Orthogonal time-frequency space communication system compatible with OFDM

Abstract

The present invention relates to an orthogonal time-frequency-space communication system compatible with OFDM. A system and method for orthogonal time-frequency-space communication and waveform generation. The method includes receiving a plurality of information symbols and encoding an N×M array containing the plurality of information symbols into a two-dimensional array of modulation symbols by extending each of the plurality of information symbols with respect to both time and frequency. The two-dimensional array of modulation symbols is then transmitted using M mutually orthogonal waveforms contained within M frequency subbands.

Description

[0001] This application is a divisional application of the invention patent application with application number 201680047494.6, application date June 27, 2016, and invention name “Orthogonal time-frequency space communication system compatible with OFDM”.

[0002] CROSS-REFERENCE TO RELATED APPLICATIONS

[0003] This application claims patent rights under 35 U.S.C. §119(e) to U.S. Provisional Application No. 62 / 185617, filed on June 27, 2015, entitled “ORTHOGONAL TIME FREQUENCY SPACE MODULATION”, U.S. Provisional Application No. 62 / 185619, filed on June 27, 2015, entitled “ORTHOGONAL TIME FREQUENCY SPACE MODULATION”, U.S. Provisional Application No. 62 / 185643, filed on June 28, 2015, and U.S. Provisional Application No. 62 / 185644, filed on December 9, 2015, entitled “OTFS COMMUNICATION SYSTEM USING MULTICARRIER MODULATORS AND DEMODULATORS", the contents of each of which are incorporated herein by reference in their entirety for all purposes. This application is a continuation-in-part of U.S. Application No. 15 / 188,946, filed on June 21, 2016, entitled “SYMPLECTIC ORTHOGONAL TIME FREQUENCY SPACE MODULATION SYSTEM,” which in turn is a continuation-in-part of U.S. Application No. 15 / 152,464, filed on May 11, 2016, entitled “ORTHOGONAL TIME FREQUENCY SPACE MODULATION SYSTEM,” which claims the benefit of priority under 35 U.S.C. §119(e) to U.S. Provisional Application No. 62 / 159,853, filed on May 11, 2015, entitled “ORTHOGONAL TIME FREQUENCY SPACE OTFS MODULATION,” and U.S. Provisional Application No. 62 / 159,853, filed on May 12, 2015, entitled “SYSTEMS AND METHODS FOR SYMPLECTIC ORTHOGONAL TIMEFREQUENCY SHIFTING MODULATION AND TRANSMISSION OF DATA” U.S. Provisional Application No.62 / 160257, U.S. Provisional Application No. 62 / 173801, filed on June 10, 2015, entitled “SYSTEMS AND METHODS FOR SYMPLECTICORTHOGONAL TIME FREQUENCY SHIFTING MODULATION AND TRANSMISSION OF DATA,” U.S. Provisional Application No. 62 / 182372, filed on June 19, 2015, entitled “OTFS A NEW MODULATION FOR 5G,” and U.S. Provisional Application No. 62 / 215663, filed on September 8, 2015, entitled “ORTHOGONAL TIMEFREQUENCY SPACE COMMUNICATION SYSTEM AND METHOD,” the contents of each of which are incorporated herein by reference in their entirety for all purposes. The present application is also a continuation-in-part of U.S. Patent Application No. 14 / 709,377, filed on May 11, 2015, entitled “MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM,” which is a continuation-in-part of U.S. Patent Application No. 13 / 927,086, filed on June 25, 2013, entitled “MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM,” which claims the benefit of priority under 35 U.S.C. §119(e) to U.S. Patent Application No. 13 / 927,086, filed on June 25, 2012, entitled “MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM,” which is a continuation-in-part of U.S. Patent Application No. 13 / 927,086, filed on June 25, 2013, entitled “MODULATION AND EQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM”, U.S. Provisional Application Serial No. 61 / 664,020, filed on March 15, 2013, entitled “MODULATION ANDEQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM”, and U.S. Provisional Application Serial No.61 / 801398, U.S. Provisional Application No. 61 / 801366, filed on March 15, 2013, entitled “MODULATION ANDEQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM”, U.S. Provisional Application No. 61 / 801435, filed on March 15, 2013, entitled “MODULATION ANDEQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM”, U.S. Provisional Application No. 61 / 801435, filed on March 15, 2013, entitled “MODULATION ANDEQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM”, U.S. Provisional Application Serial No. 61 / 801,994, filed on March 15, 2013, and entitled “MODULATION ANDEQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM”, and U.S. Provisional Application Serial No. 61 / 801,968, filed on March 15, 2013, and entitled “MODULATION ANDQUALIZATION IN AN ORTHONORMAL TIME-FREQUENCY SHIFTING COMMUNICATIONS SYSTEM”, the contents of each of which are incorporated herein by reference in their entirety for all purposes. Technical Field

[0004] The present disclosure relates generally to communication protocols and methods, and more particularly to methods of modulation and related processing of signals for wireless and other forms of communication. Background Art

[0005] Fourth-generation (4G) wireless networks have served the public well, providing ubiquitous access to the Internet and enabling explosive growth in mobile applications, smartphones, and sophisticated data-intensive applications such as mobile video. This continues the evolution of cellular technology, with each new generation bringing substantial benefits to the public, enabling significant advances in productivity, convenience, and quality of life.

[0006] Looking ahead to the demands that increasing and diverse data usage will place on existing networks, it is becoming increasingly clear to industry players that current 4G networks will not be able to support the foreseen data usage needs. This is in part because data traffic has and continues to grow at an exponential rate. In addition, new applications coupled with the relentless expansion of mobile video (such as immersive reality and remote robotics) are expected to overwhelm the carrying capacity of current network systems. One of the goals of 5G system design is to be able to economically scale the capacity of the network in dense urban settings (e.g., up to 750Gbps / Km 2 ), which is impossible to achieve using commercially deployed technologies.

[0007] In addition to being able to handle larger amounts of data, next generation systems will need to improve the quality of data delivery in order to support expected future applications. The public increasingly expects wireless networks to provide a near "wired" experience for unrestricted users. This can translate into requirements of, for example, 50+Mbps across the entire coverage area (i.e., even at the cell edge), which will require the implementation of advanced interference mitigation techniques.

[0008] Another aspect of the quality of user experience is mobility. Due to the Doppler effect, the throughput of current wireless networks tends to decrease significantly as the speed of movement increases. Future 5G systems are designed to increase the supported speeds up to 500Km / h not only for high-speed trains and aviation, but also to support numerous new automotive applications for vehicle-to-vehicle and vehicle-to-infrastructure communications.

[0009] While support for increased and higher quality data traffic is necessary for wireless networks to continue to support user demand, operators are also exploring new applications that will enable new revenue and innovative use cases. These include the automotive and smart infrastructure applications discussed above. Other expected applications include the deployment of ultra-reliable networks for public safety, the use of cellular networks to support the decline of the PSTN, and so on. In addition, 5G networks are expected to usher in the deployment of a large number of Internet-connected devices, also known as the Internet of Things (IoT). However, existing networks are not designed to support a large number of connected devices and very low traffic per device. Summary of the invention

[0010] In one aspect, the present disclosure is directed to a method for transmitting data over a communication channel. The method includes receiving a plurality of information symbols. The method also includes encoding an N×M array containing the plurality of information symbols into a two-dimensional array of modulation symbols by spreading each of the plurality of information symbols with respect to both time and frequency. The two-dimensional array of modulation symbols is then transmitted using M mutually orthogonal waveforms contained within M frequency subbands.

[0011] Encoding may also include transforming the NxM array into an array of filtered OFDM symbols using at least one Fourier transform and filtering process, and transforming the array of filtered OFDM symbols into an array of OTFS symbols using at least one two-dimensional Fourier transform.

[0012] The encoding can also be done according to the following relationship:

[0013]

[0014]

[0015] Wherein x[l,k], k=0,…,N-1, l=0,…,M-1 represents an N×M array containing the plurality of information symbols, X[n,m], n=0,…,N-1, m=0,…,M-1 represents a two-dimensional array of modulation symbols, and W tr [n,m] is the window function, and b k,l [n,m] represents a set of basis functions.

[0016] The present disclosure also relates to an automated method for wireless communication over a damaged data channel. The method includes receiving a plurality of data symbols. The method also includes encoding an N×M two-dimensional array containing the plurality of data symbols into a two-dimensional array of modulation symbols by spreading each of the plurality of data symbols using a set of cyclic time-shifted and frequency-shifted basis functions. The two-dimensional array of modulation symbols is transmitted using M mutually orthogonal wireless waveforms contained within M frequency subbands.

[0017] Encoding may also include transforming the NxM array into an array of filtered OFDM symbols using at least one Fourier transform and filtering process, and transforming the array of filtered OFDM symbols into an array of OTFS symbols using at least one two-dimensional Fourier transform.

[0018] The encoding may also include encoding at least one N×M array of two-dimensional data symbols onto at least one symplectic-like analysis compatible manifold distributed on a column time axis of length T and a row frequency axis of length F, thereby generating at least one information manifold. In addition, the encoding may include transforming at least one information manifold according to a two-dimensional symplectic-like Fourier transform, thereby generating at least one two-dimensional Fourier transformed information manifold.

[0019] In another aspect, the present disclosure relates to a method for generating a waveform for transmission over a communication channel. The method includes receiving a plurality of information symbols and creating a plurality of modulation symbols by modulating one of a plurality of two-dimensional basis functions in a time-frequency plane using each of the plurality of information symbols. Each of the plurality of two-dimensional basis functions is uniquely associated with one of the plurality of information symbols. The method also includes generating a transmit waveform composed of a plurality of pulse waveforms, each of the plurality of pulse waveforms corresponding to a combination of one of the plurality of modulation symbols and one of a time-translated and frequency-modulated version of a basic transmit pulse.

[0020] In one implementation, each of the multiple time-shifted and frequency-modulated versions of a basic transmit pulse is associated with a time shift of one of N times T and a frequency modulation of one of M times Δf, where the total duration of the transmit waveform is NT seconds and the total bandwidth is MΔf Hz. The basic transmit pulse may have characteristics that are orthogonal to the shifts of times T and the modulations of times Δf.

[0021] In another aspect, the present disclosure relates to a communication device, which includes a wireless transmitter, a processor, and a memory including a program code executable by the processor. The program code includes code for causing the processor to receive a plurality of information symbols. The program code also includes code for causing the processor to create a plurality of modulation symbols by modulating one of a plurality of two-dimensional basis functions on a time-frequency plane using one of the plurality of information symbols. Each of the plurality of two-dimensional basis functions is uniquely associated with one of the plurality of information symbols. The program code also causes the processor to generate a transmit waveform consisting of a plurality of pulse waveforms, each of which corresponds to a combination of one of the plurality of modulation symbols and a time-shifted and frequency-modulated version of a basic transmit pulse. The program code also includes code for causing the processor to provide the transmit waveform to the wireless transmitter.

[0022] The present disclosure is also directed to a method for receiving one or more modulation waveforms at a communication receiver. The method also includes performing matched filtering of samples of the one or more modulation waveforms relative to a received pulse to produce estimated time-frequency modulation symbols. Each of the estimated time-frequency modulation symbols corresponds to a modulation of one of a plurality of orthogonal two-dimensional basis functions by one of a plurality of information symbols. The method also includes projecting estimates of the time-frequency modulation symbols onto the plurality of orthogonal two-dimensional basis functions to obtain estimates of the plurality of information symbols.

[0023] In one implementation, the method may further include performing windowing and periodization operations with respect to the estimated time-frequency modulation symbols. In addition, the projection operation may include performing a symplectic Fourier transform operation with respect to the periodic sequence composed of the estimates of the time-frequency modulation symbols.

[0024] In yet another aspect, the present disclosure relates to a communication device comprising a wireless receiver configured to receive one or more modulated waveforms, a processor, and a memory including a program code executable by the processor. The program code includes code for causing the processor to receive samples of one or more modulated waveforms from the wireless receiver. The code also includes code for causing the processor to match filter the samples of the one or more modulated waveforms relative to the received pulse to generate estimated time-frequency modulation symbols. Each of the estimated time-frequency modulation symbols corresponds to a modulation of one of a plurality of orthogonal two-dimensional basis functions by one of a plurality of information symbols. The program code also includes code for causing the processor to project estimates of the time-frequency modulation symbols on a plurality of orthogonal two-dimensional basis functions to obtain estimates of the plurality of information symbols.

[0025] In one implementation, the program code may also include code for performing windowing and periodization operations relative to the estimated time-frequency modulation symbol. In addition, the code may include code for causing the processor to perform a symplectic Fourier transform operation relative to a periodic sequence composed of estimates of the time-frequency modulation symbol.

[0026] The present disclosure is also directed to a method for providing a modulated signal that can be used in a signal transmission system. The method includes performing a two-dimensional time-frequency transform of a data frame containing a plurality of information symbols to a plane of time-frequency modulation symbols. The method also includes generating the modulated signal by performing a Heisenberg transform using the plane of time-frequency modulation symbols. In one implementation, performing the two-dimensional time-frequency transform requires performing an inverse time-frequency symplectic transform and a windowing operation.

[0027] In another aspect, the present disclosure relates to a signal modulation method in which a data frame containing a plurality of information symbols is transformed into a plane of time-frequency modulation symbols by using a two-dimensional time-frequency transform. The method includes using an OFDM modulator to generate a modulated signal based on the plane of time-frequency modulation symbols. In one implementation, performing the two-dimensional time-frequency transform requires performing an inverse time-frequency symplectic transform and a windowing operation.

[0028] The present disclosure also relates to a method for providing a modulated signal that can be used in a signal transmission system. The method includes performing a two-dimensional time-frequency transform of a data frame containing a plurality of information symbols to a plane of time-frequency modulation symbols. The method also includes using a multi-carrier filter bank (MCFB) modulator to generate a modulated signal based on the plane of time-frequency modulation symbols. In one implementation, performing the two-dimensional time-frequency transform includes performing an inverse time-frequency symplectic transform and a windowing operation.

[0029] In another aspect, the present disclosure describes a transmitter apparatus comprising an OTFS preprocessing unit. The OTFS preprocessing unit is configured to perform a two-dimensional time-frequency transform of a data frame containing a plurality of information symbols to a plane of time-frequency modulation symbols. The transmitter apparatus comprises an OFDM or multi-carrier filter bank modulator configured to generate a modulated signal based on the plane of time-frequency modulation symbols. In one implementation, the two-dimensional transform comprises an inverse time-frequency symplectic transform and a windowing operation.

[0030] The present disclosure also relates to a method for receiving a modulated signal. The method includes: receiving a plurality of signal components of the modulated signal, and generating a plane of estimated time-frequency modulation symbols by performing an OFDM demodulation operation using the plurality of signal components. The method also includes providing an estimated data frame by performing an inverse of a two-dimensional time-frequency transform relative to the plane of estimated time-frequency modulation symbols. In one implementation, performing the inverse of the two-dimensional time-frequency transform includes performing a windowing operation and a sigmoid Fourier transform.

[0031] In another aspect, the present disclosure is directed to a method for receiving a modulated signal. The method includes receiving a plurality of signal components of the modulated signal, and generating a plane of estimated time-frequency modulation symbols by performing one of an OFDM demodulation operation and an MCFB demodulation operation using the plurality of signal components. The method also includes providing an estimated data frame by performing an inverse of a two-dimensional time-frequency transform relative to the plane of the estimated time-frequency modulation symbols. In one implementation, performing the inverse of the two-dimensional time-frequency transform includes performing a windowing operation and a sigmoid Fourier transform.

[0032] In another aspect, the present disclosure relates to a receiver device comprising a receiver front end configured to receive a plurality of signal components of a modulated signal. The receiver comprises one of an OFDM demodulator and an MCFB demodulator configured to generate a plane of estimated time-frequency modulation symbols based on the plurality of signal components. An OTFS post-processing unit is operable to provide an estimated data frame by performing an inverse of a two-dimensional time-frequency transform relative to the plane of the estimated time-frequency modulation symbols. In one implementation, the inverse of the two-dimensional time-frequency transform corresponds to a windowing operation and a sigmoid Fourier transform. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] To better understand the nature and purpose of various embodiments of the present invention, reference should be made to the following detailed description in conjunction with the accompanying drawings, in which:

[0034] Figure 1A Examples of wireless communication systems that may exhibit time / frequency selective fading are illustrated.

[0035] Figure 1B Provides the ability to Figure 1A A high-level representation of a conventional transceiver used in a wireless communication system.

[0036] Figure 2A The time-varying impulse response of an accelerating reflector in a channel represented by a one-dimensional channel model in the (τ, t) coordinate system is shown.

[0037] Figure 2B The same channel is shown represented using a time-invariant impulse response in a delay-Doppler (τ, v) coordinate system.

[0038] Figure 3 is a block diagram of components of an exemplary OTFS communication system.

[0039] Figure 4 Represents the conceptual implementation of the Heisenberg transform in the OTFS transmitter and the Wigner transform in the OTFS receiver.

[0040] Figure 5A An exemplary embodiment of OTFS modulation is illustratively shown, including the transformation of the time-frequency plane to the Doppler-delay plane.

[0041] Figure 5B Indicates the relationship between sampling rate, delay resolution and time resolution in an OTFS communication system.

[0042] Figure 5C The signal transmission between various domains in the OTFS communication system is illustrated.

[0043] Figure 5DThe notation used in representing the signals present at the various processing stages in the OTFS transmitter and receiver is plotted.

[0044] Figure 6 Discrete impulses in the OTFS domain for channel estimation purposes are shown.

[0045] Fig. 7A and 7B Two different basis functions belonging to different users are illustrated, where each basis function spans the entire time-frequency frame.

[0046] Figure 7C and 7D An illustration of two-dimensional expansion with respect to both time and frequency in an OTFS communication system is provided.

[0047] Fig. 7E Members of a possible set of such mutually orthogonal two-dimensional basis functions are illustrated.

[0048] Figure 8 and 9 It is illustrated that multiple users are multiplexed in the time-frequency domain by allocating different resource blocks or subframes to different users in a staggered manner.

[0049] Fig.10 Components of an exemplary OTFS transceiver are illustrated.

[0050] Fig.11 A comparison of the bit error rates (BER) predicted by simulations of a TDMA system and an OTFS system is illustrated.

[0051] Fig.12 is a flow chart representing operations performed by an exemplary OTFS transceiver.

[0052] Fig.13 The OTFS modulator is illustrated as acting as an orthogonal map deployed to transform a two-dimensional time-frequency matrix into a transmitted waveform.

[0053] Fig.14 The operation of an OTFS demodulator that transforms a received waveform into a two-dimensional time-frequency matrix according to an orthogonal mapping is illustrated.

[0054] Fig.15 The pulse train included in the pulse waveform generated by the OTFS modulator is exemplarily shown.

[0055] Fig.16 Depicted is a two-dimensional discriminant feedback equalizer configured to perform a least mean square (LMS) equalization process.

[0056] Figures 17A-17DAn OTFS transmitter and receiver are depicted, along with the operation of each with respect to the associated time-frequency grid.

[0057] Fig.18A and 18B OTFS communication over a communication channel characterized by a two-dimensional delay-Doppler impulse response is illustratively shown.

[0058] Fig.19A The transmission of a two-dimensional Fourier transformed information manifold represented by an N×M structure over M frequency bands during N time periods of duration Tμ is illustrated.

[0059] Fig.19B Another perspective of the image domain and transform domain dual grids that can be used for the symplectic OTFS method described herein is illustrated.

[0060] Fig. 20 An example of M filtered OTFS bands being transmitted simultaneously according to respective smaller time slices Tμ is shown.

[0061] Fig.21 Additional examples of OTFS waveforms transmitted according to various smaller time slices Tμ are provided.

[0062] Fig. 22 A block diagram representation of an exemplary process for OTFS transmission and reception is provided.

