Wireless system implementation

Abstract

A method of digital communications includes transmitting, by a transmitting device, a transmission waveform to one or more receiving devices using one or more time and frequency resources of a transmission medium. The time and frequency resources are divided into physical resource blocks (PRBs) that are configured according to a PRB configuration scheme. Each PRB is generated from a corresponding orthogonal time frequency space (OTFS) frame. Each OTFS frame is generated from symbols along delay and Doppler dimensions using an OTFS framing scheme.

Description

PCT Patent Application Attorney Docket No. 119314.8128.WO00WIRELESS SYSTEM IMPLEMENTATION CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to U. S. Provisional Application No. 63 / 748,908, filed on January 23, 2025, the disclosure of which is hereby incorporated by reference herein in its entirety.TECHNICAL FIELD

[0002] The present document relates to digital communication.BACKGROUND

[0003] Due to an explosive growth in the number of wireless user devices and the amount of wireless data that these devices can generate or consume, current wireless communication networks are fast running out of bandwidth to accommodate such a high growth in data traffic and provide high quality of service to users.

[0004] Various efforts are underway in the telecommunication industry to come up with next generation of wireless technologies that can keep up with the demand on performance of wireless devices and networks. Many of those activities involve situations in which a large number of user devices may be served by a network.SUMMARY

[0005] This document discloses techniques for a wireless system that uses orthogonal time frequency space modulation (OTFS) technology.

[0006] In one aspect, the present document discloses a new physical layer for digital data communication in which OTFS technology is used. Such a system may include time-frequency physical resource blocks (PRBs) as building blocks, where PRBs have configurational dimensions into which an OTFS frame with configurable delay-Doppler dimensions may be mapped. Each PRB and its OTFS frame may be configured in both domains according to a specific user’s channel characteristics.

[0007] In one example aspect, a method of digital communications includes transmitting, by a transmitting device, a transmission waveform to one or more receiving devices using one or more time and frequency resources of a transmission medium, wherein the time and frequency resources are divided into PRBs that are configured according to a PRB configuration scheme: wherein each PRB is generated from a corresponding OTFS frame; wherein each OTFS frame is generated from symbols along delay and Doppler dimensions using an OTFS framing scheme.1184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0008] In another example aspect, a method of digital communications includes receiving, by a receiving device, a transmission waveform using time and frequency resources of a transmission medium, and processing the transmission waveform to extract pilots or information bits carried by the transmission waveform; wherein the time and frequency resources are divided into PRBs according to a PRB configuration scheme; wherein each PRB corresponds to an OTFS frame; wherein each OTFS frame comprises symbols along delay and Doppler dimensions according to an OTFS framing scheme.

[0009] In another example aspect, a method of digital communication, comprising: generating information symbols by mapping source information bits for one or more user devices into the information symbols; generating OTFS frames from the information symbols and / or pilot symbols according to an OTFS framing scheme; transforming the OTFS frames to PRBs in time-frequency domain according to a PRB configuration scheme; and generating a transmission waveform of a time slot by superpositioning the PRBs.

[0010] In another example aspect, a method of wireless communication, comprising: transmitting a transmission waveform to one or more receiving devices, wherein the transmission waveform is generated by multiplexing transmissions to the one or more receiving devices, wherein each transmission includes of one or more PRBs, wherein each PRB is mapped to time and frequency resources, wherein each PRB corresponds to an OTFS frame, wherein each OTFS frame is generated from symbols assigned to resources in the delay and Doppler dimensions.

[0011] In another example aspect, a system for digital communication includes a transmitting device (e.g., network device) that implements an above-described transmitting method and one or more receiving devices (e.g., wireless devices) that implement the above described receiving method.

[0012] In some embodiments, the duration and offset of the PRB are determined according to the latency requirements of the specific transmission. In some embodiments, the carrier bandwidth is divided into multiple PRBs by providing a tight fit into the carrier bandwidth, where number of PRBs may be represented as NPRB= ⌊BW / ΔfPRB⌋. In some embodiments, the multiple PRBs may be coded with a sameIAJPRBIforward error correction (FEC) code. The OTFS transmission in a slot may comprise a superposition of all PRBs.

[0013] With respect to features of PRBs, the following features may be adopted: (a) PRB is mapped to a frequency location within the carrier bandwidth, (b) the PRB has a configurable offset within a transmission slot, (c) PRB has a fixed bandwidth and configurable duration. In some embodiments, the following values may be used to provide compatibility with present day wireless systems:■ BWPRB= 168 kHz■ ΔfPRB= 180 kHz2184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00■ Tslot= 1 msec■ TPRB= ^- Tslot, nT= 1,2,...,14■Tgffset=°f^et' ^slot’noffset=■■■,13.

[0014] In some embodiments, an OTFS frame may have a delay-Doppler grid with M ■ N = NRE= BWpRB■ TPRB= 12 ■ nT.

[0015] In some embodiments, OTFS uses a sine pulse for the Doppler and a Root-Raised Cosine pulse for delay. In some embodiments, delay pulse roll-off factor= 0.07 may be used.

[0016] In some embodiments, an OTFS frame may be generated by performing Fourier transform on the Doppler dimension. One of two pulse configurations may be used for delay-Doppler filtering. In some embodiments, a delay filter may be used for upsampling. In some embodiments, frequency and time adjustments may be performed at carrier sampling rate.

[0017] Various types of OTFS schemes are also disclosed, including standard OTFS, minimum period (MP) OTFS on Doppler dimension or MP-OTFS on delay dimension, faster-than-Nyquist (FTN) OTFS. Furthermore, receiver-side processing for each of the variants of OTFS are disclosed.

[0018] In another example aspect, a wireless communication apparatus that implements one or more of the above-described methods is disclosed. The apparatus may include one or more processors configured to control the apparatus to implement one or more of the described methods.

[0019] In yet another example aspect, a computer-readable storage medium that stores processorexecutable code for one or more of the above-described methods is disclosed.

[0020] These, and other, features are described in this document.DESCRIPTION OF THE DRAWINGS

[0021] Drawings described herein are used to provide a further understanding and constitute a part of this application. Example embodiments and illustrations thereof are used to explain the technology rather than limiting its scope.

[0022] FIG. 1A shows an example of a wireless communication system.

[0023] FIG. IB shows a simplified wireless network with two wireless devices.

[0024] FIG. 1C shows an example of a delay-doppler grid.

[0025] FIG. 2 illustrates periodicity in delay and Doppler domains.

[0026] FIG. 3 illustrates examples of physical resource blocks (PRBs).

[0027] FIG. 4 illustrates an OTFS frame mapping to PRB example.3184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0028] FIG. 5 shows an example of transmitter processing for multiple PRBs, coded together with the same forward error correction (FEC) code. Note, that the delay-Doppler configurations and PRB durations are not mandated to be the same.

[0029] FIG. 6 depicts a time window implementation example of the Doppler pulse, for (a) a rectangular window (sine Doppler pulse) and (b) a general time- window with a roll-off Pv.

[0030] FIG. 7 depicts an OTFS waveform generation example for pulse configuration #1 and a general time window. Here, WTis a rectangular window covering exactly MN samples.

[0031] FIG. 8 depicts an OTFS waveform generation example for pulse configuration #2 with a general ptime window. Here, WTis a rectangular window covering exactly MN - samples.

[0032] FIG. 9 illustrates pilot configurations examples for a standard OTFS scheme.

[0033] FIG. 10 depicts a waveform generation example for MP-OTFS on Doppler, using pulse configuration #1 and a general time window.

[0034] FIG. 11 depicts a waveform generation example for MP-OTFS on Doppler, using pulse configuration #2 and a rectangular time window.

[0035] FIG. 12 depicts pilot allocation examples for MP-OTFS on Doppler dimension.

[0036] FIG. 13 depicts a waveform generation example for MP-OTFS on delay, using pulse configuration #1 and a general time window.

[0037] FIG. 14 depicts pilot allocation examples for MP-OTFS on delay.

[0038] FIG. 15 depicts an example of multiple pilots along delay for MP-OTFS on delay, to enable channel estimation in high mobility.

[0039] FIG. 16 depicts a pilot location staggering example between multiple PRBs, to reduce the overall PAPR (peak to average power ratio).

[0040] FIG. 17 illustrates examples of sampling theory for different cases.

[0041] FIG. 18 depicts an FTN-OTFS delay-Doppler grid example. The narrower box (with dimension N) represents MP-OTFS on Doppler without FTN and the wider box (with dimension N') represents MP-OTFS on Doppler with FTN.

[0042] FIG. 19 depicts a waveform generation example for FTN-OTFS using MP-OTFS on Doppler, pulse configuration #2 and a rectangular time window.

[0043] FIG. 20 depicts a receiving process example for OTFS with pulse configuration #1.

[0044] FIG. 21 depicts a receiving process example for OTFS with pulse configuration #2.4184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0045] FIG. 22 depicts an interpolated channel response example. The circles represent the pilot locations, and the line is the interpolated channel response between the pilots in both amplitude and phase.

[0046] FIG. 23 depicts a receiver side processing example for FTN-OTFS with pulse configuration #2 and a rectangular time window.

[0047] FIG. 24 is a block diagram of an example hardware platform.

[0048] FIG. 25A-25D are flowcharts of example methods of digital communications.

[0049] FIG. 26 pictorially depicts relationship between time, frequency and Zak domains.

[0050] FIG. 27 shows an example of a hexagonal lattice. The hexagon at the center region encloses Voronoi region around the zero lattice point. The two lattice points with arrow decoration are the basis of the maximal rectangular sub-lattice.

[0051] FIG. 28 pictorially depicts the periodic and quasi-periodic nature of an information grid in the Zak domain.

[0052] FIG. 29 is a graphical representation of an OTFS waveform.

[0053] FIG. 30 is a graphical representation of filtered OTFS waveform.

[0054] FIG. 31 is a graphical comparison of transmit waveforms of a single QAM (quadrature amplitude modulation) symbol using OTFS and OTFS-MC (multicarrier).

[0055] FIG. 32 pictorially depicts an example in which the Zak domain and the time / frequency Zak transforms realizing the signal space realization lying in between time and frequency realizations.

[0056] FIG. 33 shows a depiction of the Zak to generalized Zak interwining transformation.

[0057] FIG. 34 shows an example of a nested system of lattices with N=3 and M=2.

[0058] FIG. 35 is a graphical depiction of an example of a chirp lattice corresponding to N = 3, M=2, a = 1. The dashed square designated clear region around zero.

[0059] FIG. 36 is a depiction of an example of an ambiguity function of a continuous Zak chirp corresponding to slope a = 1.

[0060] FIG. 37 depicts example waveforms in a pulse-tone generation.

[0061] FIG. 38 shows an example of generation of a Pulse-tone waveform.

[0062] FIG. 39A and 39B show examples of pilot symbols in a delay-Doppler plane.

[0063] FIG. 40 shows an example of a delay-Doppler plan in which symbols that contain information bits rate show.

[0064] FIG. 41 shows another example of pilot symbols in a delay-Doppler plane.5184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0065] FIG. 42 shows an example of an orthogonal time frequency space (OTFS) waveform having a power-boosted pilot signal.

[0066] FIG. 43A shows an example of a transmission method in which a delay-Doppler grid is transformed to a time-frequency grid using a Symplectic Fast Fourier Transform (SFFT).

[0067] FIG. 43B shows an example of a transmission method in which a delay-Doppler grid is transformed to a time-frequency sub-grid.

[0068] FIG. 43C shows an example of transmission method in which a delay-Doppler grid is transformed to an OTFS waveform using a Zak transform over the Doppler dimension.

[0069] FIG. 43D shows an example of a reception method in which Inverse SFFT (ISFFT) is used to recover information bits from a received waveform.

[0070] FIG. 43E shows an example of a reception method in which Inverse SFFT (ISFFT) is used to recover information bits from an OTFS sub-grid of a received waveform.

[0071] FIG. 43F shows an example of a reception method in which an inverse Zak transform over the time dimension is used to recover information bits from a received waveform.

[0072] FIG. 44 shows a block diagram of an example iterative receiver apparatus.

[0073] FIG. 45 is a block diagram showing an example of a multi-level transmission system.

[0074] FIG. 46 shows a block diagram of an example iterative receiver apparatus that uses multi-level decoding.

[0075] FIG. 47 is a block diagram showing an example 2-D iterative equalizer.DETAILED DESCRIPTION

[0076] To make the purposes, technical solutions and advantages of this disclosure more apparent, various embodiments are described in detail below with reference to the drawings. Unless otherwise noted, embodiments and features in embodiments of the present document may be combined with each other.

[0077] Section headings are used in the present document to improve readability of the description and do not in any way limit the discussion or the embodiments to the respective sections only. Furthermore, certain standard-specific terms are used for illustrative purpose only, and the disclosed techniques are applicable to any wireless communication systems.

[0078] 1. Introduction to wireless communication environment

[0079] The wireless or time-variant nature of the communication channel poses several challenges in designing a transmission protocol suitable for wireless communication scenarios. These days, users expect their wireless devices to work everywhere and in a variety of mobile or stationary situations.6184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0080] The time-variant nature of a wireless network and the expectation by users of a reliable, high-bandwidth network connection at any time and in any place creates a tension between the required amount of transmission resources a wireless network needs for overhead signal communications (e.g., for calibrating a wireless channel) and allocating as much transmission bandwidth to user data as possible. Deployments of user devices and network devices having multiple antennas makes this problem becomes even more challenging because wireless networks may need to calibrate wireless channel to / from each antenna of a multi-antenna device.

[0081] The techniques described in the present application allow for calibration of uplink or downlink wireless network connections using various techniques that provide operational advantages as further described throughout the present document.

[0082] 2. Example wireless systems

[0083] FIG. 1 A shows an example of a wireless communication system 100 in which a transmitter device 102 transmits signals to a receiver 104. The signals may undergo various wireless channels and multipaths, as depicted. Some reflectors such as buildings and trees may be static, while others such as cars, may be moving scatterers. The transmitter device 102 may be, for example, a user device, a mobile phone, a tablet, a computer, or another Internet of Things (loT) device such as a smartwatch, a camera, and so on. The receiver device 104 may be a network device such as the base station. The signals transmitted from the base station to the transmitter 102 may experience similar channel degradations produced by static or moving scatterers. The techniques described in the present document may be implemented by the devices in the wireless communication system 100. The terms “transmitter” and “receiver” are simply used for convenience of explanation and as further described herein, depending on the direction of transmission (uplink or downlink), the network station may be transmitting or receiving, and user device may be receiving or transmitting.

[0084] FIG. IB shows a simplified wireless network to highlight certain aspects of the disclosed technology. A transmitter transmits wireless signals to a receiver in the wireless network. Some transmissions in the network, variously called as downlink or downstream transmissions, a network-side node such as a base station acts as a transmitter of wireless signals and one or more user devices act as the receiver of these wireless signals. For some other transmissions, as depicted in FIG. IB, the direction of transmission may be opposite. Such transmissions are often called uplink or upstream transmissions. For such transmissions, one or more user devices act as transmitters of the wireless signals and a network-side node such as the base station acts as the receiver of these signals (as depicted in FIG. IB). Other type of transmissions in the network may include device-to-device transmissions, sometimes called direct or sideband transmissions. While the present document primarily uses the terms “downlink” and “uplink”7184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00for the sake of convenience, similar techniques may also be used for other situations in which transmissions in two directions are performed - e.g., inbound or incoming transmissions that are received by a wireless device and outbound or outgoing transmissions that are transmitted by a wireless device. For example, downlink transmissions may be inbound transmissions for a user device, while outbound transmissions for a network device. Similarly, uplink transmission may be inbound transmissions for a network device while outbound transmissions from a wireless device. Therefore, for some embodiments, the disclosed techniques may also be described using terms such as “inbound” and “outbound” transmission without importing any 3GPP-specific or other wireless protocol-specific meaning to the terms “uplink” and “downlink.”

[0085] In frequency division multiplexing (FDM) networks, the transmissions to a base station and the transmissions from the base station may occupy different frequency bands (each of which may occupy continuous or discontinuous spectrum). In time division multiplexing (TDM) networks, the transmissions to a base station and the transmissions from the base station occupy a same frequency band but are separated in time domain using a TDM mechanism such as time slot-based transmissions. Other types of multiplexing are also possible (e.g., code division multiplexing, orthogonal time frequency space, or OTFS, multiplexing, spatial multiplexing, etc.). In general, the various multiplexing schemes can be combined with each other. For example, in spatially multiplexed systems, transmissions to and from two different user devices may be isolated from each other using directional or orientational difference between the two end points (e.g., the user devices and a network station such as a base station).

[0086] 3. Introduction

[0087] This document describes different configurations and variations of Orthogonal Time Frequency and Space (OTFS), specifically based on the Zak transform (Zak-OTFS). Section 7 (Zak-OTFS) provides additional details of Zak-OTFS.

[0088] Referring to FIG. 1C, signal processing for OTFS is done in a two-dimensional space called delay-Doppler, which is defined by box of dimensions (periods) vpalong Doppler, and rpalong delay, always satisfying vp■ rp— 1. Along the delay period, we assign M grid points, spaced by the delay resolution, AT, and along Doppler we assign N grid points spaced by the Doppler resolution, Av.Typically, these resolutions satisfy Aτ ≥ 1 / BW and Av ≥ 1 / T, where BW is the bandwidth of the signal and T is its duration. Therefore, within the delay-Doppler box, we have an M x N grid, where resource elements (data symbols, pilots, zero-power guard, etc.) may be assigned. FIG. 1C illustrates the delay-Doppler grid, with all its parameters.8184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0089] In the delay-Doppler domain, this fundamental delay-Doppler box is infinitely extended in a quasi-periodic manner, which means that it is periodic along the Doppler dimension and periodic with an additional phase, applied to all its resource elements, along the delay dimension, as illustrated in FIG. 2.

[0090] Defining the delay-Doppler grid resource elements by the discrete variable x[m, n], where m = 0,1,..., M — 1 and N = 0,1,..., N — 1, the equivalent continuous delay-Doppler signal isM-l N-lx(T.v) = £ £m=0 n=0where Sqp(r, v) is the quasi-periodic extension of a delay-Doppler delta function defined asto otherwise

[0091] FIG. 2 illustrates the concept of delay-Doppler quasi-periodicity. The phase within each box,ej2nvkTPjs app| jet[to au qieelements within the box and it is a function of the Doppler shift of each element (v = ndv) and the box index along delay (k).

