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WO2025166301A1 - Pilot signal processing in the delay-doppler domain - Google Patents

Pilot signal processing in the delay-doppler domain

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Publication number
WO2025166301A1
WO2025166301A1 PCT/US2025/014206 US2025014206W WO2025166301A1 WO 2025166301 A1 WO2025166301 A1 WO 2025166301A1 US 2025014206 W US2025014206 W US 2025014206W WO 2025166301 A1 WO2025166301 A1 WO 2025166301A1
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WIPO (PCT)
Prior art keywords
delay
doppler
signal
pulse
otfs
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PCT/US2025/014206
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French (fr)
Inventor
Shlomo Selim Rakib
Yoav Hebron
Ronny Hadani
Shachar Kons
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Cohere Technologies Inc
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Cohere Technologies Inc
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Publication of WO2025166301A1 publication Critical patent/WO2025166301A1/en
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Classifications

    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L27/00—Modulated-carrier systems
    • H04L27/26—Systems using multi-frequency codes
    • H04L27/2601—Multicarrier modulation systems
    • H04L27/2626—Arrangements specific to the transmitter only
    • H04L27/2627—Modulators
    • H04L27/2639—Modulators using other transforms, e.g. discrete cosine transforms, Orthogonal Time Frequency and Space [OTFS] or hermetic transforms
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B1/00—Details of transmission systems, not covered by a single one of groups H04B3/00 - H04B13/00; Details of transmission systems not characterised by the medium used for transmission
    • H04B1/69—Spread spectrum techniques
    • H04B1/7163—Spread spectrum techniques using impulse radio
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L27/00—Modulated-carrier systems
    • H04L27/26—Systems using multi-frequency codes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L5/00—Arrangements affording multiple use of the transmission path

Definitions

  • BACKGROUND [003] 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. [004] 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 [005] This document discloses techniques that may be used by communication networks to achieve several operational improvements. [006] In one example aspect, a method of transmitting a signal is disclosed.
  • the method includes generating a two-dimensional delay-Doppler signal comprising a sum of an information signal and a reference signal, wherein the sum is performed in delay-Doppler domain; and generating a transmission waveform from the two-dimensional delay-Doppler signal.
  • the method includes generating a transmission waveform as a summation of a first signal component and a second signal component, wherein the sum is performed in a time domain; wherein the first signal component comprises a reference signal, wherein the second signal component is generated by transforming a two-dimensional information signal from a delay-Doppler domain to a time domain, 170688394.5 Attorney Docket No.: 119314.8121.WO00 [008] In yet another aspect, a method of receiving a signal is disclosed.
  • the method includes receiving a received signal over a transmission channel, determining, by processing the received signal in a delay-Doppler domain, an estimate of cross-correlation between the received signal and a known reference signal that makes up the received signal, and estimating a delay-Doppler domain channel characteristic of the transmission channel based on the estimate of cross-correlation.
  • another method of transmitting a signal includes generating a transmission waveform as a summation of a first signal component and a second signal component, wherein the first signal component comprises an information signal modulated with a standard pulse-tone, and wherein the second signal component comprises a rotated pulse-tone.
  • FIG.1 shows an example communication network.
  • FIG.2 shows a simplified example of a wireless communication system in which uplink and downlink transmissions are performed.
  • FIG.3 shows an example OTFS modulation block diagram with a transmitter and a receiver.
  • FIG.4 shows an example flowchart for OTFS processing.
  • FIG.5 shows an example arrangement of reference signal (RS) antenna ports multiplexed in the delay-Doppler domain.
  • RS reference signal
  • FIG.6 shows an example pictorial depiction of the relationship between time, frequency, and ZAK domains.
  • FIG.7 shows an example pictorial depiction of the periodic and quasi-periodic nature of an information grid in the Zak domain.
  • FIG.8 shows an example of a graphical representation of an OTFS waveform.
  • FIG.9 shows an example block diagram of an OTFS communication system.
  • FIG.10 illustrates components of an example OTFS transceiver.
  • FIG.11 shows a block diagram of an example iterative receiver apparatus.
  • FIG.12 shows a block diagram of an example iterative receiver apparatus that uses multi-level decoding.
  • FIG.13 shows a block diagram of an example 2-D iterative equalizer.
  • FIG.14 depicts an example of an iterative decoder in which a single forward error correction (FEC) code is processed.
  • FIG.15 shows an example of an iterative decoder architecture for multi-level-coding (MLC) FEC.
  • FIG.16 is an illustration of an example channel equation.
  • FIG.17 shows an example of an iterative decoder architecture for improved channel estimation, through the decoder’s iterations.
  • FIG.18 depicts a two-antenna multi-input multi-output (MIMO) example.
  • MIMO multi-input multi-output
  • FIG.19 shows a pictorial depiction of a pulse-tone waveform.
  • FIG.20 depicts a visual example of a pulse-tone as a delay-Doppler domain waveform.
  • FIG.21 shows a pictorial representation of a pulse-tone being invariant under the Heisenberg Uncertainty Principle (HUP).
  • HUP Heisenberg Uncertainty Principle
  • FIG.22 is a pictorial depiction of a quasi-periodic pulse.
  • FIG.23 is a pictorial depiction of a quasi-periodic pulse – work around HUP.
  • FIG.24 depicts an example operation of a Zak transform implementation.
  • FIG.25 depicts an example of communication with pulse-tones.
  • FIG.26 shows a depiction of a crystalline regime along the delay-Doppler domain.
  • FIG.27 illustrates three fundamental signal representations.
  • FIG.28 shows examples of channel coupling in the TDM/FDM regimes.
  • FIG.29 shows examples of channel coupling in a crystalline regime.
  • FIG.30 shows a visual representation of aliasing versus the crystallization condition.
  • FIG.31 shows visual representations of aliasing in the FDM and TDM regimes.
  • FIG.32 shows a visual representation of a crystalline regime.
  • FIG.33 shows an example of radar sensing in a crystalline regime.
  • FIG.34 depicts an example of a joint communication and sensing system with a pair of unbiased pulse-tones.
  • FIG.35 shows an example of a transmitter-side method of reference signal overlaying an information signal.
  • FIG.36 shows an example of a transmitter-side method of reference signal overlaying an information signal.
  • FIG.37 shows an example of distortions caused by a channel.
  • FIG.38 shows an example of a hardware platform.
  • FIGS.39A–39D are flowcharts for example methods of facilitating digital communication.
  • FIG.1 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 (IoT) 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.
  • FIG.2 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.
  • 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.
  • the direction of transmission may be reversed.
  • Such transmissions are often called uplink or upstream 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.2).
  • Other type of transmissions in the network may include device-to-device transmissions, sometimes called direct or sideband transmissions.
  • downlink transmissions may be inbound transmissions for a user device, while outbound transmissions for a network device.
  • uplink transmission may be inbound transmissions for a network device while outbound transmissions from a wireless device.
  • 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.”
  • FDM frequency division multiplexing
  • TDM time division multiplexing
  • 5G solutions are expected to include massive MIMO antenna arrays of up to 256 antenna elements. These require a large number of reference signals to be multiplexed in order 170688394.5 Attorney Docket No.: 119314.8121.WO00 to simultaneously estimate the channels to and from the UEs.
  • the reference signal overhead is a crucial component in determining the overall throughput of the link.
  • Reference signals are typically placed in the time-frequency domain to assist the receiver in estimating the channel, generally in a coarse (regular or irregular) grid, as in LTE. Multiple antenna ports are multiplexed on the same coarse grid using different (ideally orthogonal) signature sequences. However, this orthogonality is often lost after transmission through the channel.
  • the present document provides, among other aspects, improvements to the OTFS technology by disclosing methods for inserting pilot or reference signals in the transmitted waveforms and correspondingly receiving signals that include pilot or reference signals, and process and use the pilot or reference signals for channel estimation. [0066] 4.1.
  • Orthogonal Time Frequency Space (OTFS) modulation is a modulation scheme whereby each transmitted symbol experiences a near-constant channel gain even in channels with high Doppler, large antenna arrays (massive MIMO), or at high frequencies such as millimeter waves.
  • OTFS modulates each information symbol onto one of a set of two- dimensional (2D) orthogonal basis functions that span the bandwidth and time duration of the transmission burst or packet.
  • the modulation basis function set of OTFS is specifically derived to combat the dynamics of the time-varying multipath channel.
  • OTFS can be implemented as a 170688394.5 Attorney Docket No.: 119314.8121.WO00 pre- and post-processing block to filtered OFDM systems, thus enabling architectural compatibility with LTE.
  • the main premise behind OTFS is to transform the time-varying multipath channel into a two-dimensional channel in the delay-Doppler domain. Through this transformation, all symbols over a transmission frame experience the same channel gain. That is because OTFS extracts the full diversity of the channel across time and frequency. This full diversity property of OTFS greatly reduces system overhead and the complexity associated with physical layer adaptation. It also presents the application layer with a robust fixed-rate channel, which is highly desirable in many of the delay-sensitive applications envisioned for 5G.
  • OTFS full diversity enables linear scaling of throughput with the number of antennas, regardless of channel Doppler.
  • OTFS enables dense and flexible packing of reference signals, a key requirement to support the large antenna arrays used in massive MIMO applications.
  • OTFS is designed so that its information symbols experience minimal cross-interference as well as full diversity in the delay-Doppler domain through appropriate design of the modulation lattice and pulse shape design in that domain.
  • OTFS as a modulation matches wireless channel characteristics through two processing steps; a transmitter first maps the two-dimensional delay-Doppler domain, where the information symbols (e.g., QAM symbols) reside (e.g., placed on a lattice or grid in the 2-D delay-Doppler domain), to the time-frequency domain through a combination of the inverse symplectic Fourier transform and windowing (e.g., each QAM symbol is spread throughout the time-frequency plane/domain). It then applies the Heisenberg transform to the time-frequency modulated signal to convert it into the time domain for transmission. A receiver performs the reverse operations.
  • the information symbols e.g., QAM symbols
  • windowing e.g., each QAM symbol is spread throughout the time-frequency plane/domain
  • OTFS works in the delay-Doppler coordinate system using a set of basis functions orthogonal to both time and frequency shifts. Both data and reference signals or pilots are carried in this coordinate system.
  • the delay-Doppler domain mirrors the geometry of the wireless channel, which changes far more slowly than the phase changes experienced in the 170688394.5 Attorney Docket No.: 119314.8121.WO00 rapidly varying time-frequency domain.
  • OTFS symbols experience the full diversity of the channel over time and frequency, trading latency for performance in high Doppler scenarios.
  • the complex baseband channel impulse response h( ⁇ ,v) characterizes the channel response to an impulse with delay ⁇ and Doppler v.
  • the received signal due to an input signal s(t) transmitted over this channel is given by .
  • (1) can also be interpreted as a linear operator ⁇ h( ⁇ ), parameterized by the impulse response h( ⁇ ,v), that operates on the input s(t) to produce the output r(t): .
  • OTFS modulation In the mathematics ⁇ h parameterized by a function h( ⁇ ,v) and operating on a function s(t) as defined in (2) is called a Heisenberg transform.
  • OTFS modulation also utilizes a Heisenberg transform on the transmitted symbols, hence the received signal becomes a cascade of two Heisenberg transforms, one corresponding to the OTFS modulation, and the other corresponding to the channel.
  • the corresponding structure of the received signal results in a near-constant gain on each of the transmitted symbols as well as a particularly simple mechanism to recover these symbols.
  • 4.1.2 OTFS Modulation [0077] OTFS modulation is comprised of a cascade of two two-dimensional transforms at both the transmitter and the receiver, as shown in FIG 3.
  • the transmitter first maps the information symbols x[n,m] residing (e.g., along the points of the reciprocal lattice ⁇ ⁇ ) in the two- Doppler domain to symbols X[n,m] (e.g., residing along the points of the lattice ⁇ ) in the time-frequency domain through a combination of the inverse symplectic Fourier transform and windowing. This cascade of operations is called the OTFS transform. Next the Heisenberg transform is applied to X[n,m] to convert the time-frequency modulated signal to the 170688394.5 Attorney Docket No.: 119314.8121.WO00 time domain signal s(t) for transmission over the channel.
  • Time-Frequency Modulation is a generic description of time-frequency modulation, which has the following components: A lattice or grid ⁇ in the time-frequency domain that is a sampling of the time and frequency axes at intervals T and ⁇ f respectively: .
  • a time-frequency modulator with these components maps the two-dimensional symbols X[n,m] on the lattice ⁇ to a transmitted waveform s(t) via a superposition of delay-and- operations on the pulse waveform gtx(t), namely (5)
  • Modulation of the OFDM transform mapping modulated symbols in the domain i.e., on each subcarrier to the transmitted signal in the time domain.
  • the received symbol X[n,m] is the same as the transmitted symbol except for a complex scale factor H[n,m], similar to OFDM transmitted through time-invariant frequency-selective fading
  • H[n,m] similar to OFDM transmitted through time-invariant frequency-selective fading
  • OTFS transforms utilize a variant of the standard Fourier transform called the Symplectic Finite Fourier Transform (SFFT). This transform is defined as follows. Let X p [n,m] denote the periodized version of X[n,m] with period (N,M).
  • SFFT Symplectic Finite Fourier Transform
  • OTFS can be defined as a time-frequency modulation with an additional pre-processing step.
  • OTFS incorporates a transmit windowing square summable function W tx [n,m] that multiplies the modulation symbols in the time-frequency domain.
  • W tx [n,m] the modulated symbols in OTFS can be defined as follows: . (21) [00101]
  • the transmitted defined in (6). (21) is called the OTFS transform, which combines an inverse symplectic transform with a windowing operation.
  • the second equation describes the Heisenberg transform of gtx(t) parameterized by the symbols X[n,m] into the transmitted signal s(t).
  • composition 170688394.5 Attorney Docket No.: 119314.8121.WO00 of these two transforms can comprise OTFS modulation, as depicted in the two transmitter blocks of FIG.3.
  • a different basis function representation useful in the OTFS demodulation process, discussed below, is as follows: [00103] The basis function b k , l [n,m] in the time-frequency domain. [00104] OTFS Demodulation: Let the receiver employ a receive windowing square summable function Wrx[n,m]. Then the demodulation operation can include the following steps: [00105] 1) Take the Wigner transform of the received signal, which yields .
  • the OTFS transmitter encodes information bits in one or more forward error correction (FEC) codes, corresponding to one or more symbol constellation levels (denoted by ⁇ ), interleave the coded bits and map them to symbols (typically QAM) which are then assigned to delay-Doppler grid elements.
  • FEC forward error correction
  • QAM symbol constellation levels
  • Some of the delay-Doppler grid elements may not be assigned with any symbols (value of zero) and others may be assigned with known symbols (pilots).
  • the OTFS modulator is applied to the delay-Doppler grid.
  • source bits e.g., data bits
  • the FEC coded outputs are interleaved through corresponding interleavers.
  • the resulting signals are mapped to symbols and mapped to a delay-Doppler grid along with pilot signals.
  • the resulting mapped signal is processed through an OTFS modulator to generate an OTFS waveform.
  • the OTFS waveform in the time domain may be considered to be a super-position of waveforms that are a combination of a pulse and a tone, called a pulse-tone, (one example being a PulsoneTM) multiplied by the grid elements ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ [00118] where ⁇ ⁇ ⁇ , ⁇ ⁇ are of a pulse-tone waveform) is defined as ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ [00119] where operation, ⁇ a ⁇ ⁇ ⁇ / ⁇ and ⁇ ⁇ ⁇ / ⁇ are the delay and Doppler grid resolutions, respectively, and ⁇ is the Dirac delta function.
  • the pulse-tones may be considered to be basis signals used for the delay-Doppler grid.
  • ⁇ ⁇ , ⁇ ⁇ ⁇ for the second term of (33), which represents an infinite delta train, with rotating by a time window: 170688394.5
  • the OTFS This equation may be implemented in multiple ways, such as using the Zak transform, a 2- dimensional transform, or using pulse-tones
  • a method of OTFS waveform generation in which the delta trains ⁇ ⁇ , ⁇ ⁇ ⁇ are multiplied by the delay-Doppler grid elements ⁇ , ⁇ is provided in an exemplary embodiment.
  • the resulting signal is combined and convolved with ⁇ ⁇ ⁇ ⁇ ⁇ to obtain the output signal.
  • the signal is composed in the delay domain.
  • the convolution operation is performed before adding the resulting signals together.
  • signal is composed in the Doppler domain.
  • pulse-tones are multiplied by grid elements and the result is combined to obtain the transmission waveform.
  • the time-domain signal can be rewritten as ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ [00126] [00127] 4.1.2.6 OTFS Receiver [00128]
  • the OTFS transform can include pre- and post-processing blocks to the OFDM modulator and demodulator in the transmitter and receiver respectively, as depicted in FIG 4. It should be noted that the OTFS modulation can also be derived as a pre- and post-processing of multicarrier systems other than OFDM (e.g., filter bank multicarrier).
  • OTFS Receiver Examples [00130]
  • the basis signals may combine certain mathematical properties of a pulse and a tone, and may be called a pulse-tone or a Pulsone.
  • Channel estimation may be performed by assigning to one or more delay-Doppler grid elements a known symbol (pilot) at the transmitter.
  • the received signal may be processed for finding out the grid elements and their values where the pilot symbol was transformed to by the channel interaction.
  • OTFS Multiplexing There are a variety of ways to multiplex several uplink or downlink transmissions in one OTFS frame. The most natural one is multiplexing in the delay-Doppler domain, such that different sets of OTFS basis functions, or sets of information symbols or resource blocks can be given to different users. Given the orthogonality of the basis functions, the users can be separated at the receiver. For the downlink, the UE (user equipment) need only demodulate the portion of the OTFS frame that is assigned to it.
  • the OTFS signals from all users extend over the whole time-frequency window, thus providing full diversity; that is, for a channel with Q clustered reflectors (Q multipath components separable in either the delay or Doppler dimension) the OTFS modulation can achieve a diversity order equal to Q. Furthermore, this full spreading is also advantageous from a Peak to Average Power Ratio (PAPR) point of view.
  • PAPR Peak to Average Power Ratio
  • PAPR peak-to-average power ratio
  • Low PAPR is an important goal for modulation/multiple access design since it reduces the maximum linear power requirements for the transmit amplifiers. This is particularly important for the uplink of cellular systems, since amplifiers in consumer devices such as handsets need to be low-cost.
  • OTFS (considered here with delay/Doppler multiplexing) can reduce uplink PAPR in two ways: (i) if a user is assigned a single Doppler frequency, then the PAPR is the same as for single-carrier transmission, i.e., significantly lower than for OFDM.
  • the packet transmission can extend over a longer 170688394.5 Attorney Docket No.: 119314.8121.WO00 period of time than in OFDM which allows to increase the maximum energy per bit under Tx power constraints. This is particularly relevant for short packets.
