Patent Description:
The following relates to methods, systems, and devices for positioning and timing using networks of beacon transmitters, which can provide combinations of high precision, resilience to co-channel interference, especially due to beacons transmitting on the same time and frequency channel, and security to jamming and spoofing attacks.

High-precision, low-latency positioning, navigation, and timing (PNT) will be critical to the success of next-generation outdoor systems for management and navigation of sUAS's and autonomous vehicles, and for next-generation indoor systems to enable the Industrial Internet of Things (IIoT) and meet the ±<NUM> meter <NUM>th percentile Z-axis requirement (under review for reduction to ±<NUM> meters) mandated by the Federal Communications Commission (FCC) for in-building E911 handset calls. In the nascent commercial drone industry, detection and avoidance of collisions between Class-<NUM> (commercial class) small unmanned aerial systems (sUAS's), and between sUAS's and other objects, is already an issue as the popularity of sUAS's grow. Solving these issues will become critical as commercial ventures, such as Amazon and Google roll out their drone delivery services, creating both a dramatic increase in the density of such vehicles, and the need for beyond-visual-line-of-sight (BVLOS) command and control (C2) procedures to navigate them to their destinations. For this reason, fast, accurate, and cross-airspace shared PNT will be a critical component of next-generation UAS traffic management (UTM) systems operating below and outside conventional air traffic management (ATM) systems, such as the UTM pilot project (UPP) under development by NASA. Given the low altitudes and radar cross-section of class-<NUM> sUAS's, which make them difficult to detect and localize using conventional ATM radars, the UTM concept relies on position reports from the sUAS's themselves, either on a regular basis during sUAS operations, or at request from UTM service suppliers (USS's) monitoring those operations. As the number and density of sUAS's grows, the timeliness and precision requirements for this positioning information will also grow.

Current PNT concepts provide this positioning information using signals received from global satellite navigation systems (GNSS), e.g., the Global Positioning System (GPS) operated and maintained by the United States, enabled by GNSS chipsets on-board the UAS's. With few exceptions (for example, <NPL>, and <NPL>," <CIT>), these systems rely on "correlative" or "matched filter" methods that detect the GNSS signals and estimate their geo-observables, e.g., their time-of-arrival (TOA) and frequency-of-arrival (FOA) observed at the receiver, by correlating the received signals against replicas of the transmitted ranging codes, and searching over trial TOA's that compensate for the time-of-flight from the satellite vehicles (SV's) to the receiver, and trial FOA's that compensate for the Doppler shift between the satellite and receiver. In "cold start" scenarios where the specific ranging codes for satellites within the probable field of view (FoV) of the receiver are unknown, and where the receiver's internal clock is not synchronized to universal time coordinates (UTC), this requires further correlating the received signal against the entire library of possible ranging codes, e.g., <NUM> ranging codes for the <NUM> GPS satellites currently in orbit, and estimating the observed TOA's and FOA's between those signals and each of those replica codes, i.e., the additional unknown timing and carrier offset induced during the reception operation. The time-to-first-fix (TTFF) required to accomplish this search can take over <NUM> seconds in absence of any prior information about the satellite codes and timing/carrier offset relative to the receiver (cold-start TTFF), and can take <NUM>-to-<NUM> seconds if code lock has been lost for several hours (warm-start TTFF) or for a short time (hot-start TTFF), and is hugely power consumptive over that time period. Moreover, a full positioning and timing (P/T) solution requires knowledge of the ephemeris (trajectory over time) of the SV's, e.g., using the GPS satellite almanac transmitted over the GPS navigation signal, which takes <NUM> minutes to download in its entirety. Even then, acquisition of at least four GPS signals is needed to provide an initial P/T solution.

In addition, GNSS ranging signals have inherent low received incident power (RIP), due to propagation from satellites in medium-Earth orbit (MEO), or (for the Indian Navic and Japanese QZSS systems) geo-synchronous orbit (GSO). For example, GPS L1 C/A signals are mandated to have an RIP of at least -<NUM> dBm at the ground, i.e., -<NUM> dB signal-to-noise ratio (SNR) over the <NUM> null-to-null bandwidth of that signal, assuming a <NUM> dB rolled-up receiver noise figure. These signals are easily suppressed by <NUM>-to-<NUM> dB in practice, e.g., due to attenuation by trees, foliage, and building walls, and can be lost entirely in valleys and urban canyons. At RIP's below -<NUM> dBm (SNR's below -<NUM> dB), acquisition can fail completely, e.g., due to suppression of the GPS signal down to or below the noise floor, even after despreading the signal down to the <NUM> bit-per-second (bps) GPS navigation signal rate (~ <NUM> dB processing gain), or due to "false lock" caused by nonzero (-<NUM> dB) cross-correlation between the <NUM>,<NUM>-chip C/A replica codes and actual codes received from the GPS SV's (for example, <NPL>). GNSS solutions are also vulnerable to systematic (or systemic?) errors due to ionospheric propagation, satellite positioning and timing errors, and (esp. for low altitude aircraft or in urban environments) specular multipath.

In re UTM applications, stable sub-meter altitude accuracy needed for GPS-based commercial aircraft landing systems, and expected to be required for accurate UAS traffic management, can require minutes to hours to achieve, even using the wide area augmentation system (WAAS) to improve positioning precision. In addition, sUAS receivers are especially vulnerable to co-channel interference (CCI), e.g., intentional jamming of sUAS platforms, or inadvertent jamming due to so-called "personal privacy devices (PPD's)", due to the large FoV of sUAS's at even modest altitudes, and typical line-of-sight (LOS) propagation between the interferers and the sUAS's. All of these issues can cause critical lapses in positioning capability at low altitude and in dense deployment scenarios, where errors of a few feet can inordinately increase the risk of sUAS's colliding with each other, or with buildings, ground vehicles, or even people.

Lastly, it is recognized that GPS based systems have difficulty providing centimeter-level horizontal precision accuracy and <NUM> millisecond tracking capability needed for autonomous vehicles. For this reason, Verizon recently announced rollout of a nation-wide network of reference stations to enable real-time kinematics (RTK), a method for enhancing reliability of GPS signals by using carrier phase, in addition TOA and FOA, as a positioning geo-observable. As described in <NPL>," <CIT>, RTK has long been used to provide precise point positioning (PPP) and timing measurements in surveying and high-accuracy timing applications; however, it requires both careful calibration of system phase offset at the GPS transmitters and the user receivers, e.g., induced by the mixer local-oscillators (LO's), cabling, and filtering modules in both devices; and a means for resolving the cycle ambiguity in the carrier phase geo-observable; hence the need for a reference network, to perform both the system calibration, and at least partially reduce cycle ambiguity. Moreover, LO phase noise induced at either end of the link can require rapid tracking of this system phase, and/or expensive LO's and/or ancillary receiver hardware to precisely control or calibrate and compensate that phase noise. For this reason, stand-alone RTK-enabled systems can take as long as <NUM> minutes to achieve precise solutions. Moreover, the approach is inherently vulnerable to time-varying multipath, e.g., Jakes Law multipath caused by vehicle motion in vicinity of near-field scatterers, which can affect stability of RTK-based solutions even when using precise reference systems. Currently available RTK systems, such as Swift Navigations' Skylark Cloud-based reference station and Starling positioning engine, claim RTK convergence in as little as <NUM> seconds, and reacquisition time of <NUM> second, however, they can only provide an <NUM>% circular error probability (CEP80) of > <NUM> centimeters, and only for fixed users (tracking capability has not been provided for either product).

