Linear precoding in OTFS frequency-selective fading channels

Abstract

A method of transmitting information symbols over an OTFS communication channel subject to frequency-selective fading comprises precoding information symbols to be transmitted, yielding precoded information symbols, and arranging the precoded information symbols into the two-dimensional delay-Doppler domain, yielding a symbol matrix in the delay-Doppler domain. The symbol matrix in the delay-Doppler domain is then converted into an OTFS time-domain signal for transmission over the OTFS communication channel. The precoding comprises multiplying the information symbols to be transmitted with a precoding matrix having a trace that equals the product of the numbers or dimensions of the OTFS delay and Doppler resource grids, and whose elements are targeted to maximise a coding gain and / or a diversity gain on the OTFS communication channel.

Description

[0001] LINEAR PRECODING IN OTFS FREQUENCY-SELECTIVE FADING CHANNELS

[0002] FIELD OF THE INVENTION

[0003] The present invention relates to orthogonal time frequency space (OTFS) modulation, in particular for transmission over multipath frequency-selective fading channels.

[0004] NOTATIONS

[0005] Throughout this specification, bold symbols represent vectors or matrices, as in xand X. Superscripts T and H respectively denote the transpose and complex conjugate transpose of a vector or matrix, diag {a} is a diagonal matrix with vector a on its diagonal, while diag {A} is a vector whose elements are from the diagonal of matrix A. ® is the Kronecker product.

[0006] BACKGROUND

[0007] The deployment and evolution of 5G networks continue to play a significant role in the connectivity of Internet-of-Things (loT) devices. 5G offers higher data rates, lower latency, and improved connectivity over previous generations of public wireless networks, which is crucial for applications in industrial settings. However, high data rates and multipath propagation give rise to frequency-selectivity of wireless channels, which critically affects the communication performance. Multipath frequency-selective fading channels presents challenges that need to be addressed by specific signal processing techniques. Orthogonal frequency division multiplexing (OFDM) is particularly attractive in practice because it can transform a frequency- selective fading channel into parallel flat-fading sub-channels with the use of a sufficiently long cyclic prefix (CP), exhibiting complexity reduction at the receiver based on a subcarrier-by-subcarrier single-tap equalization. However, the performance of OFDM degrades significantly when channel nulls occur as it cannot exploit multipath diversity. Also, OFDM cannot guarantee symbol detectability. For improving the performance in frequency-selective fading channels coded OFDM is used, which employs error-correction codes such as convolutional codes, trellis- coded modulation (TCM), turbo codes and low density parity check (LDPC), before performing the actual OFDM processing. However, such coded OFDM schemes often incur high complexity, large decoding delay and potential transmission rate loss. In addition, the standard design paradigms of coded OFDM systems make it difficult to achieve maximum diversity gain.

[0008] These drawbacks give rise to the need for more efficient precoding OFDM schemes to exploit all of the available diversities in the mobile channel.

[0009] A linear constellation precoded OFDM system for multicamer transmissions over multipath frequency-selective fading channels proposed by Z. Liu, Y. Xin, and G. Giannakis, in "Linear constellation precoding for OFDM with maximum multipath diversity and coding gains," IEEE Trans. Commun., vol. 51 , no. 3, pp. 416-427, March 2003, exhibits no essential decrease in transmission rate. Specifically, a linear constellation precoder is developed to maximize both the diversity and coding gains in OFDM systems under frequency-selective fading channels. To simplify the decoding, an optimal subcarrier grouping scheme is further considered without sacrificing the diversity and coding gains.

[0010] In addition, multiple-input multiple-output (MIMO) systems employing the OFDM technique have been considered to further improve performance. Specifically, spacetime coded OFDM, space-frequency coded OFDM, and space-time-frequency coded OFDM have been investigated to achieve space diversity, joint space- frequency diversity, and joint space-time-frequency diversity, respectively.

[0011] However, to avoid inter-block interference, a CP of length larger than or equal to the maximal channel delay must be inserted per OFDM block at the transmitter and discarded at the receiver, leading to a lower spectral efficiency caused by the more significant CPs.

[0012] Another main limitation associated with OFDM is the high peak-to-average power ratio (PAPR) of the transmitted signals and high sensitivity to carrier frequency offset (CFO).