  • OTFS can achieve a superior trade-off between PAPR and performance compared to SC-FDMA, even in time-invariant channels. While SC-FDMA can have low PAPR during the active signal duration, the overall PAPR is only small if the signal has a duty cycle close to unity, which in turn requires that (due to the small packet size) it utilizes only a single (or very few) subcarriers.
  • OTFS can obtain full spreading in time and frequency while keeping the PAPR low.
  • OTFS reference signals or pilots are carried in the delay-Doppler domain as impulses to probe the channel. Each pilot has a space reserved around it to account for the maximum delay and Doppler spread of the channel. Like the information symbols, the pilots experience the same time and frequency diversity of the channel over the full observation bandwidth and time.
  • FIG.5 shows an example of such an arrangement of RS antenna ports (e.g., as RS impulses) in the delay-Doppler domain. Notice that each antenna port RS in FIG.5 is generally affected by a different channel.
  • the multiplexed reference signals are sampled in the time-frequency domain according to a selected coarse grid that does not overlap with the data grid points. This enables the observation window for estimating the channel from the reference signals to be independent of the data. Importantly, it also allows OTFS reference signals to be utilized for both OTFS as well as any multicarrier modulation, including OFDM and other proposed 5G waveforms. [00143] Because the reference signals are multiplexed in the delay-Doppler plane, which mirrors the geometry of the wireless channel, they can be very densely packed, based on the delay and Doppler characteristics of the channels.
  • further efficiency can be 170688394.5 Attorney Docket No.: 119314.8121.WO00 obtained with knowledge of channel conditions for different users or groups of users by flexibly assigning the users or groups different pilot spacing in the delay-Doppler domain. [00144] 5. Zak transforms [00145] Wireless devices may attempt to join a network while the channel between the wireless device and a base station may be impaired both in delay and in Doppler domains due to the movement of the wireless device and multi-path echoes between the wireless device and the base station. In a similar manner, the theoretical framework for operation of radars in detecting objects that could be moving, also benefits from waveforms that show similar robustness properties as the random access waveforms in the wireless domain.
  • 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.
  • a time-Zak transform may be used to convert from Zak domain to time domain.
  • a frequency-Zak transform may be used to convert from the Zak domain to the frequency domain.
  • the Fourier transform (or its inverse) may be used to convert between the time and frequency domains.
  • a Zak signal is a function ⁇ ⁇ ⁇ , ⁇ ⁇ of two variables.
  • the variable ⁇ is called delay and the variable ⁇ is called ⁇ ⁇ ⁇ , ⁇ ⁇ is assumed to be periodic along ⁇ with period ⁇ r and quasi-periodic along ⁇ ⁇ r .
  • Zak domain signals are related to time and frequency domain signals through canonical transforms ⁇ t and ⁇ f called the time and frequency Zak transforms.
  • the time and frequency Zak transforms are linear transformations: ⁇ t : ⁇ z ⁇ L 2 ⁇ t ⁇ ⁇ , (36) ⁇ f : ⁇ z ⁇ L 2 ⁇ f ⁇ ⁇ , (37) 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00149]
  • Zak factorization This factorization is sometimes referred to as the Zak factorization.
  • the Zak embodies the combinatorics of the fast Fourier transform algorithm.
  • the precise 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 integration 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.6.
  • FIG.6 The different signal domains and the transformations connecting between them are depicted in FIG.6.
  • the information bits are encoded on the delay-Doppler domain as a Zak signal x ⁇ ⁇ , ⁇ ⁇ and transmitted through the rule: OTFS ⁇ x ⁇ t ⁇ w ⁇ x ⁇ ⁇ , ⁇ ⁇ ⁇ , (38) [00151] where w ⁇ ⁇ x ⁇ ⁇ , ⁇ ⁇ w ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ called twisted convolution (to be explained in the present . The conversion to the physical time domain is done using the Zak transform. [00152] 5.1 Zak Theory [00153] In this section we describe the Zak realization of the signal space. A Zak realization depends on a choice of a parameter.
  • a delay-Doppler lattice is an integral span of a pair of linear independent vectors g 1, g 2 ⁇ V .
  • the associated lattice is the set: ⁇ ⁇ ⁇ a1 g 1 ⁇ a 2 g 2 : a 1 , a 2 ⁇ ⁇ , (39) 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00156]
  • the vectors g 1 and g 2 are called the lattice basis vectors. It is convenient to arrange the basis vectors as the first and second columns of a matrix G , i.e.,: ⁇
  • the volume of the definition the area of the fundamental domain which is equal to the absolute value of of G. Every lattice admits a symplectic reciprocal lattice, aka orthogonal complement lattice that we denote by ⁇ ⁇ .
  • the definition of ⁇ ⁇ is: ⁇ ⁇ v ⁇ V : ⁇ ⁇ v , ⁇ ⁇ ⁇ for every ⁇ ⁇ ⁇ , (41) [00158]
  • ⁇ is . say that ⁇ is critically sampled if ⁇ ⁇ ⁇ ⁇ .
  • 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.
  • the standard example of a critically sampled rectangular lattice is ⁇ rec ⁇ ⁇ ⁇ , generated by the unit matrix: G ⁇ 1 0 ⁇ rec ⁇ ⁇ , (44) ⁇ 0 1 ⁇ ⁇ 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00160]
  • An important example of critically sampled lattice that is not rectangular is the hexagonal lattice ⁇ ⁇ , generated by the basis matrix: G ⁇ a a 2 ⁇ hex ⁇ ⁇ ⁇ , (45) ⁇ 0 a ⁇ 1 ⁇ [00161]
  • the interesting attribute of the hexagonal lattice is that among all critically sampled lattices it has the longest distance between neighboring points.
  • a Zak signal is a function ⁇ :V ⁇ that satisfies the following quasi periodicity condition: ⁇ ⁇ v ⁇ ⁇ ⁇ ⁇ exp ⁇ j 2 ⁇ ⁇ v , ⁇ ⁇ ⁇ ⁇ v ⁇ , (47) [00165] for every v ⁇ V and . r , form: ⁇ ⁇ ⁇ ⁇ k ⁇ r, ⁇ ⁇ l ⁇ r ⁇ ⁇ exp ⁇ j 2 ⁇ k ⁇ r ⁇ ⁇ ⁇ ⁇ , v ⁇ , (48) [00166] that is to say r and quasi-periodic function along the delay dimension with quasi period ⁇ r .
  • 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.
  • the associated with the point ⁇ Consequently, the extended action of an impulse function h ⁇ ⁇ V ⁇ is given by: z . ⁇ , is given by twisted convolution of the impulse h with the waveform ⁇ .
  • the time Zak transform is integration along the Doppler dimension (taking the DC component) for every point of time.
  • the frequency Zak 170688394.5 Attorney Docket No.: 119314.8121.
  • WO00 transform is Fourier transform along the delay dimension.
  • the formulas of the inverse transforms are as follows: ⁇ ⁇ 1 t ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ exp ⁇ ⁇ j 2 ⁇ r n ⁇ ⁇ ⁇ ⁇ n ⁇ r ⁇ , (56) n ⁇ ⁇ [00176] for every and we will denote it by ⁇ ⁇ ⁇ t .
  • P ⁇ 1 ⁇ p ⁇ is given by: P ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ n ⁇ r ⁇ p ⁇ ⁇ ⁇ n ⁇ r ⁇ ⁇ ⁇ [00181]
  • P ⁇ r ⁇ 1 for every a, ⁇ , which means that it is of constant modulo 1 with a regular step function along ⁇ with constant step given by the Doppler coordinate ⁇ .
  • 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.
  • the OTFS transceiver structure depends on the choice of the following parameters: a critically sampled lattice ⁇ ⁇ ⁇ r,0 ⁇ ⁇ ⁇ ⁇ 0, ⁇ r ⁇ , a filter function w ⁇ ⁇ V ⁇ and an 170688394.5 Attorney Docket No.: 119314.8121.WO00 information grid specified by N , M ⁇ .
  • the filter function factorizes as w ⁇ , ⁇ ⁇ w ⁇ ⁇ ⁇ ⁇ w ⁇ ⁇ ⁇ ⁇ where the delay and Doppler factors are square root Nyquist with ⁇ ⁇ ⁇ and ⁇ ⁇ ⁇ r M respectively.
  • the modulated transmit waveform as: ⁇ ⁇ x ⁇ ⁇ ⁇ ⁇ z ⁇ w ⁇ ⁇ ⁇ x ⁇ P ⁇ (61) [00184] To summarize: 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. [00185] To better understand the structure of the transmit waveform we apply few simple algebraic manipulations to (61).
  • the bare OTFS waveform takes the form: x ⁇ P ⁇ ⁇ ⁇ exp ⁇ j 2 ⁇ m ⁇ K ⁇ n N ⁇ M ⁇ ⁇ ⁇ K ⁇ r ⁇ n ⁇ ⁇ , (64) ⁇ K [00188] In words, of pulse rate ⁇ r ⁇ ⁇ ⁇ 1 r where the shift is determined by the delay parameter n and the modulation is determined Doppler parameter m.
  • V ⁇ 2 be the delay-Doppler plane equipped with the standard symplectic form ⁇ : ⁇ ⁇ v1, v 2 ⁇ ⁇ 1 ⁇ 2 ⁇ ⁇ 2 ⁇ 1 , [00194] for every v1 ⁇ ⁇ ⁇ 1, ⁇ 1 ⁇ and . the polarization form: v 2 ⁇ ⁇ 1 ⁇ 2.
  • ⁇ the symplectic orthogonal complement of ⁇ defined by: ⁇ ⁇ v ⁇ V : ⁇ ⁇ v , ⁇ ⁇ ⁇ for every ⁇ ⁇ , [00199]
  • ⁇ ⁇ ⁇ family of lattices ⁇ ⁇ ⁇ 1 ⁇
  • ⁇ ⁇ ⁇ ⁇ L 2 or, equivalently, the number of points in the quotient group ⁇ / ⁇ ⁇ is equal to we introduce a discrete variant of the twisted convolution operation between functions on the lattice ⁇ .
  • a continuous Zak signal is a function ⁇ :V ⁇ that satisfies the quasi-periodicity condition: ⁇ ⁇ v ⁇ 1 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ v, ⁇ 1 ⁇ ⁇ ⁇ ⁇ v ⁇ , [00203] for every v ⁇ V and ⁇ 1 ⁇ ⁇ ⁇ , ⁇ ⁇ and ⁇ ⁇ ⁇ k ⁇ , l ⁇ ⁇ then 1 r r the condition takes the form: ⁇ ⁇ ⁇ ⁇ k ⁇ r, ⁇ ⁇ l ⁇ r ⁇ ⁇ ⁇ k ⁇ r , [00204] Given a pair of Zak product as: ⁇ 1 , ⁇ 2 ⁇ ⁇ ⁇ 1 ⁇ v ⁇ ⁇ 2 ⁇ v ⁇ dv , ⁇ [00205] We denote the Hilbert ⁇ ⁇ ⁇ V ⁇ 1, ⁇ ⁇ .
  • the Wigner transform of the rank one operator ⁇ 2 ⁇ 1 is the function ⁇ 1 , ⁇ 2 :V ⁇ given by: ⁇ 1, ⁇ 2 ⁇ v ⁇ ⁇ ⁇ ⁇ v ⁇ ⁇ 1 , ⁇ 2 , [00207] for every v ⁇ V , where ⁇ is called the cross-ambiguity function of the signals ⁇ 1 ⁇ 2 ⁇ , we denote the cross-ambiguity function simply by ⁇ ⁇ and refer to it as the ambiguity function of the signal ⁇ .
  • the conversion between the Zak to the time domain is carried through the Zak transform ⁇ : ⁇ L2 ⁇ t ⁇ ⁇ , given by: ⁇ r ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ d ⁇ , 0 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00208] for every ⁇ .
  • ⁇ n , m be the unique quasi-periodic extension of the delta function supported on the ⁇ n ⁇ , m ⁇ ⁇ ⁇ , for 0 ⁇ n ⁇ N ⁇ 1 and 0 ⁇ m ⁇ M ⁇ 1 , i.e.,: ⁇ n , m ⁇ ⁇ ⁇ ⁇ m ⁇ ⁇ ⁇ k ⁇ r ⁇ ⁇ ⁇ n ⁇ ⁇ ⁇ k ⁇ r , m ⁇ ⁇ ⁇ l ⁇ r ⁇ , l r ⁇ [00209]
  • Direct phase modulated, infinite delta pulse train 802 (see FIG.8), given by: ⁇ ⁇ ⁇ n, m ⁇ ⁇ ⁇ ⁇ ⁇ mk M ⁇ ⁇ ⁇ n ⁇ ⁇ ⁇ k ⁇ r ⁇ , k ⁇ ⁇ [00210] 5.3.3 Discrete [00211] The continuous Zak theory admits a (finite)
  • a discrete Zak signal is a function ⁇ : ⁇ that satisfies the following quasi-periodicity condition: ⁇ ⁇ ⁇ ⁇ ⁇ 1 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ 1 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , [00212] for every ⁇ ⁇ and ⁇ n ⁇ ⁇ , m ⁇ ⁇ ⁇ and ⁇ 1 ⁇ ⁇ k ⁇ r , l ⁇ r ⁇ then the condition takes the form: ⁇ ⁇ n ⁇ ⁇ k ⁇ r , m ⁇ ⁇ ⁇ l ⁇ r ⁇ ⁇ ⁇ mk ⁇ ⁇ r ⁇ ⁇ ⁇ n ⁇ ⁇ , m ⁇ ⁇ ⁇ ⁇ [00213] Given as: ⁇ 1 , ⁇ 2 L ⁇ 1 ⁇
  • the discrete Wigner transform of the rank one operator ⁇ 2 ⁇ 1 is the function ⁇ ⁇ given by: ⁇ 1, ⁇ 2 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 1 , ⁇ 2 L , [00216] for every ⁇ ⁇ , where is called the discrete cross- ambiguity function of the signals ⁇ 1 and L( ⁇ ⁇ ) ⁇ ⁇ L ⁇ ⁇ ⁇ for every ⁇ ⁇ and ⁇ ⁇ , it follows that ⁇ 1, ⁇ 2 with respect to the sub-lattice ⁇ ⁇ , i.e.,: ⁇ 1, ⁇ 2 ( ⁇ ⁇ ⁇ ⁇ ) ⁇ ⁇ ⁇ 1 , ⁇ 2 ⁇ ⁇ ⁇ , [00217] for every ⁇ ⁇ and denote the discrete cross- ambiguity function by ⁇ ⁇ and refer to it as function of ⁇ .
  • the transform ⁇ is called embedding and it sends a discrete function ⁇ : ⁇ to 170688394.5 Attorney Docket No.: 119314.8121.WO00 the generalized function (distribution) on V given by the following super-position of delta functions: ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ [00221] The sampling and to induced transforms between the corresponding Hilbert spaces of signals.
  • Such a function takes the form: w ⁇ , ⁇ ⁇ w ⁇ ⁇ ⁇ ⁇ w ⁇ ⁇ ⁇ ⁇ ⁇ , [00226] for every ⁇ , ⁇ ⁇ .
  • the w on a Zak signal ⁇ is carried through the Heisenberg transform, i.e.,: ⁇ w ⁇ ⁇ ⁇ w ⁇ ⁇ ⁇ w ⁇ ⁇ ⁇ ⁇ ⁇ , [00227]
  • the above equation signal and the Heisenberg transform While the relationship is described as a sequence of mathematical steps, in general, 170688394.5 Attorney Docket No.: 119314.8121.WO00 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.
  • Theorem 5.2 (Main theorem of filter theory). Given a discrete Zak signal ⁇ L and a Heisenberg filter w ⁇ ⁇ V ⁇ , the following relation holds: ⁇ ⁇ w ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ P ⁇ , ⁇ ⁇ function of the waveform ⁇ w is obtained from the ambiguity function of the sequence ⁇ through shaping with a pulse (whose shape depends on the particular value of ⁇ ). a sense, the design of an 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 ⁇ .
  • P ⁇ a desired pulse shape
  • N , M ⁇ are coprime odd integers.
  • FIG.9 is a block diagram of components of an exemplary OTFS communication system 300.
  • the system 300 includes a transmitting device 310 and a receiving device 330.
  • the transmitting device 310 and the receiving device 330 include first and second OTFS transceivers 315-1 and 315-2, respectively.
  • the OTFS transceivers 315-1 and 315-2 communicate, either unidirectionally or bidirectionally, via communication channel 320 in the manner described herein.
  • the system 300 may comprise a wireless communication system
  • the communication channel may comprise a wired communication channel such as, for example, a communication channel within a fiber optic or coaxial cable.
  • the communication channel 320 may include multiple pathways and be characterized by time/frequency selective fading.
  • FIG.10 illustrates components of an exemplary OTFS transceiver 400.
  • the OTFS transceiver 400 can be used as one or both of the exemplary OTFS transceivers 315 illustrated in the communication system 300 of FIG.9.
  • the OTFS transceiver 400 includes a transmitter module 405 that includes a pre-equalizer 410, an OTFS encoder 420 and an OTFS modulator 430.
  • the OTFS transceiver 400 also includes a receiver module 455 that includes a post- equalizer 480, an OTFS decoder 470 and an OTFS demodulator 460.
  • the components of the OTFS transceiver may be implemented in hardware, software, or a combination thereof. The disclosed OTFS methods will be described in view of the various components of the transceiver 400.
  • a method of OTFS communication involves transmitting at least one frame of data ([D]) from the transmitting device 310 to the receiving device 330 through the communication channel 320, such frame of data comprising a matrix of up to N 2 data elements, N being greater than 1.
  • the method comprises convolving, within the OTFS transceiver 315-1, the data elements of the data frame so that the value of each data element, when transmitted, is 170688394.5 Attorney Docket No.: 119314.8121.WO00 spread over a plurality of wireless waveforms, each waveform having a characteristic frequency, and each waveform carrying the convolved results from a plurality of said data elements from the data frame [D].
  • the OTFS transceiver 315-2 receives and deconvolves these wireless waveforms thereby reconstructing a replica of said at least one frame of data [D].
  • the convolution process is such that an arbitrary data element of an arbitrary frame of data ([D]) cannot be guaranteed to be reconstructed with full accuracy until substantially all of these wireless waveforms have been transmitted and received.
  • iterative equalization and decoding of multi-level encoded symbols exchange extrinsic information between the equalizer and the FEC decoder to achieve close to optimal performance, as shown in FIG.11 for an OTFS receiver 400.
  • the extrinsic information may include a priori knowledge of which transmission resources (e.g., time slots of subcarriers) use which particular FEC.