[0092] The OTFS modulated signal in delay-Doppler is defined ass(τ, ν) = ρtx*σx(τ,ν)where ptxis a two-dimensional pulse (filter) defined in the delay-Doppler domain, and *„ is the twisted-convolution operation, defined asa b(r, v) = $ ej2πν′(τ−τ′)a(j',v'}b(j — r',v — v'^dr'dv'

[0093] The equivalent OTFS modulated signal in the time domain, is computed using the inverse Zak transform∫νs(t) = z−1{S(τ, ν)} ≜ ∫S(τ, ν)dν2o

[0094] A wireless channel, h, can be defined as a super-position of Nhreflections, each associated with a complex gain gi, a Doppler shift νiand a time shift τi, where i = 1,2,..., Nhh(τ,ν) = Σ gi· δqp(τ − τi, ν − νi)i=l

[0095] The receive OTFS signal in the time domain, after the channel interaction isyh(t) = Σgi· ej2πν(t−τ)· s(t − τi)1=1

[0096] To get its equivalent delay-Doppler signal, we apply the Zak transform9184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00yh(τ, ν) = z{yh(t)} ≜ (1 / N) Σ e−j2πνkτyh(t − τ − kτp)fc ——00followed by a two-dimensional filter (pulse), prxy( ) = prx*ayh^,v')

[0097] The channel interaction, can be derived directly in the delay-Doppler domainy(τ,ν) = ρrx*σh(τ,ν) *σs(τ,ν)which yields the simple delay-Doppler channel equationy(τ,ν) = heff(τ,ν) *σx(τ,ν)whereheff(τ, ν) = ρrx*σh(τ,ν) *σρtxis the effective channel response.

[0098] For the discrete received delay-Doppler elements, we sample y(r, v) on the grid points y[m,n] = y(mAu,nAv)

[0099] 4. OTFS System examples

[0100] This section describes the building blocks of a system based on OTFS.

[0101] 4.1 OTFS Time-Frequency Physical Resource Block examples

[0102] An OTFS time-frequency Physical Resource Block (PRB) has a bandwidth of BWPRBand a duration of TPRB. In this current system specification, the PRB bandwidth is fixed to BWPRB= 168 kHz. The time dimension is divided into transmission slots of duration Tslot— 1 msec, and each PRB is transmitted within one slot (TPRB< Tslot). A PRB is defined by the triplet (nf, nT, nOffset)', where nfis the index along frequency, nTis the PRB duration index, and n0^setis the PRB time offset index within the slot.

[0103] For a carrier with a center frequency, fc, and a bandwidth BW. the frequency index nf- = 0,1,..., NPRB— 1, defines the PRB center frequency, relative to fcfoffset= (nf− (NPRB− 1) / 2) · fPRBwhere NPRB= ⌊BW / ΔfPRB⌋ and fPRB= 180 kHz.

[0104] The time slot of each PRB is further divided into 14 equal time slices and within the duration of the slot, the PRB is offset in time by Toffset= (noffset / 14) · Tslotand has a duration of TPRB= — 14 ■ Tsiot. Note, that noffset= 0,1,...,13 and nT= 1,2,...,14, and they must satisfy noffset+ nT≤ 14. An illustration example of different PRBs is given in FIG. 3.10184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0105] In some embodiments, each time-frequency PRB consists of a single OTFS delay-Doppler frame, as described in the next section. In some embodiments, a given OTFS frame may be mapped to multiple PRBs. FIG. 3 depicts a PRB allocation example in which there is a one-to-one mapping between PRBs and OTFS frames. In this example, BW = 1 MHz, which gives NPRB= 5. The PRBs are defined by the triplets (nf,nT,noffset): PRBUO = (3,14,0), PRB#1 = (2,3,0), PRB#2 = (2,5,6) and PRB#3 = (0,5,4).

[0106] 4.2 OTFS delay-Doppler Frame examples

[0107] The time-frequency PRB, with bandwidth BWPRBand duration TPRB, defines the OTFS delay- Doppler grid resolution1 1AT = — =,,, = 5.95 useeBWPRB168 kHz1 ( 14 14 14 )Av = - = 1, —, — —, 14 [kHzTPRB 1 13 12 2 J

[0108] The delay-Doppler grid has a total number of Resource Elements (RE)RE=B PRB ' TPRB=12 ' nT

[0109] and the delay-Doppler grid dimensions satisfyM ■ N = NRE

[0110] Note that there is complete flexibility in defining the OTFS grid dimensions per each PRB, while satisfying the above condition. This enables customization of the OTFS frame in each PRB to a specific receiving user and its channel conditions. Thus, for the same time-frequency PRB dimensions, the OTFS frame can be optimized for high mobility, or for static channel conditions. Also, by selecting the PRB duration within the slot, the Doppler resolution Av is controlled along with the delay-Doppler dimensions which allows further optimization of the OTFS configuration for the channel conditions of each specific user. An example of such mapping is shown in FIG. 4.

[0111] The two-dimensional delay-Doppler pulse may be defined by orthogonal pulses along delay and Doppler. For this, there are two configurations, which result in a slightly different implementation of the OTFS transmitter and receiver:Pulse Configuration #1: pw= pT*apvPulse Configuration #2: p(2>= pv*apT

[0112] In Pulse Configuration #1, we first apply the Doppler pulse, and then the delay pulse. In the new Pulse Configuration #2, we first apply the delay pulse and then the Doppler pulse. From a performance perspective, these two configurations are identical.11184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0113] In the current definition of the OTFS system specification, the following is being used:Pv= JTPRB■ sinc(y ■ TPRB)PT= ■ RRC^T ■ BWPRB)wheresin(7T%)sinc(x) = - 71Xsin(7rx(l — / ?)) + 4 / ?x COS(TTX(1 + / ?))RRCp(x) =7T%(1 — (4 / ?x)2)and (3T= 0.07. Note, that BWPRB■(1 + = fPRB. Also, it is assumed thatPTX Ptx

[0114] The OTFS delay-Doppler frame is generated by first assigning resource elements in the delay -Doppler grid, deriving X(T, V, and then applying to it the two-dimensional filter using a twisted-convolutionSframefa Ptx *o X(T, V^

[0115] Then, the OTFS frame is converted to the time domain using the inverse Zak transform SframeCO—% {•‘'frame Ctv)}and shifted in time and frequency to the PRB locationSPRBW = sframe(t - Toffset) ■e12^( / c+fo / / set)(t-To / / set)

[0116] The overall transmitted OTFS waveform in a slot, is a super-position of all the different timefrequency PRBsSslot(t) = SPRB i(t)i

[0117] A data transmission may consist of multiple PRBs, bounded together with the same FEC code, as illustrated in in FIG. 4 and FIG. 5. In some embodiments, parity bits may be computed over a single OTFS frame. Alternatively, parity bits of the FEC may be computed over information bits that are carried over multiple OTFS frames.

[0118] 4.3 OTFS Frames with Different Pulse Configurations

[0119] 4.3.1 Generating the M / V-Periodic time vector

[0120] This step is common to all different OTFS frame types, described below.12184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0121] The discrete delay-Doppler grid, defined by x[m, n], with m = 0,1,..., M — 1 and n =0,1,..., N - 1, is transformed to a discrete delay-Time variable, xDT[m, n], by applying an inverse Fourier transform to the Doppler dimension:N-lxDT[m, n] = —= x[m, k\ej2Tink^Ny / N k=0

[0122] This is equivalent to applying an IFFT operation to each row of x. The MlV-periodic vector is generated from infinite repetitions of column- wise vectorization of xDTxT= vec(xDT)X.'PXMN- periodic xTXT

[0123] 4.3.2 OTFS with Pulse Configuration #1

[0124] In this configuration, the Doppler pulse is first applied and then the delay pulse. The Doppler pulse may be applied as a window in the time domain, computed from the inverse Fourier transform of the Doppler pulse

[0125] The time window discrete coefficients are multiplied element-by-element with the samples ofXMN- periodic, generating a vector, XWT, which will typically have (1 + / ?v) ■ MN samples with significant energy, where / ?vis the roll-off factor of the Doppler filter.

[0126] In the current definition of the OTFS system specification, / ?v= 0, as pvis a sine function, and WTis a rectangular window of size MN and amplitude one, covering exactly one period in xMN-periodic. The outcome of applying the Doppler pulse, or time window, is simply x^ = xT. The time window for the general case (602) and for a rectangular window (604) are illustrated in FIG. 6.

[0127] The discrete baseband signal XWTis critically sampled with a baseband sampling rate_ MNfs,bb 771PRB

[0128] To enable multiplexing of multiple PRBs in time and frequency, the baseband OTFS frame must be upconverted to the carrier sampling rate, fs. The up-conversion sampling rate ratio is defined by P fsQ fs,bb13184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0129] The delay pulse, which is applied next, is utilized to perform both filtering and up-conversion operations. Let, pT, be the discrete delay filter derived from sampling the continuous delay filter, p at a sampling rate of Q ■ fs.

[0130] Applying the delay filter consists of the following steps:1. Up-sampling: zero-padding, XWT, by inserting P — 1 zeros, after each vector elementx^O], 0, -, >xwT+ &) ■ MN - l], 0,...,0,p-i P-I p-i2. Filtering: convolving the discrete filter it with XWTx = PT* XWT3. Down- sampling to fs: decimating the result by a factor of Q4. x[z] = x[Qz]

[0131] The outcome of the delay filter is the OTFS frame, sampled at fsand ready to be time and frequency shifted to the PRB location. FIG. 7 illustrates the process of generating the OTFS waveform for pulse configuration #1. In the figures, the up arrow and down arrow indicate upsampling and downsampling operations.

[0132] 4.3.3 OTFS with Pulse Configuration #2

[0133] In this configuration, the delay pulse is first applied and then the Doppler pulse. Following the notation of the previous section, we apply the discrete delay filter, pT, sampled at Q ■ fsto xMN-periodicusing the following steps:1. Up-sampling: zero-padding, xMN-periodic, by inserting P — 1 zeros, after each vector elementXMN— periodic ■■■ ixMN-periodic [b], 2. 0 ixMN-periodic [1]» ■■■,0 ■■■ >XMN— periodic [MN 1], 0,...,0,...P-1 p-i P-1 2. Filtering: convolving the discrete filter it with xMN-periodicD P * -MN-periodic3. Down-sampling to fs: decimating the result by a factor of Q1. XD[Z] = XD[QZ]

[0134] Next, the time window is appliedx = WTQxDwhere © denotes the element-by-element multiplication. Note, that the time window, WT, now covers p(1 -I- / ?v) ■ MN ■ - samples, with significant energy.14184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0135] In the current definition of the OTFS system specification, whereis a rectangular window, pMN ■ - samples, corresponding to one M / V-period of the signal, are taken from the output of the delay filter. FIG. 8 illustrates the process of generating the OTFS waveform for pulse configuration #2.

[0136] 4.4 OTFS Frame Configurations for Different Scenarios

[0137] On top of the selected pulse configuration, OTFS can be configured for different use cases, depending on the channel conditions and required complexity.

[0138] 4.4.1 Features of Standard OTFS

[0139] The standard OTFS frame is typically suitable for doubly spread channels and / or users with high mobility. It is also more efficient for a low number of spatial layers, due to the pilot structure. Illustrations of the waveform generation for standard OTFS are shown in FIG. 7 and FIG. 8 for the two pulse configurations. In this Standard OTFS configuration, entire rows (Doppler dimension) may be allocated for the use of channel estimation. One row per spatial layer consists of a pilot symbol and other rows may be left empty (zero-power) acting as guard bands for the spread of the effective channel response. FIG. 9 shows examples for possible pilot configurations for this mode for one and two pilots (one and two spatial layers). Note that the pilots’ power may be boosted, utilizing the non-allocated power at the guard area. The location of pilots within the delay-Doppler grid may vary from one PRB to another, to reduce the overall PAPR.

[0140] In FIG. 9, the shaded gray areas 902 represent data, the white areas (904) represent guard (zeropower) and the squares 906, 908 represent pilot symbols for different spatial layers. (910) is an example for a pilot configuration for a single spatial layer. The power of pilot can be boosted to include the guard area, P1= 3N. (920) is an example for a pilot configuration for two spatial layers. The pilot power here is boosted to P12= V5N.

[0141] Due to the pilot structure, it is assumed for this configuration that N, M > 1. Other than that, any setting, which satisfies M ■ N = NRE, and supports the desired pilot structure is valid.

[0142] The choice of the PRB duration, TPRB, which defines the Doppler resolution (Av = 1 / TPRB), and the delay and Doppler dimensions, provides flexibility in accommodating different delay and Doppler spreads in various effective channel conditions.

[0143] 4.4.2 Minimum-Period (MP) OTFS on Doppler

[0144] OTFS can be configured to have a minimum delay period, by setting the delay period to be equal to the delay resolution,= Ar, which results in M = 1 and N = NRE. The entire OTFS frame is assigned to the Doppler dimension (one row). In the current implementation of the specification, this configuration may be more suitable for low-mobility users, as the Doppler resolution ranges from 1 —15184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO0014 kHz, depending on the time allocation, TPRB. It may also be more suitable for transmissions with a higher number of spatial layers, as pilot packing efficiency may be better than standard OTFS.Illustrations of the waveform generation for this case are given in FIG. 10 and FIG. 11.

[0145] FIG. 10 shows a waveform generation example for MP-OTFS on Doppler, using pulse configuration #1 and a general time window.

[0146] FIG. 11 shows a waveform generation example for MP-OTFS on Doppler, using pulse configuration #2 and a rectangular time window.

[0147] To enable estimation of the effective channel response at the receiver side, multiple pilot symbols may be allocated (evenly or non-evenly) along the Doppler dimension, as seen in FIG. 12. Typically, the power of the pilots for this mode is equal to the power of the data symbols, as boosting their power will reduce the power of the data symbols. It is also possible to apply techniques such as cover codes to these pilots, to reduce their received noise.

[0148] Note that MP-OTFS on Doppler has similarities to OFDM modulation, as it requires a single IFFT operation of size N, to generate xT. However, unlike OFDM, the FFT size is related to the PRB size and not to the entire carrier bandwidth. After the IFFT, the two-dimensional pulse may be applied in any order (Pulse Configuration #1 or #2). A receiver for this configuration is discussed in a later section of this document.

[0149] FIG. 12 shows pilot allocation examples for MP-OTFS on Doppler. Data symbols are represented in longer rectangular swaths (shaded gray), zero-power in empty symbols, and pilots in shaded black and pattern- shaded symbols. In 1202, the pilots of a single spatial layer are allocated evenly along the Doppler dimension. In 1204, the pilots of a single spatial layer are allocated unevenly along the Doppler dimension. In 1206, the pilots of two spatial layers are allocated orthogonally along the Doppler dimension

[0150] 4.4.3 Minimum-Period (MP) OTFS on Delay

[0151] OTFS can be configured to have a minimum Doppler period, by setting the Doppler period to be equal to the Doppler resolution, vp= Av, which results in N = 1 and M = NRE. The entire OTFS frame is assigned to the delay dimension (one column). In the current implementation of the specification, this configuration may be more suitable for low-mobility users, as the Doppler ranges from 1 — 14 kHz, depending on the time allocation, TPRB. This configuration may also be more suitable for transmissions with a higher number of spatial layers, as pilot packing efficiency may be better than standard OTFS. An illustration of the waveform generation for this case is given in FIG. 13.

[0152] For low mobility user, to enable channel estimation at the receiver, one pilot symbol with optional guards before and after, is assigned to each spatial layer, as shown in FIG. 1416184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0153] To enable channel estimation at the receiver, one pilot symbol with optional guards before and after, is assigned to each spatial layer, as shown in FIG. 14. If guard symbols are used, the power of the pilot can be boosted to include them.

[0154] FIG. 14 shows pilot allocation examples for MP-OTFS on delay. Data symbols are represented in gray, zero-power guard in white, and pilots in black and pattern-shaded. In 1402, a single pilot for a single spatial layer is allocated along with a guard symbol before and after it. In 1404, two orthogonal pilots for two spatial layers are allocated. The power of the pilots may be boosted to include the guard area, resulting in(a) P±= x / 3 and (b) Pi 2— V5.

[0155] For users with higher mobility, more pilots can be allocated along the delay dimension, as seen in FIG. 15.

[0156] Note that this configuration has similarities to Single Carrier modulation, and it does not require any IFFT operations, as xT= x. MP-OTFS on delay, has low PAPR, and may be suitable for power-constrained transmissions. To further reduce the PAPR of multiple PRBs, it is recommended to stagger the pilot locations of the different PRBs, as seen in FIG. 16, which shows pilot location staggering example between multiple PRBs, to reduce the overall PAPR.

[0157] A receiver for this configuration is discussed in a later section of this document.

[0158] 4.4.4 Faster- Than-Nyquist OTFS

[0159] When one of the orthogonal pulses, defining the OTFS two-dimensional filter, has a roll-off greater than zero, it is possible utilize it to gain more capacity by applying Faster-Than-Nyquist (FTN) signaling.

[0160] In the current system specification, we can only apply this method to the delay filter having / ?T> 0, because the Doppler filter has a zero roll-off factor. In FTN-OTFS, it is also recommended to use it with Pulse Configuration #2, where the delay filter is applied before the Doppler filter.

[0161] Considering this case, of a delay filter having, / ?T> 0, we can increase the capacity of the OTFS signal if we increase the sampling rate of the baseband signal going into the delay filter by a factor of at least (1 + / ?Tand allocate more power to the roll-off parts of the spectrum. Error! Reference source not found. 17 illustrates this concept. Increasing the sampling rate of the input to the filter is equivalent to shortening the delay resolution to AT < 1 / BW, which leads to Inter-Sample-Interference (ISI). To overcome this, precoding is applied at both transmitter and receiver. Note, that although the sampling rate at the input to the filter is increased, the bandwidth of the signal remains the same.17184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0162] Starting from standard OTFS or MP-OTFS, with delay-Doppler grid of dimensions M x IV, we set AT' = a AT, where a > *, satisfying 0 < a < 1. Keeping the delay dimension the same, M' — M, the delay period is also shortened by a factor of aT'P= M!AT!= aM AT = aTpwhile the Doppler period is extended by a factor of 1 / ar1 1 1V p r ^PHTPaTp aHwhich also yields an extended Doppler dimension, v' 1 vv1A = —, = — - = -NAv a Av a

[0163] The new FTN-OTFS has a grid with dimensions M x N' and more resource elements1N RE — M ■ N — ~NREand it can deliver more information in the same bandwidth and duration. However, the processing must also take care of the transmitter introduced 1ST An illustration of FTN-MP-OTFS on Doppler compared with MP-OTFS on Doppler is shown in FIG. 18.

[0164] FIG. 17 shows sampling theory for different cases. In (1702) a bandlimited signal with zero rolloff is sampled exactly at fo, which is the Nyquist rate, and the replicas of the signal are orthogonal. In (1704) a bandlimited signal with a non- zero roll-off is sampled at the Nyquist rate having some aliasing at the edges of the band. In (1706) a bandlimited signal is sampled at a faster than Nyquist rate and the replicas of the signal are orthogonal.

[0165] FIG. 18 shows an FTN-OTFS delay-Doppler grid example. The narrower box (with dimension N) represents MP-OTFS on Doppler without FTN and the wider box (with dimension N') represents MP-OTFS on Doppler with FTN

[0166] Following similar processing described in Section 8 for a Single Carrier waveform, the ISI function of the delay filter is defined asgW) = J pT(t')pT(t' - t)dt'and the equivalent 1S1 vector, g, has elements defined bygi = g t - aAv ■ i)for i = 0,1,..., N'RE— 1. Let, G be a Toeplitz matrix generated from the vector g18184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO009o 9i 9N'RE-I9N'RE-I 9ORE-29I 92 9o.

[0167] An Eigenvalue Decomposition is applied to factorize the matrix GG = VAVHwhere, V is a unitary matrix and A is a diagonal matrix, with elements A = [A0, A1,, AN' / ?£._1]7’ in a descending order. Note, that V and A are a function of the delay pulse and the factor a, which are constant per configuration and can be pre-computed.