[0063] Fig.23 An exemplary structure of a limited OTFS modulation mapping is illustratively shown.

[0064] Fig.24A and 24B The standard communication dot matrix and the reciprocal of the standard communication dot matrix are illustrated respectively.

[0065] Fig.25 A standard communication torus is shown by way of example.

[0066] Fig.26 A standard communications finite torus is shown by way of example.

[0067] Fig. 27 An exemplary structure of an OTFS modulation mapping is illustrated.

[0068] Fig.28 The frequency domain interpretation of the OTFS modulation block is illustrated.

[0069] Fig.29A and 29B One way in which the symplectic OTFS method can work in a transmitter and receiver system is illustrated.

[0070] Fig.29C The characteristics of OTFS pre-processing that allow compatibility with OFDM modulation systems are illustrated.

[0071] Fig.29D Further details of OTFS pre-processing operations compatible with OFDM modulation systems are illustrated.

[0072] Fig.30 Alternative methods of sending and receiving data on a channel are shown.

[0073] Fig.31 The effects of Doppler and time delay caused by the channel on the double grids in the image domain and the transform domain are shown.

[0074] Fig.32 An example of interleaving is shown.

[0075] Fig.33 Another example of interleaving is shown, where frames of the same size are interleaved on a frequency staggered basis.

[0076] Fig.34 Another example of interleaving is shown, where frames of variable size are interleaved on a temporal basis.

[0077] Fig.35 The OTFS pre-processing steps within the transmitter module and the OTFS post-processing steps within the receiver module are depicted.

[0078] Fig.36 A block diagram of an OTFS transmitter according to an embodiment is provided.

[0079] Fig.37 An OTFS receiver configured to demodulate OTFS modulated data received over a wireless link is depicted.

[0080] Fig.38 An example of how an active OTFS relay system may work between an OTFS sender and a receiver is shown. DETAILED DESCRIPTION

[0081] As discussed below, an embodiment of orthogonal time-frequency-space (OTFS) modulation involves sending each information symbol by modulating a two-dimensional (2D) basis function on the time-frequency plane. In an exemplary embodiment, the set of modulation basis functions is specifically derived to best represent the dynamics of a time-varying multipath channel. In this way, OTFS transforms the time-varying multipath channel into a time-invariant delay-Doppler two-dimensional convolution channel. This effectively eliminates the difficulty of tracking time-varying fading, such as in communications involving high-speed vehicles.

[0082] OTFS increases the coherence time of the channel by orders of magnitude. It uses AWGN codes that are well studied on average channel SNR to simplify signaling on the channel. More importantly, due to the inherently accurate and efficient estimation of channel state information (CSI), it enables linear scaling of throughput with the number of antennas in mobile vehicle applications. In addition, since the delay-Doppler channel representation is very compact, OTFS enables large-scale MIMO and beamforming with CSI at the transmitter for four, eight and more antennas in mobile vehicle applications. The CSI information required in OTFS is part of what is needed to track time-varying channels.

[0083] As will be appreciated from the following discussion, one feature of OTFS is that a single QAM symbol can be distributed over multiple time and / or frequency points. This is a key technology to increase processing gain and in-building penetration for IoT deployments and PSTN replacement applications. Scaling in the OTFS domain allows scaling over wider bandwidths and durations while maintaining a stationary channel that does not need to be tracked over time.

[0084] These benefits of OTFS will become apparent once the basic concepts behind OTFS are understood. There is a rich mathematical basis for OTFS that leads to several variations; for example, it can be combined with OFDM or with multi-carrier filterbanks. Before proceeding with a detailed discussion of OTFS, various shortcomings of communication systems predicted on a one-dimensional channel model are first described.

[0085] Figure 1A An example of a wireless communication system 100 that may exhibit time / frequency selective fading is illustrated. System 100 includes a transmitter 110 (e.g., a cell phone tower) and a receiver 120 (e.g., a cell phone). The scenario shown in FIG. 1 includes multiple paths (multipaths) that a signal transmitted from transmitter 100 travels before reaching receiver 100. A first path 130 reflects off a tree 132, a second path 140 reflects off a building 142, and a third path 150 reflects off a second building 152. A fourth path 160 reflects off a moving car 162. Because paths 130, 140, 150, and 160 all travel different distances and attenuate or fade at different levels and at different frequencies, when conventionally configured, receiver 120 may drop out of service or at least suffer from low throughput due to destructive interference of the multipath signals.

[0086] Now go to Figure 1B , which provides Figure 1A1. A high-level representation of a conventional transceiver 200 used in a wireless communication system 100 of the present invention. For example, the transceiver 200 may operate according to established protocols for time division multiple access (TDMA), code division multiple access (CDMA), or orthogonal frequency division multiple access (OFDM) systems. In conventional wireless communication systems such as TDMA, CDMA, and OFDM, a multipath communication channel 210 between a transmitter 204 and a receiver 208 is represented by a one-dimensional model. In these systems, a one-dimensional representation of an impulse response of the communication channel is used to characterize the channel distortion. The transceiver 200 may include a one-dimensional equalizer 220 configured to remove, at least in part, this estimated channel distortion from a one-dimensional output data stream 230 produced by the receiver 208.

[0087] Unfortunately, there are several fundamental problems with using one-dimensional channel models. First, the one-dimensional channel models employed in existing communication systems are non-stationary; that is, the symbol distortion effects of the communication channel vary from symbol to symbol. In addition, when the channel is modeled in only one dimension, it is possible that some received symbols have significantly less energy than other symbols due to "channel fading." Finally, one-dimensional channel state information (CSI) occurs randomly, and much of it is estimated by interpolating between channel measurements taken at specific points, making the information inherently inaccurate. These problems are only exacerbated in multi-antenna (MIMO) communication systems. As discussed below, embodiments of the OTFS method described herein can be used to substantially overcome the fundamental problems caused by using a one-dimensional channel model.

[0088] Multipath fading channels are usually modeled one-dimensionally as convolutional channels in baseband using time-varying impulse responses

[0089]

[0090] where s(t) and r(t) represent the complex baseband channel input and output, respectively, and where is the complex baseband time-varying channel response.

[0091] This representation, while general, does not give us any insight into the behavior and variability of the time-varying impulse response. A more useful and insightful model that is also often used for Doppler multipath double-fading channels is

[0092] r(t)=∫∫h(τ,v)e j2πv(t-τ) s(t-τ)dvdτ (2)

[0093] In this representation, the received signal is a superposition of reflected copies of the transmitted signal, where each copy is delayed by the path delay τ, frequency shifted by the Doppler frequency shift v, and weighted by a time-independent delay-Doppler impulse response h(τ, v) for that τ and v. In addition to the intuitiveness of this representation, equation (2) maintains the generality of equation (1). In other words, it can represent complex Doppler trajectories such as accelerating vehicles, reflecting objects, etc. If we express the time-varying impulse response as a Fourier expansion with respect to the time variable t, we can see that

[0094]

[0095] Substituting (3) into (1), we obtain equation (2) after some processing. As an example, Figure 2A shows the time-varying impulse response for accelerating reflectors in the (τ, t) coordinate system, while Figure 2B The same channel represented as a time-invariant impulse response in the (τ, v) coordinate system is shown.

[0096] An important feature revealed by these two figures is how compact the (τ, v) representation is compared to the (τ, t) representation. This has important implications for channel estimation, equalization, and tracking, as will be discussed later.

[0097] Note that while h(τ, v) is effectively time-independent, the operation on s(t) is still time-varying, as can be seen by the effect of an explicit complex exponential function of time in equation (2). In implementation, the disclosed modulation scheme contemplates appropriate selection of orthogonal basis functions that render the effects of the channel truly time-independent in the domain defined by those basis functions. The proposed scheme has the following high-level summary.

[0098] First, let us consider a set of orthonormal basis functions indexed by τ,v that are orthogonal to translation and modulation, i.e.,

[0099]

[0100]

[0101] Let us consider the transmitted signal as a superposition of these basis functions

[0102] s(t)=∫∫x(τ,v)φ τ，v (t)dτdv (5)

[0103] where the weights x(τ, v) represent the information-bearing signal to be transmitted. After the transmitted signal of (5) passes through the time-varying channel of equation (2), we obtain the superposition of the delayed and modulated versions of the basis functions, which results from (4)

[0104] r(t)=∫∫h(τ,v)e j2πv(t-τ) s(t-τ)dvdτ

[0105] =∫∫φ τ，v (t){h(τ,v)*x(τ,v)}dτdv (6)

[0106] where * denotes a two-dimensional convolution. Equation (6) can be considered as a generalization of the convolution relation for linear time-invariant systems, where one-dimensional exponentials are used as basis functions. Note that the terms in brackets can be expressed for each basis function φ τ，v (t) is recovered at the receiver by matched filtering. In this way, a two-dimensional channel relationship is established in the (τ, v) domain

[0107] y(τ,v)=h(τ,v)*x(τ,v) (7)

[0108] where y(τ, v) is the receiver 2D matched filter output. Also note that in this domain, the channel is described by a time-invariant 2D convolution.

[0109] A final different interpretation of the wireless channel will also be useful below. Let us consider s(t) and r(t) as square-integrable functions The elements of the Hilbert space of . Then equation (2) can be interpreted as A linear operator on , which acts on an input s(t), parameterized by an impulse response h(τ, v) and produces an output r(t):

[0110]

[0111] Note that while the operator is linear, it is not time invariant. If Doppler is not present, i.e., if h(v,τ) = h(0,τ)δ(v), then Eq. (2) reduces to a time invariant convolution. Note also that while, for a time invariant system, the impulse response is parameterized by one dimension, in the time varying case we have a two dimensional impulse response. While in the time invariant case the convolution operator produces a superposition of delays of the input s(t) (hence the parameterization is along the one dimensional delay axis), in the time varying case we have a superposition of delay and modulation operations, as seen in Eq. (2) (hence the parameterization is along the two dimensional delay and Doppler axes). This is the main difference that causes the time varying representation to be non-commutative (in contrast to the convolution operation which is commutative) and complicates the processing of time varying systems.

[0112] An important point in equation (8) is that the operator Π h(·) can be compactly parameterized by a two-dimensional function h(τ, v), providing an efficient, time-independent channel description. Typical channel delay spread and Doppler spread are very small fractions of the symbol duration and subcarrier spacing of a multicarrier system.

[0113] The representation of the time-varying system defined by equations (2) and (8) can be characterized as a Heisenberg representation. In this regard, it can be shown that each linear operator (equation (8)) can be parameterized by a certain impulse response as in equation (2).

[0114] OTFS modulation over Doppler multipath channels

[0115] The temporal variation of the channel introduces significant difficulties in wireless communications related to channel acquisition, tracking, equalization, and transmission of channel state information (CSI) to the transmit side for beamforming and MIMO processing. We develop in this paper a modulation domain based on a set of standard orthogonal basis functions, whereby we can transmit information symbols and whereby the information symbols experience a static, time-invariant, two-dimensional channel for the duration of a packet or burst transmission. In that modulation domain, the channel coherence time is increased by orders of magnitude and the problems associated with channel fading in the time or frequency domain in SISO or MIMO systems are significantly reduced.

[0116] Figure 3 3 is a block diagram of components of an exemplary OTFS communication system 300. As shown, the system 300 includes a transmitter 310 and a receiver 330. The transmitting device 310 and the receiving device 330 include a first OTFS transceiver 315-1 and a second OTFS transceiver 315-2, respectively. The OTFS transceivers 315-1 and 315-2 communicate unidirectionally or bidirectionally on a communication channel 320 in the manner described herein. Although in the exemplary embodiments described herein, the system 300 may include a wireless communication system, in other embodiments, the communication channel may include a wired communication channel, such as a communication channel within an optical fiber or coaxial cable. As described above, the communication channel 320 may include multiple paths and be characterized by time / frequency selective fading.

[0117] The components of the OTFS transceiver may be implemented in hardware, software, or a combination thereof. For hardware implementation, the processing unit may be implemented in one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, other electronic units designed to perform the above functions, and / or combinations thereof.

[0118] Reference now Figure 5C, provides a pictorial view of two transforms that make up an exemplary form of OTFS modulation. It shows at a high level the signal processing steps required at a transmitter (such as transmitter 310) and a receiver (such as receiver 330). It also includes the parameters that define each step, which will become apparent as we further reveal each step. In addition, Figure 5D Block diagrams of the different processing stages at the transmitter and receiver are shown, and the notations that will be used for the various signals are established.

[0119] We first describe the transform that relates the waveform domain to the time-frequency domain.

[0120] Heisenberg Transformation

[0121] In this section, we aim to construct an appropriate transmit waveform that carries the information provided by the symbols on a grid in the time-frequency plane. Our intention in developing this modulation scheme is to transform the channel operations into equivalent operations in the time-frequency domain by exploiting two important properties: (i) the channels are orthogonal on the time-frequency grid; and (ii) the channel time variations are simplified on the time-frequency grid and can be accounted for with additional transformations. Fortunately, these goals can be achieved with a scheme that is very close to the well-known multicarrier modulation techniques, as explained next. We will start with a general framework for multicarrier modulation and then give examples of OFDM and multicarrier filter bank implementations.

[0122] Let us consider the following components of time frequency modulation:

[0123] A lattice or grid on the time-frequency plane, ie, samples on the time axis with a sampling period T and on the frequency axis with a sampling period Δf.

[0124]

[0125] Packet bursts with a total duration of NT seconds and a total bandwidth of MΔfHz

[0126] The set of modulation symbols X[n,m] we wish to send in this burst, n=0,...,N-1, m=0,...,M-1

[0127] Send pulse g tr (t), which has the property of being orthogonal to the translation by T and the modulation by Δf (generally required if the receiver uses the same pulses as the transmitter)

[0128]

[0129] Given the above components, the time-frequency modulator is a Heisenberg operator on the lattice Λ, that is, it modulates the pulse waveform g by tr The superposition of the delay and modulation operations of (t) maps the two-dimensional symbol X[n, m] to the transmitted waveform

[0130]

[0131] More formally

[0132]

[0133] We use Π X (·) denotes the “discrete” Heisenberg operator parameterized by discrete values ​​X[n,m].

[0134] Note the similarity of (12) to the channel equation (8). This is not a coincidence, but rather because we apply a modulation effect that mimics the channel effect, so the end effects of the concatenation of modulation and channel are easier to handle at the receiver. This is not uncommon, for example, the simplest form of linear modulation (for a time-invariant channel) is the convolution of the transmit pulse g(t) with a delta train of QAM information symbols sampled at baud rate T.

[0135]

[0136] In the present case for a time-varying channel, we convolve and modulate the transmit pulse (refer to channel equation (2)) with a two-dimensional delta string that samples the time-frequency domain at a certain baud rate and subcarrier spacing.

[0137] Sampling rate and pulse g in the time-frequency domain tr The bandwidth and duration of (t) (i.e., its time-frequency localization) are related. In order for the orthogonality condition of (10) to hold for a frequency separation Δf, the time interval must be T ≥ 1 / Δf. The critical sampling case of T ≥ 1 / Δf is generally unrealistic and refers to the limiting case, e.g., an OFDM system with a cyclic prefix length equal to zero or g tr (t) is equal to the filter bank of the ideal Nyquist pulse.

[0138] Some examples illustrate these principles:

[0139] Example 1: OFDM Modulation : Let us consider a system with M subcarriers and symbol length T OFDM , cyclic prefix length T CP and subcarrier spacing 1 / T OFDMOFDM system. If we substitute the symbol duration T = T in equation (11) OFDM +T CP , number of symbols N = 1, subcarrier spacing Δf = 1 / T OFDM , and g tr (t) is a square window that limits the duration of the subcarrier to symbol length T, then

[0140]

[0141] Then we get the OFDM formula

[0142]

[0143] Technically, the pulses of equation (14) are not orthogonal, but are orthogonal to the receive filter (where the CP samples are discarded).

[0144] Example 2: Single Carrier Modulation : If we substitute M = 1 subcarrier, T equals the baud period, and g tr (t) is equal to the square root raised cosine Nyquist pulse, then equation (11) is reduced to single carrier modulation.

[0145] Example 3: Multicarrier Filter Bank (MCFB) :If g tr (t) is a square-root raised cosine Nyquist pulse with excess bandwidth α, T equals the baud period and Δf = (1 + α) / T, then equation (11) describes the MCFB.

[0146] It may be counterintuitive to express the modulation operation as a Heisenberg transform as in equation (12). That is, modulation is usually perceived as a transformation of the modulation symbols X[m,n] to the transmitted waveform s(t). When applied to the prototype transmit filter response g tr (t) (see equation (12)), the Heisenberg transform instead uses X[m,n] as the weights / parameters of the operator that produces s(t). Although counterintuitive, this formula is useful in pursuing an abstraction of the modulation-channel-demodulation cascade effect in a two-dimensional domain where the channel can be described as time-invariant.

[0147] Attention turns to the processing required at the receiver side to get back from the waveform domain to the time-frequency domain. Since the received signal undergoes a cascade of two Heisenberg transforms (one through the effects of the modulation and one through the effects of the channel), it is natural to wonder what the end-to-end effect of this cascade is. The answer to this question is given by the following result:

[0148] Proposition 1: Let the two Heisenberg transforms defined in equations (8) and (2) be replaced by the impulse response h 1 (τ, v), h 2 (τ, v) parameterized and applied in cascade to the waveform

[0149] So

[0150]

[0151] where h(τ, v) = h 2 (τ,v)⊙h 1 (τ, v) is defined by the following convolution and modulation operations h 1 (τ, v), h 2 “Twisted” convolution of (τ, v)

[0152] h(τ, v) = ∫∫h 2 (τ′,v′)h 1 (τ-τ′,vv′)e j2πv′(τ-τ′)t dτ′dv′ (17)

[0153] Applying the above results to the concatenation of the modulation and channel Heisenberg transforms of (12) and (8), it can be shown that the received signal is given by the Heisenberg transform

[0154] r(t)=Π f (g tr f(t))+v(t)=∫∫f(τ,v)e j2πv(t-τ) g tr (t-τ)dvdτ+v(t) (18)

[0155] where v(t) is additive noise and f(τ,v), (the impulse response of the combined transform) is given by the warped convolution of X[n,m] and h(τ,v)

[0156]

[0157] This result can be considered as an extension of the single carrier modulation case, where the received signal through a time-invariant channel is given by the convolution of the QAM symbol with a composite pulse, which is the convolution of the transmitter pulse with the channel impulse response.

[0158] With this result established, we are ready to examine exemplary receiver processing steps.

[0159] Receiver Processing and Wigner Transform

[0160] Typical communication system designs generally require the receiver to perform a matched filtering operation to take the inner product of the received waveform and the transmitter pulse (appropriately delayed or otherwise distorted by the channel). In this case, we have used a batch of delayed and modulated transmit pulses, and typically perform a matched filtering operation on each of them.

[0161] Figure 4 provides a conceptual view of this process. At the transmitter, for each symbol we send, we modulate a set of M subcarriers, and at the receiver we perform matched filtering on each of these subcarrier pulses. We define the receiver pulse g r (t) and takes the inner product with a batch of delayed and modulated versions of the receiver pulse. The receiver pulse g r (t) is in many cases the same as the transmitter pulse, but we keep a separate notation to cover the receiver pulse g r (t) Some cases where it is not identical to the transmitter pulse (especially in OFDM where CP samples must be discarded).

[0162] While this approach yields sufficient statistics for data detection in the case of an ideal channel, concerns may be raised here for the case of non-ideal channel effects. In this case, sufficient statistics for symbol detection are obtained by matched filtering with the channel-distorted information-carrying pulses (assuming the additive noise is white and Gaussian). However, in many well-designed multicarrier systems (e.g., OFDM and MCFB), the channel-distorted version of each subcarrier signal is just a scalar version of the transmitted signal, allowing matched filter design that is channel-independent and uses the original transmitted subcarrier pulses. We will make these statements briefer and more precise, and examine the conditions required to make them true.