[0013] More recently, OTFS modulation has been proposed as a promising alternative physical (PHY)-layer modulation scheme to traditional OFDM for high mobility communications. Unlike OFDM, OTFS multiplexes information symbols in the delay- Doppler domain and exhibits performance advantages since it can exploit the diversity gain coming from both the channel delays and Doppler shifts. OTFS can effectively simplify channel estimation and symbol detection at wireless receivers by utilizing the property of a quasi-stationary sparse channel in the delay-Doppler domain for high-mobility communication scenarios. In addition, only one CP is required for an entire OTFS frame containing multiple OFDM blocks, leading to a high spectral efficiency compared to traditional OFDM systems. More importantly, OTFS has a smaller PAPR and is less sensitive to CFO than OFDM. Therefore, OTFS can enable a much more reliable and robust transmission over harsh wireless environments.

[0014] A number of studies and patents have already discussed OTFS. US Patent no. 11 ,777,566 B2 to R. Patchava, J. Ma, M. Soltani, and X. Zhang discloses using an OTFS precoding scheme to improve the system performance for a sounding reference signal. US patent application no. 2023 / 0155,761 A1 to the same inventors discloses demodulation reference signal (DMRS) precoding in high-Doppler scenarios. US patent application no. 2023 / 0327,827 A1 to the same inventors discloses using OTFS precoding for the tracking reference signal. US patent application no. 2023 / 0189,692 A1 to the same inventors discloses OTFS precoding of the physical sidelink control channel (PSCCH) and physical sidelink shared channel (PSSCH) channel in high-Doppler scenarios. US patent application no. 2023 / 0189,265 A1 to the same inventors discloses OTFS precoding of control channel and shared channel communications.

[0015] A method for co-existence between an OTFS modulation system and a Long Term Evolution (LTE) system is proposed in US patent no. 11 ,817,922 B2 to C. I. Casas, J. Delfeld, Y. Hebron, and S. Kons, where distinct precoding, i.e. , beamforming, is applied to different user groups with LTE scheme and OTFS modulation, respectively.

[0016] In US patent no. 11 ,296,919 B2 to J. Delfield a precoding scheme is proposed that adds a perturbation signal to the transmitted OTFS signal, where the expected interference plus noise is minimized. However, the channel state information (CSI) is required at the transmitter with complex channel tracking and predicting methods.

[0017] The diversity performance analysis of uncoded and coded OTFS systems over high- mobility doubly-selective fading channels have been respectively analysed and evaluated by G. Surabhi, R. M. Augustine, and A. Chockalingam, in "On the diversity of uncoded OTFS modulation in doubly-dispersive channels," IEEE Trans. Wireless Commun., vol. 18, no. 6, pp. 3049-3063, Jun. 2019, by P. Raviteja, Y. Hong,

[0018] E. Viterbo, and E. Biglieri, in "Effective diversity of OTFS modulation,” IEEE Wireless Commun. Lett., vol. 9, no. 2, pp. 249-253, Feb. 2020, and by S. Li, J. Yuan, W. Yuan, Z. Wei, B. Bai, and D. W. K. Ng, in "Performance analysis of coded OTFS systems over high-mobility channels,” IEEE Trans. Wireless Commun., vol. 20, no. 9, pp. 6033-6048, Sep. 2021.

[0019] However, attainability of the OTFS full diversity order in multipath frequency-selective fading channels has not been investigated nor has been proven theoretically in the literature. By performing pairwise error probability (PEP) analysis, the upper-bound of the diversity order of OTFS transmissions over randomly frequency-selective fading channels can be found. The PEP analysis also reveals that the original OTFS system cannot always guarantee full exploitation of the embedded diversity in such multipath frequency-selective channel scenario.

[0020] It is, therefore, desirable to propose efficient methods in OTFS systems that can guarantee both performance, i.e. , maximal diversity and coding gains, and high spectral efficiency, i.e., high data rate, in frequency-selective fading channels.

[0021] SUMMARY OF THE INVENTION

[0022] This need is addressed by the method of claim 1 and the apparatus of claim 9. A corresponding computer program product and computer-readable medium is provided in claims 10 and 11 , respectively. Advantageous embodiments and developments of the method and apparatus are provided in the respective dependent claims. The present invention will be discussed using the system model presented below. Figure 1 shows a schematic block diagram of a general OTFS transmission system in which precoded OTFS signals are transmitted over multi-path frequency-selective fading channels.