[0168] To generate the FTN-OTFS signal, we start with the discrete delay-Doppler grid, defined by x[m, n], with m = 0,1,..., M — 1 and n = 0,1,..., N' — 1, and transform it to a discrete delay-Time variable, xDT[m, n], by applying an inverse Fourier transform to the Doppler dimensionNr-lxDT[m, n] = —= y x[m, k]ej27mk / N'9N' Z_ik=0

[0169] This is equivalent to applying an IFFT operation to each row of x. Next, we vectorize xDTcolumn-wisexT= vec(xDT)and power scale the result using the diagonal elements of A

[0170] The result is precoded by VYFTN _ —VV. XT.and then infinitely periodizedAV,^FTNVFTNX-MN'-periodicAV,^FTN

[0171] The same delay filter, pT, is used for FTN-OTFS, as in Standard / MP-OTFS. However, it has a higher input rate. To accommodate that, the new up-conversion rate is computed by selecting an up-sampling ratio P' > aP and keeping the down-sampling ratio the same, Q' = Q.

[0172] Applying the delay filter consists of the following steps:1. Upsampling: zero-padding, xMNr-periodic, by inserting P' — 1 zeros, after each vector element19184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00■X-MNi-periodic ■■■, XMNr periodic [0], 0,...,0, Xy*,! r-periodic [1]< ■■■,0 > ■■■ > %MN— periodic [MN Pi-1 Pi-1 1], 0,...Pr-l 2. Filtering: convolving the discrete filter it with xMNr-periodicXQ P * Xf^ N-periodic3. Down-sampling to fsdecimating the result by a factor of Q2. xD[z] = xD[Qz]

[0173] Next, the time window is appliedx = WTQXDp

[0174] When WTis a rectangular window, MN ■ - samples, are taken from the filter’s output, which correspond to one period of the signal.

[0175] As stated earlier, FTN-OTFS can be applied to standard OTFS or MP-OTFS and the pilot schemes for these configurations may be the same. An illustration of the waveform generation for MP-OTFS on Doppler is given in FIG. 19.

[0176] FIG. 19 shows waveform generation example for FTN-OTFS using MP-OTFS on Doppler, pulse configuration #2 and a rectangular time window

[0177] The receiver processing for FTN-OTFS is described in another section in this document.

[0178] 5. Receiver-side processing examples

[0179] At a receiver, the received transmission waveform is processed to recover information bits for the user device that includes the receiver.

[0180] 5.1 Receiver side Processing for Different Pulse Configurations

[0181] The receiver side consists of reciprocal operations to the transmit side, which takes a time domain signal and computes its delay-Doppler domain representation.

[0182] 5.1.1 Pulse Configuration #1

[0183] In this configuration, the delay filter is applied first to the received signal and then the Doppler pulse as a time window. Note, that for a real and symmetric filter, pxx= px, and WTis also the same on both transmit and receive sides. Before the filter, the incoming samples at a sampling rate, fs, are up sampled by a factor Q, by inserting Q — 1 zeros between every two samples. After the convolution with the receive delay filter, the output is decimated by a factor of P, resulting in a baseband sample rate,Next, the time receive window, WTis applied resulting in a vector, yWr, with (1 + / ?v) ■ MN samples. If?v> 0, the edges are folded and added to the center, as also illustrated in FIG. 20.20184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00fywylX? + i + MN] 0 < i < Np ywT,foided[i] = }Wr[N / ? + i] + < o Np< i < MN - Np[Np+ i - MN] MN - Nfl< i < MN

[0184] where Np = (1 + / ?v) / 2 samples. When / ?v= 0, no folding is required and ywT, folded = ywT-The MN samples of ywT, folded are then rearranged in an M x N grid column- wise and then FFT is applied to each row. The output is y [m, n], which is the discrete received delay-Doppler domain.

[0185] FIG. 20 shows receiving process example for OTFS with pulse configuration #1.

[0186] 5.1.2 Pulse Configuration #2

[0187] In this configuration, the time window is applied first to the received signal and then the delay pfilter. Applying the time window, results in an output vector of (1 +v)MN- samples with significant energy. This vector is periodically extended, upsampled by a factor of Q, filtered by the delay pulse and then decimated by a factor of P to a baseband sampling rate. From output of the filter, MN samples are taken and rearranged in an M x N grid column-wise and then FFT is applied to each row. The output is y[m, n], which is the discrete received delay-Doppler domain. This process is illustrated in FIG. 21.

[0188] 5.2 Receiver side Processing for OTFS Frame Configurations

[0189] Receiver side processing for OTFS includes channel estimation and equalization of the received data symbols. From the equalized data symbols, log-likelihood ratios (LLR) can be derived for the FEC decoder to compute estimations of the transmitted information bits. Note that the LLRs from multiple PRBs can be concatenated or interleaved together to one FEC codeword.

[0190] 5.2.1 Standard OTFS

[0191] Methods and techniques for channel estimation and equalization are disclosed in Section 9.

[0192] 5.2.2 Minimum-Period (MP) OTFS on Doppler

[0193] Channel estimation for this configuration is based on interpolation of the estimated channel response on the pilot locations to the entire Doppler dimension. Note, that the channel response on the Doppler dimension is the Fourier transform of the effective channel response hD— T[hef-f.

[0194] For a single layer, letbe a set of pilot symbols transmitted on the Doppler indexes set {k}^. The estimated channel response at the pilot locations is h<ki=, and the entire channel response is computed from interpolatingto all N indexes, as shown in the example in FIG. 21.

[0195] With the interpolated channel response vector, h, a single tap equalizer may be applied to y to equalize it. For example, MMSE equalization21184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00hlXt= - — = - Vihi - h^ + N071where Nois the estimated noise variance and i — 0,1,..., N — 1.

[0196] When more spatial layers are transmitted, channel estimation can be easily extended to MIMO schemes, in a similar way to OFDM. If cover codes were applied at the transmitter between different pilots, they should be uncovered before channel interpolation.

[0197] FIG. 22 shows an interpolated channel response example. The circles represent the pilot locations, and the line is the interpolated channel response between the pilots in both amplitude and phase

[0198] 5.2.3 Minimum-Period (MP) OTFS on Delay

[0199] In this configuration, which has a single element along Doppler (IV = 1), the effective channel response excited from the pilot symbol, spreads along delay. For the current implementation of the specification the delay resolution is AT = 5.95 psec. For channels having most of their reflection’s energies on delays smaller than that, it may be sufficient to use the channel response measured on the single pilot and apply it to all data symbols. Let be the transmitted pilot symbol at delay index p and h = yp■ <£*. Then, the MMSE (maximum mean square error) estimated symbols areh*xt= - Vih - h* + N0

[0200] Note that this scheme has very low complexity (no FFT and no interpolation) and a single channel estimation is applied to all symbols. For more complicated channels, where the effective channel response spreads into the guard symbols before and after the pilot as well, more complicated equalization schemes may be used, such as the ones used in standard OTFS.

[0201] 5.2.4. Faster- Than- Nyquist OTFS

[0202] The receiver side operations for FTN-OTFS are reciprocal to the transmission. The processing is the same as non FTN-OTFS until the output of the delay filter. There, the signal is downsampled by a factor of P' and a vector, ypTN, with MN' samples is extracted. Next, a post-coder is applied to the received signalyT= VH■ yTFTNand power scaling is appliedVryLi] ■ yrtd22184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0203] Finally, the MN' samples of yT^ are taken and rearranged in an M x N' grid column-wise and an FFT is applied to each row. The output is y[m, n], the discrete received signal in the delay-Doppler domain. An example of this process is illustrated in FIG.23.

[0204] FIG. 23 shows receiver side processing example for FTN-OTFS with pulse configuration #2 and a rectangular time window.

[0205] Once the delay-Doppler grid elements are computed, channel estimation and equalization techniques that are used for non FTN-OTFS can be applied.

[0206] 6. Examples of Delay-Doppler Configurations for Different PRB Durations

[0207] Table 1 lists the valid configurations for the delay-Doppler dimensions fitting into one timefrequency PRB. To reduce the number of configurations and to simplify the FFT implementations, the Doppler dimension, N, is kept even, except for the case N = 1, where MP-OTFS is used.

[0208] One advantageous aspect of the below-mentioned numerology is that, from a higher level perspective, the available resources may look compatible or similar to existing wireless protocols such as the 5G New Radio (NR) protocol.Table 1 Valid delay-Doppler configurations for one time- frequency PRB as a function of nTnTNRE Standard OTFS (M, N) MP-OTFS (M, N) 1 12 (2,6), (3,4), (6,2) (1,12), (12,1) 2 24 (2,12), (3,8), (4,6) (6,4), (12,2) (1,24), (24,1) 3 36 (2,18), (3,12), (6,6), (9,4), (18,2) (1,36), (36,1) 4 48 (2, 24), (3,16), (4,12), (6,8), (8,6), (12,4), (24,2) (1,48), (48,1) 5 60 (2,30), (3,20), (5,12), (6,10), (10,6), (15,4), (30,2) (1,60), (60,1) 6 72 (2,36), (3,24), (4,18), (6,12), (9,8), (12,6), (18,4), (36,2) (1,72), (72,1) 7 84 (2,42), (3,28), (6,14), (7,12), (14,6), (21,4), (42,2) (1,84), (84,1) 8 96 (2,48), (3,32), (4,24), (6,16), (8,12), (12,8), (16,6), (24,4), (48,2) (1,96), (96,1) 9 108 (2,54), (3,36), (6,18), (9,12), (18,6), (27,4), (54,2) (1,108), (108,1) 10 120 (2,60), (3,40), (4,30), (5,24), (6,20), (10,12), (12,10), (15,8), (20,6), (30,4), (60,2) (1,120), (120,1) 11 132 (2,66), (3,44), (6,22), (22,6), (11,12), (33,4), (66,2) (1,132), (132,1) 12 144 (2,72), (3,48), (4,36), (6,24), (8,18), (9,16), (12,12), (18,8), (24,6), (36,4), (72,2) (1,144), (144,1) 13 156 (2,78), (3,52), (6,26), (13,12), (26,6), (39,4), (78,2) (1,156), (156,1)14 168 (2,84), (3,56), (4,42), (6,28), (7,24), (12,14), (14,12), (21,8), (28,6), (42,4), (84,2) (1,168), (168,1)

[0209] 7. Zak-OTFS

[0210] Signal transmissions in a wireless network may be represented by describing the waveforms in the time domain, in the frequency domain, or in the delay-Doppler domain (e.g., Zak domain). Because these three represent three different ways of describing the signals, signal in one domain can be converted into signal in the other domain via a transform. For example, a time-Zak transform may be used to convert from Zak domain to time domain. For example, a frequency-Zak transform may be used to23184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00convert from the Zak domain to the frequency domain. For example, the Fourier transform (or its inverse) may be used to convert between the time and frequency domains.

[0211] 7.0 Introduction to OTFS modulation from Zak theoretic point

[0212] Next few sections explain the OTFS modulation from the Zak theoretic point of view. This line of exposition push to the forefront the independent status of OTFS as a novel modulation technique and reveals its unique mathematical attributes. This, in contrast to the alternative approach of presenting OTFS as a preprocessing step over multi-carrier (MC) modulation which somehow obscures the true nature of OTFS and also sacrifice some of its unique strengths. We focus our attention on the following core theoretical topics:

[0213] (1) Heisenberg theory.

[0214] (2) Zak theory.

[0215] (3) OTFS modulation.

[0216] (4) Symplectic Fourier duality relation between OTFS and Multi Carrier modulations which is a particular case of the general relation between Radar theory and communication theory.

[0217] Before proceeding into a detailed development, it is beneficial to give a brief outline. In signal processing, it is traditional to represent signals (or waveforms) either in time or in the frequency domain. Each representation reveals different attributes of the signal. The dictionary between these two realizations is the Fourier transform:(0.1) FT: L2( / e x) ^ L2(f e x).

[0218] Interestingly, there is another domain where signals can be naturally realized. This domain is called the delay Doppler domain. For the purpose of the present discussion, this is also referred to as the Zak domain. In its simplest form, a Zak signal is a function <p(T, v) of two variables. The variable T is called delay and the variable V is called Doppler. The function (p r,v is assumed to be periodic along V with period Vrand quasi-periodic along f with period Tr. The quasi periodicity condition is given by:(0.2) <p(T+nTr,v + mvr) = exp(j27inv-Tr)<p(T,v),

[0219] for every n,m E TL. The periods are assumed to satisfy the Nyquist condition Tr• Vr= 1. Zak domain signals are related to time and frequency domain signals through canonical transforms Z24184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00and Zicalled the time and frequency Zak transforms. In more precise terms, denoting the Hilbert space of Zak signals by 7Z, the time and frequency Zak transforms are linear transformations:(0.3) 3t: F; A L,( / G X).(0.4) 2f:

[0220] The pair Zxand Ziestablishes a factorization of the Fourier transform FT = Zx° Z{]1. This factorization is sometimes referred to as the Zak factorization. The Zak factorization embodies the combinatorics of the fast Fourier transform algorithm. The precise formulas for the Zak transforms will be given in the sequel. At this point it is enough to say that they are principally geometric projections: the time Zak transform is i ntegration along the Doppler variable and reciprocally the frequency Zak transform is integration along the delay variable. The different signal domains and the transformations connecting between them are depicted in FIG. 26.

[0221] We next proceed to give the outline of the OTFS modulation. The key thing to note is that the Zak transform plays for OTFS the same role the Fourier transform plays for OFDM. More specifically, in OTFS, the information bits are encoded on the delay Doppler domain as a Zak signal %(T, v) and transmitted through the rale:(0.5) OTFS(x) = Zt(w*, X(T, V)),

[0222] where W *oX(T, V) stands for two-dimensional fdtering operation with a 2D pulse W(T, V) using an operation *acalled twisted convolution (to be explained in the present document). The conversion to the physical time domain is done using the Zak transform. Formula (0.5) should be contrasted with the analogue formulas in case of frequency division multiple access FDMA and time division multiple access TDMA. In FDMA, the information bits are encoded on the frequency domain as a signal x(f) and transmitted through the rule:(0.6) FDMA(x) = FT(w(f ) * x(f )),

[0223] where the filtering is done on the frequency domain by linear convolution with a ID pulse w(f) (in case of standard OFDM w(f) is equal an sine function ). The modulation mapping is the Fourier transform. In TDMA, the information bits are encoded on the time domain as a signal x(t) and transmitted through the rale:25184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00(0.7) TDMA(x) = Id(w(r) * %(?)),

[0224] where the filtering is done on the time domain by linear convolution with a ID pulse w(i). The modulation mapping in this case is identity.

[0225] 7.1 Heisenberg Theory

[0226] In this section we introduce the Heisenberg group and the associated Heisenberg representation. These constitute the fundamental structures of signal processing. In a nutshell, signal processing can be cast as the study of various realizations of signals under Heisenberg operations of delay and phase modulation.

[0227] 7.1.1 The Delay Doppler plane.

[0228] The most fundamental structure is the delay Doppler plane V = R2equipped with the standard symplectic form:(1-1)

[0229] for every Vj = (T), Vl) and V2— (f2, V2). Another way to express co is to arrange the vectors Vj and v2asthe columns of a 2 x 2 matrix so that CO ( Vj, V2) is equal the additive inverse of the matrix determinant.ty(vpv2) = -det t)V2

[0230] The symplectic form is anti-symmetric (O (, v2) =— co( v2, Vj ), thus, in particular v) = 0 for every v e V. We also consider the polarization form:(1.2) ^(V1A2) =V1T2,

[0231] for every Vj = ) and t’2= (f2, V2). We have that:(1.3) ft (vpv2) - P (v2,v1) = CO (vpv2),

[0232] The form / ? should be thought of as "half" of the symplectic form. Finally, we denote by { / (z) = exp(2> Tzz) is the standard one-dimensional Fourier exponent.

[0233] 7.1.2 The Heisenberg group.26184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0234] The polarization form 0 gives rise to a two step unipotent group called the Heisenberg group. As a set, the Heisenberg group is realized as Heis = V X S1where the multiplication rule is given by:(1.4) (v1,z1)-(v2,z2) = (v1+v2,exp(j2^?(v1,v2))z1z2),

[0235] One can verify that indeed rule (1.4) yields a group structure: it is associative, the element (0, 1) acts as unit and the inverse of the element ( V, ) is given by:(v,z)-1=(-v,exp(j2^(v,v))1)

[0236] Most importantly, the Heisenberg group is not commutative. In general,( t’|,7| )( V’2,"2) A ( V’2, 42) • ( V’|, Z| ). The center consists of all elements of the form (0,z), ZG S^. The multiplication rule gives rise to a group convolution operation between functions:(1.5) MA(V) = j exp(j2^ / ?(v1,v2)) / 21(v1) / f2(v2)= Jexp ( / 2zz)Z? ( t’\ v — v,z) ) / q (v,) / t2(v- v')dv',v'

[0237] for every pair of functions / l,G C(V). We refer to (1.5) as Heisenberg convolution or twisted convolution. We note that a twisted convolution differs from linear convolution through the additional phase factor exp( (vpv2)) •

[0238] 7.1.3 The Heisenberg representation

[0239] The representation theory of the Heisenberg group is relatively simple. In a nutshell, fixing the action of the center, there is a unique (up-to isomorphism) irreducible representation. This uniqueness is referred to as the Stone-von Neumann property. The precise statement is summarized in the following theorem:

[0240] Theorem 1.1 (Stone-von-Neumann Theorem). There is a unique (up to isomorphism) irreducible Unitary representation 71: Heis U (7Y) such that f(0,z) = zldw.

[0241] In concrete terms, the Heisenberg representation is a collection of unitary operators ^■(V)G U (7Y), for every VG V satisfying the multiplicativity relation:(1.6) 'T'(t'i)0^(v2) = exp( J2^(V1, V2))^‘(V1+V2),27184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0242] for every tq, v2& V. In other words, the relations between the various operators in the family are encoded in the structure of the Heisenberg group. An equivalent way to view the Heisenberg representation is as a linear transform II: C ( V ) — > Op ( 7 ), taking a function h € C ( V ) and sending it to the operator n( / t)e Op(H) given by:(1.7) n(7z) = j h(v)dv,

[0243] The multiplicativity relation (1.6) translates to the fact that II interchanges between Heisenberg convolution of functions and composition of linear transformations, i.e.,d-8) n( ) =n(11) on( / i2),

[0244] for everyC (V). Interestingly, the representation it, although is unique, admits multitude of realizations. Particularly well known are the time and frequency realizations, both defined on the Hilbert space of complex valued functions on the real line TL = (R). For every xe R, we define two basic unitary transforms:(1.9) Lx(<p}(y) =H.10) Mx(<p)(y) = exp(j27rxy)(p(y),

[0245] for every (f)G 7Y. The transform Lxis called delay by X and the transform Mxis called modulation by X. Given a point V = (f, v) £ V we define the time realization of the Heisenberg representation by:(l. H) ^t(v) >^ = 4 «Mv(^),

[0246] where we use the notation t> to designate the application of an operator on a vector. It is usual in this context to denote the basic coordinate function by t (time). Under this convention, the righthand side of (1.11) takes the explicit form exp (j‘2#V (f - T)) (p{t -. Reciprocally, we define the frequency realization of the Heisenberg representation by:(1.12) (v) > (p = M_To L^tp),

[0247] In this context, it is accustom to denote the basic coordinate function by f (frequency). Under this convention, the right-hand side of (1.12) takes the explicit form exp (— j'2.7Tf )^(f — v). By28184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00Theorem 1.1, the time and frequency realizations are isomorphic in the sense that there is an intertwining transform translating between the time and frequency Heisenberg actions. The intertwining transform in this case is the Fourier transform:(1.13) FT(^)(f ) = j exp(-j2^ff)^(f)tZf,

[0248] for every (p& l. The time and frequency Heisenberg operators?Ft(v,z) and 71i(h,z) are interchanged via the Fourier transform in the sense that:(1.14) FTo^t(v) = ^f(v)oFT,

[0249] for every v G V. We stress that from the point of view of representation theory the characteristic property of the Fourier transform is the interchanging equation (1.14).