[0163] In practical embodiments of OTFS receivers, such matched filtering can be implemented in the digital domain using FFT or polyphase transform for OFDM and MCFB, respectively. However, for the purposes of this discussion, we will perform the matched filtering by taking the inner product of the received waveform with a delayed and modulated version of the receiver pulse for an arbitrary time and frequency offset (τ, v). <g r (t-τ)e j2 πv(t-τ) , r(t)> to consider this generalization of matched filtering. Although this is probably not necessarily the actual implementation, it allows us to generalize Figure 4 The operation of can be viewed as a two-dimensional sampling of this more general inner product.

[0164] Let us define the inner product

[0165]

[0166] function is called the cross-ambiguity function, and if sampled (on the lattice Λ) at τ = nT, v = mΔf, then the matched filter output is produced, i.e.,

[0167]

[0168] The ambiguity function is related to the inverse of the Heisenberg transform (ie, the Wigner transform). Figure 4 provides an intuitive sense of it, as the receiver appears to be reversing the operation of the transmitter. More formally, if we obtain cross-ambiguity or the transmitted and received pulses And using it as the impulse response of the Heisenberg operator, we obtain the orthogonal cross-projection operator

[0169]

[0170] In other words, if used in the Heisenberg representation, then the coefficients from the matched filter will provide the best approximation to the original y(t) in terms of least squared error.

[0171] A key issue to be addressed is the relationship between the matched filter output Y[n,m] (or more generally, Y(τ,v)) and the transmitter input X[n,m]. We have already established in (18) that the input to the matched filter r(t) can be expressed as a Heisenberg representation with an impulse response f(τ,v) (plus noise). The output of the matched filter then has two contributions

[0172]

[0173] The last term is the contribution of noise, which we express as The first term on the right-hand side is the matched filter output for an input consisting of a superposition of delayed and modulated versions of the transmitted pulse (noise-free). We next establish that this term can be expressed as a warped convolution of the 2D impulse response f(τ, v) with the cross-ambiguity function (or 2D cross-correlation) of the transmitted and received pulses.

[0174] The following theorem summarizes the key results.

[0175] Theorem 1 : (Basic time-frequency domain channel equation). If the received signal can be expressed as

[0176] Π f (gtr (t)) = ∫∫f(τ, v)e j2πv(t-τ) g tr (t-τ)dvdτ (23)

[0177] Then the signal and the received pulse g tr The cross ambiguity of (t) can be expressed as

[0178]

[0179] Recall from (19) that f(τ,v)=h(τ,v)⊙X[n,m], that is, the composite impulse response is itself the distorted convolution of the channel response and the modulation symbols.

[0180] Substituting f(τ, v) from (19) into (22), we obtain the end-to-end channel description in the time-frequency domain

[0181]

[0182] where y(τ, v) is the additive noise term. Equation (25) provides an abstraction of the time-varying channel in the time-frequency plane. It shows that the output of the matched filter at any time and frequency point (τ, v) is given by the warped convolution of the delay-Doppler impulse response of the channel with the impulse response of the modulation operator and the warped convolution of the cross-blur (or two-dimensional cross-correlation) function of the transmitted and received pulses.

[0183] Evaluating equation (25) on the lattice Λ, we obtain the matched filter output modulation symbol estimate

[0184]

[0185] To gain more intuition for equations (25) and (26), let us first consider the case of an ideal channel, i.e., h(τ, v) = δ(τ)δ(v). In this case, by direct substitution, we obtain the convolution relationship

[0186]

[0187] To simplify equation (27), we will exploit the orthogonality of the ambiguity function. Since we use different transmit and receive pulses, we modify the orthogonality condition for the transmit pulse design described in (10) to a biorthogonality condition

[0188]

[0189] Under this condition, only one term survives (27) and we obtain

[0190] Y[n,m]=X[n,m]+V[n,m] (29)

[0191] where V[n,m] is additive white noise. Equation (29) shows that under ideal channel conditions, the matched filter output does indeed recover the transmitted symbols (plus noise). More interesting is of course the case of non-ideal time-varying channel effects. We next show that even in this case, channel orthogonality is maintained (no intersymbol or intercarrier interference), and the channel complex gain distortion has a closed-form expression.

[0192] The following theorem is summarized as a consequence of the generalization of (29).

[0193] Theorem 2 : (End-to-end time-frequency domain channel equation):

[0194] If h(τ, v) has the max , v max ) is defined by a finite support, and if for τ∈(nT-τ max , nT+τ max )、v∈(mΔf-v max , mΔf+v max )have That is, the biorthogonality property of the ambiguity function of (28) is true in a neighborhood of each grid point (mΔf, nT) of the lattice Λ that is at least as large as the support of the channel response h(τ, v), then the following equation holds

[0195] Y[n,m]=H[n,m]X[n,m]

[0196] H[n,m]=∫∫h(τ,v)e j2πvnT e -j2π(v+mΔf)τ dvdτ (30)

[0197] If the ambiguity functions are only approximately biorthogonal in the neighborhood of Λ (by continuity), then (30) is only approximately true. Equation (30) is the fundamental equation describing the channel behavior in the time-frequency domain. This is fundamental to understanding the nature of the channel and its variations along the time and frequency dimensions.

[0198] Some observations are presented in turn for equation (30). As mentioned earlier, there is no interference across X[n,m] in time n or frequency m.

[0199] • The end-to-end channel distortion in the modulation domain is a (complex) scalar quantity that needs to be equalized.

[0200] If there is no Doppler, that is, h(τ, v) = h(τ, 0)δ(v), then equation (30)

[0201] become

[0202] Y[n,m]=X[n,m]∫h(τ,0)e -j2πmΔfτ dτ

[0203] =X[n,m]H(0,mΔf) (31)

[0204] This is a well-known multi-carrier result, where each sub-carrier symbol is multiplied by the frequency response of the time-invariant channel evaluated at that sub-carrier frequency.

[0205] If there is no multipath, i.e., h(τ, v) = h(0, v)δ(τ), then equation (30) becomes

[0206] Y[n,m]=X[n,m]∫h(v,0)e j2πvnT dτ (32)

[0207] Note that the fading experienced by each subcarrier over time nT has a complex representation as a weighted superposition of exponentials. This is a major complexity in the design of wireless systems with mobility like LTE; it requires the transmission of pilots and continuous tracking of the channel, which becomes more difficult the higher the vehicle speed or Doppler bandwidth.

[0208] Some examples of this general framework are provided below.

[0209] Example 3 : (OFDM modulation). In this case, the basic transmit pulse is given by (14) and the basic receive pulse is

[0210]

[0211] That is, the receiver zeroes the CP samples and applies a square window to the symbols including the OFDM symbols.It is noteworthy that in this case, the biorthogonality property holds exactly along the time dimension.

[0212] Example 4 :(MCFB modulation). In the multi-carrier filter bank g tr (t) = g r (t) = g(t). There are several designs for the basic pulse g(t). The square root raised cosine pulse provides good localization along the frequency dimension at the expense of poor localization along the time dimension. If T is much larger than the support of the channel in the time dimension, then each subchannel sees a flat channel and the biorthogonality property approximately holds.

[0213] In summary, one of the two transforms that define OTFS has now been described. Specifically, an explanation has been provided on how a transmitter and receiver apply appropriate operators on the elementary transmit and receive pulses and orthogonalize the channel according to equation (30). Examples have also been provided to illustrate how the choice of elementary pulses affects the time and frequency positioning of the transmitted modulation symbols and the quality of the achieved channel orthogonalization. However, equation (30) shows that channels in this domain, while free of inter-symbol interference, suffer from fading in both the time and frequency dimensions via a complex superposition of linear phase factors.

[0214] In the following we start from equation (30) and describe the second transformation that defines OTFS; we will show how this transformation defines an information domain in which the channel does not fade in any dimension.

[0215] 2D OTFS Transform

[0216] Note that the time-frequency response H[n,m] in (30) is related to the channel delay-Doppler response h(τ,v) by an expression similar to that of the Fourier transform. However, there are two important differences: (i) the transform is two-dimensional (along delay and Doppler) and (ii) the exponents defining the transform for the two dimensions have opposite signs. Despite these difficulties, equation (30) guides the use of complex exponentials as basis functions on which information symbols are modulated; and simply transmitting the superposition of these modulated complex exponential bases in the time-frequency domain. As discussed below, this approach exploits the Fourier transform properties and effectively converts a multiplicative channel in one Fourier domain into a convolutional channel in another Fourier domain.

[0217] Given the difficulties with equation (30) mentioned above, we need to develop a suitable version of the Fourier transform and the associated sampling theory results. Let us start with the following definitions:

[0218] Definition 1: Symplectic Discrete Fourier Transform : A two-dimensional sequence that can be summed given squares We define

[0219]

[0220] Note that the above 2D Fourier transform (called the symplectic discrete Fourier transform) differs from the more well-known Cartesian Fourier transform in that the exponential functions in each of the two dimensions have opposite signs. This is necessary in this case because it matches the behavior of the channel equation.

[0221] It is further noted that the resulting x(τ, v) is periodic, with a period (1 / Δf, 1 / T). This transformation defines a new two-dimensional plane, which we will call the Delay-Doppler plane, and which can represent the maximum delay of 1 / Δf and the maximum Doppler of 1 / T. A one-dimensional periodic function is also called a function on a circle, while a 2D periodic function is called a function on a torus (or donut). In this case, x(τ, v) is defined on a torus Z, which has a circumference (dimension) of (1 / Δf, 1 / T).

[0222] The periodicity of x(τ, v) (or the sampling rate in the time-frequency plane) also defines a lattice in the delay-Doppler plane, which we call the reciprocal lattice

[0223]

[0224] Points on the reciprocal lattice have the property that the exponent in (34) is an integer multiple of 2π.

[0225] The inverse transform is given by:

[0226]

[0227] Where c = TΔf.

[0228] Next we define a sampled version of x(τ, v). Specifically, we want to take M samples in the delay dimension (with intervals of 1 / MΔf) and N samples in the Doppler dimension (with intervals of 1 / NT). More formally, a denser version of the reciprocal lattice is defined such that

[0229]

[0230] We define a discrete periodic function with period (1 / Δf, 1 / T) on this dense lattice, or equivalently define a function on a discrete torus with these dimensions

[0231]

[0232] These functions are related to discrete periodic functions on the lattice Λ, or equivalently, to functions on the discrete torus, via the Fourier transform relation

[0233] Z 0 ={(nT, mΔf), m=0,..., M-1, n=0,...N-1,} (39)

[0234] We wish to derive an expression for sampling equation (34) on the lattice of (38). First, we start with the following definition.

[0235] Definition 2: Symplectic Finite Fourier Transformation: If X p [k, l] is periodic, with period (N, M), so we define

[0236]

[0237] It should be noted that x p [m,n] is also periodic, with period [M,N], or equivalently, it is defined on the discrete torus Formally, SFFT(X[n,m]) is derived from Linear transformation of .

[0238] Now let us consider generating x p [m, n] is the sampled version of (34), i.e., We can then show that (40) still holds, where X p [m, n] is the periodization of X[n, m], with period (N, M)

[0239]

[0240] This is similar to the effect that sampling in one Fourier domain produces aliasing in the other domain.

[0241] The inverse discrete (symplectic) Fourier transform is given by

[0242]

[0243] where l = 0, ..., M-1, k = 0, ..., N-1. If the support of X[n, m] is time-frequency restricted to Z 0 (There is no aliasing in (41)), then for n, m∈Z 0 Existence X p [n,m]=X[n,m], and the inverse transform (42) restores the original signal.

[0244] The SDFT is called "discrete" because it represents the signal using a discrete set of indices, while the SFFT is called "finite" because it represents the signal using a finite set of indices.

[0245] In this context, the important property of the symplectic Fourier transform is that it transforms multiplicative channel effects in one domain into circular convolution effects in the transformed domain. This is summarized in the following proposition:

[0246] Proposition 2 :make is a periodic 2D sequence.

[0247] then

[0248] SFFT(X 1 [n,m]*X 2 [n, m]) = SFFT(X 1 [n, m])·SFFT(X 2 [n,m]) (43)

[0249] where * denotes a 2D circular convolution. With this framework established, we are ready to define OTFS modulation.

[0250] Discrete OTFS modulation : Consider a set of NM QAM information symbols that we wish to transmit arranged on a 2D grid x[l,k] (k=0,...,N-1, l=0,...,M-1). We will consider x[l,k] to be two-dimensionally periodic with period [N,M]. In addition, assume a multicarrier modulation system defined by

[0251] · A lattice on the time-frequency plane, i.e., sampling on the time axis with a sampling period of T and sampling on the frequency axis with a sampling period of Δf (see equation (9)).

[0252] • A burst of packets with a total duration of NT seconds and a total bandwidth of MΔf Hz.

[0253] The transmit and receive pulses g satisfying the biorthogonality property of (28) tr (t),

[0254] Multiply the transmitted windowed square summation function of the modulation symbol in the time-frequency domain

[0255] Through a set of basis functions b k，l [n, m] and a set of modulation symbols X[n, m] associated with the information symbol x[k, l], n = 0, ..., N-1, m = 0, ..., M-1

[0256]

[0257]

[0258] The basis function b k，l [n,m] is related to the inverse symplectic Fourier transform (see equation (42)).

[0259] Given the above components, we define the discrete OTFS modulation via the following two steps:

[0260] X[n,m]=W tr[n,m]SFFT -1 (x[k,l])

[0261] s(t)=Π X (g tr (t)) (45)

[0262] The first equation in (45) describes the OTFS transform, which combines the inverse symplectic transform with a windowing operation. The second equation describes the modulation symbol via the pair g parameterized by X[n,m]. tr (t) is transmitted by the Heisenberg transform of X[n,m]. A more explicit formula for the modulation step is given by equations (42) and (11).

[0263] Although the expression of the OTFS modulation via the symplectic Fourier transform reveals important properties, it is easier to understand the modulation via equation (44), namely, by modulating the 2D basis function b in the time-frequency plane k，l [n, m] to send each information symbol x[k, l].

[0264] Discrete OTFS demodulation : Let us assume that the transmitted signal s(t) experiences channel distortion according to (8) and (2), resulting in r(t) at the receiver. In addition, let the receiver adopt the receive windowed square summable function W r [n, m]. The demodulation operation then consists of the following steps:

[0265] (i) Match filtering the received pulse, or more formally, evaluating the ambiguity function (Wigner transform) on Λ to obtain an estimate of the time-frequency modulation symbol

[0266]

[0267] (ii) Windowing and periodizing Y[n, m]

[0268] Y w [n, m] = W r [n,m]Y[n,m]

[0269]

[0270] (iii) and for the periodic sequence Y p [n,m] applies the symplectic Fourier transform

[0271]

[0272] As we discussed earlier, the first step of the demodulation operation can be interpreted as a matched filtering operation in the time-frequency domain. The second step is there to ensure that the input to the SFFT is a periodic sequence. If a trivial window is used, then this step can be skipped. The third step can also be interpreted as the projection of the time-frequency modulation symbols onto the orthogonal basis functions.

[0273]

[0274]

[0275] The discrete OTFS modulation defined above points to an efficient implementation via discrete and periodic FFT type processing. However, it may not provide insight into the temporal and bandwidth resolution of these operations in the context of two-dimensional Fourier sampling theory. Next we introduce continuous OTFS modulation and describe the more practical discrete OTFS as a sampled version of continuous modulation.

[0276] Continuous OTFS modulation : Consider a two-dimensional periodic function x(τ, v) with period [1 / Δf, 1 / T] that we wish to transmit. The choice of period may seem arbitrary at this point, but the rationale for its selection will become clear after the following discussion. In addition, assume that the multicarrier modulation system defined by

[0277] ·Point matrix on the time-frequency plane, i.e., sampling on the time axis with a sampling period of T and sampling on the frequency axis with a sampling period of Δf (see equation (9)).

[0278] The transmit and receive pulses g satisfying the biorthogonality property of (28) tr (t),

[0279] Multiply the transmit windowing function of the modulation symbols in the time-frequency domain

[0280] Given the above components, we define the continuous OTFS modulation in the following two steps:

[0281] X[n,m]=W tr [n,m]SDFT -1 (x(τ,v))

[0282]

[0283] The first equation describes the inverse discrete time-frequency sigmoid Fourier transform [see equation (36)] and the windowing function, while the second equation describes the transmission of the modulation symbols via the Heisenberg transform [see equation (11)].

[0284] Continuous OTFS demodulation : Assume that the transmitted signal s(t) undergoes channel distortion according to (8) and (2), resulting in r(t) at the receiver. In addition, let the receiver adopt the receive windowing function The demodulation operation then consists of two steps:

[0285] (i) Evaluate the ambiguity function (Wigner transform) on Λ to obtain an estimate of the time-frequency modulation symbol

[0286]

[0287] (ii) Window the modulation symbols and apply the symplectic Fourier transform

[0288]

[0289] Note that there is no periodization of Y[n,m] in (51) and (52) because the SDFT is defined on a non-periodic square-summable sequence. The required periodization step in discrete OTFS can be understood as follows. Suppose we wish to recover the transmitted information symbols by performing continuous OTFS demodulation and then sampling on a delay-Doppler grid.

[0290]

[0291] Since it is generally impractical to perform a continuous symplectic Fourier transform, we consider whether we can obtain the same result using the SFFT. The answer is that if the input sequence is first periodized (aliased), then the SFFT process will produce exactly the samples we want. See also equations (40) and (41).

[0292] We have now described each step of an exemplary form of OTFS modulation. We have also discussed how the Wigner transform at the receiver inverts the Heisenberg transform at the transmitter [see Eqs. (27), (29)], and have made similar discussions for the forward and inverse symplectic Fourier transforms.

[0293] Figure 5A The operations involved in OTFS-based communication are exemplarily shown, including the transformation from the time-frequency plane to the Doppler-delay plane. In addition, Figure 5A and 5BIndicates the relationship between sampling rate, delay resolution, and time resolution. Figure 5A In the first operation, the Heisenberg transform translates the time-varying convolution channel in the waveform domain to an orthogonal but still time-varying channel in the time-frequency domain. For a total bandwidth BW and M subcarriers, the frequency resolution is Δf = BW / M. For a total frame duration T f and N symbols, the time resolution is T = T f / N.

[0294] In the second operation, the SFFT transform translates the time-varying channel in the time-frequency domain to a time-invariant channel in the delay-Doppler domain. The Doppler resolution is 1 / T f , and the delay resolution is 1 / BW, where Doppler resolution and delay resolution are decoupled from each other. As in classical spectrum analysis, the choice of window can provide a trade-off between mainlobe width (resolution) and sidelobe suppression.

[0295] refer to Figure 5C , provides an example of signal transmission between various domains in an OTFS communication system. Specifically, Figure 5C Signal transmission on (i) an actual physical channel having a signal transmission waveform, (ii) a time-frequency domain, and (iii) a delay-Doppler domain is illustrated.

[0296] Figure 5D Schematic notation used in representing signals present at various processing stages (eg, encoding, modulation, decoding, demodulation) in an OTFS transmitter and receiver is illustrated.

[0297] Channel equation in OTFS domain

[0298] We will now provide a mathematical characterization of the end-to-end signal relationship in an OTFS system when a non-ideal channel is located between the transmitter and the receiver. Specifically, this section will demonstrate how to transform the time-varying channel in (2) and (8) into a time-invariant convolutional channel in the delay-Doppler domain.

[0299] Proposition 3 :Consider arranging in a period of A set of NM QAM information symbols in a 2D periodic sequence x[l, k] of . The sequence x[k, l] undergoes the following transformation:

[0300] • Modulate it using the discrete OTFS modulation of equation (45).

[0301] • It is distorted by the delay-Doppler channel of equations (2) and (8).

[0302] • It is demodulated by discrete OTFS demodulation of equations (46) and (48).

