Using spread pulses in wireless communication

Abstract

Methods, apparatus, systems for digital communication are described. One example method includes generating, by a transmitter apparatus, a transmission waveform by applying a discrete filter to an input signal in a delay-Doppler domain, wherein the discrete filter is periodic in delay domain and in Doppler domain; and controlling a transceiver to transmit the transmission waveform over a channel.

Description

CROSS-REFERENCE TO RELATED APPLICATION

[001] This application is claims priority to U.S. Provisional Application No. 63 / 567,776, filed on March 20, 2024, 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 wireless networks to achieve several operational improvements. In particular, methods, apparatus, and systems that use spread pulse based digital communication are disclosed.

[006] In one example aspect, a method of digital communication is disclosed. The method includes generating, by a transmitter apparatus, a transmission waveform by applying a discrete filter to an input signal in a delay-Doppler domain, wherein the discrete filter is periodic in delay domain and in Doppler domain; and controlling a transceiver to transmit the transmission waveform over a channel.

[007] In another aspect, another method of digital communications is disclosed. The method includes generating a transmission waveform comprising L layers, wherein each of the L layers comprises a corresponding data and a corresponding spread pilot, wherein, forthe each ofthe L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain, and wherein L is a positive integer; and controlling a transceiver to transmit the transmission waveform over a channel.

[008] In yet another aspect, a method of wireless communication is disclosed. The method includes generating K transmission waveforms, each of the K transmission waveforms comprising L layers, wherein each of the L layers comprises a corresponding data and a corresponding spread pilot, wherein, for the each of the L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain, and wherein K and L are positive integers and K > L; and transmitting, the each of the K transmission waveforms using a corresponding transmission antenna.

[009] In yet another aspect, a method of digital communication is disclosed. The method includes receiving a received signal comprising L layers, each of the L layers comprising corresponding data and a corresponding spread pilot over a channel, wherein L>=1, and wherein, for the each of the L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain; and extracting data from at least some of the L layers of the received waveform by subtracting, for each extracted layer, the corresponding spread pilot from the received signal and by performing a channel equalization based on a channel sensing using the corresponding spread pilot.

[0010] In yet another aspect, an apparatus is disclosed. The apparatus is configured to implement an above-described method.

[0011] In yet another example aspect, a wireless system in which one or more of the above-described methods are implemented is disclosed.

[0012] These, and other, features are described in this document. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] FIG. 1 shows an example communication network.

[0014] FIG. 2 shows a simplified example of a wireless communication system in which uplink and downlink transmissions are performed.

[0015] FIG. 3 shows examples of future applications of wireless technologies.

[0016] FIGS. 4A and 4B show examples of pulses that are used as basis signals for wireless communication.

[0017] FIG. 40 is a visual depiction example of rotations undergone by a delay-Doppler domain pulse.

[0018] FIG. 5 is an example of a quasi-periodic delay Doppler domain pulse.

[0019] FIG. 6 depicts an example of channel distortions undergone by a pilot signal.

[0020] FIG. 7 is a block diagram of an example of a Zak transform based wireless system including a transmitter and a receiver.

[0021] FIG. 8 is a pictorial depiction of effects of a doubly spread channel.

[0022] FIG. 9 is a graphical example of predictability achieved in a crystalline region of wireless communications.

[0023] FIG. 10 is a graph showing example results for model-free and model-dependent implementations.

[0024] FIG. 11A-B are flowcharts of different example implementations of an integrated sensing and communication with point pulse-tone (SCPP) and spread pulse-tone.

[0025] FIG. 12 shows an example heat map of results obtained when using a pilot signal without a guard band.

[0026] FIG. 13 shows an example heat map of results obtained when using a pilot signal with a guard band.

[0027] FIG. 14 is a graph of example results obtained for an SCPP implementation.

[0028] FIG. 15 graphically depicts examples of peak-to-average power ratio (PAPR).

[0029] FIG. 16 shows an example of Zak-OTFS transceiver processing.

[0030] FIG. 17 shows an example of a Zak-OTFS carrier waveform.

[0031] FIG. 18 graphically depicts examples where chirp filters rotate the period lattice.

[0032] FIG. 19A graphically depicts an example time domain (TD) realization of a point pulse-tone.

[0033] FIG. 19B graphically depicts a magnitude of the cross-ambiguity between an example transmitted point pilot signal and an example received point pilot signal.

[0034] FIG. 19C graphically depicts a magnitude of the cross-ambiguity between an example transmitted point pilot signal and an example received signal.

[0035] FIG. 19D graphically depicts a magnitude of the cross-ambiguity between an example transmitted pilot signal and a data component of an example received signal.

[0036] FIG. 19E graphically depicts energy profiles of example delay-Doppler domain pilot signals.

[0037] FIG. 19F graphically depicts a TD realization of an example spread pilot signal.

[0038] FIG. 19G is a plot showing the complementary CDF of the instantaneous-to-average-power ratio (IAPR) for example spread and point pilots.

[0039] FIG. 19H-19J graphically depicts the support of a self-ambiguity function of example spread pulse-tones.

[0040] FIG. 19K graphically depicts example heat maps for an effective discrete delay-Doppler domain channel filter.

[0041] FIG. 19L graphically depicts example heat maps for a cross-ambiguity.

[0042] FIG. 19M graphically depicts example heat maps for a cross-ambiguity.

[0043] FIG. 19N graphically depicts example heat map

[0044] FIG. 20A is an example plot of bit error rate (BER) of uncoded 4-QAM as a function of increasing Doppler spread.

[0045] FIG. 20B is an example plot of BER of uncoded 4-QAM as a function of increasing pilot to data power ratio.

[0046] FIG. 20C is an example plot of normalized mean squared error (NMSE) as a function of pilot to data power ratio.

[0047] FIG. 20D is an example plot of NMSE as a function of increasing pilot to data power ratio.

[0048] FIG. 20E is an example plot of BER of uncoded 4-QAM as a function of increasing pilot to data power ratio.

[0049] FIG. 20F is an example plot of signal to interference ratio (SIR) as a function of increasing pilot to data power ratio.

[0050] FIG. 20G shows a graph of an example BER performance.

[0051] FIG. 20H shows a graph of an example BER performance.

[0052] FIG. 20I is a graph showing example effective throughput results.

[0053] FIG. 20J are graphs showing examples of spreading operations in time domain (TD) and delay-Doppler(DD) domain.

[0054] FIG. 20K is a graph showing PAPR result examples.

[0055] FIG. 21 is a graphical example of a sensing waveform.

[0056] FIG. 22 graphically depicts examples of effect of spreading.

[0057] FIG. 23 graphically depicts examples of effect of spreading.

[0058] FIG. 24 shows an example relationship among filters in discrete DD domain.

[0059] FIG. 25 shows an example of multiplexing data with a spread pilot signal.

[0060] FIG. 26 is a graphical depiction of an example of sensing accuracy as a function of pulse dynamic range (PDR).

[0061] FIG. 27 is a graph showing example interference from a residual spread pilot.

[0062] FIG. 28 shows a graph of an example bit error rate (BER) performance.

[0063] FIG. 29 shows a graph of an example BER performance as a function of maximum Doppler spread.

[0064] FIG. 30 is a graph showing examples of geometric criteria for filter design.

[0065] FIG. 31 is a graph showing example performance of using spread pilots.

[0066] FIG. 32 is a graph showing example results obtained in a turbo-aided communication and sensing implementation.

[0067] FIG. 33 shows an example of a hardware platform.

[0068] FIGS. 34A-34D are flowcharts for example methods of facilitating digital communication. DETAILED DESCRIPTION

[0069] 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.

[0070] 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 and / or digital communication systems.

[0071] 1. Introduction - wireless communication environment examples

[0072] The wireless or time-variant nature of the communication channel poses several challenges in design 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. Presently, the communication industry is working on a next generation of digital communication technology, sometimes called 6G (sixth generation) wireless technology.

[0073] At the same time, Artificial Intelligence Al and Communications: Machine Learning has revolutionized image and natural language processing - data-driven discovery has revolutionized bioinformatics. If Machine Learning is to revolutionize wireless then it needs to work at the speed of wireless.

[0074] It is expected that the next generation wireless systems will adopt Integrated Sensing and Communication (ISAC) in which signals used to transmit data interfere with signals used to sense Input / Output relations.

[0075] 2. Example wireless systems

[0076] 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 (loT) device such as a smartwatch, a camera, and so on. The receiver device 104 may be a network device such as the base station. The signals transmitted from the base station to the transmitter 102 may experience similar channel degradations produced by static or moving scatterers. The techniques described in the present document may be implemented by the devices in the wireless communication system 100. The terms “transmitter” and “receiver” are simply used for convenience of explanation. As further described herein, depending on the direction of transmission (uplink ordownlink), the network station may be transmitting or receiving, and / or the user device may be receiving or transmitting.

[0077] 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 ordownstream 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 a 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.”

[0078] 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.

[0079] 3. Brief introduction to use of Spread pulses and delay Doppler representations

[0080] FIG. 3 shows examples of future applications of wireless technologies such as with Artificial Intelligence (Al) and Communication along with Integrated Sensing and Communication (ISAC) for 6G and beyond. As described above, with Al & Communication, machine learning algorithms have revolutionized image and natural language processing, but if they are to revolutionize wireless then they need to learn at the speed of wireless. With ISAC, coexistence in the same subframe can increase effective throughput but requires minimizing interference between sensing and data transmission. These are some of the research challenges for6G, which the present document aims to address.

[0081] Huge demand for high-speed data led to the transition from code division multiple access CDMA to orthogonal frequency division multiplexing OFDM, and to the opportunity to think about the benefits of measuring and adapting to instantaneous channel gains.

[0082] Recently, new communication technology, called orthogonal time frequency space (OTFS) was mainly investigated in the context of communication - e.g., using the OTFS waveform for conducting robust and error-free communication through all kinds of complicated channel conditions. The present document discloses use of the OTFS waveform in a broader context of joint communication and sensing. The present document not only analyzes the advantages of OTFS as a way for doing communication, but also as a way fordoing radar imaging of the environment and going even beyond that and analyzing it as a way for doing radar and communication jointly on the same band in a very efficient and precise manner.

[0083] In 6G, there is a possibility that radar theory will converge with communication. OTFS technology offers this advantage, as is further disclosed in the present document. As further disclosed in the present document, a waveform that combines desirable properties, called pulse-tone, may be used as a basic building block for digital communication.

[0084] In particular, a spread waveform called “spread pulse-tone” is disclosed in the present document. This waveform exhibits all the good properties of OTFS and can also be used like a radar signal for channel sensing. Furthermore, the present document also discloses ways to achieve a control over the peak to average power ratio (PAPR), to allow implementation of the disclosed technology in a practically efficient manner.

[0085] In one advantageous aspect, spread pulse-tone signals have weak correlation with each other. So, they interfere with one another in a code division multiple access CDMA kind of manner. Systems can actually put spread pulse-tone signals on top of one another, conduct whatever thing you want to do with each of them, and then separate them in the receiver like in the case of CDMA. One way to look at the spreading is as a kind of code which is associated with the pulse-tone.

[0086] The disclosed waveforms can offer at least the following three distinct operational advantages. First, the waveform is amenable to joint communication and sensing. Second, the waveform represents a large family of waveforms without sacrificing anything on the level of performance, while opening up new capabilities of multiplexing. Third, because this waveform behaves like separable coding (e.g., CDMA) because of their weak correlation with each other, they can be used simultaneously for data transmission and sensing, without causing any significant interference to each other.

[0087] In the next generation of wireless technology, low latency of communication may be an important feature. Low latency also enables moving functions like scheduling towards the cloud. Also, it would be desirable for a machine learning application and Al application to operate on a physical layer that is predictable and has a long-time horizon. The disclosed technology offers such a solution because a representation of the propagation environment in delay Doppler can provide such features. You cannot get a representation that changes any slower because it's change captured at the speed at which physics changes. And this ability allows to on a 1 millisecond frame to acquire the environment and acquire the data; that is an enabler of machine learning.

[0088] 4. Introduction to pulse-tone waveforms

[0089] FIGS. 4A and 4B show examples of pulses that can be used as basis signals for wireless communication.

[0090] In FIG. 4A, an example of separating sensing and communications is depicted with a pilot signal (e.g., point pulse-tone x=xs), where the Cross-Ambiguity can be represented as Ay,x[k,l].

[0091] In some embodiments, separate Zak-OTFS subframes may be dedicated to sensing and data transmission / communication. A transmitter may transmit a pilot, the receiver may read off the I / O relation for the entire subframe from the response to a single pilot, and then use this estimate to equalize the data in a subsequent data subframe. For example, as depicted in FIG. 4A, the discrete input / output (I / O) can be read off from the response to a single pilot signal transmitted through a linear time variant (LTV) channel, and the channel estimate can be used to recover data in a subsequent Zak-OTFS subframe.

[0092] How does this work? - embodiments can use linear time invariant LTI channel models as depicted in FIG. 4B. A time domain TD pulse is a geometric mode of the LTI channel - the I / O relation is made up of delays that move the pulse about in time. The I / O relation can be read off from the response to a single TD pulse. After waiting longer than the delay spread, embodiments can read off the I / O relation, and equalize the data. Because delay-Doppler profile of a channel typically is static over long time periods, this approach can provide predictability, which means that the I / O relation does not change with time. And predictability can enable a model-free mode of operation such as under the crystallization condition where delay spread < delay period.

[0093] FIG. 4C is a visual depiction example of rotations undergone by a delay-Doppler domain pulse. In FIG. 4C, a pulse in the delay-Doppler domain is shown, which can, for example, have a delay spread tp = 50 ps and a Doppler spread vp = 20 KHz.

[0094] Defining a pulse in the delay-Doppler domain runs into the problem that the Heisenberg Uncertainty Principle implies that it is not possible to simultaneously localize a signal in delay and Doppler (DD). However, it is possible to “cheat” or work around the Heisenberg Uncertainty Principle by starting with a pulse in the DD domain and extending it quasi-periodically.

[0095] The DD domain pulse is in fact a configuration of infinitely many pulses which repeat at integer multiples of the delay period tp along the delay axis and at integer multiples of the Doppler period vp along the Doppler axis. Refer to the box of width tp and height vp as the fundamental period, and take vp tp = 1. [e.g., the fundamental domain can be defined by delay period tp and the Doppler period vp.]

[0096] The phase of the pulse changes when the pulse location shifts by an integer multiple of tp along the delay axis, but there is no change in phase when the pulse location shifts by an integer multiple of vp along the Doppler axis.

[0097] Accordingly, the DD realization of a TD signal is a quasi-periodic function.

[0098] FIG. 5 is an example of a quasi-periodic delay-Doppler domain pulse.

[0099] How to get from a DD domain pulse to a TD pulse-tone waveform? - This may be achieved by applying an (inverse time) Zak transform. That is, an (inverse time) Zak transform may be applied to get a TD pulse-tone from a Quasi-Periodic DD Domain Pulse. There is also an inverse (frequency) Zak transform that transforms a DD domain pulse to a frequency domain FD pulse-tone. Additional details are provided in following section 5.

[00100] Here a DD domain pulse is located at (to , vo) within the fundamental period. The DD domain pulse is spread along the delay axis over a length 1 / B ( / .e., B-1) and is spread along the Doppler axis over a length 1 / T ( / .e., T1). The TD realization x(t) can be obtained by applying the Zak transform to the DD domain pulse xdd (t, v). The TD realization x(t) is a pulse train of finite duration T, with each pulse in the train spread overtime duration 1 / B. Consecutive pulses of the pulse train are separated by the delay period tp- moving the pulse location to along the delay axis displaces the TD pulse-tone in time. The pulse train is modulated by a sinusoid of frequency vo- moving the pulse location vo along the Doppler axis displaces the modulating tone in frequency.

[00101] TDM (time division multiplexing) is a limiting case. As the delay period tp grows, the TD pulses grow further apart, the tone structure disappears, and only a single TD pulse at to remains.

[00102] FDM (frequency division multiplexing) is a limiting case. As the delay period tp shrinks, the pulses grow closer and closer together in time, and in the limit, the pulse structure disappears and only the tone remains. Pulse-tones parametrized by tp and vp interpolate between TDM and FDM; that is, they live on the hyperbola vp tp = 1. What value of tp optimizes predictability?

[00103] Pulse-tones are optimal as time- and band- limited signals because there is almost no overlap between two DD pulses whose delay domain locations differ by 1 / B or whose Doppler locations differ by 1 / T.

[00104] In the example depicted in FIG. 5, B= 1 / 100ns = 10MHz, T = 1 / 1 kHz = 1ms, N = 50ps / 100ns - 500, and M - 20kHz / 1kHz = 20, with vp - 20 KHz (Doppler Spread) and tp = 50 ms (Delay Spread).

[00105] FIG. 6 depicts an example of channel distortions undergone by a pilot signal.

[00106] The input-output response can be given by twisted convolution - like regular convolution, but with a twist - see section 5 below for details.

[00107] Embodiments will want the action of a doubly spread wireless channel on a pulse-tone to be predictable and geometric. Geometric may mean that for a channel path delay, the DD domain pulse is simply translated along the delay axis by an amount equal to the path delay; and it may be advantageous that for channel path Doppler shift, the DD domain pulse is simply translated along the Doppler axis by an amount equal to the Doppler shift.

[00108] Pulse-tone waveforms live on the hyperbola tp vp = 1, and accordingly the action of the doubly spread channel on a pulse-tone is predictable if the fundamental period captures the delay and Doppler spreads of the channel.

[00109] The fundamental period captures the channel spreads when the delay domain period tp is greater than the channel path delay spread and the Doppler domain period vp is greater than the channel path Doppler spread. When these conditions are met, we are in the crystalline regime and the action of a doubly spread channel on a pulse-tone is predictable. In some embodiments, such as FIG. 6, this can allow the predictability of LTV channels, where wireless channel dynamics can lead to replicas.

[00110] In the example of FIG. 6, the input-output response can be given by the twisted convolution: Y(i,v)=h(T, v) *ctX(t, v), where= £-=1hi8(r —       — Vf).

[00111] FIG. 7 is a block diagram of an example of a Zak transform based wireless system including a transmitter and a receiver. The example Zak transform based wireless system of FIG. 7 supports signal processing in Zak-OTFS.

[00112] The continuous output is the twisted convolution of the input with the effective DD channel filter. Quasi-periodic signals are shown in gray scale.

[00113] The discrete output is obtained by sampling the continuous output on the information grid.

[00114] The discrete output is the discrete twisted convolution of the discrete input with the discrete effective DD channel filter.

[00115] For the example system of FIG. 7, the Zak-OTFS I / O relation can be represented as: ydd\T’v) = v) *<7 v) wtx(T, v) xdd(r, v) = hdd(r, v) xdd(r, v).

[00116] FIG. 8 is a pictorial depiction of effects of a doubly spread channel (e.g. doubly spread channels acting on pulse-tones)

[00117] The interaction of a doubly spread channel with a TD pulse-tone can be predictable and geometric.

[00118] In the crystalline regime, the delay domain period tp is greater than the channel path delay spread, and the Doppler domain period vp is greater than the path Doppler spread: tp > delay spread and vp > Doppler spread.

[00119] When communicating with pulse-tone (e.g., Pulsone™) in the crystalline regime, the input-output relations can be governed by twisted convolution, in which the I / O relation for the entire subframe can be learned directly, without learning the channel, by reading off the response to a single transmitted pilot.

[00120] Unpredictability results from aliasing in the Delay-Doppler domain. Aliasing occurs when the channel spreads are bigger than the Delay-Doppler periods of the pulse-tone.

[00121] FIG. 9 is a graphical example of predictability achieved in a crystalline region / regime of wireless communications.

[00122] In this example, a doubly spread channel is provided and M=N=4 is fixed to have 16 bins.

[00123] The average power of the received discrete DD domain signal for varying tp , vp is plotted on the hyperbola tp vp = 1.

[00124] As the delay domain period shrinks, aliasing in delay and frequency selectivity in FDM results.

[00125] As the Doppler domain period shrinks, aliasing in Doppler and time selectivity in TDM results.

[00126] There is a sweet spot in the middle of the hyperbola. In the sweet spot, the fundamental period can capture the channel spread. When the fundamental period captures the channel spread, there is no fading. Here, the received power profile is flat like the surface of a crystalline solid and this is called the crystalline regime.

[00127] In the crystalline regime, there is no fading and the input-output (I / O) relation is predictable.

[00128] As described above, DD domain aliasing controls predictability - as the delay domain period shrinks, aliasing in delay and frequency selectivity in FDM results; as the Doppler domain period shrinks, aliasing in Doppler and time selectivity in TDM results; and in the crystalline regime, there is no fading and the input-output (I / O) relation is predictable.

[00129] FIG. 10 is a graph showing example results for model-free and model-dependent implementations (e.g., Model Free vs Model-Dependent). Here BER performance in the crystalline regime is explored where the I / O relation is predictable and communications can take place effectively even when it is not possible to estimate the channel accurately. In various embodiments, pulse-tone waveforms can support model-free operation in the crystalline regime when it is not possible to learn the channel. Although not shown in the BER performance example results, additional advantages such as improvements in filtering, root raised cosine versus sine, and extending the region of reliable operation are also achievable in the crystalline regime.