[0250] 7.2. Zak Theory

[0251] In this section we describe the Zak realization of the signal space. A Zak realization depends on a choice of a parameter. This parameter is a critically sampled lattice in the delay Doppler plane. Hence, first we devote some time to get some familiarity with the basic theory of lattices. For simplicity, we focus our attention on rectangular lattices.

[0252] 7.2.1 Delay Doppler Lattices.

[0253] A delay Doppler lathee is an integral span of a pair of linear independent vectors gj, g2G V. In more details, given such a pair, the associated lathee is the set:(2.1) A ~{algl+ a2g2: a{,a2Z},

[0254] The vectors gYand g2are called the lathee basis vectors. It is convenient to arrange the basis vectors as the first and second columns of a matrix G. i.e.,:(2-2) G =glg2,

[0255] referred to as the basis matrix. In this way the lathee A = G (Z2), that is, the image of the standard lattice under the matrix G. The volume of the lattice is by definition the area of the fundamental domain which is equal to the absolute value of the determinant of G. Every lattice admits a symplectic reciprocal lathee, aka orthogonal complement lathee that we denote by A1. The definition of A1is:29184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00(2.3) A1= {VG V: ty (V,2) G Z for everyA},

[0256] We say that A is under-sampled if A C A1. we say that A is critically sampled if A = A1. Alternatively, an under-sampled lattice is such that the volume of its fundamental domain is > 1. From this point on we consider only under-sampled lattices. Given a lattice A, we define its maximal rectangular sub-lattice as Ar= Zr,. © Xb. where:(2.4) Tr= arg min{'r > 0: (r,0)G A],(2.5) vr= argmin{v > 0: (0,v)G A],

[0257] When either Tror Vr, are infinite, we define Ar= {0}. We say a lathee A is rectangular if A = Ar. Evidently, a sub-lattice of a rectangular lathee is also rectangular. A rectangular lathee is under-sampled if TrVr> 1. The standard example of a critically sampled rectangular lathee is Arec= Z © Z, generated by the unit matrix:(2.6)

[0258] An important example of critically sampled lathee that is not rectangular is the hexagonal lathee Ahex, see FIG. 27, generated by the basis matrix:a a / 2

[0259] where a = 2 / V3 The interesting attribute of the hexagonal lathee is that among all critically sampled lathees it has the longest distance between neighboring points. The maximal rectangular sub-lathce of Ahexis generated by g and 2g2— g. see the two lathee points decorated with arrow heads in FIG. 27. From this point on we consider only rectangular lathees.

[0260] 7.2.2 Zak waveforms

[0261] A Zak realization is parametrized by a choice of a critically sampled lathee:(2.8) A = Z(-rr,0)©Z(0,vr),

[0262] where Tr• Vr= 1. The signals in a Zak realization are called Zak signals. Fixing the lattice A, a Zak signal is a function (p: V — > C that satisfies the following quasi periodicity condition:30184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00(2.9) $>(v + 2) = exp(j2^5(v,2))^(v),

[0263] for every ve V and G A. Writing λ = (kτr, lνr), condition (2.9) takes the concrete form:(2.10) <p(r+kTr, V + lvr) = exp(j27TVkTr)g)(T,v),

[0264] that is to say that <p is periodic function along the Doppler dimension with period Vrand quasi-periodic function along the delay dimension with quasi period Tr. In conclusion, we denote the Hilbert space of Zak signals by 7Z.

[0265] 7.2.3 Heisenberg action

[0266] The Hilbert space of Zak signals supports a realization of the Heisenberg representation. Given an element u G V, the corresponding Heisenberg operator 71z(l / ) is given by:(2.11) {^z(«) > ^}(v) = exp( jl7ifi(u,v-u}}(p(y - w),

[0267] for everyφ ∈ Hz. In words, the element u acts through two-dimensional shift in combination with modulation by a linear phase. The Heisenberg action simplifies in case the element u belongs to the lattice. A direct computation reveals that in this case the action of u - e A takes the form:(2.12) {TTZ(2) > ^}(v) = exp(j‘2^zp(2,v))^(v),

[0268] In words, the operator 7TZ( ) is multiplication with the symplectic Fourier exponent associated with the point A,. Consequently, the extended action of an impulse function h &C(V) is given by:(2.13) {nz(h) > ^}(v) = j h(u)[7Tz(M) > <p](y)duueV= j i / 7^j27iP(ii,v -u}^h{u)(p{y -u)du,ueV

[0269] for every φ ∈ Hz. In fact,ΠZ(h) ▷ φ = h *σφ, that is to say that the extended action is given by twisted convolution of the impulse h with the waveform (p.

[0270] 7.2.4 Zak transforms31184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0271] There are canonical intertwining transforms converting between Zak signals and time / frequency signals, referred to in the literature as the time / frequency Zak transforms. We denote them by:(2.14) Zt: Hz→ L2(t ∈ ℝ),(2.15) Zf:

[0272] As it turns out, the time / frequency Zak transforms are basically geometric projections along the reciprocal dimensions, see FIG. 28. The formulas of the transforms are as follows:(2.16) Zt(φ)(t) = ∫φ(t,ν)dν,* u(2.17)= * P 0 exp(-j2^ / r)^(r, / )JT,

[0273] for every φ ∈ Hz. In words, the time Zak transform is integration along the Doppler dimension (taking the DC component) for every point of time. Reciprocally, the frequency Zak transform is Fourier transform along the delay dimension. The formulas of the inverse transforms are as follows: (2.18) Z;\(p){T,v) = ^2 exp (-7'2^4 / 7^77)^(7+ 777,. ),neZ(2.19) z;\(p)(T,v) = ^&^[j27iT{yrn + v}}(p{y + nvr),

[0274] for every φ ∈ L2(ℝ). From this point on we will focus only on the time Zak transform and we will denote it by Z = Z. As an intertwining transform Z interchanges between the two Heisenberg operators,7z(v,z) and,7, (t',z). i.e..:(2.20) Z ° 7TZ( v) = 7Tt(v) ° Z,

[0275] for every v e V. From the point of view of representation theory the characteristic property of the Zak transform is the interchanging equation (2.20).

[0276] 7.2.5 Standard Zak signal

[0277] Our goal is to describe the Zak representation of the window function:1 Q<t< Tr(2.21) P(t)=0 otherwise32184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0278] This function is typically used as the generator waveform in multi-carrier modulations (without CP). A direct application of formula (2.18) reveals that P = Z-1( / ?) is given by:(2.22) P(T,v) = YJV{ynTr) p[T-nTr),neZ

[0279] One can show that P aTr,bVr= 1 for every a,b<^ ^0,1), which means that it is of constant modulo 1 with phase given by a regular step function along T with constant step given by the Doppler coordinate v. Note the discontinuity of P as it jumps in phase at every integer point along delay. This phase discontinuity is the Zak domain manifestation of the disconti nuity of the rectangular window p at the boundaries.

[0280] 7.3 OTFS

[0281] The OTFS transceiver structure depends on the choice of the following parameters: a critically sampled lattice Λ = Z(τr,0) ⊕ Z(0, νr), a filter function H'G C( V ) and an information grid specified by N, M e N. We assume that the filter function factorizes as W(T, V) = wT(T) WV(P) where the delay and Doppler factors are square root Nyquist with respect to AT = Tr / N andA V = Vr / M respectively. We encode the information bits as a periodic 2D sequence of QAM symbolsX = mA v] with periods (A, M). Multiplying x by the standard Zak signal P we obtain a Zak signal X P. A concrete way to think of X P is as the unique quasi periodic extension of the finite sequence x[nAr,mAv] where n = 0,and m = 0,... M-l. We define the modulated transmit waveform as:(3.1) Ad(x) =- Z(w*ax - P),

[0282] To summarize: the modulation rule proceeds in three steps. In the first step the information block x is quasi-periodized thus transformed into a discrete Zak signal. In the second step, the bandwidth and duration of the signal are shaped through a 2D filtering procedure defined by twisted convolution with the pulse w. In the third step, the filtered signal is transformed to the time domain through application of the Zak transform. To beter understand the structure of the transmit waveform we apply few simple algebraic manipulations to (3.1). First, we note that, being an intertwiner (Formula (2.20)), the Zak transform obeys the relation:33184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00(3.2) z(nz(W) *A%• P) = nt(w) > Z(x- P),

[0283] Second, we note that the factorization H’(T, v) = wr(T) WV(V) can be expressed as twisted convolution w = wT*o. wv. Hence, we can write:(3-3) nt(vv) > Z(x-P) = IIt(wr*awv) > Z(x P)=nt (Vt’r) >{ntH) > P)}= wr*{Wt-Z(JC-P)},

[0284] where Wt= FT-1(wν) and * stands for linear convolution in ti me. We refer to the waveform Z(x- P) as the bare OTFS waveform. We see from Formula (3.3) that the transmit waveform is obtained from the bare waveform through windowing in time followed by convolution with a pulse. This cascade of operations is the time representation of 2D filtering in the Zak domain. It is beneficial to study the structure of the bare OTFS waveform in the case X is supported on a single grid point (aka consists of a single QAM symbol), i.e., X = (rtAT,mAv). In this case, one can show that the bare waveform takes the form:(3.4) Z(x- P) = ^exp( + nj N} / M + nAr),K

[0285] In words, the bare waveform is a shifted and phase modulated infinite delta pulse train of pulse rate νr= τr-1where the shift is determined by the delay parameter n and the modulation is determined by the Doppler parameter m. Bare and filtered OTFS waveforms corresponding to a single QAM symbol are depicted in FIG. 29 and FIG. 30 respectively. We next proceed to describe the demodulation mapping. Given a received waveform <prx, its de-modulated image y — M is defined through the rule:(3-5) P(^rJ = vt.'* ^ Z-|(^rx).

[0286] where it* is the matched filter given by w* (v) = exp(— j2jl / 3(v, v)) w(— v). We often incorporate an additional step of sampling y at ( nA T, mAh’) for n = 0,..., N — 1 andm = 0,..., M-1.

[0287] 7.3.1 OTFS channel model34184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0288] The OTFS channel model is the explicit relation between the input variable x and the output variable y in the presence of a channel H. We assume the channel transformation is defined asH = IIt( / z) whereh = v) is the delay Doppler impulse response. This means that given a transmit waveform ^>|x, the received waveform (p^ — II {(p^ ^ is given by:(3.6) ^rx(() = J h(T,v)exp(j27rv(t-T))<ptx(t)dTdv,-r,v

[0289] If we take the transmit waveform to be (ptx= d (x) then direct computation reveals that:(3.7) y = M= w**f7z-1(nt( / t)>z(w*f7x-p))= vt* *o. h*a2Z oZ(w*ax-P)- w* h *aw x-P,

[0290] If we denote hw=> V* *o. h *aW then we can write the input-output relation in the form: (3.8) y = hw*ax-P,

[0291] The delay Doppler impulse hwrepresents the filtered channel that interacts with the QAM symbols when those are modulated and de-modulated through the OTFS transceiver cycle. One can show that under some mild assumptions hwis well approximated by h * w ' where * stands for linear convolution andw(2)= w★* w is the linear auto-correlation function. In case the channel is trivial, that is h = δ(0,0), we get that hw= w★*σw ~ w(2), thus after sampling we get (an approximate) perfect reconstruction relation:(3.9) y[nΔτ, mΔν] ~ x[nΔτ, mΔν],

[0292] for every n = 0,..., N — 1 and m = 0,..., M — 1.

[0293] 7.4. Symplectic Fourier Duality

[0294] In this section we describe a variant of the OTFS modulation that can be expressed by means of symplectic Fourier duality as a pre-processing step over critically sampled MC modulation. We refer to this variant as OTFS-MC. For the sake of concreteness, we develop explicit formulas only for the case of OFDM without a CP.

[0295] 7.4.1 Symplectic Fourier transform35184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0296] We denote by / .2 (V) the Hilbert space of square integrable functions on the vector space V. For every v G V we define the symplectic exponential (wave function) parametrized by v as the function: V -> C given by:(4.1) y / v(u) = exp( j'2^ty(v,w)),

[0297] for every u G V. Concretely, if V = (T, V) and It = (?', / ) thenl / fv(w) — exp( jl7r(vT — r / )). Using symplectic exponents we define the symplectic Fourier transform as the unitary transformation SF: Lj ( V) — > (V) given by:(4.2) SF(g)(v) = j? / v(v / )g (v / )dv'v'= jexp(-j2^®(v,v'

[0298] The symplectic Fourier transform satisfies various interesting properties (much in analogy with the standard Euclidean Fourier transform). The symplectic Fourier transform converts between linear convolution and multiplication of functions, that is:(4.3) SF(g1* g2) = SF(g1) · SF(g2),

[0299] for every g1, g2∈ L2(V). Given a lattice A c V, the symplectic Fourier transform maps sampled functions on A to periodic function with respect to the symplectic reciprocal lattice A. That is, if g is sampled and G = SF (g) then+ = G(v) for every l’G V and1G A1. This relation takes a simpler form in case A is critically sampled since A1= A. Finally, unlike its Euclidean counterpart, the symplectic Fourier transform is equal to its inverse, that is SF = SF1.

[0300] 7.4.2 OTFS-MC.

[0301] The main point of departure is the definition of the filtering pulse w and the way it applies to the QAM symbols. To define the MC filtering pulse we consider sampled window function W: A — C on the lattice A = Z Tr© Z Vr. We define w to be the symplectic Fourier dual to W:(4.4) VV = SF(W),

[0302] By definition, w is a periodic function on V satisfying w(v + λ) = w(v) for every V G V and G A. Typically, W is taken to be a square window with 0 / 1 values spanning over a certain36184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00bandwidth B = M - Vrand duration T = N • Tr. In such a case, w will turn to be a Dirichlet sine function that is Nyquist with respect to the grid Av M— TLAT © ZA V, where:(4.5) AT = Tr / N,(4.6) Av = Vr / M,

[0303] More sophisticated windows designs can include tapering along the boundaries and also include pseudo-random scrambling phase values. As before, the bits are encoded as a 2D periodic sequence of QAM symbols X = x[nA,mA v] with period (N, M). The transmit waveform is defined through the rule:(4.7) MMC(X) = Z((W*X)-P),

[0304] In words, the OTFS-MC modulation proceeds in three steps. First step, the periodic sequence is filtered by means of periodic convolution with the periodic pulse w. Second step, the filtered function is converted to a Zak signal by multiplication with the Zak signal P. Third step, the Zak signal is converted into the physical time domain by means of the Zak transform. We stress the differences from Formula (3.1) where the sequence is first multiplied by P and then filtered by twisted convolution with a non-periodic pulse. The point is that unlike (3.1), Formula (4.7) is related through symplectic Fourier duality to MC modulation. To see this, we first note that IV * X = SF( W • X ) where X = SF ( x). This means that we can write:(4.8) (w*x)-P = YWWX(X)^ -PZe A / IGA

[0305] where the first equality is by definition of the Symplectic Fourier transform and the second equality is by Formula (2.12). We denote Xw= W • X. Having established this relation we can develop (4.7) into the form:37184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00(4.9) ■^MC (X) zJx. WzfqiH)

[0306] where the third equality is the intertwining property of the Zak transform and the fourth equality is by definition= 3(P). In case of OFDM without CP, the pulse p is given by the square window along the interval [0, Tr]. Consequently, the last expression in (4.9) can be written explicitly as:k,l(4-W) = > l[0,rr]k,l= ^xw(kTr, lvr) exp ( jlrivr(t-kvr)) K(A+1)r,,k,l

[0307] The last expression of (4.10) can be recognized as MC modulation of the (windowed) sequence of Fourier coefficients Xw. It is interesting to compare the transmit waveforms of OTFS and OTFS-MC corresponding to single QAM symbols. The two structures are depicted in FIG. 31. The main structural difference is the presence of discontinuities at the grid points Zf,. in the case of OTFS-MC.

[0308] 7.5.0 Introduction to OTFS Transceiver Operations from Realization Theory Perspective

[0309] We introduce yet another mathematical interpretation of the OTFS transceiver from the point of view of realization theory. In a nutshell, in this approach one considers the signal space of waveforms as a representation space of the Heisenberg group or equivalently as a Hilbert space equipped with collection of Heisenberg operators, each associated with a different point in the delay Doppler plane. This representation space admits multitude of realizations. The two standard ones are the time and frequency realizations and they are related through the one-dimensional Fourier transform. In communication theory the TDMA transceiver structure is naturally adapted to the time realization as QAM symbols are multiplexed along the time coordinate while the OFDM transceiver structure is naturally adapted to the frequency realization as the QAM symbols are multiplexed along the frequency coordinate. The main observation is that, there is a canonical realization lying in between the time and38184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00frequency realizations, called the Zak realization. Interestingly, waveforms in Zak realization are represented as functions on a two-dimensional delay Doppler domain satisfying certain quasi-periodicity condition. The main message of this note is that the Zak realization is naturally adapted to the OTFS transceiver. Viewing the OTFS transceiver from this perspective extenuates its novel and independent standing among the other existing transceiver structures. For convenience, we summarize in the following table the main formulas presented in this note:QPp(v + A) = p( / 7(v, A))?re(A) ^ ^(v)Z-Heis(v0) > <p(v) = ] / (-^(vo> Vo ))^(^(vo^))^(v - Vo)Z-Heis (lattice) 7? (2,c(2)) > ^>(v) = ^(<p(2,v))^(v)Zak to time^time,c (^)(0 = \V' (p(t, V) dVtime to ZakZak to freqfreq to ZakA-Zak to Zak 2e, A<p) (r,v) = (p0(T,v)Zak to A-Zak2e',e W (w) = • i / N) ^(r + i / N,v)Z-std windowPstd (w) = Wr )

[0310] where the Q abbreviate Quasi and Z abbreviate Zak.