[0303] The estimated sequence obtained after demodulation is The input QAM sequence x[m,n] and the windowed impulse response h w The sampled version of the two-dimensional periodic convolution of (·) gives

[0304]

[0305]

[0306] where h w (τ′, v′) represents the circular convolution of the channel response and the windowing function

[0307] h w (τ′, v′) = ∫∫e -j2πvτ h(τ,v)w(τ′-τ,v′-v)dτdv (55)

[0308] More precisely, as can be seen in the equation, the window w(τ, v) is related to a slightly modified version of the channel impulse response (via the complex exponential) e -j2πvτ h(τ, v) is circularly convolved. The windowing function w(τ, v) is the symplectic Fourier transform of the time-frequency window W[n, m]

[0309]

[0310] And where W[n,m] is the product of the send window and the receive window.

[0311] W[n,m]=W tr [n,m]W r [n,m] (57)

[0312] In many cases, the windows in the transmitter and receiver are matched, i.e., W tr [n, m] = W 0 [n, m] and Therefore W[n, m] = |W 0 [n,m]| 2 ,

[0313] The effect of a window is to produce an obfuscated version of the original channel at a resolution that depends on the span of available frequency and time samples. If a rectangular (or trivial) window is considered, i.e., when n=0, ..., N-1, m=-M / 2, ..., M / 2-1 and the rest are equal to zero, then its SDFTw(τ,v) in (56) is a two-dimensional Dirichlet kernel with bandwidth inversely proportional to N and M.

[0314] The window function has several other uses. The system can be designed with a window function that is intended to randomize the phase of the transmitted symbols. This randomization may be more important for pilot symbols than for symbols carrying data. For example, if adjacent cells use different window functions, the problem of pilot contamination is avoided.

[0315] Channel Estimation in OTFS Domain

[0316] There are many different ways to design a channel estimation scheme for an OTFS system, with various implementation options and details.

[0317] A straightforward way to perform channel estimation requires sending a sounding OTFS frame containing a discrete delta function in the OTFS domain, or equivalently, sending a set of unmodulated carriers in the time-frequency domain. From a practical point of view, the carriers can be modulated with known (say, BPSK) symbols that are removed at the receiver, as is common in many OFDM systems.

[0318] Figure 6 Discrete impulses in the OTFS domain are shown that can be used for channel estimation purposes, for example, in the form of pilot signals. Figure 6 In the example of , the remaining portion of the M×N delay-Doppler plane includes information symbols arranged in a grid. In a typical implementation, M×N may be, for example, 1024×256 or 512×16.

[0319] However, this approach may be wasteful since the range of the channel response is only a small fraction of the full range of the OTFS frame (1 / T, 1 / Δf). For example, in LTE systems 1 / T ≈ 15 kHz and the maximum Doppler shift f d，max Usually one to two orders of magnitude smaller. Similarly, 1 / Δf ≈ 67usec, and the maximum delay spread τ max is also one to two orders of magnitude smaller. Therefore, we can dedicate a much smaller region of the OTFS frame to channel estimation, while the rest of the frame carries useful data. More specifically, for a channel with support (±f d，max , ±τ max ) channel, we need a length of (2f d，max / T,τ max / Δf) of OTFS subframe.

[0320] In the case of multi-user transmission, each UE may have its own channel estimation subframe located in a different part of the OTFS frame. However, the size of such a channel estimation subframe may be relatively limited. For example, if τ max is 5% of the range of the delay dimension, and fd，max It is 5% of the Doppler dimension, so the channel estimation subframe only needs to be 5%×5%=0.25% of the OTFS frame.

[0321] Importantly, although the channel estimation symbols are confined to a small portion of the OTFS frame, they actually probe the entire time-frequency domain via the corresponding two-dimensional time-frequency basis functions associated with these symbols.

[0322] A different approach to channel estimation is to place pilot symbols on a subgrid in the time-frequency domain. The key problem in this approach is to determine a pilot density that is sufficient for channel estimation without introducing aliasing. Assume that for some integer n 0 、m 0 , the pilot occupies a subgrid (n 0 T, m 0 Δf). Recall that for this grid, the SDFT will be periodic, with period (1 / n 0 T, 1 / m 0 Δf). Then, applying the aliasing results discussed previously to this grid, we obtain the alias-free Nyquist channel support region (±f d，max , ±τ max )=(±1 / 2n 0 T, ±1 / 2m 0 Δf). Given the maximum support of a channel, the density of pilots can be determined from this relationship. The pilot subgrid should be extended to the entire time-frequency frame so that the resolution of the channel is not compromised.

[0323] OTFS access: multiplexing more than one user

[0324] There are various ways to multiplex several uplink transmissions or downlink transmissions in one OTFS frame. Here we briefly review the following multiplexing methods:

[0325] OTFS delay-Doppler domain multiplexing

[0326] Multiplexing in the time-frequency domain

[0327] Multiplexing in code spread spectrum domain

[0328] Multiplexing in the spatial domain

[0329] 1. Multiplexing in the delay-Doppler domain: This is probably the most natural multiplexing scheme for downlink transmission. Different users are given different sets of OTFS basis functions or groups of information symbols or resource blocks. Given the orthogonality of the basis functions, the users can be separated at the UE receiver. The UE only needs to demodulate the portion of the OTFS frame that is assigned to it.

[0330] Attention shifts Fig. 7A and 7B , Fig. 7A and 7B A pair of different, illustrative basis functions that can be used in OTFS communications are illustrated. Compared to conventional communication systems, in an OTFS system, even small subframes or resource blocks in the OTFS domain will be transmitted over the entire time-frequency frame via a two-dimensional basis function and will experience an average channel response. Fig. 7A and 7B This is illustrated by showing two different basis functions belonging to different users.Thus, there is no channel resolution impairment for each user, regardless of the resource block or subframe size.

[0331] Figure 7C and 7D The two-dimensional spreading with respect to both time and frequency in an OTFS communication system is generally illustrated. As shown, one or more QAM symbols 710 may be placed in a two-dimensional information domain. Each QAM symbol in the information domain is multiplied by a two-dimensional (i.e., time and frequency domain) basis function 720. In a typical OTFS communication system, the transmitted signal includes a set of QAM symbols spread across both time and frequency by a set of corresponding two-dimensional basis functions that are mutually orthogonal with respect to both time and frequency.

[0332] Fig. 7E Members of a possible set of such mutually orthogonal two-dimensional basis functions are illustrated.

[0333] In the uplink direction, transmissions from different users experience different channel responses. Therefore, different subframes in the OTFS domain will experience different convolved channels. This can potentially introduce inter-user interference at the edge where two user subframes are adjacent, and a guard gap will be required to eliminate it. To avoid this overhead, different multiplexing schemes can be used in the uplink as explained next.

[0334] 2. Multiplexing in the time-frequency domain: In this approach, resource blocks or subframes are allocated to different users in the time-frequency domain. Figure 8 This is illustrated for the case of three users. Figure 8 As shown, the first user (U1) occupies the entire frame length, but only occupies half of the available subcarriers. The second user (U2) and the third user (U3) occupy the other half of the subcarriers and divide the total length of the frame between them.

[0335] Note that in this case, each user uses a slightly different version of the described OTFS modulation. One difference is that each user i modulates the subframe (Ni , M i )N i ≤N,M i ≤M performs SFFT. This reduces the resolution of the channel, or in other words, reduces the range of the time-frequency plane over which each user will experience variations in its channel. On the other hand, this also gives the scheduler the opportunity to schedule users in the part of the time-frequency plane where their channels are best.

[0336] If it is desired to extract the maximum diversity of the channel and distribute users across the entire time-frequency frame, users can be multiplexed via interleaving. In this case, one user occupies a subsampled grid of the time-frequency frame, while another user occupies another subsampled grid adjacent to it. Fig. 9 Shown with Figure 8 , but interleaved in the subcarrier dimension. Of course, it is also possible to interleave in the time dimension and / or in both dimensions. The degree of interleaving or the degree of grid subsampling per user is only limited by the extension of the channel that must be accommodated.

[0337] 3. Multiplexing in the time-frequency spreading code domain: Assume that it is desirable to design a random access PHY and MAC layer where users can access the network without the need for elaborate RACH and other synchronization procedures. The need for such a system to support Internet of Things (IoT) deployments has been recognized. OTFS can support such a system by assigning a different two-dimensional window function designed as a randomizer to each user. In this embodiment, the windows of different users are designed to be approximately orthogonal to each other and to the time and frequency offsets. Each user then transmits on only one or a few basis functions and uses the window as a means to randomize interference and provide processing gain. This results in a very simplified system that may be attractive for low-cost, short-burst type IoT applications.

[0338] 4. Multiplexing in the spatial domain: Finally, similar to other OFDM multi-carrier systems, a multi-antenna OTFS system can support multiple users transmitting on the same basis function across the entire time-frequency frame. The users are separated by appropriate transmitter and receiver beamforming operations.

[0339] Exemplary Implementation of OTFS Communication System

[0340] As discussed above, an embodiment of orthogonal time-frequency space (OTFS) modulation consists of a cascade of two transforms. The first transform maps the two-dimensional plane where the information symbols reside (and which may be referred to as the delay-Doppler plane) to the time-frequency plane. The second transform transforms the time-frequency domain to the waveform time domain that actually constructs the transmitted signal. This transform may be considered a generalization of a multi-carrier modulation scheme.

[0341] Fig.10 Components of an exemplary OTFS transceiver 1000 are shown. OTFS transceiver 1000 may be used as Figure 3 The OTFS transceiver 1000 includes one or two of the exemplary OTFS transceivers 315 shown in the communication system 300 of FIG. 1000. The OTFS transceiver 1000 includes a transmitter module 1005, which includes a pre-equalizer 1010, an OTFS encoder 1020, and an OTFS modulator 1030. The OTFS transceiver 1000 also includes a receiver module 1055, which includes a post-equalizer 1080, an OTFS decoder 1070, and an OTFS demodulator 1060. The components of the OTFS transceiver can be implemented in hardware, software, or a combination thereof. For hardware implementations, the processing unit can be implemented in one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, other electronic units designed to perform the above functions, and / or combinations thereof. The disclosed OTFS method will be described from the perspective of the various components of the transceiver 1000.

[0342] Reference again Figure 3 In one aspect, the method of OTFS communication involves sending at least one data frame ([D]) from a sending device 310 to a receiving device 330 via a communication channel 320, such data frame including up to N 2The method includes convolving the data elements of the data frame within the OTFS transceiver 315-1 so that the value of each data element is spread over a plurality of wireless waveforms when transmitted, each waveform having a characteristic frequency, and each waveform carries the convolution result of a plurality of said data elements from the data frame [D]. In addition, during the transmission process, the frequencies of the plurality of wireless waveforms are cyclically shifted multiple times so that the value of each data element is transmitted as a plurality of cyclically frequency-shifted waveforms transmitted multiple times. At the receiving device 330, the OTFS transceiver 315-2 receives and deconvolves the wireless waveforms, thereby reconstructing a copy of the at least one data frame [D]. In an exemplary embodiment, the convolution process is such that any data element of any data frame ([D]) is guaranteed to be completely accurately reconstructed until substantially all of these wireless waveforms have been transmitted and received.

[0343] Fig.11 A comparison of the bit error rate (BER) predicted by simulations for a TDMA system and an OTFS system is shown. Both systems use a 16QAM constellation. The simulation models a Doppler spread of 100 Hz and a delay spread of 3 milliseconds. As can be seen from the figure, for the same signal-to-noise ratio (SNR), the OTFS system provides a much lower BER than the TDMA system.

[0344] Follow now Fig.12 , Fig.12 1 is a flow chart representing operations performed by the OTFS transceiver 1200, wherein the OTFS transceiver 1200 may be implemented as, for example, the OTFS transceiver 1000 ( Fig.10 ). The OTFS transceiver 1200 includes a transmitter and a receiver, wherein the transmitter includes a modulator 1210 and the receiver includes a demodulator 1220 and a two-dimensional equalizer 1230. In operation, the transmitter of the OTFS transceiver 1200 receives a two-dimensional symbol stream in the form of an N×N matrix of symbols, which may be referred to as a TF matrix hereinafter:

[0345] x∈C N×N

[0346] like Fig.13 As shown, in one embodiment, the modulator 1210 is used as an orthogonal map arranged to transform the two-dimensional TF matrix into the following transmission waveform:

[0347] φ t =M(x)=∑x(i,j)φ i，j φ i，j ⊥φ k，l

[0348] refer to Fig.14, the demodulator 1220 transforms the received waveform into a two-dimensional TF matrix according to an orthogonal mapping to generate an output stream:

[0349]

[0350] In one embodiment, the OTFS transceiver 1200 can be characterized by a plurality of variable parameters, including, for example, delay resolution (i.e., digital time "ticks" or clock increments), Doppler resolution, processing gain factor (block size), and orthogonal basis functions. These variable parameters can each be expressed as follows.

[0351] Delay resolution (digital time ticks):

[0352]

[0353] Doppler resolution:

[0354]

[0355] Processing gain factor (block size):

[0356] N>0

[0357] C Nx1 The standard orthogonal basis (spectral shape) of is:

[0358] U={u 1 ,u 2 , ..., u N}

[0359] like Fig.12 As shown, during operation, the modulator 1210 obtains the TF matrix x∈C N×N And transform it into a pulse waveform. In one embodiment, the pulse waveform includes a pulse train defined according to the Heisenberg representation and the spectrum shape:

[0360]

[0361] where b 1 , b 2 ...b N exist Fig.15 And, according to the Heisenberg relationship:

[0362] Π(h*x)=Π(h)·Π(x), in particular

[0363] Π(δ (τ，o) *x)=L τ ·Π(x)

[0364] Π(δ (0，w) *x)=Mw ·Π(x)

[0365] Heisenberg said it provides:

[0366] Given by the following formula

[0367]

[0368] Where L t and M w denote the cycle time and frequency offset respectively, and can be expressed as:

[0369] L τ ∈C N×N :

[0370] M w ∈C N×N :

[0371] The demodulator 1220 obtains the received waveform and transforms it into a TF matrix y∈C defined by the Wigner transform and the spectral shape N×N :

[0372]

[0373] Main properties of M and D (Stone-von Neumann theorem):

[0374] D(h a M(x)) = h*x, where

[0375] h(τ, w)≈a(τΔT, wΔF)

[0376] like Fig.16 As shown, the equalizer 1230 can be implemented as a two-dimensional discriminant feedback equalizer, which is configured to perform a least mean square (LMS) equalization process such that:

[0377]

[0378] The equalizer 1230 exploits the following property of OTFS: In the time-frequency plane, all bits experience the same distortion during propagation through the channel. By "de-blurring" the bits using the same distortion they experienced in the channel, the signal sent into the channel is revealed.

[0379] Transmitter Grid and Receiver Bin Structure

[0380] Now turn your attention to Figures 17A-17D, which depicts an OTFS transmitter 102 and receiver 104 to which reference will be made when describing the transmission and reception of OTFS waveforms. More specifically, Figure 17B-Figure 17D The transmission and reception of the OTFS waveform is shown relative to a time-frequency transmitter grid or series of bins and a corresponding time-frequency receiver grid or series of bins. As will be discussed below, the receiver 104 will generally operate relative to a finer time-frequency receive grid than the time-frequency transmit grid associated with the transmitter 102.

[0381] Now go to Fig.17A , a transmitter 102 and a receiver 104 are separated by an impaired wireless data channel 100 including one or more reflectors 106. As shown, the reflectors 106 may reflect or otherwise impair the waveforms (112, 114a, 114b) as they travel through the data channel 100. These reflectors may be inherently caused by the channel 100 (see, for example, the limited channel h of FIG. 18). eqv，f )’s two-dimensional (2D) channel state representation.

[0382] Fig.18A and 18B OTFS communication over a communication channel characterized by a two-dimensional delay-Doppler impulse response is illustratively shown. Fig.18A and 18B The collection of bar graphs of can represent a two-dimensional impulse response that implements a finite modulation equivalent channel, a transmitted information vector x containing OTFS QAM symbols, and a received information vector y containing received OTFS symbols. An advantage of the embodiment of the OTFS communication system shown by Figures 18A and 18B is that the two-dimensional channel model is fixed and all symbols experience the same distortion. As a result of remaining fixed for the necessary duration, the OTFS channel model is non-fading, and each symbol can be spread across the entire duration and bandwidth and experience all diversity branches of the channel. The deterministic properties of the two-dimensional OTFS channel reflect the geometry (distance and speed) of the channel reflectors.

[0383] In one embodiment, the transmitter 102 includes a transmitter processor 102p to pack the input data into at least one N×M array of data symbols. This array of data symbols is then transmitted using an encoding process according to the OTFS modulation technique described herein. The transmitted OTFS waveform is received by a receiver 104 including a receiver processor 104p. In one embodiment, the receiver processor 104p utilizes information related to the 2D state of the channel 100 to enable the OTFS waveforms to be decoded and the transmitted data symbols to be recovered. Specifically, the receiver processor 104p may decode and extract the plurality of data symbols using the inverse of the OTFS encoding process. Alternatively, the signal may be corrected for data channel impairments after the receiver has decoded and extracted the plurality of data symbols.

[0384] In some embodiments, OTFS data transmission can be achieved by transforming an N×M array of input data symbols into at least one block or array of filtered OFDM symbols. This can be done, for example, using a one-dimensional Fourier transform and a filtering process or algorithm. Various types of two-dimensional Fourier transforms can then be used to transform this block or array of filtered OFDM symbols into at least one block or array of OFTS symbols. These results will typically be stored in the transmitter memory 102m. The stored results can then be transmitted on the wireless frequency sub-band by various methods. For example, in one embodiment, a transmitter 102c using a series of M narrowband filter banks can be used. In this implementation, the transmitter 102c generates a series of M mutually orthogonal waveforms transmitted over at least N time intervals.

[0385] In one embodiment, gaps or "guard bands" may be imposed in both time and frequency to minimize the possibility of unintentional crosstalk between the various narrowband filters and time intervals prior to transmission. Depending on the characteristics of the data channel, any such gaps or guard bands may be increased or decreased or set to zero as circumstances warrant.

[0386] Alternatively, the OTFS encoding process may encode an N×M array of data symbols onto a manifold compatible with symplectic analysis. These symbols may be distributed over a column time axis of length T and a row frequency axis of length F, thereby generating at least one information manifold for storage in the transmitter memory 102m.

[0387] The information manifold effectively holds information corresponding to the input data symbols in a form that enables the input data symbols to be subsequently transformed according to a desired OTFS transform operation (e.g., symplectic 2D Fourier transform, discrete symplectic 2D Fourier transform, finite symplectic Fourier transform, etc.) In some embodiments, the data symbols may also be expanded before being held within the information manifold.

[0388] The OTFS processor 102p may then transform the information manifold according to a 2D symplectic Fourier transform. Such a transform may be implemented using any of the previously discussed symplectic 2D Fourier transforms, discrete symplectic 2D Fourier transforms, and finite symplectic Fourier transforms. This operation produces at least one 2D Fourier transformed information manifold, which may be stored in the transmitter memory 102m.

[0389] The OTFS transmitter 102c will typically transmit the at least one 2D Fourier transformed information manifold as a series of "M" simultaneous narrowband waveforms, each series at successive time intervals until the entire 2D Fourier transformed information manifold has been transmitted. For example, the transmitter processor 102p may operate on all frequencies and times (often on a column-by-column basis in time) of the 2D Fourier transformed information manifold. The transmitter processor 102p may select a given column at position n (where n may vary from 1 to N) and transmit a column of a certain width according to a time slice of duration proportional to Tμ, where μ=1 / N. Those frequencies in the column slice of the 2D Fourier transformed information manifold (e.g., the frequencies corresponding to the transmit time slice) may then be passed through a set of at least M different, non-overlapping narrowband frequency filters. This produces M mutually orthogonal waveforms. The processor 102p may then cause these resulting filtered waveforms to be transmitted as a plurality of at least M mutually orthogonal waveforms at different transmission time intervals (eg, one column at a time) until the entire 2D Fourier transformed information manifold has been transmitted.

