WO2025219973A1 - Linear precoding in otfs time-selective fading channels - Google Patents

Abstract

A method of transmitting information symbols over an OTFS communication channel subject to time-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

2024P01254WO -1- _{METHOD OF PROCESSING SIGNALS FOR WIRELESS COMMUNICATION OVER} TIME-SELECTIVE FADING CHANNELS FIELD OF THE INVENTION The present invention relates to orthogonal time frequency space (OTFS) modulation, in particular for transmission over multipath time-selective fading channels. NOTATIONS Throughout this specification, bold symbols represent vectors or matrices. _{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. Ä is the Kronecker product.} BACKGROUND The deployment and evolution of 5G networks continue to play a significant role in _{the connectivity of Internet-of-Things (IoT) 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. The demand for the} variety and the number of mobile devices along with the high-mobility communication applications such as high-speed railways, low earth orbit (LEO) satellite communications and vehicle-to-everything (V2X) have increased tremendously. However, carrier frequency offsets and mobility between the transmitter and receiver induced Doppler frequency spreads introduce time-selectivity in wireless channels, which affects critically communication performance. Modelling temporal channel variations and coping with Doppler time-selective fading channels present challenges _{that stimulate research and development in 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 due to the severe inter-} carrier-interference (ICI) caused by the channel Doppler spread in time-selective 2024P01254WO -2- _{fading channels. Coded OFDM, by using error-correction codes such as} convolutional codes, trellis-coded modulation (TCM), turbo codes and low density parity check (LDPC), are usually invoked before the OFDM processing, in order to _{deal with the frequency-selective fading channels and achieve better performance.} 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 maximal diversity gain and hard to extend for time-selective fading channels. _{These drawbacks give rise to the need for more efficient coding and modulation} schemes that improve the reliability of information transmission over rapidly fading wireless links. A linear constellation precoded OFDM system for multicarrier 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. 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. X. Ma, G. Leus, and G. Giannakis, in "Space-time-Doppler block coding for correlated time-selective fading channels," IEEE Trans. Signal Process., vol.53, no. 6, pp.2167-2181, June 2005, have developed a space-time-Doppler coded system that guarantee the maximum possible space-Doppler diversity, along with the largest coding gains in time-selective fading channels. 2024P01254WO -3- X. Ma and G. Giannakis, in "Maximum-diversity transmissions over doubly selective wireless channels," IEEE Trans. Inf. Theory, vol.49, no.7, pp.1832-1840, July 2003, propose a block precoded transmission for single-carrier communications to guarantee the maximum diversity gain in doubly-selective fading channels. However, to achieve the full diversity, in the first cited reference a cyclic prefix (CP)/zero padding (ZP) guard interval must be inserted per block at the transmitter _{and discarded at the receiver, and in the second citation the spreading technique is} applied, both leading to a lower spectral efficiency caused by the more significant CPs/ZPs or lower spreading gain. _{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. _{A number of studies and patents have already discussed OTFS. US Patent no. 11,777,566 B1 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 publication no. 2023/0155,761 A1 to the same inventors} discloses demodulation reference signal (DMRS) precoding in high-Doppler _{scenarios. US patent publication no. 2023/0327,827 A1 to the same inventors discloses using OTFS precoding for the tracking reference signal. US patent application nos. 2023/0198,692 A1 and 2023/0189265 A1 to the same inventors} disclose OTFS precoding of the physical sidelink control channel (PSCCH) and 2024P01254WO -4- _{physical sidelink shared channel (PSSCH) channel, respectively, in high-Doppler} scenarios. 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. _{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. B. C. Pandey, S. K. Mohammed, P. Raviteja, Y. Hong, and E. Viterbo, in "Low complexity precoding and detection in multi-user massive MIMO OTFS downlink," IEEE Trans. Veh. Tech., vol.70, no.5, pp.4389-4405, May 2021, propose a low-complexity multi-user precoding and detector for downlink massive MIMO-OTFS systems. However, the channel state information (CSI) is required at the transmitter, with complex channel tracking and predicting methods. 