[00130] Channels are becoming infinitely complicated, and it is becoming more and more challenging to learn them in the time available. Also, importantly, the effect of the channel on the modulation may matter more than the channel itself. This effect can be captured in the I / O relation, and it is predictable.

[00131] Operating model-free can comprise estimating the I / O relation everywhere in the Zak-OTFS subframe from the response to a single pilot waveform.

[00132] Model-free performance can be very close to performance with perfect channel state information CSI.

[00133] FIG. 11A is an example flowchart of different implementations of an integrated sensing and communication with point pulse-tones (SCPP) [e.g., integrated sensing and communication with point pulse-tones].

[00134] The present document considers the Veh-A channel model comprising 6 channel paths first because it is representative of real propagation environments, second because it is very difficult to make the model-dependent mode of operation work.

[00135] The model-dependent approach limits the number of paths and constrains delay and Doppler shifts.

[00136] The model-free approach was described in the previous section and it can be similar in spirit to estimating the taps of an effective LTI channel.

[00137] When the crystallization condition holds, the taps of the effective channel filter hefr[k, I] can be read off from the response to a single point pilot pulse-tone.

[00138] This has been shown to work better than the model-dependent approach. However, in some embodiments, it may not be optimal and in this section the (maximum-likelihood) ML estimator is introduced.

[00139] Given the effective channel estimate, an estimate for the received pilot can be formed, the estimate can be subtracted from the received signal, and the data can be recovered.

[00140] It will be shown that when point pulse-tones are used for sensing and for data transmission then it may be necessary to divide DD domain resources between the two functions. A guard band may need to be introduced around the point pilot, and data is not transmitted within this guard band.

[00141] Also, an example flowchart of an implementation of an integrated sensing and communication with point and spread pulse-tones is provided in FIG. 11B.

[00142] FIG. 12 shows an example heat map of results obtained when using a pilot signal without a guard band.

[00143] A point pilot xP and a data signal Xd are transmitted, and a signal y comprising a contribution yP from the pilot and a contribution ydfrom the data can be received.

[00144] The ML estimator can be the cross-ambiguity between yP and xP observed within the guard band which suppresses interference to yP from yP.

[00145] If a guard band is not included, then interference from data could completely obscure the effective channel taps preventing channel sensing.

[00146] In some embodiments, in the crystalline regime, the ML estimate for the tap heff[k, Z] can be given by the Cross-ambiguity function AyXp[k,Z] A [k, I] = Av x [Zc, Z] + A [Zc, Z] y>xpL ’ J yp’xpL ' J yd'xpL J

[00147] When there is no guard band, data interference could prevent channel sensing by completely obscuring the target term x [Zc, Z], JP’^P

[00148] Hence, guard bands may be needed for examples such as this example for FIG. 12.

[00149] In the example of FIG. 12, |AyAp[fc,Z]| is represented for Veh-A channel with vmax = 815 Hz, Doppler period 30 KHz, M = 31,TV = 37, and RRC pulse shaping (J3T =   = 0.6).

[00150] FIG. 13 shows an example heat map of results obtained when using a pilot signal with a guard band.

[00151] Here, the guard band is a 7x7 rectangle centered at the pilot location.

[00152] It creates a little window around the pilot location where there is very little interference from the data signal yd.

[00153] Hence, data interference preventing channel sensing can be suppressed in the example of FIG. 13. That is, guard bands can suppress data interference.

[00154] For the example of FIG. 13, Ayri [k, Z] represents cross-ambiguity between the received data and the transmitted pilot.

[00155] In the example of FIG. 13, \AydXp[k, Z]| is represented forVeh-A channel with vmax - 815 Hz, Doppler period 30 KHz, M = 31,7V = 37, and RRC pulse Shaping ( / ?T = / 3V = 0.6).

[00156] FIG. 14 is a graph of example results obtained for an SCPP implementation.

[00157] The graph shows example results for Uncoded 4-QAM BER vs. for different sensing and communication cases with a Point pulse-tone based on a Veh-A channel with Doppler period 30 KHz, M = 31,TV = 37],

[00158] The graph provides example results for the following cases: S|C & C|S vs. S|C & C|S .

[00159] S stands for sensing, and C stands for data communications.

[00160] The notation S|C means that sensing takes place with interference from data, and the notation S|C bar means that sensing takes place without interference from data.

[00161] The BER for uncoded 4-QAM as a function of Doppler spread is plotted in FIG. 14.

[00162] The case S|C bar & C|S bar is the baseline considered in the previous section, where separate Zak-OTFS subframes are dedicated to sensing and to data transmission. These are the two bottom curves with the gray circle and triangle markers. Note that, in this example, the RRC filter is better than the sine filter at confining energy to the guard band.

[00163] The case S|C & C|S is integrated sensing and communications with point pulse-tones.

[00164] In this example, it is shown that when vmax > 1 KHz, the channel response to a point pilot extends beyond the guard band regardless of the choice of filter. Here (when vmax > 1KHz), data interference can severely compromise BER performance because the channel response to a point pilot (e.g., pulse-tone) extends beyond the guard band regardless of the choice of filter.

[00165] FIG. 15 graphically depicts examples of peak-to-average power ratio (PAPR).

[00166] An example TD realization of a point pulse-tone with Doppler period 30 KHz, M = 31,7V = 37, and RRC pulse shaping ( / ?T =   = 0.6) is depicted.

[00167] The examples of PAPR for different cases are shown in the provided Complementary CDF (CCDF) [complementary cumulative distribution function] of the Instantaneous to Average Power Ratio (IAPR) plot with RRC pulse shaping ( / ?T - pv - 0.6).

[00168] Peak to Average Power Ratio (PAPR) can be an unattractive feature of the pulse-tone waveform. In some embodiments, it can be about 15dB and may require highly linear power amplifiers which are typically power inefficient. For high PAPR, highly linear power amplifiers are typically required.

[00169] 5. Zak-OTFS and Integration of Sensing and Communication (ISAC)

[00170] The Zak-OTFS input / output (I / O) relation is predictable and non-fading when the delay and Doppler periods are greater than the effective channel delay and Doppler spreads, a condition which is referred to as the crystallization condition. The filter taps can be read off from the response to a single Zak-OTFS point (impulse) pulse-tone waveform, and the I / O relation can be reconstructed fora sampled system that operates under finite duration and bandwidth constraints. Predictability opens up the possibility of a model-free mode of operation.

[00171] The time-domain realization of a Zak-OTFS point pulse-tone is a pulse train modulated by a tone, hence the name, pulse-tone, with Pulsone™ being an example implementation. The Peak-to-Average Power Ratio (PAPR) of a Pulse-tone is about 15 dB, and a general method for constructing a spread Pulse-tone for which the time-domain realization has a PAPR of about 5~6 dB is described. The spread Pulse-tone can be constructed by applying a type of discrete spreading filter to a Zak-OTFS point pulse-tone.

[00172] The self-ambiguity function of the point pulse-tone is supported on the period lattice AP. By applying a discrete chirp filter, a spread pulse-tone with a self-ambiguity function that is supported on a rotated lattice can be obtained. If the channel satisfies the crystallization conditions with respect to the rotated lattice then the effective DD domain filter taps can be read off from the cross-ambiguity between the channel response to the spread pulse-tone and the transmitted spread pulse-tone.

[00173] If, in addition, the channel satisfies the crystallization conditions with respect to the period lattice AP, then in an OTFS frame consisting of a spread pilot pulse-tone and point data pulse-tones, after cancelling the received signal corresponding to the spread pulse-tone, the channel response to any data pulse-tone can be recovered. This translates integration of communication and sensing within a single OTFS frame into geometric properties of a lattice AP used for data transmission and a rotated lattice used for sensing. The spread pilot pulse-tone looks like noise to the point data pulse-tones, and it is this incoherence that makes it possible to integrate communications and sensing without time-sharing delay-Doppler resources. Integrated sensing and communication can increase effective throughput.

[00174] Pulses in the delay-Doppler domain can enable sensing, since they are geometric modes that are moved around by the Linear Time-Variant (LTV) channel.

[00175] Brief Introduction for Zak-OTFS and ISAC

[00176] In 6G propagation environments, as we encounter Doppler spreads measured in KHz, especially for non-terrestrial networks, it is becoming more and more difficult to estimate channels, and the standard channel-model-dependent approach to wireless communication is starting to break down. The present document describes a physical layer that is designed to integrate sensing and communications (ISAC) in challenging propagation environments, typical of 6G networks.

[00177] 4G and 5G wireless communication networks use OFDM. Here, a cyclic prefix is used to create shared eigenfunctions of the group of time shifts, and we have shared eigenfunctions because the time shift group is commutative. Note that pulses in the time domain serve as geometric modes that are moved around by the Linear Time-Invariant (LTI) channel. In the present document, we discuss and describe doubly-spread channels, where there are no shared eigenfunctions because delay shifts and Doppler shifts do not commute. Nevertheless, pulses in the delay-Doppler domain enable sensing, since they are geometric modes that are moved around by the Linear Time-Variant (LTV) channel. These pulses are the Orthogonal Time Frequency Space (OTFS) carrier waveforms, their time domain (TD) realizations are called pulse-tones as mentioned above.

[00178] A pulse in the delay-Doppler DD domain is a quasi-periodic localized function, defined by a delay period tp and a Doppler period vP. We have shown that the Zak-OTFS Input I Output (I / O) relation is predictable and non-fading when the delay period tp is greater than the effective channel delay spread, and the Doppler period vp is greater than the effective Doppler spread. We refer to this condition as the crystallization condition with respect to the period lattice and speak of operating in the crystalline regime. We have explained how non-predictability and fading result from aliasing in the DD domain, and why the crystallization condition prevents aliasing. In the crystalline regime we have shown that the effective DD domain channel filter taps can simply be read off from the response to a single Zak-OTFS point pulsetone.

[00179] Why OTFS ratherthan OFDM? The first reason is that 6G propagation environments are changing the balance between time-frequency methods characteristic of OFDM and delay-Doppler methods. In OFDM, once the I / O relation is known, equalization is relatively simple, at least when there is no inter-carrier interference. However, acquisition of the I / O relation is non-trivial and model-dependent. In contrast, equalization is more involved in OTFS, due to intersymbol interference, but acquisition of the I / O relation can be simple and model-free (it can be read off from the response to a single point pulsetone). Acquisition becomes more critical in 6G, as Doppler spreads measured in KHz make it more and more challenging to estimate channels.

[00180] The second reason is the integration of sensing and communication. The present document describes how to spread an OTFS point pulse-tone so that sensing is simple, and so that pilot waveform and data can coexist in the same OTFS subframe.

[00181] Sensing is introduced through a discussion of practical limitations on pilot design with a point pulse-tone. As mentioned in other sections, a pulse-tone is a pulse train modulated by a tone, and this is a waveform with high PAPR. Also, when we use the same lattice for communication and sensing we end up dedicating delay Doppler resources to either sensing or communication. For example, when we combine point data symbols with a point (impulse) pilot in a single OTFS subframe, we need to avoid interference between data symbols and pilot. The standard approach is to introduce a guard band, but this is an overhead that reduces spectral efficiency, and the greater the channel spread, the greater is the overhead. This tradeoff motivates our use of different lattices for communications and sensing.

[00182] We also need to solve the issue of high PAPR which plagues multicarrier modulation. For example, OFDM signals are generated by adding many subcarrier components, and they can have high peak values in the time domain. As a consequence, OFDM systems suffer from high PAPR compared with single carrier systems. Many techniques have been proposed for reducing PAPR in OFDM but none avoid degrading BER performance. Over the past several years several variants of OTFS have been reported in the literature. A multicarrier approximation to Zak-OTFS, which we refer to as MC-OTFS, has been the focus of most research attention so far. MC-OTFS suffers from high PAPR, techniques for PAPR reduction have been presented, and again none avoid degrading BER performance.

[00183] Zak-OTFS modulation starts with a quasi-periodic Dirac-delta DD domain pulse, and we apply a DD domain pulse shaping filter. The time-domain (TD) representation of the filtered signal is a TD pulse-tone with time duration T and bandwidth B inversely proportional to the Doppler and delay spread respectively of the filter. We transmit information using non-overlapping DD domain pulses spaced 1 / 7 apart along the delay axis and 1 / 5 apart along the Doppler axis. Since each pulse repeats quasi-periodically, there are M =        = Btp pulse locations along the delay axis and N = vp!(MT) = Tvp pulse locations along the Doppler axis. The number of distinct non-overlapping information carriers is the timebandwidth product BT-MN, and the pulse locations are the points in the information lattice.

[00184] A discrete quasi-periodic DD domain signal is periodic along both delay and Doppler axes with period MN (e.g.. M*N). In a later section, we define a discrete DD domain filter to be a discrete periodic DD domain function with period MN along both delay and Doppler axes (M2N2 degrees of freedom). A later section develops the fundamentals of filtering in the discrete DD domain. We describe how we obtain a spread pilot by applying a discrete spreading filter to a DD domain signal localized at a point in the information lattice. This is a general method. In the present document, however, we also discuss and describe using a discrete chirp filter which distributes energy equally to all lattice points. Different lattice points correspond to pulse-tones with different delays and Doppler shifts; hence we observe a TD signal with almost constant amplitude. As it will be further discussed, the PAPR of the spread pulse-tone is about 5 dB, significantly less than the PAPR of the point pulse-tone, which is about 15 dB.

[00185] A later section describes how we recover the effective channel filter by de-spreading the received signal. We can estimate the taps of the discrete effective channel filter from the discrete crossambiguity function between the received DD signal and the transmitted spread pilot signal. The selfambiguity function of a point pulse-tone is a rectangular lattice Az,p, and we show that the self-ambiguity function of the spread pulse-tone is obtained by rotating Ap to obtain e.g., a lattice A*. If the channel satisfies the crystallization conditions defined by Ap then we show that we are able to accurately estimate the discrete effective channel taps using a point pulse-tone. If the channel satisfies the crystallization conditions defined by e.g. A* then we show that we are able to accurately estimate the discrete effective channel taps using the spread pulse-tone. Embodiments described in the present document for spreading the pilot are connected to the dynamics of the sensing environment.

[00186] Integration of sensing and communication (ISAC) is central to 6G; hence we consider embodiments involving transmission of pilot and data within a single OTFS subframe with no division of DD domain resources between sensing and communication. Data can be carried by point pulse-tones, the data signal interferes with channel estimation using a spread pulse-tone, and this interference can adversely affects the estimation of certain discrete effective channel filter taps. We describe how the significance of this interference depends on the ratio of pilot power to data power. After equalization, the spread pulse-tone interferes with data carried by the point pulse-tones, and again the degree of interference depends on the ratio of pilot power to data power. The present document identifies embodiments on how to choose this ratio to accomplish integration of sensing and communication within a single OTFS subframe and avoid time-sharing of delay-Doppler resources.

[00187] Passive radar is an important instance of integrated sensing and communication. Passive radar exploits readily available, non-cooperative sources of radio energy to measure reflections from the environment and targets of interest. When a suitable illuminator is available, covert surveillance becomes possible, without the need for deployment and operation of a dedicated transmitter. For example, terrestrial digital television transmissions (DVB-T) provide an especially attractive opportunity for radar. We describe how to design the illuminator in a passive radar to be a geometric mode of the radar scene (a spread pulse-tone). We describe how this choice can simplify acquisition of the radar scene, since the scene can be simply read off from the received signal, unlike the DVB-T system. The radar application also benefits from the noise-like characteristics of the spread pulse-tone which increase the fraction of energy on target.

[00188] Pilot Design: In the present document, we describe constructing a spread pulse-tone, where the self-ambiguity function of the spread pulse-tone is supported on e.g. a lattice A* obtained by rotating the period lattice Ap associated with the data-bearing pulse-tones. We show that if the channel satisfies the crystallization conditions with respect to A* then the effective DD domain filter taps can simply be read off from the channel response. If the channel also satisfies the crystallization conditions with respect to the Ap then given the I / O response at one point in the OTFS subframe, it is possible to predict the I / O response at all other points in the subframe. We also show that the spread pulse-tone is a noise-like waveform with excellent PAPR.

[00189] Integrating Sensing and Communication (ISAC): We have translated integration of communication and sensing into geometric properties of a lattice Ap used for data transmission and a rotated lattice e.g., A* used for sensing. The data pulse-tones look like noise to the sensing pulse-tones, and we have demonstrated that this incoherence makes it possible to integrate sensing and communication without the loss in spectral efficiency associated with time-sharing delay-Doppler resources. We have also demonstrated other advantages that ISAC can offer such as increase in effective throughput by avoiding time-sharing delay-Doppler resources. For example, an important application is the design of a passive radar system.

[00190] System Model

[00191] A time-domain (TD) pulse is an ideal waveform for delay-only channels (where paths induce zero Doppler shift) since it is possible to separate signals received along different paths according to their path length / distance. Similarly, a frequency-domain (FD) pulse is an ideal waveform for Doppler-only channels since it is possible to separate signals received along different paths according to the Doppler shift induced on the transmitted signal. However, neither a TD pulse nor a FD pulse is suited for doubly-spread channels where paths induce both delay and Doppler shift.

[00192] Here, it is described how a pulse in the delay-Doppler (DD) domain is matched to doubly-spread channels. A pulse in the DD domain is a quasi-periodic localized function, defined by a delay period tp and Doppler period vp = 1 / tp. In the period lattice Np = ^nrp,myp^n,m g , there is only one pulse within the fundamental region Do = ^(r,v)|o < i < ^,0 < y < yp|, and there are infinitely many replicas along the delay and Doppler axes given by + nrp, v + mup) = ej2^rWTp               (1) for all n,m G Z. Only quasi-periodic DD domain functions can have a TD representation. When viewed in the time domain, this function can be realized as a pulse train modulated by a tone (see e.g. FIG. 17), hence the name Pulsone™. The DD domain pulse is the carrier waveform forZak-OTFS modulation.

[00193] 5.1 .A. Zak-OTFS Modulation

[00194] Zak-OTFS modulation with modulation parameters (rp, vp), vp = l / rp is considered here. The transmitted TD Zak-OTFS frame is limited to a time duration T=Ntp and bandwidth B=Mvp. FIG. 16 illustrates Zak-OTFS transceiver processing.

[00195] Let x[k,l],k = 0,1,-• -,M — 1,1 = 0,1,• • -, N — 1 , denote the BT = MN information symbols, each having unit average energy, i.e., = 1. These MN information symbols are encoded into a discrete DD domain information signal Xdd[-,] given by .Tdd[& + nMy I + mN] = e^2" N x[k. I] (2) for all k = 0,1,- • - ,M -1,1 = 0,1,- • ■, N — 1, n,m e Z. From this it follows that j’dd[k + nM, I + mN] =        x^[k. I]          (3) for all k,l,n, m e Z . In other words, the information signal Xdd[h / ] is quasi-periodic, with period Af along the delay axis and N along the Doppler axis. The encoding of the MN information symbols to Xdd[£, / ] is carried out as in (2), so that Xdd[A / ] is quasi-periodic, since only quasi-periodic DD functions have TD realizations.

[00196] The information signal Xdd[^, / ] can then be converted to a continuous quasi-periodic DD signal by lifting it to the information lattice Add = {(kTp / M, lvplN^\k,l e z|, i.e. ^d(r, v)                 ” kr.plM) - lvp / N}       W

[00197] This signal is quasi-periodic in the continuous DD domain with periods tp and vp respectively along the delay and Doppler axis, i.e., for all n, m e Z Xdd(r + nrp,v + mup) =                       (5)

[00198] Note, in the present document, 5 with a square bracket, i.e., 6[] is used to denote a Dirac-delta function in a discrete domain and 6() is used to denote a Dirac-delta function in a continuous domain.

[00199] Pulse shape filtering in the DD domain can be implemented by twisted convolution. Twisted convolution of Xdd(x, v) with a pulse shaping filter Wt^r, v) gives *              A wtx(r,v) ^dd(r, A         (6)

[00200] Since twisted convolution preserves quasi-periodicity,             is also quasi-periodic. Also, appropriate pulse shaping guarantees that the transmit TD Zak-OTFS modulated signal has time duration T and bandwidth B.

[00201] Note, in the present document, twisted convolution is denoted by *o. Twisted convolution between two DD functions a(r, v) and 6(t, v) can be given by =       % b(r,v) = j ja(r\v'}b(r -d,v -v'}ej2!TVk~T ^dr'dv'. Twisted convolution is associative, i.e., 6(t, v) *o (c(t, v) *o J(t. v)) = (6(t, v) *o c(t. v)) *o d(r, v). However it is not commutative, i.e., a{r, v) *o b(t, v) 6(t, v) *o a(r, v).

[00202] The inverse Zak-transform of (t, v) gives the transmit TD Zak-OTFS modulated signal Std(0                                                      (7)

[00203] Equivalently, the transmit TD signal Std(t) can be given by std(t) = Z^1      (r, / / )) A7-1 A’-l = (8) A—0 / =0 where is the carrier waveform for the (£, / )-th information symbol x[k, / ]. FIG. 17 provides an illustration of the Zak-OTFS carrier waveform. In the DD domain it is a quasi-periodic pulse. In both the TD / FD domains, the carrier waveform is a pulse train modulated by a tone and is therefore called a TD / FD pulsetone.