[0311] 7.6. The Zak realization

[0312] 7.6.1 Zak waveforms

[0313] See previous discussion in 7.2.2. In this section we describe a family of realizations of the Heisenberg representation that simultaneously combine attributes of both time and frequency. These are known in the literature as Zak type or lattice type realizations. A particular Zak realization is parametrized by a choice of an Heisenberg lattice (Λ, ε) where A is critically sampled. A Zak waveform is a function (p: V — > C that satisfies the following quasi periodicity condition:(2.1) ^(V + 2) = 6(2)] / ( / 7(V,2))^(V)39184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0314] There is an alternative formulation of condition (2.1) that is better suited when considering generalizations. The basic observation is that the map e defines a one dimensional representationπε: Λ × S1→ U(ℂ) satisfying the extra condition that πε(0, z) = z. This representation is given by πε(λ, z) = ε(λ)-1z. Indeed, verify that:(2.2)—(( ‘ (A’ ^2 ))’

[0315] In addition, we have that 7Te^A,e^A)) = 1 implying the relation Im e G ker 71. Hence πεis in fact a representation of the finite Heisenberg group Heis(Λ,ε) = Λ × S1 / Im ê. Using the representation 77e, we can express (2.1) in the form:(2.3) ^(v + / l) = ^( / 7(v,2))|^'e(2)~1> ^(v)},

[0316] We denote the Hilbert space of Zak waveforms byH(V, πε) or sometimes for short by.15 He. For example, in the rectangular situation where A = Arand e = 1, condition (2.1) takes the concrete form(p(r +ktr, V + / V;.) = y / (vkTr)^(T,v), thatis, (p is periodic function along the Doppler dimension (with period Vr) and quasi-periodic function along the delay dimension. Next, we describe the action of the Heisenberg group on the Hilbert space of Zak waveforms. Given a Zak waveform (p L(, and an element (U, Z)G Heis, the action of the element on the waveform is given by:(2.4) {7re(ii,z) > ^}(v) = z-y / {fi{it,v -u))(p{y -u),In addition, given a lattice point A e A, the action of the elementc ( A) = (zl, c ( ) ) takes the simple form:(2.5) {jTe(A,6(2)) > ^}(v) = e(A)ys( / 3(A,v-A))p(v-A)= €(A)€(-A)^(jS(A,-A))^(at(A,v))^(v)= yf(a)(A,v))(p(v],40184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0317] where in the first equality we use (2.4), in the second equality we use (2.1) and the polarization equation (1.3) and in the third equality we use (1.13). To conclude, we see that7le(e (A)) is given by multiplication with the symplectic Fourier exponent associated with the point. As usual, the representation gives rise to an extended action by functions on V. Given a function h G C ( V ), its action on a Zak waveform(2.6) = j / ?( / . / ) { / A (w) > p](v)duueV= j y / (j3(u,v -u)}h(u)(p(y -u)du,ueV

[0318] From the last expression we conclude that II ( ) (p = h ® (p, namely, the extended action is realized by twisted convolution of the impulse h with the waveform (p.

[0319] 7.6.2 Zak transforms

[0320] See also section 7.2.4. By Theorem 1.1, the Zak realization is isomorphic both to the time and frequency realizations. Hence there are intertwining transforms interchanging between the corresponding Heisenberg group actions. These intertwining transforms are usually referred to in the literature as the time / frequency Zak transforms and we denote them by:(2.7) 3time>e: He^?<ime= C(fG R),^2.8) ^freq.e: We^ Kq= C(f e R),

[0321] As it turns out, the time / frequency Zak transforms are basically geometric projections along the reciprocal dimensions, see FIG. 26. Formally, this assertion is true only when the maximal rectangular sublattice Az. = Z(rr,0)©Z(0, Vr) is non-trivial, i.e., when the rectangular parameters T,., Vr< °°. Assuming this condition holds, let N =r• V. denote the index of the rectangular sublattice Arwith respect to the full lattice A, i.e., N = [ Az.: A]. For example, when A = Arecwe have Tr= Vr= 1 and N = 1. When A = Ahex, we have Tr= a and Vr= 2 / a and consequently N = 2. Without loss of generality, we assume that c|Az. = 1.

[0322] Granting this assumption, we have the following formulas:(2-9) ^time. W M =* u41184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00(2.10)

[0323] We now proceed describe the intertwining transforms in the opposite direction, which we denote by:(2- ID 2tnnc:(2-12) -2), freq: ^4eq ^ ^c >

[0324] To describe these we need to introduce some terminology. Letand Z?treqdenote the time and frequency realizations of the Heisenberg representation of the group Heis (A^.,1), / / rcqe CNare the unique (up to multiplication by scalar) invariant vectors under the action of Aethrough and 7^ respectively. The formulas of (2.11) and (2.12) are:v-i (2-13) ^.tirne (^)(M = E Ebtlme+ >1)) + Tr(k / N + 71)),k=0 weZ(2 14) Ze.freq(^)(f,v) = ^MEE^ [k] p(rvr(k / N + n)) p(v + vr( k / N + 77)),k=0 neZ,

[0325] In the rectangular situation where A — Ar, and 6 — 1, we have N = 1 and / ?time=req= 1. Substituting these values in (2.13) and (2.14) we get:(245)neZ(2-16) ^,fcq(^)(AV) = K^)EK^MV + 77V,.),HGZ

[0326] In addition, in the hexagonal situation where A = Ahexand e = ehex, we have N = 2, Tr— U, Vr— 2a1and = (l,z), bfreq= (1,— z). Substituting these values in (2.11) and (2.12) we get:(2-17)tl[ne(^)(r,v) = 2>(-van)^(f- + an)neZ+z^2 y<(-va(l / 2 + n)) (p{r + a(l / 2 + n)),neZ42184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00(2.18) 3efreq(^)(T,y) - ^(n')2( / (2ra’1n)^(v + 2a-1n)WGZ-iys (n / ) j / (2ra-1(n + 1 / 2)) (p{y + 2a~\n +1 / 2))

[0327] Furthermore, one can show thatZtiias t°Zt foc FT hence the pair of Zak transforms constitute a square root decomposition of the Fourier transform, reinforcing the interpretation of the Zak realization as residing between the time and the frequency (see FIG. 32). As mentioned before, the characteristic property of the Zak transform is that it interchanges between the Heisenberg group actions:

[0328] Proposition 2.1. We have:(2-19) (U z) > ^me,e - (2-20) 2freq e(^(v,z)> ^) = ^req(v,z) > Zfreq e(^),for every (p& TCeand (v, Z) G Heis.

[0329] Example 2.2. As an example we consider the rectangular latticeAr= φ ∈ Hz., 0) © Z(0,l / Tr) and the trivial Heisenberg character € = 1. Under these choices, we describe the Zak realization of the window function:.. f l 0 < t < Ta-2,).: •f 0 otherwise

[0330] This function is typically used as the generator filter in multi-carrier modulations (without CP). A direct application of formula (2.15) reveals thatP = Ztinie( / ?) is given by:(2.22) P(T, V) = 2^(1 / Z7rr)l9(r _ z7rr)’neZ

[0331] One can show that P(«Tr, / ? / T;. ) = l for every a, b G [0, 1), which means that it is of constant modulo 1 with phase given by a regular step function along T with constant step given by the Doppler coordinate V. Note the discontinuity of P as it jumps in phase at every integer point along delay. This phase discontinuity is the Zak domain manifestation of the discontinuity of the rectangular window p at the boundaries.

[0332] 7.6.3 The generalized Zak realization43184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0333] For various computational reasons that arise in the context of channel equalization we need to extend the scope and consider also higher dimensional generalizations of the standard scalar Zak realization. Specifically, a generalized Zak realization is a parametrized by an under-sampled Heisenberg lattice ( A,e). Given this choice, we fix the following structures:

[0334] Let Heis (A,c) = A1X, S'I / A(, be the finite Heisenberg group associated with (A,c), see Formula (1.15). Let A'2= |A: AX| be the index of A inside A1. Finally, letbe the finite dimensional Heisenberg representation of Heis ( A,e). At this point we are not interested in any specific realization of the representation.

[0335] A generalized Zak waveform is a vector valued function (p: V — » CNthat satisfy the following, quasi-periodicity condition:(3.1) #>(v + 2)

[0336] for every V e V and G A1. Observe that when the lattice A is critically sampled, we have N = 1 and condition (3.1) reduces to (2.3). In the rectangular situation where A = Ar, 6 = 1 we can take = 2r'""e, thus the quasi-periodicity condition (3.1) takes the explicit form:(3.2) (p T + k / vr, v + 1 / Tr) = y / k / vr){i / / (kl / N)M_lL_kt> (p(r, v)},

[0337] where we substituteV — and — klvr,l / Tr. In particular, we see from (3.2) that the nth coordinate of (p satisfies the following condition along Doppler:

[0338] (3.3) <pn(T, V+l / Tr)=^-nl / N)<pn(T,v),

[0339] for every (^,v) G V and I e IL. We denote by 7e— 7 (V,7k ) the Hilbert space of generalized Zak waveforms. We now proceed to define the action of Heisenberg group on 7t. The action formula is similar to (2.4) and is given by:

[0340] (3.4) (v, z) > ^}(v') = z • v))^(v'- v),44184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0341] for every (p?eand (v, z) G Heis. Similarly, we havepre(2,e (2 )) > = ^(<y(2,v))^(v), for every 2G A.

[0342] 7.6.3.1 Zak to Zak intertwining transforms

[0343] The standard and the generalized Zak realizations of the Heisenberg representation are isomorphic in the sense that there exists a non-zero intertwining transform commuting between the corresponding Heisenberg actions. To describe it, we consider the following setup. We fix a critically sampled Heisenberg lathee (A,c) and a sub-lattice A' c A of index N. We denote by c' the restriction of 6 to the sub-lattice A. Our goal is to describe the intertwining transforms (See FIG. 33): (3.5)(3.6) Z,:

[0344] We begin with the description of Z -. Lete CAbe the unique (up-to multiplication by scalar) invariant vector under the action of the subgroup Ve= c ( V) C Heis ( A,6 ) through the representation, namely, satisfies the condition:^(2,c(2))K = C

[0345] for every 2 G A. Given a generalized Zak waveform (p te', the transformed waveform ^,e'W is given by:(3-7) Zy(^)(v) = (^,^(v)},

[0346] for every V G V. In words, the transformed waveform is defined pointwise by taking the inner product with the invariant vector. We proceed with the description of Z^e. To this end, we define the Hilbert space of sampled Zak waveforms. A sampled Zak waveform is a function0: A / J_— > C satisfying the following discrete version of the quasi-periodicity condition (2.3):(3.8) 0(8 + 2) = (2)”1> 0(8} >

[0347] for every 3 G A'1and 2 G A. We denote the Hilbert space of sampled Zak waveforms by 7 (A,_1', 71 ). One can show that T, ( A / -L, 71 ) is a finite dimensional vector space of dimension45184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00[A: A'X] =[A': A] = N. The Hilbert space of sampled Zak waveforms admits an action of the finite Heisenberg group Heis ( A,6 ). This action is a discrete version of 2.4) given by:(3.9) > 0}(£') = zip( / 3(8,8' - 8y}( / ){8' -,

[0348] for every ( / ) G Tt ( A'1, 71e), and points 8.8' A'1. We can now define the intertwining transform Zr’e.Given a Zak waveform (p& Lethe transformed generalized waveform (f) =((?>) is a function on V taking values in the A dimensional Hilbert space ( A'L,7Te^ — CN, defined by:(3 10) ^'(v)(^) = ^(-^(v,< J))^(v + ^),

[0349] for every v G V and A'1. For the sake of concreteness, it is beneficial to describe in detail the rectangular situation. We consider a rectangular lattice A = Arwith trivial embedding 6 — 1 and the sublattice A =Z(fr,0) © Z(0, NVr). Evidently, we have [ A': A] = N. For these particular choices, the structures described above take the following concrete form:• The finite Heisenberg group associated with (A,e) is given by:Heis(A,c) = A’L / AxS1- S'1,• The finite Heisenberg representation of Heis (A,e), is given by:^(z) = z,• The orthogonal complement lattice of A is given by:A'1= Z(rr / A,0)©Z(0,vr),• The finite Heisenberg group associated with ( A,6 ) is given by:Heis (A', e ) = A' / A x S1- Z / N x Z / A x S1,• The finite Heisenberg representation of Heis ( A,6 ), is given by 2T '=, where:7r^me(kTr / N,lvr,z) = zLkMl,• The invariant vector under7T^ = 71^ (A,ez(A)), 2 G A is given by:46184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0350] Substituting in Formula (3.7), we get:(3 H)Ze,e'(^)G’) - (^(O),^(V))= Po(v)

[0351] In words, the conversion from generalized to standard Zak waveforms is “simply” to take the zero coordinate at each point V e V. In the opposite direction, given a Zak waveform7Y( V,. T ) its restriction to the lattice A,_Lis periodic with respect to translations by elements of A, hence is a function on the quotient group A^ / A = Z / V, i.e., a vector in C(Z JV). Substituting in Formula (3.10), we get that:y / (-VTrlN)(p(T+TrlN,v) ^A(^)(AV) =^(-PTr(1-W))^(f + Tr(1-1 / A),v)

[0352] for every <p& 7Y(V,^"e) and (T, V)G V.

[0353] 7.7. Introduction to radar waveform design in the Zak realization

[0354] In the subsequent sections, a general systematic method for radar waveform design that is based on the Zak representation of discrete sequences and continuous signals (aka waveforms) is described. Along the way we develop the theory of sampling and filtering using the formalism of the Heisenberg group. We conclude with an example of a particular family of compressed radar waveforms based on discrete Zak sequences. These waveforms enjoy uniform temporal power profile and thumb-tack like ambiguity function with a clean punctured region around the origin whose dimensions are free parameters.

[0355] 7.7.1. Set-up for radar waveform design

[0356] Let V = R be the delay Doppler plane equipped with the standard symplectic form G)^1’^2)—^1 A—^2^1 >

[0357] for every— (ij, kj) and v2— (A, F2). Let fi be the polarization form:^(VIA2) =V1 2.47184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0358] Using the form [3 we introduce a binary operation between functions on V called twisted convolution. To simplify notations, we denote by= exp( j27Tz) the standard Fourier exponent. Given a pair of functions hvh G C(V), we define their twisted convolution to be:Ma MV) = f (AVPV2)) MV1 ) MV2 ),

[0359] We fix a critically sampled lattice Aj C V. We assume Aj is rectangular of the form:Aj -7Lr®7Lvr,

[0360] such thatr• V = 1. We fix a rectangular superlattice AD Aj of the form:A = ZAT® ZAV,

[0361] where AT = Tr / N and Av = Vr / M. We denote by L = [A: A the index of Atas a sub-lattice of A. It is easy to verify that L = N • M. This number also counts the number of points in the finite quotient group A / Ax. In addition, we denote by A1the symplectic orthogonal complement of A defined by:A1={VE V: tw(v,2) e Z for every G A},

[0362] We have A1cA1. Overall, we defined a nested family of lattices (see Figure 34):A1cAj cA,

[0363] We note that |^A: A1J = L2or, equivalently, the number of points in the quotient groupA / A is equal to L. Finally, we introduce a discrete variant of the twisted convolution operation between functions on the lattice A. Given a pair of functions / ?,, / ?2G ® ( A), we define their twisted convolution to be:

[0364] 7.7.2. Continuous Zak signals

[0365] In classical signal processing there are two fundamental domains of signal realizations: the time domain and the frequency domain. Each of these domains reveals complementary attributes and the conversion between these two realizations is carried through the Fourier transform. As it turns out, there48184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00is another fundamental domain called the Zak domain. A continuous Zak signal is a functionE>: V — > C that satisfies the quasi-periodicity condition:0(v + ^ ) = ^(^(v,^ ))0(v),

[0366] for every V G Vand € A(. Concretely, if we take V — ( T, P ) and = ( kTr, lVr) then condition (2.1) takes the form:<I>(T + kTr, V + lvr) = y / (kvrr)( A v),

[0367] Given a pair of Zak signals,,< P2G 7 we define their inner product as:(3> I,< D2) = j 01(v) -02(v)^v,V / Aj

[0368] We denote the Hilbert space of continuous Zak signals by 7Y = C ( V / Aj,?). We equip 7Y with an Heisenberg action defined by the operator valued transform II: C(V) — > End(T ) defined by:n( / ?.) > 0= / 7*O. < E>,

[0369] for every 0G ~C and C(V). We refer to II as the Heisenberg transform. The Heisenberg transform admits an inverse called the Wigner transform. Given a pair of Zak signals,0, G T~L, die Wigner transform of die rank one operator |02)(0] 1 is die function-> C given by:(v) = (^(v)01,02),

[0370] for every VG V, whereA(I') =. The function (2.5) is called the cross-ambiguity function of the signals 0, and 02■ In case 0j = 02= 0, we denote the cross-ambiguity function simply by A and refer to it as the ambiguity function of the signal 0. The conversion between the Zak domain to the time domain is carried through the Zak transform Z 7 L2(( R), given by:VrZ(0) =049184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0371] for every $6 C. We conclude this section with an example of an explicit Zak signal and its time domain realization. Let <$>r mbe the unique quasi-periodic extension of the delta function supported on the lattice point (rtAT,7rtAv), for 0 < n < N — 1 and 0 < m < M — 1, i.e.,:<t>nm= ^^(mAv -kTr)6(nAT + kTr,mAv + lvr)kJ= ^Y(mkvr-TjM)8(nAT + kTr,mAv + lvr)kJ= ^^{mk / M) 8(nAT+kTr,mAv + lvr),kJ

[0372] Direct calculation reveals that the Zak transform of n mis a time shifted, phase modulated, infinite delta pulse train (see Figure 29), given by:. ) = y(mk / M ) <5(HAT + kTr),

[0373] 7.7.3. Discrete Zak signals

[0374] The continuous Zak theory admits a (finite) discrete counterpart which we proceed to describe. The development follows the same lines as in the previous section. We use lower case letters to denote discrete Zak signals. A discrete Zak signal is a function: A — > C that satisfies the following quasi-periodicity condition:

[0375] for every Z e Aand ApConcretely, if we take — (rtAf, / rtAv) and — kTr, lvr) then condition (3.1) takes the form:0(nAT + kTr,mAv + lvr) = ys(mkAvTr)0(nAT,mAv)= yf{mkvrTr / M]( / ){nAT,mAv)= iy(mk / M)0(nAT,mAv),

[0376] Given a pair of discrete Zak signals,^2e 7Lwe define their inner product as:zteA / Aj50184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0377] We denote the Hilbert space of discrete Zak signals by 7L— C( A / A,?). One can show that dim 7L= L. We equip A). with an action of the finite Heisenberg group, expressed through the transform IIL: 7 (A) -A End(77 / ), given by:nL( / ?.)>^= / ?.*CT^,

[0378] for every ( / ) G 'HLand / 1G C (A). We refer to H as the discrete Heisenberg transform. The discrete Heisenberg transform admits an inverse called the discrete Wigner transform. Given a pair of discrete Zak signals< / \, G lL, the discrete Wigner transform of the rank one operator | <4 ) (^11 is the function A^: A — > C given by:

[0379] for every 2 G A, where ^ (2) = nz(<7(2)). The function (3.5) is called the discrete cross-ambiguity function of the signals ( / and < / >2. Since TL(2 + 2 ) = 71L(2) for every 2 G A and 21G A1, it follows that A^ is a periodic with respect to the sub-lattice A1, i.e.,:

[0380] for every 2 G A and 21G A1. When=2= we denote the discrete crossambiguity function by A^ and refer to it as the discrete ambiguity function of ( / ).