  • the equalizer 402 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 410 and OTFS transformation module 412. Then, these symbols are converted to bit likelihoods that are FEC decoded.
  • An inverse OTFS transform module 404 may apply inverse OTFS transform and a symbol demapper 406 may recover bits from modulation symbols.
  • the error-rate performance of the scheme 400 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 the equalizer has less information to improve the equalization.
  • the iterative receiver 550 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.12, allows the equalizer to receive in earlier iterations priors, which are dominant from the constellation symbols point of view and better improve the equalization.
  • 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 170688394.5 Attorney Docket No.: 119314.8121.WO00 met.
  • the received signal may be equalized by the equalizer 402.
  • the equalized signal may undergo an inverse OTFS transform (404), and the symbols from the resulting transformed signal may be demapped for decoding by multiple different FECs FEC1 to FECq (modules 558a to 558q).
  • FIG.13 is a block diagram of an example embodiment of an iterative 2-D equalizer 501.
  • the 2-D Iterative equalizer illustrated in FIG.13, iterates between the 2-D equalizer 503 and the FEC MAP decoder 505, by passing information from one to the other.
  • the MAP decoder After several iterations, the MAP decoder outputs estimation on the information bits.
  • 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.
  • a received grid element is denoted by ⁇ , ⁇ , where ⁇ ⁇ 0,1, ... , ⁇ ⁇ 1 and ⁇ ⁇ 0,1, , ... , ⁇ ⁇ 1.
  • the channel estimation module extracts the channel response h, from the channel estimation area in the delay-Doppler grid.
  • a delay-Doppler equalizer generates A Posteriori probability estimation of the data symbols, ⁇ ⁇ ⁇ ⁇ , based on ⁇ , h and the a priori probability ⁇ ⁇ ⁇ ⁇ ⁇ , which is fed-back from a previous iteration of the decoder.
  • a symbol demapper module computes bit Log-Likelihoods Ratios (LLRs), ⁇ , from the a posteriori probability, ⁇ ⁇ ⁇ ⁇ .
  • LLRs Log-Likelihoods Ratios
  • Extrinsic LLRs are derived by subtracting from ⁇ , the a priori LLRs, ⁇ ⁇ , computed in the previous iteration. The extrinsic LLRs may be deinterleaved and then they are fed into the FEC for decoding. If decoding is successful, the decoded information bits are passed to the next module following the iterative decoder for further processing.
  • the FEC will output coded bit LLRs, which may be interleaved and then fed into the symbol mapper as, ⁇ ⁇ .
  • the symbol mapper computes symbol a priori symbol probabilities, 170688394.5 Attorney Docket No.: 119314.8121.WO00 ⁇ ⁇ ⁇ , for the next iteration. Iterations are terminated, when there is a successful decoding in the FEC, or some other criterion is met, such as maximum number of iterations. [00258] If the transmission processing was based on MLC, the basic iterative decoder is modified to accommodate it as well, as illustrated in FIGS.12 and 15.
  • FIGS.11 and 14 show examples of iterative decoder architectures for a single FEC.
  • FIGS.12 and 15 show examples of iterative decoder architectures for multi-level coding (MLC) FEC.
  • the transmitter may allocate a unique pilot symbol in the channel estimation area of the delay-Doppler grid, at location ⁇ ⁇ , ⁇ ⁇ ⁇ .
  • this pilot symbol will be convolved with the channel response and thus allow computing it from the received delay-Doppler grid elements, ⁇ .
  • ⁇ ′ be the received delay-Doppler grid elements, at the channel estimation area (or some part of this area) and zero otherwise, cyclically shifted to the location of the pilot.
  • ⁇ ⁇ ⁇ , ⁇ , ... ⁇ ⁇ ⁇ , ⁇ , ⁇ , ... ⁇ be a set indexes in the delay-Doppler grid of ⁇ ⁇ , which satisfy:
  • the channel response, h is a vector of these received values: h ⁇ ⁇ h ⁇ ⁇ ,h ⁇ , ... , h
  • FIG.16 is an illustration of the channel equation.
  • the a posteriori probability equalizer computes for each delay-Doppler data symbol, ⁇ ⁇ ⁇ , ⁇ ⁇ , the estimated symbol probabilities ⁇ , ⁇ ⁇ Pr ⁇ , ⁇ ⁇ ⁇
  • ⁇ is the set of symbol constellation points of which the symbols were selected at the transmitter and ⁇ ⁇ ⁇ .
  • the a posteriori probability can be also computed and approximated as ⁇ ⁇ , ⁇ ⁇ Pr ⁇
  • the extrinsic LLRs of symbol ⁇ , ⁇ are computed as ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ log , ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ does the inverse permutation of the interleaver. These modules are optional.
  • FEC Forward error correction
  • the FEC decoder may also be an iterative decoder for codes such as low-density parity check (LDPC) codes or Turbo codes.
  • the symbol mapper converts the coded bits LLRs, computed by the FEC, ⁇ ⁇ , to constellation symbols probability vectors, ⁇ , ⁇ , where its ⁇ -th element ( ⁇ ⁇ 0,1, ... ,
  • the guard area will be as small as possible.
  • the following method allows using a smaller guard area, while refining the channel estimation through the decoder iterations, by removing interference from data symbols that have already been estimated.
  • the symbol mapper output probabilities for the data symbols, ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ is fed to the channel estimation module as well.
  • FIG. 18 depicts a transmitter example, with two antenna ports.
  • Two different streams of delay-Doppler symbols, x ⁇ and x ⁇ are mapped to two delay-Doppler grids, leaving the channel estimation area empty, except for a pilot symbol.
  • the pilot symbols are separated enough, that the channel response from each one of them does not overlap.
  • the receiver may have a delay-Doppler grid for each receive antenna port, ⁇ ⁇ , ⁇ , ... , ⁇ .
  • Two modules are modified to accommodate MIMO: Channel estimation [00293] 6.5.1 MIMO channel estimation [00294]
  • Channel response vectors are derived similarly to the SISO case, for each combination o f transmit and receive antenna.
  • the channel equation can be written in a matrix format: ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ [00295] or in a ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ [00296] where, ⁇ ⁇ and ⁇ , ⁇ h′ ⁇ , ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ h ⁇ ⁇ ⁇ ⁇ ⁇ [00297] 6.5.2 MIMO [00298]
  • the A Posteriori probability equation changes to a matrix form: ⁇ ⁇ , ⁇ ⁇ Pr ⁇ , ⁇ ⁇ ⁇
  • ⁇ ⁇ ⁇ is the set of symbol constellation points of which ⁇ ⁇ data symbols were selected at the transmitter, for each delay-Doppler grid element and ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ .
  • the a priori and a posteriori probability, ⁇ and ⁇ ⁇ ⁇ also use a matrix notation.
  • An example, for a MIMO MAP equalizer, using an iterative message-passing approach can be found in existing implementations (e.g., Ramachandran et al. “MIMO-OTFS in High-Doppler Fading Channels: Signal Detection and Channel Estimation”).
  • 7.1 OTFS Waveform - Pulse-tones [00301] The OTFS waveform is constructed from symbols assigned to a grid in a two- dimensional domain called the delay-Doppler.
  • the grid is characterized by a Doppler period ⁇ ⁇ , typically satisfying ⁇ ⁇ 2 ⁇ ⁇ , where ⁇ is the maximum expected Doppler shift, and a delay period, ⁇ ⁇ 1/ ⁇ .
  • the grid has ⁇ ⁇ ⁇ ⁇ ⁇ elements along Doppler and ⁇ ⁇ ⁇ ⁇ ⁇ elements along delay, where ⁇ of the OTFS signal and ⁇ is its duration.
  • the 170688394.5 typically quadrature amplitude modulation QAM
  • the 170688394.5 Attorney Docket No.: 119314.8121.
  • WO00 grid may include pilot symbols used for channel detection and estimation.
  • An OTFS waveform can be generated using pre-defined basis signals called Pulsones.
  • the OTFS waveform in the time domain can be considered to be a super-position of pulse-tones multiplied by the grid elements ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ , ⁇ [00303] where ⁇ , ⁇ are tone is defined as ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ [00304] where a operation, ⁇ ⁇ F ⁇ is the inverse Fourier transform of a pulse in the Doppler domain, ⁇ ⁇ ⁇ / ⁇ and ⁇ ⁇ ⁇ / ⁇ are the delay and Doppler grid resolutions, respectively, and ⁇ is the Dirac delta function.
  • the pulse-tones can be considered to be basis signals used for the delay-Doppler grid.
  • a pulse-tone is depicted in FIG.19.
  • This OTFS waveform carrier has the mathematical properties of invariance under time, delay and Doppler shifts. That is, the pulse-tone can remain invariant under the operations of time, delay and Doppler shift. Because of the invariance, when modeling different wireless channels, the underlying coefficients representing the channel remain stable.
  • FIG.20 depicts a visual mathematical example of a pulse-tone as a delay- Doppler domain waveform that is invariant under the Heisenberg uncertainty principle as depicted in FIG.21. The pulse is mathematically quasi-periodic as depicted along the delay and Doppler axes.
  • FIG.22 shows an illustrative representation of a quasi-periodic pulse in the delay- Doppler domain.
  • FIG.23 shows an example of a quasi-periodic pulse with invariance under the Heisenberg uncertainty principle [HUP] within a region defined by the delay period [ ⁇ ⁇ ⁇ and the Doppler period [ ⁇ ⁇ ⁇ as shown [e.g., quasi-periodic pulse – work around HUP].
  • FIG.24 depicts the operation of a Zak transform based implementation in which a pulse-tone is modeled as a time domain pulse realization of a quasi-periodic pulse in the delay-Doppler domain.
  • FIG.25 shows an example representation of a communication using pulse-tones within a region defined by the delay period [ ⁇ ⁇ ⁇ and the Doppler period [ ⁇ ⁇ ⁇ (e.g., crystalline regime).
  • FIG.26 shows a period curve with a depiction of a crystalline regime along the delay-Doppler two dimensional plane where pulse-tone based communication proves to be an efficient 170688394.5 Attorney Docket No.: 119314.8121.WO00 communication technique.
  • the crystalline regime can be the optimal regime for communication and Radar applications.
  • FIG.27 shows the following three fundamental signal representations of time, frequency, and delay-Doppler and example transforms such as Zak transforms to go from one to another.
  • the complexity of the Zak transform is half the complexity of the FFT.
  • 7.2 OTFS Waveform – Crystalline regime [00308]
  • FIG.28 shows examples of channel coupling in the TDM and FDM regimes. The coupling of the channel and the waveform in the TDM and FDM regimes is selective (i.e., fading and unpredictable).
  • FIG.29 depicts examples of channel coupling in the crystalline regime in which delay spread is less than a delay period, ⁇ ⁇ 1/ ⁇ , and Doppler spread is less than a Doppler period, ⁇ , typically satisfying ⁇ ⁇ 2 ⁇ ⁇ , where ⁇ is the maximum expected Doppler shift.
  • ⁇ 500 ⁇ ⁇ ⁇ ⁇ 20 ⁇ and ⁇ ⁇ 5 ⁇ ⁇ ⁇ ⁇ 50 ⁇ .
  • the crystalline regime is between the TDM regime and the FDM regime. In the crystalline regime the channel coupling of the doubly spread channel with the OTFS waveform crystallizes – i.e., predictable and non-fading.
  • the coupling of the channel and the waveform in the TDM/FDM regime is selective – fading and unpredictable.
  • Aliasing which causes time selectivity in the TDM regime and frequency selectivity in the FDM regime, is the root cause. Fading and unpredictability occur in regions of self interaction as illustrated in FIG.30 and FIG.31. When the crystallization condition holds, there is no self interaction.
  • the effects of aliasing is the corruption of communication signals, which are corrected in the crystalline regime. That is, the optimal regime for communication is the crystalline regime, shown in FIG.32, where the channel coupling crystallizes and is predictable and non-fading.
  • OTFS in the crystalline regime is advantageous as superior BER performance is achievable under perfect CSI and under non-perfect CSI (i.e., under model-free mode of operation). Also, within the crystalline regime, OTFS is optimal for Radar sensing as high resolution detection of the delay-Doppler characteristics of reflectors can be achieved with no ambiguity. An example of Radar sensing in the crystalline regime is shown in FIG.33. Additionally, within the crystalline regime, OTFS is optimal for joint communication and sensing with a pair of unbiased pulse-tones (e.g., a Crystaline rotated pulse-tone and a standard crystalline standard pulse-tone) as depicted in FIG.34, allowing simultaneous high communication throughput and high resolution sensing.
  • a pair of unbiased pulse-tones e.g., a Crystaline rotated pulse-tone and a standard crystalline standard pulse-tone
  • an information signal e.g., information bits
  • a crystalline standard pulse- tone to a generate a first signal component.
  • the first signal component can then be summed 170688394.5 Attorney Docket No.: 119314.8121.WO00 (e.g., combined/added) together with a second signal component comprising a crystalline rotated pulse-tone to generate a signal to be transmitted over a channel.
  • the pair of crystalline standard pulse-tone and crystalline rotated pulse-tone are unbiased.
  • the signal transmitted over the channel can be provided as an input to an interference subtraction unit and as an input to a radar sensing unit.
  • the radar sensing unit can then output a radar image and also send the radar image to the interference subtraction unit.
  • the crystalline rotated pulse-tone can also be provided as an input to the interference subtraction unit. With the crystalline rotated pulse-tone, the radar image, and the signal transmitted over the channel provided as inputs to the interference subtraction unit, the interference subtraction unit can provide an output signal to a data detection unit that will then output the information signal (e.g., the information bits).
  • the information signal e.g., the information bits.
  • FIG.35 shows one example embodiment (on transmitter side) for adding pilot signals to a transmission waveform.
  • An information signal may be generated in a delay-Doppler domain (e.g., as QAM or QPSK symbol array).
  • a reference signal e.g., Ref. Signal (ROM/Gen.) x rs [m,n]
  • ROM/Gen. Ref. Signal
  • x rs [m,n] may be added to the information signal [e.g., Information DD x[m,n]] in the delay- Doppler domain, represented by the summation (sigma) block in FIG.35.
  • the resulting signal may be transformed into a time domain signal s(t) using one of several techniques to achieve an OTFS transformation of the signal.
  • FIG.36 shows another arrangement (on transmitter side) for adding a reference signal (e.g., Ref. Signal (ROM/Gen.) to the information signal (e.g.,, x[m,n]) to generate a transmission waveform s(t).
  • a reference signal e.g., Ref. Signal (ROM/Gen.
  • the information signal e.g., an array of symbols
  • the reference signal may be added to the resulting signal in the time domain to generate the transmission waveform s(t).
  • the reference signal may be generated in real-time using a waveform generator or may be pre-stored in a memory and read back during the process of combining with the information signal.
  • the reference signal may have the following properties: [00316] [1] The reference signal spreads over the entire ⁇ ⁇ ⁇ delay-Doppler grid. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00317] [2] Super-imposed over the data. [00318] [3] Satisfies: [00319] ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ ⁇ ⁇ reference signal after the OTFS transform.
  • FIG.37 pictorially depicts an example of effect of a transmission channel on a transmitted signal.
  • the depicted example shows a two-path channel (e.g., two discrete paths in the channel).
  • the grid shows a grid in a two-dimensional domain (e.g., delay-Doppler) with the top-left position showing identity (no distortion) coefficient.
  • two coefficients represent the distortion caused by the channel... h[1,1] and h[3,2].
  • the reference signal contribution at the [3,2] location comes from the reference signal at the [0, 0] location due to the operation of h[1,1] and h[3,2].
  • a receiver-side implementation can perform channel estimation by cross- correlating the received signal (with an underlying delay-Doppler domain representation) with a conjugate of the known reference signal to obtain an estimated channel response at coordinates [k,l].
  • the phase in the equation below may be used to compensate for the channel response rotation as a function of delay-Doppler coordinates.
  • For side can include cross correlating a received delay-Doppler data, ⁇ , ⁇ , with the conjugate of a known transmitted reference signal ⁇ ⁇ ⁇ , ⁇ ⁇ ⁇ , to obtain the estimated channel response at delay-Doppler coordinates ⁇ , ⁇ ; 170688394.5 Attorney Docket No.: 119314.8121.WO00 and the additional phase can be used to compensate for the channel response rotation as a function of the delay-Doppler coordinates.
  • FIG.38 is a block diagram representation of a wireless hardware platform 800 which may be used to implement the various methods described in the present document.
  • the hardware platform 800 may be incorporated within a base station or a user device.
  • the hardware platform 800 includes one or more processors 802, a memory 804 (this may be optional and in some cases the memory may be internal to the processor) and at least one transceiver circuitry 806.
  • the processor may execute instructions, e.
  • the memory 804 and/or the transceiver circuitry 806 may be partially or completely contained within the processor(s) 802 (e.g., same semiconductor package).
  • the following solutions may be preferably implemented by some embodiments.
  • some solutions may be as follows. [00333] 1.
  • a method of transmitting a signal comprising: generating (902) a two-dimensional delay-Doppler signal comprising a sum of an information signal and a reference signal, wherein the sum is performed in delay-Doppler domain; and generating (904) a transmission waveform from the two-dimensional delay-Doppler signal.
  • a transmission waveform is generated by performing an orthogonal time frequency space transform (OTFS) on the two-dimensional delay-Doppler signal.
  • OTFS orthogonal time frequency space transform
  • the OTFS transform comprises a forward or an inverse symplectic Fourier transform.
  • the method of solution 1, wherein the transmission waveform is generated by applying a Zak transform. [00337] 5. The method of solution 1, wherein the transmission waveform is generated by applying a two-dimensional transform. [00338] 6. The method of solution 1, wherein the transmission waveform is generated by applying a pulse-tone waveform in the delay-Doppler domain. [00339] 7. The method of solution 6, wherein the pulse-tone waveform comprises a quasi- periodic pulse. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00340] 8.
  • a method of transmitting a signal comprising: generating (912) a transmission waveform as a summation of a first signal component and a second signal component, wherein the sum is performed in a time domain; wherein the first signal component comprises a reference signal, wherein the second signal component is generated by transforming a two-dimensional information signal from a delay- Doppler domain to a time domain.
  • the second signal component is generated by performing an orthogonal time frequency space (OTFS) transform.
  • OTFS orthogonal time frequency space
  • the method of solution 10, wherein the second signal component is generated by applying a two-dimensional transform. [00347] 14. The method of solution 10, wherein the second signal component is generated by applying a pulse-tone waveform in the delay-Doppler domain. [00348] 15. The method of solution 14, wherein the pulse-tone waveform comprises a quasi- periodic pulse. [00349] 16. The method of solution 14, wherein the pulse-tone waveform is implemented in a portion of the delay-Doppler domain where a delay spread is less than a delay period and a Doppler spread is less than a Doppler period. [00350] 17.