[00204] In FIG. 17, the (k, / )-th quasi-periodic DD domain pulse located at /        \ __ / p lvp \ V0>vo / —           / anc* its TD / FD realizations referred to as TD / FD pulse-tone. The TD pulse-tone comprises of a finite duration pulse train modulated by a TD tone. The FD pulse-tone comprises of a finite bandwidth pulse train modulated by a FD tone. The location of the pulses in the TD / FD pulse train and the frequency of the modulated TD / FD tone is determined by the location of the DD domain pulse (to, vo). The time duration and bandwidth of a pulse-tone are inversely proportional to the characteristic width of the DD domain pulse along the Doppler axis and the delay axis, respectively. As rp —► x, the TD pulse-tone approaches a single TD pulse which is suited for delay-only channels. Similarly, as vp —► oo, the FD pulsetone approaches a single FD pulse which is suited for carrying information in Doppler-only channels. Zak-OTFS is therefore a family of modulations parameterized by rp that interpolates between TD pulse modulation (TDM) and FD pulse modulation (FDM).

[00205] 5.1 .B. Zak-OTFS Receiver

[00206] The received TD signal can be given by Hd (t) = J J ^phy (r, / / ) Sld(t - t) 67277^( / .-^) dr                               (9) where hphy(r, v) is the channel DD spreading function and wtd(0 is the AWGN at the receiver. The DD representation>>dd(T. v) of rtd(0 is given by jydd (r,v) = Zt (rtd ( / )). Matched filtering with the receive DD pulse w„(t, v) results in the signal y&T(T^) =         *a                do)

[00207] Note that (^, v) 's quasi-periodic because twisted convolution preserves quasiperiodicity. Sampling (r, v) on the information lattice Add gives i, (Z- / 1   —         (.__— ^P]                (11) (y for all k,l e Z.

[00208] 5.1 .C. Zak-OTFS I / O Relation

[00209] Combining all equations above, from (4) to (11), gives the I / O relation ydd ~ ^eff pA *<7          +                  02) where the effective channel he{f\k, / ] is given by _      _                        /               / 'fl               IS \ / / df M =3 r = k-—, v = 1-- , where A     . W     N / w^tm)                            (13)

[00210] In (12), nM[k, / ] is the discrete DD domain noise signal. Also, in (12), *CT denotes twisted convolution between two discrete DD domain functions / 7eff[^, / ] and Xdd[k, / ] which is explicitly given by = <14> MIMZ

[00211] Quasi-periodicity of Vdd[^ / ] implies that the transmitted information symbols x^k, / ],k = 0,1,-1,1 = 0,I,---, TV-1 may be detected using the symbols _ydd [kj],k = 0,1, • • -,M -1,1 = 0,1,- ••, N -1. This yields the matrix-vector form of the I / O relation where the MA x 1 vector of these MN received symbols is given by the product of an MN x MN effective channel matrix with the MN x 1 vector of information symbols. It is possible to detect transmitted Zak-OTFS information symbols using techniques developed for MIMO equalization since the I / O relations share the same form.

[00212] In the present document, for DD domain pulse shaping only sine and root raised cosine (RRC) pulses are considered which are respectively given by wtx(r,v) — VBT sinc(TB) sinc(vT),            (15) and wtx(T.v) — VBTrrc8_ (Br) rrc^ (Tv)’           (16) where for 0 < ^ < 1 - 3)) + \>xcos(ttx(1 + ,3))           fin rrc„ (x =-------------------——— --------— ■            ’ (I - ( I       )2)

[00213] Note that / / = 0 coincides with a sine filter. Choosing p > 0 trades energy leakage outside the main lobe and expansion in time / bandwidth. The receiver pulse shaping filter for both sine and RRC waveforms can be given by wrx(r,v) =        wlx(-T,-v).           (18)

[00214] 5.2 Zak-OTFS with Point Pilot pulse-tone

[00215] The Zak-OTFS I / O relation is predictable and non-fading when the delay period tp is greater than the effective channel delay spread (the spread of v) along the delay axis) and the Doppler period is greater than the effective channel Doppler spread (the spread of hefpry) along the Doppler axis). This is the crystallization condition. In the crystalline regime (when the crystallization condition holds), it is possible to read off the taps ofthe effective DD domain channel filter / ?eff[V] from the response to a single Zak-OTFS pulse-tone. This pilot waveform can be referred to as a point pilot since it’s DD domain realization is a localized quasi-periodic pulse. While simple and effective, this is not the maximum likelihood (ML) estimate. In this section, the ML method for estimating the taps of ZieffR, / ] in terms of ambiguity functions is described.

[00216] To begin, the model-dependent and model-free approaches to acquiring the I / O relation in the crystalline regime are reviewed. In the model-dependent approach, a model is imposed on the DD spreading function hpby(r, v), typically by prescribing a finite number of paths and constraining their delay and Doppler shifts. Given this model, the receiver estimates APhy(T,v). The accuracy of this estimate is limited by the time and bandwidth constraints on the pilot signal and by any mismatch between the channel model and the physical channel.

[00217] In the model-free approach, the receiver can estimate the taps ofthe effective DD domain channel filter he^k,!] from the response to a pilot waveform. As mentioned above, this approach is similar in spirit to estimating the taps of an effective discrete-time LTI channel instead of estimating the continuoustime impulse response ofthe underlying physical channel.

[00218] Here and throughout this present document, a Veh-A channel model which consists of six channel paths is considered. The delay-Doppler spreading function can be given by fj z —.1 where hi, t;, and v, respectively denote the gain, delay, and Doppler shift of the z-th channel path. Table-I lists an example power-delay profile for the six channel paths. In an embodiment, the channel delay spread is Tjnax = max( Tt. — mitT = 2.5jUS. For a maximum Doppler shift ofvraaLY = 815 Hz, the Doppler spread is 2 x 0.815 = 1.63 KHz. The Doppler shift of the / -th path can be modeled as Vi=vmaxcos(6,), where the variables = 1,2,-- -,6 are independent and distributed uniformly in the interval [0, 2a-). TABLE I: Power Delay Profile of Doubly-spread Veh-A Channel. Path no. / 1 2 3 4 5 6 Rei. Delay t, ( / is) 0 0.31 0.71 1.09 1.73 2.51 E [Aj2 Rei. Power r J (dB) EfK] K ’ 0 -1 -9 -10 -15 -20

[00219] Veh-A channel is focused on first because it is representative of real propagation environments, and second because it can be very difficult to make the model-dependent mode of operation work. As an example, for a channel bandwidth B = 0.96 MHz, the delay domain resolution is MB ® 1.04^, and the first three paths introduce delay shifts in the interval [0, 0.71J / Z5- which is less than the delay domain resolution. These paths are therefore not separable, and so cannot be estimated accurately. One can choose a higher bandwidth so that the resolution is smaller than the path difference between any two paths thereby making the paths separable. However, this would only work for a particular path delay profile. In real scenarios, the path delays can change, however bandwidth cannot be increased indefinitely.

[00220] By contrast, model-free operation is always feasible irrespective of whether the paths are separable or not. FIG. 10 compares BER performance of model-dependent and model-free modes of operation. We consider the Veh-A channel with vp = 15 KHz, B = 0.96 MHz, T = 1.6 ms. Separate Zak-OTFS subframes can be dedicated to channel sensing and data transmission in order to focus on the difference between model-dependent and model-free modes of operation. It is emphasized that this is NOT integrated sensing and communication. In the model-free mode, the taps of Aeff[£, / ] are estimated directly from the received DD domain symbols received in response to a point pulse-tone situated in the middle of 7?0. In the model-dependent mode, the receiver first estimates the parameters (t;, v>, h,) of the underlying physical channel, which is then used to estimate the taps of h^k,I\. MMSE equalization of the matrix-vector form of the Zak-OTFS I / O relation can be employed to detect information symbols at the receiver. It is observed that BER performance of the model-dependent mode exhibits a high error floor. This is due to inaccurate estimates of the parameters of the underlying physical channel, which is a consequence of insufficient subframe bandwidth and duration. On the other hand, BER performance of the model-free mode of Zak-OTFS can be considerably better, only slightly worse than performance with perfect knowledge of the Zak-OTFS I / O relation. For example, in the crystalline regime, BER performance of the model-free mode of Zak-OTFS is only slightly worse than performance with perfect knowledge of the I / O relation.

[00221] FIG. 10 provides BER performance of Zak-OTFS in model-dependent and model-free mode of operation. In the crystalline regime (vp = 15 KHz), BER performance of the model-free mode of Zak-OTFS is only slightly worse than performance with perfect knowledge of the I / O relation.

[00222] Herein, this present document focuses on the model-free mode of operation in the crystalline regime. It is now described how to integrate sensing and communication within a single Zak-OTFS subframe. The discrete DD domain point pulse-tone indexed by (kp, lp) can be given by trp.dd^, I] =              ~   ~ nM] 6\l — lp — mN]- (20) n.mEZ

[00223] Quasi-periodicity results in impulses at (kp + nM, lp + mN) for n,m e Z . In the absence of data, the received response can be given by h^\k kp nM, l-l,p- mN] N2^------ n.mEZ + / zdd • I]                                                                                          (21) where the last equation follows from (14).

[00224] The support set 5 of / ] can consist of all pairs (k, I) for which heS[k, / ] / 0. The crystallization condition is satisfied when the delay spread of S is less than the delay period M and the Doppler spread is less than the Doppler period N. In the crystalline regime, the taps ofh^k, / ] can be read off from (n, m) = (0, 0) term in (21). A previously presented method was simply to read off the point pulse- J Ltt tone response within the support set of h^k-kp, l-lp\ and multiply by S 1     . This method can give a much better estimate than the model-dependent approach, but it is not the ML estimation method. The ML estimator can be given by the samples of the cross-ambiguity between the received point pulse tone j^>Cid[K / ] and the point pulse-tone xp.dd[£,Z], when observed only within the support set <S of heAk, / ]. Given (4,Z)e<S we have = Ayp.Xp[k,l] = E                                                  (22) k'=Q r=c>

[00225] The cross-ambiguity function A^k, / ] between any two quasi-periodic discrete DD signals a\k. / ] and b[k, / ] can be defined as in section 5.6 (under The Discrete Ambiguity Function). It follows from Theorem 6 in section 5.6 (under The Discrete Ambiguity Function) that the noise-free cross-ambiguity Av % [^]'s given by .vp [k, I] — hRff[k^ Z] *0- x4.Ep,Tp [k, Z] (23) where Ax x [kJ] is the self-ambiguity function of the point pulse-tone xp>dd[£, / ]- From section 5.6 (under Self-ambiguity Function of the Point Pilot Signal) it follows that the self-ambiguity function of a point pulsetone can be given by 6[k - nM] Nl - mN] (24) from which it is clear that it is supported on the period lattice Ap, where Ap MZ x NZ — {(nM, mN) | n, m G Z}.       (25)

[00226] Using (24) in (23) gives M!Jp,Xp [kJ] — .   (n M. I p - m N k p ) — hQ^[k, I] *<r y^?'7^            d[k — nM] 6[l — mN] I: .11' : 'j where the (n, m) = (0, 0) term is simply heff[k, / ]. In the crystalline regime, the supports of terms in the RHS do not overlap, and the (0, 0) term is simply equal to Ay x [kJ] for (kj)eS.

[00227] FIG. 18 shows points on the information lattice A^, the period lattice Ap and the dual of the information lattice A^. If the delay and Doppler coordinates of these lattices are scaled down by tpIM and vpIN respectively, then Add may be identified with Z and A^ with MNZ. xMNl. Now consider any shift(nMN, mMN) of a quasi-periodic signal Xdd[A / ] in the discrete DD domain. It follows from (3) that [k + nMN, I 4- mMN] = e?27r *     [k, I] xM[kJ].                   (27)

[00228] In FIG. 18, points on the information lattice Add, period lattice Ap and the dual of information lattice Add are shown by black, light gray and dark gray dots respectively. Adjacent points on the information lattice are separated by At = tpIM and Av = vp / Nalong the delay and Doppler axis respectively. Adjacent points on the dual of the information lattice are separated by MNAt=Ntp and MNAv=Mvp along the delay and Doppler axis respectively.

[00229] Signals that are quasi-periodic with respect to the information lattice are periodic with respect to the dual of the information lattice. In section 5.3, filters in the discrete DD domain are introduced where this interplay will be important.

[00230]

[0042] We conclude this Section by discussing two practical issues that limit the use of point sensing pulse-tones.

[00231] 5.2.A High Peak-to-Average-Power-Ratio (PAPR)

[00232] FIG 19A depicts a TD realization of a point pulse-tone with Doppler period vp = 30 KHz, M= 31, N = 37. Point pulse-tone is located at (kp, lp) = ((A / + 1) / 2, (N + 1) / 2). Only part of the entire TD pulse-tone is shown here, with samples taken every    seconds where B = Mvp = 930 KHz. RRC pulse shaping with A = A = 0.6.

[00233] FIG. 19A displays the magnitude of the TD realization of a point sensing pulse-tone. We observe a train of narrow pulses, exhibiting sharp peaks at the pulse locations, leading to high PAPR. This requires the use of highly linear power amplifiers which are typically power-inefficient.

[00234] 5.2.B Interference Between Data and Sensing pulse-tones

[00235] Data and sensing pulse-tones can combine at the transmitter to produce the noise-free DD domain signal given by data pulsones            pilot pulsone (28)

[00236] First it will be shown that the received data signal results from the action of a discrete DD domain filter 1                 / 52 52 x^k^ / zi I h^k-% ^k -kf^1 ~ A-'—0 1^0         \ (29) on the discrete quasi-periodic pulse at the origin, denoted by x0.dd[&, / ]. Then it will be shown that the received pilot pulse-tone signal results from the action of a second discrete DD domain filter ^,ha[[k. I] *„ (6[k - kp]6[Z - Zj)              (30) on the same quasi-periodic pulse x.ddA. / ]. This description makes it clear that data and sensing pulsetones will interfere at the receiver. It is emphasized that it is only possible to express this interference in terms of operators because signal processing takes place in the DD domain where twisted convolution is associative.

[00237] An equivalent expression for the DD domain information signal in (2) is that the data symbol is carried by a discrete quasi-periodic DD domain pulseat (k'J') so that 1 A i / ^d,ddpMi == y yy h'yyy k'-O l'-() \ yy ei2n~ d[k — k' — nM] -1 / - mN] j Quasi-periodic DD pulse atik'J.') M-l A'-l          /                                 \ = E E xum / w -   - (A ♦„        .        on k' = 0 l'=0         \                                    /

[00238] Step (a) follows from the fact that a quasi-periodic DD pulse at (k'J’) is simply x0.dd[&, / ] shifted by ) , ie. 52       8[k — k' — nM] 8[l — I1 - mN] n - (8[k - k']8[l - I'])     52 d^k - nM^ ~ n. m £ £ ------------------------------------------------------------------------ = x0M[k-l\

[00239] Since the pilot signal xp.dd[£, / ] is a discrete quasi-periodic DD domain pulse at (kp,lp) it follows that ;£p.dd[£ ^] — [k              £]) *cr ^O.ddf&i = £ e32”’^ $[k ~ kf, ~ nM] S[l               W

[00240] Since twisted convolution is associative, it follows from (31) that w i a ■                                   \ 52 53’^ W “ k'W ~ Zo4M} (34) fc'=0 / '=0                            / ^d,dd[£ ~ so that the received noise-free signal is given by i \ / Nph'Qff [k, k *a '£p,dd [^; (35)

[00241] Suppose that separate Zak-OTFS subframes are dedicated to channel sensing and data transmission. The ML estimate for the taps of h^k, / ] is simply the cross-ambiguity function of the received pilot with the transmitted pilot (see (129)), as shown in FIG. 19B.

[00242] FIG. 19B graphically depicts magnitude of the cross-ambiguity between the transmitted point pilot signal and the received point pilot signal, i.e., for EVA channel with vmax = 815 Hz. M=3i,N=31, ^ = 30 KHz; RRC pulse shaping = = 0.6; (kpJp) = ((M+ 1) / 2, (N+ 1) / 2). The support set of / ?eff[£, / ] is approximately the region within the white rectangle.

[00243] Now suppose that we integrate channel sensing and data transmission within the same Zak-OTFS subframe for which the received discrete noise-free DD signal is given by (35). The ML estimates for the taps of / ?en| / x / ] are now given by the cross-ambiguity function (36)

[00244] The second term in (36) is the ML estimate for the taps of h^k, / ] in the absence of data. FIG. 19C illustrates how the first term, which represents interference from the data signal, can obscure the second term. Within the first term, Ax^ x [£, / ] is the cross-ambiguity function between the quasi-periodic pulse at the origin and the quasi-periodic pilot signal.

[00245] FIG. 19C graphically depicts the magnitude of the cross-ambiguity between the transmitted point pilot signal and the received signal which includes contributions from pilot and data ( Ay x ). In FIG. 19C: EVA channel with vmax = 815 Hz. M= 31, N= 37, vp = 30 KHz; RRC pulse shaping pr=pv = 0.6; (kp, Ip) = ((M + 1) / 2, (N + 1) / 2). In the example of FIG. 19C, interference prevents channel sensing.

[00246] The cross-ambiguity functions Ax^ x and Ax x are both supported on the period lattice. This leads to interference between data and pilot signals at the receiver which prevents channel sensing by completely obscuring the second term. FIG. 19D illustrates how the second term can be made visible by introducing a guard band Q around the pilot signal, in this case a 7x7 rectangle. Guard bands cannot be avoided when the cross-ambiguity functions A .. \kJ\ and Ax x [£, / ] are both supported on the same lattice.

[00247] In the example of FIG. 19D, data / information symbols are not transmitted in a guard region Q (e.g., a 7x7 rectangle around the location (kp, lp) = ((M + 1) / 2, (N+ 1) / 2) of the pilot signal) which is the location of the point sensing signal. FIG. 19D graphically depicts the magnitude of the cross-ambiguity between the transmitted pilot signal and the data component of the received signal, i.e., A v [^] (see (36)). In FIG. 19D: Veh-A channel with v„iax = 815 Hz. M = 31, N= 37, vp = 30 KHz; RRC pulse shaping pT = [E = 0.6. In the example of FIG. 19D, interference preventing channel sensing has been suppressed.

[00248] In section 5.4, it will be described how to filter the point pilot signal so that Ax x can be supported on a lattice different from the period lattice. It will be shown that it is possible to choose the filter so that the cross-ambiguity function Ax° x [£, / ] appears as noise to the self-ambiguity function of the filtered pilot signal. We seek a filter that results in cross-ambiguity functions that are xp-p L J unbiased or incoherent. This can eliminate the need for a guard band at the cost of reducing the SNR of channel sensing. In addition, the energy of the filtered pilot signal will be spread uniformly in the discrete DD domain, leading to a TD realization with excellent PAPR.

[00249] Finally, we define the data signal power to noise power ratio, the pilot signal power to noise power ratio, and the pilot signal power to data signal power ratio.

[00250] From (31), it follows that the average energy of the data signal is Ed, i.e. (37)

[00251] Since the point pilot signal xp>dd[^, / ] is a discrete quasi-periodic DD pulse localized at (kp, lp), from (33) it follows that the energy of the point pilot signal is Ep, i.e. ,W I \    : fe=0 1=0 Ep (38)

[00252] AWGN «td(0 at the receiver has PSD No Watt / Hz. The average total energy of the discrete DD domain noise signal 7] (see (12)) can be given by nw-i a - i Lk=0 1=0 MNN0 (39)

[00253] The effective channel gain is Z MM and therefore the received energy of (k,l)&S data / information pulse-tones and that of the pilot pulse-tone are Ed X |MV]|2 and (k,iys Ep |^eff |X / ]| respectively. Hence the ratio of the power of the received data pulse-tones to the (kN^S noise power (data SNR) can be given by Pd (40) MNNq and the ratio of the power of the received pilot pulse-tone to that of noise (i.e., pilot SNR) can be given by Ev £ \EM\2 (kEes               (41) MN No

[00254] The ratio of the power of the received data pulse-tones to the power of the received pilot pulse-tones can therefore be given by X which is subsequently referred to as the pilot power to data power ratio (PDR).

[00255] 5.3. Filtering in the Discrete Delay Doppler Domain

[00256] Here, we describe how to design a filter in the discrete DD domain to spread a point pilot signal over the MN pulse-tones (M*N pulse-tones) located on the information grid, so that each pulse-tone contributes a fraction of the spread pilot energy. The discrete DD domain filter ws\E / ] can act on the point pilot xPdd|^. / ] by twisted convolution to produce a spread pilot signal x5;dd[^, / ] given by ^s.ddlM] = ™s[kJ] *a xp,dd[M]-            (42) where xPjdd[£, / ] is given by (20). Note that xs.dd[£. / ] is quasi-periodic since twisted convolution preserves quasi-periodicity.