[0390] In one embodiment, gaps or "guard bands" may be imposed in both time and frequency to minimize the possibility of unintentional crosstalk between the various narrowband filters and time intervals prior to transmission. Depending on the characteristics of the data channel, any such gaps or guard bands may be increased or decreased or set to zero as circumstances warrant.

[0391] Each OTFS receiver 104 may then receive a channel-convolved version of the 2D Fourier transformed information manifold transmitted by transmitter 102. Due to the distortion introduced by channel 100, the M narrowband waveforms originally transmitted at M original frequencies may now include more than M narrowband waveforms at different frequency ranges. Moreover, because the transmitted OTFS waveforms hit various reflectors 106, the originally transmitted signal and its reflections may be received at different times. Therefore, each receiver 104 will generally supersample or oversample each received waveform on a time-frequency grid having a finer mesh than the time-frequency grid associated with transmitter 102. This oversampling process is accomplished by Figure 17B-Figure 17D These figures depict a receiver time-frequency grid with smaller time and frequency increments than the transmitter OTFS grid.

[0392] Each OTFS receiver 104 operates to receive the transmitted 2D Fourier transformed information manifold at time slices that are generally less than or equal to the duration of the transmission time interval employed by the transmitter 102. In one embodiment, the receiver 104 uses a receiving bank of at least M different, non-overlapping narrowband frequency filters to analyze the received waveform. The receiver then typically stores a decomposed approximation (a channel-convolved version) of the originally transmitted 2D Fourier transformed information manifold in a receiver memory 104m.

[0393] Once the waveform transmitted by transmitter 102 has been received, receiver 104 corrects for the convolution effects of channel 100 in order to recover an estimate of the data symbols originally transmitted. Receiver 104 may implement these corrections in a variety of ways.

[0394] For example, the receiver 104 may transform the received waveform into an initial approximation of the originally transmitted information manifold using the inverse of the 2D symplectic Fourier transform used by the transmitter 102. Alternatively, the receiver 104 may first correct the channel-convolved approximation of the transmitted 2D Fourier transformed information manifold (stored in the receiver memory) using information related to the 2D channel state. After such correction, the receiver 104 may then use the inverse of the 2D symplectic Fourier transform employed at the transmitter 102 to generate the received information manifold and subsequently extract the estimated data symbols.

[0395] Although the OTFS method described herein inherently spreads any given data symbol across the entire time-frequency plane associated with the transmitter, in some embodiments, it is useful to implement an additional spreading operation to ensure that the transmitted data symbols are evenly distributed. Such spreading operations may be performed by the transmitter processor 102p before or after encoding the N×M 2D array of input data symbols onto a symplectic analysis compatible manifold. For this purpose, a plurality of spreading functions may be used, such as a 2D chirp operation. Where such spreading operations are implemented at the transmitter 102, the receiver 104 will utilize the inverse of the spreading operation to decode and extract the data symbols from each received information manifold.

[0396] FIG. 19 shows a 2D Fourier transformed information manifold represented by an N×M structure transmitted on M frequency bands during N time periods of duration Tμ. In this example, each of the M frequency bands is represented by a given row, and each different time period is represented by a given column. In the embodiment of FIG. 19 , it is assumed that the OTFS transmitter is configured to transmit the OTFS signal on an allocated bandwidth comprising the M frequency bands without a guard interval. The bandwidth (ω) of each of the M frequency bands 0 ) is 1 / Tμ. Thus, if it is desired to send all N columns of information within a minimum time interval N*Tμ, then M must have a bandwidth no greater than 1 / Tμ, and the bandwidth used by all M filtered OTFS bands cannot exceed M / T, where T is the total amount of time used to send all N columns of the 2D Fourier transformed information manifold.

[0397] At the receiver 104, different non-overlapping narrowband frequency filter banks, generally similar to the filter banks used by the transmitter 102, may be used to receive each 2D Fourier transformed information manifold. Again, the receiver time slices and receive filter banks will generally operate at a finer granularity; that is, the receiver will typically operate over a smaller frequency bandwidth and shorter time slices but over a typically wider overall frequency and time range. Therefore, the receiver bin structure will preferably oversample the transmit banks of different, non-overlapping narrowband frequency filters and corresponding transmit time slices previously used by the transmitter.

[0398] As can be appreciated with reference to Figure 19, the OTFS transmitter typically transmits the resulting filtered waveform (in this example, over all rows and successive columns) until the entire 2D Fourier transformed information manifold has been transmitted. However, the transmitter may transmit successive columns (time slices) continuously and continuously (i.e., without any time gaps in between) more as a series of continuous longer duration waveforms, or alternatively, the transmitter may place some time interval between each successive column, thereby producing a more distinct series of waveform bursts.

[0399] In other words, the transmitter may transmit the resulting filtered waveform as either: 1) a plurality of at least M simultaneously transmitted mutually orthogonal waveforms at different consecutive transmit time intervals; or 2) a plurality of OTFS data or OTFS pilot bursts, comprising at least M simultaneously transmitted mutually orthogonal waveform bursts at different transmit intervals separated by at least one isolation time interval.

[0400] Fig. 20 An example of M filtered OTFS bands transmitted simultaneously according to various smaller time slices Tμ is shown. The repeated curve shape is based on The center frequency for each filtered band is shown. One of the transmitted frequency bandwidth bins is shown in more detail, having a size of 1 / T and a duration of T*μ. Again, as discussed previously, in a preferred embodiment, the OTFS receiver will use oversampling, and therefore a finer granularity of bins, which can still extend over a wider time and frequency range in order to capture signals with a high degree of delay or Doppler shift.

[0401] In other words, in some embodiments, the non-overlapping narrowband frequency filters used at the transmitter may be configured to pass the AND filter function from each 2D Fourier transformed information manifold proportional to the frequency, where j is the square root of -1, t corresponds to a given time slice of duration Tμ selected from the 2D Fourier transformed information manifold, and k corresponds to a given row position in the given 2D Fourier transformed information manifold, where k varies between 1 and M. In this example, the bandwidth ω is in Hz 0 It can be proportional to 1 / T, and T=M / (allowed wireless bandwidth).

[0402] As can be seen from Figures 19 and Fig. 20 As recognized in , each 2D Fourier transformed information manifold can have a total dimension NT according to the time axis. μ, and may have a total dimension M / T along the frequency axis, and each "cell" or "bin" in each 2D Fourier transformed information manifold may have a total dimension proportional to Tμ along the time axis and proportional to 1 / T along the frequency axis.

[0403] Fig.21 Another example of an OTFS waveform transmitted according to each smaller time slice Tμ is provided. Fig.21 In the illustration, the modulation degree or amplitude of each waveform as a function of time is also shown.

[0404] In some embodiments, it may be useful to modulate the transmitted wireless OTFS waveform with an underlying modulation signal that allows the receiver to distinguish where on the original 2D time and frequency grid a given received signal originated. This may, for example, help an OTFS receiver distinguish between various types of received signals, as well as distinguish between direct signals and various time-delayed and / or frequency-shifted reflected signals. In these embodiments, the grid, bin, or lattice location of the originally transmitted OTFS waveform may be distinguished by determining time and frequency-related parameters of the received waveform. For example, in the "symplectic" implementation currently discussed, each "row" of the 2D Fourier transformed information manifold is passed through a plurality of bins according to a plurality of time-delayed and / or frequency-shifted parameters such as the received waveform. A narrowband filter that operates with parameters like “kω 0 The " term may enable the receiver to distinguish any given incoming OTFS waveform by its originating "column" position "t". In this case, the receiver should also be able to determine the bin (grid, lattice) position of each received waveform by determining both its t (time-related) and k (frequency-related) values. These values ​​can then be used during subsequent deconvolution of the received signal.

[0405] If further distinguishability of the bins (grid lattice) of the received OTFS signal's time and frequency origin is desired, additional time and / or frequency varying modulation schemes may also be applied to the OTFS signal prior to transmission to allow the OTFS receiver to further distinguish the bin (grid, lattice) origins of the individual received signals.

[0406] In alternative embodiments, the information manifold or the 2D Fourier transformed information manifold may be sampled and modulated using Dirac comb methods. The Dirac comb used by these methods may be, for example, a periodic tempered distribution constructed from a Dirac delta function.

[0407] Now the attention is directed to Fig. 22, which provides a block diagram representation of an exemplary process 2200 for OTFS transmission and reception according to the present disclosure. The process 2200 begins with the packaging of data for transmission and its optional precoding for correcting known channel impairments (stage 2210). The material is then processed by a 2D Fourier transform (such as a symplectic Fourier transform, a discrete symplectic Fourier transform, or a finite symplectic Fourier transform) (stage 2220). After this processing, the results are then passed through a filter bank (FB) and transmitted over a series of time intervals Tμ (stage 2230). The transmitted wireless OTFS waveforms are then passed through a communication or data channel (C), where they are subject to various distortions and signal impairments (stage 2240). At the receiver, the received waveform is received at each time interval according to the filter bank (stage 2250). The receiver filter bank (FB*) can be an oversampling filter bank (FB*) that operates according to an oversampling duration that may be a fraction of the original time interval Tμ. This oversampling allows for better analysis of the received signal with higher resolution for the time delays and frequency shifts caused by the channel. At stage 2260, the received material is analyzed by an inverse 2D Fourier transform (2D-FT) (again, which can be an inverse symplectic Fourier transform, an inverse discrete symplectic Fourier transform, or an inverse finite Fourier transform). The result can then be further corrected for channel distortion using, for example, 2D channel state information (stage 2270). In other embodiments, stage 2270 may precede stage 2260.

[0408] Further mathematical characterization of OTFS modulation and derivation of a two-dimensional (2D) channel model

[0409] In the following, we further develop the OTFS communication paradigm focusing on the central role played by the Heisenberg representation and the 2D symplectic Fourier transform. The main technical result of this development is the exact derivation of the 2D channel model for OTFS.

[0410] 0. Introduction

[0411] Orthogonal Time Frequency Space is a new type of modulation scheme that can be implemented by communication transceivers and converts a dynamic one-dimensional wireless medium into a static two-dimensional local ISI channel by placing the time and frequency dimensions equally. The main advantages of OTFS transceivers over conventional transceivers are as follows:

[0412] 1. Fading. Eliminate both time and frequency selective fading.

[0413] 2. Diversity. Extract all diversity branches in the channel.

[0414] 3. Stationarity. All symbols experience the same distortion.

[0415] 4.CSI. Perfect and efficient channel state information (CSI).

[0416] In a sense, the OTFS transceiver establishes a virtual line through the communication medium, allowing conventional wired DSP techniques to be applied in the wireless domain. Embodiments of the OTFS transceiver are based on principles from representation theory, inductively constructed from classical Fourier theory. At an operational level, OTFS can be roughly characterized as applying a two-dimensional Fourier transform to a filtered OFDM symbol block. OTFS is a true two-dimensional time-frequency modulation and can combine both two-dimensional time-frequency filtering and two-dimensional equalization techniques. Below we provide the formal mathematical expansion of the OTFS transceiver, focusing on the rigorous derivation of the two-dimensional channel model.

[0417] OTFS and Lattice

[0418] We first choose an undersampled time-frequency lattice, i.e., a two-dimensional lattice with density less than or equal to 1. The undersampling condition is essential for perfect reconstruction, however, it appears to limit the delay-Doppler resolution of the channel acquisition. In contrast, radar theory is equivalent to choosing an oversampled time-frequency lattice with density greater than or equal to 1, where the oversampling condition is essential to maximize the delay-Doppler resolution of the target measurement. It turns out that the symplectic (two-dimensional) Fourier transform is intertwined between the communication and radar lattices. The OTFS communication paradigm is to multiplex the information symbols on the oversampled high-resolution radar lattice and convert them back to communication coordinates using the symplectic Fourier transform and two-dimensional filtering. This allows OTFS to get the best of both worlds - high-resolution delay-Doppler channel state measurements without sacrificing spectral efficiency. In particular, the OTFS channel model can be thought of as a high-resolution delay-Doppler radar image of the wireless medium.

[0419] Wireless Channel

[0420] To understand OTFS, it is helpful to think of the wireless channel as a mathematical object. Let H = L 2(R) represents the vector space of "physical" waveforms defined in the time domain. The physics of the wireless medium is governed by the phenomenon of multipath reflections, i.e., the transmitted signal propagates in the atmosphere and reflects from various objects around it. Some of these objects (possibly including the transmitter and the receiver) are moving at non-zero speeds. Therefore, (under some mild "narrowband" assumptions), the received signal is a superposition of the time delay and Doppler shift of the transmitted signal, where the time delay is caused by the excess distance traversed by the reflected wave, and the Doppler shift is caused by the relative speed between the reflecting object and the transmitting and / or receiving antennas. Mathematically, this is equivalent to the fact that the wireless channel can be represented as a linear transformation C:H→H, which is implemented for each transmitted waveform as a weighted superposition of many time delays and Doppler shifts, i.e.:

[0421]

[0422] From equation (0.1), it can be seen that the channel C is determined by a function h that depends on two variables τ and v, called delay and Doppler. The (τ, v) pair can be viewed as a plane V = R 2 , (called the Delay-Doppler plane). Therefore, h is a two-dimensional (delay-Doppler) impulse response that characterizes the wireless channel. However, it should be remembered that this terminology is misleading because the action of h given by (0.1) is not a convolution action.

[0423] Decline

[0424] A fundamental physical phenomenon characteristic of wireless channels is fading. The fading phenomenon corresponds to a localized attenuation in the energy distribution of the received signal measured in a specific dimension. Two types of fading are usually considered: time selective fading and frequency selective fading. The first is caused by the destructive superposition of Doppler shifts, and the second is caused by the destructive superposition of time delays. Since a wireless channel consists of a combination of both time delays and Doppler shifts, it exhibits both types of fading. Mitigating fading phenomena is an important motivation behind the development of OTFS transceivers.

[0425] Heisenberg said

[0426] A key observation is that the delay-Doppler channel representation given in Eq. (0.1) is an application of a fundamental mathematical transformation, called the Heisenberg representation, that transforms between functions on the delay-Doppler plane V and linear operators on the signal space H. To see this, for each Let's use L τ and M v to represent the operation of time delay by τ and Doppler frequency shift by v, that is:

[0427]

[0428]

[0429] Using this terminology, we can rewrite the channel equation (0.1) as follows:

[0430]

[0431] Let us define the Heisenberg representation as a transformation of a function a:V→C into a linear operator Π(a):H→H, given by:

[0432]

[0433] We call the function a the delay-Doppler impulse response of the operator Π(a). From this perspective, we see that the wireless channel is the application of the Heisenberg representation to a specific function h on the delay-Doppler plane. This higher level of abstraction establishes the mapping Π as a fundamental object underlying wireless communications. In fact, the correspondence The classical correspondence between a stationary linear system and a one-dimensional impulse response is generalized to the case of arbitrary time-varying systems (also called linear operators). In this regard, the main property of the Heisenberg representation is its translation between a composition of linear operators and a warped convolution operation between the corresponding impulse responses. In more detail, if:

[0434] A=Π(a),

[0435] B=Π(b),

[0436] Then we have:

[0437]

[0438] in* t is a non-commutative distortion of the 2D convolution. Equation (0.4) is crucial for the derivation of the 2D channel model, a characteristic feature of OTFS transceivers.

[0439] OTFS transceiver and 2D channel model

[0440] The OTFS transceiver provides a mathematical transformation that has the effect of converting a fading wireless channel into a stationary two-dimensional convolutional channel. We call this property the two-dimensional channel model.

[0441] Formally, an OTFS transceiver can be characterized as a pair of linear transformations (M, D), where M is called the modulation mapping and D is called the demodulation mapping and is the inverse of M. According to the OTFS paradigm, information bits are encoded as complex-valued functions on V with respect to a lattice called the reciprocal communication lattice is periodic. Note that the term “reciprocal” is used to imply that Λ ⊥ and a more general lattice Λ (called the primitive communication lattice). If we use Represents Λ ⊥ The vector space of (periodic function on V), then OTFS modulation is a linear transformation:

[0442]

[0443] Geometrically, one can think of information as being transmitted relative to the lattice Λ ⊥ Function on a two-dimensional periodic domain (ring) obtained by folding V. Respectively, the demodulation map is a linear transformation acting in the opposite direction, namely:

[0444]

[0445] The precise mathematical meaning of the two-dimensional channel model is that, given the information function We have:

[0446]

[0447] Where * represents the periodic convolution on the torus, and the function c is the reciprocal lattice Λ of the delayed Doppler impulse response h of the wireless channel ⊥ The periodization of is:

[0448]

[0449] Equations (0.7) and (0.8) encode the exact way in which the OTFS transceiver interacts with the wireless channel.

[0450] The remainder of this explanation of the OTFS method and OTFS transceiver is organized as follows:

[0451] Section 1 discusses several basic mathematical structures associated with the delay-Doppler plane V. We begin by introducing the symplectic form on V, which is an antisymmetric variant of the more familiar Euclidean form used in classical signal processing. We then discuss lattices as two-dimensional discrete subdomains of V. We focus our attention on the construction of reciprocal lattices. Reciprocal lattices play a key role in the definition of the OTFS transceiver. We then proceed to the dual object of the lattice, called the torus, which is a two-dimensional periodic domain obtained by folding the plane with respect to the lattice.

[0452] Section 2 discusses the symplectic Fourier transform, which is a variant of the two-dimensional Fourier transform defined in terms of the symplectic form on V. We discuss three variants of the symplectic Fourier transform: the continuous symplectic Fourier transform, the discrete symplectic Fourier transform, and the finite symplectic Fourier transform. We explain the relationship between these variants.

[0453] Section 3 discusses the Heisenberg representation and its inverse, the Wigner transform. In short, the Heisenberg representation is a structure that encodes the exact algebraic relationship between the operations of time delay and Doppler shift. We relate the Wigner transform to the more familiar notation of ambiguity functions and cross-ambiguity functions. We conclude with a formulation of the basic channel equation.

[0454] Section 4 discusses successive variants of the OTFS transceiver. We begin by specifying the parameters that define the OTFS transceiver. We then proceed to define the modulation and demodulation mappings. We conclude this section by deriving a two-dimensional channel model from first principles.

[0455] Section 5 discusses finite variations of OTFS transceivers. In short, finite variations are obtained from continuous variations by sampling the reciprocal torus along finite uniformly distributed sub-torus surfaces. We define finite OTFS modulation and demodulation mappings. We then formulate a finite version of the two-dimensional channel model, explaining that the finite two-dimensional impulse response is a restriction of the continuous two-dimensional impulse response to a finite sub-torus surface. We end this section with an explicit interpretation of the modulation formula in terms of classical DSP operations.

[0456] 1. Delay-Doppler plane

[0457] 1.1 Symplectic plane

[0458] The delay-Doppler plane is a two-dimensional vector space on the real numbers. Specifically, we take V = R 2, where the first coordinate is the delay, denoted by τ, and the second coordinate is the Doppler, denoted by v. The delay-Doppler plane is equipped with an intrinsic geometric structure encoded by the symplectic form (also called symplectic inner product or symplectic pairing). The symplectic form is a pairing defined by the deterministic formula ω: V × V → R:

[0459]

[0460] where v = (τ, v) and v′ = (τ′, v′). Note that, in contrast to its Euclidean counterpart, the symplectic form is antisymmetric, i.e., for every v, v′∈V, ω(v, v′) = -ω(v′, v). Therefore, the symplectic product of a vector with itself is always equal to zero, i.e., for every v∈V, ω(v, v) = 0. It turns out that the architecture of time and frequency is governed by the symplectic structure.