  • a method implemented at a receiver-side comprising: receiving (922) a received signal over a transmission channel, determining (924), by processing the received signal in a delay-Doppler domain, an estimate of cross-correlation between the received signal and a known reference signal that makes up the received signal, 170688394.5 Attorney Docket No.: 119314.8121.WO00 and estimating (926) a delay-Doppler domain channel characteristic of the transmission channel based on the estimate of cross-correlation.
  • a method of transmitting a signal (e.g., method 930 depicted in FIG.39D), comprising: generating (932) a transmission waveform as a summation of a first signal component and a second signal component, wherein the first signal component comprises an information signal modulated with a standard pulse-tone, and wherein the second signal component comprises a rotated pulse-tone.
  • a transmission waveform as a summation of a first signal component and a second signal component, wherein the first signal component comprises an information signal modulated with a standard pulse-tone, and wherein the second signal component comprises a rotated pulse-tone.
  • a digital communication apparatus comprising one or more processor electronics and one or more transceiver electronics, wherein the transceiver is configured to receive or transmit a signal under control of the one or more processor electronics, and wherein the one or more processor electronics is configured to implement a method recited in any of solutions 1 to 22.
  • the hardware platform disclosed in FIG.38 may be used for the implementation of the above methods.
  • [00358] 24. A computer-readable storage medium having code stored thereon, the code, upon execution by one or more processors, causing the one or more processors to implement a method recited in any one or more of solutions 1-22.
  • OTFS operation in the crystalline regime can be advantageous for various communication applications such as Radar sensing and simultaneous joint communication and sensing applications as high communication throughput and high resolution 170688394.5 Attorney Docket No.: 119314.8121.
  • WO00 sensing/detection can be achieved in the crystalline regime where non-fading and predictability of the channel may be improved.
  • OTFS is a universal family of waveforms admitting TDM and FDM as limits.
  • the present document further discloses operational trade- offs and design choices available to implementors of OTFS technology.
  • crystalline regime is introduced.
  • OTFS is optimal for communication, the channel coupling crystallizes – becoming non-fading and predictable.
  • Theory and simulations demonstrate superior performance under perfect CSI and superior performance under non-perfect CSI (Model free).
  • OTFS is optimal for Radar sensing.
  • the disclosed waveform facilitate high resolution detection of delay-Doppler characteristics of the reflectors, with no (or suppressed) ambiguity.
  • OTFS is optimal for joint communication and sensing, e.g., using a pair of unbiased Pulsones (or pulse- tones) and provides simultaneous high communication throughput and high-resolution sensing.
  • 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 them.
  • 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., code that 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, that is generated to encode information for transmission to suitable receiver apparatus. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00364]
  • a computer program also known as a program, software, software application, script, or code
  • 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.
  • 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.
  • 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.
  • 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.
  • mass storage devices for storing data
  • 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.

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Abstract

A method implemented at a receiver-side in a digital communication network includes receiving a received signal over a transmission channel, determining, by processing the received signal in a delay-Doppler domain, an estimate of cross-correlation between the received signal and a known reference signal that makes up the received signal, and estimating a delay-Doppler domain channel characteristic of the transmission channel based on the estimate of cross-correlation.

Description

Attorney Docket No.: 119314.8121.WO00 PILOT SIGNAL PROCESSING IN THE DELAY-DOPPLER DOMAIN CROSS-REFERENCE TO RELATED APPLICATION [001] This application claims priority to U.S. Provisional Application No.63/548,700, filed on February 1, 2024, entitled “PILOT SIGNAL PROCESSING IN DELAY DOPPLER DOMAIN”, the disclosure of which is hereby incorporated by reference herein in its entirety. TECHNICAL FIELD [002] The present document relates to digital communication. BACKGROUND [003] 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. [004] 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 [005] This document discloses techniques that may be used by communication networks to achieve several operational improvements. [006] In one example aspect, a method of transmitting a signal is disclosed. The method includes generating a two-dimensional delay-Doppler signal comprising a sum of an information signal and a reference signal, wherein the sum is performed in delay-Doppler domain; and generating a transmission waveform from the two-dimensional delay-Doppler signal. [007] In yet another aspect, another method of transmitting a signal is disclosed. The method includes generating a transmission waveform as a summation of a first signal component and a second signal component, wherein the sum is performed in a time domain; wherein the first signal component comprises a reference signal, wherein the second signal component is generated by transforming a two-dimensional information signal from a delay-Doppler domain to a time domain, 170688394.5 Attorney Docket No.: 119314.8121.WO00 [008] In yet another aspect, a method of receiving a signal is disclosed. The method includes receiving a received signal over a transmission channel, determining, by processing the received signal in a delay-Doppler domain, an estimate of cross-correlation between the received signal and a known reference signal that makes up the received signal, and estimating a delay-Doppler domain channel characteristic of the transmission channel based on the estimate of cross-correlation. [009] In yet another aspect, another method of transmitting a signal is disclosed. The method includes generating a transmission waveform as a summation of a first signal component and a second signal component, wherein the first signal component comprises an information signal modulated with a standard pulse-tone, and wherein the second signal component comprises a rotated pulse-tone. [0010] In yet another example aspect, a wireless system in which one or more of the above- described methods are implemented is disclosed. [0011] These, and other, features are described in this document. DESCRIPTION OF THE DRAWINGS [0012] FIG.1 shows an example communication network. [0013] FIG.2 shows a simplified example of a wireless communication system in which uplink and downlink transmissions are performed. [0014] FIG.3 shows an example OTFS modulation block diagram with a transmitter and a receiver. [0015] FIG.4 shows an example flowchart for OTFS processing. [0016] FIG.5 shows an example arrangement of reference signal (RS) antenna ports multiplexed in the delay-Doppler domain. [0017] FIG.6 shows an example pictorial depiction of the relationship between time, frequency, and ZAK domains. [0018] FIG.7 shows an example pictorial depiction of the periodic and quasi-periodic nature of an information grid in the Zak domain. [0019] FIG.8 shows an example of a graphical representation of an OTFS waveform. [0020] FIG.9 shows an example block diagram of an OTFS communication system. [0021] FIG.10 illustrates components of an example OTFS transceiver. [0022] FIG.11 shows a block diagram of an example iterative receiver apparatus. [0023] FIG.12 shows a block diagram of an example iterative receiver apparatus that uses multi-level decoding. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [0024] FIG.13 shows a block diagram of an example 2-D iterative equalizer. [0025] FIG.14 depicts an example of an iterative decoder in which a single forward error correction (FEC) code is processed. [0026] FIG.15 shows an example of an iterative decoder architecture for multi-level-coding (MLC) FEC. [0027] FIG.16 is an illustration of an example channel equation. [0028] FIG.17 shows an example of an iterative decoder architecture for improved channel estimation, through the decoder’s iterations. [0029] FIG.18 depicts a two-antenna multi-input multi-output (MIMO) example. [0030] FIG.19 shows a pictorial depiction of a pulse-tone waveform. [0031] FIG.20 depicts a visual example of a pulse-tone as a delay-Doppler domain waveform. [0032] FIG.21 shows a pictorial representation of a pulse-tone being invariant under the Heisenberg Uncertainty Principle (HUP). [0033] FIG.22 is a pictorial depiction of a quasi-periodic pulse. [0034] FIG.23 is a pictorial depiction of a quasi-periodic pulse – work around HUP. [0035] FIG.24 depicts an example operation of a Zak transform implementation. [0036] FIG.25 depicts an example of communication with pulse-tones. [0037] FIG.26 shows a depiction of a crystalline regime along the delay-Doppler domain. [0038] FIG.27 illustrates three fundamental signal representations. [0039] FIG.28 shows examples of channel coupling in the TDM/FDM regimes. [0040] FIG.29 shows examples of channel coupling in a crystalline regime. [0041] FIG.30 shows a visual representation of aliasing versus the crystallization condition. [0042] FIG.31 shows visual representations of aliasing in the FDM and TDM regimes. [0043] FIG.32 shows a visual representation of a crystalline regime. [0044] FIG.33 shows an example of radar sensing in a crystalline regime. [0045] FIG.34 depicts an example of a joint communication and sensing system with a pair of unbiased pulse-tones. [0046] FIG.35 shows an example of a transmitter-side method of reference signal overlaying an information signal. [0047] FIG.36 shows an example of a transmitter-side method of reference signal overlaying an information signal. [0048] FIG.37 shows an example of distortions caused by a channel. [0049] FIG.38 shows an example of a hardware platform. [0050] FIGS.39A–39D are flowcharts for example methods of facilitating digital communication. 170688394.5 Attorney Docket No.: 119314.8121.WO00 DETAILED DESCRIPTION [0051] 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. [0052] 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. [0053] 1. Introduction – wireless communication environment examples [0054] 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. [0055] The relative movement of transmitters and receivers with respect to each other cause signal distortions such as varying channel delay, Doppler and/or angular spread, signal degradation due to ground clutter, sea clutter, and so on. Another example of signal degradation is flat fading in which an entire channel occupied by a transmission signal will experience fading or attenuation that may be relatively constant across the channel. In practice, a transmission scheme may need to be designed to fit within a certain link budget, maximum power constraint, or linearity of electronics used for transmitting or receiving signals. [0056] 2. Example wireless systems [0057] FIG.1 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 (IoT) 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. As further described herein, depending on the 170688394.5 Attorney Docket No.: 119314.8121.WO00 direction of transmission (uplink or downlink), the network station may be transmitting or receiving, and/or the user device may be receiving or transmitting. [0058] FIG.2 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.2, the direction of transmission may be reversed. 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.2). 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” for 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.” [0059] 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. [0060] 3. Technical advantages of the disclosed technology [0061] 5G solutions are expected to include massive MIMO antenna arrays of up to 256 antenna elements. These require a large number of reference signals to be multiplexed in order 170688394.5 Attorney Docket No.: 119314.8121.WO00 to simultaneously estimate the channels to and from the UEs. The reference signal overhead is a crucial component in determining the overall throughput of the link. [0062] Reference signals (RSs) are typically placed in the time-frequency domain to assist the receiver in estimating the channel, generally in a coarse (regular or irregular) grid, as in LTE. Multiple antenna ports are multiplexed on the same coarse grid using different (ideally orthogonal) signature sequences. However, this orthogonality is often lost after transmission through the channel. Understanding the effects of the channel in the delay-Doppler domain can lead to significant improvements in RS multiplexing overhead when multiplexing antenna port RSs in the delay-Doppler domain. [0063] 4. Signal transmission/reception using orthogonal time frequency space (OTFS) technology. [0064] Recently, co-inventors of this patent document and Applicant have conceived and implemented signal transmission and reception techniques using a new paradigm in which transmitted signals are shaped in a two-dimensional domain to achieve more robust and denser transmission performance. Details of the modulation, the signal waveform and reception and transmission techniques for receiving signals based on the OTFS technology are provided below. It is noted that, although the description often uses wireless transmission medium as an example transmission channel, same technology may also be deployed on wired channels such as copper wire, fiber, coaxial cable and the like. [0065] The present document provides, among other aspects, improvements to the OTFS technology by disclosing methods for inserting pilot or reference signals in the transmitted waveforms and correspondingly receiving signals that include pilot or reference signals, and process and use the pilot or reference signals for channel estimation. [0066] 4.1. OTFS (Orthogonal Time Frequency Space) technology [0067] Orthogonal Time Frequency Space (OTFS) modulation is a modulation scheme whereby each transmitted symbol experiences a near-constant channel gain even in channels with high Doppler, large antenna arrays (massive MIMO), or at high frequencies such as millimeter waves. In particular, OTFS modulates each information symbol onto one of a set of two- dimensional (2D) orthogonal basis functions that span the bandwidth and time duration of the transmission burst or packet. The modulation basis function set of OTFS is specifically derived to combat the dynamics of the time-varying multipath channel. OTFS can be implemented as a 170688394.5 Attorney Docket No.: 119314.8121.WO00 pre- and post-processing block to filtered OFDM systems, thus enabling architectural compatibility with LTE. [0068] The main premise behind OTFS is to transform the time-varying multipath channel into a two-dimensional channel in the delay-Doppler domain. Through this transformation, all symbols over a transmission frame experience the same channel gain. That is because OTFS extracts the full diversity of the channel across time and frequency. This full diversity property of OTFS greatly reduces system overhead and the complexity associated with physical layer adaptation. It also presents the application layer with a robust fixed-rate channel, which is highly desirable in many of the delay-sensitive applications envisioned for 5G. Moreover, full diversity enables linear scaling of throughput with the number of antennas, regardless of channel Doppler. In addition to OTFS's full diversity benefits, since the delay-Doppler channel representation is very compact, OTFS enables dense and flexible packing of reference signals, a key requirement to support the large antenna arrays used in massive MIMO applications. Moreover, OTFS is designed so that its information symbols experience minimal cross-interference as well as full diversity in the delay-Doppler domain through appropriate design of the modulation lattice and pulse shape design in that domain. [0069] OTFS as a modulation matches wireless channel characteristics through two processing steps; a transmitter first maps the two-dimensional delay-Doppler domain, where the information symbols (e.g., QAM symbols) reside (e.g., placed on a lattice or grid in the 2-D delay-Doppler domain), to the time-frequency domain through a combination of the inverse symplectic Fourier transform and windowing (e.g., each QAM symbol is spread throughout the time-frequency plane/domain). It then applies the Heisenberg transform to the time-frequency modulated signal to convert it into the time domain for transmission. A receiver performs the reverse operations. The modulated symbols in OTFS, after transmission through the channel, exhibit a constant channel gain over each frame due to the extraction of full time and frequency diversity in the demodulation process. [0070] 4.1.1 The Delay-Doppler Channel [0071] OTFS works in the delay-Doppler coordinate system using a set of basis functions orthogonal to both time and frequency shifts. Both data and reference signals or pilots are carried in this coordinate system. The delay-Doppler domain mirrors the geometry of the wireless channel, which changes far more slowly than the phase changes experienced in the 170688394.5 Attorney Docket No.: 119314.8121.WO00 rapidly varying time-frequency domain. OTFS symbols experience the full diversity of the channel over time and frequency, trading latency for performance in high Doppler scenarios. [0072] The complex baseband channel impulse response h(τ,v) characterizes the channel response to an impulse with delay τ and Doppler v. The received signal due to an input signal s(t) transmitted over this channel is given by . (1) [0073] An important typically there are only a small number of channel reflectors with associated Dopplers, far fewer parameters are needed for channel estimation in the delay-Doppler domain than in the time- frequency domain. This has important implications for channel estimation, equalization and tracking. [0074] Note that (1) can also be interpreted as a linear operator ^h(∙), parameterized by the impulse response h(τ,v), that operates on the input s(t) to produce the output r(t): . (2) [0075] In the mathematics ^h parameterized by a function h(τ,v) and operating on a function s(t) as defined in (2) is called a Heisenberg transform. OTFS modulation also utilizes a Heisenberg transform on the transmitted symbols, hence the received signal becomes a cascade of two Heisenberg transforms, one corresponding to the OTFS modulation, and the other corresponding to the channel. The corresponding structure of the received signal results in a near-constant gain on each of the transmitted symbols as well as a particularly simple mechanism to recover these symbols. [0076] 4.1.2 OTFS Modulation [0077] OTFS modulation is comprised of a cascade of two two-dimensional transforms at both the transmitter and the receiver, as shown in FIG 3. The transmitter first maps the information symbols x[n,m] residing (e.g., along the points of the reciprocal lattice Λ^) in the two- Doppler domain to symbols X[n,m] (e.g., residing along the points of the lattice Λ) in the time-frequency domain