[0381] 7.7.4. Sampling theory on the Zak domain

[0382] The focus of sampling theory is to describe the relation between the continuous and discrete cross -ambiguity functions. To this end, we denote by C (V) the vector space of generalized functions on V. The main assertions are stated in terms of two basic transforms:5: C(V) — > C(A),1: 3(A) — > C'(V),

[0383] The transform s is called sampling and it sends a function on V to its samples on the lattice A. The transform I is called embedding and it sends a discrete function: A — > C to the generalized function (distribution) on V given by the following super-position of delta functions:TeA51184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0384] The sampling and embedding transforms give rise to induced transforms between the corresponding Hilbert spaces of continuous and discrete Zak signals. We denote the induced transforms by the same names, i.e.,:s': H — > HL,I: HL— > H',

[0385] where = C?( V / At, / ?) denotes the vector space of generalized Zak signals (distributions). Given a function Tie C(A), we denote by hLits periodization with respect to the sublattice A1cA, i.e.,: / i1G A1

[0386] for every G A. The main technical statement is summarized in the following theorem.

[0387] Theorem 4.1 (Main Theorem of Sampling Theory). The following two relations hold:

[0388] (1) Sampling relation. For every.<t>2G 7Y we have:A(4> J), S(4>2) “1S( A>1,4>2)AX ’

[0389] (2) Embedding relation. For every G 'HLwe have:

[0390] In plain language, the sampling relation asserts that the discrete cross-ambiguity function of sampled continuous signals is the sampled (and periodized) cross-ambiguity function of the continuous signals. The embedding relation asserts that the continuous cross-ambiguity function of embedded discrete signals is the embedding of the cross-ambiguity function of the discrete signals.

[0391] 7.7.5. Filter theory

[0392] Filter theory gives means to convert a discrete sequences to continuous waveforms. We define an Heisenberg filter to be a function W G C ( V ). We say the filter w is factorizable if it can be written as iv = IV. *awywhere WTis a distribution supported on the delay axis and wyis a distribution supported on the Doppler axis. Note that such a function takes the form:

[0393] for every T, V G R. The manner of operation of a filter w on a Zak signal is carried through the Heisenberg transform, i.e.,:52184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00o,, =n(w) > 0 = w 0,

[0394] The above equation shows a relationship between Zak signal and the Heisenberg transform. While the relationship is described as a sequence of mathematical steps, in general, implementations need not explicitly perform these steps, but may use numerical methods to compute end results without having to compute and store any intermediate results.

[0395] Time domain interpretation of effects of Heisenberg transform

[0396] To get some intuition, it is beneficial to interpret the effect of Heisenberg filtering in the time domain by exploring the structure of Z (0W). Assuming w is factorizable, one can show that:

[0397] where Wt= FT1(wF) and * stands for linear convolution. We see that Heisenberg filtering amounts to a cascade of first applying a window in time followed by a window in frequency, aka, convolution with a pulse (see Figure 30). The main technical statement of this section describes the relation between the discrete and continuous ambiguity functions. The result will follow from the following general proposition.

[0398] Proposition 5.1. Given a pair of Zak signals 0,,026 T~t and corresponding pair of Heisenberg filters M'1, W2G C(V), the following relation holds:

[0399] where vv* (r ) = ( / / ( / ? ( r. r)) w2(-v) is the Heisenberg conjugate function.

[0400] In the case 0, = 02= 0 where 0 = z(^)and = w2= w the statement of the proposition describes the relation between the discrete ambiguity function of the sequence ( / ) and the continuous ambiguity function of the waveform 0W. The result is summarized in the following theorem.

[0401] Theorem 5.2 (Main theorem of filter theory). Given a discrete Zak signal € fiLand a Heisenberg filter HA ), the following relation holds:= P.. / leA

[0402] where Pv= W*aW* for every’ VG V.53184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0403] In plain language, the theorem asserts that the ambiguity function of the waveformis obtained from the ambiguity function of the sequence through shaping with a pulse(whose shape depends on the particular value of ). In a sense, the design of an optimal Radar waveform involves two aspects. The first concerns the design of a finite sequence of a desired discrete ambiguity function and the second concerns the design of a Heisenberg filter w of a desired pulse shape P for various values of 2.

[0404] 7.7.6. Zak theoretic chirp waveforms

[0405] In this section we describe a particular family of compressed Radar waveforms based on discrete chirp sequences in the Zak domain. These waveforms enjoy uniform temporal power profile and thumbtack like ambiguity function. The construction assumes the following set-up. We assumeN, M G N are coprime odd integers. We let <7 G (Z / A) be an invertible element in the ring of integers modulo N. We denote by l / / N: Z / A — > C the finite Fourier exponent l / Nfl) =l / n / N).

[0406] We define the discrete Zak signal ch = chaG LLas:m = 0modMch(7tAr,mAy) =0 otherwise

[0407] for every n,m G TL. We refer to ch as the discrete Zak chirp of order A and slope a. We next explore the structure of the discrete ambiguity function. Ach. To that end, we introduce the sublattice AflC A (see Figure 35), given by:Aa={(nA'T, kMAv): k = a-n mod A],

[0408] Theorem 6.1. The discrete ambiguity function.4chis supported on the latticea.Moreover:1 2 1ylch(7iAr, WAv) = ^7V— an A,2 )

[0409] for every (n,k) such that k = a - n mod A.

[0410] A direct consequence of Theorem 6.1 is that lcllvanishes on the non-zero points of the interval Ir= [— 'Z’. / 2,'T(. / 2]x[— Pr / 2, Pr / 2], which we refer to as the "clean" region. Next, we fix a filter function W G C ( V ) and define the continuous Zak chirp Ch = Cha we PL as:54184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00Ch = H’*o. z(ch),

[0411] By the main theorem of filter theory (Theorem 5.2) we know that the continuous ambiguity function AlChis related to the discrete ambiguity function lchthrough the equation:A, = X-AAAA4eA=TleA,,

[0412] where Pv= VV *f7*o. W* for every ve V. Assuming the pulses P are well localized for every A G Aan 21r, the continuous ambiguity function ^4Chwill have a thumbtack shape with a clean region around zero coinciding with the interval Ir(see Figure 36). In case the numbers N, M » 1, choosing he filter function w to be square root Nyquist with respect to the lathee A ensures1S wc" localized for every A G A n 21r.

[0413] 8. Single Carrier Waveform - Precoded FTN Signaling

[0414] This section introduces the general system model of a linearly precoded FTN (faster-than- Nyquist) signaling scheme and reviews the conventional SVD-precoded FTN signaling scheme (singular-value decomposition precoded FTN signaling scheme).

[0415] Notation'. We use upper- and lower-case bold-faced letters to represent matrices and vectors, respectively. (-)r, (•), and (-) denote the transpose, conjugate, and the conjugate transpose of (•), respectively. The determinant and trace operations of a matrix X are represented by |X| and trace {X}, respectively. A diagonal matrix whose diagonal elements are given by a vector of x is denoted by diag{x]. IE[ ] represents the expectation operation. C. N (m,v) denotes the Gaussian distribution having a mean of m and a variance of V. I (x;y) is mutual information between a vector of x and y. A differential entropy is denoted by he(•).

[0416] 8.1 General System Model of Precoded FTN Signaling

[0417] It is assumed that the transmission of N -length complex-valued Gaussian symbolsS = [50,51,-- -, 5A,_1]rG < CNhaving an average symbol power of55184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00E. =E|W = 1 (z = 0, ■ ■ •, - 1) -1The Gaussian symbols are precoded using a linear precodingmatrix F G 'PN / N, and hence the precoded symbols X = [x0, • ■ •, x;v-1] G C'Vare given by:X “ FS, (8.1)

[0418] Then, the precoded symbols are band-limited with the aid of an RRC filter h ( / ) with the roll-off factor fd. Note that a sine filter corresponds to the RRC filter, having the roll-off factor of ft = 0. Finally, the precoded FTN signals are transmitted with an FTN symbol interval of T = as follows:x(t} ~ \ / T'TQ ~~ nT),(8.2)

[0419] where the coefficient j TQnormalizes the power of an FTN signaling block so that it remains equal to that of its Nyquist-criterion-based counterpart which has tire same block interval as and fewer information bits than those of the FTN signaling scheme. Furthermore, note that the total transmit energy per block is given by EN= E x(r)r dt, while the precoding matrix F is designedfor maintaining the average energy consumption ENto be constant.

[0420] Assuming an additive white Gaussian noise (AWGN) channel, the received FTN signals are passed through a matched filter h* ( —? ), and are represented by:n (8.3)

[0421] where we have g (t) = J / t(<^) / T — t}d% and — t)d<%, while n t)is a complex-valued random variable that obeys the Gaussian distribution of CJ\f (0, No).1Here, idealistic Gaussian symbols are considered because the main focus here is the derivation of a unified information-theoretic bound, rather than error-rate calculations, assuming a specific modulation scheme.56184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0422] Ignoring the effects of IBI for simplicity, the i th received sampleyi— y ( iT) (z = 0, • • •, N — 1) can be expressed byyi — v(8.4)

[0423] where / / (zT)(z — 1) represent the colored noise, with a correlation of E[7( / r)<7,(”>r)] =A'os (G — m) r). Furthermore, the received sample blocky = [ Vo ’ Vi ’ ’ ’ ’ ’ fisrepresented by- VrToGFs + (8.6)

[0424] where G e RWxAis the FTN-induced ISI matrix, with a Toeplitz structure, the first column of which is given by andT] = |y(0),--. Moreover, G is a positive-definite matrix.

[0425] 8.2 Conventional SVD-Precoded FTN Signaling

[0426] In previous studies, the precoding matrix F was calculated based on the SVD. More specifically, the matrix G in (8.6), representing FTN-induced ISI, is factorized into: G = UAVr, where U G RWxWand V G RWxWare orthogonal matrices, and A G RjVxWis a diagonal matrix, composed of the descending-order singular values, ■ • •, ] of G. Note that since the matrix G is real and symmetric, the relationship U = V holds, which corresponds to the eigenvalue decomposition of G. Therefore, the received samples of (8.6) can be rewritten by / - - - y = y T. ZQVAV ‘‘. F S -f- 7?,

[0427] Then, by setting the precoding matrix F to the orthogonal matrix fp V and noting the relationship VVr= I, where P is the power factor and I is the identity matrix, (7) can be further simplified to:57184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00y = / Pr7oVAs 4- / .(88)

[0428] Furthermore, multiplying the weight matrix Vrby y in (8.8) yields the following diagonalized signal representation:Vd = VTy e CN(8.9)= / PrTo AS + r)l;,(8 10)T

[0429] where 77v=V 77. Note that ydin (8.10) is modeled as N independent parallel substreams.The related noise components 7] = V TJ in (8.10) are no longer correlated. The i th component of 7[ has variance, as follows:E[W ] = VTE[WH]V(8 I I)— A,7. (8.12) ” Ao A,(813)

[0430] where IE J = N0G. Hence, the conventional SVD-precoded FTN signaling schemeallows the FTN symbols of (8.10) to be demodulated based on low-complexity symbol -by-symbol maximum-likelihood detection. It should be noted that SVD is carried out offline before transmission since G is determined uniquely by T and [3.

[0431] 9. OTFS - Channel Estimation and Equalization

[0432] 9.0 Mathematical description of an OTFS Waveform

[0433] An OTFS data frame is allocated on a delay-Doppler grid. Let us denote the number of Doppler elements by N and the number of delay elements by M. The grid spacing is AT and Av in delay and Doppler, respectively, with periods rp= M T, where A = 1 / BIV and vp= / VAv, Av = 1 / T, where BW is the signal’s bandwidth, T is the signal’s duration and Tp■ vp= 1.

[0434] Each data symbol (typically a quadrature amplitude modulation QAM symbol), x(n, m), where n = 0,..., N — 1 and M = 0,..., M — 1, is carried over a waveform called a pulse-tone waveform (e.g., Pulsone™, a combination of a pulse train and a tone). The construction of a pulse-tone (e.g., Pulsone™ )58184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00is described as follows. Let 8qpmt) be a quasi -periodic Dirac delta train for data symbol x(n, m), defined as:bqp’m)(t) = x(n, m) ■ Xk=-<x> eJ2nn&vkTv6(t - mAx - kip) (9.3)= x(n, m) ■ Efct-oo e]2rmk / N8(t — / 8x(m + kM)) (9.4)

[0435] A pulse-tone signal is generated by multiplying this modulated delta train by a time window function, Wt, which is the inverse Fourier transform of the Doppler pulse, = F~l{pv}. and then convolving it with a delay pulse, pt^n’mKQ = pT^ [wt- 6^m\t)] (9.5)

[0436] This concept is illustrated in FIG. 37. As shown in FIG. 37, from left to right, a delta train shown with time has the horizontal axis is passed through a time window of a certain shape and then convolved with a Doppler pulse (indicated by the * operation) to generate a desired delay-Doppler pulse 3700.

[0437] For all the data symbols on the delay-Doppler grid, we can compute the OTFS waveform as:OTFS(t) = 2^-12"; J (t) (9.6)

[0438] Equivalently, we can first compute the delta train for all the data symbols and then apply the time window and delay pulse convolution:<5qp(t) = S^o1S^1oCm)(t) (9.7) OTFS(t) = p^ [Wt- Sqp(t}] (9.8)

[0439] An example of an OTFS waveform in shown in FIG. 38. As depicted in the top waveform, the OTFS waveform includes a train of pulses separated in time by rpsuch that the phases of each subsequent pulse are rotated with respect to each other, as indicated by the curved arrows above each pulse. The lower graph shows a corresponding waveform.

[0440] After the interaction with the channel, the received OTFS signal, y, is processed in a reversed order to the transmitter. First it is convolved with a pulse, pkx. then it is multiplied by a time window, WfX, and finally an inverse Zak transform is applied to the received data, to obtain the received delay-Doppler grid elements:y' = Wkx■ [pkx* y] COy(n, m) = — _m&T_k = -<x>= ^Sfc=-oo e~'27Tnk,'Ny'(t — 8x(m + kM)) (9.9)

[0441] 9.1 Pulse shaping an OTFS Waveform for channel estimation59184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0442] An OTFS waveform may include a known reference signal for channel estimation, such as a known data symbol, called as a pilot symbol. Typically, the pilot symbol will be cyclically surrounded by empty grid elements (value of zero) that will be used at the receiver, as a channel estimation area and possibly a guard area from interfering data symbols. An example of such an arrangement is illustrated in FIG. 39A. In FIGS. 39A and 39B, a frame of transmission resources is depicted along the Delay (vertical) and Doppler (horizontal) dimensions. The corner regions “O” are reserved for pilot, however, in general, pilots may be inserted anywhere within the frame in a predetermined position known to both the transmitter-side and the receiver-side.

[0443] After the interaction of the waveform with a wireless channel, replicas of the pilot symbol (with a complex gain) will appear at the receiver in the channel estimation area (corresponding to the physical reflectors of the channel). Detection of these replicas forms a channel estimation and enables equalization of the received signal. However, at low Signal to Noise Ratios (SNR), it will be very difficult to distinguish between replicas of the pilot symbol and noise at the channel estimation area. For example, at 0 dB SNR, the strongest replica will have the same energy as the noise. Therefore, to enable adequate channel estimation, the power of the pilot symbol must be boosted, such that the pilot replicas can be detected in the presence of noise. Note, that in the creation of the waveform, the unassigned transmission power to the grid elements of the channel estimation area, may be used to boost the power of the pilot. For example, for a delay-Doppler grid with an average data symbol power of 1 and NCEgrid elements reserved for channel estimation (including the pilot location), the pilot symbol may be assigned as:Pilot = N^ (9.10)

[0444] An example of such an arrangement is illustrated in FIG. 39B.

[0445] FIG. 40 shows an example of a delay-Doppler plane in which symbols that contain information bits are shown. In FIG. 40, the horizontal axis is the delay axis and the vertical axis is Doppler axis. Each dot in the graphs shows a modulation symbol that comprises information bits. One example of a symbol 4002 in the delay domain is shown by a tall rectangle that comprises all modulated bits having a same delay, but different Doppler values. Another example of a symbol 4004 in the Doppler domain is shown by a wide rectangle that comprises all modulated bits have a same Doppler value, but different delay values. The example depicted in FIG. 40 shows a delay-Doppler grid where N = 16 (elements along Doppler direction) and M = 512 (elements along delay direction). Here, delay resolution is reciprocal of channel bandwidth. The Doppler resolution is inversely proportional to the frame time used for the communication.60184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0446] If we examine the OTFS waveform, generated from a power-boosted pilot, we can see that the pulse train has a boosted pulse at a delay location corresponding to the location of the pilot, as seen in FIG. 41. This waveform suffers from a high Peak-to- Average-Power-Ratio (PAPR), which is undesirable. For example, in FIG. 41, the high peaks 4101 are shown to have an amplitude or signal power that is greater than the “typical” pulse peaks, e.g., 4103. Depending on run-time conditions and values of N and M, the high PAPR may amount to 6 to 12 dB greater than the nominal pulse power.

[0447] To reduce the PAPR, while keeping the power of the pilot boosted, the OTFS waveform must be spread over time. Unlike, the spread spectrum (or spread time) techniques, this spreading does not extend the bandwidth or the duration of the OTFS waveform. One possible method to achieve this is to convolve the OTFS waveform with a spreading signal, such as a chirp. This is equivalent to convolving the delay pulse with a spreading signal,resulting in a new combined delay pulse:pTSpread= P * 1 (9.11)

[0448] For example, a chirp spreading signal,= e / 27r“f2, where a is a spreading constant, as shown in FIG. 42 (compared to FIG. 37). The bandwidth of the signal remains the same, but the data and pilot symbols are spread over the entire duration of the OTFS signal, resulting in a lower PAPR signal.

[0449] At the receiver, the inverse of the spreading signal is applied to regain the non-spread waveform.For example, a conjugated chirp signal is applied at the receiver. Afterwards, the receiver operations are the same. Alternatively, the receiver may use a combined pulse, p^pread'Rxinstead of pRx.

[0450] 9.2 Examples of OTFS transmission and reception

[0451] The dimensions of the channel estimation area (e.g., the pilot signal regions depicted in FIGS.39A and 39B) depend on the expected channel response and its delay and Doppler spreads. Within the channel estimation area, pilot symbols may be placed. A pilot symbol has a known value, and its power may be larger than the other data symbols.

[0452] The delay-Doppler grid may be transformed to a transmission waveform in one of the following methods:1. Transformation to a time-frequency equivalent grid via a discrete Symplectic Fast Fourier Transform (SFFT). This method creates a time-frequency grid, like the one used for OFDM modulation. This OTFS transformed grid may be multiplexed with other time-frequency elements in the OFDM grid, as shown in FIG. 43A, thus allowing multi-user data multiplexing. Then, an OFDM waveform may be generated using an inverse Fourier transform (IFFT) over each OFDM symbol.61184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO002. Transformation to a time-frequency equivalent sub-grid via discrete SFFT, as shown in FIG. 43B. The sub-grid is part of a larger time-frequency grid and has M elements along frequency and N elements along time. Then, an OFDM waveform may be generated using an inverse Fourier transform (IFFT) over each OFDM symbol.3. Direct transformation to the time domain using a Zak transform over the Doppler dimension of the grid (after extending the grid in a quasi-periodic manner and applying a two-dimensional transmission pulse), as shown in FIG. 43C.