[0023] Without loss of generality, the information streams x e AMNxlare drawn from a finite modulation alphabet A, e.g., PSK and QAM symbols, where M and N represent the numbers of resource grids along the OTFS delay and Doppler dimensions, respectively. After linear precoding, the transmitted OTFS symbols x e (CMNxlcan be obtained as x = Vx, (1) where V e (CM / VxM / vwill be designed later on, targeting guaranteeing maximum diversity and large coding gains.

[0024] The information symbols x e (CMNxlare then arranged into the two-dimensional delay-Doppler plane X e i.e. , X = invec(x). By using the inverse symplectic finite Fourier transform (ISFFT), the delay-Doppler symbols X e (CMxNare converted into the time-frequency domain X e

[0025] X = FMXF», (2) where FMe (CMxMand FNe (CNxNare the normalized M-point and N-point fast Fourier transform (FFT) matrices, respectively.

[0026] Next, the time domain signal s e (CMNxlis generated by applying a Heisenberg transform with a transmit pulse gtx(t), where T (seconds) and A / = (Hz) are determined to be larger than the maximum channel delay spread and maximal Doppler frequency shift, respectively. The symbol spaced sampling interval is Ts= l / M&f.

[0027] To overcome the inter-frame interference, a cyclic prefix (CP) is added in front of the generated time domain signal with a length no shorter than the maximal channel delay spread. The resulting time domain signal enters the multipath frequency selective fading channels characterized by a finite-impulse response h = [ / i[0], / i[l], ■■■,h [L - I]]7e (CLxl, where h[p] is the complex gain for the p-th channel tap.

[0028] At the OTFS receiver, the received signal r e (CMNxlis obtained after removing the CP, where n e (CMNxl~ CMO'NoI) is the received noise and the notation [-]mdenotes the mod-m operation.

[0029] The received time domain signal r is then subjected to a Wigner transform, i.e. , the inverse of a Heisenberg transform, with a receive pulse grx(t) to generate the time- frequency domain signal,

[0030] Finally, the delay-Doppler domain signal can be recovered by applying a symplectic finite Fourier transform (SFFT) to the time-frequency signal Y e (CMxNas

[0031] Y = F^YFN. (6) For simplicity, it is assumed that a rectangular pulse is applied for both #tx(t) and grx(t) 'nthe above steps, for which the end-to-end input-output relationship of the OTFS transmission in the delay-Doppler domain is given by where to e (CMxNis the noise at the output of the SFFT and

[0032] To summarize, the input-output relationship of equation (7) can be vectorized column-wise into where 4>(x) = (Fw0 IM)F^Ndiag {FMN(F^ 0 IM)Vx}FMNxLand the noise term is omitted for simplicity.

[0033] It is assumed that the channel state information (CSI) is known at the receiver but not at the transmitter. Given the received signal y, it is the object to decode x with maximum diversity, i.e. , exploiting multipath to the maximum possible, and large coding gains. These goals will be achieved by carefully designing the precoding matrix V e cMNxMN, as will be described further below.

[0034] Prior to designing the precoding matrix V it will be necessary to derive the performance criteria for the precoded OTFS systems, and also to determine the maximum achievable diversity and coding gains for such systems. Exact bit-error rate (BER) performance analysis would be desirable, but is difficult, if not impossible to achieve. Instead, the invention resorts to the pairwise error probability (PEP) analysis based on the maximum likelihood (ML) detector, which can provide a good approximation for BER at high signal-to-noise ratio (SNR).

[0035] Assuming perfect CSI is available at the receiver, the conditional PEP, i.e. , the probability of transmitting x but erroneously deciding on x, is given by where Q(x) is the tail distribution function of the standard Gaussian distribution and P = denotes the signal-to-noise ratio (SNR).