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, _{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. However, attainability of the OTFS full diversity order in time-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} 2024P01254WO -5- the diversity order of OTFS transmissions over randomly time-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 time- selective channel scenario. 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 Doppler time-selective fading channels.} SUMMARY OF THE INVENTION This need is addressed by the method of claim 1 and the apparatus of claim 11. A corresponding computer program product and computer-readable medium is _{provided in claims 12 and 13, respectively. Advantageous embodiments and} developments of the method and apparatus are provided in the respective dependent claims. _{The present invention adopts the basis expansion model (BEM) to capture the time- selective fading channels and proposes a linear precoding scheme for the OTFS system based on algebraic number theory tools. By performing pairwise error} probability (PEP) analysis, the proposed linear precoded OTFS system can achieve the maximal diversity and potential coding gains for wireless transmissions over _{Doppler time-selective fading channels. In addition, the proposed linear precoding} design for OTFS does not require the channel state information (CSI) at the transmitter and can be used for arbitrary-number system dimensions. Further, the proposed linear precoded OTFS system guarantees symbol detectability irrespective _{of the underlying constellation and regardless of channel nulls, and does not reduce} the transmission rate. It turns out that the proposed linear precoding design for OTFS _{system can exploit the maximal Doppler diversity gain to improve the receiver} performance compared to original non-precoded as well as the existing phase _{rotation OTFS schemes, and is less sensitive to carrier frequency offset (CFO) than} traditional OFDM systems. The present invention will be discussed in greater detail using the system model _{presented below. Figure 1 shows a schematic block diagram of a general OTFS} 2024P01254WO -6- _{transmission system in which precoded OTFS signals are transmitted over high- mobility time-selective fading channels. Without loss of generality, the information streams퐱 ∈ 픸��×� are drawn from a finite modulation alphabet픸, e.g., PSK and QAM symbols, where푀 and푁 represent the} numbers of resource grids along the OTFS delay and Doppler dimensions, _{respectively. After linear precoding, the transmitted OTFS symbols퐱� ∈ ℂ��×� can be} obtained as 퐱_{� = 퐕퐱, (1) where퐕 ∈ ℂ��×�� will be designed later on, targeting guaranteeing maximum} diversity and large coding gains. _{The information symbols퐱� ∈ ℂ��×� are then arranged into the two-dimensional delay-Doppler plane퐗 ∈ ℂ�×�, i.e.,퐗 = invec(퐱�). By using the inverse symplectic finite Fourier transform (ISFFT), the delay-Doppler symbols퐗 ∈ ℂ�×� are converted into the time-frequency domain퐗� ∈} 퐗_{� = 퐅 � �퐗퐅�, (2) where퐅 ∈ ℂ�×� and퐅 ∈ �×� � � ℂ are the normalized푀-point and푁-point fast Fourier} transform (FFT) matrices, respectively. _{Next, the time domain signal퐬 ∈ ℂ��×� is generated by applying a Heisenberg} transform with a transmit pulse푔_��(푡), _{where푇 (seconds) and Δ푓 = (Hz) are determined to be larger than the maximum} channel delay spread and maximal Doppler frequency shift, respectively. The symbol _{spaced sampling interval is푇� = 1/푀Δ푓.} 2024P01254WO -7- _{The resulting time-domain signal enters the high-mobility time-selective fading channels characterized through BEM with where푄 = 2�푁푓‾���� is the order of BEM basis functions,푓‾��� = 푓���/Δ푓, with푓���} being the maximum Doppler frequency.휔 �� � � =_{���푞 −���� denotes the푞-th BEM modeling frequency and푐� is the푞-th BEM channel coefficient. ⌈⋅⌉ represents the round-up operation. Here, the channel ℎ[푐] changes along with the time index푐 and} the Doppler spread is controlled by the maximum Doppler frequency푓_���, i.e., the Doppler spread may consist of multiple Doppler shifts which are no larger than the _{maximum Doppler frequency. Note that the BEM facilitates the development and} analysis of the diversity for OTFS systems over time-selective fading channels. _{At the OTFS receiver, the received signal퐫 ∈ ℂ��×� is obtained as 푟[푐] = ℎ[푐]푠[푐] +푛[푐],푐 = 0,1, ⋯ ,푀푁 − 1, (5) where퐧 ∈ ℂ��×� ∼ is the noise signal at the receiver. The received time domain signal퐫 is then subjected