through a combination of the inverse symplectic Fourier transform and windowing. This cascade of operations is called the OTFS transform. Next the Heisenberg transform is applied to X[n,m] to convert the time-frequency modulated signal to the 170688394.5 Attorney Docket No.: 119314.8121.WO00 time domain signal s(t) for transmission over the channel. The reverse operations are performed in the receiver, mapping the received time signal r(t) first to the time-frequency domain through the Wigner transform (the inverse of the Heisenberg transform), and then to the delay-Doppler domain for symbol demodulation/detection. [0078] 4.1.2.1 Time-Frequency Modulation Below is a generic description of time-frequency modulation, which has the following components: A lattice or grid Λ in the time-frequency domain that is a sampling of the time and frequency axes at intervals T and Δf respectively: . (3) A packet burst with total A set of modulated symbols X[n,m], n = 0,…, N – 1, m = 0,…, M – 1 that is to be transmitted over a given packet A transmit pulse gtx(t) (and associated receive pulse grx(t)) whose inner product is bi- orthogonal with respect to translations by time T and frequency Δf, i.e. . (4) [0079] Note that the eliminates cross-symbol interference in symbol reception. [0080] A time-frequency modulator with these components maps the two-dimensional symbols X[n,m] on the lattice Λ to a transmitted waveform s(t) via a superposition of delay-and- operations on the pulse waveform gtx(t), namely (5) [0081] Modulation of the OFDM transform mapping modulated symbols in the domain (i.e., on each subcarrier) to the transmitted signal in the time domain. Similar to interpreting the channel operation (1) as a Heisenberg operator (2) applied to the transmitted signal s(t), the modulation 170688394.5 Attorney Docket No.: 119314.8121.WO00 of (5) can also be interpreted as a Heisenberg operator ^X( ∙) with parameters X[n,m] that is applied to the pulse shape gtx(t) as . (6) [0082] This interpretation is useful Heisenberg operators, one associated given the following property of such a cascade. [0083] Given a pair of Heisenberg operators parameterized by h1 and h2 and applied in cascade to a waveform g(t), we have , (7) where h(τ,v) = h2(τ,v) * σh1(τ,v) is (8) [0084] Applying the operators (6) and (1) yields the received signal , (9) where v(t) is the combined transform given by the twisted convolution of X[n,m] and h(τ,v): [0085] With this [0086] 4.1.2.2 Reception of Time-Frequency Modulated Signals: Sufficient Statistics and Channel Distortion [0087] The sufficient statistic for symbol detection based on the received signal is obtained by matched filtering of the received signal r(t) with the channel-distorted, information-carrying 170688394.5 Attorney Docket No.: 119314.8121.WO00 pulses (assuming that the additive channel noise is white and Gaussian). The matched filter first requires computation of the cross-ambiguity function Agrx,r(τ,v), defined as follows. . (11) [0088] The cross- correlation function. It is theory, indicating the implicit link between communication theory and radar. This function, when sampled on the lattice Λ, i.e., at τ = nT and at v = mΔf (i.e., at integer multiples of time T and frequency Δf), yields the matched filter output . (12) [0089] The operation of (12) Heisenberg transform of (5) and is a generalization of the inverse OFDM transform mapping the received OFDM signal to modulated symbols. [0090] We now establish the relationship between the matched filter output Y[n,m] and the transmitter input X[n,m]. We have already established in (9) that the input to the matched filter r(t) can be respect to a Heisenberg operator ^f(gtx(t)) parameterized by the impulse response f(τ,v) and operating on the pulse shape grx(t) (plus noise). The output of the matched filter then has two contributions: . (13) [0091] The last term on V(τ,v) = Agrx,v(τ,v), while the first term is the output of the matched filter corresponding to the of noise. A key result shows that this term can be expressed as the twisted convolution of the two-dimensional impulse response f(τ,v) with the function Agrx,gtx(τ,v): . (14) [0092] Substituting (10) end-to-end channel description as follows: . (15) 170688394.5 Attorney Docket No.: 119314.8121.WO00 [0093] Evaluating Eq. (15) on the lattice Λ the matched filter output estimate of the modulation symbols is obtained as follows . (16) [0094] In the case of an + V[n,m] where V[n,m] is recovers the transmitted symbols (plus noise) under ideal channel [0095] The for more general channels is now considered. In this case, if h(τ,v) has finite support bounded by (τmax, vmax) and if Agrx,gtx(τ,v) = 0 for τ ^ (nT – τmax, nT + τmax), v ^ (mΔf – vmax, mΔf + vmax), then Y[n,m] = H[n,m]X[n,m] for . (17) [0096] Note that for X[n,m] in either time n or frequency m. Hence, the received symbol X[n,m] is the same as the transmitted symbol except for a complex scale factor H[n,m], similar to OFDM transmitted through time-invariant frequency-selective fading Note that if the ambiguity function is only approximately bi-orthogonal in the neighborhood of (T,Δf) then there is some minimal cross-symbol interference. The fading H[n,m] in (17) that each symbol X[n,m] experiences has a complicated expression as a weighted superposition of exponentials. In the next section the specific transforms associated with OTFS as a time-frequency modulation are described, and how these transforms result in constant channel gains for each information symbol. [0097] 4.1.2.3 OTFS Modulation and Demodulation [0098] With the background of the previous two sections, the specific transforms associated with OTFS modulation and demodulation that result in a near-constant channel gain across symbols can now be defined. OTFS transforms utilize a variant of the standard Fourier transform called the Symplectic Finite Fourier Transform (SFFT). This transform is defined as follows. Let Xp[n,m] denote the periodized version of X[n,m] with period (N,M). The SFFT transform of Xp[n,m] is then defined as xp(k,l) = SFFT(Xp[n,m]) for 170688394.5 Attorney Docket No.: 119314.8121.WO00 [0099] The inverse transform is given by Xp[n,m] = SFFT–1(x[k,l]) for  , (19) where l = 0,…, M – 1, k = = 0 ≤ m ≤ M – 0 ≤ n ≤ N – then Xp[n,m] = X[n,m] for (n,m) ^ Z0 and the inverse signal X[n,m]. For X1[n,m] and X2[n,m] periodic 2D sequences , where denotes two- of the conventional discrete Fourier transform. Now OTFS can be defined as a time-frequency modulation with an additional pre-processing step. [00100] OTFS modulation: Consider a set of QAM information symbols arranged on a 2D grid x[k,l], k = 0,…, N – 1, l = 0,…, M – 1 that is to be transmitted. Further, assume a time-frequency modulation system defined by the lattice, packet burst, and bi-orthogonal transmit and receive pulses described in Section 4.1.2.1. In addition to these components, OTFS incorporates a transmit windowing square summable function Wtx[n,m] that multiplies the modulation symbols in the time-frequency domain. Given the above components, the modulated symbols in OTFS can be defined as follows: . (21) [00101] The transmitted defined in (6). (21) is called the OTFS transform, which combines an inverse symplectic transform with a windowing operation. The second equation describes the Heisenberg transform of gtx(t) parameterized by the symbols X[n,m] into the transmitted signal s(t). The composition 170688394.5 Attorney Docket No.: 119314.8121.WO00 of these two transforms can comprise OTFS modulation, as depicted in the two transmitter blocks of FIG.3. [00102] A different basis function representation useful in the OTFS demodulation process, discussed below, is as follows: [00103] The basis function bk , l[n,m] in the time-frequency domain. [00104] OTFS Demodulation: Let the receiver employ a receive windowing square summable function Wrx[n,m]. Then the demodulation operation can include the following steps: [00105] 1) Take the Wigner transform of the received signal, which yields . (23) [00106] 2) Apply the the time-frequency function YW[n,m] and then periodize the result to obtain the periodic (N,M) signal Yp[n,m]: [00107] 3) Apply the . (25) [00108] The last step can be symbols onto the two-dimensional orthogonal basis functions bk , l(n,m) as follows: 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00109] The estimated sequence xˆ ^ k , l ^ of information symbols obtained after demodulation can be given by the two-dimensional periodic convolution of the input QAM sequence x[n,m] and a sampled version of the windowed impulse response hw(∙): , (27) where (28) for hw(v′,τ′) the . (29) [00110] In (29) the of the time-frequency window W[n,m], defined as , (30) for W[n,m] = Wtx[n,m] the window W its support over time and frequency, hw(^,^) more closely approximates the response h(^,^). [00111] From (27) it can be seen that over a given frame, each demodulated symbol xˆ ^ l , k ^ for a given l and k experiences the same channel gain hw(0,0) on the transmitted symbol x[l,k]. Moreover, cross-symbol interference nearly vanishes if . [00112] Whether this and the window design. When cross-symbol interference is present an equalizer may be 170688394.5 Attorney Docket No.: 119314.8121.WO00 needed at the receiver to extract the full diversity of the channel. Common equalization architectures can be employed for this purpose. [00113] 4.1.2.4 OTFS Transmitter Examples [00114] In some embodiments, the OTFS transmitter encodes information bits in one or more forward error correction (FEC) codes, corresponding to one or more symbol constellation levels (denoted by ^^), interleave the coded bits and map them to symbols (typically QAM) which are then assigned to delay-Doppler grid elements. Some of the delay-Doppler grid elements may not be assigned with any symbols (value of zero) and others may be assigned with known symbols (pilots). Finally, the OTFS modulator is applied to the delay-Doppler grid. [00115] In an example of generating an OTFS waveform: from left to right, source bits (e.g., data bits) are input to a number of FEC stages, each operating at a corresponding code rate. The FEC coded outputs are interleaved through corresponding interleavers. The resulting signals are mapped to symbols and mapped to a delay-Doppler grid along with pilot signals. The resulting mapped signal is processed through an OTFS modulator to generate an OTFS waveform. [00116] A.2.5 OTFS Modulator Examples [00117] The OTFS waveform in the time domain, may be considered to be a super-position of waveforms that are a combination of a pulse and a tone, called a pulse-tone, (one example being a Pulsone™) multiplied by the grid elements ேି^ ெି^ ^^^^^^ ^ ^ ∙ ^^^^,^^^^^^ [00118] where ^^^^^,^^^ are of a pulse-tone waveform) is defined as ^ ^^^^,^^^^^^ ≜ ^^ ∗ ∙ ^ ^^^ଶగ^∆ఔ^ఛ ^^Δ^^ [00119] where operation, ఔ a Δ^^ ൌ ^^^/^^ and ∆^^ ൌ ^^^/^^ are the delay and Doppler grid resolutions, respectively, and ^^^∙^ is the Dirac delta function. In general, the pulse-tones may be considered to be basis signals used for the delay-Doppler grid. [00120] Here, we will use the notation ∆ ^^,^^ ^ ^^^^ for the second term of (33), which represents an infinite delta train, with rotating by a time window: 170688394.5 Attorney Docket No.: 119314.8121.WO00 ^ ∆ ^^,^^ ^ ^^^^ ≜ ^^௧^^^^ ∙ ^ ^^^ଶగ^∆ఔ^ఛ^^^൫^^ െ ^^Δ^^ െ ^^^^^൯ [00121] The OTFS This equation may be implemented in multiple ways, such as using the Zak transform, a 2- dimensional transform, or using pulse-tones [00122] A method of OTFS waveform generation in which the delta trains ∆ ^^,^^ ^ ^^^^ are multiplied by the delay-Doppler grid elements ^^^^^,^^^ is provided in an exemplary embodiment. The resulting signal is combined and convolved with ^^ఛ ^^^^ to obtain the output signal. Here, the signal is composed in the delay domain. [00123] In an exemplary embodiment, the convolution operation is performed before adding the resulting signals together. In other words, signal is composed in the Doppler domain. [00124] In an exemplary embodiment, pulse-tones are multiplied by grid elements and the result is combined to obtain the transmission waveform. [00125] Note, that there may be other equivalent implementation of equations (32)-(34). For example, the time-domain signal can be rewritten as ெି^ ேି^ ^ ^^^^^^ ൌ ^^ ∗ ^ ^ ^^^^^, ^^^ ∙ ^ ^^^ଶగ^∆ఔ^ఛ^^^൫^^ െ ^^Δ^^ െ ^^^^^൯ [00126] [00127] 4.1.2.6 OTFS Receiver [00128] In terms of implementation, the OTFS transform can include pre- and post-processing blocks to the OFDM modulator and demodulator in the transmitter and receiver respectively, as depicted in FIG 4. It should be noted that the OTFS modulation can also be derived as a pre- and post-processing of multicarrier systems other than OFDM (e.g., filter bank multicarrier). [00129] 4.1.2.7 OTFS Receiver Examples [00130] Here, we give further details on equalization in the delay-Doppler domain, when using OTFS with a set of predefined basis signals. For example, the basis signals may combine certain mathematical properties of a pulse and a tone, and may be called a pulse-tone or a Pulsone. [00131] Channel estimation may be performed by assigning to one or more delay-Doppler grid elements a known symbol (pilot) at the transmitter. At the receiver, the received signal may be processed for finding out the grid elements and their values where the pilot symbol was transformed to by the channel interaction. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00132] Let’s consider a pilot symbol that was assigned at the transmitter to the grid location ൫^^^,^^^൯, and received after the channel interaction at grid locations ^^^^ ,^^^ ^ with received values ℎ^, for ^^ ൌ 1,2, … ,Ω. [00133] To equalize symbols other than the pilot, a receiver will the estimated channel response obtained pilot to other locations on the grid, by rotating it: ℎ^^^^^, ^^^ ൌ ℎ^ ∙ ^^ ^ଶగ∙^൫^,^,^^,^^,ே,ெ൯ (34.2) [00134] where ^^^∙^ is a dimensions ^^ ൈ ^^. [00135] 4.1.3 OTFS Multiplexing [00136] There are a variety of ways to multiplex several uplink or downlink transmissions in one OTFS frame. The most natural one is multiplexing in the delay-Doppler domain, such that different sets of OTFS basis functions, or sets of information symbols or resource blocks can be given to different users. Given the orthogonality of the basis functions, the users can be separated at the receiver. For the downlink, the UE (user equipment) need only demodulate the portion of the OTFS frame that is assigned to it. It is noteworthy, however, that the OTFS signals from all users extend over the whole time-frequency window, thus providing full diversity; that is, for a channel with Q clustered reflectors (Q multipath components separable in either the delay or Doppler dimension) the OTFS modulation can achieve a diversity order equal to Q. Furthermore, this full spreading is also advantageous from a Peak to Average Power Ratio (PAPR) point of view. In the uplink direction, transmissions from different users experience different channel responses. Hence, the different subframes in the OTFS domain will experience a different channel. This can potentially introduce inter-user interference at the edges where two user subframes are adjacent, and would e.g., require guard gaps to eliminate it. [00137] 4.1.4 PAPR (peak-to-average power ratio) [00138] Low PAPR is an important goal for modulation/multiple access design since it reduces the maximum linear power requirements for the transmit amplifiers. This is particularly important for the uplink of cellular systems, since amplifiers in consumer devices such as handsets need to be low-cost. OTFS (considered here with delay/Doppler multiplexing) can reduce uplink PAPR in two ways: (i) if a user is assigned a single Doppler frequency, then the PAPR is the same as for single-carrier transmission, i.e., significantly lower than for OFDM. (ii) in conjunction, due to the spreading operation, the packet transmission can extend over a longer 170688394.5 Attorney Docket No.: 119314.8121.WO00 period of time than in OFDM which allows to increase the maximum energy per bit under Tx power constraints. This is particularly relevant for short packets. [00139] It is noteworthy that especially for short packets, OTFS can achieve a superior trade-off between PAPR and performance compared to SC-FDMA, even in time-invariant channels. While SC-FDMA can have low PAPR during the active signal duration, the overall PAPR is only small if the signal has a duty cycle close to unity, which in turn requires that (due to the small packet size) it utilizes only a single (or very few) subcarriers. However, such an approach, which is also used by LTE, leads to low frequency diversity and thus inferior performance; furthermore, for very short packets, SC-FDMA might still have a duty cycle less than unity even for such a configuration. OTFS, on the other hand, can obtain full spreading in time and frequency while keeping the PAPR low. [00140] 4.1.5 OTFS Reference Signals [00141] OTFS reference signals or pilots are carried in the delay-Doppler domain as impulses to probe the channel. Each pilot has a space reserved around it to account for the maximum delay and Doppler spread of the channel. Like the information symbols, the pilots experience the same time and frequency diversity of the channel over the full observation bandwidth and time. The interaction with the channel results in a 2D convolution of the delay-Doppler impulse response with the pilot – the effects of which are local, that is a delta in the delay-Doppler domain will be spread only to the extent of the support of the channel in the delay and in the Doppler dimensions. This fact provides the blueprint for multiplexing antenna ports in this domain, i.e., represent each antenna port sequence as an RS impulse, and space the impulses sufficiently far apart so that when the impulses are spread by the channel they still do not overlap, or overlap minimally. FIG.5 shows an example of such an arrangement of RS antenna ports (e.g., as RS impulses) in the delay-Doppler domain. Notice that each antenna port RS in FIG.5 is generally affected by a different channel. [00142] The multiplexed reference signals are sampled in the time-frequency domain according to a selected coarse grid that does not overlap with the data grid points. This enables the observation window for estimating the channel from the reference signals to be independent of the data. Importantly, it also allows OTFS reference signals to be utilized for both OTFS as well as any multicarrier modulation, including OFDM and other proposed 5G waveforms. [00143] Because the reference signals are multiplexed in the delay-Doppler plane, which mirrors the geometry of the wireless channel, they can be very densely packed, based on the delay and Doppler characteristics of the channels. In some embodiments, further efficiency can be 170688394.5 Attorney Docket No.: 119314.8121.WO00 obtained with knowledge of channel conditions for different users or groups of users by flexibly assigning the users or groups different pilot spacing in the delay-Doppler domain. [00144] 5. Zak transforms [00145] Wireless devices may attempt to join a network while the channel between the wireless device and a base station may be impaired both in delay and in Doppler domains due to the movement of the wireless device and multi-path echoes between the wireless device and the base station. In a similar manner, the theoretical framework for operation of radars in detecting objects that could be moving, also benefits from waveforms that show similar robustness properties as the random access waveforms in the wireless domain. [00146] As discussed above, 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 to convert 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. [00147] In its simplest form, a Zak signal is a function ^ ^ ^, ^ ^ of two variables. The variable ^ is called delay and the variable ^ is called ^ ^ ^, ^ ^ is assumed to be periodic along ^ with period^ r and quasi-periodic along ^ ^ r . The quasi periodicity condition is given by: ^ ^ ^ ^n ^r, ^ ^ m ^ r ^ ^ exp ^ j 2 ^ n ^ ^ ^ r ^ ^ ^ ^ , ^ ^ , (35) [00148] for every n , . are r r = 1. Zak domain signals are related to time and frequency domain signals through canonical transforms ^ t and ^ f called the time and frequency Zak transforms. In more precise terms, denoting of Zak signals by ^ z , the time and frequency Zak transforms are linear transformations: ^t : ^ z ^ L 2 ^ t ^^ ^ , (36) ^f : ^ z ^L 2 ^ f ^^ ^ , (37) 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00149] The pair ^ t and ^ f establishes a factorization of the Fourier transform FT = ^ ^ 1 t ^ ^ ^ f ^ . This factorization is sometimes referred to as the Zak factorization. The Zak embodies the combinatorics of the fast Fourier transform algorithm. The precise 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 integration 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.6. [00150] 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 x ^ ^, ^ ^ and transmitted through the rule: OTFS ^x ^^^t^ w ^^ x ^ ^ , ^ ^ ^ , (38) [00151] where w^ ^ x ^ ^, ^ ^ w ^ ^, ^ ^ ^ ^ called twisted convolution (to be explained in the present . The conversion to the physical time domain is done using the Zak transform. [00152] 5.1 Zak Theory [00153] 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. [00154] 5.1.1 Delay-Doppler Lattices [00155] A delay-Doppler lattice is an integral span of a pair of linear independent vectorsg 1, g 2^ V . In more details, given such a pair, the associated lattice is the set: ^ ^ ^a1 g 1 ^ a 2 g 2 : a 1 , a 2 ^^ ^ , (39) 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00156] The vectors g 1 and g 2 are called the lattice basis vectors. It is convenient to arrange the basis vectors as the first and second columns of a matrix G , i.e.,: ^ | | ^ G^ ^ ^ ^ ^ [00157] referred to as the basis matrix. , of the standard lattice under the matrix G. The volume of the definition the area of the fundamental domain which is equal to the absolute value of of G. Every lattice admits a symplectic reciprocal lattice, aka orthogonal complement lattice that we denote by ^ ^ . The definition of ^ ^ is: ^^ ^^v ^ V :^ ^ v , ^ ^ ^^ for every ^ ^^ ^ , (41) [00158] We say that ^ is . say that ^ is critically sampled if ^ ^ ^ ^ . 