[0036] Note that C = (4>(x) - (x))H((x) - C>(x)) is a Hermitian matrix, its rank and the nonzero eigenvalues being defined respectively. Hence, where U is a unitary matrix,

[0037] Substituting equation (11 ) in equation (10), the conditional PEP can be written as Since h is obtained by multiplying a unitary matrix with h, it has the same distribution as that of h. The elements in h are assumed to be independent and identically distributed complex Gaussian random variables. Considering h ~ the final

[0038] PEP is calculated by averaging equation (12) over the channel statistics and given by where E[-] represents the expectation operation. At high SNRs (i.e. , p oo ), equation (13) can be further simplified as

[0039] From the above analysis, it can be concluded that the system diversity order is determined by R, which could be as high as the number of resolvable paths of the channel. (nf=i^i)Rstands for the pairwise coding gain to control how this PEP shifts relative to the benchmark error-rate curve of (p / 4EyR.

[0040] Accounting for all possible pairwise errors, the diversity and coding gains, respectively, are defined herein as Because the system performance depends on both Gdand Gc, it is important to maximize both Gdand Gc. By checking the dimensionality of C, it is clear that the maximum diversity gain Gd max= L is achieved if and only if the matrix C has full rank for Vx x. When the maximum diversity gain Gd max= L is achieved, the coding gain becomes (15) = IE[hhH]. Equation (15) implies that Gcis a function of the minimum determinant where 0j is the i-th row of 0.

[0041] It is readily apparent that the diversity gain Gdand the coding gain Gcare both depending on the choice of the precoding matrix V. Without a proper precoding matrix V, it is not possible to achieve the maximum diversity and coding gains, leading to significant performance loss. Further, at high SNR it is reasonable to maximise the diversity gain first, because it determines the slope of the log-log bit- error rate (BER)-SNR curve. Within the class of Vs that achieve Gd max, the coding gain Gcshould be maximized as much as possible afterwards.

[0042] In the general precoding setup, it is only ensured that Tr(VVH) = MN to normalize the total transmit power, but no structural constraints are imposed on V. The design criteria for V can be summarized as follows. 1 . Maximize the diversity gain: Develop a matrix V with Tr(WH) = MN such that, where 0] is the i-th row of 0, with 0 — FM / V (F / V 0 IM)V.

[0043] 2. Maximize the coding gain: Develop a matrix V with Tr(VVH) = MN to maximize

[0044] It is noted that when criterion 2 is satisfied, criterion 1 will be automatically satisfied.

[0045] For finding a solution that satisfies criterion 2 a Vandermonde / unitary matrix 0 is constructed using algebraic number theory. Specifically, 0 can be expressed as a Vandermonde matrix where ft is a normalization factor chosen to guarantee the power constraint Tr(WH) = MN, and the selection of elements, or parameters, MN, for example

[0046] If MN = 2d(d > 1), the parameter akis determined as

[0047] If MN = 3 x 2d(d > 0), the parameter akis specified as (21)

[0048] If MN = 2dx 3t(d > 1, t > 1), the parameter akis given by (22) After determining 0, the precoding matrix is given by (23)

[0049] The following section discusses the performance of the proposed precoded OTFS systems for frequency-selective fading channels. In the considered simulations the carrier frequency is centred at 4GHz and the subcarrier spacing is A / = 15kHz. A typical urban channel model as presented “Study on Channel Model for Frequencies From 0.5 to 100 GHz", Standard 3GPP TR 38.901 , 2017 is adopted with exponential power delay profile. It is assumed that the perfect channel knowledge is available at the receiver and QPSK modulation is applied.

[0050] In figure 2 the effectiveness of the proposed precoding results for OTFS systems with ML detector is examined for different numbers of resolvable paths. A delay-Doppler plane with M= 4 and N= 2 is considered, where the complexity of the ML detector is still acceptable. It is obvious that the BER performance improves as the number of resolvable paths L increases for both the precoded, non-precoded and conventional phase rotation OTFS systems. This is due to the fact that high diversity gain can be obtained for better performance with large value of L. It is also apparent that the traditional non-precoded and phase rotation OTFS systems cannot exploit the full diversity, leading to significant performance loss at high SNR. However, the proposed precoded OTFS system outperforms the traditional non-precoded and phase rotation ones, and can achieve the potential maximal diversity and coding gains to improve the system performance. This confirms the targeted effectiveness of the proposed precoded OTFS systems.

[0051] As the complexity of the ML detector grows exponentially with the system dimension, it cannot be directly applied to practical large-dimensional systems due to the intolerable computational burden.