In response to these issues, a number of alternative navigation (AltNav) solutions have been advanced over the years. These include Locata's "LocataLite" beaconing system, as described in <NPL>, and <NPL>, and NextNav's Metropolitan Beacon System (MBS), described in <NPL>, and <NPL>, which has been incorporated into LTE Release <NUM>, both of which employ DSSS ranging signals and correlative despreading methods at the receiver; Satelles' Iridium-based system, e.g., <NPL>," <CIT>, which exploits narrowband (~ <NUM>) signals transmitted from low-Earth orbiting (LEO) Iridium SV's; and systems exploiting "signals of opportunity" (SOP's), e.g., <NPL>," <CIT>, which exploit cellular and broadcast television signals transmitted from known positions with known time-synchronized signal components. These solutions address some, but not all of GNSS vulnerabilities, and possess weaknesses of their own. In particular, the pseudolite, LocataLite, and NextNav systems are highly vulnerable to "near-far" interference caused by extreme differences in pathloss between transmit nodes. Mitigation of this issue requires either excessive integration time to separate co-channel signals using correlative methods, or transmission of signals over widely separated frequency channels or time slots to avoid it entirely. NextNav's system, for example, separates signals into ten <NUM> time slots separated by <NUM> second in time, which requires continuous reception over <NUM>-<NUM> seconds for a cold-start TTFF and <NUM> second for a warm-start TTFF, and even then provides an initial median horizontal positioning accuracy (CEP50) of <NUM> meters in outdoor environments (Van Grass, slides <NUM>) and <NUM> meters in optimized "local" environments, e.g., campuses, malls, and warehouse-like areas (Van Grass, slides <NUM>). Similarly, although Locata has reported centimeter-level accuracy for its <NUM> Mcps ranging system (Rizos), that accuracy requires time-hopping its signal by a factor of <NUM> (Cheong), yielding the same TTFF as GPS systems.

Although Satelles' system can exploit the much higher RIP and Doppler shift afforded by Iridium's network of LEO SV's, the Iridium signal requires at least one <NUM> second (<NUM>-frame) superframe, and typically two-to-three superframes (<NUM>-<NUM> seconds), to acquire and obtain satellite ephemeres from the Iridium Ring Channel. Moreover, the <NUM> xGPS signal bandwidth provides an inherently poor TOA geo-observable estimate on a per-slot basis, requiring many minutes to provide < <NUM> ns timing synchronization, e.g., as SV's come into the receivers' FoV.

Other solutions use Bluetooth Low Energy (BLE) beacons, LTE position reference signals (PRS's), and <NUM>-based positioning systems. None of these systems can provide the accuracy and latency required for next-generation 5GNR systems, e.g., <NUM> meter XYZ location accuracy, < <NUM> second TTFF and <NUM> latency, and <NUM> meter/second XYZ velocity accuracy. Nor can they meet the FCC's <NUM> goal of ± <NUM> meter <NUM>% Z-Axis handset positioning accuracy for E911 applications.

Aspects of the disclosure can overcome these issues, using resilient distributed positioning networks (RDPN), in which multiple network-provisioned co-channel navigation beacons are transmitted from a network of nodes (e.g., network nodes) to users on a common frequency channel; network-provisioned co-channel beacons are transmitted from users and received at network nodes; or navigation beacons transmitted from network notes, transponded through users, and received by network nodes. The disclosure describes aspects and features that overcome vulnerabilities of existing PNT systems. These aspects and features include (but are not limited to) the following:.

These features can eliminate the need for time slotting or hopping to avoid near-far interference, and allow geo-observables to be determined with high precision over much lower TTFF than competing methods. This precision and TTFF advantage can be traded against multiple system parameters, e.g., latency, available bandwidth, available power, etc., to meet the needs of the network or the users.

The disclosed RDPN aspects can be implemented in at least three network topologies:.

In some aspects, the RDPN's also deploy <NUM>-<NUM> calibration receivers, also connected to the NOC, to bring the network nodes into time and carrier synchronization, and/or provide time/carrier offsets used by the NOC during geolocation operations. The calibration receivers can also be used to locate network nodes, e.g., when they are first deployed in a theater of operations for the network. Other aspects perform these operations at the network nodes, using beacon calibration information provided over a low-rate data link.

This approach can provide a number of benefits not shared by any competing GNSS or non-GNSS method, including (but not limited to) any of the following:.

A first aspect relates to a method for transmitting beacon signals from network nodes to network users, transmitting beacon signals from the network users to the network nodes, and/or transponding beacon signals to and from the network nodes through the network users using bent-pipe transponders. The method comprises inducing at least one of spectral redundancy and temporal redundancy in the beacon signals; and exploiting the at least one of spectral redundancy and temporal redundancy to separate received beacon signals at the network users, the network nodes, or a central processing site.

The method of the first aspect may further comprise determining geo-observables from separated beacon transmissions. The method may further comprise determining positioning and/or timing from the geo-observables. The beacon signals may be separated with precision dictated by the power of the beacon signals above a receiver noise floor, and irrespective of other beacon signals received at the same time and frequency.

A second aspect relates to a method for transmitting beacon signals from network nodes to network users, transmitting beacon signals from the network users to the network nodes, or transponding beacon signals to and from the network nodes through the network users using bent-pipe transponders. The method comprises inducing at least one of spectral redundancy and temporal redundancy in each of the beacon signals, thereby enabling a receiver to exploit the at least one of spectral redundancy and temporal redundancy to separate multiples ones of the beacon signals in a snapshot of received signals.

A third aspect relates to a method, comprising generating a snapshot of a received plurality of beacon transmissions, each of the plurality of beacon transmissions having at least one of spectral redundancy and temporal redundancy; and exploiting the at least one of spectral redundancy and temporal redundancy to separate multiples ones of the beacon transmissions in the snapshot.

In the third aspect, the generating and the exploiting may be performed at a network user or a network node. The generating may be performed at the network user and the exploiting may be performed at a NOC. The generating may be performed at the network node and the exploiting may be performed at the NOC. The generating may be performed at the network user and the exploiting may be performed at the network node.

