), i.e., P * , is also optimal because it has the exact same SE. For example, as in the case of Figure 10, T = 6, T = 2 and M = 4, from equation (14) P *= 8, but in equation (8) we note that all values ​​P = {7, 8, 9} result in ζ = 16. Similarly, all values ​​P = {10, 11, 12, 13} result in n T = 6, T = 3 and M = 4, which is the maximum SE, ζ = 17.

[0155] With this in mind, turning to the results obtained in Figure 11, we see that the degradation in BER when upscaling P is very slight, and in fact becomes smaller as T increases as shown in Figure 11, which is a small and fair price to pay for nearly doubling the spectral efficiency of the system. The slight degradation in BER observed when optimally upscaling the ratio P / T towards SE is the result of a corresponding decrease in the sparsity of the vectorized received signal, which in systems with higher diversity and coding gain is n T ,T,or both tend to become less severe as a result. This trend is indeed visible in Figure 11, as the gap between the BER curves narrows as T=2 increases to T=3.

[0156] FIG. 12 shows a flow diagram of the proposed computer-implemented method for selecting optimal index vectors for spatial modulation.

[0157] This method uses the number of symbols P, the number of symbol slots T, and the number of transmit antennas n. T The method starts with input. The first step generates an empty set of valid vectors. The second step adds a random seed vector. Then it runs a routine to find the least frequently used index in the set. The next step selects a vector with the index and adds it to the set. It then checks whether the set has reached its maximum size. If not, it proceeds to find the least frequently used index in the set. If the set has reached its maximum size, the method ends.

[0158] Furthermore, the proposed decoder also provides great flexibility in transmitter development, since it does not require design constraints such as block diagonal or orthogonal transmission schemes, but only the inherent sparsity of spatial modulation schemes. In other words, the proposed decoder can accommodate many different coding structures (flexibility).

[0159] The decoder can be used in any MIMO system where the number of antennas is large and therefore ML approaches would be infeasible. The scheme can be used to improve the efficiency of cellular networks and V2X communications (eMBB).

[0160] This application is directed to a system with n transmit antennas. T The proposed method focuses on the design of a new transmitter and receiver for the QSM scheme, focusing on performance optimization in terms of SE, diversity, and coding gain, as well as the number of transmission instances T and the number of coded M-ary symbols P. The motivation for this application is that in order to achieve SE optimality, the QSM scheme can achieve n scalability, which is not possible with the SotA scheme. T , T, and P. On the transmitter side, the newly proposed OS-QSM scheme differs from the SotA alternative in that its covariance matrix is ​​designed based on FDFR STBS, and the covariance matrix index selection is performed using a novel greedy algorithm, ensuring that all space-time resources of the transmitter are evenly utilized across multiple transmissions. On the receiver side, the proposed technique benefits from a novel ISTA-based receiver, relying on the sparse structure of QSM signaling, eliminating the combined nature of existing ML- or SD-based approaches and further enabling system scaling in terms of feasibility. Indeed, a complexity analysis was performed and showed that the complexity of the proposed GB-ISTA receiver scales with T and n, with P as the exponent. T In contrast to ML and SD detectors, which have geometric complexity for T, are not feasible in scaled scenarios, and are therefore cubic for T, quadratic for P, and n TSimulation results in a scaling setting not previously shown in the relevant literature support both the high performance and feasibility of the proposed OS-QSM scheme and GB-ISTA receiver.

Claims

1. 1. A computer-implemented method for optimal index vector selection for spatial modulation using multiple transmit antennas, comprising: each of the plurality of transmit antennas represents an in-phase spatial constellation symbol in an in-phase spatial constellation and an orthogonal spatial constellation symbol in an orthogonal spatial constellation; determining a transmit antenna to be activated among the plurality of transmit antennas based on the index vector; the spatial modulation includes mapping source data to the in-phase spatial constellation symbols and the orthogonal spatial constellation symbols represented by the multiple transmit antennas; The method comprises: The number of symbols P, the number of symbol slots T, and the number of transmit antennas nT are input. In the first step, an empty set of valid vectors is generated, In the second step, a random seed vector is added to the set, Then, proceed with a routine to find the least frequently used index in said set; In the next step, the index vector having an index is selected and added to the set; Check whether the set has reached its maximum size, and if not, proceed to find the least frequently used index in the set, or stop if the set has reached its maximum size. A method characterized by:

2. A computer program comprising computer executable instructions which, when executed on a computer, cause the computer to carry out the method of claim 1.

3. A computer readable medium storing the computer program of claim 2.

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