to a Wigner transform, i.e., the inverse of a Heisenberg transform, with a receive pulse푔��(푡) to generate the time-} frequency domain signal, Finally, the delay-Doppler domain signal can be recovered by applying a symplectic _{finite Fourier transform (SFFT) to the time-frequency signal퐘� ∈ ℂ�×� as 퐘 = 퐅� �퐘�퐅�. (7)} 2024P01254WO -8- _{For simplicity, it is assumed that a rectangular pulse is applied for both푔��(푡) and 푔��(푡) in the above steps, for which the end-to-end input-output relationship of the} OTFS transmission in the delay-Doppler domain is given by _{where흎 ∈ ℂ��×� is the noise at the output of the SFFT and} To summarize, the input-output relationship in equation (8) can be rewritten as _{It is assumed that the channel state information (CSI) is known at the receiver but not at the transmitter. Given the received signal퐲, it is the object to decode퐱 with} 2024P01254WO -9- _{maximum diversity, i.e., exploiting multipath to the maximum, and large coding gains.} These goals will be achieved by carefully designing the precoding matrix _{퐕 ∈ ℂ��×��, as will be described further below. Prior to designing the precoding matrix퐕 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). Assuming perfect CSI is available at the receiver, the conditional PEP, i.e., the _{probability of transmitting퐱 but erroneously deciding on퐱̂, is given by where푄(푥) is the tail distribution function of the standard Gaussian distribution and 휌 = � �� denotes signal-to-noise ratio (SNR). Note that퐂 = (횽(퐱) −횽(퐱̂))�(횽(퐱) −횽(퐱̂)) is a Hermitian matrix, its rank and the nonzero eigenvalues being defined respectively. Hence, where퐔 is a unitary matrix,퐡̅ = 퐔�퐡 and횺 = diag{휆�,휆�, ⋯ ,휆�}.} Substituting equation (11) in equation (11), the conditional PEP can be written as 2024P01254WO -10- _{Since퐡̅ is obtained by multiplying a unitary matrix with퐡, it has the same distribution as that of 퐡. The elements in퐡̅ are assumed to be independent and identically distributed complex Gaussian random variables. Considering퐡̅ ∼ 퐈�, the} final PEP is calculated by averaging equation (13) over the channel statistics and given by � _{where피[⋅] represents the expectation operation. At high SNRs (i.e.,휌 → ∞ ),} equation (14) can be further simplified as 2024P01254WO -11- From the above analysis, it can be concluded that the system diversity order is determined by푅, which could be as high as the number of resolvable paths of the _channel. ⁽ _∏ ^� � �_��휆� ^)� stands for the pairwise coding gain to control how this PEP shifts _{relative to the benchmark error-rate curve of (휌/4(푄 + 1))��.} Accounting for all possible pairwise errors, the diversity and coding gains, _{respectively, are defined herein as} Because the system performance depends on both퐺_� and퐺_�, it is important to maximize both퐺_� and퐺_�. By checking the dimensionality of 퐂, it is clear that the _{maximum diversity gain퐺�,��� = 푄 + 1 is achieved if and only if the matrix퐂 has full rank for ∀퐱 ≠ 퐱̂. When the maximum diversity gain퐺�,��� =푄 + 1 is achieved, the} coding gain becomes � � 퐺_{� = m} ^�� _{��� 퐱�i퐱n̂  det(퐑�)} ^�det(퐂) _{, (16) = 피[퐡퐡�]. Equation (16) implies that퐺� is a function of the minimum} determinant 퐺_{‾� = m퐱�in퐱̂   det((횽(퐱) −횽( = m}퐱�_in퐱̂_{det(퐁�퐁) det(d} = _{min  det(diag{(�퐅� �� ⊗��퐈��)�퐕 (퐱 −퐱̂)})�} 퐱�퐱̂ 횯 �� = _m퐱�_in퐱̂_� ��� _{where휽� � is the푖-th row of 횯, with횯 = 퐅��(퐅� � ⊗퐈�)퐕. It is readily apparent that the diversity gain퐺� and the coding gain퐺� are both depending on the choice of the precoding matrix퐕. Without a proper precoding matrix퐕, 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} 2024P01254WO -12- maximize 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 achieve퐺�,���, the coding} gain퐺_� should be maximized as much as possible afterwards. _{In the general precoding setup, it is only ensured that Tr(퐕퐕�) = 푀푁 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(퐕퐕�) = 푀푁 such that, where휽� � is the푖-th row of 횯, with횯 = 퐅��(퐅� � ⊗퐈�)퐕. 2. Maximize the coding gain: Develop a matrix퐕 with Tr(퐕퐕�) = 푀푁 to maximize} �� 퐺_{‾ = min  �    � � � � 퐱�퐱̂ 휽�(퐱 −퐱̂)� . (19)} ��� It is noted that when criterion 2 is satisfied, criterion 1 will be automatically satisfied. _{For finding a solution that satisfies criterion 2 a Vandermonde/unitary matrix횯 is constructed using algebraic number theory. Specifically,횯 can be expressed as a} Vandermonde matrix _{where훽 is a normalization factor chosen to guarantee the power constraint Tr(퐕퐕�) = 푀푁 is observed, and the selection of elements, or parameters,} depends on푀푁, for example I_{f 푀푁 = 2�(푑 ≥ 1), the parameter훼� is determined as} 2024P01254WO -13- 훼 _����� � _{= 푒 ����,푘 = 1,2, ⋯ ,푀푁. (21) If 푀푁 = 3 × 2�(푑 ≥ 