In a fourth aspect, a method comprises receiving a snapshot of a received plurality of beacon transmissions, each of the plurality of beacon transmissions having at least one of spectral redundancy and temporal redundancy; and exploiting the at least one of spectral redundancy and temporal redundancy to separate multiples ones of the beacon transmissions in the snapshot.

A NOC may be configured to perform the method of the third aspect, wherein the snapshot is generated by a network user and received by the NOC via a wireless network; or wherein the snapshot is generated by a network node and received by the NOC via a backhaul network.

A fifth aspect relates to a method, comprising receiving a plurality of beacon transmissions to produce a received signal, each of the plurality of beacon transmissions having at least one of spectral redundancy and temporal redundancy; and generating a snapshot of the received signal, wherein the snapshot retains the at least one of spectral redundancy and temporal redundancy; and wherein the at least one of spectral redundancy and temporal redundancy is exploitable for separating the plurality of beacon transmissions. A network user or a network node may be configured to perform the method of the fifth aspect.

A sixth aspect relates to a method comprising synthesizing multitone beacon signals, wherein subcarrier spacing and symbol duration of the multitone beacon signals are selected according to an expected range of time-of-arrival and frequency-of-arrival for network users; inducing at least one of spectral or temporal redundancy on the multitone beacon signals; and transmitting the multitone beacon signals to the network users.

A seventh aspect relates to a method comprising receiving multiple multitone beacon signals; and exploiting spectral redundancy in the multiple multitone beacon signals to use code nulling or Class-C linear minimum-mean-square error methods to separate the multiple multitone beacon signals.

Some aspects relate to an apparatus, comprising at least one processor and at least one memory in electronic communication with the at least one processor, and instructions stored in the at least one memory. The instructions executable by the at least one processor may perform the method of any of the above aspects.

Some aspects relate to a computer program product, comprising a computer readable hardware storage device (such as a non-transitory computer-readable memory) having computer-readable program code stored therein, wherein the program code contains instructions executable by one or more processors of a computer system for performing any of the methods of the above aspects.

Some aspects relate to an apparatus comprising a means for performing each step in any of the methods of the above aspects.

An eight aspect relates to an apparatus, comprising a means for transmitting beacon signals from network nodes to network users, a means for transmitting beacon signals from the network users to the network nodes, and/or a means for transponding beacon signals to and from the network nodes through the network users using bent-pipe transponders. The apparatus further includes a means for inducing at least one of spectral redundancy and temporal redundancy in beacon transmissions; and a means for exploiting the at least one of spectral redundancy and temporal redundancy to separate received beacon transmissions at the network users, the network nodes, or a central processing site. The apparatus may further comprise a means for determining geo-observables from separated beacon transmissions, and a means for determining positioning and/or timing from the geo-observables.

The means for transmitting beacon signals from network nodes to network users can include geographically distributed network nodes at calibrated locations, which can be communicatively coupled to a means for central processing. The means for transmitting beacon signals may include an RDPN or an RDTN. Exemplary network nodes include fixed outdoor transmitters co-located with cellular transmission towers or <NUM> access points; indoor transmitters coexisting with <NUM> WLAN or <NUM> Bluetooth or Zigbee networks; or standalone transmitters. The means for central processing can include a NOC, e.g., a 5GNR MEC or a USS, which provisions each of the network nodes with configuration data or time symbols over a means for communicating data between the network nodes and the means for central processing. The means for communicating data can include an Ethernet-based network, a PLC network, an <NUM> WLAN, an <NUM> Zigbee or Bluetooth network, and/or a <NUM>, <NUM> LTE, or <NUM> cellular network. The means for transmitting beacon signals can include computer processors and computer-readable memory that programs the processors to generate and/or transmit the beacon signals.

The means for transmitting beacon signals from the network users to the network nodes can comprise a wireless communications apparatus onboard a user device configured to receive navigation signals or beacon configuration information from the means for central processing over an ancillary wireless communication link, and transmit beacon signals in the FoV of network nodes configured to receive the beacon signals. Furthermore, the means for transmitting beacon signals can comprise means for generating the beacons signals. The means for transmitting beacon signals can include computer processors and computer-readable memory that programs the processors to generate and/or transmit the beacon signals.

The means for transponding beacon signals can comprise wireless communication transceivers that receive, condition, and retransmit the beacon signals without otherwise processing them. Network receivers (e.g., network nodes) in the FoV of the users then capture and backhaul snapshots of those retransmitted to the means for central processing, which can compute a P/T solution from the snapshots, and transmit the solution to the users over the ancillary wireless communication link. Thus, the means for transponding beacon signals can further comprise a wireless receiver for the ancillary wireless communication link. The means for transponding beacon signals can include computer processors and computer-readable memory that programs the processors to receive and transmit the beacon signals, and optionally, to receive the P/T solution. The means for transponding beacon signals may comprise an RDXN.

The means for inducing can include a modulator configured to perform subcarrier spreading modulation, such as SCSS modulation. In one example, an inner code is replicated over multiple clusters, each of which is modulated by one element of an outer code. Spectral redundancy can be achieved by spreading a narrowband signal with a wideband signal. The means for inducing can include a modulator configured to repeat a time symbol. Time symbols may be organized in slots, with multiple repetitions per slot. The means for inducing may include a computer processor and computer-readable memory that programs the processor to perform the spreading and/or repetition of symbols. A software-defined radio is one example of such a processor. The means for inducing may comprise a multitone modulator, which can employ a DFT, IDFT, FFT, IFFT, polyphase filter, and/or a discrete filter bank.

The means for exploiting can comprise any apparatus or computer program product having instructions that implement a resilient detection operation for excising CCI in a snapshot. The means for exploiting can comprise a subcarrier demodulator that eliminates inter-symbol interference and inter-subcarrier interference. The means for exploiting can perform code nulling or Class-C linear minimum-mean-square error (LMMSE) operations to separate co-channel received beacon signals with quality limited only by the received SNR of those signals, rather than the received SIR of those signals. The means for exploiting can further include spatial/polarization diverse antenna arrays at their transmitters or receivers, which allow for copy-aided DF methods to determine the DOA of the beacon signals, and copy-enhanced DF methods to determine the DOA of jammers. The means for exploiting may further comprise a means for channelizing.

The means for channelizing can comprise a DFT, such as a sparse DFT, a windowed DFT, or a combination thereof. Equivalent structure, such as filters configured for snapshot channelization, may be used. The means for channelizing can comprise any apparatus or computer program product having instructions that channelize a snapshot. In one aspect, the means for channelizing removes an estimated coarse (cold-start) or fine (warm/hot start) observed LO offset, and removes timing offset, if necessary. The means for channelizing may separate the snapshot into frequency subcarriers and time symbols using a windowed DFT.

A ninth aspect relates to an apparatus, comprising a means for generating a snapshot of a received plurality of beacon transmissions, each of the plurality of beacon transmissions having at least one of spectral redundancy and temporal redundancy; and means for exploiting the at least one of spectral redundancy and temporal redundancy to separate multiples ones of the beacon transmissions in the snapshot.