[0052] Therefore, in embodiments of the present invention, a different detector is used. In the following, the BER performance of the proposed precoded OTFS in large- dimension systems is simulated, where M = 8 and N = 16. In place of the ML detector the practical low-complexity advanced Memory AMP detector discussed by Y. Ge, L. Liu, S. Huang, D. G. G., Y. L. Guan, and Z. Ding, in "Low-complexity memory AMP detector for high-mobility MIMO-OTFS SCMA systems," 2023 IEEE International Conference on Communications Workshops (ICC Workshops), May 2023, pp. 807-812, is adopted to further verify the advantage of the proposed precoding results for OTFS systems compared to the traditional non-precoded and phase rotation ones.

[0053] From the results of the simulation shown in figure 3 it can be observed, like for the ML detector discussed with reference to figure 2, that the BER performance of both precoded, non-precoded and existing phase rotation OTFS systems improves as L increases since the potential higher diversity can be exploited from a larger number of independent resolvable paths. Still, the proposed precoded OTFS system outperforms the traditional non-precoded and phase rotation ones by using the practical low complexity detectors. Specifically, the precoded OTFS system shows respectively around 4 dB and 3 dB gains over the unprecoded and existing phase rotation OTFS systems at BER = 10-5under L = 4.

[0054] In view of the foregoing discussion, in accordance with a first aspect of the invention, a method of transmitting information symbols over an OTFS communication channel subject to frequency-selective fading is presented. The information symbols, which in typical use cases represent binary data, are represented by information about an amplitude and / or a phase, which information is carried in a symbol vector. The method comprises receiving the information symbols, i.e. , the symbol vector, to be transmitted, and precoding the received information symbols, yielding precoded information symbols. The precoded information symbols are arranged into the two- dimensional delay-Doppler domain, yielding a symbol matrix in the delay-Doppler domain, and are converted from the delay-Doppler domain into a corresponding time- frequency domain matrix. Next, the time-frequency domain matrix is converted into a time-domain signal, which is then transmitted over the OTFS communication channel. Transmitting may be carried by a transmitter, e.g., a radio transmitter, comprising radio frequency (RF) circuitry such as amplifiers, mixers, filters and one or more antennas. In accordance with the invention the precoding comprises multiplying the received information symbols with a precoding matrix designed targeted to maximise a coding gain and / or a diversity gain on the OTFS communication channel, prior to arranging the precoded information symbols into the two-dimensional delay- Doppler domain.

[0055] In one or more embodiments of the method, precoding comprises, for determining the precoding matrix, determining one or more candidate precoding matrices that each maximises the diversity gain and, if two or more candidate precoding matrices that each maximises the diversity gain are determined, determine, from said two or more candidate precoding matrices that each maximises the diversity gain, the precoding matrix that maximises the coding gain.

[0056] In one or more embodiments of the method, precoding comprises, for determining the precoding matrix, determining a precoding matrix that maximises the coding gain.

[0057] In one or more embodiments of the method, precoding comprises, for determining the precoding matrix, determining a Vandermonde matrix, and a normalisation factor that imposes the maximum transmit power constraint. The elements, or parameters, of the Vandermonde matrix may be determined in dependence of the numbers or dimensions of the OTFS delay and Doppler resource grids.

[0058] In one or more embodiments of the method, converting the symbol matrix from the delay-Doppler domain into a corresponding time-frequency domain matrix comprises subjecting the symbol matrix in the delay-Doppler domain to an inverse symplectic fast Fourier transform, ISFFT.

[0059] In one or more embodiments of the method, converting the time-frequency domain matrix into a time-domain signal comprises subjecting the time-frequency domain matrix to a Heisenberg transform.

[0060] In one or more embodiments the method further comprises adding a cyclic prefix, CP, prior to transmitting the time-domain signal over the OTFS communication channel. In accordance with a second aspect of the invention an apparatus configured for executing the method in accordance with the first aspect of the invention is presented. The apparatus is generally configured for transmitting signals over an OTFS communication channel, and comprises one or more antennas, radio frequency circuitry associated with the one or more antennas, e.g., amplifiers, filters, mixers and the like, and further comprises one or more microprocessors, and volatile and non-volatile memory functionally associated and coupled with the one or more microprocessors. The aforementioned components and elements are functionally and / or communicatively coupled via one or more signal and / or data lines and / or buses. The non-volatile memory stores computer program instructions which, when executed by the one or more microprocessors, configures the apparatus to execute the method in accordance with the first aspect of the invention and / or to control components and / or elements of the apparatus accordingly.