[0461] 1.1.1 Functions on a plane

[0462] We use C(V) to denote the vector space of complex-valued functions on V. We use * to denote the linear convolution operation of functions on V. Given a pair of functions f, g∈C(V), for each v∈V, their convolution is defined as:

[0463]

[0464] 1.2 Dot Matrix

[0465] lattice Yes and Z 2 Isomorphic commutative subgroups are defined as follows:

[0466]

[0467] where v 1 , v 2 ∈V are linearly independent vectors. In other words, Λ includes the vector v 1 and v 2 All integer linear combinations of . Fig.23 . Vector v 1 and v 2 is called the generator of the lattice. By definition, the volume of Λ is the volume of the elementary domain. It can be shown that:

[0468] vol(Λ)=|ω(v 1 , v 2 )|. (1.3)

[0469] When vol(Λ)≥1, the lattice is said to be undersampled, and when vol(Λ)≤1, the lattice is said to be oversampled. Finally, if vol(Λ)=1, then the lattice is said to be critically sampled.

[0470] Example 1.1 (Standard communication dot matrix). Fixed parameters T ≥ 0 and μ ≥ 1. Let:

[0471]

[0472] We have vol(Λ T，μ )=μ. We will T，μ It is called the standard communication dot matrix.

[0473] 1.2.1 Reciprocal lattice. Given a lattice Its orthogonal complementary lattice is defined as:

[0474] Λ ⊥ ={v∈V:ω(v,λ)∈Z for each λ∈Λ} (1.5)

[0475] In other words, Λ ⊥ consists of all vectors in V such that their symplectic pairing with every vector in Λ is an integer. It can be shown that Λ ⊥ Actually, it is a lattice. We will ⊥ is called the reciprocal lattice of Λ. It can be shown that:

[0476] vol(Λ ⊥ )=1 / vol(Λ), (1.6)

[0477] This implies that if and only if Λ ⊥ When oversampled, Λ is undersampled. This means that the reciprocity between the coarse (undersampled) lattice and the fine (oversampled) lattice is interchangeable. Another property involves how the lattice inclusion relation behaves under reciprocity. Given a lattice and sub-lattice The composed pairs show that the inclusion relationship between the reciprocals is reversed, that is:

[0478]

[0479] Example 1.2 Consider the standard communication lattice Λ T，μ . Its inverse is given by:

[0480]

[0481] See also Fig.24A and 24B , which respectively show the standard communication lattice and the reciprocal of the standard communication lattice. In fact, we have:

[0482]

[0483] It should be noted that vol(Λ T，μ ) ⊥ =1 / μ, which means that when the original lattice becomes more sparse, the reciprocal lattice becomes more dense.

[0484] 1.2.2 Functions on the lattice. We use C(Λ) to represent the vector space of complex-valued functions on the lattice. We use R Λ :C(V)→C(Λ) represents a typical constraint mapping. For each f∈C(V) and λ∈Λ, R Λ :C(V)→C(Λ) is given by:

[0485] R Λ (f)(λ)=f(λ),

[0486] We use * to denote the convolution operation between functions on Λ. Given f, g∈C(Λ), for each λ∈Λ, their convolution is defined as:

[0487]

[0488] 1.3 Torus The torus Z is a two-dimensional periodic field that constitutes the geometric dual of the lattice Λ. Formally, Z is a continuous group obtained as the quotient of the vector space V divided by the lattice Λ, that is:

[0489] Z=V / Λ. (1.10)

[0490] Specifically, by definition, a point z∈Z is a coset of Λ in V, i.e., for some v∈V:

[0491] z=v+Λ, (1.11)

[0492] An alternative (although less canonical) way to construct Z is to glue opposite faces of the fundamental domain of Λ. Geometrically, Z has the shape of a "torus" obtained by folding the plane V relative to the lattice Λ. We call Z the torus associated with Λ, or sometimes the dual of Λ. Note that the torus is the two-dimensional counterpart of the circle, where the latter is obtained by folding the plane V relative to the one-dimensional lattice The folding line R is obtained.

[0493] Example 1.3 (Standard communication torus). Fig.25 As shown in the figure, the standard communication matrix Λ T，μ The associated torus is given by:

[0494]

[0495] Geometrically, Z T，μis the Cartesian product of two circles; one with diameter Tμ and the other with diameter 1 / T. We call Z T，μ It is a standard communication torus.

[0496] 1.3.1 Functions on the torus. We denote by C(Z) the vector space of complex-valued functions on the torus Z = V / Λ. Functions on Z are naturally equivalent to functions f:V→C, which are periodic with respect to translations according to the elements of the lattice Λ, i.e., for every v∈V and λ∈Λ:

[0497] f(v+λ)=f(v), (1.13)

[0498] Therefore, the vector space of functions on Z coincides with the subspace of Λ-periodic functions on V, that is, C(Z) = C(V) Λ . Hence, we have the natural periodic mapping R Λ :C(V)→C(Z), for every f∈C(V) and v∈V, R Λ :C(V)→C(Z) is given by:

[0499]

[0500] We use * to denote the circular convolution operation of a function on Z. Given a pair of functions f, g∈C(Z), for every v∈V, their convolution is defined as:

[0501]

[0502] Note that the integration over the torus Z is equivalent to the integration over the fundamental domain of the lattice Λ.

[0503] 1.4 Finite torus Finite Torus Z 0 is the same as the dot matrix and sub-lattice The domains associated with the pair. Formally, Z 0 is the lattice Λ divided by the sublattice Λ 0 The finite set defined by the quotient of is:

[0504] Z 0 =Λ / Λ 0 . (1.16)

[0505] Specifically, point z∈Z 0 It is the Λ in Λ 0 Coset, that is, for some λ∈Λ:

[0506] z=λ+Λ 0 , (1.17)

[0507] Geometrically, Z0 For a continuous torus Z = V / Λ 0 Finite uniform sampling of , because we have a natural inclusion relation:

[0508] Λ / Λ 0 °V / Λ 0 . (1.18)

[0509] Example 1.4 (Standard communication finite torus). Consider the standard communication lattice Λ T，μ . Fixed positive integer n, m∈N ≥1 . Let (Λ T，μ ) n，m is a sublattice defined by:

[0510]

[0511] and The associated finite torus is given by (see Fig.26 ):

[0512]

[0513] The conclusion is as follows: a finite torus is isomorphic to the Cartesian product of two cyclic groups; one of which is of order n and the other of order m. We call It is a standard communication finite torus.

[0514] 1.4.1 Functions on a finite torus. We use C(Z 0 ) represents a finite torus Z 0 =Λ / Λ 0 The vector space of complex-valued functions on Z. 0 The function on is naturally equivalent to the function f:Λ→C, which is equivalent to the function f:Λ→C according to the sublattice Λ 0 The translation is periodic, that is, for every λ∈Λ and λ 0 ∈Λ 0 :

[0515] f(λ+λ 0 )=f(λ), (1.21)

[0516] Therefore, the vector space C(Z 0 ) and Λ on Λ 0 The subspace of periodic functions is consistent, that is, Therefore, for every f∈C(Λ) and λ∈Λ, we have the natural periodic mapping given by

[0517]

[0518] We use * to represent Z 0 Finite cyclic convolution operation on functions. Given a pair of functions f, g∈C(Z 0 ), for each v∈V, their convolution is defined as:

[0519]

[0520] It should be noted that in the finite torus Z 0 The upper sum is equivalent to the sublattice Λ in the superlattice Λ 0 The sum is calculated over the basic domain of .

[0521] 1.4.2 Reciprocity between finite tori. Given a finite tori Z 0 =Λ / Λ 0 , we use Z ⊥ Reciprocal pair The associated finite torus is:

[0522]

[0523] We call is a reciprocal finite torus. Although they are different sets, it can be shown that, in fact, Z 0 and As finite groups are isomorphic.

[0524] Example 1.5 Consider the standard communication lattice Λ T，μ and sub-lattice As shown above, The associated finite torus is isomorphic to:

[0525] Z 0 ; Z / Zn×Z / Zm.

[0526] The reciprocal lattice is given by:

[0527]

[0528]

[0529] Therefore, the reciprocal finite torus is given by:

[0530]

[0531] We see Z 0 and As finite groups are isomorphic, since two groups are isomorphic (although in a different order) to the Cartesian product of two cyclic groups, one of which is of order n and the other of which is of order m.

[0532] 2. Symplectic Fourier Transform

[0533] In this section, we will introduce a variant of the two-dimensional Fourier transform associated with the symplectic form, called the symplectic Fourier transform. For each z∈R, let ψ:R→C x Represents the standard complex exponential function:

[0534] ψ(z)=e 2πiz , (2.1)

[0535] 2.1 Characteristics of the Symplectic Fourier Transform

[0536] The symplectic Fourier transform is a variant of the two-dimensional Fourier transform associated with the symplectic form ω. Formally, for every f∈C(V) and u=(t,f), the symplectic Fourier transform is a linear transformation SF:C(V)→C(V) defined by the following rule:

[0537]

[0538] We call the coordinates of the transformed domain (t, f) time and frequency respectively.

[0539] In general, the inverse transform of (2.2) is given by:

[0540]

[0541] However, since ω is antisymmetric, we have SF -1 =SF. That is, the symplectic Fourier transform is equal to its inverse.

[0542] 2.1.1 Interchange Property. The symplectic Fourier transform interchanges between function multiplication and function convolution, as formulated in the following proposition.

[0543] Proposition 2.1 (Interchangeability). For every f, g∈C(V), the following conditions hold:

[0544] SF(f·g)=SF(f)*SF(g),

[0545] SF(f*g)=SF(f)·SF(g), (2.4)

[0546] In fact, the commutative properties are inferred from more fundamental properties involving two-dimensional translation and symplectic modulation operations.

[0547] Translation: given a vector v 0 ∈V, defined by v 0 The translation is a linear translation For each f∈C(V), it is given by:

[0548]

[0549] Modulation: Given a vector v 0 ∈V, defined by v 0 Symplectic modulation is linear modulation For each f∈C(V), it is given by:

[0550]

[0551] Perhaps the most fundamental property of the symplectic Fourier transform is that it interchanges between translation and symplectic modulation. This property is formulated in the following proposition.

[0552] Proposition 2.2 (Translation and symplectic modulation interchangeably). The following conditions hold:

[0553] For each v 0 ∈V:

[0554]

[0555]

[0556] 2.2 Discrete Symplectic Fourier Transform

[0557] The discrete symplectic Fourier transform is related to a function of two discrete variables and a function of two continuous periodic variables. The formal definition assumes that The choice of dot matrix. is a reciprocal lattice and let Z ⊥ Represents and Λ ⊥ The associated torus is:

[0558] Z ⊥ =V / Λ ⊥ .

[0559] We call Z ⊥ is the reciprocal torus. The discrete symplectic Fourier transform is a linear transformation SF given by for every f∈C(Λ) and u∈V Λ :C(Λ)→C(Z ⊥ ):

[0560]

[0561] where c is the normalization coefficient, taken as c = vol(Λ). Note that for a fixed value of λ∈Λ, the function ψ(-ω(u,λ))f(λ) is periodic with respect to the reciprocal lattice and is therefore a function on the reciprocal torus. Inverse transformation For each f∈C(Λ) it is given by:

[0562]

[0563] It should be noted that for the torus Z ⊥ The integration is equivalent to the integration of the lattice Λ ⊥ The basic domain of the integration.

[0564] 2.2.1 Discrete Interchange Property. As formulated in the following proposition, the discrete symplectic Fourier transform interchanges between function multiplication and function convolution.

[0565] Proposition 2.3 (Discrete interchangeability property). The following condition holds for every f, g∈C(Λ):

[0566] SF Λ (f·g)=SF Λ (f)*SF Λ (g), (2.9)

[0567]

[0568] Where * represents periodic convolution.

[0569] 2.2.2 Compatibility with continuous transforms. The continuous symplectic Fourier transform and the discrete symplectic Fourier transform are compatible. This compatibility relationship is expressed in the following theorem.

[0570] Theorem 2.4 (Discrete-continuous compatibility relation). We have:

[0571]

[0572]

[0573] In other words, equation (2.11) provides: take the continuous Fourier transform of the function f and then transform it relative to the reciprocal lattice Λ ⊥ Periodizing the translation of is the same as first restricting f to the lattice Λ and then taking the discrete Fourier transform.

[0574] 2.3 Finite Symplectic Fourier Transform

[0575] The finite symplectic Fourier transform involves functions of two finite periodic variables. The formal definition assumes that the lattice and sub-lattice We use Z 0 represents the finite torus associated with this pair, namely:

[0576] Z 0 =Λ / Λ 0 .

[0577] Let Λ ⊥ and is the corresponding reciprocal lattice. We use Z ⊥ represents the finite torus associated with this reciprocal lattice pair, namely:

[0578]

[0579] For every f∈C(Z 0 )and The finite symplectic Fourier transform is a linear transformation defined by the following rules

[0580]

[0581] Where c is the normalization coefficient, taken as c = vol(Λ). and λ∈Λ, the inverse transform Given by:

[0582]

[0583] where c 0 is the normalization coefficient, taken as c 0 =vol(Λ 0 ).

[0584] 2.3.1 Finite Interchange Property. As formulated in the following proposition, the finite symplectic Fourier transform interchanges between function multiplication and function circular convolution.

[0585] Proposition 2.5 (Discrete interchange property). The following condition is true for every f, g∈C(Z 0 ) Established:

[0586]

[0587]

[0588] Where * represents finite circular convolution.

[0589] It should be noted that the normalization coefficient c / c in equation (2.15) 0 Equal to the finite torus Z 0 The number of points in .

[0590] 2.3.2 Compatibility with discrete transforms. The discrete symplectic Fourier transform and the finite symplectic Fourier transform are compatible. The compatibility relation is formulated in the following theorem.

[0591] Theorem 2.6. We have:

[0592]

[0593]

[0594] In plain language, equation (2.17) states that taking the discrete symplectic Fourier transform of a function f on the lattice Λ and then restricting it to the reciprocal lattice and first relative to the sublattice Λ 0 It is the same to periodize f by a translation of and then take the finite Fourier transform.

[0595] Example 2.7. Consider the standard communication lattice Λ T，μ and sublattice (Λ T，μ ) n，m . We have the following isomorphism:

[0596] Z 0 ; Z / nZ×Z / mZ,

[0597]

[0598] With respect to these implementations, the finite symplectic Fourier transform and its inverse have the following specific form:

[0599]

[0600]

[0601] where k∈[0,m-1],l∈[0,n-1] in the first equation, and K∈[0,n-1],L∈[0,m-1] in the second equation. Note that the Fourier indices have negative signs due to symplectic pairing.

[0602] 3Heisenberg Theory

[0603] Let H be the Hilbert space of square-integrable complex functions on the real line R. We denote the parameter of the line by t, and call it time. The inner product on H is given by the standard formula:

[0604]

[0605] We call H the signal space, and functions in the signal space are called waveforms. Heisenberg theory deals with the mathematical structure behind the intricate interplay between the time and frequency dimensions. In short, the theory studies the algebraic relationship between two fundamental operations on functions: time delay and Doppler shift.

[0606] 3.1 Time Delay and Doppler Shift

[0607] The operation of time delay and Doppler shift establishes two one-parameter systems of unitary transformations on H.

[0608] 3.1.1 Time delay. Given a real parameter τ∈R, for each f∈H and t∈R, the operation of time delay τ is given by the linear transformation L given by τ :H→H

[0609] L τ (f)(t)=f(t-τ), (3.2)

[0610] It can be shown that L τ is a unitary transformation, i.e., it preserves the inner product for every f, g∈H:

[0611] <L τ f,L τ g>=<f,g> ,

[0612] Moreover, the transformation system {L τ :τ∈R} for every τ 1 , τ 2 ∈R satisfies:

[0613]

[0614] In particular, the time delay operations are interchangeable, i.e.,

[0615] 3.1.2 Doppler shift. Given a real parameter v∈R, for every f∈H and t∈R, the Doppler shift v is calculated by the linear transformation M given by v :H→H:

[0616] M v (f)(t)=ψ(vt)f(t), (3.3)

[0617] Recall that ψ represents the standard complex exponential function ψ(z) = e 2πiz It can be shown that M v is a unitary transformation, i.e., it preserves the inner product for every f, g∈H:

[0618] <M v f,M v g>=<f,g> ,

[0619] Moreover, the transformation system {M v :v∈R} for every v 1 , v 2 ∈R satisfies:

[0620]

[0621] In particular, the time delay operations are interchangeable, i.e.,

[0622] 3.2 Heisenberg representation

[0623] The Heisenberg representation is a mathematical structure that unifies the two operations of time delay and Doppler shift. The main difficulty is that these operations are not interchangeable with each other, but rather they satisfy the following condition:

[0624] L τ M v =ψ(-τv)M v L τ . (3.4)

[0625] The starting point is to consider a uniform delay-Doppler linear transformation for every pair of real parameters τ, v∈R:

[0626] π(τ, v) = L τ M v , (3.5)

[0627] In this representation, consider ordered pairs (τ, v) as points in the delay-Doppler plane V. It can be shown that, due to this composition, π(τ, v) is a unitary transformation. For each f∈C(V), the two-dimensional transformation system {π(v):v∈V} defines a linear transformation Π:C(V)→Hom(H,H) given by:

[0628] П(f)=∫ v∈V f(v)π(v)dv, (3.6)

[0629] where the range of Π is the vector space of linear transformations from H to itself, which we denote by Hom(H,H). In other words, the mapping Π takes a function on the delay-Doppler plane and sends it to a linear transformation given by a weighted superposition of delay-Doppler transforms, where the weights are specified by the values ​​of this function. The mapping Π is called the Heisenberg representation. A fundamental fact that we cannot prove is that the mapping Π is essentially an isomorphism of vector spaces. It therefore admits the inverse Π known as the Wigner transform -1 :Hom(H,H)→C(V). For each A∈Hom(H,H) and v∈V, the Wigner transform is given by:

[0630] Π -1 (A)(v)=Tr(π(v) H A). (3.7)

[0631] The Heisenberg representation and the Wigner transform should be thought of as "coordinate transformations" (which can be represented by matrices) that convert between functions on the delay-Doppler plane and linear transformations on the signal space. In summary, the linear transformation A∈Hom(H,H) admits a unique expansion as a superposition of delay-Doppler transforms. The coefficients in this expansion are given by the function a=Π -1 (A) is given. Function a is called the delay-Doppler impulse response of transform A. The Heisenberg formalism generalizes the classical framework of time-invariant linear systems to time-varying linear systems. It should be noted that in the former, the time-invariant linear transform admits a unique expansion as a superposition of time delays, and the coefficients in this expansion constitute the classical impulse response.

[0632] 3.2.1 Blur function. The formula for the Wigner transform of a general linear transformation (Eq. (3.7)) is quite abstract. Fortunately, for specific types of linear transformations, the Wigner transform takes a more explicit form. For example, we are given a waveform g∈H with unit norm ||g||=1. Let P g represents the orthogonal projection on the one-dimensional subspace with a span of g. For each P g Given by:

[0633]

[0634] Proposition For every v∈V, P g The Wigner transform admits the following formula:

[0635] Π -1 (P g )(v)=<π(v)g,g>, (3.9)

[0636] Indicates A g =Π -1 (P g ) and call this function the fuzziness function of g. We have:

[0637] Π(A g )=P g. (3.10)

[0638] The above equation means that A g is the operator P g The coefficients in the delay-Doppler expansion of θ are given in the Heisenberg interpretation of the ambiguity function.

[0639] 3.2.2 Cross ambiguity function. The cross ambiguity function is to generalize the ambiguity function to two waveforms g1 , g 2 ∈H, where g 1 is assumed to be unit norm. For each make represents the following rank one linear transformation on H:

[0640]

[0641] Proposition For every v∈V, The Wigner transform admits the following formula:

[0642]

[0643] express And call this function g 1 and g 2 The cross ambiguity function of . We have:

[0644]

[0645] Therefore, according to the Heisenberg interpretation, the cross-ambiguity function is the operator The coefficient in the delay-Doppler spread of .