[00257] More generally, we develop the fundamentals of filtering in the discrete DD domain. We start from the theory of linear time invariant (LTI) systems. Here it is well known that linear convolution of a discrete-time periodic signal with a discrete-time filter is equivalent to periodic linear convolution of the periodic signal with the periodic extension of the filter. Thus, two discrete-time filters with identical periodic extensions are equivalent. If the period is L, then the discrete-time filter is specified by a vector in CL . In the discrete DD domain, quasi-periodic signals are periodic with period MN along both delay and Doppler axes (see (27)). We now show that twisted convolution of a quasi-periodic discrete DD signal with a discrete filter / ] is equivalent to a MN-periodic twisted convolution [i.e., M*N-periodic twisted convolution] of that quasi-periodic signal with the MN-periodic extension of the discrete filter. Thus, two discrete DD domain filters with identical periodic extensions are equivalent. Since the period is MN along both delay and Doppler axes, a discrete DD domain filter can be specified by a vector in C .

[00258] Given an arbitrary discrete DD domain filteras[k, / ], we can define its ATA-periodic extension a [k, / ] by a[kj] - as\k + nMN,l + mMN]- n,mel

[00259] Note thato[£, / ] is periodic with period MN along the delay and Doppler axes. The extension a[k, / ] acts on a quasi-periodic signal b\k. I] by ATA-periodic twisted convolution, denoted by ®a. More precisely ilk. I] b\k,l] ■ \i\ i                                                 , =           a[kM]b[k~                                (44) k / —0

[00260] Theorem 1: The twisted convolution of a discrete DD filter as[k, / ] with a discrete quasi-periodic signal b[k, / ] coincides with the ATA-periodic twisted convolution of its ATA-periodic extension a\k. / ] with b[k, Z], i.e. as[A:,Z] b[k, I] — a[A’, / ]®a 6[A:, / ].          (45) Proof: \Ne start from the definition of twisted convolution, then view delay and Doppler indices modulo ATA, then simplify using the definition of an ATA-periodic extension (43). The detailed derivation of (45) is given by (46). (46)

[00261] Returning to (42), we apply Theorem 1 to obtain 3\dd[^H               I] ^p,dd[^'- ^] = w[k,l] ®a X^dd[k,l]             (47) where w[k,l] 4 w4k + nMN, I + mMN] n,m& (48) is the DD domain A / ? / -periodic extension of ws[k, / ]. We substitute forxp>dd[£, I] in (47) using (20) to obtain the spread DD pilot signal expression given by (49). (49)

[00262] MN-periodic discrete chirp filter w[k, I]: We define ...... ...... MN ' (50) for k,l G Z, and we refer to q € Z as the slope parameter. In the next section, we will show that the constant factor M(MN) in (50) results in a spread pilot signal xs.dd[£, / ] with unit energy. We will observe that |xsdd [^, / ]| is almost constant (say |xsdd [^, / ]| ~ a for all k, 0 when M and N are odd primes, and when q is relatively prime to both M and N. Since the total energy of the spread pilot is A / -LV-1           2                                   ____ 1=S [A; / ]| x MNa1, we conclude a ~ 1 / a / AW . Hence the peak amplitude of the point k=0 1=0 pilot signal (1 at location (k, I) = (kp, lp)) is about times higher than that of the spread pilot signal. FIG. 19E compares the energy profiles of point and spread pilot signals.

[00263] In the example of FIG. 19E, energy profiles of DD domain pilot signals are provided. M= 31, N = 37 and the total DD domain energy is normalized to 1 for both point pilot and spread pilot signals, that M-EV-1          2  A / -LV-1          2 is, |acsdd \k, / ]| = ^|xpdd [^]| = 1 ■ The point pilot takes the value 0 at all locations except £=0 / =0                  £=0 / =0 (kp, lp) where the magnitude is 1. The squared magnitude of each spread pilot symbol is about 1 / (ACV), i.e., about 101ogio(A / AO = 30.6 dB below the peak squared-magnitude of a point pilot DD signal.

[00264] 5.4 Zak-OTFS with Spread Sensing pulse-tone

[00265] 5.4.A Reducing PAPR

[00266] In Section 5.3, we constructed a spread pulse-tone using a discrete chirp filter, and we compared the energy profiles of point and spread pulse-tones in the discrete DD domain. Recall that the TD realization of the point pulse-tone located at (kp, lp) consists of narrow TD pulses at time instances t = {nrp +kp-^j,n G Z where the spread of each pulse is approximately rp / M, which is the inverse bandwidth MB (See FIG. 19A). FIG. 19F shows the TD realization of the spread pulse-tone constructed in Section 5.3. This TD realization is the sum of all MN pulse-tones located on the information grid and the constant term M(MN) appearing in (50) reduces the amplitude of each point pulse-tone in the sum. The N pulse-tones located at a given kp interfere destructively, and the point pulse-tones at different locations kp result in trains of TD pulses at different locations. This explains why the TD realization of the spread pulsetone is less peaky than the TD realization of the point pulse-tone shown in FIG. 19A.

[00267] FIG. 19F graphically depicts a TD realization std (t = k / (AB^,k g Z of a spread pilot signal with Doppler period vp = 30 KHz, M= 31, N= 37. Discrete chirp filter with q = 3. Only part of the entire TD realization is shown here, with samples taken every seconds where B =Mvp = 930 KHz. RRC pulse      v) with roll-off factors,   = ■ = 0.6. Again, observe that the TD realization is not as peaky as that of the point pulse-tone shown in FIG. 19A.

[00268] FIG. 19G provides a complementary CDF (CCDF) plot of IAPR, where M= 31, N = 37; RRC pulse v) with roll-off factors, = pv = 0.6; Discrete chirp filter with q = 3; and PDR pplpd = 10 dB.

[00269] FIG. 19G displays the complementary CDF of the instantaneous-to-average-power ratio (IAPR) for example spread and point pilots. In the absence of data, the IAPR ofthe spread pulse-tone does not exceed 5 dB, whereas the IAPR ofthe point pilot is almost 15 dB. With data alone, the IAPR is similar to that of Gaussian noise. In Section 5.5 we choose a pilot to data power ratio (PDR) of 10 dB to integrate sensing and communication within a single Zak-OTFS subframe. For both spread pilot with data and point pilot with data, the IAPR rarely exceeds 7 dB. However, for a higher PDR of 25 dB, while the IAPR of spread pilot with data rarely exceeds 9 dB, the IAPR of point pilot with data can be as high as 12 dB. In various embodiments, spreading can reduce the PAPR ofthe transmitted signal.

[00270] 5.4.B Achieving Predictability

[00271] Here, we show that the self-ambiguity function ofthe spread pulse-tone is supported on a lattice that is obtained by applying a linear transformation to the period lattice that depends on the slope q ofthe discrete chirp. When the effective channel filter satisfies the crystallization condition with respect to the rotated lattice A,, we will describe how to estimate the taps of / ] from the crossambiguity function ofthe received pilot and the transmitted pilot. Here, we will focus on sensing in the absence of data and noise, but in section 5.5 we will describe methods for channel sensing in the presence of data and noise.

[00272] The spread pilot signal xs.dd|A / ] = w[h / ] ©„ xp.dd[£, / ] is given by (47). Here w\k. / ] is the spreading filter and xp,dd[^ / ] is the point pilot signal. In the absence of data, the received pilot signal can be given by £ / s,dd— ^eff P© ^s,dd[^?^] + ^dd■      (51)

[00273] The ML estimate of helT [£, / ] is then given by the cross-ambiguity between the received spread pilot and the transmitted spread pilot. It follows from (113) in section 5.6 (under The Discrete Ambiguity Function) that Aeff[A', I] — zl.;     [ / c. Z] A / I .VI k1^ 1'^0 (52) for (£, / ) e <S. We apply Theorem 6 from section 5.6 (under The Discrete Ambiguity Function) to obtain (in the absence of data and noise) *cr ,xs ■                (53) where Ax x [£, / ] is the self-ambiguity function of the spread pilot xs.dd[£ / ] (cf. (23)).

[00274] Henceforth, we restrict our attention to the spread pulse-tone ^s.dd[A’J] = U)[kJ] ®a%dd[M             (54) j Q (£^ +1 ) arising from the discrete chirp filter           =              MN Since ambiguity functions are periodic along both delay and Doppler axes with period MN, i.e., Ax x [£+ p,MN,l + p2MN] = AXi JM]. for all ^, / >2 e Z , it suffices to characterize the support set of Ax x [£, / ] modulo MN along both axes. Theorem 2 below shows that the self-ambiguity function Ax x \k,l\ is supported on a lattice.

[00275] Theorem 2: Given odd primes N, Aland q relatively prime to both Aland N '1 ,[2qk-l]M = 0 <     ,               -2<i)].v =0’ 0 , otherwise (55) where (2q) 1 is the unique inverse of   modulo MN.

[00276] Note, for any integer a and positive integer M, [a] / w denotes the unique smallest non negative integer which is congruent to a modulo M.

[00277] Theorem 2 implies that the self-ambiguity function Ax v [£,Z] is non-zero if and only if there exist integers n, m such that I = nA'I + 2qk . k - 01 + mN, 0 & [(2( / )^-2( / 1,^,. (56)

[00278] We rewrite (56) as (1 - 2q0)k - OnM + mN (1 - 2q0)l = nA! + 2qmN ■ (57)

[00279] It follows that the self-ambiguity function A s supported on a lattice A, given by 'e r 1 2q A ■                            (58)

[00280] Note, in (58), arithmetic is interpreted as modulo MN.

[00281] We show that the lattices A^, A, have the same fundamental volume by showing that they have the same density. We show the densities are the same by showing that the number of A, points in the rectangle Q bounded by (0, 0), (MN, 0), (Q,MN) and (MN, MN) is the same as the number of A^ points.

[00282] Since 0 = [(2q)~X-2q\MN,(\-2qO) = Aq2 modulo MN. Since M and N are odd primes and q is relatively prime to both M and N, ^q2 is relatively prime to MN. Therefore, (1 - 2q0) is relatively prime to MN, i.e., (1 — 2q0^ 0 modulo .VW and hence, the linear transformation non-singular modulo MN. Hence the number of Aq points in £1 equals the number of Ap points in Q.

[00283] From (113) in section 5.6 (under The Discrete Ambiguity Function) we know that the total energy of the spread pulse-tone is simply the value of its cross-ambiguity at the origin, i.e., -W^o] = £ £Kdd[Mp =. r               <69) k=0 l-o

[00284] From (55) it follows that the self-ambiguity function Ax x [£, / ] is given by 1 2q N.-NmN. where (kt, h) are the lattice points of Aq. Using (60) in (53) gives (ki .1 j) G Aq = h^k, I] +            - k’’1 -                      *” ( / vUOGA^^Ud^aM))

[00285] The first term in the R.H.S. of the last equation above is the discrete channel filter h^k, / ] itself. The other terms in the R.H.S. correspond to the other lattice points not at the origin. Therefore, accurate estimation of the taps of / ] may not be possible if for any non-zero lattice point (ki, 1,) / (0, 0) the support set of the corresponding term CJ ’he^ ^k — k^l — lqJe MN overlaps with the support set of h^lk, / ]. We now show that if the channel satisfies the crystallization condition with respect to Aq, then we can accurately estimate the taps of heS[k, / ].

[00286] Example 1: The dark gray dots in FIG. 19H mark the points of the lattice Aq, i.e., support of the self-ambiguity function of the spread pulse-tone with M - 11, IM - 13, q = 5, (kP, lP) = (0, 0). The fundamental period / region of Ag is a parallelogram with area 143 = 11 x 13 = MN bounded by the points (0, 0), (3, 19), (11,22) and (8, 3). The lattice Aq is generated by (3, 19) and (8, 3). The cross-ambiguity between the received spread pulse-tone and the transmitted spread pulse-tone is supported on the union of the gray rectangles (these rectangles correspond to the terms in the RHS of (61)). Since the rectangles do not overlap, there is no DD domain aliasing (i.e., the channel satisfies the crystallization condition with respect to Aq), and we can accurately estimate hett[k, / ] from the response received within the gray rectangle with the black border. In other words, the support of 7?etf[A, / ] is limited to a rectangular region with delay spread (kmax - kmm) and Doppler spread {Imax — Imin) shown as the gray rectangle with the black border.

[00287] That is, in FIG. 19H, the dark gray dots mark the support of the self-ambiguity function of the spread pulse-tone with M = 11, N= 13, q = 5, (kp, lp) = (0, 0). The support of / 7ea{Ar, / ] is limited to a rectangular region with delay spread (kmax - kmi„) and Doppler spread (lmax - lmi„) shown as the gray rectangle with the black border.

[00288] Example 2: The dark gray dots in FIG. 191 mark the support of the self-ambiguity function of the spread pulse-tone with M= 11,7V= 13, ^ = 4, (kp, lP) = (0, 0). The image of the fundamental period / region is a parallelogram bounded by (0, 0), (24, 5), (5, 7) and (29, 12). The support of hef^k, / ] is the same as that in FIG. 19H and is again shown as the gray rectangle with the black border. This rectangle overlaps the gray rectangles located at (5, 7) and (-5, -7) and taps of Am] A. / ] located in the overlapped regions cannot be estimated accurately. By contrast, taps of hes[k, / | located in the non-overlapped region can be estimated accurately.

[00289] That is, in FIG. 191, the dark gray dots mark the lattice supporting the self-ambiguity function of the spread pulse-tone with M= 11, N= 13, q = 4, (kp, lp) = (0, 0). The lattice A9 is generated by the vectors (24, 5) and (5, 7). The channel spread is the same as that in Figure 19H. The support of h^k, / ] is limited to the gray rectangle with the black border.

[00290] Note that the size and shape of the gray rectangle does not change with q since it depends only on the delay Doppler spreading function Aphy(r, v) of the underlying physical channel, the pulse shaping filters wJt, v), w'n(r, v), and the information grid A^ (which in turn depends only on T and B, and is independent of (¾ vp)).

[00291] Previously, we have emphasized that the (strong) crystallization condition is satisfied when the channel delay spread is less than the delay period and the channel Doppler spread is less than the Doppler period. We have also previously described a weaker mathematical condition that eliminates DD domain aliasing. When we use discrete chirp filters to construct spread pulse-tones, the self-ambiguity function is supported on a transformed lattice Ag. The significance of the weaker mathematical condition is that there are channels for which aliasing-free acquisition of Aeff[£. / ] may not be possible with the period lattice Ap, but becomes possible with the transformed lattice A^.

[00292] Consider a spread pulse-tone where the self-ambiguity function Ax x [£, / ] is supported on a lattice A,. Define the support S^. z of heff[k - k, I - A] by =          \heff[k — kijl — lt] ^ ()}■

[00293] The weaker mathematical condition that eliminates DD domain aliasing is 5(0,0) n ( [J <%,(,)] = 0'        <63)

[00294] Observe that the set z is obtained by translating 5J() 0 by the lattice point (kJ,) e A?. Note also that when (63) is satisfied, two distinct taps in 0) cannot differ by a lattice point in A^.

[00295] Lemmal: Let           (^p^), betwodistincttaps in *~^o 0) ■ If the weaker crystallization condition (63) is satisfied then Ar T \k' - k' - I'] - 0-                  (64) u'' y 'A' b L &           _L 1   £          1 J

[00296] Proof: The self-ambiguity function As.s[k, I] is supported on the lattice Aq and (k2 —k[, l2 —l[) £ A when (63) is satisfied. This is because, if (k2 — k[, l2 — were to be some (^, / JgA^ ((^,7,)^(0,0)) , then [ki + k[, It +1'^ & S^k which is not possible since (ki + k[, / , + Zj) = {k'2,l2 ) G <S(0 0) and this point cannot lie both in <S(0 0| and S^k as (63) is satisfied.

[00297] Example 3: We consider a channel with support set <SJQ 0) for which the delay spread and         the         Doppler         spread A, = (max,, / -min,,       / ) satisfy At A / >MN = BT(MN = (Btp}(Tv„) = Bl}. For a given B and T, it is not possible to choose periods M and N such that At < M and A? < N, so the channel does not satisfy the (strong) crystallization condition with respect to the period lattice Ap. Here we show that aliasing free acquisition of h^k, / ] may still be possible if the channel is underspread ( c (0,0) <MN). The support set of the effective channel filter is shown as the gray rectangle with the black border in FIG. 19J. The dotted rectangle in FIG. 19J is the smallest rectangle with axes parallel to the delay and Doppler axes that circumscribes <S(0 ()). The crystallization condition is not satisfied with respect to the period lattice Ap since the area of the dotted rectangle At A is greater than MN. However the (weaker) crystallization condition (63) is satisfied with respect to Aq because the gray rectangles are disjoint. Hence, we can accurately estimate the effective channel taps h^k, / ].

[00298] That is, in FIG. 19J, the dark gray dots mark the lattice Aq supporting the self-ambiguity function of the spread pulse-tone with M = 11, N = 13, q = 3, (kp, lp) = (0, 0). The support <S(00) of the effective channel taps is shown as the gray rectangle with the black border. The dotted rectangle is the smallest rectangle with axes parallel to the delay and Doppler axes that circumscribes <S(0 .

[00299] Example 4: We consider the Veh-A channel with the power delay profile given in Table-1 of section 5.2 and with path Doppler shifts vi V2 Vmax? V3 VmaxN. V4 Vm^xN. V5 v^^ / 4,      v^^ / 4. We now illustrate the importance of filter design by comparing heat maps of the cross-ambiguity Ay x [£, / ] (see (53)) between the received spread pulse-tone and the transmitted spread pulse-tone, for spread pulse-tones defined by chirp filters with <7 = 3 and ^=36. Before we study these comparisons, through FIG. 19K we illustrate the heat map for the effective discrete DD domain channel filter hea{k, / ]. It is observed that the effective channel Doppler spread is higher for vmax = 12 KHz than for vmax = 815 Hz. FIG. 19K provides heat map of | / zeff for Veh-A channel, RRC pulse shaping filter ( / ?T=0V = 0.6), Doppler period vp = 30 KHz, M= 31, N= 37.

[00300] We first considerthe spread pulse-tone defined by the chirp filter with ^ = 3. FIG. 19L shows the heat map of x for vmax = 815 Hz and vmax = 12 KHz [for chirp filter with q = 3 and Veh-A channel described in FIG. 19K for both cases]. In both cases, the support sets Sk t , (ki,li) eA9 of hes[k-ki, l-lt] do not overlap and the (weaker) crystallization condition is satisfied.

[00301] Next we consider the spread pulse-tone defined by the chirp filter with q = 36. FIG. 19M shows the heat map of for vmax = 815 Hz and vmax = 12 KHz [for chirp filter with q = 36 and Veh-A channel described in FIG. 19K for both cases]. When vmax = 815 Hz, the support sets do not overlap. When vmax = 12 KHz, the support sets do overlap and the (weaker) crystallization condition is not satisfied. This compromises the accuracy of channel estimation, which in turn degrades BER performance (see Section 5.5 for more details).

[00302] 5.5 Integrated Sensing and Communication (ISAC)

[00303] 5.5.A ISAC with Spread Sensing pulse-tone

[00304] When we integrate channel sensing and data transmission in the same Zak-OTFS subframe, the spread pulse-tone used for channel sensing interferes with the point pulse-tones used for data transmission. In this section, we describe howto design the spread pulse-tone so that interference is noiselike. This property can eliminate the need for a guard band at the cost of reducing the SNR for channel sensing. Noise-like interference translates to a spread pulse-tone with energy that is almost uniformly distributed over the discrete DD domain. This explains why the PAPR of the spread pulse-tone is significantly lower than that of the point pulse-tone.

[00305] The discrete DD domain transmit signal *dd[£. / ] is given by E(i lCd,dd [k, / ] 4“ \ / -Ep ^s.dd Q is the sum of a data term and a sensing term. The quasi-periodic data signal xddd [kJ], kJ e Z isgiven by 1                                   I > h M.dd[M =                                              (66) yM N

[00306] The terms x[kj],k — 0,1 — 0,1,are the MN information symbols, each with unit average energy. The average energy of the data signal, i.e., M-W-l r        , LLE MM £=0 / =0 L is Ed MN M-l Ai 0 / = 0 Ed- (67) 100307] It follows from (59) that the average energy of the sensing signal is / M \ A i              \ E» £ YM'NM =£A \ k-0 / =0             / (68)

[00308] We define PDR = EJEu to be the ratio of pilot power to data power. It follows from (12) that the received discrete DD domain signal can then be given by yddlk.l] ■■■■• heff[kd] xdd[k,l] + ndd[M — \ / -^d (^eff[A> Z] *cr M,dd[A'5Z]) Received data signal M                   Z] *tr -7 s,dd[k, / ]) 4~ ndd[A , Z] Received sensing signal (69) where Aetr[^, / ] is the effective discrete DD domain channel filter.

[00309] Channel sensing: Here we suppose that the (weaker) crystallization condition (63) holds, and we describe how to estimate the effective channel filter hes{k, Z] from the received signaly^[k, Z], Recall from (52), that in the absence of data, the ML estimate [&, / ] is given by the cross-ambiguity function between the received signal and the transmitted spread pulse-tone.