The means for generating the snapshot can comprise any apparatus or computer program product having instructions that, when directed to, collects a snapshot, e.g., based on prompts from the means for central processing, or at scheduled snapshot collection times. The means for generating can include a receiver front-end configured to receive beacon signals, a frequency down-converter, and an ADC, as well as other radio components. The means for generating may comprise an SDR.

A tenth aspect relates to an apparatus, comprising a means for receiving a snapshot of a received plurality of beacon transmissions, each of the plurality of beacon transmissions having at least one of spectral redundancy and temporal redundancy; and means for exploiting the at least one of spectral redundancy and temporal redundancy to separate multiples ones of the beacon transmissions in the snapshot.

The means for receiving the snapshot can comprise an ancillary wireless communication receiver configured to receive snapshots transmitted by network users on the ancillary wireless communication link. The ancillary wireless communication receiver may be an <NUM> WLAN, <NUM> Zigbee, Bluetooth, <NUM>, <NUM> LTE, or <NUM> cellular receiver. The means for receiving the snapshot can comprise a receiver coupled to a beacon communication bus, which connects the NOC with the network nodes, and the receiver may be an Ethernet, PLC, optical fiber, <NUM> WLAN, <NUM> Zigbee, Bluetooth, <NUM>, <NUM> LTE, or <NUM> cellular receiver.

An eleventh aspect relates to an apparatus, comprising a means for receiving a plurality of beacon transmissions to produce a received signal, each of the plurality of beacon transmissions having at least one of spectral redundancy and temporal redundancy; and means for generating a snapshot of the received signal, wherein the snapshot retains the at least one of spectral redundancy and temporal redundancy; and wherein the at least one of spectral redundancy and temporal redundancy is exploitable for separating the plurality of beacon transmissions.

The means for receiving the plurality of beacon transmissions can comprise a receiver front-end of a radio configured to receive transmitted beacon signals. The means for receiving can include a frequency down-converter and an ADC, as well as other radio components. In some aspects, the means for receiving comprises an SDR. The means for receiving may comprise a network user's beacon receiver configured to receive beacon transmissions, such as beacon signals transmitted from network nodes. The means for receiving may comprise network nodes configured to receive beacon transmissions from network users. The means for receiving may comprise network nodes in an RDRN and/or network users in an RDTN or an RDXN.

A twelfth aspect relates to an apparatus, comprising a means for synthesizing multitone beacon signals, wherein subcarrier spacing and symbol duration of the multitone beacon signals are selected according to an expected range of time-of-arrival and frequency-of-arrival for network users; means for inducing at least one of spectral or temporal redundancy on the multitone beacon signals; and means for transmitting the multitone beacon signals to the network users.

The means for synthesizing the multitone beacon signals can include a multitone modulator, which can employ a DFT, IDFT, FFT, IFFT, polyphase filter, and/or a discrete filter bank. The means for synthesizing may include at least one processor and at least one memory in electronic communication with the at least one processor, and instructions stored in the at least one memory to perform multitone signal generation. A software-defined radio is one example of such a processor.

A thirteenth aspect relates to an apparatus, comprising a means for receiving multiple multitone beacon signals; and a means for exploiting spectral redundancy in the multiple multitone beacon signals to use code nulling or Class-C linear minimum-mean-square error methods to separate the multiple multitone beacon signals.

The means for receiving multiple multitone beacon signals can include a multitone demodulator, which can employ a DFT, IDFT, FFT, IFFT, polyphase filter, and/or a discrete filter bank. The means for receiving may include at least one processor and at least one memory in electronic communication with the at least one processor, and instructions stored in the at least one memory to perform multitone demodulation. A software-defined radio is one example of such a processor.

A fourteenth aspect relates to an apparatus, comprising a means for transmitting beacon signals from network nodes to network users, a means for transmitting beacon signals from the network users to the network nodes, and/or a means for transponding beacon signals to and from the network nodes through the network users using bent-pipe transponders. The apparatus further includes a means for inducing at least one of spectral redundancy and temporal redundancy in the beacon signals, thereby enabling a receiver to exploit the at least one of spectral redundancy and temporal redundancy to separate multiples ones of the beacon signals in a snapshot of received signals.

Each of the figures is provided for the purpose of illustration and description only, and not as a definition of the limits of the claims.

<FIG> illustrates a resilient distributed transmitter network (RDTN) that can be instantiated using the system introduced here. In this network, beacons are transmitted from a network of L network nodes <NUM> on a known frequency channel. Exemplary network nodes <NUM> include fixed outdoor transmitters co-located with cellular transmission towers or <NUM> access points; indoor transmitters coexisting with <NUM> WLAN or <NUM> Bluetooth or Zigbee networks; or standalone transmitters. The network nodes <NUM> are connected to a network operations center (NOC) <NUM>, e.g., a 5GNR multi-access edge computer (MEC), or a UAS service supplier (USS), which provisions each of those network nodes <NUM> with separate configuration data or time symbols over a secure communication link. In alternate instantiations, the network nodes <NUM> choose their own configuration and apprise the NOC <NUM> site of that choice. The transmitted beacons are received by M users <NUM>, e.g., Class-<NUM> sUAS's, as shown in this FIG. , ground vehicles, cellular user equipment (UE's), <NUM> STA's, <NUM> Zigbee or Bluetooth devices, or Internet of Thing devices, etc., each of which collect snapshots of data containing a superposition of the beacons transmit from network nodes <NUM> within their field of view (FoV). In one aspect, the users <NUM> backhaul those snapshots to the NOC <NUM> over at least one wireless communication transceiver <NUM>, e.g., a cellular LTE, <NUM>, or <NUM> network or an <NUM> wireless local-area network (WLAN), connected to the users <NUM> and the NOC <NUM>. In alternate aspects, the users <NUM> may perform positioning and timing operations themselves, using network node <NUM> locations and configuration data provided through the wireless communication transceiver <NUM>, and transmit results of those operations to the NOC <NUM> over the wireless communication transceiver <NUM>. In some aspects, different users <NUM> with varying capabilities and network permissions, including time-varying capabilities and permissions, may implement one aspect or the other.

<FIG> illustrates an alternate resilient distributed receiver network (RDRN) that can be instantiated using the system introduced here. In this network, beacons are transmitted from the users <NUM>, based on beacon configuration data transmitted to the users <NUM> from the NOC <NUM> over a wireless communication transceiver <NUM>. Snapshots can be transmitted on a continuous or user-<NUM> initiated basis, e.g., after detection of PNT outage conditions, or upon request by the NOC <NUM>. The snapshots are then received at network nodes <NUM>, and backhauled to the NOC <NUM> from the network nodes <NUM>.