[0453] FIG. 43 A shows an example of transmission method 1, where a delay-Doppler grid is transformed to a time-frequency grid using a Symplectic Fast Fourier Transform (SFFT). This transformed grid, denoted as “#3 OTFS”, is multiplexed with the data of other OFDM users (denoted as #1, #2 and #4) in the overall OFDM time-frequency grid. An inverse Fast Fourier Transform (IFFT) may be applied to the OFDM symbols to generate the transmission waveform.

[0454] FIG. 43B shows an example of transmission method 2, where a delay-Doppler grid is transformed to a time-frequency sub-grid with N elements along the time dimension and M elements along the frequency dimension, using a Symplectic Fast Fourier Transform (SFFT). Note, that the sub-grid may not take all the time-frequency resources and other sub-grids may be also allocated for other delay- Doppler transformations (possibly of different users).

[0455] FIG. 43C shows an example of transmission method 3, where a delay-Doppler grid is transformed to an OTFS waveform using the Zak transform over the Doppler dimension.

[0456] 9.3 Examples of receiver-side signal processing

[0457] At a receiver, the received waveform is transformed back to delay-Doppler for further processing.This transformation depends on how the waveform was transmitted:1. A waveform of transmission method 1, is first transformed to a time-frequency grid using a Fast Fourier Transform (FFT) and then the OTFS section of the grid is extracted and converted to delay-Doppler via the Inverse Symplectic Fast Fourier Transform (ISFFT). An example for this is given in FIG. 43D.2. A waveform of transmission method 2, is first transformed to a time-frequency grid using a Fast Fourier Transform (FFT) and then the OTFS sub-grid is extracted and converted to delay-Doppler via the Inverse Symplectic Fast Fourier Transform (ISFFT). An example for this is given in FIG. 43E.3. A waveform of transmission method 3, is transformed directly to delay-Doppler via an inverse Zak transform over the time dimension, as shown in FIG. 43F. Afterwards, a receive two-dimensional pulse may be applied to it.

[0458] FIG. 43D depicts a receiver processing example for a waveform generated by transmission method 1. The received waveform is transformed to a time-frequency grid using a Fast Fourier62184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00Transform (FFT) and the OTFS part (denoted as “#3 OTFS”) is extracted and transformed to delay- Doppler via an Inverse Symplectic Fast Fourier Transform (ISFFT).

[0459] FIG. 43E depicts a receiver processing example for a waveform generated by transmission method 2. The received waveform is transformed to a time-frequency grid using a Fast Fourier Transform (FFT) and the OTFS sub-grid is extracted and transformed to delay-Doppler via an Inverse Symplectic Fast Fourier Transform (ISFFT).

[0460] FIG. 43F depicts a receiver processing example for a waveform generated by transmission method 3. The received OTFS waveform is arranged in a grid N X M elements and transformed to delay-Doppler via the inverse Zak transform over the time dimension. It is noted that in the description of OTFS signal generation and reception as described in the present document, while reference signals are not specifically described, in practical systems, some resources may be allocated to various reference signals for monitoring or calibration of the channel between a transmitter and a receiver.

[0461] In some embodiments, the described embodiments include receiver signal processing that can be configured to implement iterative equalization and decoding of multi-level encoded symbols (in Section 9.4) and iterative two-dimensional (2-D) equalization (in Section 9.5).

[0462] 9.4 Examples of iterative equalization and decoding for multi-level encoding

[0463] In general, iterative receivers exchange extrinsic information between the equalizer and the FEC (forward error correction) decoder to achieve close to optimal performance, as shown in FIG. 44 for an OTFS receiver 4400. The extrinsic information may include a priori knowledge of which transmission resources (e.g., time slots of subcarriers) use which particular FEC. For example, the equalizer 4402 uses prior information on the data symbols coming from the FEC feedback path to improve the equalization of the symbols. This feedback path comprises a symbol mapper 4410 and OTFS transformation module 4412. Then, these symbols are converted to bit likelihoods that are FEC decoded. Several iterations are performed until all the source data is decoded correctly, or until some other stopping criteria is met. An inverse OTFS transform module 4404 may apply inverse OTFS transform and a symbol demapper 4406 may recover bits from modulation symbols.

[0464] Compared to other techniques described next, the error-rate performance of the scheme 4400 may be degraded. One reason for the degradation may be because of the mixture of bits with different level of reliability in every FEC codeword that is being decoded. The constellation bits with low reliability make it harder for the FEC decoder to converge to the correct codeword and therefore, the feedback to tlie equalizer has less information to improve tlie equalization.63184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0465] When multi-level encoding is applied at the transmitter (e.g., as shown in FIG. 45), the iterative receiver 4600, in each decoding iteration, decodes only a part of the constellation bits. It typically starts with the most reliable bits and then proceeds in the next iterations to less reliable ones. This scheme, shown in FIG. 46, allows the equalizer to receive in earlier iterations priors, which are dominant from the constellation symbols point of view and better improve the equalization. When the FEC has successfully decoded one level, it switches to decode the next one. The receiver continues to iterate until all levels have been decoded successfully or until some other stopping criteria is met. The most reliable bits are often bits that are used to decide the "macro" region within the constellation map where a symbol lies-e.g., the quadrant in which a constellation symbol of a 4 or 8 QAM signal lies, followed by sub-quadrant within the quadrant, and so on. Thus, as shown in FIG. 46 the received signal may be equalized by the equalizer 4602. In the forward path, the equalized signal may undergo an inverse OTFS transform (4604), and the symbols from the resulting transformed signal may be demapped for decoding by multiple different FECs FEC 1 to FECq (modules 4658a to 4658q). In the feedback path, the decoded symbol (bit) outputs of the FEC modules may be mapped to symbols (4610) and transformed into OTFS domain signals (symbols) for feedback to the equalizer 4602. As described above, in some implementations, different forward error correction codes are used for symbols from the multiple symbols corresponding to header and payload portions of the bits from the signal.

[0466] 9.5 Examples of iterative 2-D equalization

[0467] FIG. 47 is a block diagram of an example embodiment of an iterative 2-D equalizer 501. The 2-D Iterative equalizer, illustrated in FIG. 47, iterates between the 2-D equalizer 503 and the FEC MAP decoder 505, by passing information from one to the other. After several iterations, the MAP decoder outputs estimation on the information bits. In various embodiments, the iteration termination criteria may be based on a total number of iterations, meeting, but not exceeding, a time budget for the iterative process, the improvement in successive iterations falling below a threshold, and so on.

[0468] 9.5.1 2-D equalizer (503)

[0469] In some embodiments, the 2-D equalizer may be implemented as an affine MMSE (minimum mean square error) equalizer, computing the Wiener estimator of XX = CY + (J - CH)X (9.16)

[0470] Herein, C = RXY^Y1and I is the identity matrix. Note that C is a function of Rx and Rw. For the first iteration there is no prior information on the symbols of X, therefore we set X=0 and Rx=I. The 2-D equalizer also computes the variance of the estimation error, denoted as RE.

[0471] 9.5.2 2-D SFFT (507)64184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0472] The estimated symbols and error variances, X and RE respectively, are transformed from the 2-D Time-Frequency grid to the 2-D Delay-Doppler grid via a 2-D Symplectic Fourier transform to x and Rerespectively.

[0473] 9.5.3 Likelihoods (509)

[0474] Likelihoods for the coded bits LE(x), are computed from the symbols x. Gaussian distribution may be assumed for x and the likelihoods can be derived from it. The probabilities for this case are P(x\x = co) <x eRz('x(9.17)

[0475] Herein, m G fl is a constellation symbol, A = — ReRxRz= AReand fj.( o, A) = coA + (1 — A)x ((18)). Note that x is defined in Equation (21). For each symbols, the extrinsic coded bits log likelihoods ratio (LLR) can be derived as(9.19)

[0476] Herein, i, j = 0,, q-1, s(co) is the constellation bits label that is associated with the constellation symbol co and P(co)j is defined in Equation (9.20).

[0477] 9.5.4 Deinterleaver (511)

[0478] The deinterleaver permutes the likelihoods LEx to L( C). These likelihoods will be used as a priori information for the MAP decoder. In some implementations this deinterleaver might be optional.

[0479] 9.5.5 MAP decoder (505)

[0480] The maximum a posteriori (MAP) decoder computes the a posteriori probabilities (APP's) of the information bits and also the extrinsic probabilities for the coded bits, which when using LLRs, are the APP's minus the a priori inputs.

[0481] 9.5.6 Interleaver (513)

[0482] The interleaver permutes the likelihoods IDC) to L(x). These likelihoods will be used as a priori information for the MAP decoder. Note that in some implementations this interleaver might be optional.

[0483] 9.5.7 Symbol mapper (515)

[0484] The symbol mapper estimates the probabilities of each constellation symbol m G fl from the likelihood values L(x):65184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00P(a>) j = — 1 + (2 ■ s(w) j - 1 ) • ta (9.20) <?i P(co)3]”[ P(^)jj=0

[0485] These probabilities are used for computing the expectation of the constellation and the variance:<7-1 X ~ ■ P(u> )1=0 (9.21) <7-1xxHi=0

[0486] 9.5.8 2-D SFFT1(517)

[0487] The 2-D Delay-Doppler domain symbols' expectation and variance x and Rx are transformed to X and Rx in the 2-D Time-Frequency domain using a 2-D Inverse Symplectic Fourier transform to transform from the delay-Doppler domain to the Time-Frequency domain. These are used as priors to the 2-D Equalizer in the next iteration. In some embodiments, the 2-D transforms used by operation 507 and 517 may be swapped. In other words, an inverse SFFT may be used in the operation 507, while an SFFT may be used in the operation 517.

[0488] In some embodiments, the iterative 2-D Equalizer may be operated so that the receiver gets side information about some resource elements in the time-frequency grid that have been "erased" (e.g., not transmitted, or not useable) and the receiver can ignore them. The receiver may skip equalization for these resources and directly use the prior estimates as outputs for the equalizer. In this case, Equation (9.16) simply becomes for these resources: X = X.

[0489] 10. Summary Tables:Carrier parameters:Parameter Commentfc Carrier frequency in [Hz]BW Carrier bandwidth in [Hz]fs Sampling rate in [samples per second]Tslot Slot duration in [sec]NpRB BWNumber of PRBs within the carrier bandwidth = — —_ IAJ PRBI _PRB parameters:66184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00Parameter CommentBlVpRBPRB bandwidth = 168 kHzTpRB PRB variable duration (function of nT)kfpRB PRB spacing = 180 kHznffrequency indexnTTime duration index, correspond to a duration of TPRB= — ■ Tsiot^-offset Time offset index, corresponds to an offset of Toffset= °^sec■ TslotOTFS parameters:Parameter CommentAT Delay resolution = — - — = 5.95 ■ 10-6[sec]BIVPRBAv 1Doppler resolution = - [Hz]M Delay grid dimensionN Doppler grid dimensionvv Doppler period in [Hz]Tp Delay period = l / vpin [sec]pvDopplerfilter = sinePTDelay filter = RRC with 7% roll-off& Delay filter roll-off factor = 0.070VDopplerfilter roll-off factor = 0

[0490] 11. Implementation examples

[0491] FIG. 24 is a block diagram representation of a hardware platform 2400 which may be used to implement the various methods described in the present document. The hardware platform 2400 may be incorporated within a base station or a user device. The hardware platform 2400 includes at least one processor 2402, a memory 2404 and a transceiver circuitry 2406. The at least one processor may execute instructions, e. g., by reading from the memory 2404, and control the operation of the transceiver circuitry 2406 and the hardware platform 2400 to perform the methods described herein. In some embodiments, the memory 2404 and / or the transceiver circuitry 2406 may be partially or completely contained within the at least one processor 2402 (e.g., a same semiconductor package).

[0492] 12. Examples of Technical solutions.

[0493] The following solutions may be adopted by preferred embodiments.

[0494] Embodiments of a transmitting side may adopt the following solutions. Here, the transmitting side may be implemented in the network (e.g., a base station) and / or a user device such as a user device that may be transmitting to one or more other user devices.67184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0495] 1. A method of digital communications (e.g., method 2510 depicted in FIG. 25 A), comprising: transmitting (2512), by a transmitting device, a transmission waveform to one or more receiving devices using one or more time and frequency resources of a transmission medium, wherein the time and frequency resources are divided into physical resource blocks (PRBs) that are configured according to a PRE configuration scheme; wherein each PRE is generated from corresponding orthogonal time frequency space (OTFS) frame; wherein each OTFS frame is generated from symbols along delay and Doppler dimensions using an OIF'S framing scheme. Additional details of this method are also disclosed in Section 4 with respect to various ways by which OTFS framing is performed and OTFS is mapped to PRBs.

[0496] 2. The method of solution 1, wherein the PRB configuration scheme and / or the OTFS framing scheme are responsive to channel characteristics of transmission channels between the transmitting device and the one or more receiving devices.

[0497] 3. The method of any of above solutions, wherein the transmission waveform is configured to occupy a carrier bandwidth of a carrier.

[0498] 4. The method of solution 3, wherein each PRB is defined by a frequency location and a bandwiddi diat is a portion of the carrier bandwidth.

[0499] 5. The method of solution 4, wherein the time resources are defined according to slots, wherein each PRB is defined by an offset within a slot and a duration, wherein the offset and / or the duration are configurable according to the PRB configuration scheme.

[0500] 6. The method of any of above solutions, wherein each OTFS frame is generated by applying a Doppler pulse along the Doppler dimension.

[0501] 7. The method of any of above solutions, wherein each OTFS frame is generated by applying a delay pulse along the delay dimension.

[0502] 8. The method of solutions 6 or 7, wherein the OTFS framing scheme specifies that the OTFS frame is generated according to a first pulse configuration in which the Doppler pulse is applied first to symbols, followed by the delay pulse.

[0503] 9. The method of solutions 6 or 7, wherein the OTFS framing scheme specifies that the OTFS frame is generated according to a second pulse configuration in which the delay pulse is applied first followed by the Doppler pulse.

[0504] 10. The method of solution 8 or 9, wherein the OTFS frame is generated by transforming a discrete delay-Doppler grid comprising symbols to a discrete delay-Time variable by applying a Fourier transform (FT) or an inverse Fourier transform to the Doppler dimension. For example, transmitting side68184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00may use FT and receiving side may use inverse FT, or alternatively, transmitting side may use inverse FT and receiving side may use FT.

[0505] 11. The method of solution 8 or 10, wherein the OTFS frame that is generated using the first pulse configuration is a result of: generating a periodic time vector by applying an inverse Fourier transform to symbols along the Doppler dimension and periodically repeating a result of the applying; applying a window in the time domain, wherein the window is computed from an inverse Fourier transform of the Doppler pulse; and applying a delay pulse to a result of the applying the window in the time domain.

[0506] 12. The method of solution 9 or 10, wherein the OTFS frame that is generated using the second pulse configuration is a result of: generating a periodic time vector by applying an inverse Fourier transform to symbols along the Doppler dimension and periodically repeating a result of the applying; applying a discrete delay filter to the periodic time vector; and applying a time window to a result of the applying the discrete delay filter.

[0507] 13. The method of solution 12, wherein the applying the discrete delay filter to the periodic time vector includes: upsampling the periodic time vector by zero-padding; convolving the discrete delay filter with a result of the upsampling; decimating a result of the convolving to tire carrier sampling rate; and applying a time domain window to a result of the decimating.

[0508] 14. The method of any of above solutions, wherein the OTFS framing scheme specifies that the OTFS frame is framed according to a standard OTFS scheme in which the symbols along delay and Doppler dimensions comprises M rows along the delay dimension and N columns along the Doppler dimension, where N and M are positive integers greater than 1.

[0509] 15. The method of solution 15, wherein one or more rows along the delay dimension are allocated to pilot signals.

[0510] 16. The method of any of solutions 1-13, wherein the OTFS framing scheme specifies that the OTFS frame is framed according to a minimum-period (MP) on Doppler OTFS scheme in which a delay period is equal to resolution along the delay dimension.

[0511] 17. The method of solution 16, wherein the OTFS frame according to the MP on Doppler OTFS scheme is configured with pilot symbols that are evenly or unevenly distributed along the Doppler dimension.

[0512] 18. The method of solution 17, wherein the pilot symbols are distributed along the Doppler dimension without guard symbols.

[0513] 19. The method of any of solutions 16-18, wherein the OTFS frame generation uses a single Fourier transform operation.69184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0514] 20. The method of any of solutions 1-13, wherein the OTFS framing scheme specifies that the OTFS frame is framed according to a minimum-period (MP) on delay OTFS scheme in which a Doppler period is equal to resolution along the Doppler dimension.

[0515] 21. The method of solution 20, wherein the OTFS frame according to the MP on delay OTFS scheme is configured with a single pilot symbol along the delay dimension with or without guard symbols.

[0516] 22. The method of any of solution 20-21, wherein the OTFS frame is generated without using a Fourier transform operation.

[0517] 23. The method of solutions 16- 22, wherein the OTFS framing scheme further includes power boosting the pilot signals, and / or using guard bands for separating the pilot signals from information signals and / or other pilots signals.

[0518] 24. The method of any of solutions 1-13, wherein the OTFS framing scheme specifies that the OTFS frame is framed according a faster-than-Nyquist (FTN) OTFS scheme in which the OTFS frame is generated using a delay filter or a Doppler filter that has a roll-off factor that is greater than zero.

[0519] 25. The method of solution 24, wherein the OTFS framing scheme specifies that a delay resolution of the OTFS frame is smaller than 1 / BW or a Doppler resolution of the OTFS frame is smaller than 1 / T, where BW represents signal bandwidth and T represents signal duration.

[0520] 26. The method of any of solutions 23-25, the FTN OTFS scheme comprises generating the OTFS frame by: transforming a delay-Doppler grid having M rows and N’ columns to a discrete delay-Time variable by applying an inverse Fourier transform along the Doppler dimension; vectorizing the delay- Time variable in a column-wise manner; power-scaling a result of the vectorizing; precoding a result of the power-scaling; and generating a periodic time vector by periodizing a result of the precoding.

[0521] 27. The method of any of above solutions, wherein PRB configuration scheme specifies that one OTFS frame is mapped to more than one PRBs.

[0522] 28. The method of any of above solutions, wherein the transmission waveform comprises a multilayer transmission and wherein pilot locations of adjacent layers are in non-overlapping positions.

[0523] Embodiments of a receiving side operation may implement the following solutions. Here, the receiving side may be implemented in the network (e.g., a base station) and / or user devices that receive signals from a network or another user device.

[0524] 29. A method of digital communications (e.g., method 2520 depicted in FIG. 25B), comprising: receiving (2522), by a receiving device, a transmission waveform using time and frequency resources of a transmission medium, and processing (2524) the transmission waveform to extract pilots or information bits carried by the transmission waveform; wherein the time and frequency resources are divided into 70184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00physical resource blocks (PRBs) according to a PRB configuration scheme; wherein each PRB corresponds to an orthogonal time frequency space (OTFS) frame; wherein each OTFS frame comprises symbols along delay and Doppler dimensions according to an OTFS framing scheme.