[00310] Theorem 3: In the presence of data, the cross-ambiguity function Ay x [kJ] between j’ddfc / ] and xs.dd[&, Z] is given by J i j           X j , X s [ k ; + E E +mli']      ~ + i' ~ i] k^O lf^Q < --------------------------------------------------------------------------------------------------------------------------------------- ^An^[kd] (70) where Ax x [A, / ] is the self-ambiguity function of the spread pulse-tone and M-l N —1                                               , , . . -‘k.-UM = E E           - k.f k'=O l'=0                                                           (71) is the cross-ambiguity between the data signal and the sensing signal.

[00311] When the crystallization condition (63) holds, we can obtain h^k, l\ by evaluating the first term on the RHS of (70) inside 0) (see (62)). For                , the cross-ambiguity function Avx [£, / ] nowreducesto -4<,.MM] = Mr[Mj + AEdhefi[k,i] 4x4,»,[k,z] + -4,,.:,.,. [ML                                                    <72)

[00312] Our estimate of heff[k. / ] is hefflk, I] for (k, I) 6 5(o,o)- (73)

[00313] Theorem 4 provides an estimate for the mean squared error contributed by the second term in (72) which measures interference to sensing from data. This error term is given by

[00314] Theorem 4: When the (weaker) crystallization condition (63) holds, interference from data to sensing contributes 21 Eki ■v-' MN (kE)&St (75) to the mean squared error of the estimate h,n [£, / ] given in (73).

[00315] FIG. 19N shows the heat map for the quantities x [^, / ]| which measure interference to sensing from data. We observe that the values are roughly of the order 101og10 ^1 / y / MN ) « —15 dB , in other words x [V]| ~ ]J4mn .

[00316] The factor of 1 / (MN) is due to the fact that x is roughly of order 1 / >JMN . In FIG. 19N, we have plotted the heat map for \A          with M= 31, N= 37, spread pulse-tone defined | xd’xs L J| by chirp filter with q = 3 and for a random realization of the information symbols [including PDR = EpIEd = 0 dB], It is observed that the values are roughly of the order 101og10 (1 / y / MN j ® -15 dB.

[00317] The last term in (72) represents the contribution of noise to channel estimation error. Since the received DD domain noise samples are i.i.d. CA / (0, Ao) distributed, the samples of An x [£, / ] are M-W-l        2 zero mean. Since L2xs dd [ V]| = 1, it follows from (70) that the variance is given by k=0 1=0 E [IAn,x*[kJ]I2] W                    (76)

[00318] It follows from (73) that the normalized mean squared estimation error (NMSE) can be given by (HAAwO where we have used the expression for Ay x yk, / Jin the R.H.S. of (72), and the mean squared values of the data interference and noise terms in (72) from (75) and (76) respectively. In (77), pd refers to data SNR (see (40)) and pp refers to the pilot SNR (see (41)).

[00319] Note, in practical implementations, while reading the estimate of / 7eff[£, / ] from Ay x [WWA e <SJ0 0), all taps of Ay x [£, / ] in may not be genuine taps of Acfft / ] primarily due to the randomness of noise and data interference. Therefore, in some embodiments, only those taps of Ay x [£, / ],(k,l) e , are considered whose magnitude exceeds a pre-determined threshold. This threshold could be 2 or 3 times the standard deviation of the estimation error (Aeff [&, / ] — heS [£, / ] ) (which can be calculated from (77)).

[00320] When the crystallization condition is satisfied, the factor 1 / (MV) in (77) significantly reduces the impact of data and noise on the accuracy of channel estimation. The factor 1 / (A£V) is present because interference between the spread pulse-tone used for sensing and the point pulse-tones used for data transmission is noise-like.

[00321] Cancellation of spread pilot followed by data detection: FIG. 11B provides another example flow chart of a method proposed for integrated sensing and communication (ISAC). That is, FIG. 11B provides an example architecture of integrated sensing and communication (ISAC) with a spread pilot pulse-tone. We first estimate the received spread pilot pulse-tone as \ / Ep         I] ^s,dd[^5 Z]-                        (78) using the estimate heS [£, / ] derived in (73). It follows from (77) that this estimate is accurate when the (weaker) crystallization condition is satisfied with respect to the lattice A?. The estimate becomes more accurate for larger M, N, that is Z^eff[k- Z] [k, Z] ~ heff[k, Z] *cr d^ddl^Z]- (79)

[00322] We then cancel this estimate from the received signal to obtain ydAd[k-.l] = VddlkJ] “                *a xs,dd[^,Z].                (BO)

[00323] When the (weaker) crystallization condition is satisfied with respect to the period lattice Ap, we can reliably recover data from the almost sensing pulse-tone free received signal va.dd[^ / ] in (80).

[00324] This approach can increase effective throughput by eliminating the need to divide physical resources between sensing and communication.

[00325] The flow chart of an example spread pilot pulse-tone based joint sensing and communication is depicted in FIG. 11B.

[00326] 5.5.B Numerical Simulations

[00327] As described in the present document, Zak-OTFS allows for ISAC with spread sensing pulse-tone where there is no sensing overhead and the transmit signal has low PAPR. In this section, we also see this is supported by numerical simulations of the BER performance of ISAC using point pulsetones and spread sensing pulse-tones.

[00328] FIG. 20A plots BER of uncoded 4-QAM as a function of increasing Doppler spread (2vmax) for the Veh-A channel described in Section 5.2 (see Table I and (19)). Point pulse-tones can be used for both channel sensing and data transmission.

[00329] In FIG. 20A, a plot for Uncoded 4-QAM BER vs vmax for ISAC with point sensing pulse-tone for Veh-A channel is provided. Data SNR and PDR are fixed atpd = 25 dB and -^- = 10 dB respectively, with vp = 30 KHz, M= 31, N= 37; fixed guard region Q (7 x 7 rectangle around (kp, lp) = ((M+ 1) / 2, (N+ 1 ) / 2)); RRC pulse shaping filter ( / ?T = pv = 0.6); and data detection with MMSE equalization.

[00330] Here, and throughout this Section, we consider channel sensing in the presence of interference from communication data (S|C) and in the absence of interference from data (S|C). Similarly, we consider data equalization in the presence of interference from sensing (C|S) and in the absence of interference from sensing (C|S).

[00331] S|C and C|S: This is the baseline (two bottom curves with the gray circle and triangle markers) where we dedicate separate Zak-OTFS subframes to channel sensing and to data transmission. Doppler domain aliasing increases with vmax, and BER increases as the accuracy of channel sensing degrades. We can reduce Doppler domain aliasing by substituting an RRC filter (gray triangles) for a sine filter (gray circles).

[00332] S|C and C|S : When we integrate channel sensing and data transmission in the same Zak-OTFS subframe (solid black curves), the point pulse-tone used for channel sensing interferes with the point pulse-tones used for data transmission. This interference severely compromises BER performance when the Doppler spread exceeds 2 KHz (ymax > 1 KHz) because the channel response to a point pulsetone extends beyond the guard band, regardless of the choice of filter. When the Doppler spread is less than 1 kHz, the choice of filter makes a significant difference. When we introduce a guard band, we divide DD domain resources between sensing and communication. By increasing the size of the guard band (sacrificing transmission rate) we can extend the range of reliable operation to higher Doppler spreads.

[00333] S|C and C|S : We integrate sensing and data transmission in the same Zak-OTFS subframe so the point pulse-tones used to transmit data interfere with the point pulse-tone used to sense the channel (S|C ). We ask what BER performance would be, if when recovering the data, there was no interference from the sensing pulse-tone (C|S). We estimate the channel within the guard band of the ISAC subframe (S | C ), then generate a data-only subframe carrying the same data as the ISAC subframe, then equalize the data using the channel estimate ( C|S ). We observed little difference in BER performance (compared to the solid black curves) and have not shown these curves in FIG. 20A.

[00334] S|C and C|S : We ask what BER performance would be if channel estimation were not subject to interference from data. FIG. 20A illustrates BER performance when we use a separate Zak-OTFS subframe for channel sensing (tighter dashed black curves with black triangles and black circles). The improvement in BER performance is significant when compared to the solid black curves. For the tighter dashed black curves with black triangles and black circles, we observed only a small improvement in BER performance when we replaced the estimated channel by perfect channel state information (CSI), and we have not shown these curves in FIG. 20A.

[00335] FIG. 20B plots BER of uncoded 4-QAM as a function of increasing pilot to data power ratio (PDR) for the Veh-A channel described in Section 5.2. We fix the data SNR pd = 25 dB, and we fix vmax = 815 Hz. By limiting the Doppler spread to 1.63 KHz, we limit the fraction of received pulse-tone energy that falls outside the guard band. Point pulse-tones are used for both channel sensing and data transmission, and we only consider RRC pulse-shaping filters.

[00336] In FIG. 20B, a plot for Uncoded 4-QAM bit error rate (BER) as a function of the ratio of the point sensing pulse-tone power to the total power of data pulse-tones (PDR) for ISAC with point Pd sensing pulse-tones is provided: for Veh-A channel; RRC pulse shaping filter = ^v = 0.6); Data SNR fixed at pd = 25 dB; Doppler period vp = 30 KHz; M= 31, N= 37; and the guard region is a 7 x 7 rectangle. Note the characteristic "U" shaped curves for integrated sensing and communication.

[00337] S|C and C|S: This is the baseline (gray curve with hollow triangles) where we dedicate separate Zak-OTFS subframes to channel estimation and data transmission. The accuracy of channel sensing increases with PDR, but DD domain aliasing in the data subframe limits BER, resulting in an error floor.

[00338] S|C and C|S : The point pulse-tone used for channel sensing interferes with the point pulse-tones used for data transmission. When PDR is small, the interference is small, and BER improves as the channel estimate becomes more accurate. When the power of the pilot pulse-tone exceeds the total power of all data pulse-tones (PDR > 0 dB), interference becomes more significant than noise, and BER degrades with increasing PDR. This explains the characteristic "U" shape of the black curve with solid black triangles.

[00339] S|C and C|S: There is no interference from the pilot pulse-tone when recovering the data, and BER is almost independent of PDR. This explains the flatness of the gray curve with solid gray triangles.

[00340] S|C and C|S: When there is a dedicated sensing subframe, the accuracy of the channel estimate improves with increasing PDR. When PDR > 0, interference from the pilot pulse-tone becomes more significant than noise, and BER degrades with increasing PDR. This explains why the black curve with hollow triangles has the same characteristic "U" shape as the black curve with solid black triangles. We observed only a small improvement in BER when we replaced the estimated channel by perfect CSI and we have not shown this curve in FIG. 20B.

[00341] The normalized mean squared error (NMSE) of the channel estimate can be given by I                    - r , 2 E ^eff[^ J] “ ^eff[& J] (kJKS '                         .                     (81) E Mk-if

[00342] FIG. 200 plots NMSE as a function of PDR for the Veh-A channel considered in FIG. 20A. We consider sine and RRC pulse shaping filters to understand the significance of the fraction of received sensing pulse-tone energy that falls outside the guard band.

[00343] In FIG. 20C, NMSE vs. PDR for ISAC with point sensing pulse-tone is provided, with NMSE of estimation of the taps of the effective channel filter Arfr[£, / ] as a function of increasing PDR for a fixed data SNR o\pj- 25 dB, RRC filter ( / 1.- / 1.- 0.6), vp = 30 KHz.

[00344] s|c : When the received pilot pulse-tone is confined within the guard band, NMSE decreases linearly with increasing PDR (gray triangles). Otherwise NMSE decreases to a floor that depends on the fraction of the energy of the received pulse-tone that lies outside the guard band (gray circles).

[00345] S|C : The limiting behavior is the same, but the floors are different (black triangles and black circles).

[00346] FIG. 20D plots NMSE as a function of increasing PDR for a spread sensing pulse-tone (q = 3) on the Veh-A channel considered in FIG. 20A. The Doppler spread 2vmax = 1.63 KHz is significantly less than the Doppler period vp = 30 KHz, but even with RRC pulse shaping filters, there is residual DD domain aliasing.

[00347] In FIG. 20D, NMSE vs. PDR for a spread sensing pulse-tone with q = 3 is provided, with Veh-A channel, RRC pulse shaping filter (pT = pv = 0.6), data SNR / ^ = 25 dB, vmax = 815 Hz, vp = 30 KHz, Af=31,Af=37.

[00348] S|C : This is the baseline (gray curve) where we dedicate separate Zak-OTFS subframes to channel estimation and data transmission. At high PDR, the pilot power to noise power ratio (PNR) is high. NMSE saturates at high PNR due to DD domain aliasing.

[00349] S|C : As PDR increases, the spread pilot becomes stronger than the combination of data interference and noise. DD domain aliasing again causes NMSE to saturate at high PDR.

[00350] FIG. 20E plots BER of uncoded 4-QAM as a function of increasing PDR. Note that with perfect CSI, the BER performance is independent of PDR (red curve).

[00351] In FIG. 20E, BER vs. PDR for a spread sensing pulse-tone with q = 3 is provided, with Veh-A channel, RRC pulse shaping filter =   = 0.6), data SNR = 25 dB, vmax = 815 Hz, vp = 30 KHz, M=31,N=37.

[00352] S|C and C|S : This is the baseline (gray curve with solid gray triangles) where we dedicate separate Zak-OTFS subframes to channel estimation and data transmission. BER performance tracks NMSE performance in FIG. 20D, improving with increasing PDR and saturating at high PDR. The gray curve with hollow triangles illustrates the case S|C & c|s where we estimate the channel in the presence of interference from data. The gray curve with hollow triangles and gray curve with solid gray triangles differ by a small PDR offset.

[00353] S|C and C|S : The spread pulse-tone used for channel sensing interferes with the point pulse-tones used for data transmission. When PDR < 10 dB, BER improves with increasing PDR. Beyond 10 dB, interference from the residual pilot pulse-tone (after cancellation) degrades BER with increasing PDR. This explains the characteristic "U" shape of the black curve with solid black triangles. When we dedicate a separate Zak-OTFS subframe to sensing (S C & C|S) we observe only a small improvement in BER, and we have not shown this curve in FIG. 20E. In this case, there is an optimal PDR which minimizes BER.

[00354] We use signal to interference ratio (SIR) to represent the ratio of the average power of the data pulse-tones to the average power of the residual sensing pulse-tones after cancellation. Figure 20F plots SIR as a function of increasing PDR for the Veh-A channel considered in FIGS. 20D and 20E.

[00355] In FIG. 20F, Signal to Interference ratio (SIR) of data power to the power of the residual spread sensing pulse-tone (after cancellation) as a function of increasing PDR (for fixed data SNR pd = 25 dB) is provided, with RRC pulse shaping filter (pT = pv = 0.6). vp = 30 KHz, Chirp filter with ^ = 3.

[00356] S|C and C|S: Sensing is performed on a separate Zak-OTFS subframe (black curve with hollow triangles). Estimation accuracy improves with increasing PDR, but the energy of the residual pilot pulse-tone increases proportionally. This is because, there are always some taps of the effective channel filter that are not estimated. We sense taps within a support set, and outside this set, taps are noisy because they are subject to DD domain aliasing. This explains why SIR decreases with increasing PDR.

[00357] S|C and C|S: Here the channel estimate is slightly inferior to the estimate obtained from a dedicated sensing subframe. Hence the SIR ratio is smaller. The effect diminishes as PDR increases because the effect of data interference on sensing diminishes.

[00358] FIG 20G plots BER as a function of increasing vmax.

[00359] FIG 20G: BER vs vmax for spread sensing pulse-tone (q = 3). Veh-A channel model, RRC pulse shaping filter = [1 = 0.6), Doppler period vp = 30 KHz, PDR = 10 dB, data SNRpd = 25 dB.

[00360] S|C and C|S : This is the baseline (gray curve with hollow gray triangles) where we dedicate separate subframes to sensing and data transmission. There is no residual pilot pulse-tone to interfere with the data pulse-tones. Channel estimation is accurate even when vmax is large. This is because the lattice A9 supporting the self-ambiguity function of the spread sensing pulse-tone satisfies the crystallization condition (63), even for large vmax. This explains why BER is excellent and almost independent of vmax. The gray curve with solid gray triangles (S|C & C|S) differs from the gray curve with hollow gray triangles by a small SNR offset. The black curve with solid gray triangles (C|S , perfect CSI) approaches the gray curve with hollow gray triangles in the limit as vmax approaches 14 KHz.

[00361] S|C and C|S : Here the spread pulse-tone used for channel sensing interferes with the point pulse-tones used for data transmission. Nevertheless, the variation in BER is small over a wide range of Doppler spreads. This is very different from BER performance for point sensing pulse-tones illustrated in FIG. 20A (BER = 4 x 10’3 at vmax = 300 Hz and BER = 2 / 10~2 at vmax = 14 KHz in FIG. 20G versus BER = 10“3 at vmax = 300 Hz and BER = 0.4 at vmax = 14 KHz in FIG. 20A). When we dedicate a separate Zak-OTFS subframe to sensing (S|c and C|S), we observe a modest improvement in BER (black curve with hollow black triangles versus black curve with solid black triangles). We conclude that spread sensing pulse-tones improve upon point sensing pulse-tones by extending the range of reliable operation to a wider range of Doppler spreads.

[00362] FIG. 20H illustrates the importance of filter design by comparing BER performance of chirp filters with q = 3 and q = 36. Accurate sensing requires crystallization with respect to the lattice Aq, and reliable data detection requires crystallization with respect to the period lattice A^. The crystallization condition is satisfied for the period lattice Ap since vmax < 15 KHz and the delay spread for the Veh-A channel is 2.5 / / 5, which is less than tp = 1¼ = 33.33 / / 5. When q = 3, crystallization conditions w.r.t. both Ap and A9 are satisfied and BER performance is almost constant for a wide range of Doppler shifts (black curve with solid black triangles). When q = 36, the crystallization condition with respect to Aq is not satisfied for Vmax > 2 KHz and BER degrades sharply (black curve with hollow triangles).

[00363] In FIG. 20H, BER vs vmax for spread sensing pulse-tones defined by chirp filters with q = 3 and q = 36 is provided, with Veh-A channel described in FIG. 20G, vp = 30 KHz, fixed PDR -^ = 10 dB, data SNRpj = 25 dB.

[00364] Effective Throughput: This is defined to be the ratio of the number of bits reliably communicated in each subframe to the available degrees of freedom. In various embodiments, only channel sensing and data transmission in the same Zak-OTFS subframe (S|C & C|S) is considered. When we use a point pulse-tone for channel sensing we need to introduce a guard band for various embodiments, and this division of DD domain resources between sensing and communication decreases effective throughput. When we use a spread pulse-tone for channel sensing we can avoid this overhead.

[00365] Assuming RRC pulse shaping filter with roll-off factors pT and pv, the number of degrees of freedom is BT(l +PP(1 + PvY Setting BER = R, the number of reliably communicated bits in each subframe is 2MN([ - H(R)) when using a spread sensing pulse-tone, compared with 2(MN- 49)(1 - H(R)) when using a point sensing pulse-tone (assuming a 7 x 7 guard band) and 2(MN- 7A)(1 -H(R)) (assuming the guard region to be a strip along the Doppler domain having width 7 along the discrete delay axis). Here H denotes the binary entropy function and we note that for small R we have H(R) ® R / In 2.

[00366] FIG. 201 plots effective throughput as a function of increasing vmax. Recall from FIG. 20H that for a spread sensing pulse-tone defined by a chirp filter with ^ = 3, BER performance is almost constant for a wide range of Doppler spreads. This is reflected in FIG. 201. Recall from FIG. 20H that when q = 36, the crystallization condition with respect to is not satisfied for vmax > 2 KHz, and the BER degrades sharply. This explains the difference in effective throughput for spread sensing pulse-tones defined by chirp filters with q = 3 and q = 36.

[00367] In FIG. 201, effective throughput (bits / sec / Hz) as a function of increasing vmax, is provided for integrated sensing and communication (S|C &C|S)with vp = 30 KHz, M= 31, N= 37, RRC pulse shaping filter ([3T = pv = 0.6), Veh-A channel as described in FIG. 20G.

[00368] Recall from FIG. 20A that for vmax > 1 KHz, the channel response to a point sensing pulsetone extends beyond the guard band regardless of the choice of pulse shaping filter. Interference between the point pulse-tone used for sensing and the point pulse-tones used for data transmission can severely compromise BER performance. This explains the characteristics of the effective throughput curve for a point sensing pulse-tone (dashed curve with solid triangles).

[00369] FIG. 201 illustrates the gain in effective throughput that results from sharing DD domain resources between sensing and communication.