0), the parameter훼� is specified as 훼 �}���� � _{= 푒 ����,푘 = 1,2, ⋯ ,푀푁. (22) If 푀푁 = 2� × 3�(푑 ≥ 1,푡 ≥ 1), the parameter훼� is given by 훼 �}���� � _{= 푒 ����,푘 = 1,2, ⋯ ,푀푁. (23)} After determining횯, the precoding matrix is given by 퐕 _{= (퐅� ⊗퐈�)횯 (24) The following section discusses the performance of the proposed precoded OTFS} systems for time-selective fading channels. In the considered simulations the carrier _{frequency is centred at 4GHz and the subcarrier spacing is Δ푓 = 15kHz. The BEM is} adopted to capture parsimoniously the time-selective fading channel, where the _{maximum user velocity is set to 60 km/h, 300 km/h and 600 km/h, respectively. It is assumed that the perfect channel knowledge is available at the receiver and QPSK modulation is applied. In figure 2 the effectiveness of the proposed precoding results for OTFS systems with ML detector is examined for different user velocities. 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 user velocity increases for both the precoded, non-precoded and conventional phase rotation} OTFS systems. This is due to the fact that high Doppler diversity gain can be _{obtained for better performance with large value of user velocity. 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. 2024P01254WO -14- 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. _{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, where푀 = 8 and푁 = 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. _{From the results of the simulation shown in figure 3 it can be observed that the BER} performance of both precoded, non-precoded and existing phase rotation OTFS _{systems improves as the user velocity increases, since the potential higher diversity can be exploited from a larger number of independent resolvable Doppler 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 4.5 dB and 3.4 dB gains over the non-precoded and existing phase rotation OTFS systems at BER = 10�� under 600 km/h.} 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 time-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- 2024P01254WO -15- 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. 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. _{In one or more embodiments of the method, precoding comprises, for determining} the precoding matrix, determining one or more candidate precoding matrices that each maximise the coding gain and, if two or more candidate precoding matrices that each maximises the coding gain are determined, selecting one for use with the method. _{In one or more embodiments of the method, precoding comprises, for determining} the candidate precoding matrix, determining one or more Vandermonde matrices _{whose elements maximise the diversity and/or the coding gain, and a normalisation} factor that imposes the maximum transmit power constraint. The elements of the Vandermonde matrix may be determined in dependence of the numbers or dimensions of the OTFS delay and Doppler resource grids. 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 2024P01254WO -16- subjecting the symbol matrix in the delay-Doppler domain to an inverse symplectic fast Fourier transform, ISFFT. 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. 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 with the one or more microprocessors. 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. 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.} 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 2024P01254WO -17- _{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). 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. 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 2024P01254WO -18- 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. 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. The method and apparatus described hereinbefore provide a linear precoded OTFS transmission system that employs a parsimonious BEM for modelling the underlying time-selective fading channels. The maximal diversity gain in such scenario can be as high as the number of bases in the BEM. The proposed linear precoded OTFS system can guarantee the maximal diversity and potential coding gains in Doppler time-selective fading channels without any transmission rate loss. 2024P01254WO -19- Simulation results demonstrate that the proposed precoded OTFS system can exploit the Doppler diversity gain to achieve better performance than original non-precoded and the conventional phase rotation OTFS systems for both optimal maximum likelihood (ML) detector and low-complexity advanced Memory approximate message passing (AMP) detector. The proposed precoding design for OTFS exhibits a practical implementation advantage