<FIG> illustrates an alternate resilient distributed transceiver network (RDXN) that can also be instantiated using the system introduced here. In this network, beacons are transmitted from network users <NUM>, using beacon time symbols or configuration data provided by NOC <NUM> and are received and directly transponded from the users <NUM> as needed by the users <NUM> or the NOC <NUM>.

<FIG> depicts two exemplary resilient distributed positioning network deployment scenarios, one intended for operation in Subband A of the Location and Monitoring Services (LMS) Band, located at <NUM>-<NUM> (<FIG>), referred to here as the "LMS Scenario", and the other intended for operation in <NUM><NUM> Channel <NUM>, located at <NUM>,<NUM>-<NUM>,<NUM> (<FIG>), a <NUM> channel in the <NUM>-<NUM> ISM band that is not used for <NUM> transmissions in the United States, referred to here as the "<NUM> Ch. <NUM> Scenario". In each scenario, <NUM> users <NUM> are deployed at <NUM>-to-<NUM> meter (<NUM>-<NUM> foot) elevations above ground level (AGL), and <NUM>-to-<NUM> meter/second (<NUM>-<NUM> mile/hour) airspeeds, consistent with class-<NUM> sUAS's, over a <NUM> (<NUM> mile) range from an areal analysis center. Assuming spherical Earth model with <NUM>/<NUM> refraction index, the users <NUM> have a <NUM> (<NUM> mile) ground horizon at maximum altitude. In <FIG> the users <NUM> are in the FoV of <NUM> fixed network nodes <NUM>, deployed at <NUM>-to-<NUM> (<NUM>-<NUM> foot) elevations AGL, i.e., with a ground horizon of <NUM> (<NUM> miles) at maximum altitude, and in a roughly hexagonal layout, consistent with sparse deployment from cell towers, over a <NUM>,<NUM> square-kilometer area. In <FIG> the users <NUM> are in the FoV of <NUM> fixed network nodes <NUM>, deployed over the same range of altitudes, and over a <NUM>,<NUM> square-kilometer area. Both layouts cover the full FoV of the users <NUM> over the range of users <NUM> and network node <NUM> elevations, allowing <NUM>,<NUM> links in the LMS Scenario, or between <NUM> and <NUM> links per user <NUM>; and <NUM>,<NUM> links in the <NUM> Ch. <NUM> Scenario, or between <NUM> and <NUM> links per user <NUM>.

<FIG> show the received incident power (RIP) of the beacons at each receiver, as a function of the time-of-arrival (TOA) and frequency-of-arrival (FOA), i.e., Doppler shift, of those received beacons, for the LMS Scenario (<FIG>) and the <NUM> Ch. <NUM> Scenario (<FIG>). In each scenario, the beacons are assumed to possess azimuthally omnidirectional antennas with <NUM> dBi gain at <NUM>° elevation and -<NUM> dBi gain beyond ±<NUM>° elevation. In the LMS Scenario, the beacons are transmitted at a <NUM> dBm power, yielding an ERP of <NUM> Watts, the maximum power allowed in this band. In the <NUM> Ch. <NUM> Scenario, the beacons are transmitted at <NUM> dBm power, yielding a <NUM> dBm EIRP, the maximum FCC Part <NUM> compliant power allowed in the <NUM>-<NUM> ISM band. Each link assumes a two-ray median pathloss, with <NUM> dB rms log-normal shadowing and (<NUM> dB mean, <NUM> dB standard deviation) Rician fading over each link.

As these FIG. s show, the link TOA's are restricted to <NUM>-<NUM> in the LMS Scenario, and between <NUM>-<NUM> in the <NUM> Ch. <NUM> Scenario, much lower than the <NUM>-<NUM> TOA range expected for GNSS signals transmitted from MEO. Similarly, the link FOA's range are restricted to ±<NUM> in the LMS Scenario and ±<NUM> in the <NUM> Ch. <NUM> Scenario, much lower than the ±<NUM> FOA range expected for L-band GNSS signals transmitted from MEO. This is an exploitable feature of the RDPN for both scenarios. At the same, the RIP of the beacons range from -<NUM> dBm to -<NUM> dBm in the LMS Scenario, and from -<NUM> dBm to -<NUM> dBm in the <NUM> Ch. <NUM> Scenario, much stronger than the nominal -<NUM> dBm GPS L1 signal strength at the Earth surface. While this is also a clear advantage for any beacon-based positioning solution, it also shows that the beacons will likely be received at positive SNR and with significant near-far interference. That is, the performance of conventional "matched filter" receivers that correlate the received signal against replicas of the transmitted beacons will be limited by the interference observed relative to each beacon, despite their high receive SNR, due to self-interference between those received beacons.

This observation is borne out in <FIG>, which show the observed TOA, FOA, and SINR of each link shown in <FIG>, for the LMS Scenario (<FIG>) and the <NUM> Ch. <NUM> Scenario (<FIG>). In both cases, the beacon transmitters assume azimuthally omnidirectional antennas with <NUM> dBi gain at <NUM>° elevation and -<NUM> dBi gain beyond ±<NUM>° elevation, and receivers with <NUM> dB rolled-up noise figure (NF) through the receiver's analog-to-digital converter (ADC), and assume the ADC's and downconversion local oscillators (LO) are locked to clocks with ±<NUM> ppm and ±<NUM> 90th percentile rate and timing offset, respectively. In this scenario, the beacon transmitters are assumed to have negligible timing and LO offset, i.e., they are employing GPS disciplined oscillators, or they are synchronized to UTC via a calibration procedure.

As these FIG. s show, the ADC output SINR ranges between -<NUM> dB and +<NUM> dB in the LMS Scenario, and between -<NUM> dB and +<NUM> dB in the <NUM> Ch. <NUM> Scenario. In fact, while all of the links are above <NUM> dB SNR in the LMS Scenario, only <NUM>% of the links are above a <NUM> dB SINR, and less than <NUM>% of the links are above a -<NUM> dB SINR. Similarly, <NUM>% of the links are above <NUM> dB SNR in the <NUM> Ch. <NUM> Scenario, but only <NUM>% of the links are above <NUM> dB SINR, and <NUM>% of the links are above a -<NUM> dB SINR. Hence, the received beacons are clearly in an interference-limited environment. This is the reason that competing systems introduce time hopping and time slotting into their beacon transmitters - in order to avoid such interference.

These results motivate the use of beacons that can both exploit the much tighter range of TOA and FOA obtaining in a ground-to-low-altitude reception geometry, and allow the use of interference excision methods that can separate the received beacons with performance gated by their (high) receive SNR, rather than their (low) receive SINR.

<FIG> depicts operations used to generate beacon time symbols for the networks shown in <FIG> and <FIG>, and for aspects in which the network nodes <NUM> are provided beacon time symbols from the NOC <NUM>. The time symbol generator first selects code indices <NUM> for each network node <NUM>, creating inner and outer code indices <NUM> <MAT>, where the code indices <NUM> point to code libraries <NUM> containing inner and outer code vectors <MAT> and <MAT>, respectively, and where <MAT> is a Ki ×<NUM> vector of phase values in radians. The code indices <NUM> can be chosen deterministically, pseudorandomly, or unpredictably, or based on environmental factors determined by the network, e.g., evidence of spoofing by malicious actors.