[0061] The method described hereinbefore may be represented by computer program instructions. Thus, in accordance with a third aspect of the present invention a computer program product comprises computer program instructions which, when executed by a microprocessor of or functionally coupled with an apparatus in accordance with the second aspect of the invention, cause the processor to carry out a method in accordance with the first aspect of the invention, and / or to accordingly control hardware components and / or elements, and / or software blocks or modules of the apparatus.

[0062] Computer program instructions, or code, for carrying out operations for embodiments may be any number of lines and may be written in any combination of one or more programming languages including an object- oriented programming language such as Python, Ruby, Java, Smalltalk, C++, or the like, and conventional procedural programming languages, such as the “C” programming language, or the like, and / or machine languages such as assembly languages. The code may execute entirely on the user’s computer, partly on the user’s computer, as a stand-alone software package, partly on the user’s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user’s computer through any type of network, including a local area network (LAN), wireless LAN (WLAN), or a wide area network (WAN), or the connection may be made to an external computer, for example, through the Internet using an Internet Service Provider (ISP).

[0063] The computer program instructions may be retrievably stored or transmitted on a computer-readable medium or data carrier. The medium or the data carrier may by physically embodied, e.g., in the form of a hard disk, solid state disk, flash memory device or the like. However, the medium or the data carrier may also comprise a modulated electro-magnetic, electrical, or optical signal that is received by the computer by means of a corresponding receiver, and that is transferred to and stored in a memory of the computer.

[0064] The described features, structures, or characteristics of the embodiments may be combined in any suitable manner. In this description, numerous specific details are provided, such as examples of programming, software modules, user selections, network transactions, database queries, database structures, hardware modules, hardware circuits, hardware chips, etc., to provide a thorough understanding of embodiments. One skilled in the relevant art will recognize, however, that embodiments may be practiced without one or more of the specific details, or with other methods, components, materials, and so forth. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of an embodiment. Reference throughout this specification to “one embodiment,” “an embodiment,” or similar language means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. Thus, appearances of the phrases “in one embodiment,” “in an embodiment,” and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment, but mean “one or more but not all embodiments” unless expressly specified otherwise. The terms “including,” “comprising,” “having,” and variations thereof mean “including but not limited to,” unless expressly specified otherwise. An enumerated listing of items does not imply that any or all of the items are mutually exclusive, unless expressly specified otherwise. The terms “a,” “an,” and “the” also refer to “one or more” unless expressly specified otherwise. Where aspects of the embodiments are described in this specification with reference to schematic flowchart diagrams and / or schematic block diagrams of methods, apparatuses, systems, and program products according to embodiments it will be understood that each block of the schematic flowchart diagrams and / or schematic block diagrams, and combinations of blocks in the schematic flowchart diagrams and / or schematic block diagrams, can be implemented by code. This code may be provided to a processor of a general-purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in the flowchart diagrams and / or block diagrams.

[0065] It should be noted that, in some implementations or embodiments, the functions noted in the exemplary embodiments shown in the figures may occur out of the order shown in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. Other steps and methods may be conceived that are equivalent in function, logic, or effect to one or more blocks, or portions thereof, shown in the figures.

[0066] The linear precoding scheme for OTFS systems based on algebraic number theory tools proposed by the present invention effectively realizes the maximal achievable multipath diversity gain and potential coding gains for wireless transmissions over multipath frequency-selective fading channels. This significantly improves the receiver performance compared to original non-precoded and the conventional phase rotation OTFS schemes, and is less sensitive to CFO than traditional OFDM systems.

[0067] The proposed precoding design for OTFS does not require the CSI at the transmitter, guarantees symbol detectability regardless of channel nulls, and can be used for an arbitrary size of system dimension without any transmission rate loss. Simulations demonstrated that the proposed precoding design for OTFS system exhibits sufficient statistic diversity of multipath frequency-selective fading channels, and outperforms the original non-precoded and conventional phase rotation OTFS systems for both an optimal ML detector and a low-complexity advanced Memory AMP detector.