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 ^ , we define its maximal rectangular sub-lattice as ^ r ^^^ r ^ ^ ^ r where: ^r ^ arg min^ ^ ^ 0 : ^ ^ ,0 ^ ^^ ^ , (42) ^ ^ [00159] When either r or r , are we . We say a rectangular if ^ ^^ r . Evidently, a sub-lattice of a rectangular lattice is also rectangular. A rectangular lattice is under-sampled if ^ r ^ r > 1. The standard example of a critically sampled rectangular lattice is ^ rec ^^ ^ ^ , generated by the unit matrix: G ^ 1 0 ^ rec ^ ^ , (44) ^ 0 1 ^ ^ 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00160] An important example of critically sampled lattice that is not rectangular is the hexagonal lattice Λ୦^^, generated by the basis matrix: G ^ a a 2 ^ hex ^ ^ ^ , (45) ^ 0 a^ 1 ^ [00161] The interesting attribute of the hexagonal lattice is that among all critically sampled lattices it has the longest distance between neighboring points. The maximal rectangular sub- lattice of Λ୦^^ is generated by ^^^ and 2^^ଶ െ ^^^. From this point on we consider only rectangular lattices. [00162] 5.1.2 Zak waveforms [00163] A Zak realization is parametrized by a choice of a critically sampled lattice: ^ ^^ ^^r,0 ^ ^ ^ ^ 0, ^ r ^ , (46) [00164] where ^ r^ ^ r = 1. The signals in a Zak realization are called Zak signals. Fixing the lattice Λ, a Zak signal is a function ^ :V ^^ that satisfies the following quasi periodicity condition: ^ ^v^ ^ ^ ^exp^ j 2 ^^ ^ v , ^ ^ ^ ^ ^ v ^ , (47) [00165] for every v^ V and . r , form: ^ ^ ^ ^k ^r, ^ ^ l ^ r ^ ^ exp ^ j 2 ^^ k ^ r ^ ^ ^ ^ , v ^ , (48) [00166] that is to say r and quasi-periodic function along the delay dimension with quasi period ^ r . In conclusion, we denote the Hilbert space of Zak signals by ^ z . [00167] B.1.3 Heisenberg action [00168] The Hilbert space of Zak signals supports a realization of the Heisenberg representation. Given an element u^ V , the corresponding Heisenberg operator ^z ^ u ^ is given by: ^^z ^u ^^ ^ ^^ v ^^ exp^ j 2 ^^ ^ u , v ^ u ^ ^ ^ ^ v ^ u ^ , (49) 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00169] for every ^^^ z . 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 = ^ ^^ takes the form: ^^z ^ ^ ^^ ^ ^^v ^^ exp^ j 2 ^^ ^ ^ , v ^ ^ ^ ^ v ^ , (50) [00170] In words, the associated with the point ^ . Consequently, the extended action of an impulse function h^^ ^ V ^ is given by: z . ^ , is given by twisted convolution of the impulse h with the waveform ^ . [00172] 5.1.4 Zak transforms [00173] 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: ^t : ^ z ^ L 2 ^ t ^^ ^ , (52) [00174] As it turns out, the are along the reciprocal dimensions, see FIG.7. The formulas of the transforms are as follows: ^t ^ ^ ^^t ^ ^ ^ ^r ^ ^ t , ^ ^ dv , (54) [00175] for every ^^^ z . 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 170688394.5 Attorney Docket No.: 119314.8121.WO00 transform is Fourier transform along the delay dimension. The formulas of the inverse transforms are as follows: ^^1 t ^^ ^^ ^ , ^ ^ ^ ^ exp ^ ^ j 2 ^^^ r n ^ ^ ^ ^ ^ n ^ r ^ , (56) n ^ ^ [00176] for every and we will denote it by ^ ^ ^ t . As an intertwining transform ^ interchanges between the two Heisenberg operators ^z ^v , z ^ and ^t ^v , z ^ , i.e.,: ^ t ^ v ^ ^ ^ , (58) [00177] for every v^ V . From the property of the Zak transform is the interchanging equation (58). [00178] 5.1.5 Standard Zak signal [00179] Our goal is to describe the Zak representation of the window function: 1 0^t ^ p ^ ^ ^ t ^ ^ ^ r , (59) [00180] This function is typically (without CP). A direct application of formula (56) reveals that P ^^^1 ^ p ^ is given by: P ^^, ^ ^ ^^ ^ ^ ^ n ^ r ^ p ^ ^ ^ n ^ r ^ ^ ^ [00181] One can show that P ^ r ^ = 1 for every a, ^ , which means that it is of constant modulo 1 with a regular step function along ^ with constant step given by the Doppler coordinate ^^. 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. [00182] 5.2 OTFS with Zak [00183] The OTFS transceiver structure depends on the choice of the following parameters: a critically sampled lattice ^ ^^ ^^r,0 ^ ^ ^ ^ 0, ^ r ^ , a filter function w^^ ^ V ^ and an 170688394.5 Attorney Docket No.: 119314.8121.WO00 information grid specified by N , M ^^ . We assume that the filter function factorizes as w ^^, ^ ^^ w^ ^ ^ ^ w ^ ^ ^ ^ where the delay and Doppler factors are square root Nyquist with ^^ ^ ^ and ^^ ^ ^ r M respectively. We encode the information bits as a periodic 2D sequence of QAM symbols x ^ x ^ n ^^, m ^ ^ ^ with periods (N, 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 ^ n^^, m ^ ^ ^ where n = 0, .., N–1 and m = 0, .., M–1. We define the modulated transmit waveform as: ^ ^ x ^ ^ ^^ ^ z ^ w ^ ^ ^ x ^ P ^ (61) [00184] To summarize: 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. [00185] To better understand the structure of the transmit waveform we apply few simple algebraic manipulations to (61). First, we note that, being an intertwiner (Formula (58)), the Zak transform obeys the relation: ^^^z ^w ^ ^^ x ^ P ^ ^ ^ t ^ w ^^ ^ ^ x ^ P ^ , (62) [00186] Second, we note ^ can as twisted convolution w^ w^ ^ ^ w ^ . Hence, ^ t ^ w ^ ^ ^ ^ x ^ P ^ ^ ^ t ^ w ^ ^ ^ w ^ ^ ^ ^ ^ x ^ P ^ (63) [00187] where Wt ^ ^ w^ ^ and * stands for linear convolution in time. We refer to the waveform ^ ^x^ P ^ as the bare OTFS waveform. We see from Formula (63) 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 170688394.5 Attorney Docket No.: 119314.8121.WO00 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^^ ^ n ^ ^, m ^ ^ ^ . In this case, one can show that the bare waveform takes the form: x^ P ^ ^^exp^ j 2 ^ m ^ K ^ n N ^ M ^ ^ ^ K ^ r ^ n ^ ^ , (64) ^ K [00188] In words, of pulse rate^r ^ ^ ^ 1 r where the shift is determined by the delay parameter n and the modulation is determined Doppler parameter m. We next proceed to describe the de-modulation mapping. Given a received waveform ^^୰^, its de-modulated image y^^ ^ ^rx ^ is defined through the rule: ^ ^^rx ^^ w^ ^ ^1 ^ ^ ^ ^ rx ^ , (65) [00189] where w^ is the , . incorporate an additional step of sampling y and m^ 0,^ , M ^ 1. waveform design in the Zak realization [00191] 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. [00192] 5.3.1. Set-up for radar waveform design [00193] Let V ^^ 2 be the delay-Doppler plane equipped with the standard symplectic form ^ : ^ ^v1, v 2 ^^ ^ 1 ^ 2 ^ ^ 2 ^ 1 , [00194] for every v1^ ^ ^ 1, ^ 1 ^ and . the polarization form: v 2 ^^ ^ 1 ^ 2. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00195] Using the form ^ we introduce a binary operation between functions on V called twisted convolution. notations, we denote by^ ^z ^^exp ^ j 2 ^ z ^ the standard Fourier exponent. Given a pair of functions h1, h 2^^ twisted convolution to be: h 1 ^^ h 2 ^ v ^ ^ ^ ^ v 1, v 2 ^ ^ h 1 ^ v 1 ^ h 2 ^ v 2 ^ , v ^ v ^ v [00196] We fix a critically . rectangular of the form: ^ 1 ^^^ r ^ ^ ^ r , [00197] such that^r ^ ^ r ^ 1. We fix a rectangular super-lattice ^ ^^ 1 of the form: ^ ^^ ^ ^ ^ ^ ^ ^ , [00198] where ^^ ^ ^ r N and ^^ ^ ^ r M . We denote by L ^ ^ ^: ^1 ^ the index of ^ 1 as a sub-lattice of ^ . It is easy to verify that L ^ N ^ M . This number also counts the number of points in the finite quotient group ^ ^ ^ 1 . In addition, we denote by ^ the symplectic orthogonal complement of ^ defined by: ^^ ^{v ^ V :^ ^ v , ^ ^ ^^ for every ^^^ } , [00199] We have ^ ^ ^ family of lattices: ^ ^ ^^ 1 ^^ , [00200] We note that ^^^: ^^ ^ ^ ^ L 2 or, equivalently, the number of points in the quotient group ^ / ^ ^ is equal to we introduce a discrete variant of the twisted convolution operation between functions on the lattice ^ . Given a pair of functions h1, h 2 ^^ ^ ^ ^ , we define their twisted convolution to be: h 1 ^^ h 2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 1, ^ 2 ^ ^ h 1 ^ ^ 1 ^ h 2 ^ ^ 2 ^ . [00201] 5.3.2 [00202] 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 170688394.5 Attorney Docket No.: 119314.8121.WO00 transform. As it turns out, there is another fundamental domain called the Zak domain. A continuous Zak signal is a function ^ :V ^^ that satisfies the quasi-periodicity condition: ^ ^v ^^1 ^ ^ ^ ^ ^ ^ v, ^ 1 ^ ^ ^ ^ v ^ , [00203] for every v^ V and ^ 1 ^ ^, ^ ^ and ^ ^ ^k ^ , l ^ ^ then 1 r r the condition takes the form: ^ ^^ ^k ^r, ^ ^ l ^ r ^ ^ ^ ^ k ^^ r , , [00204] Given a pair of Zak product as: ^ 1, ^ 2 ^ ^ ^ 1 ^ v ^ ^^ 2 ^ v ^ dv , ^ [00205] We denote the Hilbert ^ ^^ ^V ^1, ^ ^ . We equip ^ with an Heisenberg action defined by the operator valued ^ End ^ ^ ^ defined by: ^ ^h ^^ ^^ h ^ ^ ^ , [00206] for every ^^^ and h^^ ^ the Heisenberg transform. The Heisenberg transform admits an inverse called the Wigner transform. Given a pair of Zak signals ^ 1, ^ 2 ^^ , the Wigner transform of the rank one operator ^ 2 ^ 1 is the function ^^ 1, ^ 2 :V ^^ given by: ^^ 1, ^ 2 ^v ^ ^ ^ ^ v ^ ^ 1 , ^ 2 , [00207] for every v^ V , where ^ is called the cross-ambiguity function of the signals ^ 1 ^^ 2 ^^ , we denote the cross-ambiguity function simply by ^ ^ and refer to it as the ambiguity function of the signal ^ . The conversion between the Zak to the time domain is carried through the Zak transform ^ : ^^ L2 ^ t ^^ ^ , given by: ^ r ^ ^ ^ ^ ^^ ^ ^ ^ , ^ ^ d ^ , 0 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00208] for every ^^^ . We conclude this section with an example of an explicit Zak signal and its time domain realization. Let ^n , m be the unique quasi-periodic extension of the delta function supported on the ^n^^, m ^ ^ ^ , for 0^n ^ N ^ 1 and 0^m ^ M ^ 1 , i.e.,: ^ n , m ^ ^ ^ ^ m ^ ^ ^ k ^ r ^ ^ ^ n ^ ^ ^ k ^ r , m ^ ^ ^ l ^ r ^ k , l r ^ [00209] Direct phase modulated, infinite delta pulse train 802 (see FIG.8), given by: ^ ^ ^ n, m ^ ^^^ ^ mk M ^ ^ ^ n ^ ^ ^ k ^ r ^ , k ^ ^ [00210] 5.3.3 Discrete [00211] 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 ^ : ^^^ that satisfies the following quasi-periodicity condition: ^ ^ ^ ^ ^1 ^ ^ ^ ^ ^ ^ ^, ^ 1 ^ ^ ^ ^ ^ ^ , [00212] for every ^ ^^ and ^n ^ ^, m ^ ^ ^ and ^1 ^ ^k ^r , l ^ r ^ then the condition takes the form: ^ ^ n ^ ^ ^ k ^ r , m ^ ^ ^ l ^ r ^ ^ ^ ^ mk ^ ^^ r ^ ^ ^ n ^ ^ , m ^ ^ ^ ^ [00213] Given as: ^ 1 , ^ 2 L ^ 1 ^ ^ ^ ^ ^ 2 ^ ^ ^ , ^ ^^ / ^ 1 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00214] We denote the Hilbert space of discrete Zak signals by ^L ^^ ^ ^ ^1, ^ ^ . One can show that dim^ L ^ L . We equip ^ L with an action of the expressed through the transform ^L : F ^ ^ ^ ^ End ^ ^ L ^ , given by: ^L ^ h ^^^ ^ h ^ ^ ^ , [00215] for every ^^^ L and h^^ as the discrete Heisenberg transform. The discrete transform admits an inverse called the discrete Wigner transform. Given a pair of discrete Zak signals ^ 1, ^ 2 ^^ L , the discrete Wigner transform of the rank one operator ^ 2 ^ 1 is the function ^ ^^ given by: ^^ 1, ^ 2 ^ ^ ^^ ^ ^ ^ ^ ^ ^ 1 , ^ 2 L , [00216] for every ^ ^^ , where is called the discrete cross- ambiguity function of the signals ^ 1 and L( ^^ ^^ ) ^ ^ L ^ ^ ^ for every ^ ^^ and ^^^^ ^ , it follows that ^^ 1, ^ 2 with respect to the sub-lattice ^ ^ , i.e.,: ^^ 1, ^ 2 (^ ^ ^ ^ ) ^ ^ ^ 1 , ^ 2 ^ ^ ^ , [00217] for every ^ ^^ and denote the discrete cross- ambiguity function by ^ ^ and refer to it as function of ^ . [00218] 5.3.4 Sampling theory on the Zak domain [00219] The focus of sampling theory is to describe the relation between the continuous and discrete cross-ambiguity functions. To this end, we denote by ^^ ^V ^ the vector space of generalized functions on V . The main assertions are of two basic transforms: s : ^ ^ V ^ ^^ ^ ^ ^ ^ , ^ , [00220] The transform s is called on V to its samples on the lattice ^ . The transform ^ is called embedding and it sends a discrete function ^ : ^^^ to 170688394.5 Attorney Docket No.: 119314.8121.WO00 the generalized function (distribution) on V given by the following super-position of delta functions: ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ , ^ ^^ [00221] The sampling and to induced transforms between the corresponding Hilbert spaces of signals. We denote the induced transforms by the same names, i.e.,: s : ^ ^^ ^ L , ^ : ^ ^^ ^ ^ , [00222] where ^^^^ ^ ^V ^1, ^ ^ of generalized Zak signals (distributions) h^^ ^ ^ ^ , we denote by h^ ^ its periodization with respect to the sub-lattice ^ ^ ^ ^ , i.e.,: h ^^ ^ ^ ^ ^ h ^ ^ ^ ^ , ^^ ^ ^ ^ ^^ ^ [00223] for every ^ ^^ . In plain 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. [00224] 5.3.5 Filter Theory [00225] Filter theory gives means to convert a discrete sequences to continuous waveforms. We define an Heisenberg filter to be a function w^^ ^ V ^ . We say the filter w is factorizable if it can be written as w^ w^ ^ ^ w ^ where w^ supported on the delay axis and w^ is a distribution on the Doppler axis. Note that such a function takes the form: w ^^, ^ ^^ w^ ^ ^ ^ w ^ ^ ^ ^ , [00226] for every^ , ^ ^^ . The w on a Zak signal ^ is carried through the Heisenberg transform, i.e.,: ^w ^ ^ ^ w ^^ ^ ^ w ^ ^ ^ , [00227] The above equation signal and the Heisenberg transform. While the relationship is described as a sequence of mathematical steps, in general, 170688394.5 Attorney Docket No.: 119314.8121.WO00 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. [00228] 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. [00229] Proposition 5.1. Given a pair of Zak signals ^ 1, ^ 2 ^^ and corresponding pair of Heisenberg filters w1, w 2^^ ^ V ^ , the following relation holds: ^ ^ w ★ , ^ 1,w1 , ^ 2 , w 2 1 ^^ ^ ^ 1 , ^ 2 ^ ^ w 2 conjugate function. ^ w the statement of the proposition describes the relation function of the sequence ^ and the continuous ambiguity function of the waveform ^ w . The result is summarized following theorem. [00232] Theorem 5.2 (Main theorem of filter theory). Given a discrete Zak signal ^^^ L and a Heisenberg filter w^^ ^ V ^ , the following relation holds: ^^ w ^^ ^ ^ ^ ^ ^ ^ P ^ , ^ ^^ function of the waveform ^ w is obtained from the ambiguity function of the sequence ^ through shaping with a pulse (whose shape depends on the particular value of ^ ). a sense, the design of an 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 ^ . [00235] 5.3.6 Zak theoretic chirp waveforms [00236] 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 assume N , M ^^ are coprime odd integers. We let a^ ^ ^ N ^ ^ be an invertible element 170688394.5 Attorney Docket No.: 119314.8121.WO00 in the ring of integers modulo N . We denote by^N :^ N ^ ^ the finite Fourier exponent ^N ^ n ^^ ^ ^ n N ^ . discrete Zak signal ch^ cha ^^ L as: ^ ch ^n^ ^ , m ^ ^ ^ ^^ ^ ^ N ^1 2 an 2 ^ m ^ 0 mod M , otherwise [00238] for every n , Zak chirp of order N and slope a . We next explore the structure of the discrete ambiguity function ^ ch . To that end, we introduce the sub-lattice ^ a ^^ given by: ^a ^^ ^ n ^^, kM ^ ^ ^ : k ^ a ^ n mod N ^ , [00239] Theorem 5.3: The discrete ambiguity function ^ ch is supported on the lattice ^ a . Moreover: ^ ^ 1 ch ^ n^^ , kM ^ ^ ^ ^ ^ N ^ an 2 ^ ^ N , ^ ^ on the non-zero points of the interval Ir ^ ^ ^ ^ r 2, ^ r 2 ^ ^ ^ ^ ^ r 2, ^ r 2 ^ , which refer to as the "clean" region. Next, we fix a filter function w^^ ^ V ^ and define the continuous Zak chirp Ch ^ Cha , w ^^ as: Ch ^ w ^ ^ ^ ^ ch ^ , [00242] By the main theorem of filter theory (Theorem 5.2) we know that the continuous ambiguity function ^ Ch is related to the discrete ambiguity function ^ ch through the equation: ^ Ch ^ ^ ^ ch ^ ^ ^ P ^ ^ , [00243] where Pv ^ w ^^ ^ ^ v ^ ^ ^ . the pulses P^ are well localized for every ^^^a ^ 2I r , the continuous ambiguity function ^ Ch will have a thumbtack shape with a clean region around zero coinciding with the interval I r . In case the numbers 170688394.5 Attorney Docket No.: 119314.8121.WO00 N , M ^ 1 , choosing he filter function w to be square root Nyquist with respect to the lattice ^ ensures P^ is well localized for every ^^^^ 2I r . [00244] 6. Iterative Decoding of an OTFS waveform in delay-Doppler domain [00245] In this section, iterative decoding of an OTFS waveform in the delay-Doppler is described. The iterative decoding consists of channel estimation in delay-Doppler, data symbol equalization and forward error correction (FEC) decoding. For simplicity, the description is provided for a Single-Input-Single-Output (SISO) system with one antenna port at the transmitter and one antenna port at the receiver. A straightforward extension for multiple antenna ports is given afterwards. [00246] FIG.9 is a block diagram of components of an exemplary OTFS communication system 300. As shown, the system 300 includes a transmitting device 310 and a receiving device 330. The transmitting device 310 and the receiving device 330 include first and second OTFS transceivers 315-1 and 315-2, respectively. The OTFS transceivers 315-1 and 315-2 communicate, either unidirectionally or bidirectionally, via communication channel 320 in the manner described herein. Although in the exemplary embodiments described herein the system 300 may comprise a wireless communication system, in other embodiments the communication channel may comprise a wired communication channel such as, for example, a communication channel within a fiber optic or coaxial cable. The communication channel 320 may include multiple pathways and be characterized by time/frequency selective fading. [00247] FIG.10 illustrates components of an exemplary OTFS transceiver 400. The OTFS transceiver 400 can be used as one or both of the exemplary OTFS transceivers 315 illustrated in the communication system 300 of FIG.9. The OTFS transceiver 400 includes a transmitter module 405 that includes a pre-equalizer 410, an OTFS encoder 420 and an OTFS modulator 430. The OTFS transceiver 400 also includes a receiver module 455 that includes a post- equalizer 480, an OTFS decoder 470 and an OTFS demodulator 460. The components of the OTFS transceiver may be implemented in hardware, software, or a combination thereof. The disclosed OTFS methods will be described in view of the various components of the transceiver 400. [00248] In one aspect a method of OTFS communication involves transmitting at least one frame of data ([D]) from the transmitting device 310 to the receiving device 330 through the communication channel 320, such frame of data comprising a matrix of up to N2 data elements, N being greater than 1. The method comprises convolving, within the OTFS transceiver 315-1, the data elements of the data frame so that the value of each data element, when transmitted, is 170688394.5 Attorney Docket No.: 119314.8121.WO00 spread over a plurality of wireless waveforms, each waveform having a characteristic frequency, and each waveform carrying the convolved results from a plurality of said data elements from the data frame [D]. Further, during the transmission process, cyclically shifting the frequency of this plurality of wireless waveforms over a plurality of times so that the value of each data element is transmitted as a plurality of cyclically frequency shifted waveforms sent over a plurality of times. At the receiving device 330, the OTFS transceiver 315-2 receives and deconvolves these wireless waveforms thereby reconstructing a replica of said at least one frame of data [D]. In the exemplary embodiment the convolution process is such that an arbitrary data element of an arbitrary frame of data ([D]) cannot be guaranteed to be reconstructed with full accuracy until substantially all of these wireless waveforms have been transmitted and received. [00249] 6.1 Iterative equalization and decoding of multi-level encoded symbols [00250] In general, iterative receivers exchange extrinsic information between the equalizer and the FEC decoder to achieve close to optimal performance, as shown in FIG.11 for an OTFS receiver 400. 