In one aspect, the phase vectors contained in the code libraries <NUM> are designed to yield library codebook signal <MAT> with low peak-to-average power ratio (PAPR), and low cross-correlation between other library codebook signals. The code libraries <NUM> can be designed to satisfy other system requirements as well.

The time symbol generator then performs an inner subcarrier construction operation <NUM> to generate K<NUM>×<NUM> inner subcarrier vector <NUM> <MAT> and an outer subcarrier construction operation <NUM> to generate K<NUM>×<NUM> outer subcarrier vector <NUM>T<NUM>(ℓ) = <MAT>. In one aspect, the vectors are generated using simple phase modulation operations, such that <MAT> and ST<NUM>(ℓ) = <MAT> where θTi(ℓ)(ki) = θki(ci(ℓ)). Other aspects apply non-uniform amplitude weightings to either or both vectors, for example, to further reduce PAPR of the transmitted beacon, reduce interference to non-beacon networks caused by beacons in selected portions of the beacon transmission band, or reduce susceptibility to non-beacon interference at network receivers operating in the beacon transmission band.

The inner subcarrier vector <NUM> and outer subcarrier vector <NUM> are then combined <NUM> to form Ksub×<NUM> full subcarrier vector <NUM> <MAT>. In one aspect, this is performed using a Kronecker product operation, such that ST(ℓ) = ST<NUM>(ℓ) ⊗ ST<NUM>(ℓ). In other aspects, this may be a more complex combining operation, for example, to improve robustness to LO frequency uncertainty at the transmitter or receiver, reduce interference to non-beacon networks caused by beacons in selected portions of the beacon transmission band, or reduce susceptibility to non-beacon interference at network receivers operating in the beacon transmission band.

The full subcarrier vector <NUM> is then passed through an optional subcarrier preemphasis <NUM> operation to generate preemphasized subcarrier vector S̃T(ℓ) = GT(ℓ) ∘ ST(ℓ), where "∘" denotes the element-wise (Shur or Hadamard) product operation, and GT(ℓ) is a Ksub×<NUM> preemphasis vector <NUM> that compensates for front-end digital-to-analog conversion (DAC), antialiasing lowpass filtering (LPF), and upconversion operations performed at network node <NUM>ℓ. The preemphasis vector <NUM> can be designed using analytic models for beacon transmission operations <NUM>; or using calibration data obtained at each network node <NUM>, for example, as described in <CIT>, and can be based on the magnitude or complex value of those beacon transmission operations <NUM>.

The preemphasized subcarriers are then passed to a multitone modulator <NUM> that transforms the subcarriers to the time domain, and is optionally quantized <NUM>, to provide an NDAC×<NUM> time symbol vector <NUM> <MAT>. The multitone modulator <NUM> can be implemented using combinations of discrete Fourier transform (DFT), inverse DFT (IDFT), fast Fourier transform (FFT), or inverse FFT (IFFT) operations, or using polyphase filtering or discrete filter bank operations.

Each time symbol vector <NUM> is then passed from the NOC <NUM> to the beacon transmitter over a beacon communication bus <NUM>. Exemplary communication networks supporting a beacon communication bus <NUM> can include Ethernet-based networks, optical networks, power-line communication (PLC) networks, <NUM> WLAN's, <NUM> Zigbee or Bluetooth networks, or <NUM>, <NUM> LTE, or <NUM> cellular networks. In the networks shown in <FIG> and <FIG>, including this aspect of the disclosure, the beacon communication bus <NUM> connects the NOC <NUM> to the network nodes <NUM>. In the network shown in <FIG>, the beacon communication bus <NUM> connects the NOC <NUM> to the users <NUM> over a wireless communication transceiver <NUM>.

<FIG> shows operations performed in this aspect of the disclosure at each beacon transmitter, e.g., the network nodes <NUM> for the networks shown in <FIG> and <FIG>, or the users <NUM> for the network shown in <FIG>. At each beacon transmitter, the NDAC×<NUM> time symbol vector <NUM> is first obtained from the beacon communication bus <NUM>. If Ksub is even, the time symbol vector <NUM> is placed through time symbol extension operation <NUM> to convert the NDAC ×<NUM> time symbol vector <NUM> to a <NUM>NDAC ×<NUM> time symbol vector <NUM> given by <MAT> and placed in local storage <NUM>. For the networks shown in <FIG> and <FIG>, the time symbol vector <NUM> in local storage <NUM> is then continuously and repeatedly cycled through beacon transmission operations <NUM>, until a new time symbol is provided by the NOC <NUM>, or it must cease transmission for other reasons, e.g., to comply with license provisions of spectra reserved for time-division-duplex (TDD) operation, or based on other instructions received from the NOC <NUM>. For the network shown in <FIG>, the time symbol vector <NUM> is transmitted from the user <NUM> over varying time interval durations, dependent for example upon the priority of positioning/timing solutions demanded by the user <NUM> or the NOC <NUM>, or user <NUM> energy conservation needs.

The operations used to perform the beacon transmission operations <NUM> are analogous to an arbitrary waveform generator (AWG). In the aspect shown in <FIG>, the operations comprise dual DAC, dual LPF, frequency shift operations using in-phase and quadrature LO's, and power amplification (PA) operations. However, many other means can be used to implement the beacon transmission operations <NUM>, e.g., superheterodyne transmitters that convert the time symbol vector <NUM> to a real intermediate-frequency (real-IF) form and perform the frequency conversion in multiple steps, and polar modulators, for example, <CIT>, which convert the time symbol vector <NUM> to polar form and separately phase modulate the time symbol vector <NUM> phase and amplitude-modulate the time symbol vector <NUM> amplitude. These conversions can be implemented at the NOC <NUM>, or at the beacon transmitters.

Typically, the DAC and LO employed in the beacon transmission operations <NUM> are locked to a system clock <NUM>, which in general, has a clock rate and timing that is offset from a common time standard, e.g., UTC. As shown in <FIG>, consistent with the networks shown in <FIG> and <FIG>, the system clock <NUM> for network node <NUM>ℓ is offset from UTC by rate offset εT(ℓ) and timing offset <MAT>, where <MAT> is the internal clock time at UTC time <MAT>. In some aspects, the system clock <NUM> is synchronized to an external time and frequency standard using an external source, e.g., a GNSS receiver (Rx) <NUM>. In some aspects, the system clock <NUM> is brought into a common time standard using network calibration methods computed at the NOC <NUM>, e.g., by providing the system clock <NUM> with clock synchronization data <NUM>, e.g., timing and rate offset estimates τ̂T(ℓ) and ε̂T(ℓ). In the network shown in <FIG>, the system clock <NUM> employed each user <NUM> can similarly be brought into synchronization with a common time standard using clock synchronization data <NUM> provided by the NOC <NUM>.