[0068] In addition to the above, the proposed linear precoding scheme for OTFS system improves the spectral efficiency compared to traditional OFDM system due to requiring only one CP for each OTFS frame.

[0069] The simulation results demonstrate that the proposed precoded OTFS system can exploit the multipath diversity gain to achieve better performance than original non- precoded and conventional phase rotation OTFS systems for both optimal ML detector and low-complexity advanced Memory approximate message passing (AMP) detector.

[0070] The linear precoding scheme in accordance with the present invention is expected to support high data rates wireless transmissions and to provide robust and ultra- reliable communications for a wide range of emerging large-scale applications, including online gaming, virtual reality (VR) and augmented reality (AR), indoor wireless local area networks (WLANs), wireless sensor networks, smart cities and remote health care. The linear precoding scheme in accordance with the present invention can also be combined with future extra-large-scale MIMO and reconfigurable intelligent surface (RIS) communication systems, 6G wireless factory for industrial 4.0 and 6G massive machine-type loT scenarios in a straightforward manner.

[0071] BRIEF DESCRIPTION OF THE DRAWING

[0072] In the following section exemplary embodiments of the invention will be described in greater detail with reference to the drawing. In the drawing,

[0073] Fig. 1 shows a schematic block diagram of an OTFS transmission system in accordance with the present invention,

[0074] Fig. 2 shows a BER performance comparison with different numbers of resolvable paths under ML detector,

[0075] Fig. 3 shows a BER performance comparison with different number of resolvable paths under Memory AMP detector, Fig. 4 shows an exemplary flow diagram of a method in accordance with the present invention, and

[0076] Fig. 5 shows a schematic block diagram of an apparatus configured for executing the method in accordance with the present invention.

[0077] Throughout the figures identical or similar elements may be referenced using the same reference designators.

[0078] DETAILED DESCRIPTION OF EMBODIMENTS

[0079] Figures 1 to 3 have been described further above and will not be discussed again.

[0080] Figure 4 shows an exemplary flow diagram of a method 100 in accordance with the first aspect of the present invention. In step 102 information symbols x to be transmitted are received. In step 104 the received information symbols x are precoded, yielding precoded information symbols x, which are subsequently arranged, in step 106, into the two-dimensional delay-Doppler domain, yielding a symbol matrix x in the delay-Doppler domain. In step 108 the symbol matrix x is converted from the delay-Doppler domain into a corresponding time-frequency domain matrix X, which is subsequently, in step 100, converted into a time-domain signal s. A cyclic prefix may be added to the time-domain signal s in step 112 before it is transmitted, in step 114, over the OTFS communication channel.

[0081] Figure 5 shows a schematic block diagram of an apparatus configured for executing the method in accordance with the present invention. The apparatus comprises one or more antennas 601 , radio frequency circuitry 602 associated with the one or more antennas 601 , and further comprises one or more microprocessors 603, and volatile 604 and non-volatile 605 memory functionally associated with the one or more microprocessors 603. The aforementioned components and elements are functionally coupled via one or more signal and / or data lines and / or buses 606. The non-volatile memory 605 stores computer program instructions which, when executed by the one or more microprocessors 603, configures the apparatus 600 to execute the method in accordance with the first aspect of the invention as described herein and / or to accordingly control components and / or elements of the apparatus.

Claims

CLAIMS1 . A method (100) of transmitting information symbols over an OTFS communication channel subject to frequency-selective fading, comprising:- receiving (102) information symbols (x) to be transmitted,- precoding (104) the received information symbols (x), yielding precoded information symbols (x),- arranging (106) the precoded information symbols (x) into the two- dimensional delay-Doppler domain, yielding a symbol matrix (x) in the delay- Doppler domain,- converting (108) the symbol matrix (x) from the delay-Doppler domain into a corresponding time-frequency domain matrix (X),- converting (110) the time-frequency domain matrix (X) into a time-domain signal (s), and- transmitting (114) the time-domain signal (s) over the OTFS communication channel, wherein precoding (104) comprises:- multiplying, prior to arranging (106) the precoded information symbols (x) into the two-dimensional delay-Doppler domain, the received information symbols (x) with a precoding matrix (V) having a trace that equals the product of the numbers or dimensions of the OTFS delay and Doppler resource grids, and whose elements are targeted to maximise a coding gain and / or a diversity gain on the OTFS communication channel.