[0646] 3.3 Heisenberg Interchange Property

[0647] The main property of the Heisenberg representation is that it interchanges between the composition of linear transformations on H and a twisted version of the convolution operation on a function on V. To define the twisted convolution operation, we consider the form β: V × V → V given by:

[0648] β(v, v′)=vτ′, (3.14)

[0649] where v = (τ, v) and v′ = (τ′, v′). The form β satisfies the “polarization” condition for every v, v′∈V:

[0650] β(v, v′)-β(v′, v)=ω(v, v′), (3.15)

[0651] Given a pair of functions f, g∈C(V), their warped convolution is defined by the following rule:

[0652]

[0653] It can be seen that the twisted convolution operation is different from the usual convolution operation (Equation (1.2)) because there is a multiplicative factor ψ(β(v 1 , v 2)). Due to this factor, twisted convolution is a non-commutative operation compared to regular convolution. This non-commutativity is inherent in the time and frequency architecture. The Heisenberg commutativity property is formulated in the theorem below.

[0654] Theorem 3.1 (Heisenberg interchangeability property). For every f, g∈C(V), we have:

[0655]

[0656] The following example is crucial to understand the motivation behind the constructions presented in this section. In short, it explains why the distortion in equation (3.16) leads to an interchangeable relationship between the time delay and Doppler shift operations, see equation (3.4).

[0657] Example 3.2 We verify equation (3.17) in a specific case. Let v = (τ, v) and v′ = (τ′, v′). Consider the delta function δ v and δ v′ On the one hand, we have:

[0658] Π(δ v )=L τ M v ,

[0659] Π(δ v′ )=L τ′ M v′ ,

[0660] And therefore:

[0661]

[0662] on the other hand:

[0663] δ v * t δ v′ =ψ(β(v,v′))δ v *δ v′

[0664] =ψ(vτ′)δ v+v′ . (3.19)

[0665] therefore:

[0666] Π(δ v * t δ v′ )=ψ(vτ′)π(v+v′). (3.20)

[0667] We therefore verified that:

[0668] 3.4 Basic Channel Equation

[0669] We end this section by formulating the basic equations associated with the following structure:

[0670] 1. Cross ambiguity function.

[0671] 2. Fuzziness function.

[0672] 3. Channel change.

[0673] 4. Distorted convolution.

[0674] This basic equation is crucial for the two-dimensional channel model discussed in the next section. Let g∈H be a unit-normalized waveform. Let h∈C(V). We use H to denote the channel transform:

[0675] H=Π(h). (3.21)

[0676] Theorem 3.3 (Basic channel equation). The following equation holds:

[0677] A g，H(g) =h* t A g . (3.22)

[0678] In other words, the basic equation (3.22) states that the cross ambiguity function of g and H(g) is the distorted convolution of the ambiguity functions of h and g.

[0679] 4. Continuous OTFS transceiver

[0680] In this section, we describe successive variants of the OTFS transceiver.

[0681] 4.1 Settings

[0682] The definition of a continuous OTFS transceiver assumes the following data:

[0683] 1. Communication dot matrix. Undersampled dot matrix:

[0684]

[0685] where for some μ ≥ 1, vol(Λ) = μ,

[0686] 2. Generator waveform. Unit standard waveform:

[0687] g∈H,

[0688] For every non-zero element λ∈Λ × , satisfying the orthogonality condition Ag (λ)=0.

[0689] 3.2D filter. Window function:

[0690] W∈C(Λ).

[0691] 3.2D filter. Window function:

[0692] W∈C(Λ).

[0693] We note that, in general, the supports of a 2D filter along the delay and Doppler dimensions are bounded by the latency and bandwidth limitations of the communication packets, respectively.

[0694] Example 4.1 A typical example of a communication matrix is ​​the standard communication matrix

[0695]

[0696] A typical example of a 2D filter is:

[0697]

[0698] Where m, n∈N ≥1 and T∈R. ​​The parameter T is called the symbol time. The real numbers nμT and m / T are the delay and bandwidth of the communication packet, respectively. Note that more complex spectral window designs will involve a certain degree of tapering around the boundaries at the expense of spectral efficiency. Finally, in the case of μ=1 (critical sampling), a simple example of an orthogonal waveform is:

[0699] g=1 [0，T] .

[0700] 4.1.1 Generalized setting. We can assume that for every λ∈Λ × The transmission waveform g satisfies the following cross-orthogonality condition t ∈H and the received waveform g r ∈H (rather than a single orthogonal waveform g) to slightly generalize the setting:

[0701]

[0702] In using a pair of g r ≠g t The tradeoff when g is lower is that more freedom is gained in the design of the shape of each waveform at the expense of lower effective SNR at the receiver. For simplicity, we will only consider the case when g r =g t and understand that all results can be easily extended to more general cases.

[0703] 4.2 Continuous OTFS modulation mapping. Let Z ⊥ Representation and lattice Λ ⊥ The associated torus, where the lattice Λ ⊥ is the reciprocal of the communication lattice. For every x∈C(Z ⊥ ), the continuous OTFS modulation mapping is a linear transformation given by:

[0704]

[0705] Roughly speaking, continuous OTFS modulation is a synthesis of the Heisenberg representation and the (inverse) discrete symplectic Fourier transform. In this respect, it combines the two inherent structures of the delay-Doppler plane. Equation (4.2) can be written more explicitly as:

[0706]

[0707] in

[0708] Fig. 27 An exemplary structure of OTFS modulation mapping is shown. It should be noted that Fig. 27 Including by using a specially designed function α∈C(Z ⊥ ) gives an additional expansion transformation. The effect of this convolution is to extend the torus Z ⊥ The energy of each information symbol is evenly distributed, thereby achieving a balanced power profile of the transmitted waveform that depends only on the total energy of the information vector x.

[0709] 4.3 Continuous OTFS Demodulation Mapping

[0710] For each The continuous OTFS demodulation mapping is a linear transformation D given by: H→C(Z ⊥ ):

[0711]

[0712] In brief, the continuous OTFS demodulation map is a combination of the discrete symplectic Fourier transform and the Wigner transform. and u∈Z ⊥ , formula (4.4) can be written more explicitly as:

[0713]

[0714] 4.4 Two-dimensional channel model

[0715] Before describing the technical details of the two-dimensional channel model for an OTFS transceiver, we will provide an overview in brief terms. First consider that in the standard one-dimensional physical coordinates of time (or frequency), the wireless channel is a combination of multipath moving reflectors that cause distortion on the transmitted signal. This distortion is due to the superposition of time delay and Doppler shift. This general distortion appears in standard physical coordinates as a fading, non-stationary inter-symbol interference pattern. In contrast, when converted to the coordinates of the OTFS modulation ring plane, the distortion becomes a static two-dimensional local ISI distortion. This is a novel and characteristic property of the OTFS transceiver. In the following we provide a rigorous derivation of this characteristic. To do this, we start by considering the simplest multipath channel H:H→H that is already a combination of time delay and Doppler shift. In our terminology, for some v 0 =(τ 0 , v 0 )∈V, the channel is given by:

[0716]

[0717] Furthermore, we assume that the vector v 0 Satisfy||v 0 || = diam(Λ), where the diameter of the lattice is by definition the radius of its Voronoi region. In other words, we assume that the vectors are smaller than the size of the lattice. Note that this assumption holds for most relevant scenarios in wireless applications. We proceed to derive the structure of the modulation equivalent channel. Let q:V→C be a quadratic exponential function given by for each v∈V:

[0718] q(v)=ψ(-β(v,v)), (4.7)

[0719] Proposition 4.2 Modulation Equivalent Channel It is the periodic convolution y=h cqv *x, where the impulse response h eqv ∈C(Z ⊥ ) is given by:

[0720]

[0721] That is, equation (4.8) states that the modulation equivalent channel is The periodic convolution of the periodic blurred version of , where the blurring pulse is composed of discrete pulses |W| 2The symplectic Fourier transform of is given by . This blurring results in a loss of resolution due to the spectral truncation imposed by the filter W. Therefore, the resolution improves with increasing window size (equivalent to longer delays and wider bandwidths). Acknowledging the validity of equation (4.8), for any function h∈C(V), the modulation equivalence for general wireless channels can be directly derived:

[0722] H=Π(h), (4.9)

[0723] Here we assume that the support of h is much smaller than the diameter of the lattice Λ. The general two-dimensional channel model is formulated in the following theorem.

[0724] theorem (Two-dimensional channel model). Modulation equivalent channel is the impulse response h given by eqv ∈C(Z ⊥ ) is a periodic convolution y = h eqv *x:

[0725]

[0726] In other words, the modulation equivalent channel is a periodic convolution with a periodic blurred version of q·h, where the blurred pulse is composed of discrete pulses |W| 2 The discrete symplectic Fourier transform of is given by .

[0727] 4.4.1 Derivation of the two-dimensional channel model. We continue to derive equation (4.8). Let x∈C(Z ⊥ ).make represents sending a signal. We have:

[0728]

[0729] in make represents the received signal. We have:

[0730]

[0731] The third equation follows from the Heisenberg property of the mapping Π (Theorem 3.1). The demodulated vector Given by:

[0732]

[0733] We now evaluate the right-hand side of (4.13) term by term. Applying the basic channel equation (Theorem 3.3), we obtain:

[0734]

[0735] Considering the limitations We have the following proposition.

[0736] Proposition We have

[0737]

[0738] where for every v∈V,

[0739] Combining equations (4.13) and (4.15), we obtain:

[0740]

[0741] This concludes the derivation of the two-dimensional channel model.

[0742] 4.5 Explicit Interpretation

[0743] We conclude this section by explaining the continuous OTFS modulation mapping in terms of classical DSP operations. We use the standard communication lattice Λ=Λ from Example 1.1 in our calculations. T，μ . Recall that for every x∈C(Z ⊥ ) is defined as:

[0744]

[0745] in Formula (4.17) can be written more explicitly as:

[0746]

[0747]

[0748] in:

[0749]

[0750] Waveform φ K It is called the Kth modulation block.

[0751] 4.5.1 Frequency Domain Interpretation Let G denote the Fourier transform of g. Equation (4.19) can be interpreted as the weighted sequence W K X K is fed into a uniform filter bank where each subcarrier is shaped by a filter G. Fig.28 .

[0752] 4.5.2 Time Domain Interpretation. Let w K and x K Represent discrete waveforms W K and XK The inverse discrete Fourier transform of . Both waveforms are periodic with period T. We have:

[0753] φ K ∝(w K *x K )·g,

[0754] Where * represents periodic convolution. Waveform x K It can be expressed as follows based on the information vector x:

[0755]

[0756] In other words, x K (t) is proportional to the Kth component of the inverse discrete Fourier transform of x along the Doppler dimension.

[0757] 5 Limited OTFS transceiver

[0758] In this section, we describe a finite variant of the OTFS transceiver. This variant is obtained from the previously described continuous variant via uniform sampling.

[0759] 5.1 Settings

[0760] The definition of a limited OTFS transceiver assumes the following:

[0761] 1. Communication dot matrix. Undersampling dot matrix:

[0762]

[0763] where, for some μ≥1, vol(Λ)=μ.

[0764] 2.Communication sub-lattice. Sub-lattice:

[0765]

[0766] 3. Generator waveform. Unit standard waveform:

[0767] g∈H,

[0768] For every λ∈Λ × , satisfying the orthogonality condition A g (λ)=0.

[0769] 4.2D filter. Window function:

[0770] W∈C(Λ).

[0771] Note that the support of a 2D filter is typically compatible with the configuration of a sub-lattice, as demonstrated in the following example.

[0772] Example 5.1 The standard nested pair of communication lattices and sublattices is:

[0773]

[0774]

[0775] Where m, n∈N ≥1 and T∈R is a parameter called symbol time. The real numbers nμT and m / T are the delay and bandwidth of the communication packet, respectively. A typical compatible 2D filter is:

[0776]

[0777] More complex spectral window designs may involve, for example, some degree of taper around the boundaries, at the expense of spectral efficiency. Finally, if μ = 1, then a simple example of an orthogonal waveform is:

[0778]

[0779] 5.2 Finite OTFS modulation mapping

[0780] make is a reciprocal nested pair. is a finite reciprocal torus. The finite OTFS modulation mapping is that for every information vector The linear transformation defined by

[0781]

[0782] Formula (5.1) can be written more explicitly as:

[0783]

[0784] in

[0785] 5.3 Limited OTFS Demodulation Mapping

[0786] The finite OTFS demodulation mapping is for each The linear transformation given by

[0787]

[0788] Formula (5.2) can be more clearly expressed for each and writing:

[0789]

[0790] Recall that the normalization coefficient c = vol(Λ).

[0791] 5.4 Finite Two-Dimensional Channel Model

[0792] Let H = П(h) be the channel transform, where h∈C(V) is assumed to have small support compared to the dimension of the communication lattice. Recall the quadratic exponential:

[0793] q(v)=ψ(-β(v,v)).

[0794] Theorem 5.2 (Finite two-dimensional channel model). Finite modulation equivalent channel is the impulse response given by The circular convolution of y = h eqv，f *x:

[0795]

[0796] Figure 18 proves the statement of this theorem. Bar graph 1810 represents the transmitted information vector x. Bar graph 1820 represents the received information vector y. Bar graph 1830 represents the 2D impulse response that implements the finite modulation equivalent channel. The received vector is related to the transmitted vector by a 2D circular convolution with the 2D impulse response. Finally, we see from equation (5.3) that the finite impulse response h eqv，f is a finite subtorus The continuous impulse response h eqv of sampling.

[0797] 5.5 Clear explanation

[0798] We conclude this section by explaining the finite OTFS modulation mapping in terms of classical DSP operations. We use the nested pair from Example 5.1 in our calculations. Recall that for each Definition of finite modulation mapping:

[0799]

[0800] in Formula (5.4) can be written more explicitly as:

[0801]

[0802]

[0803] in:

[0804]

[0805] Waveform φ KIt is called the Kth modulation block.

[0806] 5.5.1 Frequency Domain Interpretation Let G denote the Fourier transform of g. Equation (5.6) can be interpreted as transforming the sequence W K X K Feeds into a unified filter bank where each subcarrier is shaped by filter G.

[0807] 5.5.2 Time Domain Interpretation. Let w K and x K Represent discrete waveforms W K and X K The inverse discrete Fourier transform of . Both waveforms are periodic with period T. We have:

[0808]

[0809] Where * represents periodic convolution. Waveform x K It can be expressed in terms of the information vector x as:

[0810]

[0811] In other words, x K is proportional to the inverse finite Fourier transform of x along the Doppler dimension.

[0812] Symmetric OTFS, OFDM compatibility and other features

[0813] 29 illustrates one way in which the symplectic OTFS method may work in a transmitter and receiver system 2900. Here, the data on the information plane (which may optionally be subjected to predistortion 2904) may then be transformed in two dimensions using an inverse 2D Fourier transform 2910 (and typically also a 2D spread function 2920) before passing through a filter bank 2930 (which may be an OFDM compatible filter bank). The various waveforms pass through a channel (C) 2940 where they are received by a filter bank 2950 (which may be an OFDM compatible filter bank), subjected to an inverse spread function 2960, an inverse 2D Fourier transform 2970 (the inverse of the previous IFFT 2910), and then equalized 2980 as needed.

[0814] Now turn your attention to Fig.29C , Fig.29C The characteristics of OTFS preprocessing that allow compatibility with OFDM modulation systems are illustrated. Fig.29CAs shown in , OTFS QAM symbols can be defined on a grid in the delay-Doppler domain. During the OTFS pre-processing step, these OTFS QAM symbols are transformed and spread on a grid in the time-frequency domain, which is the same domain in which the OFDM QAM symbols are defined.

[0815] Fig.29D Further details of the OTFS preprocessing operation compatible with OFDM modulation systems are illustrated. Fig.29D As shown in , the OTFS QAM symbol can be represented as the product of two linear phases. In this regard, the frequency in the time domain corresponds to the Doppler parameter of the OTFS QAM symbol. Similarly, the frequency in the frequency domain corresponds to the delay parameter of the symbol.

[0816] Fig.30 Alternative methods of sending and receiving data over a channel are shown.

[0817] Use with other forms of communication

[0818] It should be noted that although wireless examples have been used throughout this disclosure, these examples are not intended to be limiting. In alternative embodiments, other media are also contemplated, such as electrical transmission or RF transmission over wires or cables, optical transmission over optical fibers, and other long-distance communication methods, including acoustic transmission of signals over air or water or solid materials.

[0819] The impact of channel interference

[0820] According to the symplectic OTFS scheme discussed herein, in the symplectic coordinate system, channel interference such as the Doppler effect will distort or transform the symplectic plane along the frequency axis as a function of the frequency shift caused by the Doppler effect, while channel interference such as time delay will distort or transform the symplectic plane along the time axis as a function of the time delay according to the speed of light. The net effect is that in the symplectic OTFS coordinate system, the channel interference time delay appears as a phase gradient on one axis, while the Doppler frequency shift appears as an amplitude modulation effect along the other axis.

[0821] Since the symplectic OTFS method is transmitted in the transform domain, channel convolution is much easier to handle because it appears as a more tractable multiplication function. One approach is to simply sample the signal finely enough in time and frequency to be able to detect the phase gradient and amplitude modulation effects caused by these channel distortions. Once these can be detected, they can be corrected and removed.

[0822] This helps solve a long-standing problem in the field. Prior to the teachings of the present disclosure, there was a lack of knowledge in the art about how to use conventional OFDM methods to correct channel distortions such as Doppler shifts and time delays. It is believed that because OFDM methods rely on sending information over multiple relatively narrow bandwidths, it is not feasible to correct such channel distortions. However, using appropriate sampling intervals to detect the phase gradient and amplitude modulation caused by channel distortion in the OFDM signal, such correction is actually possible.

[0823] Fig.31 The effect of Doppler and time delay caused by the channel on the dual grids of the image domain and the transform domain is shown.

[0824] Interleaving, and compatibility with legacy OFDM methods

[0825] It is possible to interleave the different information planes using a symplectic approach. A very useful aspect of the symplectic version of OTFS is that in some embodiments, the symplectic OTFS filter bank can be set up to be compatible with, for example, previous OFDM standards such as the popular cellular 4G / LTE standards. At the same time, previous OFDM standards such as the 4G / LTE standards also have a medium access control (MAC) protocol that allows control of timing and interleaving.

[0826] Here, for example, an example of interleaving is to send only a certain column time width of the entire symplectic field over a certain range of frequency bands during a first time interval, send the others, and then send another column time width of the entire symplectic field over a certain range of frequency bands at a later time interval. Other forms of interleaving (such as frequency-based interleaving) are also possible.

[0827] Fig.32 An example of interleaving is shown.

[0828] Fig.33 Another example of interleaving is shown, where frames of the same size are interleaved on a frequency staggered basis.

[0829] Fig.34 Another example of interleaving is shown, where frames of variable size are interleaved on a temporal basis.

[0830] Backward compatibility with OFDM methods

[0831] In certain embodiments, the Symplectic OFDM approach can coexist with conventional OFDM approaches on the same frequency and time, and in fact can even be used to improve the efficiency of conventional OFDM approaches.

[0832] In such an embodiment, the Symplectic OTFS method may be viewed as feeding a signal to an OFDM modulator or otherwise precoding a signal that is subsequently provided to an OFDM modulator. In the case of interleaving, this same OFDM modulator may be driven using conventional OFDM symbols during some time intervals and using OTFS signals during other time intervals. In this regard, the Symplectic OTFS method may be viewed as an improved front end of an OFDM modulator (on the transmit side). For example, an OTFS transceiver or transmitter may be characterized and implemented as a signal pre-processing module inserted before an OFDM modulator within a signal transmission system. Within a signal reception system, an OTFS receiver may be located after an OFDM receiver to implement signal post-processing.