In some aspects consistent with the networks shown in <FIG> and <FIG>, the network nodes <NUM> are provisioned with the code library <NUM>, and the NOC <NUM> sends it the code indices <NUM> to be used to generate time symbol vector <NUM>. Alternately, each network node <NUM> can select its own code indices <NUM> and communicate it back to the NOC <NUM>. These aspects greatly reduce the data rate needed to communicate between the network nodes <NUM> and NOC <NUM>, but increase operation security of the network, by requiring the beacons to possess copies of the code libraries <NUM>.

<FIG> illustrates an exemplary subcarrier frequency layout assumed here, and illustrates the spectral redundancy imposed in aspects where a Kronecker product operation is used to construct the full subcarrier vector <NUM> from simple BPSK inner subcarrier vectors <NUM> and BPSK outer subcarrier vectors <NUM> (<FIG>); and illustrates the temporal structure of the beacon for an aspect in which the time symbols are organized into slots with Nrep per slot (<FIG>). In <FIG>, the inner subcarrier vector <NUM> is a <NUM>-element vector [+<NUM> +<NUM> -<NUM> -<NUM> +<NUM>]T, while the outer subcarrier vector <NUM> is a <NUM>-element vector [+<NUM> -<NUM> -<NUM> +<NUM>]T. The inner and outer subcarrier combining operation <NUM> then yields a <NUM>-element full subcarrier vector <NUM> in which the <NUM>-phase inner subcarrier vector <NUM> is replicated over <NUM> clusters, each of which is modulated by one element of the outer subcarrier vector <NUM>. Beacon ℓ generated by the multitone modulator <NUM> then has complex baseband representation <MAT> at the PA input in the beacon transmission operations <NUM>, where the subcarrier frequencies are given by <MAT> and where <MAT> is the subcarrier spacing. Assuming the internal subcarrier structure shown here, then sT(ℓ)(t) = sT<NUM>(ℓ)(t)sT<NUM>(ℓ)(t), where <MAT> and where inner and outer subcarrier frequencies f<NUM>(k<NUM>) and f<NUM>(k<NUM>) are given by <MAT> <MAT> respectively, such that Ksub = K<NUM>K<NUM>, f(K<NUM>k<NUM> + k<NUM>) = f<NUM>(k<NUM>) + f<NUM>(k<NUM>) and θT(ℓ)(K<NUM>k<NUM> + k<NUM>) = θT<NUM>(ℓ)(k<NUM>) + θT<NUM>(ℓ)(k<NUM>).

This beacon can be interpreted as a stacked-carrier spread spectrum (SCSS) signal, in which a narrowband signal sT<NUM>(ℓ)(t) with bandwidth K<NUM>fsym and period <NUM>(K<NUM>=<NUM>)mod2Tsym is spread by a wideband signal sT<NUM>(ℓ)(t) with bandwidth Ksubfsym and period <NUM>(K<NUM>-<NUM>)mod <NUM> Tsym/K<NUM>. The resultant beacon possesses massive spectral redundancy, both between clusters (replication of inner code ST<NUM>(ℓ) over K<NUM> independent clusters), and within clusters (replication of outer code ST<NUM>(ℓ) over K<NUM> subcarriers within each cluster).

<FIG> further illustrates the additional temporal redundancy in the beacon over each symbol repetition. If the time symbol possesses an even number of subcarriers, then the subcarriers will be offset from the frequency origin by a factor of fsym/<NUM>, similar to LTE SC-FDMA uplink signals. In this case, successive time symbol repetitions needed to be inverted to preserve signal energy within each subcarrier. If the time symbol possesses an odd number of subcarriers, then this offset is removed, similar to LTE OFDM downlink signals, and this successive inversion is not needed. The replication induces temporal redundancy in either case.

Table <NUM> lists exemplary beacon generation and transmission parameters compatible with TOA and FOA ranges expected for Class-<NUM> sUAS's, and for network geometries shown in <FIG>. The key parameter providing compatibility with Class-<NUM> sUAS's is the <NUM> symbol duration Tsym, which encompasses the full range of TOA's shown in <FIG>, driven by the maximum altitude allowed for Class-<NUM> sUAS's. The resultant <NUM> subcarrier frequency separation is also compatible with the range of FOA's shown in <FIG>, driven by the maximum airspeed allowed for Class-<NUM> sUAS's. In each case, the beacon bandwidth fits into the channel allocation for each band. The <NUM>-element inner subcarrier vector <NUM> dimension chosen for the <NUM> Ch. <NUM> Scenario allows up to <NUM> co-channel beacons to be separated using linear-algebraic signal separation methods. The <NUM>-element inner subcarrier vector <NUM> dimension chosen for the LMS Scenario - driven largely by the much narrower <NUM> bandwidth constraint in this subband - only allows up to <NUM> co-channel beacons to be separated using linear-algebraic methods, hence a sparser set of co-channel beacons can be deployed. The narrower bandwidth and sparser deployment degrades the geolocation capability of this network. However, the higher transmit power requirement compensates for much of this performance loss. It should be noted that correlative techniques would be strongly interference-limited in this case, and could not exploit any of the power advantage available in this subband.

Assuming <NUM> bits per in-phase (I) and quadrature (Q) rail at the output of the quantizer <NUM>, a single time symbol vector <NUM> requires transmission of <NUM>,<NUM> bytes (<NUM> KB) over the beacon communication bus <NUM> for the LMS Scenario, and <NUM> KB for the <NUM> Ch. <NUM> Scenario. Assuming the time symbol vectors <NUM> are updated once per second, the NOC <NUM> requires the beacon communication bus <NUM> to support a <NUM> kbps link for the LMS scenario, and a <NUM> kbps link for the <NUM> Ch. <NUM> Scenario. These rates are achievable in low-cost networks.

<FIG> show the frequency and time response of two <NUM> Ch. <NUM> beacons with parameters consistent with Table <NUM>. <FIG> shows the flat beacon frequency response over the <NUM> signal passband, with negligible sidebands. <FIG> shows the close-in spectral distribution of the beacons - essentially, a line spectrum with <NUM> line separation and flat line power - and illustrates the layout of the <NUM> clusters within the signal. <FIG> show the instantaneous power of the beacon, and demonstrates the effect of optimized code phasings on the transmitted signal, one of the features of the disclosure. In <FIG>, the beacon is generated using inner subcarrier vector <NUM> and outer subcarrier vector <NUM> code phases with uniform-random phase distribution. This beacon has a PAPR of <NUM> dB, consistent with a bandlimited complex-Gaussian waveform. In <FIG>, the other beacon is generated using <NUM>-element inner subcarrier vector <NUM> and <NUM>-element outer subcarrier vector <NUM> code phases taken from <NUM>,<NUM>-member code libraries <NUM>, in which the phases are optimized to minimize kurtosis of their underlying time series. This beacon has a PAPR of <NUM> dB.