2. The method (100) of claim 1 , wherein precoding (104) comprises, for determining the precoding matrix (V), determining one or more candidate precoding matrices (V) that each maximises the diversity gain and, if two or more candidate precoding matrices (V) that each maximises the diversity gain are determined, determining from said two or more candidate precoding matrices (V) that each maximises the diversity gain, the precoding matrix (V) that maximises the coding gain.

3. The method (100) of claim 1 , wherein precoding (104) comprises, for determining the precoding matrix (V), determining one or more candidateprecoding matrices (V) that each maximise the coding gain and, if two or more candidate precoding matrices (V) that each maximises the coding gain are determined, selecting one for use with the method.

4. The method (100) of claim 2 or 3, wherein precoding (104) comprises, for determining the candidate precoding matrix (V), determining one or more Vandermonde matrices (0) whose elements maximise the diversity and / or the coding gain, and an associated normalisation factor (J3) that imposes the maximum transmit power constraint.

5. The method (100) of claim 4, wherein the elements of the Vandermonde matrix (0) are determined in dependence of the numbers or dimensions of the OTFS delay and Doppler resource grids.

6. The method (100) of claims 4 or 5, wherein any candidate precoding matrix (V) for which the row-wise product of- a Vandermonde matrix (0) that is formed from the product of the candidate precoding matrix (V), a Fourier matrix (FMN) having a number of points equal to the product of the numbers or dimensions of the OTFS delay (M) and Doppler (N) resource grids, and the Kronecker product of the conjugate transpose of a Fourier matrix (F^) having a number of points equal to the number of OTFS Doppler resource grids (N) and an identity matrix (IM) having a number of points equal to the number of OTFS delay resource grids (M), and-the difference between a transmitted symbol and a symbol that has a non- zero probability for being erroneously false detected, has a non-zero absolute value is a precoding matrix (V) that maximises the diversity gain.

7. The method (100) any one or more of claims 4 to 6, wherein any candidate precoding matrix (V) for which the product over squared absolute values of the rows of a Vandermonde matrix (0) that is formed from the product of the candidate precoding matrix (V), a Fourier matrix (FMN) having a number of points equal to the product of the numbers or dimensions of the OTFS delay(Af) and Doppler (N) resource grids, and the Kronecker product of the conjugate transpose of a Fourier matrix (F^) having a number of points equal to the number of OTFS Doppler resource grids (N) and an identity matrix (IM) having a number of points equal to the number of OTFS delay resource grids (M) is minimal is a precoding matrix (V) that maximises the coding gain.

8. The method (100) of any one or more of the preceding claims, wherein converting (108) the symbol matrix (x) from the delay-Doppler domain into a corresponding time-frequency domain matrix (X) comprises subjecting the symbol matrix (x) in the delay-Doppler domain to an inverse symplectic fast Fourier transform, ISFFT.

9. The method (100) of any one or more of the preceding claims, wherein converting (110) the time-frequency domain matrix (X) into a time-domain signal (s) comprises subjecting the time-frequency domain matrix (X) to a Heisenberg transform.

10. The method (100) of any one or more of the preceding claims, further comprising:- adding (112) a cyclic prefix, CP, prior to transmitting (114) the time-domain signal (s) over the OTFS communication channel.

11. Apparatus (600) configured for transmitting signals over an OTFS communication channel, comprising one or more antennas (601), radio frequency circuitry (602) associated with the one or more antennas (601 ), one or more microprocessors (603), and volatile (604) and non-volatile (605) memory functionally associated with the one or more microprocessors (603), the aforementioned components and elements being functionally coupled via one or more signal and / or data lines and / or buses (606), wherein the non- volatile memory (605) stores computer program instructions which, when executed by the one or more microprocessors (603), configures the apparatus (600) to execute the method in accordance with one or more of claims 1 to 8 and / or to control components and / or elements of the apparatus (600) accordingly.

12. Computer program product comprising computer program instructions which, when executed by a microprocessor, cause microprocessor and / or control hardware components of an apparatus configured for transmitting signals over an OTFS communication channel in accordance with claim 9 to execute the method (100) of one or more of claims 1 to 8.

13. Computer readable medium or data carrier retrievably transmitting or storing the computer program product of claim 10.

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