[00370] Summary for Section 5. Zak-OTFS and ISAC

[00371] We started from a parametric family of pulse-tone waveforms that can be matched to the delay and Doppler spreads of different propagation environments. We reviewed how the (point) pulse-tone signal in the time domain realizes a quasi-periodic localized function on the delay-Doppler (DD) domain. The characteristic structure of a pulse-tone is a train of pulses modulated by a tone, a signal with unattractive peak-to-average power ratio (PAPR). We reviewed system performance in the crystalline regime where the delay period of the pulse-tone is greater than the delay spread of the channel, and the Doppler period of the pulse-tone is greaterthan the Doppler spread of the channel. When channel sensing and data transmission take place in separate subframes, the point pulse-tone used to sense the channel does not interfere with the point pulse-tones used to transmit data. We have reviewed how the I / O relation of the sampled communication system can be read off from the response to the point pulse-tone used for channel sensing. In the present document, we have described filtering in the discrete delay-Dopplerdomain. We have described and demonstrated that it is possible to construct spread waveforms with desirable characteristics by applying a chirp filter in the discrete DD domain to a point pulse-tone. One desirable characteristic is low PAPR, about 5 dB for the exemplar spread pulse-tone, compared with about 15 dB for the point pulse-tone. A second desirable characteristic is the ability to read off the I / O relation of the sampled communication system provided a second crystallization condition is satisfied. We have described and demonstrated how to integrate sensing and communication within a single Zak-OTFS subframe through the combination of a spread pulse-tone used for channel sensing and point pulse-tones for data transmission. The filter in the discrete DD domain enables coexistence by minimizing interference between sensing and data transmission. We have described and demonstrated that sharing DD domain resources in this way increases effective throughput compared with traditional approaches that use guard bands to divide DD domain resources between sensing and communication.

[00372] 5.6 Zak-OTFS and ISAC Additional Information

[00373] Zak Transform and Quasi-periodicity

[00374] The definition of a DD domain pulse depends on a delay period rp, and a Doppler period vp, where the two periods are reciprocal, that is vp tp = 1. The Zak transform, denoted Zt, provides a unitary equivalence between TD signals and a subclass of quasi-periodic DD domain signals. The Zak transform of a TD signal x(t) is given by X(T + kTp) k&

[00375] The quasi-periodicity condition is that, for any n,m e TL + nTp, v + mvp) =                           (83) We observe thatxddO", v) is periodic along the Doppler axis with period vp, and that it is quasi-periodic along the delay axis with period tp.

[00376] The inverse Zak transform, denoted by Z~l, of a quasi-periodic DD domain signal XddCr v) is given by = Z^^Xd^T,^ = J Xdd( / ,(84) 0

[00377] TD pulse-tones: Carrier Waveforms for Zak-OTFS

[00378] The following theorem describes the Zak-OTFS carrier waveforms (pulse-tones).

[00379] Theorem 5: The transmit TD signal s^(t) is given by MO = z<r[ ^^(t.p)^ M —1 .V-l = (85) A;=0 / =0 where (0is the carrier waveform associated to the (A, / )-th information symbol x[k, Z], given by

[00380] The FD representation (i.e., Fourier transform) of the TD carrier waveform sx^i (f) is given by Huu( / ) = -‘HauCO dt ^^wtx(r, p) er^fr XkjXf - v) dr dp           (87) where XkAf) A I e-W dt is the Fourier transform ofxA. / (t). The DD domain representation of the (k, / )-th carrier waveform is •\idj.u(" / / ) wtx(r,p) *CT ;rdduu(^zy)            (89) where •''dd. / .j(~? / ) =52 r''“ ' (T ~ nTp ~               mi'p - \                        J X                     * J x 7 x 7 (90) / k lvp \ is a quasi-periodic Dirac-delta impulse at I ^~T-~ I ■ (see Proof after the following).

[00381] Remarks: From (89) and (90) it follows that the DD representation of the Zak-OTFS carrier waveform is given by (91) which is a quasi-periodic pulse in the DD domain. The representative pulse within the fundamental period 'Dq , is located at {kTpIM, lvp!N) (see FIG. 17 that represents a Zak-OTFS carrier waveform).

[00382] We now make fundamental observations about the TD and FD realization of the Zak-OTFS carrier waveform.

[00383] Carrier waveforms having unlimited time and bandwidth: For v) = 3(r) 3(v), it follows from (86) that ^kd(t) =                          (92)

[00384] For (k, 1) = (0, 0), the TD carrier waveform is therefore = x^p 52   ~nT^             (93) which is simply an infinite train of Dirac-delta impulses where the w-th impulse is located at t=mp. The (k, / )-th carrier waveform staxi (0 = Xu (t) is nothing but ^.0.0(0 with a Doppler shift of lvp / N and a delay shift of kip / M, since it can be checked that = ®o,o(t-krp / M)                   w , ,     .                       Doppler shill delay shin                     *1

[00385] Note that the (k, / )-th carrier waveform is the TD realization of a quasi-periodic DD domain / k Tp tv? \ impulse at I “O-,” I (see RHS of (90)). The (k, Z)-th carrier waveform, which we referto as a TD pulse tone, is simply a train of impulses where the w-th impulse is located at t=mp + kzp'M, modulated by a / vn             , / 2^ ,( (t-kr M) complex tone / sinusoid e                 (see (86)).

[00386] It is clear from (88) that the FD representation of the (k, / )-th TD pulse-tone is an FD impulse -jlTrfkr IM train (with m-th impulse at / = mvp + lvp!N} modulated by a FD tone &            . We refer to this waveform as a FD pulse-tone. When w^t. v) = 3(t) 3(v), it follows from (86) and (88) that the corresponding carrier waveforms have infinite time duration and bandwidth.

[00387] Carrier waveforms with limited time and bandwidth: The carrier waveforms can be limited in time duration and bandwidth by choosing an appropriate wtx(r, v). If the delay spread of wtx(r, v) is approximately MB, and the Doppler spread is approximately MT, then the TD pulse-tones have time duration T and bandwidth B. It follows from (86) that s^i (0 is simply x*,; (t) spread by MB in the time domain. Given the structure ofxv (t), it is clear that the TD pulse-tone s^i ( / ) consists of a train of narrow pulses modulated by a tone, with each narrow pulse having spread MB and adjacent pulses separated by tp seconds. Similarly, from the integral expression in (87) it follows that the FD representation of the (k, 1)-th carrier waveform, i.e., sflW ( / ), is simplyXj ( / ) spread by MT in the frequency domain, since the Doppler domain spread of v) is 1 / 7’. Given the structure ofXu ( / ) in (88), it follows from the integral expression in (87) that the FD representation of the carrier waveform, i.e., sfd;fcZ ( / ) consists of a train of narrow FD pulses modulated by a FD tone, with each narrow pulse having spread MT and adjacent pulses separated by vp Hz. Therefore, 5^, / (0 also has a pulse-tone structure in the FD, and is referred to as a FD pulsetone. FIG. 17 illustrates the TD and FD representations of the Zak-OTFS carrier waveform.

[00388] Time and Band-limited pulse-tones

[00389] For factorizable pulse shaping waveform wtx(r, v) = wi(t) w2(v), it follows from (86) that the (k, Z)-th carrier waveform is given by yy Wi(r) w2^) Xkj\t - t) ! dr du xkjdt - 7-) H L>i / - r) dr (107) where wyt) ~                 du              (1°8) is the inverse Fourier transform of the factor vr2(v) along the Doppler domain. Let ^2^.,1^  = XkJi(t)W2(t)-             (109)

[00390] Then, X2,k.i (t) has time duration limited by the duration of W2(t). Using (109) in (107) we obtain = W - t) dr =          * X2.k,l(t)                     (110) where * denotes the usual TD convolution. Hence, the Fourier transform of 5td,tz ( / ) is Sfd,kj( / ) = ^(std.k.iZZ)) = [                                                   (111) where Wi(f) and TG,*, / ( / ) are the Fourier transforms ofwi(f) and xw ( / ), respectively, given by W) =  [ w}xt)e~:i27rftdt. X2.kj(f) = I e~^*              (’12)

[00391] It is clear from (111) that the carrier waveform will be bandwidth limited to the bandwidth of wi(-). For example, choosing = y / Hsinc^Bd) limits the bandwidth of the carrier waveforms to exactly B Hz. In general, for a given bandwidth constraint B, wi(r) must have a spread of approximately 1 / 5 along the delay domain. Similarly, from (109) it follows that the duration of xzja ( / ) can be limited to approximately T seconds by choosing the factor pulse w2(y) to have a spread of approximately 1 / 7 along the Doppler domain (for example, with w2 (v) = sinc(vT), %2M ( / ) is limited exactly to the TD interval [—772. 772]). Therefore, X2,k,i(t) = Xk,i ( / ) W2Q) contains only T / tp - N number of Dirac-delta impulses of the infinite impulse train Xk,i (0- Also, due to the TD convolution in (110), in the carrier waveform A'tdx / ( / ) these N impulses are spread over a duration of approximately MB seconds since the spread of wi(t) is MB. Therefore, under a finite duration and bandwidth constraint, the (k, / ) carrier waveform consists of narrow pulses at t = mp + krpIM where the width of each pulse is MB = tp / M (since M= Btp). Note that tp / M is also the time between the location of the w-th pulses of the (k, Z)-th and the (k+ 1, / )-th carrier waveforms.

[00392] The Discrete Ambiguity Function

[00393] The discrete cross-ambiguity function Aa.b\k, / ] between two discrete quasi-periodic DD domain functions a[k, / ] and b[k, / ] can be given by M-l N-l k'^0 I'—O

[00394] The sum in the R.H.S. may be taken over any discrete delay period A / and discrete Doppler period N.

[00395] Theorem 6: Let a[k, / ] and b[k. / ] be quasi-periodic DD functions related by a[k, I] = p[Ar? Z]                                 (114)

[00396] Then, the cross ambiguity between a[k, / ] and another quasi-periodic DD function c[k, / ] is given by the twisted convolution between g[k, / ] and the cross ambiguity between b[k, / ] and c[k, / ], i.e. A / - i 2V-1 k'=0 (•'—0 ~            *(7 xA / j.c[k; • (115)

[00397] Lemma 2: The discrete cross-ambiguity between any two quasi-periodic functions a[k. I] and b[k, / ] is invariant w.r.t. the period over which the sum in the R.H.S. of (113) is computed, i.e., for any ko -i- M —11 o + N --1                                          z Au, Vo fl = 52 52 h* [fc / “ k-l> ~~ Z1 k’^ka ik : = 52 52^’^*^ fc^O t’-Q (116)

[00398] Proof: k mod AI — k A~ AI kf AI M . I'mod N 4 (117)

[00399] Since both a[k, / ] and b[k, / ] are quasi-periodic DD functions, it follows from (2) that a k' mod M + Af l! mod A' + — a [kf mod M , d mod ;V] e (118) and b k>modAl-k + Ad MA'modN-lk = b [k1 mod AI — k, 1’ mod N — / ] e1 (119)

[00400] Substituting (118) and (119) in (117) gives k' — I a i kf mod M , l! mod A"] '=lo X bv [Vrnod M — k, fmod N — (120)

[00401] The R.H.S. above depends on k’ and 1' only through k' mod M and 1' mod N. Since the summation variables k' and / ' span a full delay and Doppler period respectively, it follows that as k' varies from kn to k0 + M- 1, k'modMtakes all possible values from 0 toM- 1. Similarly, as / 'varies from Zo to Io+N- 1, / 'mod / V takes all possible values from 0 to N- 1. Therefore, replacing summation variable k' and / 'with k = k'modAf and I = / 'mod N respectively, we get (121) A—0 1=0 which is same as the expression in (113).

[00402] Lemma 3: For any two quasi-periodic discrete DD domain functions a[k, / ] and b[k,l] Ab,— A^jX—k, —1] MN. (122)

[00403] Proof: From the definition of Aa h \k, / ] in (113) it follows that -*]=52 52 A z'i w + k’l'+fl k+M-l l + N-1                         „ = (-:1^7^22 52 k=k    1 = 1 =Ab^[kJ] =             Ab,a [Ad I]                                               (123) where in the second step the summation variables k',l', are changed to k = (kr + k} and I = ( / ' + / ) respectively.

[00404] Self-ambiguity Function of the Point Pilot Signal

[00405] It follows from the definition ofthe ambiguity function (113) that the self-ambiguity of the point pilot signal is given by (124), (124)

[00406] Here, step (a) results from substituting forxp>dd[^, / ] using (20). For a non-zero sum in step (a), from the location ofthe Dirac-delta impulses in the R.H.S., it follows that k' = kp modM anti Qd - k) = kp modM. This is only possible if k is an integer multiple of M. Similarly, I’ = lp modNanti {1' - I) = lp mod N. This is only possible if I is an integer multiple of N, in other words the self-ambiguity function Ax x [k, / ] is non-zero for only the DD points (nM,mN),n,m G Z, which is simply the period lattice. Again, from step (a) it is also clear that, for (k, I) = (nM, mN), the only non-zero contribution in the sum in the R.H.S. is from the term corresponding to «i = (kp modM- kp) / M, n2 = ni- n, mi = (lp modN- lp) / Nand m2 = mi~m. The value of Ax x InM,mN\ is simply €      &         6 J 1 evaluated for the values of «i, «2,mentioned above. This leads to the final expression of Ax x [£, / ] in step (b).

[00407]

[00408] Symmetry of Cross-ambiguity First MN (125)

[00409] Using this expression for a[k, / ] in the R.H.S. ofthe first step in (115) we get k.MM .o_ lik'-k) e    MN (126)

[00410] We change variable k to k" = k' — k , variable I to 7" = 1' — I , and apply Lemma 2 to obtain (127)

[00411] Substituting (127) into (126) we obtain Aa^k, I] = 52 g [k, I] A^k -Ml-I] kkkM — g[kA] >VAC[&,Z]. (128)

[00412] Proof of Theorem 3

[00413] The expression ofthe received signalydd[£, / ] in (69) consists of the sum of the received data signal, pilot signal and noise. The cross-ambiguity ofthe received signal with the spread pilot signal xs,ddfcZ] is therefore the sum of, (i) the cross-ambiguity of the received spread pilot signal I] O xs,ddlk> Z] with xs,dd[£, / |. Applying Theorem 6 (with g[k,l] = hca[k,F\, a[k,l] = 4Epheff\k, Z] *o- xs.dd[M], b[k,l] = c\k.l\ =xs,dd[£, / ]) gives the first term in the R.H.S. of (70), (ii) the crossambiguity of the received data / information signal ylE^heff[k, Z] xddd[k, Z] with xs>dd[£, / ]- Applying Theorem 6 (with g[k,l\ = heii[kj\, a[k,l\ = y[E^heff[k, Z] *a xddd[k, Z], b\kj\ = xd.dd[£,Z], c\k,l\ = xs,dd[^ / ]) gives the second term in the R.H.S. of (70), and (iii) the cross-ambiguity between the noise signal na[k,I\ and xS;dd[£, / ], gives the third term in the R.H.S. of (70).

[00414] Derivation of the Maximum Likelihood (ML) Estimate of ^effI^-ZI

[00415] The maximum likelihood (ML) estimate of / reff[£,Z] can be given by M-l ZV-l h^zl\ = arg min V             - h[kJ} ®s.dd[A:, l]f L k—0 1=0 (129) where j\dd[£, Z] is the received pilot signal given by (51). Essentially, the ML estimate is such that it minimizes the energy of the estimation error signal enty, / ] = (ys,dd[£, / ] - h[k, / ] *<7XS dd[£, / ]). Since both Ts,dd[^, Z] and x.jd|r. Z] are quasi-periodic and twisted convolution conserves quasi-periodicity, it follows that the error signal is also quasi-periodic. The expression in (129) then follows from the fact that the energy of M-\ N-\         2 any discrete quasi-periodic DD domain signal e^k, Z] is given by LL ^dd ^]| ■ k=0 1=0

[00416] The expression for h[k, Z] *axSjdd[^ Z] follows from the definition of twisted convolution and is given by (130) (130) Further analysis of (129) gives (131) M- IjV—1            0

[00417] The first term ^|j / sdd [^]| does not depend on h[k, / ] and is therefore irrelevant £=0 / =0 to the minimization. Next, in (132) it is shown that in the crystalline regime, the term A / -1W-1 £=0 / =0 xs dd [A, 11 is equal to the total energy of the taps of h\k. l\. Using this in (131) gives (133) (133)

[00418] Proof of Theorem 2

[00419] We prove (55) fora spread pilot obtained by filtering a point pilot located at (kp, Ip). For simplicity, we consider (kp. lp) = (0, 0), but the result is the same for any choice of point pilot. From (49), the spread pilot corresponding to (kp, lp) = (0, 0) is given by n=0 m=0 (134)

[00420] From (113), the self-ambiguity function of the spread pilot is given by M-iA—1 I'— Q (135)

[00421] By Lemma 2, we can rewrite (135) by extending the summation over k’ to k' = 0,1, ■ ■ ■ , MN - 1 and / ’to / ’ = 0,1, ■ ■ ■,MN-), i.e. (136)

[00422] Substituting forx;,dd|F. / ’] using (134) gives I] k!=0 I'=0 n i=0 n 2=0 m i = 0 m 2=0 w[k' — niM, 1' — miN] ,                                   ,                                                              ,r, w* [k — k — mM, I — I — m^N] N e J~'‘ N (137)

[00423] Substituting for the chirp filter using (50) gives (138) = i when (53).(148) hold. zero otherwise =i when ; 140), (Ml) hold, zero otherwise (138)

[00424] In (138), the inner summation over k’ vanishes unless (2qk - / ) = 0 (modM) so that • L ..r, [k J] — 0, if (2qk — I) 0 (mod M)          (13

[00425] IfA / divides (2qk - / ), then for every Wi = 0, 1, ■ ■ ■ , N- 1, there is a unique n2 e {0, 1, ■ ■ ■ ,N - 1} such that (2qk — I) + 2q{n2 — rii)M = 0 (mod MN) that is given by «2 = m + (2q) —— L                       - J A                      (141) The inner summation vanishes unless (k, / ) satisfy 2qk — I ™ 0 (mod M) 1                          -             7                                             (142) and (ni, n2) satisfy (141). When these conditions are satisfied, the inner summation equals 1.

[00426] We need only evaluate the inner summation in (138) over I' , for (k, / ) satisfying (142) and {ni, n2) satisfying (141). This summation vanishes unless 2q (I + (m2 — mi )AT) + (ni — nz)M — 0 (mod A / Ar) (143) It follows from (141) that M(n2~~nx) =          (modMN) (144) Setting 0 = |(2c / )' - 2q] mn, it follows from (143) and (144) that 2q(m2 —        = 01 — k (modulo MAr)           (145) so that k 01 = 0 (modulo A7)                               (146)

[00427] Note that (142) and (146) are the two arithmetic conditions appearing in Theorem 2 as (55). Thus the inner summation over I' vanishes unless {k. t) satisfy (55). In addition, it follows from (144) that ,    _ i I — 2qk 712 — H] = (2g)   --—-- (mod A ) dl '            ,                         (147) and it follows from (145) that 2q{m.2 — mi) — ------ modulo M. A                                     (148)

[00428] These conditions appear in (138). We assume (k, I) satisfy (55), and rewrite (138) as the product of two terms in (149). We simplify the sum over m by substituting for njvl using (144) to obtain (150). We simplify the sum over mi by substituting for 2qm2N using (145) to obtain (151). Combining (150) and (151) in (149) we obtain Theorem 2. m; =0, m2 satisiies( 148) =i since (l ~2<jfe)=o (modukw >. sees 55) (151)

[00429] Average Energy of DP Noise Samples

[00430] We follow the noise through the Zak-OTFS signal processing architecture described in Section 5. At the receiver, the Zak transform Ztconverts AWGN ntd(t) with PSD No to its DD representation nad(x, v) = Zt (fitd(t)). Pulse shaping with wrx(r,v) results in favj = v)*a ndd(t, v) which is then sampled on the information grid Add to give the discrete DD domain quasi-periodic noise signal ndd[k, / ].

[00431] Here, we show that the expected energy of each noise sample ndd[k. / ] satisfies E[|ndd[fc, I] |2] — No. We consider receive pulse shaping filters v) that factor as w„(T> v) = w;(t) w2(v) where w / (t), w2(v) are unit energy pulses along the delay and Doppler axes respectively. We assume wi(f), w2(y) satisfy the Nyquist no-ISI criterion for information spacing 1 / B and 1 / T along the delay and Doppler domain respectively ( / 11^(7)12 dr = J|w2(v)|2 dv = 1).

[00432] The DD representation of AWGN ntd(t) is given by

[00433] (152) The pulse shape filtered noise is given by ' dr dz4153)

[00434] The noise signal ndd [k, / ] is therefore given by where HU) = Iw^e^dv J                                          (155)

[00435] is the inverse Fourier transform of W20. Step (a) follows from (153) after using (152) to substitute forMdd(T,v). Step (b) follows from the identity T = Ntp after changing the integration variable from t ' to t .

[00436] It follows from (154) that tidd[k, / ] is zero mean and that E[|ndd[^< ^]|2] 's given by (156). (156)

[00437] Step (a) in (156) follows from the fact that the autocorrelation E[ntd(t) ntd - t)] = N06(t). Next, we observe that for large M, the spread of w;(t) along the delay axis is significantly less than tp (since M = Brp, the spread MB = tp / M « tp). Hence forni / n2, the delay domain supports of w;(t7) and + (n2 - n^Tp) do not overlap. Step (b) in (156) now follows from the fact that we can ignore all contributions to the summation for which ni 112.