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 402 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 410 and OTFS transformation module 412. 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 404 may apply inverse OTFS transform and a symbol demapper 406 may recover bits from modulation symbols. [00251] Compared to other techniques described next, the error-rate performance of the scheme 400 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 the equalizer has less information to improve the equalization. [00252] When multi-level encoding is applied at the transmitter, the iterative receiver 550, 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.12, 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 170688394.5 Attorney Docket No.: 119314.8121.WO00 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.12 the received signal may be equalized by the equalizer 402. In the forward path, the equalized signal may undergo an inverse OTFS transform (404), and the symbols from the resulting transformed signal may be demapped for decoding by multiple different FECs FEC1 to FECq (modules 558a to 558q). In the feedback path, the decoded symbol (bit) outputs of the FEC modules may be mapped to symbols (410) and transformed into OTFS domain signals (symbols) for feedback to the equalizer 402. 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. [00253] 6.2 Iterative two-dimensional (2-D) equalization [00254] FIG.13 is a block diagram of an example embodiment of an iterative 2-D equalizer 501. The 2-D Iterative equalizer, illustrated in FIG.13, 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. [00255] 6.3 Example implementations of iterative decoding of OTFS in delay-Doppler [00256] 6.3.1 Overview [00257] The iterative decoder, illustrated in FIG.14, uses for an input the received delay-Doppler grid of dimensions ^^ ൈ ^^. A received grid element is denoted by ^^^^^, ^^^, where ^^ ൌ 0,1, … , ^^ െ 1 and ^^ ൌ 0,1, , … , ^^ െ 1. First, the channel estimation module, extracts the channel response ℎ, from the channel estimation area in the delay-Doppler grid. Then, a delay-Doppler equalizer generates A Posteriori probability estimation of the data symbols, ^^^^^^, based on ^^, ℎ and the a priori probability ^^^ ^^^^, which is fed-back from a previous iteration of the decoder. A symbol demapper module, computes bit Log-Likelihoods Ratios (LLRs), ^^, from the a posteriori probability, ^^^^^^. Extrinsic LLRs are derived by subtracting from ^^, the a priori LLRs, ^^^, computed in the previous iteration. The extrinsic LLRs may be deinterleaved and then they are fed into the FEC for decoding. If decoding is successful, the decoded information bits are passed to the next module following the iterative decoder for further processing. If decoding is not successful, the FEC will output coded bit LLRs, which may be interleaved and then fed into the symbol mapper as, ^^^. The symbol mapper computes symbol a priori symbol probabilities, 170688394.5 Attorney Docket No.: 119314.8121.WO00 ^^^^^^^, for the next iteration. Iterations are terminated, when there is a successful decoding in the FEC, or some other criterion is met, such as maximum number of iterations. [00258] If the transmission processing was based on MLC, the basic iterative decoder is modified to accommodate it as well, as illustrated in FIGS.12 and 15. The LLRs from the symbol demapper are split into the different levels, optionally deinterleaved and then fed to the different FEC decoders. The coded bits LLRs output of the different FEC decoders are optionally interleaved and fed back to the symbol mapper. [00259] FIGS.11 and 14 show examples of iterative decoder architectures for a single FEC. [00260] FIGS.12 and 15 show examples of iterative decoder architectures for multi-level coding (MLC) FEC. [00261] 6.3.2 Channel Estimation [00262] For each antenna port, the transmitter may allocate a unique pilot symbol in the channel estimation area of the delay-Doppler grid, at location ^^^^,^^^൧. At the receiver, this pilot symbol will be convolved with the channel response and thus allow computing it from the received delay-Doppler grid elements, ^^. [00263] More specifically, let ^^′ be the received delay-Doppler grid elements, at the channel estimation area (or some part of this area) and zero otherwise, cyclically shifted to the location of the pilot. Let Ω ൌ ^Ω^,Ωଶ, … ^ ൌ ^^^^^, ^^^^, ^^^ଶ, ^^ଶ^, … ^ be a set indexes in the delay-Doppler grid of ^^ᇱ, which satisfy: |^^′^^^^ , ^^^^|ଶ ^ ^^ [00264] where ^^ is a threshold defined ^^ ൌ 1,2, … , |Ω|. [00265] The channel response, ℎ, is a vector of these received values: ℎ ൌ ^ℎ ᇱ ^,ℎଶ, … , ℎ|ஐ|൧ ൌ ^^^ ^^^^,^^^^, ^^ᇱ^^^ଶ,^^ଶ^, … ,^^′^^^|ஐ|, ^^|ஐ|൧^ [00266] and the |ஐ| ^^^^^^ ^^^ 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00267] Note, that ℎ^, corresponds to the location of the pilot symbol (non-ISI term). Each received element in the delay-Doppler grid, ^^, is connected to |Ω| different data symbols, ^^, through the channel response, ℎ, as illustrated in FIG.16. [00268] FIG.16 is an illustration of the channel equation. [00269] 6.3.3 Delay-Doppler equalization [00270] The a posteriori probability equalizer, computes for each delay-Doppler data symbol, ^^ ^ ^^, ^^ ^ , the estimated symbol probabilities ^^^^^^^^^^, ^^^^ ൌ Pr^^^^^^, ^^^ ൌ ^^^|^^, ℎ^ [00271] Where, ^^, is the set of symbol constellation points of which the symbols were selected at the transmitter and ^^^ ∈ ^^. [00272] The a posteriori probability can be also computed and approximated as ^^^^^^^^^^, ^^^^ ൌ Pr^^^|^^^^^,^^^ ൌ ^^^, ℎ^Pr^^^^^^,^^^ ൌ ^^^^ ≅ Pr^^^^^^,^^^ ൌ ^^^^ ^ Pr^^^^^^ᇱ,^^′^|^^^^^,^^^ ൌ ^^^ ,ℎ^ [00273] where, the corresponding received delay-Doppler elements, ^^^^^′,^^′^, are connected to data symbol ^^ ^ ^^, ^^ ^ , through the channel response, ℎ. [00274] In some existing implementations (e.g., Raviteja et al., “Low-Complexity Iterative Detection for Orthogonal Time Frequency Space Modulation”), an iterative message-passing solution to this problem is given for the case of uniform priors, Pr^^^^^^, ^^^ ൌ ^^^^ ൌ ^ |^|, and ideal channel estimation. The approach presented in this document, uses non-uniform priors, which are fed back from the FEC, along with a channel response that is directly extracted from the received signal. [00275] Note, that the iterative decoder may consist of components that are also being individually processed iteratively, such as the equalizer and the FEC decoder. The configuration of number of iterations for each one of these components and the iterative decoder itself, is an optimization parameter of the design. [00276] 6.3.4 Symbol demapper [00277] The symbol demapper module converts symbol probabilities into bit LLRs. Each constellation point, ^^^, is associated with a known bit labeling, ^^^ ൌ ^^^^^0^, ^^^^1^, … , ^^^^^^ െ 1^^, 170688394.5 Attorney Docket No.: 119314.8121.WO00 where ^^^^^^^ ∈ ^0,1^, ^^ ൌ 0,1, … , ^^ െ 1, and ^^ ൌ logଶ|^^| . The extrinsic LLRs of symbol ^^^^^, ^^^ are computed as ∑ ொ ൭ ^ୀ^ ^ ^ ^^^^^^^^^^,^^^^ ^^^^^ ∙ ^^^ ∙ ^^ ^ ^^^ ^ ^^^ ൌ log ,^ ೖ ^ ୀ^ ொ ^ െ ^^^^^^ ∙ ^^^ ∙ ^^ ^ ^^^ ^ ^^^ does the inverse permutation of the interleaver. These modules are optional. [00280] 6.3.6 Forward error correction (FEC) [00281] Forward error correction is applied to the LLRs. The FEC decoder may also be an iterative decoder for codes such as low-density parity check (LDPC) codes or Turbo codes. If the FEC decoder is successful in decoding, it passes the corrected information bits to the next module following the iterative receiver. If not, it generates LLRs for the coded bits, which are passed to the interleaver module. [00282] 6.3.7 Symbol mapper [00283] The symbol mapper converts the coded bits LLRs, computed by the FEC, ^^^, to constellation symbols probability vectors, ^^^^^^^^^, ^^^^, where its ^^-th element (^^ ൌ 0,1, … , |^^| െ 1) is computed as: ொି^ 1 ^ ^ ^ ^^^^^^ ∙ ^^^ ∙ ^^ ^ ^^^ ^ ^^^ ^^^ ^ ^^ ^ ^^ ^ ^ ^ ^ ^^ ^ ∙ ^ ^ [00284] [00285] The purpose of the guard area in the channel estimation part of the delay-Doppler grid, is to avoid interference from the data symbols to the channel response, and vice versa. For a better spectral efficiency, it is desired that the guard area will be as small as possible. The following method allows using a smaller guard area, while refining the channel estimation through the decoder iterations, by removing interference from data symbols that have already been estimated. In this iterative decoder architecture, illustrated in FIG.17, the symbol mapper output probabilities for the data symbols, ^^^ ^ ^^ ^ ^^, ^^ ^^ , is fed to the channel estimation module as well. Each data symbol, ^^^^^,^^^, that “leaks” into, ^^^^^,^^^, in the channel estimation area, through the channel response is estimated as |େ| ^ 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00286] and then subtracted from ^^^^^,^^^: |ஐ| ^^^^^^,^^^ ൌ ^^^^^, ^^^ െ ^ℎ^ ∙ ^̅^^^^^ െ ^^^^mod^^, ^^^ െ ^^^^mod^^^ [00287] Then, [00288] channel estimation, through the decoder’s iterations. [00289] 6.5 Iterative decoder for multiple-input multiple-output (MIMO) [00290] The iterative decoder described in the previous sections can be easily extended to support multiple antenna ports at the receiver and the transmitter, also known as, Multiple-Input- Multiple-Output (MIMO). At the transmitter side, a different delay-Doppler grid may be transmitted each one of the ^^௧௫ antenna ports. Each delay-Doppler grid should have a unique pilot symbol at the channel estimation area. The different pilot symbols should be separated enough, to prevent from their received channel responses to overlap. An example for this, for two antenna ports (^^௧௫ ൌ 2), is given in FIG. 18. [00291] FIG.18 depicts a transmitter example, with two antenna ports. Two different streams of delay-Doppler symbols, x^^^ and x^^^ are mapped to two delay-Doppler grids, leaving the channel estimation area empty, except for a pilot symbol. The pilot symbols are separated enough, that the channel response from each one of them does not overlap. [00292] Then, the receiver may have a delay-Doppler grid for each receive antenna port, ^^^^^, ^^^^^, … , ^^^ே^^ି^^. Two modules are modified to accommodate MIMO: Channel estimation [00293] 6.5.1 MIMO channel estimation [00294] Channel response vectors are derived similarly to the SISO case, for each combination of transmit and receive antenna. These vectors are denoted as, ℎ′^ఈ,ఉ^, where ^^ ൌ 0,1, … , ^^^௫ െ 170688394.5 Attorney Docket No.: 119314.8121.WO00 1 and ^^ ൌ 0,1, … , ^^௧௫ െ 1, and each one has หΩ ^ఈ,ఉ^ห elements. The channel equation can be written in a matrix format: ^^ ^^^^ ^^,^^ ^ ^ ⋮ ^ ^ [00295] or in a ≅ ^ ^^^ ∙ ^^^^^^ ^^^ [00296] where, Ψ ൌ and ^ఈ, ^ ℎ′^ఈ,ఉ^ ^^^ , ^^ ^ ∈ Ω^ఈ,ఉ^ ℎ ఉ ൌ ^ ^ ^ ^^^^^^^^ [00297] 6.5.2 MIMO [00298] The A Posteriori probability equation changes to a matrix form: ^^^^^^^^^^, ^^^^ ൌ Pr^^^^^^,^^^ ൌ ^^^|^^,^^^ [00299] Where, ^^ே^^, is the set of symbol constellation points of which ^^௧௫ data symbols were selected at the transmitter, for each delay-Doppler grid element and ^^ ே ^ ∈ ^^ ^^. The a priori and a posteriori probability, ^^^^^^ and ^^^^^^^^ also use a matrix notation. An example, for a MIMO MAP equalizer, using an iterative message-passing approach can be found in existing implementations (e.g., Ramachandran et al. “MIMO-OTFS in High-Doppler Fading Channels: Signal Detection and Channel Estimation”). [00300] 7.1 OTFS Waveform - Pulse-tones [00301] The OTFS waveform is constructed from symbols assigned to a grid in a two- dimensional domain called the delay-Doppler. The grid is characterized by a Doppler period ^^^, typically satisfying ^^^ ^ 2 ∙ ^^ௗ, where ^^ௗ is the maximum expected Doppler shift, and a delay period, ^^^ ൌ 1/^^^. The grid has ^^ ^ ^^^^ ∙ ^^^ elements along Doppler and ^^ ^ ^^^^ ∙ ^^^^^ elements along delay, where ^^^^ of the OTFS signal and ^^ is its duration. On top of the information bearing symbols (typically quadrature amplitude modulation QAM), the 170688394.5 Attorney Docket No.: 119314.8121.WO00 grid may include pilot symbols used for channel detection and estimation. An OTFS waveform can be generated using pre-defined basis signals called Pulsones. [00302] As described above, the OTFS waveform in the time domain, can be considered to be a super-position of pulse-tones multiplied by the grid elements ேି^ ெି^ ^^^^^^ ൌ ^ ^ ^^^^^, ^^^ ∙ ^^^^,^^^^^^ [00303] where ^^^^^,^^^ are tone is defined as ^^^^,^^^^^^ ≜ ^^ఛ^^^^ ∗ ^^^௧^^^^ ∙ ^ ^^^ଶగ^∆ఔ^ఛ^^^൫^^ െ ^^Δ^^ െ ^^^^^൯൩ [00304] where a operation, ^^௧^^^^ ൌ ℱି^^^^ఔ^ is the inverse Fourier transform of a pulse in the Doppler domain, Δ^^ ൌ ^^^/^^ and ∆^^ ൌ ^^^/^^ are the delay and Doppler grid resolutions, respectively, and ^^^∙^ is the Dirac delta function. In general, the pulse-tones can be considered to be basis signals used for the delay-Doppler grid. [00305] A pulse-tone is depicted in FIG.19. This OTFS waveform carrier has the mathematical properties of invariance under time, delay and Doppler shifts. That is, the pulse-tone can remain invariant under the operations of time, delay and Doppler shift. Because of the invariance, when modeling different wireless channels, the underlying coefficients representing the channel remain stable. FIG.20 depicts a visual mathematical example of a pulse-tone as a delay- Doppler domain waveform that is invariant under the Heisenberg uncertainty principle as depicted in FIG.21. The pulse is mathematically quasi-periodic as depicted along the delay and Doppler axes. FIG.22 shows an illustrative representation of a quasi-periodic pulse in the delay- Doppler domain. FIG.23 shows an example of a quasi-periodic pulse with invariance under the Heisenberg uncertainty principle [HUP] within a region defined by the delay period [^^^^ and the Doppler period [^^^^ as shown [e.g., quasi-periodic pulse – work around HUP]. FIG.24 depicts the operation of a Zak transform based implementation in which a pulse-tone is modeled as a time domain pulse realization of a quasi-periodic pulse in the delay-Doppler domain. FIG.25 shows an example representation of a communication using pulse-tones within a region defined by the delay period [^^^^ and the Doppler period [^^^^ (e.g., crystalline regime). In this example, B = 1/100ns = 10MHz, T = 1/1kHz = 1ms, N = 50μs/100ns = 500, and M = 20kHz/1kHz = 20. FIG.26 shows a period curve with a depiction of a crystalline regime along the delay-Doppler two dimensional plane where pulse-tone based communication proves to be an efficient 170688394.5 Attorney Docket No.: 119314.8121.WO00 communication technique. For example, the crystalline regime can be the optimal regime for communication and Radar applications. [00306] Further, FIG.27 shows the following three fundamental signal representations of time, frequency, and delay-Doppler and example transforms such as Zak transforms to go from one to another. The complexity of the Zak transform is half the complexity of the FFT. [00307] 7.2 OTFS Waveform – Crystalline regime [00308] FIG.28 shows examples of channel coupling in the TDM and FDM regimes. The coupling of the channel and the waveform in the TDM and FDM regimes is selective (i.e., fading and unpredictable). FIG.29 depicts examples of channel coupling in the crystalline regime in which delay spread is less than a delay period, ^^^ ൌ 1/^^^, and Doppler spread is less than a Doppler period, ^^^, typically satisfying ^^^ ^ 2 ∙ ^^ௗ, where ^^ௗ is the maximum expected Doppler shift. In this particular example, ^^^ ൌ 500^^^^ ^ ^^^ ൌ 20^^^^^^ and ^^^ ൌ 5^^^^ ^ ^^^ ൌ 50^^^^. The crystalline regime is between the TDM regime and the FDM regime. In the crystalline regime the channel coupling of the doubly spread channel with the OTFS waveform crystallizes – i.e., predictable and non-fading. On the contrary, as described above with reference to FIG.28, the coupling of the channel and the waveform in the TDM/FDM regime is selective – fading and unpredictable. Aliasing, which causes time selectivity in the TDM regime and frequency selectivity in the FDM regime, is the root cause. Fading and unpredictability occur in regions of self interaction as illustrated in FIG.30 and FIG.31. When the crystallization condition holds, there is no self interaction. The effects of aliasing is the corruption of communication signals, which are corrected in the crystalline regime. That is, the optimal regime for communication is the crystalline regime, shown in FIG.32, where the channel coupling crystallizes and is predictable and non-fading. OTFS in the crystalline regime is advantageous as superior BER performance is achievable under perfect CSI and under non-perfect CSI (i.e., under model-free mode of operation). Also, within the crystalline regime, OTFS is optimal for Radar sensing as high resolution detection of the delay-Doppler characteristics of reflectors can be achieved with no ambiguity. An example of Radar sensing in the crystalline regime is shown in FIG.33. Additionally, within the crystalline regime, OTFS is optimal for joint communication and sensing with a pair of unbiased pulse-tones (e.g., a Crystaline rotated pulse-tone and a standard crystalline standard pulse-tone) as depicted in FIG.34, allowing simultaneous high communication throughput and high resolution sensing. [00309] In FIG.34, an example for joint communication and sensing is provided where an information signal (e.g., information bits) can be modulated with a crystalline standard pulse- tone to a generate a first signal component. The first signal component can then be summed 170688394.5 Attorney Docket No.: 119314.8121.WO00 (e.g., combined/added) together with a second signal component comprising a crystalline rotated pulse-tone to generate a signal to be transmitted over a channel. The pair of crystalline standard pulse-tone and crystalline rotated pulse-tone are unbiased. The signal transmitted over the channel can be provided as an input to an interference subtraction unit and as an input to a radar sensing unit. The radar sensing unit can then output a radar image and also send the radar image to the interference subtraction unit. The crystalline rotated pulse-tone can also be provided as an input to the interference subtraction unit. With the crystalline rotated pulse-tone, the radar image, and the signal transmitted over the channel provided as inputs to the interference subtraction unit, the interference subtraction unit can provide an output signal to a data detection unit that will then output the information signal (e.g., the information bits). [00310] Along with the advantages of using OTFS in the crystalline regime, OTFS offers other advantages such as requires no cyclic prefix overhead, suffers no inter-carrier interference, scrambling without loss of capacity (structured CDMA), processing gain, and security communication. [00311] 8. Pilot signal processing examples [00312] FIG.35 shows one example embodiment (on transmitter side) for adding pilot signals to a transmission waveform. An information signal may be generated in a delay-Doppler domain (e.g., as QAM or QPSK symbol array). A reference signal [e.g., Ref. Signal (ROM/Gen.) xrs[m,n]] may be added to the information signal [e.g., Information DD x[m,n]] in the delay- Doppler domain, represented by the summation (sigma) block in FIG.35. The resulting signal may be transformed into a time domain signal s(t) using one of several techniques to achieve an OTFS transformation of the signal. [00313] FIG.36 shows another arrangement (on transmitter side) for adding a reference signal (e.g., Ref. Signal (ROM/Gen.) to the information signal (e.g.,, x[m,n]) to generate a transmission waveform s(t). Here, the information signal (e.g., an array of symbols) in the delay- Doppler domain may be converted into a time domain waveform using one of several techniques for OTFS transformation. The reference signal may be added to the resulting signal in the time domain to generate the transmission waveform s(t). [00314] In the above-described signal generation techniques, the reference signal may be generated in real-time using a waveform generator or may be pre-stored in a memory and read back during the process of combining with the information signal. [00315] In various embodiments, the reference signal may have the following properties: [00316] [1] The reference signal spreads over the entire ^^ ൈ ^^ delay-Doppler grid. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00317] [2] Super-imposed over the data. [00318] [3] Satisfies: [00319] ^^^^^^^, ^^^ ∙ ^^∗ ^^ ^^^, ^^^ ൌ ா ^ೞ ெே ^^ reference signal after the OTFS transform. [00322] FIG.37 pictorially depicts an example of effect of a transmission channel on a transmitted signal. The depicted example shows a two-path channel (e.g., two discrete paths in the channel). The grid shows a grid in a two-dimensional domain (e.g., delay-Doppler) with the top-left position showing identity (no distortion) coefficient. In the depicted example, two coefficients represent the distortion caused by the channel… h[1,1] and h[3,2]. As depicted in the example, the reference signal contribution at the [3,2] location comes from the reference signal at the [0, 0] location due to the operation of h[1,1] and h[3,2]. This is represented by the following equation: ^^^^^,^^^ ൌ ℎ^1,1^^^^^^^^ െ 1,^^ െ 1^ ^ ℎ^3,2^^^^^^^^ െ 3,^^ െ 2^ ^ ^^^,^ [00323] Some additional details are provided below with respect to the example in FIG.37. represents the noise and interference from data symbols. [00325] Furthermore, a receiver-side implementation can perform channel estimation by cross- correlating the received signal (with an underlying delay-Doppler domain representation) with a conjugate of the known reference signal to obtain an estimated channel response at coordinates [k,l]. The phase in the equation below may be used to compensate for the channel response rotation as a function of delay-Doppler coordinates. 1 ℎ^^^^, ^^^ ൌ ^^^^^^, ^ ^ ∗ ^ ^ ି^ଶగ^^^ି^^/ெே ^^^^ ^ ^^^^ ^^ െ ^^, ^^ െ ^^ ^^ [00327] For side can include cross correlating a received delay-Doppler data, ^^^^^,^^^, with the conjugate of a known transmitted reference signal ^^^^^^^ െ ^^, ^^ െ ^^^, to obtain the estimated channel response at delay-Doppler coordinates ^^^, ^^^; 170688394.5 Attorney Docket No.: 119314.8121.WO00 and the additional phase can be used to compensate for the channel response rotation as a function of the delay-Doppler coordinates. [00328] Also, for example, the cross-correlation may be used to determine a sparse channel representation in delay-Doppler domain. [00329] 9. Examples and implementations of the disclosed technology [00330] FIG.38 is a block diagram representation of a wireless hardware platform 800 which may be used to implement the various methods described in the present document. The hardware platform 800 may be incorporated within a base station or a user device. The hardware platform 800 includes one or more processors 802, a memory 804 (this may be optional and in some cases the memory may be internal to the processor) and at least one transceiver circuitry 806. The processor may execute instructions, e. g., by reading from the memory 804, and control the operation of the transceiver circuitry 1806 and the hardware platform 800 to perform the methods described herein. In some embodiments, the memory 804 and/or the transceiver circuitry 806 may be partially or completely contained within the processor(s) 802 (e.g., same semiconductor package). [00331] The following solutions may be preferably implemented by some embodiments. [00332] With reference to FIG.35, some solutions may be as follows. [00333] 1. A method of transmitting a signal (e.g., method 900 of FIG.39A), comprising: generating (902) a two-dimensional delay-Doppler signal comprising a sum of an information signal and a reference signal, wherein the sum is performed in delay-Doppler domain; and generating (904) a transmission waveform from the two-dimensional delay-Doppler signal. [00334] 2. The method of solution 1, wherein the transmission waveform is generated by performing an orthogonal time frequency space transform (OTFS) on the two-dimensional delay-Doppler signal. [00335] 3. The method of solution 2, wherein the OTFS transform comprises a forward or an inverse symplectic Fourier transform. [00336] 4. The method of solution 1, wherein the transmission waveform is generated by applying a Zak transform. [00337] 5. The method of solution 1, wherein the transmission waveform is generated by applying a two-dimensional transform. [00338] 6. The method of solution 1, wherein the transmission waveform is generated by applying a pulse-tone waveform in the delay-Doppler domain. [00339] 7. The method of solution 6, wherein the pulse-tone waveform comprises a quasi- periodic pulse. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00340] 8. The method of solution 6, wherein the pulse-tone waveform is implemented in a portion of the delay-Doppler domain where a delay spread is less than a delay period and a Doppler spread is less than a Doppler period. [00341] 9. The method of claim 8, wherein an operating regime of the pulse-tone waveform is between a frequency division multiplexing (FDM) regime and a time division multiplexing (TDM) regime. [00342] With reference to FIG.36, some implementations may be as follows. [00343] 10. A method of transmitting a signal (e.g., method 910 depicted in FIG.39B), comprising: generating (912) a transmission waveform as a summation of a first signal component and a second signal component, wherein the sum is performed in a time domain; wherein the first signal component comprises a reference signal, wherein the second signal component is generated by transforming a two-dimensional information signal from a delay- Doppler domain to a time domain. [00344] 11. The method of solution 10, wherein the second signal component is generated by performing an orthogonal time frequency space (OTFS) transform. [00345] 12. The method of solution 10, wherein the second signal component is generated by applying a Zak transform. [00346] 13. The method of solution 10, wherein the second signal component is generated by applying a two-dimensional transform. [00347] 14. The method of solution 10, wherein the second signal component is generated by applying a pulse-tone waveform in the delay-Doppler domain. [00348] 15. The method of solution 14, wherein the pulse-tone waveform comprises a quasi- periodic pulse. [00349] 16. The method of solution 14, wherein the pulse-tone waveform is implemented in a portion of the delay-Doppler domain where a delay spread is less than a delay period and a Doppler spread is less than a Doppler period. [00350] 17. The method of solution 16, wherein an operating regime of the pulse-tone waveform is between a frequency division multiplexing (FDM) regime and a time division multiplexing (TDM) regime. [00351] 18. A method implemented at a receiver-side (e.g., method 920 depicted in FIG.39C), comprising: receiving (922) a received signal over a transmission channel, determining (924), by processing the received signal in a delay-Doppler domain, an estimate of cross-correlation between the received signal and a known reference signal that makes up the received signal, 170688394.5 Attorney Docket No.: 119314.8121.WO00 and estimating (926) a delay-Doppler domain channel characteristic of the transmission channel based on the estimate of cross-correlation. [00352] 19. The method of solution 18, wherein the determining the estimate of cross-correlation comprises determining the cross-correlation by using a phase shift representing a channel response rotation as a function of delay-Doppler coordinates. [00353] With reference to FIG.34, some implementations may be as follows. [00354] 20. A method of transmitting a signal (e.g., method 930 depicted in FIG.39D), comprising: generating (932) a transmission waveform as a summation of a first signal component and a second signal component, wherein the first signal component comprises an information signal modulated with a standard pulse-tone, and wherein the second signal component comprises a rotated pulse-tone. [00355] 21. The method of solution 20, wherein the standard pulse-tone and the rotated pulse- tone are unbiased and implemented in a portion of the delay-Doppler domain where a delay spread is less than a delay period and a Doppler spread is less than a Doppler period. [00356] 22. The method of solution 20, wherein each of the standard pulse-tone and the rotated pulse-tone comprises a quasi-periodic pulse. [00357] 23. A digital communication apparatus comprising one or more processor electronics and one or more transceiver electronics, wherein the transceiver is configured to receive or transmit a signal under control of the one or more processor electronics, and wherein the one or more processor electronics is configured to implement a method recited in any of solutions 1 to 22. For example, the hardware platform disclosed in FIG.38 may be used for the implementation of the above methods. [00358] 24. A computer-readable storage medium having code stored thereon, the code, upon execution by one or more processors, causing the one or more processors to implement a method recited in any one or more of solutions 1-22. [00359] Several challenges exist for the upcoming 6G network architecture such as stemming from the use of higher frequencies, wider Doppler spreads, and increased difficulties to model, sense, and equalize. Many of the stringent requirements of 6G may be met by OTFS technology. In particular, in some implementations, a channel may be represented in the delay- Doppler domain. Such a representation lends itself to being able to represent a channel in a sparse manner, requiring very few coefficients that are stable with respect to variations in the coordinates. In addition, OTFS operation in the crystalline regime can be advantageous for various communication applications such as Radar sensing and simultaneous joint communication and sensing applications as high communication throughput and high resolution 170688394.5 Attorney Docket No.: 119314.8121.WO00 sensing/detection can be achieved in the crystalline regime where non-fading and predictability of the channel may be improved. [00360] It will be appreciated by those of skill in the art that the present document discloses operational considerations and implementation of wireless transmitters and receivers using OTFS waveforms. As disclosed herein, in one aspect, OTFS is a universal family of waveforms admitting TDM and FDM as limits. The present document further discloses operational trade- offs and design choices available to implementors of OTFS technology. To this end, a concept called “crystalline regime” is introduced. Within the crystalline regime OTFS is optimal for communication, the channel coupling crystallizes – becoming non-fading and predictable. Theory and simulations demonstrate superior performance under perfect CSI and superior performance under non-perfect CSI (Model free). [00361] As disclosed herein, within the crystalline regime OTFS is optimal for Radar sensing. In one aspect, the disclosed waveform facilitate high resolution detection of delay-Doppler characteristics of the reflectors, with no (or suppressed) ambiguity. [00362] It will be appreciated by those skilled in the art that within the crystalline regime OTFS is optimal for joint communication and sensing, e.g., using a pair of unbiased Pulsones (or pulse- tones) and provides simultaneous high communication throughput and high-resolution sensing. [00363] 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 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., code that 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, that is generated to encode information for transmission to suitable receiver apparatus. 170688394.5 Attorney Docket No.: 119314.8121.WO00 [00364] 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. [00365] 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). [00366] 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. [00367] 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 170688394.5 Attorney Docket No.: 119314.8121.WO00 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 sub-combination. 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. [00368] 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. 170688394.5

Claims

Attorney Docket No.: 119314.8121.WO00 WHAT IS CLAIMED IS: 1. A method of transmitting a signal, comprising: generating a two-dimensional delay-Doppler signal comprising a sum of an information signal and a reference signal, wherein the sum is performed in delay-Doppler domain; and generating a transmission waveform from the two-dimensional delay-Doppler signal. 2. The method of claim 1, wherein the transmission waveform is generated by performing an orthogonal time frequency space (OTFS) transform on the two-dimensional delay-Doppler signal. 3. The method claim 2, wherein the OTFS transform comprises an inverse symplectic Fourier transform. 4. The method of claim 1, wherein the transmission waveform is generated by applying a Zak transform. 5. The method of claim 1, wherein the transmission waveform is generated by applying a two- dimensional transform. 6. The method of claim 1, wherein the transmission waveform is generated by applying a pulse- tone waveform in the delay-Doppler domain. 7. The method of claim 6, wherein the pulse-tone waveform comprises a quasi-periodic pulse. 8. The method of claim 6, wherein the pulse-tone waveform is implemented in a portion of the delay-Doppler domain where a delay spread is less than a delay period and a Doppler spread is less than a Doppler period. 9. The method of claim 8, wherein an operating regime of the pulse-tone waveform is between a frequency division multiplexing (FDM) regime and a time division multiplexing (TDM) regime. 10. A method of transmitting a signal, comprising: generating a transmission waveform as a summation of a first signal component and a second signal component, wherein the summation is performed in a time domain; wherein the first signal component comprises a reference signal, and wherein the second signal component is generated by transforming a two-dimensional information signal from a delay-Doppler domain to a time domain. 170688394.5 Attorney Docket No.: 119314.8121.WO00 11. The method of claim 10, wherein the second signal component is generated by performing an orthogonal time frequency space (OTFS) transform. 12. The method of claim 10, wherein the second signal component is generated by applying a Zak transform. 13. The method of claim 10, wherein the second signal component is generated by applying a two-dimensional transform. 14. The method of claim 10, wherein the second signal component is generated by applying a pulse-tone waveform in the delay-Doppler domain. 15. The method of claim 14, wherein the pulse-tone waveform comprises a quasi-periodic pulse. 16. The method of claim 14, wherein the pulse-tone waveform is implemented in a portion of the delay-Doppler domain where a delay spread is less than a delay period and a Doppler spread is less than a Doppler period. 17. The method of claim 16, wherein an operating regime of the pulse-tone waveform is between a frequency division multiplexing (FDM) regime and a time division multiplexing (TDM) regime. 18. A method implemented at a receiver-side, comprising: receiving a received signal over a transmission channel, determining, by processing the received signal in a delay-Doppler domain, an estimate of cross-correlation between the received signal and a known reference signal that makes up the received signal, and estimating a delay-Doppler domain channel characteristic of the transmission channel based on the estimate of cross-correlation. 19. The method of claim 18, wherein the determining the estimate of cross-correlation comprises determining the cross-correlation by using a phase shift representing a channel response rotation as a function of delay-Doppler coordinates. 20. A method of transmitting a signal, comprising: generating a transmission waveform as a summation of a first signal component and a second signal component, wherein the first signal component comprises an information signal modulated with a 170688394.5 Attorney Docket No.: 119314.8121.WO00 standard pulse-tone, and wherein the second signal component comprises a rotated pulse-tone. 21. The method of claim 20, wherein the standard pulse-tone and the rotated pulse-tone are unbiased and implemented in a portion of the delay-Doppler domain where a delay spread is less than a delay period and a Doppler spread is less than a Doppler period. 22. The method of claim 20, wherein each of the standard pulse-tone and the rotated pulse-tone comprises a quasi-periodic pulse. 23. A digital communication apparatus comprising one or more processor electronics and one or more transceiver electronics, wherein the one or more transceiver electronics is configured to receive or transmit a signal under control of the one or more processor electronics, and wherein the one or more processor electronics is configured to implement a method recited in any of claims 1 to 22. 24. A computer-readable storage medium having code stored thereon, the code, upon execution by one or more processors, causing the one or more processors to implement a method recited in any one or more of claims 1-22. 170688394.5
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