with sufficient diversity of the Doppler time- _{selective fading channels, outperforming the original non-precoded and the} conventional phase rotation OTFS systems. The proposed method and apparatus are expected to support high data rates wireless transmissions, and provide high speed robust and ultra-reliable communications for a wide range of emerging large-scale applications, including online gaming, virtual reality (VR) and augmented reality (AR), high-speed railway systems, low earth orbit (LEO) satellite communications, unmanned aerial vehicle _{(UAV) communication and vehicle-to-everything (V2X) networks. The proposed} method and apparatus 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 internet-of-things (IoTs) scenarios in} a straightforward manner. BRIEF DESCRIPTION OF THE DRAWING In the following section the invention will be described with reference to the drawings, in which _{Fig. 1 shows a schematic block diagram of an OTFS transmission system in} accordance with the present invention, _{Fig. 2 shows a BER performance comparison with different user velocities under} ML detector, _{Fig. 3 shows a BER performance comparison with different user velocities under} Memory AMP detector, _{Fig. 4 shows an exemplary flow diagram of a method in accordance with the} present invention, and _{Fig. 5 shows a schematic block diagram of an apparatus configured for executing} the method in accordance with the present invention. 2024P01254WO -20- Throughout the figures identical or similar elements may be referenced using the same reference designators. DETAILED DESCRIPTION OF EMBODIMENTS _{Figures 1 to 3 have been described further above and will not be discussed again.} 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 symbols퐱�, 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 matrix퐗^�, 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. 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

2024P01254WO -21- CLAIMS _{1. A method (100) of transmitting information symbols over an OTFS communication channel subject to time-selective fading, comprising: - receiving (102) information symbols (x) to be transmitted, - precoding (104) the received information symbols (x), yielding precoded} information symbols (퐱�), - _{arranging (106) the precoded information symbols (퐱�) 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 (퐗} ^� _{), - converting (110) the time-frequency domain matrix (퐗�) 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 (퐱�) 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 m_{atrices (V) that each maximises the diversity gain, the precoding matrix (V)} that maximises the coding gain. _{4. The method (100) of claim 2 or 3, wherein precoding (104) comprises, for} determining the candidate precoding matrix (V), determining one or more 2024P01254WO -22- V_{andermonde matrices (횯) whose elements, or parameters, maximise the} diversity and/or the coding gain, and an associated normalisation factor (훽) that imposes the maximum transmit power constraint. _{5. The method (100) of claim 4, wherein the elements, or parameters, of the Vandermonde matrix (횯) 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 (횯) that is formed from the product of the candidate} precoding matrix (V), a Fourier matrix (퐅_��) having a number of points equal t_{o 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 (퐅_� ^�) having a number of points equal to the number of OTFS Doppler resource grids (N) and an identity matrix (퐈_�) 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, h_{as 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 r_{ows of a Vandermonde matrix (횯) that is formed from the product of the} candidate precoding matrix (V), a Fourier matrix (퐅_��) 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 (퐅_� ^�) having a number of points equal to the number of OTFS Doppler resource grids (N) and an identity matrix (퐈_�) 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.} 2024P01254WO -23- _{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 (퐗^�) comprises subjecting the s_{ymbol 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 (퐗^�) into a time-domain signal (s) comprises subjecting the time-frequency domain matrix (퐗^�) 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 e_{xecuted 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 h_{ardware components of an apparatus configured for transmitting signals over an OTFS communication channel in accordance with claim 11 to execute the method (100) of one or more of claims 1 to 10.} 2024P01254WO -24- _{13. Computer readable medium or data carrier retrievably transmitting or storing} the computer program product of claim 12.