[0525] 30. The method of solution 29, wherein the processing the transmission waveform includes: extracting information bits by decoding a PRB, wherein the decoding the PRB comprises recovering information symbols by processing a corresponding OTFS frame by generating, from the received transmission waveform, a delay-Doppler domain signal representation.

[0526] 31. The method of solution 30, wherein the generating comprises: upsampling the received transmission waveform by a factor Q; applying a delay filter to a result of the upsampling; decimating an output of the delay filter by a factor P, and applying a time domain window to an output of the decimating.

[0527] 32. The method of solution 31, wherein the applying the time domain window includes performing window folding.

[0528] 33. The method of solution 30, wherein the processing the transmission waveform includes a sequence of operations comprising: applying a time window to the received transmission waveform; periodically extending an output of the applying; upsampling an output of the periodically extending by a factor Q; filtering a result of the upsampling by a delay pulse; decimating a result of the filtering by a factor P; and recovering the OTFS frame from a result of the decimating.

[0529] 34. The method of any of solutions 29-33, wherein each OTFS frame comprises a minimumperiod (MP) on Doppler OTFS frame in which a delay period is equal to resolution along the delay dimension.

[0530] 35. The method of solution 34, further including: estimating a channel response of the transmission medium by interpolating channel responses at pilot locations along the Doppler dimension.

[0531] 36. The method of any of solutions 29-33, wherein each OTFS frame comprises a minimumperiod (MP) on Delay OTFS frame in which a Doppler period is equal to resolution along the Doppler dimension.

[0532] 37. The method of solution 36, further including estimating a channel response of the transmission medium to be a channel response obtained by processing a single pilot symbol along the delay dimension.

[0533] 38. The method of any of solutions 29-33, wherein each OTFS frame comprises a faster-than-Nyquist (FTN) OTFS scheme in which the OTFS frame is based on a delay filter or a Doppler filter that has a roll-off factor that is greater than zero.71184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0534] 39. The method of solution 38, further including: downsampling an output of the delay filter by a factor P’; applying a post-coder and a power scaling to a result of the downsampling; and generating an estimate of OTFS symbols of the OTFS frame from a result of the post-coder and the power scaling.

[0535] Section 5 discloses additional examples of receiver-side processing.

[0536] Transmitter- side embodiments may implement the following solutions.

[0537] 40. A method of digital communication (e.g., method 2530 depicted in FIG. 25C), comprising: generating (2532) information symbols by mapping source information bits for one or more user devices into the information symbols; generating (2534) orthogonal time frequency space (OTFS) frames from the information symbols and / or pilot symbols according to an OTFS framing scheme; transforming (2536) the OTFS frames to physical resource blocks (PRBs) in time-frequency domain according to a PRB configuration scheme; and generating (2538) a transmission waveform of a time slot by superpositioning the PRBs.

[0538] 41. The method of solution 40, wherein the mapping comprises producing parity bits by applying a forward error correction (FEC) scheme to the source information bits and mapping the source information bits along with the parity bits into the information symbols.

[0539] 42. The method of any of solutions 40-41, wherein the FEC scheme specifies that parity bits are computed over multiple OTFS frames.

[0540] Additional features of the method 2530 and the PRB configuration scheme and OTFS framing scheme have been described with reference to solutions 3 to 28.

[0541] Transmitter- side embodiments may implement the following solutions. Additional details are disclosed in Section 4.

[0542] 43. A method of wireless communication (e. g., method 2540 depicted in FIG. 25D), comprising: transmitting (2542) a transmission waveform to one or more receiving devices, wherein the transmission waveform is generated by multiplexing transmissions to the one or more receiving devices, wherein each transmission includes of one or more physical resource blocks (PRBs), wherein each PRB is mapped to time and frequency resources, wherein each PRB corresponds to an orthogonal time frequency space (OTFS) frame, wherein each OTFS frame is generated from symbols assigned to resources in the delay and Doppler dimensions.

[0543] 44. The method of solution 43, wherein the transmissions to the one or more receiving devices are multiplexed by upsampling the one or more PRBs to carrier sampling rate and shifting in time or frequency.

[0544] Additional features of the method 2530 have been described with reference to solutions 3 to 28.

[0545] A digital communication system may be implemented as follows.72184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0546] 45. A digital communication system, comprising: a network device configured to implement a method recited in any of solutions 1 to 28; and one or more wireless devices configured to implement a method recited in any of solutions 29 to 44.

[0547] 46. A wireless communication apparatus comprising one or more processors and a transceiver, wherein the one or more processors cause the wireless communication apparatus to perform the method recited in any of solutions 1 to 44.

[0548] 47. A system comprising a plurality of wireless communication apparatus, each apparatus comprising one or more processors that are configured to implement the method recited in any of solutions 1 to 44.

[0549] 13. Conclusion

[0550] It will be appreciated by those of skill in the art that the present document discloses various embodiments of transmit-side implementations, receive-side implementations and communication systems that are based on OTFS modulation and multiplexing technologies. Some of the disclosed features include the following:

[0551] 1) OTFS system consisting of a time-frequency PRB with configurable dimensions, into which an OTFS frame with a configurable delay-Doppler dimensions, is mapped.o Where each PRB and its OTFS frame are configured in both domains according to a specific user’ s channel characteristicso Where the duration and offset of the PRB are determined according to the latency requirements of the specific transmissiono Where the carrier bandwidth is divided into multiple PRBs■ Tight fit into carrier bandwidth: NPRB—LA / P^BJo Multiple PRBs are coded under the same FEC codeo The OTFS transmission in a slot is a super position of all PRBs

[0552] 2) Various PRB definitionso Where the PRB is mapped to a frequency location within the carrier bandwidtho Where the PRB has a configurable offset within a transmission sloto Where the PRB has a fixed bandwidth and configurable durationo Specific PRB numbers■ BWPRB= 168 kHzΔfPRB = 180 kHzTslot= 1 msec73184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00■ TPRB= -^ Tslot, nT= 1,2,...,14*Toffset= noffset / 14 · Tslot, noffset= 0,1,...,13

[0553] 3) Various Delay-Doppler processing definitionso OTFS frame has a delay-Doppler grid with M ■ N = NRE= BWPRB■ TPRB= 12 ■ nTo OTFS uses a sine pulse for the Doppler and a Root-Raised Cosine pulse for delay■ Delay pulse roll-off factor βτ= 0.07

[0554] 4) Various techniques for OTFS Frame Generationo Fourier transform on Doppler dimensiono Periodic extensiono Delay-Doppler filtering■ The process in Pulse Configuration #1■ The process in Pulse Configuration #2o Using delay filter for up-samplingo Frequency and time adjustment at carrier sampling rate

[0555] 5) Features of Standard OTFSo Pilot structureo Pilot boosting and location staggering between PRBs

[0556] 6) Features of MP-OTFS on Dopplero Single Doppler rowo Single Fourier transformo Generating processo Pilot structure

[0557] 7) Features of MP-OTFS on delayo Single delay columno No Fourier transformo Pilot Structureo Pilot boosting and location staggering between PRBs

[0558] 8) Features of FTN-OTFSo Reduced delay resolution AT < 1 / BVf1o New delay-Doppler dimensions N'RE= M · N' = 1 / α NREo Eigenvalue decomposition of delay filter ISIo Waveform generation process74184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00■ Additional power scaling and precoding■ Delay filter’s up-sampling with P' > aP a down-sampling with Q' = Q o A Similar process with a Doppler pulse that has a roll-off greater than zero

[0559] 9) Receiver Side-Processing for receiving OTFS signals

[0560] 10) Receiving process foro Pulse Configuration #1■ Window foldingo Pulse Configuration #2o Delay filter’s down-sampling

[0561] 11) Receiving process for MP-OTFS on Dopplero Channel estimation by interpolating the pilots along Dopplero MMSE (minimum mean square error) equalizationo MIMO

[0562] 12) Receiving process for MP-OTFS on Delayo Channel estimationo Equalization - single tap and multiple tap

[0563] 13) Receiving process for FTN-OTFSo Receiver processing■ Delay filter’s down-sampling with P' > aP a down-sampling with Q' — Q■ Post-coding■ Power scaling

[0564] The disclosed and other embodiments, modules and the functional operations described in this document can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this document and their structural equivalents, or in combinations of one or more of them. The disclosed and other embodiments can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer readable medium for execution by, or to control the operation of, data processing apparatus. The computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more of them. The term “data processing apparatus” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code75184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A propagated signal is an artificially generated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, which is generated to encode information for transmission to suitable receiver apparatus.

[0565] A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.

[0566] The processes and logic flows described in this document can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).

[0567] Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read -only memory or a random access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Computer readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.76184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00

[0568] While this patent document contains many specifics, these should not be construed as limitations on the scope of an invention that is claimed or of what may be claimed, but rather as descriptions of features specific to particular embodiments. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment.Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or a variation of a subcombination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results.

[0569] Only a few examples and implementations are disclosed. Variations, modifications, and enhancements to the described examples and implementations and other implementations can be made based on what is disclosed.77184723746.3

Claims

1. PCT Patent Application Attorney Docket No. 119314.8128.WO00WHAT IS CLAIMED IS:

1. A method of digital communications, comprising:transmitting, by a transmitting device, a transmission waveform to one or more receiving devices using one or more time and frequency resources of a transmission medium,wherein the time and frequency resources are divided into physical resource blocks (PRBs) that are configured according to a PRB configuration scheme;wherein each PRB is generated from a corresponding orthogonal time frequency space (OTFS) frame;wherein each OTFS frame is generated from symbols along delay and Doppler dimensions using an OTFS framing scheme.

2. The method of claim 1, wherein the PRB configuration scheme and / or the OTFS framing scheme are responsive to channel characteristics of transmission channels between the transmitting device and the one or more receiving devices.

3. The method of claim 1 or 2, wherein the transmission waveform is configured to occupy a carrier bandwidth of a carrier.

4. The method of claim 3, wherein each PRB is defined by a frequency location and a bandwidth that is a portion of the carrier bandwidth.

5. The method of claim 4, wherein the time resources are defined according to slots, wherein each PRB is defined by an offset within a slot and a duration, wherein the offset and / or the duration are configurable according to the PRB configuration scheme.

6. The method of claim 5, wherein each OTFS frame is generated by applying a Doppler pulse along the Doppler dimension.

7. The method claim 6, wherein each OTFS frame is generated by applying a delay pulse along the delay dimension.78184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO008. The method of claims 6 or 7, wherein the OTFS framing scheme specifies that the OTFS frame is generated according to a first pulse configuration in which the Doppler pulse is applied first to symbols, followed by the delay pulse.

9. The method of claims 6 or 7, wherein the OTFS framing scheme specifies that the OTFS frame is generated according to a second pulse configuration in which the delay pulse is applied first followed by the Doppler pulse.

10. The method of claim 8 or 9, wherein the OTFS frame is generated by transforming a discrete delay-Doppler grid comprising symbols to a discrete delay-Time variable by applying a Fourier transform or an inverse Fourier transform to the Doppler dimension.

11. The method of claim 8 or 10, wherein the OTFS frame that is generated according to the first pulse configuration is a result of:generating a periodic time vector by applying an inverse Fourier transform to symbols along the Doppler dimension and periodically repeating a result of the applying;applying a window in the time domain, wherein the window is computed from an inverse Fourier transform of the Doppler pulse; andapplying a delay pulse to a result of the applying the window in the time domain.

12. The method of claim 9 or 10, wherein the OTFS frame that is generated according to the second pulse configuration is a result of:generating a periodic time vector by applying an inverse Fourier transform to symbols along the Doppler dimension and periodically repeating a result of the applying;applying a discrete delay filter to the periodic time vector; andapplying a time window to a result of the applying the discrete delay filter.

13. The method of claim 12, wherein the applying the discrete delay filter to the periodic time vector includes:upsampling the periodic time vector by zero-padding;convolving the discrete delay filter with a result of the upsampling;decimating a result of the convolving to the carrier sampling rate; andapplying a time domain window to a result of the decimating.79184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO0014. The method of any of claims 1 to 13, wherein the OTFS framing scheme specifies that the OTFS frame is framed according to a standard OTFS scheme in which the symbols along delay and Doppler dimensions comprises M rows along the delay dimension and N columns along the Doppler dimension, where N and M are positive integers greater than 1.

15. The method of claim 14, wherein one or more rows along the delay dimension are allocated to pilot signals.

16. The method of any of claims 1 to 13, wherein the OTFS framing scheme specifies that the OTFS frame is framed according to a minimum-period (MP) on Doppler OTFS scheme in which a delay period is equal to resolution along the delay dimension.

17. The method of claim 16, wherein the OTFS frame according to the MP on Doppler OTFS scheme is configured with pilot symbols that are evenly or unevenly distributed along the Doppler dimension.

18. The method of claim 17, wherein the pilot symbols are distributed along the Doppler dimension without guard symbols.

19. The method of any of claims 16-18, wherein the OTFS frame generation uses a single Fourier transform operation.

20. The method of any of claims 1-13, wherein the OTFS framing scheme specifies that the OTFS frame is framed according to a minimum-period (MP) on delay OTFS scheme in which a Doppler period is equal to resolution along the Doppler dimension.

21. The method of claim 20, wherein the OTFS frame according to the MP on delay OTFS scheme is configured with a single pilot symbol along the delay dimension with or without guard symbols.

22. The method of any of claim 20-21, wherein the OTFS frame is generated without using a Fourier transform operation.80184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO0023. The method of claims 16- 22, wherein the OTFS framing scheme further includes power boosting the pilot signals, and / or using guard bands for separating the pilot signals from information signals and / or other pilots signals.

24. The method of any of claims 1-13, wherein the OTFS framing scheme specifies that the OTFS frame is framed according a faster-than-Nyquist (FTN) OTFS scheme in which the OTFS frame is generated using a delay filter or a Doppler filter that has a roll-off factor that is greater than zero.

25. The method of claim 24, wherein the OTFS framing scheme specifies that a delay resolution of the OTFS frame is smaller than 1 / BW or a Doppler resolution of the OTFS frame is smaller than 1 / T, where BW represents signal bandwidth and T represents signal duration.

26. The method of any of claims 23-25, the FTN OTFS scheme comprises generating the OTFS frame by:transforming a delay-Doppler grid having M rows and N ’ columns to a discrete delay-Time variable by applying an inverse Fourier transform along the Doppler dimension;vectorizing the delay-Time variable in a column-wise manner;power-scaling a result of the vectorizing;precoding a result of the power-scaling; andgenerating a periodic time vector by periodizing a result of the precoding.

27. The method of any of claims 1 to 26, wherein the PRB configuration scheme specifies that one OTFS frame is mapped to more than one PRBs.

28. The method of any of claims 1 to 27, wherein the transmission waveform comprises a multi-layer transmission and wherein pilot locations of adjacent layers are in non-overlapping positions.

29. A method of digital communications, comprising:receiving, by a receiving device, a transmission waveform using time and frequency resources of a transmission medium, andprocessing the transmission waveform to extract pilots or information bits carried by the transmission waveform;wherein the time and frequency resources are divided into physical resource blocks (PRBs) according to a PRB configuration scheme;81184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO00wherein each PRB corresponds to an orthogonal time frequency space (OTFS) frame; wherein each OTFS frame comprises symbols along delay and Doppler dimensions according to an OTFS framing scheme.

30. The method of claim 29, wherein the processing the transmission waveform includes:extracting information bits by decoding a PRB, wherein the decoding the PRB comprises recovering information symbols by processing a corresponding OTFS frame by generating, from the received transmission waveform, a delay-Doppler domain signal representation.

31. The method of claim 30, wherein the generating comprises:upsampling the received transmission waveform by a factor Q;applying a delay filter to a result of the upsampling;decimating an output of the delay filter by a factor P, andapplying a time domain window to an output of the decimating.

32. The method of claim 31, wherein the applying the time domain window includes performing window folding.

33. The method of claim 30, wherein the processing the transmission waveform includes a sequence of operations comprising:applying a time window to the received transmission waveform;periodically extending an output of the applying;upsampling an output of the periodically extending by a factor Q;filtering a result of the upsampling by a delay pulse;decimating a result of the filtering by a factor P; andrecovering the OTFS frame from a result of the decimating.

34. The method of any of claims 29 to 33, wherein each OTFS frame comprises a minimum-period (MP) on Doppler OTFS frame in which a delay period is equal to resolution along the delay dimension.

35. The method of claim 34, further including:estimating a channel response of the transmission medium by interpolating channel responses at pilot locations along the Doppler dimension.82184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO0036. The method of any of claims 29 to 33, wherein each OTFS frame comprises a minimum-period (MP) on Delay OTFS frame in which a Doppler period is equal to resolution along the Doppler dimension.

37. The method of claim 36, further including:estimating a channel response of the transmission medium to be a channel response obtained by processing a single pilot symbol along the delay dimension.

38. The method of any of claims 29 to 33, wherein each OTFS frame comprises a faster-than-Nyquist (FTN) OTFS scheme in which the OTFS frame is based on a delay filter or a Doppler filter that has a rolloff factor that is greater than zero.

39. The method of claim 38, further including:downsampling an output of the delay filter by a factor P’;applying a post-coder and a power scaling to a result of the downsampling; andgenerating an estimate of OTFS symbols of the OTFS frame from a result of the post-coder and the power scaling.

40. A method of digital communication, comprising:generating information symbols by mapping source information bits for one or more user devices into the information symbols;generating orthogonal time frequency space (OTFS) frames from the information symbols and / or pilot symbols according to an OTFS framing scheme;transforming the OTFS frames to physical resource blocks (PRBs) in time-frequency domain according to a PRB configuration scheme; andgenerating a transmission waveform of a time slot by superpositioning the PRBs.

41. The method of claim 40, wherein the mapping comprises producing parity bits by applying a forward error correction (FEC) scheme to the source information bits and mapping the source information bits along with the parity bits into the information symbols.

42. The method of any of claims 40-41, wherein the FEC scheme specifies that parity bits are computed over multiple OTFS frames.83184723746.3PCT Patent Application Attorney Docket No. 119314.8128.WO0043. A method of wireless communication, comprising:transmitting a transmission waveform to one or more receiving devices,wherein the transmission waveform is generated by multiplexing transmissions to the one or more receiving devices,wherein each transmission includes of one or more physical resource blocks (PRBs), wherein each PRB is mapped to time and frequency resources,wherein each PRB corresponds to an orthogonal time frequency space (OTFS) frame, wherein each OTFS frame is generated from symbols assigned to resources in the delay and Doppler dimensions.

44. The method of claim 43, wherein the transmissions to the one or more receiving devices are multiplexed by upsampling the one or more PRBs to carrier sampling rate and shifting in time or frequency.

45. A digital communication system, comprising:a network device configured to implement a method recited in any of claims 1 to 28; and one or more wireless devices configured to implement a method recited in any of claims 29 to 44.

46. A wireless communication apparatus comprising one or more processors and a transceiver, wherein the one or more processors cause the wireless communication apparatus to perform the method recited in any of claims 1 to 44.

47. A system comprising a plurality of wireless communication apparatus, each apparatus comprising one or more processors that are configured to implement the method recited in any of claims 1 to 44.84184723746.3

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