[00438] The spread of the pulse W2(v) along the Doppler domain is roughly MT. Therefore the spread of its inverse Fourier transform HU) is roughly T = Ntp. Since w2(y) satisfies the Nyquist no-ISI criterion along the Doppler domain with symbol spacing 1 IT, it follows that its inverse Fourier transform (i.e., W2W) must satisfy nGX                                         (157)

[00439] for all t. The sine and RRC pulses used in this paper are examples of Nyquist pulses. Using (157), we show that for all t (158)

[00440] Step (a) follows from the identity T=Ntp. Step (b) results from replacing the summation index n by (mN + q) where q = n (mod N). Combining (156) and (158) we obtain = see (158) (3) ,r A (b) = iNQTp—r — Ao. *                                                                         (159)

[00441] Step (a) follows from the fact that w7(t) has unit energy, and step (b) follows the identity T = Ntp.

[00442] Proof of Theorem 4

[00443] In this Appendix we calculate the expected interference from data transmission to sensing. We start by combining (66) and (71) to give (160). The information symbols x[k, Z] have unit energy and are statistically independent. Hence the mean squared interference energy is given by (161). Through (162), we simplify the inner summation in (161) using Lemma 2 (the self-ambiguity function i4Xs Xs[ / c, I] does not depend on the period over which the summation is carried out). We now obtain (163) by substituting (162) in (161). Step (a) in (163) follows from Lemma 1 when the (weak) crystallization condition is satisfied. (160) 6. Examples of filters in the discrete DD domain and coexistence of sensing and communication in the same Zak-0TFS subframe

[00444] In this section, examples of symplectic transform operations are provided. In various embodiments, these example transform operations can be used for filters in the discrete delay-Doppler domain.

[00445] Radar engineers construct waveform libraries for tracking applications by manipulating self-ambiguity functions.

[00446] Taking a waveform $ , displacing it in delay and Doppler, and taking the inner product with the original waveform gives the self-ambiguity function in various embodiments. When the absolute value of the self-ambiguity function is high at some location, then targets at this location can be obscured. Radar engineers have learned that chirping the waveform (|) produces a new waveform 4>q and a new self-ambiguity function can be obtained by rotating the original self-ambiguity function. For example, chirping transforms the self-ambiguity surface, making it possible to track a greater diversity of radar targets.

[00447] The volume under the self-ambiguity function is fixed by Moyal’s Identity (as the square of the signal energy) but it is possible to make more targets visible by rotating the self-ambiguity function.

[00448] Chirping or Linear Frequency Modulation (LFM) is a particular example of a symplectic transformation. Example symplectic transforms in radar from Howard et al. are provided below. Displacement operator: D(t,v) (p(t) = e~VT eivt(p(t + t) Self-ambiguity function:           = (0, D(t, v), 0) Symplectic transform: For example, Linear Frequency Modulation (LFM) Chirp with slope g: U(q\. <p(t) -> (pqQt) = Symplectic transforms rotate self-ambiguity functions A<j>q(j,v) = {(pq,D(T,v),<pq) = {(p,U(qY D(t,v) U(q),(p} D |v Q| - 4with sq lzq ?]

[00449] Examples of discrete delay-Doppler domain filters in communications are described and provided below.

[00450] Communication in the discrete DD domain is revisited - the counterpart of LFM modulation is an MN-periodic discrete chirp filter [i.e., M*N-periodic discrete chirp filter],

[00451] The dimensions of the delay-Doppler grid can be chosen according to the channel characteristics and the available bandwidth and frame duration. The parameters M, N are directly related to the delay-Doppler grid periods (tau_p, nu_p) [e.g., (tp , vp)] and can be selected such that the OTFS is in its crystalline regime.

[00452] This filter can act on a point pulse-tone by twisted convolution to produce a spread pulsetone Xs.

[00453] The spread pulse-tone Xs is quasi-periodic, hence periodic with period MN along the delay and Doppler axes.

[00454] In the LTI world, when a linear filter is applied to a signal with period L, the result can be given by periodic linear convolution of the periodic signal with the periodic extension of the filter. According to the notation used in the present document, w_s[k,l] ( / .©., ws[k, / ]) is a discrete filter defined over the delay-Doppler grid and w[k,l] is its MN-periodic [i.e., M*N-periodic] extension of it. Note, that there are two periodicities here. The discrete filter w_s[k, I] is quasi-periodic in the delay-Doppler with respect to the dimensions M, N. On top of that, w[k,l] is an MN-periodic extension of w_s[k,l] that satisfies: w[k,l] = sum(w_s[k + n*MN, I + m*MN) for all integers n, m.

[00455] In the LTV world, there is a twist! The subscript s indicates the spreading operation.

[00456] In the LTI world there are L complex degrees of freedom and in the LTV world there are M2N2 complex degrees of freedom.

[00457] In various embodiments, a MN-Periodic Discrete Chirp Filter with slope q (coprime to MN) can be represented as [as also discussed in the previous section] . 1 ^(.k2+l2) w[Zc, / ] =---ej27T mn ’fork,lE% MN

[00458] The spread pilot (e.g., spread pulse-tone) can be represented as: xsdd[k,l] = ws[ / c,Z] *axPidd[k,l], where xsdd is quasi-periodic, xsdd is periodic with period MN along delay and Doppler axis.

[00459] MM-Periodic twisted convolution of xp dd [k, Z] with w[k, Z](the MN-periodic extension of the discrete filter) can be represented as: WI . I1 a| sC, / 1

[00460] A theorem provides: ws[k,l] ^ffXpdd[k,l] = w[k,l] % xp4d[k,l]

[00461] For degrees of freedom, a discrete DD domain filter can be specified by a vector in £M2N2

[00462] FIG. 18 graphically depicts examples where chirp filters rotate the period lattice.

[00463] Three lattices play an important role - the first is the information lattice (e.g., Ndd) where the data lives, the second is the period lattice Ap, and the third is the dual of the information lattice (e.g., Ndd). The information lattice can be thought of as ZxZ, the period lattice as MZxNZ, and the dual of the information lattice as MNZxMNZ.

[00464] The self-ambiguity function of a point pulse-tone (e.g., point pulse-tone xP) is supported on the period lattice Ap. For example, Ax x [k,r\ is supported on the period lattice Ap.

[00465] The self-ambiguity function of a spread pulse-tone (e.g., spread pulse-tone xs) 4XsAs[k, / ] is supported on a rotated lattice Aq.

[00466] A theorem (e.g., Theorem 2 from section 5.4.B) provides that the ambiguity function A^JM] is supported on a rotated lattice Aq given by (1 - 2q0)A(; = Ap where 0 = (2q)-1 - 2q(mod MN) .

[00467] FIG. 20J are graphs also showing examples of spreading operations in time domain (TD) and delay-Doppler (DD) domain (e.g., spreading in TD and DD domain).

[00468] The effect of spreading in the DD domain is to distribute energy equally across all point pulse-tones, resulting in a unit energy signal with energy about 1 / MN at each location on the information grid. For example, Energy |xs,dd[ / c, Z]|2

[00469] The effect of spreading in the TD is to produce a noise-like waveform that is much less peaky than a point pulse-tone, where the peaks in the spread TD waveform can be an order of magnitude smaller than the peaks in the point TD waveform.

[00470] The examples of FIG. 20J are based on chirp filter with slope q = 3.

[00471] FIG. 20K is a graph showing peak to average power ratio (PAPR) result examples. Here PAPR is compared. Complementary CDF (CCDF) [complementary cumulative distribution function] of the Instantaneous to Average Power Ratio (IAPR) with RRC transmit filter with roll-off factors - 0.6) is provided for different cases. Here, RRC stands for root-raised cosine filter. The chirp can be applied as a discrete filter on the delay-Doppler symbols. After that, a non-discrete 2-D filter (RRC) can be applied in delay-Doppler, to shape the OTFS waveform in time and frequency. In the result examples of FIG. 20K, it can be seen that performance of Spread pulse-tone is about 5 dB, point pulse-tone is about 15 dB, Data + spread pulse-tone is about 8 dB, and Data + point pulse-tone is about 12 dB. That is, a summary of the IAPR is as follows: Spread pulse-tone ~ 5 dB vs. Point pulse-tone ~ 15 dB, and Data + spread pulse-tone ~ 8 dB vs. Data + point pulse-tone ~ 12 dB.

[00472] FIG. 21 is a graphical example of a sensing waveform. Here the geometry of sensing with spread pulse-tones is described. The q value can affect the grid rotation directly. A q value needs to be selected that will not result in a rotation that causes aliasing and takes it out of the crystalline regime. In this example, M=11, N=13, q=5 with pilot location: (kp,lp) = (0,0).

[00473] The dark gray dots mark the rotated lattice Aq - this is the support of the self-ambiguity function of the spread pulse-tone.

[00474] The gray rectangle with the black border, denoted by Q, marks the support of the effective discrete DD domain channel (e.g., heff[k, I])-

[00475] Observe that the translates of Q by lattice points in Aq are disjoint (e.g., translates of Q by Aq do not overlap), which means that there is no DD domain aliasing. Hence, it is possible to accurately estimate the channel on the information grid (the lattice Add or Ap) from the response received within Q; here, the crystallization condition holds with respect to the rotated lattice Aq.

[00476] FIG. 22 graphically depicts examples of effect of spreading. The depicted example heat maps of |heff[fc, / ]| are represented for Veh-A channel with RRC pulse shaping filter ( / ?T =   = 0.6), Doppler period vp = 30 KHz, M = 31, N = 37. Here, the spread pulse-tone (e.g., Pulsone) is defined by the chirp filter with q=3. The bottom heat maps are example heat maps of the received spread pulse-tone defined by the chirp filter with q=3.

[00477] In these examples, Xs is sent and ys is received and the cross-ambiguity between ys and Xs is used to visualize the effective channel.

[00478] Good geometry means that the translates of the support set do not overlap and the crystallization condition is satisfied with respect to the rotated lattice Aq. For example, good geometry is indicated in the examples of FIG. 22 with a chirp filter with q=3.

[00479] FIG. 23 graphically depicts examples of effect of spreading. The depicted example heat maps of | / ieff[ / c, Z]| are represented for Veh-A channel with RRC pulse shaping filter = pv = 0.6), Doppler period vp = 30 KHz, M = 31, N = 37.Here, the spread pulse-tone (e.g., Pulsone) is defined by the chirp filter with q=36. The bottom heat maps are example heat maps of the received spread pulsetone defined by the chirp filter with q=36.

[00480] In an example of FIG. 23, when vmax- 815 Hz, the geometry is good - the translates of the support set do not overlap and the crystallization condition is satisfied with respect to the rotated lattice Aq.

[00481] In the example of FIG. 23, when vmax = 12 KHz, the geometry is bad - the translates of the support set overlap, the crystallization condition is not satisfied with respect to the rotated lattice Aq, and DD domain aliasing compromises accuracy of channel sensing.

[00482] FIG. 24 shows an example relationship among filters in discrete DD domain. The diagram captures the different components in Zak-OTFS transceiver processing. When separate Zak-OTFS subframes are dedicated to sensing and communication, transceiver signal processing can involve the components shown in the continuous DD domain and the time domain.

[00483] In this Section the discrete DD domain was focused on. It was described how filters in the discrete DD domain define spread waveforms with low PAPR - about 5 dB compared with about 15 dB fora point pulse-tone (e.g., Pulsone).

[00484] It was shown that it is possible to acquire the effective channel from the response to a single pilot when the crystallization condition is satisfied with respect to the rotated lattice Aq That is, it was described that it is possible to construct spread waveforms with desirable characteristics by applying a chirp filter in the discrete DD domain to a point pulse-tone. It was also described that it is possible to read off the I / O relation provided a second crystallization condition is satisfied with respect to Aq.

[00485] 7. Examples of ISAC in the same Zak-OTFS subframe

[00486] Simulation results that illustrate BER performance as a function of key systems parameters such as pilot power, data power and maximum Doppler shift are provided.

[00487] FIG. 25 shows an example of multiplexing data with a spread pilot signal. This is an example of integrating sensing and communication with a spread pulse-tone. A spread pilot Xs and a point data signal Xd are transmitted. Here, the data can interfere with the spread pilot.

[00488] In the discrete DD domain, a signal ydd[k, Z] is received, ydd [k,i] = yd,dd + ys,dd + ndd [k,i],

[00489] The signal ydd [k, Z] comprises a contribution ys dd [k, I] from the pilot and a contribution yddd [k, Z] fr°m the data and a noise term (ndd [k, Z]) which can be neglected for simplicity.

[00490] In the absence of noise, ydd[k, Z] can be represented as ydd [k> t]     Ed (^eff [fr» t] Xddd [k, Z]) + y[ Ep (Zleff [k, Z]    Xsdd [k, Z]).

[00491] The cross-ambiguity between y and xs comprises a first term that can be used to make the effective channel visible and a second term that describes the interference from the data signal.

[00492] The cross-ambiguity AyXs[k, I] between ydd[k, I] and xsdd[k, I] can be represented as Ay,Xs[k, Z] = jE^(heff[k, Z] / l^jZc, Z]) + jE^(heff[k, Z] *CT AXcbXs[k, Z]).

[00493] The mathematical effect of data interference, which can be noise like, is described below.

[00494] The crystallization condition holds with respect to the rotated lattice Aq.

[00495] The translates of the channel support S<o. o; by lattice points in Aq are disjoint. This means that there is no DD domain aliasing, and it is possible to accurately estimate the channel on the information grid (the lattice Add or Ap) from the response received within S<o,o). E                  4XdAs[k,Z]|2] = ^Z(fc,()eS(00)|heff[k,Z]|2

[00496] Interference from data translates to an estimation error that is independent of the location [k, I], This contribution to the mean squared error appears as an SNR (signal to noise ratio) hit, and the factor MN in the denominator makes this SNR hit less significant.

[00497] The channel can be estimated. For example, estimate hef{[k,l] by reading off A^Jk, Z] in %o) > 4y.xs[k, Z] Ephe^\k, Z] + Ed(keff [k, Z] *f7 AX / X^[k, Z]) fteff[k,l]=^^ for (k, Z) e S(0 0)

[00498] Then estimate y^dd[k, Z], and then subtract the estimate for the received pilot from ydd[k,l] to obtain an estimate for ydidd[k,l]. (The estimate for yddd[k,I] = yddd[k,l]).

[00499] Then recover the data from this estimate yddd [k, 1] •

[00500] FIG. 26 is a graphical depiction of an example of sensing accuracy as a function of pulse dynamic range (PDR).

[00501] In the subsequent figures, a standard Veh-A channel will be referred to. This standard channel will be described as a standard Veh-A channel with Doppler period vp = 30 KHz, M = 31, N = 37, and RRC pulse shaping filter (j3T = pv = 0.6).

[00502] Here, in this FIG. 26, sensing accuracy (NMSE) is plotted as a function of the ratio of pilot energy to data energy (PDR). NMSE can be given by M..OC, Erfc.pesl^efftM]-Seff[k,l]| NMot:     _     , r          .

[00503] Here, in FIG. 26, Data SNR pd = 25 dB and vmax = 815 Hz.

[00504] The gray curve is separate sensing and communication (S| C bar or S|C), and as PDR increases the pilot overwhelms the noise. In this example (S|C), there is no interference from data.

[00505] The black curve is sensing and communications in the same Zak-OTFS subframe (S|C), and as PDR increases the pilot (e.g., spread pilot) overwhelms the combination of data interference and noise.

[00506] NMSE saturates at high PDR because of DD domain aliasing.

[00507] FIG. 27 is a graph showing example interference from a residual spread pilot. Here, the interference is measured from the residual spread pilot. In this example, Data SNR pd = 25 dB and vmax = 815 Hz for a standard Veh-A channel. And, in this example, SIR represents the ratio of data power to residual spread pilot power (after cancellation).

[00508] When the effective channel is estimated, it is estimated within a support region - taps outside this region are noisy and subject to DD domain aliasing. That is, not all effective channel taps are estimated: channel taps within a support region are read off as taps outside this region are subject to DD domain aliasing and are noisy.

[00509] Sensing accuracy can improve with increasing PDR but so does the power of the interference from the residual pilot. That is, sensing accuracy can improve with PDR, but residual spread pilot energy increases proportionately.

[00510] FIG. 28 shows a graph of an example bit error rate (BER) performance. Here, BER performance of uncoded 4-QAM is plotted as a function of the ratio of pilot energy to data energy (PDR). In this example, Data SNR pd = 25 dB and vmax = 815 Hz fora standard Veh-A channel.

[00511] When CSI is perfect, so is subtraction of the received pilot, and BER performance is independent of PDR [e.g., C | S, perfect CSI: BER performance independent of PDR].

[00512] The dashed curve with the light gray triangle markers [middle curve] shows the baseline -separate sensing and communications (S|C bar& C|S bar) where performance tracks sensing accuracy (NMSE) [e.g., S | C & C | S : BER performance tracks sensing accuracy (NMSE)].

[00513] The dashed curve with the black triangle markers [top curve] is sensing and communications in the same Zak-OTFS subframe (S|C & C|S). Here, sensing accuracy (NMSE) improves as PDR increases, but so does the interference from the residual pilot pulse-tone after cancellation. The result is a “U” shaped curve with an optimal PDR [e.g., S | C & C | S : Characteristic ”U” shape and optimal PDR],

[00514] In example FIG. 28, when PDR < 10 dB, BER improves as PDR increases. When, PDR > 10 dB, interference from residual spread pilot degrades BER as PDR increases.

[00515] FIG. 29 shows a graph of an example BER performance as a function of maximum Doppler spread. Here, BER performance of uncoded 4-QAM is plotted as a function of the maximum Doppler spread. In this example, Data SNR pd = 25 dB and PDR =10 db for a standard Veh-A channel.

[00516] The gray curve [bottom curve] shows the baseline - separate sensing and communications (S|C bar & C|S bar) where there is no residual pilot pulse-tone to interfere with data transmission.

[00517] The black curve (black triangles) is sensing and communications in the same Zak-OTFS subframe (S|C & C|S) - the residual pilot interferes with data transmission but the variation in BER is small over a wide range of Doppler spreads.

[00518] This is very different from the performance of point pilots as the variation is much greater for point pilot pulse-tones.

[00519] Spread pilots can extend the range of reliable operation to a wider range of Doppler spreads.

[00520] FIG. 30 is a graph showing examples of geometric criteria for filter design. In this example, Data SNR pd = 25 dB and PDR =10 db for a standard Veh-A channel.

[00521] The slope q of the chirp filter that defines the spread filter is very important. Here, slopes q=3 and q=36 are considered and BER performance is compared.

[00522] Reliable data detection requires crystallization with respect to the lattice Ap, and that is the case in the example of FIG. 30 - Independent of slope q : vmax < vp = 30 KHz , rmax = 2.5 ps < tp

[00523] Accurate sensing requires crystallization with respect to the lattice Aq. In the example of FIG. 30, it is satisfied for q=3, but not for q=36 when Vmax > 2 KHz.

[00524] FIG. 31 is a graph showing example performance of using spread pilots. In this example, Data SNR pd = 25 dB and PDR =10 db for a standard Veh-A channel.

[00525] Effective throughput is the ratio of the number bits reliably communicated in each subframe to the available degrees of freedom.

[00526] In various embodiments, only sensing and communications in the same Zak-OTFS subframe (S|C & C|S) is considered.

[00527] DD domain resources are divided between sensing and communications when point pilots are used and DD domain resources are shared when spread pilots are used. It will be appreciated by skilled artisans that sharing increases effective throughput. For example, sharing DD domain resources can increase effective throughput compared with traditional approaches that use guard bands to divide DD domain resources between sensing and communications.

[00528] Hence, in various embodiments, the use of spread pilots can help maximize effective throughput.

[00529] FIG. 32 is a graph showing example results obtained in a turbo-aided communication and sensing implementation. In this example, Data SNR pd = 25 dB and PDR =10 db for a standard Veh-A channel.

[00530] In various embodiments, the effective channel is estimated, and then the received spread pilot is estimated. Then the data can be recovered after subtracting the estimate for the received pilot from the received signal.

[00531] Now take the estimated data, then estimate the received data signal, and then improve the estimate for the effective channel by subtracting the estimate for the received data signal from the received signal.

[00532] This can then be repeated as many times as is useful. This is turbo aided sensing and communication; and, in various embodiments, two turbo iterations suffice to match the performance of separate sensing and communication across a wide range of Doppler shifts.

[00533] In some embodiments, subtracting the received spread pilot from the received signal can include computing the twisted convolution of the received spread pilot and a result of the channel sensing.

[00534] 8. Examples and implementations of the disclosed technology

[00535] FIG. 33 is a block diagram representation of a hardware platform 3300 which may be used to implement the various methods described in the present document. The hardware platform 3300 may be incorporated within a base station ora user device. The hardware platform 3300 includes at least one processor 3302, an optional memory 3304 (this may be optional because in some cases the memory may be internal to the processor) and a transceiver circuitry 3306. The processor(s) may execute instructions, e. g., by reading from the memory 3304, and control the operation of the transceiver circuitry 3306 and the hardware platform 3300 to perform the methods described herein. The transceiver circuitry 3306 may be used, for example, for transmitting or receiving signals, as disclosed herein. In some embodiments, the memory 3304 and / or the transceiver circuitry 3306 may be partially or completely contained within the processor 3302 (e.g., same semiconductor package). The hardware platform 3300 may further include one or more antennas to perform multi-layer communication as disclosed herein. The hardware platform 3300 may be used to implement the transmitter apparatus or the receiver apparatus described herein.