[0833] This approach advantageously enables compatibility with popular legacy OFDM methods (such as 4G / LTE) to be retained while facilitating the use of OTFS techniques to correct channel distortions. This facilitates easy transition from, for example, legacy 4G / LTE systems to new, higher-capability "5G" systems based on the OTFS method described herein.

[0834] OTFS is a new modulation technique with many benefits and a strong data foundation. From an implementation point of view, its added benefits are compatibility with OFDM and requiring only incremental changes in transmitter and receiver architecture.

[0835] More specifically, recall that the embodiment of OTFS consists of two steps. The Heisenberg transform (which brings the time-frequency domain to the waveform domain) is already implemented in today's systems in the form of OFDM / OFDMA. In the conception used in this article, this corresponds to a prototype filter g(t) as a rectangular pulse. Other filtered OFDM and filter bank variants have been proposed for 5G, which can also be accommodated in this general framework using different choices of g(t).

[0836] The second step in this embodiment of OTFS is based on a two-dimensional Fourier transform (SFFT). Fig.35 As illustrated in , this can be characterized as a pre-processing step within the transmitter module and a post-processing step within the receiver module.

[0837] refer to Fig.35 And as discussed above, the OTFS pre-processing performed in the transmitter module may include performing an inverse sigmoid Fourier transform and a windowing operation. Similarly, the OTFS post-processing implemented in the receiver module may include performing a sigmoid Fourier transform and another windowing operation.

[0838] Now turn to Fig.36, a block diagram of an OTFS transmitter 3600 according to an embodiment is provided. The transmitter 3600 includes a digital processor 3604 (which may be a microprocessor), a digital signal processor, or other similar device. The digital processor accepts as input a data frame 3608 processed by an OTFS preprocessing block 3616 in the manner discussed above to produce a matrix of time-frequency modulated symbols. These time-frequency modulated symbols are then provided to an OFDM or MCFB modulator 3620, and the resulting waveform is filtered by a transmitter filter 3624. The filtering result is then received by a digital-to-analog converter (DAC) 3630. The baseband output of the DAC 3630 is up-converted to the radio frequency band within the RF unit 3640 to produce an analog radio frequency waveform. The waveform is then passed to the OTFS receiver, where the waveform is as will be referred to below. Fig.37 is received and demodulated as described.

[0839] Now turn your attention to Fig.37 , Fig.37 An OTFS receiver 3700 is depicted that is configured to demodulate OTFS modulated data received over a wireless link. A received signal (not shown) corresponding to a channel-impaired version of the radio frequency signal transmitted by the OTFS transmitter 3600 may be received, for example, by an antenna of the OTFS receiver 3700. Because of signal noise, impairments, or distortion caused by the communication channel, the received signal will typically not include an exact copy of the transmitted signal. The received signal is amplified by the RF unit 3704 and down-converted from the radio frequency band to baseband. The baseband signal is then digitized within an analog-to-digital converter (ADC) 3708 and filtered within a receiver filter 3712. The receiver includes a digital processor 3720 (which may be a microprocessor), a digital signal processor, or other similar device. The digital processor 3720 includes an OFDM or MCFB modulator 3728 that accepts the digitized waveform from the receiver filter 3712 and generates an estimated time-frequency modulation symbol. The digital processor 3720 also includes an OTFS post-processing block 3736 that is operable to generate an estimated data frame 3750 in the manner discussed above.

[0840] OTFS Relay

[0841] Fig.38An example of one manner in which an active OTFS relay system 3800 may operate between an OTFS transmitter and receiver is illustrated. As shown, an active OTFS relay 3810 including an OTFS transceiver is in wireless communication between an OTFS transmitter 3830 and an OTFS receiver 3840, wherein the OTFS transceiver includes an OTFS relay transmitter 3820 and an OTFS relay receiver 3822. During operation, an OTFS modulated signal transmitted by the OTFS transmitter 3830 is received and demodulated by the OTFS relay receiver 3822. The resulting recovered data is then used by the OTFS relay transmitter 3820 to generate an OTFS modulated relay signal, which is received and demodulated by the OTFS receiver 3840.

[0842] Selected Benefits of OTFS

[0843] As discussed above, OTFS modulation causes the wireless signal to experience a fixed, deterministic and non-fading channel interaction. Specifically, all symbols experience the same channel and the two-dimensional channel impulse response reveals deterministic channel geometry. This allows for coherence and aggregation of multipath energy so that all diversity branches can be fully utilized. Importantly, the deterministic nature of the channel is essentially invariant and highly tolerant to the mobility of the transmitter and receiver within the channel.

[0844] Compared to many conventional modulation techniques, OTFS requires only loose time and frequency synchronization. This is in part because time and / or frequency misalignment is captured from the acquired channel state information and can be compensated using equalization.

[0845] Of particular importance for next generation 5G communication systems is that OTFS systems can be efficiently scaled for use with higher order MIMO techniques. This is possible in part due to the timely, accurate, and compact nature of OTFS channel state information and the low burden required to acquire it. OTFS is also well suited for high frequency (e.g., millimeter wave) spectrum of the type contemplated for use in 5G systems. For example, OTFS is insensitive to the higher relative Doppler spread and frequency offset associated with these higher frequencies.

[0846] OTFS also provides interleaved variable delay with adjustable frame size and coordinated multi-point deployment through the acquisition of accurate channel state information. In addition, interference suppression can be distributed rather than centralized.

[0847] Although various embodiments have been described above, it should be understood that they are given by way of example only and not limitation. They are not intended to be exhaustive or to limit the claims to the exact form disclosed. In fact, many modifications and variations are possible in light of the above teachings. The embodiments are selected and described in order to best explain the principles of the described systems and methods and their practical applications, thereby enabling others skilled in the art to best utilize the described systems and methods and various embodiments with various modifications suitable for the intended specific use.

[0848] In the case where the above method indicates certain events that occur in a certain order, the ordering of certain events can be modified. In addition, if possible, certain events can be executed simultaneously in a parallel process, as well as sequentially as described above. Although various modules in different devices are shown as being located in the processor of the device, they can also be located / stored in the memory of the device (e.g., software modules), and can be accessed and executed by the processor. Thus, the specification is intended to cover all such modifications and changes of the disclosed embodiments that fall within the spirit and scope of the appended claims.

[0849] For the purpose of explanation, the foregoing description uses specific terms to provide a thorough understanding of the claimed systems and methods. However, it will be apparent to those skilled in the art that these specific details are not necessary to practice the systems and methods described herein. Therefore, the foregoing descriptions of specific embodiments of the described systems and methods are given for the purpose of illustration and description. They are not intended to be exhaustive or to limit the claims to the precise form disclosed; obviously, many modifications and variations are possible in view of the above teachings. The selection and description of the embodiments are to best explain the principles of the described systems and methods and their practical applications, thereby enabling other technical personnel in the art to best utilize the described systems and methods and various embodiments with various modifications suitable for the intended specific use. It is intended that the following claims and their equivalents define the scope of the systems and methods described herein.

[0850] The various methods or processes outlined herein may be encoded as software that can be executed on one or more processors using any of a variety of operating systems or platforms. In addition, such software may be written using any of a variety of suitable programming languages ​​and / or programming or scripting tools, and may also be compiled into executable machine language code or intermediate code that is executed on a framework or virtual machine.

[0851] Examples of computer code include, but are not limited to, microcode or microinstructions, such as machine instructions generated by a compiler, code for generating web services, and files containing higher-level instructions executed by a computer using an interpreter. For example, embodiments may be implemented using an imperative programming language (e.g., C, Fortran, etc.), a functional programming language (Haskell, Erlang, etc.), a logical programming language (e.g., Prolog), an object-oriented programming language (e.g., Java, C++, etc.), or other suitable programming languages ​​and / or development tools. Additional examples of computer code include, but are not limited to, control signals, encryption codes, and compression codes.

[0852] In this regard, various inventive concepts may be embodied as a computer-readable storage medium (or multiple computer-readable storage media) (e.g., computer memory, one or more floppy disks, compressed disks, optical disks, magnetic tapes, flash memory, circuit configurations in field programmable gate arrays or other semiconductor devices, or other non-transitory media or tangible computer storage media) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments of the present invention discussed above. The one or more computer-readable media may be removable so that the one or more programs stored thereon may be loaded into one or more different computers or other processors to implement various aspects of the present invention as discussed above.

[0853] The term "program" or "software" is used in this article in a general sense to refer to any type of computer code or computer executable instruction set that can be used to program a computer or other processor to implement various aspects of the embodiments as discussed above. In addition, it should be recognized that, according to one aspect, one or more computer programs that perform the methods of the present invention when executed do not need to reside on a single computer or processor, but can be distributed in a modular manner among multiple different computers or processors to implement various aspects of the present invention.

[0854] Computer executable instructions can be in many forms, such as program modules, that are executed by one or more computers or other devices. Generally speaking, program modules include routines, programs, objects, components, data structures, etc. that perform specific tasks or implement specific abstract data types. Typically, the functionality of the program modules can be combined or distributed in various embodiments as desired.

[0855] Moreover, the data structure can be stored in a computer-readable medium in any suitable form. To simplify the description, the data structure can be shown as having fields that are related by position in the data structure. This relationship can also be achieved by assigning storage with a position in a computer-readable medium that conveys the relationship between the fields for the fields. However, any suitable mechanism can be used to establish the relationship between the information in the fields of the data structure, including by using pointers, tags or other mechanisms that establish the relationship between the data elements.

[0856] Moreover, various inventive concepts may be embodied as one or more methods, examples of which have been provided. The actions performed as part of a method may be ordered in any suitable manner. Thus, embodiments may be constructed in which the actions are performed in an order different from the order shown, which may include performing some actions simultaneously, although shown as sequential actions in the illustrative embodiments.

[0857] As defined and used herein, all definitions should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0858] As used herein in the specification and claims, the indefinite articles "a" and "an" should be understood to mean "at least one" unless explicitly indicated to the contrary.

[0859] As used herein in the specification and claims, the phrase "and / or" should be understood to mean "either or both" of the elements so connected, i.e., elements that exist jointly in some cases and separately in other cases. Multiple elements listed with "and / or" should be interpreted in the same manner, i.e., "one or more" of the elements so connected. In addition to the elements specifically identified by the "and / or" clause, other elements may optionally be present, whether related or unrelated to those specifically identified elements. Thus, as a non-limiting example, when used in conjunction with open language such as "comprises", a reference to "A and / or B" may refer to only A (optionally including elements other than B) in one embodiment; to only B (optionally including elements other than A) in another embodiment; to both A and B (optionally including other elements) in yet another embodiment; and so on.

[0860] As used in this specification and claims, "or" should be understood to have the same meaning as "and / or" defined above. For example, when separating items in a list, "or" or "and / or" should be interpreted as inclusive, that is, including at least one element, but also including more than one element in a plurality of elements or a list of elements, and optional other unlisted items. Only items that are clearly indicated as opposite, such as "only one of..." or "exactly one of...", or, when used in the claims, "consisting of..." will refer to exactly one element in a plurality of elements or a list of elements. In general, the term "or" used herein will be interpreted as indicating an exclusive alternative only when there is an exclusive term (such as "any one", "one of...", "only one of..." or "exactly one of...") in front. When used in the claims, "consisting essentially of..." will have its ordinary meaning used in the field of patent law.

[0861] As used herein in the specification and claims, the phrase "at least one" referring to a list of one or more elements should be understood to mean at least one element selected from any one or more elements in the list of elements, but not necessarily including at least one of each element specifically listed in the list of elements, and not excluding any combination of elements in the list of elements. This definition also allows that elements other than the elements specifically identified in the list of elements referred to by the phrase "at least one" may optionally be present, regardless of whether they are related or unrelated to those specifically identified elements. Thus, as a non-limiting example, "at least one of A and B" (or equivalently, "at least one of A or B", or equivalently, "at least one of A and / or B") may refer to at least one (optionally including more than one) A without B (and optionally including elements other than B) in one embodiment; to at least one (optionally including more than one) B without A (and optionally including elements other than A) in another embodiment; to at least one (optionally including more than one) A and at least one (optionally including more than one) B (and optionally including other elements) in yet another embodiment; and so on.

[0862] In the claims and the foregoing description, all transitional phrases, such as "comprising," "including," "carrying," "having," "containing," "involving," "maintaining," "comprising," and the like are to be understood as open-ended, i.e., meaning including, but not limited to. Only the transitional phrases "consisting of" and "consisting essentially of" shall be closed or semi-closed transitional phrases, respectively, as set forth in Section 2111.03 of the U.S. Patent Office Manual of Patent Examining Procedures.

Claims

1. A method for providing a modulated signal that can be used in a signal transmission system, the method comprising: performing a two-dimensional time-frequency transformation of a data frame comprising a plurality of information symbols into a plane of time-frequency modulation symbols; as well as generating the modulated signal by performing a Heisenberg transform using a plane of time-frequency modulation symbols, The performing of the two-dimensional time-frequency transform comprises: transforming the data frame into an array of filtered symbols using at least one Fourier transform and filtering process; as well as The array of filtered symbols is transformed into an array of OTFS symbols using at least one two-dimensional Fourier transform, wherein the array of OTFS symbols corresponds to the plane of time-frequency modulation symbols. 2 . The method of claim 1 , wherein performing the two-dimensional time-frequency transform comprises performing an inverse time-frequency symplectic transform and a windowing operation.

3. A method for providing a modulated signal that can be used in a signal transmission system, the method comprising: performing a two-dimensional time-frequency transformation of a data frame comprising a plurality of information symbols into a plane of time-frequency modulation symbols; as well as generating the modulated signal based on a plane of time-frequency modulation symbols using an OFDM modulator, The performing of the two-dimensional time-frequency transform comprises: transforming the data frame into an array of filtered symbols using at least one Fourier transform and filtering process; as well as The array of filtered symbols is transformed into an array of OTFS symbols using at least one two-dimensional Fourier transform, wherein the array of OTFS symbols corresponds to the plane of time-frequency modulation symbols. 4 . The method of claim 3 , wherein performing the two-dimensional time-frequency transform comprises performing an inverse time-frequency symplectic transform and a windowing operation.

5. A method for providing a modulated signal that can be used in a signal transmission system, the method comprising: performing a two-dimensional time-frequency transformation of a data frame comprising a plurality of information symbols into a plane of time-frequency modulation symbols; as well as generating the modulated signal based on a plane of time-frequency modulation symbols using a multi-carrier filter bank MCFB modulator, The performing of the two-dimensional time-frequency transform comprises: transforming the data frame into an array of filtered symbols using at least one Fourier transform and filtering process; as well as The array of filtered symbols is transformed into an array of OTFS symbols using at least one two-dimensional Fourier transform, wherein the array of OTFS symbols corresponds to the plane of time-frequency modulation symbols.

6. The method of claim 5, wherein performing the two-dimensional time-frequency transform comprises performing an inverse time-frequency symplectic transform and a windowing operation.

7. A transmitter device, the device comprising: an OTFS pre-processing unit configured to perform a two-dimensional time-frequency transformation of a data frame containing a plurality of information symbols into a plane of time-frequency modulation symbols; as well as An OFDM modulator configured to generate a modulated signal based on a plane of time-frequency modulation symbols, The performing of the two-dimensional time-frequency transform comprises: transforming the data frame into an array of filtered symbols using at least one Fourier transform and filtering process; as well as The array of filtered symbols is transformed into an array of OTFS symbols using at least one two-dimensional Fourier transform, wherein the array of OTFS symbols corresponds to the plane of time-frequency modulation symbols.

8. The transmitter device of claim 7, wherein the two-dimensional time-frequency transform comprises an inverse time-frequency symplectic transform and a windowing operation.

9. A transmitter device, the device comprising: an OTFS pre-processing unit configured to perform a two-dimensional time-frequency transformation of a data frame containing a plurality of information symbols into a plane of time-frequency modulation symbols; as well as a multi-carrier filter bank modulator configured to generate a modulated signal based on a plane of time-frequency modulation symbols, The performing of the two-dimensional time-frequency transform comprises: transforming the data frame into an array of filtered symbols using at least one Fourier transform and filtering process; as well as The array of filtered symbols is transformed into an array of OTFS symbols using at least one two-dimensional Fourier transform, wherein the array of OTFS symbols corresponds to the plane of time-frequency modulation symbols.

10. The transmitter device of claim 9, wherein the two-dimensional time-frequency transform comprises an inverse time-frequency symplectic transform and a windowing operation.

11. A method for receiving a modulated signal, the method comprising: receiving a plurality of signal components of the modulated signal; generating a plane of estimated time-frequency modulation symbols by performing an OFDM demodulation operation using the plurality of signal components; as well as Providing an estimated data frame by performing an inverse of a two-dimensional time-frequency transform on the plane of estimated time-frequency modulation symbols, wherein performing an inverse of the two-dimensional time-frequency transform comprises: projecting estimated symbols from said plane of estimated time-frequency modulation symbols onto a plurality of orthogonal two-dimensional basis functions to obtain said estimated data frame, and Wherein said projecting comprises performing a symplectic Fourier transform operation on a periodic sequence of estimated symbols from a plane of said estimated time-frequency modulation symbols.

12. The method of claim 11, wherein performing an inverse of the two-dimensional time-frequency transform comprises performing a windowing operation and the symplectic Fourier transform operation.

13. A method for receiving a modulated signal, the method comprising: receiving a plurality of signal components of the modulated signal; generating a plane of estimated time-frequency modulation symbols by performing an MCFB demodulation operation using the plurality of signal components; as well as Providing an estimated data frame by performing an inverse of a two-dimensional time-frequency transform on the plane of estimated time-frequency modulation symbols, wherein performing an inverse of the two-dimensional time-frequency transform comprises: projecting estimated symbols from said plane of estimated time-frequency modulation symbols onto a plurality of orthogonal two-dimensional basis functions to obtain said estimated data frame, and Wherein said projecting comprises performing a symplectic Fourier transform operation on a periodic sequence of estimated symbols from a plane of said estimated time-frequency modulation symbols.

14. The method of claim 13, wherein performing an inverse of the two-dimensional time-frequency transform comprises performing a windowing operation and the symplectic Fourier transform operation.

15. A receiver device, the device comprising: a receiver front end configured to receive a plurality of signal components of a modulated signal; an OFDM demodulator configured to generate a plane of estimated time-frequency modulation symbols based on the plurality of signal components; and An OTFS post-processing unit operable to provide an estimated data frame, the OTFS post-processing unit performing an inverse of a two-dimensional time-frequency transform on a plane of the estimated time-frequency modulation symbols, wherein performing the inverse of the two-dimensional time-frequency transform comprises: projecting estimated symbols from said plane of estimated time-frequency modulation symbols onto a plurality of orthogonal two-dimensional basis functions to obtain said estimated data frame, and Wherein said projecting comprises performing a symplectic Fourier transform operation on a periodic sequence of estimated symbols from a plane of said estimated time-frequency modulation symbols.

16. The receiver apparatus of claim 15, wherein the inverse of the two-dimensional time-frequency transform comprises a windowing operation and the symplectic Fourier transform operation.

17. A receiver device, the device comprising: a receiver front end configured to receive a plurality of signal components of a modulated signal; an MCFB demodulator configured to generate a plane of estimated time-frequency modulation symbols based on the plurality of signal components; and An OTFS post-processing unit operable to provide an estimated data frame, the OTFS post-processing unit performing an inverse of a two-dimensional time-frequency transform on a plane of the estimated time-frequency modulation symbols, wherein performing the inverse of the two-dimensional time-frequency transform comprises: projecting estimated symbols from said plane of estimated time-frequency modulation symbols onto a plurality of orthogonal two-dimensional basis functions to obtain said estimated data frame, and Wherein said projecting comprises performing a symplectic Fourier transform operation on a periodic sequence of estimated symbols from a plane of said estimated time-frequency modulation symbols.

18. The receiver apparatus of claim 17, wherein the inverse of the two-dimensional time-frequency transform comprises a windowing operation and the symplectic Fourier transform operation.

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