[00536] In some embodiments, method of digital communication includes generating, by a transmitter apparatus, a transmission waveform comprising a data portion and a pilot portion and operating a transceiver of the transmitter apparatus to transmit the transmission waveform over the channel. The pilot portion comprises a single pulse-tone pilot signal that allows a receiver to estimate a channel on which the transmission waveform is transmitted. Additional details of such a method are provided with reference to the point-pulse pilot technology disclosed herein.

[00537] In some embodiments, a digital communication method includes receiving, by a receiver apparatus, over a channel, a received signal corresponding to a transmission waveform transmitted over the channel, the transmission waveform comprising a data portion and a pilot portion wherein the pilot portion comprises a single pulse-tone pilot signal; and estimating the channel based on a cross-ambiguity function between the received signal and the single pulse-tone pilot signal.

[00538] In the disclosed embodiments, the single pulse-tone pilot signal comprises a time domain signal that is represented in a delay-Doppler domain as a quasi-periodic pulse, wherein the delay-Doppler domain is defined by a delay period tp and the Doppler period vp.

[00539] The above-disclosed embodiments may use a point pulse pilot, as disclosed in the present document.

[00540] In some embodiments, a method of data communications, implemented by a transmitter apparatus, includes generating a first intermediate signal by combining information symbols in a delay-Doppler domain with a quasi-periodic impulse train in the Delay-Doppler domain; generating a second intermediate signal by performing a twisted convolution of the first intermediate signal with a transmit pulse shaping filter; generating a transmission waveform by applying inverse Zak transform to the second intermediate signal; and causing a transmitter electronics to transmit the transmission waveform over a channel. Additional features of such a method are described with respect to the Zak OTFS implementations disclosed in the present document.

[00541] In some embodiments, a method of digital communications includes, receiving, by a receiver apparatus from a transmitter apparatus a transmission waveform over a channel, generating a first intermediate signal by applying a Zak transform to the transmission waveform, generating a second intermediate signal by performing a twisted convolution between the first intermediate signal and a receiver pulse shaping filter, recovering information signal bits from the second intermediate signal.

[00542] In the above-disclosed methods, the twisted convolution may be performed in the delay-Doppler domain. Some examples are provided with reference to, e.g., FIG. 7.

[00543] In some embodiments, a digital communication method includes, generating, by a transmitter apparatus, a transmission waveform comprising a data portion and / or a pilot portion in a delay-Doppler domain; wherein the pilot portion comprises a pulse-tone pilot signal; and operating a transceiver to transmit the transmission waveform over the channel. Additional features of this method are described with respect to ISAC using point pulse-tone waveforms.

[00544] In some embodiments, a method of digital communication includes, receiving, by a receiver apparatus, a received signal representing a transmission waveform transmitted over a channel, wherein the transmission waveform comprises a data portion and / or a pilot portion that are combined together in a delay-Doppler domain, wherein the pilot portion comprises a pulse-tone pilot signal; determining an estimate ofthe channel using the pilot portion, wherein the determining is performed based on a channel sensing operation, estimating the data portion by subtracting, from the received signal, an estimate ofthe pilot portion determined from the channel sensing operation, and recovering information bits from the data portion. Additional features of this method are described with respect to ISAC using point pulse-tone waveforms.

[00545] In the above-disclosed methods, the channel sensing operation may be performed by calculating a cross-ambiguity function between the received signal and the pulse-tone pilot signal.

[00546] In some embodiments, the data portion and the pilot portion are separated by a guard band in the delay-Doppler domain.

[00547] In some embodiments, no guard band is used for separating the data portion and the pilot portion in the delay-Doppler domain.

[00548] In some embodiments, the transmission waveform is organized in subframes comprising one or more of: (1) a subframe for sensing in which the pilot portion is transmitted, (2) a subframe for communication in which the data portion is transmitted, or (3) a subframe for joint sensing and communication in which both the data portion and the pilot portions are transmitted.

[00549] The above-described methods may be used for joint sensing and communication in which transmissions are organized as either separate frames for sensing and communications, or joint frames for sensing and communication. Here, a point pilot may be used.

[00550] In some embodiments, a digital communication method includes generating, by a transmitter, a transmission waveform comprising a data portion and / or a pilot portion in a delay-Doppler domain and operating a transceiver to transmit the transmission waveform over the channel. In some embodiments, the pilot portion comprises a spread pulse-tone pilot signal.

[00551] In some embodiments, a digital communication method includes receiving, by a receiver apparatus, a received signal representing a transmission waveform transmitted over a channel, wherein the transmission waveform comprises a data portion and / or a pilot portion that are combined together in a delay-Doppler domain, wherein the pilot portion comprises a spread pulse-tone pilot signal, determining an estimate ofthe channel using the pilot portion, wherein the determining is performed based on a channel sensing operation, estimating the data portion by subtracting, from the received signal, an estimate ofthe pilot portion determined from the channel sensing operation, and recovering information bits from the data portion.

[00552] In the above-disclosed methods, the channel sensing operation may be performed by calculating a cross-ambiguity function between the received signal and the pulse-tone pilot signal. In some embodiments, no guard band is used for separating the data portion and the pilot portion in the delay-Doppler domain. In various embodiments, the transmission waveform is organized in subframes comprising one or more of: (1) a subframe for sensing in which the pilot portion is transmitted, (2) a subframe for communication in which the data portion is transmitted, or (3) a subframe for joint sensing and communication in which both the data portion and the pilot portions are transmitted.

[00553] In some embodiments, the spread pulse-tone pilot signal is generated by applying a chirp filter to a pulse-tone signal. In some embodiments, the spread pulse-tone pilot signal is quasi periodic in the delay-Doppler domain.

[00554] In some embodiments, a digital communication method can include receiving, by a receiver apparatus, a received signal comprising multiple layers, each layer comprising corresponding data and a corresponding spread pilot over a channel, wherein, for each layer, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain; and extracting data from at least some of the layers of the received waveform by subtracting, for each extracted layer, the corresponding spread pilot from the received signal and by performing a channel equalization based on a channel sensing using the corresponding spread pilot. In some embodiments, the extracting can include performing the subtracting, the channel equalization and the channel sensing over multiple iterations. In some embodiments, subtracting the corresponding spread pilot signal from the received signal can include computing a twisted convolution of the corresponding spread pilot signal and a result of the channel sensing.

[00555] 9. Example technical solutions

[00556] The technical solutions disclosed in the present document provide a discrete filter that is periodic in M*N on both delay and Doppler dimension. One can think of the M x N basic delay-Doppler grid, as being extended N times along delay and M times along Doppler, creating an (M*N) x (N*M) grid. The filter is defined over this extended grid, which is then extended periodically to have infinite number of (M*N) x (N*M) duplicates along both axes.

[00557] For the waveforms defined in the present document, the delay-Doppler grid is infinite, but can be described using a basic grid of M x N, where signals that are quasi-periodic. This means that these signals are periodic on the Doppler dimension (every N grid elements, the signal repeats) and nonperiodic on the delay dimension (repeat every M grid elements but with some phase rotation).

[00558] The present document also defines a filter in delay-Doppler. The filter is periodic with periodicity of M*N on both dimensions. This means that it repeats itself after M*N delay-Doppler grid elements.

[00559] Applying a filter to a signal (such as a delta), will result in a filtered signal that is also M*N periodicon both dimensions.

[00560] The following technical solutions may be implemented by some preferred embodiments.

[00561] With respect to the use of a spread pilot signal, the following solutions may be used.

[00562] 1. A method of digital communication (e.g., method 3410 depicted in FIG. 34A), comprising: generating (3402), by a transmitter apparatus, a transmission waveform by applying a discrete filter to an input signal in a delay-Doppler domain, wherein the discrete filter is periodic in delay domain and in Doppler domain; and controlling (3404) a transceiver to transmit the transmission waveform over a channel. For example, one or more processors of the transmitter apparatus may control the transceiver to cause a transmission to occur.

[00563] 2. The method of solution 1, wherein the discrete filter is M*N periodic in the delay- Doppler domain, where M and N are positive integers. In this document M*N represents a multiplication of numbers M and N.

[00564] 3. The method of solution 2, wherein M and N represent delay and Doppler dimensions of a delay-Doppler grid in the delay-Doppler domain, corresponding to the delay period and the Doppler period.

[00565] 4. The method of solutions 1-3, wherein the input signal comprises a quasi-periodic impulse signal in the delay-Doppler domain, and wherein the applying comprises using an M*N-periodic twisted convolution.

[00566] 5. The method of solution 4, wherein an intermediate signal resulting from the applying is spread over the delay-Doppler domain as an M*N-periodic signal in the delay-Doppler domain.

[00567] 6. The method of solution 5, wherein the transmission waveform is generated by applying an inverse Zak transform to the intermediate signal.

[00568] 7. The method of any of solutions 1-6, wherein the discrete filter comprises a chirp signal in the delay-Doppler domain.

[00569] 8. The method of solution 7, wherein the chirp signal is defined by a slope parameter q, and wherein the slope parameter q is configured to avoid aliasing in the delay-Doppler domain after the transmission waveform is transmitted over the channel.

[00570] Some solutions, e.g., as listed below, may be used to perform a multi-layer communication.

[00571] 9. A method of digital communication (e.g., method 3420 depicted in FIG. 34B), comprising: generating (3422) a transmission waveform comprising L layers, wherein each of the L layers comprises a corresponding data and a corresponding spread pilot, wherein, for the each of the L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain, and wherein L is a positive integer; and controlling (3424) a transceiver to transmit the transmission waveform over a channel.

[00572] 10. The method of solution 9, wherein the data and the spread pilot are overlapping in the delay-Doppler domain.

[00573] 11. The method of solution 9, wherein the data occupies entirety of the delay-Doppler domain, and the spread pilot occupies entirety of the delay-Doppler domain.

[00574] 12. The method of any of solutions 9-11, wherein the controlling comprises weighting transmission energies of the data and the spread pilot.

[00575] 13. The method of any of solutions 9-12, wherein the transmission waveform is configured for an integrated sensing and communication operation.

[00576] 14. The method of any of solutions 9-13, wherein L is greater than 1, and wherein the each of the L layers uses a corresponding different spread pilot that is generated using a different discrete filter.

[00577] 15. The method of any of solutions 9-14, wherein the filtered pulse-tone signal is generated by applying a discrete filter to an input signal in the delay-Doppler domain, wherein the input signal comprises a quasi-periodic impulse signal in the delay-Doppler domain, and wherein the applying comprises using an M*N-periodic twisted convolution, where M and N are positive integers representing delay and Doppler domain dimensions of a basic delay-Doppler grid.

[00578] The following solutions may be implemented by embodiments that use multiple antennas.

[00579] 16. A method of wireless communication (e.g., method 3430 depicted in FIG. 34C), comprising: generating (3432) K transmission waveforms, each of the K transmission waveforms comprising L layers, wherein each of the L layers comprises a corresponding data and a corresponding spread pilot, wherein, forthe each of the L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain, and wherein Kand Lare positive integers and K > L; and transmitting (3434), the each of the K transmission waveforms using a corresponding transmission antenna.

[00580] 17. The method of solution 16, wherein spread pilots used for at least some of the K transmission waveforms are different from each other.

[00581] 18. The method of any of solutions 16-17, wherein each spread pilot is obtained by applying a corresponding discrete filter to a corresponding input signal in a delay-Doppler domain, wherein the discrete filter is M*N periodic in the delay-Doppler domain, where M and N are positive integers.

[00582] 19. The method of solution 18, wherein M and N represent delay and Doppler dimensions of a delay-Doppler grid along the delay-Doppler domain.

[00583] 20. The method of solution 19, wherein the corresponding input signal comprises a quasi- periodic impulse signal in the delay-Doppler domain, and wherein the applying comprises using an M*N-periodic twisted convolution.

[00584] 21. The method of any of solutions 16-20, wherein each of the corresponding discrete filter comprises a corresponding chirp signal in the delay-Doppler domain.

[00585] 22. The method of solution 21, wherein each of the corresponding chirp signal is defined by a corresponding slope parameter q, and wherein the corresponding slope parameter q is configured to avoid aliasing after the corresponding transmission waveform is transmitted over the channel.

[00586] The following solutions may be implemented by receiver apparatus.

[00587] 23. A digital communication method (e.g., method 3440 depicted in FIG. 34D), comprising: receiving (3442) a received signal comprising L layers, each of the L layers comprising corresponding data and a corresponding spread pilot over a channel, wherein L>=1, and wherein, for the each of the L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain; and extracting (3444) data from at least some of the L layers of the received waveform by subtracting, for each extracted layer, the corresponding spread pilot from the received signal and by performing a channel equalization based on a channel sensing using the corresponding spread pilot.

[00588] 24. The method of solution 23, wherein the channel sensing is performed by computing a cross-ambiguity function between the extracted layer of the received signal and a priori knowledge of the corresponding spread pilot.

[00589] 25. The method of any of solutions 23-24, wherein the subtracting the corresponding spread pilot comprises computing a twisted convolution of the corresponding spread pilot and a result of the channel sensing.

[00590] 26. The method of any of solutions 23-25, wherein the extracting comprises performing the subtracting, the channel equalization and the channel sensing over multiple iterations.

[00591] 27. The method of any of solutions 23-26, further including, performing an integrated sensing and communication operation.

[00592] 28. An apparatus for digital communications, comprising at least one processor and a transceiver, wherein the transceiver is configured to transmit or receive signals under control of the at least one processor, and the at least one processor is configured to implement a method recited in any of solutions 1 to 27.

[00593] 29. A non-transitory computer readable 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 of solutions 1 to 27.

[00594] 10. Conclusion

[00595] It will be appreciated that the present document discloses techniques for Zak-OTFS pulse-tone shaping in delay-Doppler (Delay / Doppler) domain and Zak-OTFS pulse-tone spreading in delay-Doppler domain and heterogenous Zak-OTFS shaped pulse-tone in delay-Doppler (different pulse shaping) for Pilots & Data.

[00596] It will also be appreciated that the present document discloses a method of separating Heterogenous Zak-OTFS shaped pulse-tone in delay-Doppler, channel sensing based on Zak-OTFS shaped pulse-tone in delay-Doppler.

[00597] The disclosed techniques may be used to implement a passive radar based on Zak-OTFS shaped pulse-tone in delay-Doppler Pilots or a radar based on delay-Doppler Pulse shaping.

[00598] It will also be appreciated that the present document discloses Turbo receivers for Zak-OTFS.

[00599] It will be appreciated by one of skill in the art that Integrated sensing and communications reduces to geometric properties of a lattice Ap used for data transmission and a lattice A^ used for sensing.

[00600] By sharing DD domain resources between sensing and communications it is possible to optimize throughput without compromising BER performance achieved by separating sensing and communication.

[00601] When the channel satisfies the crystallization condition with respect to the lattice Aq the effective DD filter taps can be read off from the response to a single spread pilot.

[00602] When the channel satisfies the crystallization condition with respect to the lattice A , then given the I / O response at one point in a Zak-OTFS subframe it is possible to predict the response at all points in the subframe.

[00603] Filters in the discrete DD domain enable integration by minimizing interference between sensing and data transmission - the data pulse-tones look like noise to the sensing pulse-tone.

[00604] Filters in the discrete DD domain can be used to design noise-like waveforms with excellent PAPR.

[00605] The disclosed and other embodiments, modules and the functional operations described in this document can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this document and their structural equivalents, or in combinations of one or more of them. The disclosed and other embodiments can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer readable medium for execution by, or to control the operation of, data processing apparatus. The computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more of them. The term “data processing apparatus” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., 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, which is generated to encode information for transmission to suitable receiver apparatus.

[00606] 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.

[00607] 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).

[00608] 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.

[00609] While this patent document contains many specifics, these should not be construed as limitations on the scope of an invention that is claimed or of what may be claimed, but rather as descriptions of features specific to particular embodiments. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination ora variation of a subcombination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results.

[00610] 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.

Claims

1. A method of digital communication, comprising:generating, by a transmitter apparatus, a transmission waveform by applying a discrete filter to an input signal in a delay-Doppler domain, wherein the discrete filter is periodic in delay domain and in Doppler domain; andcontrolling a transceiver to transmit the transmission waveform over a channel.

2. The method of claim 1, wherein the discrete filter is M*N periodic in the delay-Doppler domain, where M and N are positive integers.

3. The method of claim 2, wherein M and N represent delay and Doppler dimensions of a delay-Doppler grid in the delay-Doppler domain, corresponding to the delay period and the Doppler period.

4. The method of claims 1-3, wherein the input signal comprises a quasi-periodic impulse signal in the delay-Doppler domain, and wherein the applying comprises using an M*N-periodic twisted convolution.

5. The method of claim 4, wherein an intermediate signal resulting from the applying is spread over the delay-Doppler domain as an M*N-periodic signal in the delay-Doppler domain.

6. The method of claim 5, wherein the transmission waveform is generated by applying an inverse Zak transform to the intermediate signal.

7. The method of claim 1, wherein the discrete filter comprises a chirp signal in the delay-Doppler domain.

8. The method of claim 7, wherein the chirp signal is defined by a slope parameter q, and wherein the slope parameter q is configured to avoid aliasing in the delay-Doppler domain after the transmission waveform is transmitted over the channel.

9. A method of digital communication, comprising:generating a transmission waveform comprising L layers, wherein each of the L layers comprises a corresponding data and a corresponding spread pilot, wherein, for the each of the L Layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain, and wherein L is a positive integer; andcontrolling a transceiver to transmit the transmission waveform over a channel.

10. The method of claim 9, wherein the data and the spread pilot are overlapping in the delay-Doppler domain.

11. The method of claim 9, wherein the data occupies entirety of the delay-Doppler domain and the spread pilot occupies entirety of the delay-Doppler domain.

12. The method of any of claims 9-11, wherein the controlling comprises weighting transmission energies of the data and the spread pilot.

13. The method of claim 12, wherein the transmission waveform is configured for an integrated sensing and communication operation.

14. The method of claim 13, wherein L is greater than 1, and wherein the each of the L layers uses a corresponding different spread pilot that is generated using a different discrete filter.

15. The method of claim 14, wherein the filtered pulse-tone signal is generated by applying a discrete filterto an input signal in the delay-Doppler domain, wherein the input signal comprises a quasi-periodic impulse signal in the delay-Doppler domain, and wherein the applying comprises using an M*N-periodic twisted convolution, where M and N are positive integers representing delay and Doppler domain dimensions of a basic delay-Doppler grid.

16. A method ofwireless communication, comprising:generating K transmission waveforms, each of the K transmission waveforms comprising L layers, wherein each of the L layers comprises a corresponding data and a corresponding spread pilot, wherein, for the each of the L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain, and wherein K and L are positive integers and K > L; andtransmitting, the each of the K transmission waveforms using a corresponding transmission antenna.

17. The method of claim 16, wherein spread pilots used for at least some of the K transmission waveforms are different from each other.

18. The method of any of claims 16-17, wherein each spread pilot is obtained by applying a corresponding discrete filterto a corresponding input signal in a delay-Doppler domain, wherein the discrete filter is M*N periodic in the delay-Doppler domain, where M and N are positive integers.

19. The method of claim 18, wherein M and N represent delay and Doppler dimensions of a delay-Dopplergrid along the delay-Doppler domain.

20. The method of claim 19, wherein the corresponding input signal comprises a quasi-periodic impulse signal in the delay-Doppler domain, and wherein the applying comprises using an M*N-periodic twisted convolution.

21. The method of claim 20, wherein each of the corresponding discrete filter comprises a corresponding chirp signal in the delay-Doppler domain.

22. The method of claim 21, wherein each of the corresponding chirp signal is defined by a corresponding slope parameter q, and wherein the corresponding slope parameter q is configured to avoid aliasing after the corresponding transmission waveform is transmitted over a channel.

23. A digital communication method, comprising:receiving a received signal comprising L layers, each of the L layers comprising corresponding data and a corresponding spread pilot over a channel, wherein L>=1, and wherein, for the each of the L layers, the corresponding data comprises symbols in a delay-Doppler domain and the corresponding spread pilot comprises a filtered pulse-tone signal that is a filtered quasi-periodic delta signal over the delay-Doppler domain; andextracting data from at least some of the L layers of the received waveform by subtracting, for each extracted layer, the corresponding spread pilot from the received signal and by performing a channel equalization based on a channel sensing using the corresponding spread pilot.

24. The method of claim 23, wherein the channel sensing is performed by computing a cross-ambiguity function between the extracted layer of the received signal and a priori knowledge of the corresponding spread pilot.

25. The method of any of claims 23-24, wherein the subtracting the corresponding spread pilot comprises computing a twisted convolution of the corresponding spread pilot and a result of the channel sensing.

26. The method of claim 23, wherein the extracting comprises performing the subtracting, the channel equalization and the channel sensing over multiple iterations.

27. The method of claim 26, further including, performing an integrated sensing and communication operation.

28. An apparatus for digital communications, comprising at least one processor and a transceiver, whereinthe transceiver is configured to transmit or receive signals under control of the at least one processor, andthe at least one processor is configured to implement a method recited in any of claims 1 to 27.

29. A non-transitory computer readable 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 of claims 1 to 27.

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