<FIG> shows a receiver system used in one aspect of the disclosure. When directed to collect a snapshot, e.g., based on prompts from the NOC <NUM> over a wireless communication transceiver <NUM>, or at scheduled snapshot collection times, the receiver system performs reception operations <NUM> to generate a downconverted and sampled data stream covering the beacon transmission band. The reception operations shown in <FIG> can comprise optional bandpass filtering (BPF), followed by low-noise amplification (LNA), in-phase and quadrature (IQ) downconversion to complex baseband, represented as multiplication of the LNA output signal by a complex LO output signal, dual lowpass filtering (LPF), and dual analog-to-digital conversion (ADC) operations, resulting in a complex IQ sampled received data stream. However, the reception operations <NUM> can be implemented in many other ways, including two-stage superheterodyne reception operations <NUM> that downconvert the beacon transmission band to real-IF representation, and hybrid analog-digital reception operations <NUM> that downconvert a larger frequency band containing the beacon transmission band, and implements a digital drop receiver to generate a reduced-bandwidth, decimated signal centered on the beacon transmission band.

When needed at scheduled intervals, or given prompts from the NOC <NUM> over a wireless communication transceiver <NUM>, the receiver system then performs a data snapshot collection <NUM> operation, which generates a snapshot <NUM> comprising the data provided by the reception operations <NUM> at a reception time and over a snapshot <NUM> time duration, shown in <FIG> as time center <MAT>, and time duration TR also included as a time-stamp with that snapshot <NUM>. In aspects with slotted beacon formats, a prefix and suffix with duration Tprefix and Tsuffix, respectively, are also collected as part of the snapshot <NUM>, in order to encompass inter-slot interference introduced by timing offset between the beacon transmitter and receiver. The snapshot <NUM> is then sent to a position/timing (P/T) solution generator <NUM> over a snapshot communication bus <NUM>. In the network shown in <FIG>, the snapshot communication bus <NUM> can be connected to the NOC <NUM> over a wireless communication transceiver <NUM>, or it can connect to a P/T solution generator <NUM> on-board the user <NUM>. In the networks shown in <FIG> and <FIG>, the snapshot communication bus <NUM> is connected to the NOC <NUM> over a link similar to links supporting the beacon communication bus <NUM>. However, the snapshot communication bus <NUM> typically requires a higher-rate link than the beacon communication bus <NUM>.

In general, the receiver LO(s) and ADC samplers used in the reception operation <NUM> are locked to a system clock <NUM> with rate offset εR and timing offset <MAT>, unique for each receiver, where <MAT> is the receiver time estimate at UTC time <MAT>.

In some aspects of the disclosure, the NOC <NUM> also provides synchronization data <NUM> that can be used to bring the receiver system clock <NUM> into synchronization for subsequent time-stamped snapshots <NUM>. In other aspects, the receiver obtains coarse synchronization information from the NOC <NUM> over a wireless communication transceiver <NUM>. In additional aspects of the disclosure, the receiver performs coarse synchronization operations to determine the approximate center frequency and (for slotted beacon formats) slot transition time of the beacons, prior to the snapshot collection <NUM>. The coarse frequency and timing information can then be used to adjust the timing carrier offset of the ADC output signal, or the receiver clock driving the LO and ADC; or simply conveyed to the NOC <NUM>, along with time-stamped snapshot <NUM>. In this last case, the frequency and timing offset is included in the time-stamped snapshot <NUM>, for use by the P/T solution generator <NUM>. The receiver may stream data to the P/T solution generator <NUM> (which may be remote or on the receiver platform itself), or may sparsely capture time-stamped snapshot <NUM> of frequency-and-timing aligned data, e.g., at the start of processing or as required/requested by the NOC <NUM>.

<FIG> shows exemplary cold-start operations used at the P/T solution generator <NUM> to resiliently detect and estimate geo-observables of co-channel beacons contained within individual received snapshots <NUM>, applicable to the network shown in <FIG>. The snapshot <NUM> is first processed to estimate the FOA centroid <NUM>, described in <FIG>, i.e., the center of the beacon FOA's observed in the downconverted snapshot <NUM>, which is roughly offset by the error between the target and actual LO in the reception operations <NUM>. This can be accomplished in a number of manners; for example, using spectral analysis tools to determine the center of the received beacon spectrum, or to exploit the nearly-rectangular shape of the beacon signals, e.g., by detecting and estimating the up-edge and down-edge of the signal spectrum. The chief goal of this step is to determine the FOA centroid <NUM> to within a fraction of a subcarrier, e.g., ±<NUM> (±<NUM>/<NUM> subcarrier) for the beacon transmission parameters given in Table <NUM>. In some aspects with slotted beacon formats, the snapshot <NUM> is optionally analyzed to estimate a timing offset estimate n̂R, e.g., by performing cross-slot time correlation operations to detect the slot transition.

The snapshot <NUM> is then frequency-shifted to remove the estimated FOA centroid <NUM>, and if needed time-shifted to remove the estimated timing offset, and channelized into subcarriers and time symbols covering the active snapshot bandwidth and duration <NUM>, creating a channelized snapshot <NUM>, described in <FIG>, defined over subcarriers and time symbols. The channelized snapshot <NUM> is then stacked into an Ndata × MDoF windowed data matrix <NUM>, described in <FIG>, where MDoF is the degrees-of-freedom (DoF) of the data matrix and Ndata is the number of data samples in the matrix, and where the beacons are redundant in the DoF dimension; and whitened over the DoF dimension, e.g., using a QR decomposition (QRD) operation <NUM>, thereby creating a whitened snapshot matrix <NUM>, described in <FIG>. In some aspects, FOA clutter and FOA centroid computation operations <NUM> are performed to generate a FOA clutter spectrum <NUM>, described in <FIG>, and used to improve detection of the received beacons, and optionally to improve the estimate of the FOA centroid <NUM>. In this case, the improved FOA centroid <NUM> estimate can also be used to regenerate <NUM> the channelized snapshot <NUM> and the whitened snapshot matrix <NUM>.

The whitened snapshot matrix <NUM> is then passed through an FFT/IFFT mechanized resilient least-square (LS) search operation <NUM> to form a least-squares (LS) TOA-FOA surface <NUM>, described in <FIG>, which provides a detection statistic as a function of candidate TOA and FOA values, and to search the LS TOA-FOA surface <NUM> to detect each beacon in the whitened snapshot matrix <NUM>, using known data dimension codes <NUM> applied during beacon time-symbol generation operations shown in <FIG>, and to determine the observed TOA and FOA, and LS SINR (estimation quality) of the detected beacons. The maximizing TOA and FOA, and the maximal LS SINR are then refined <NUM>, for example, using polynomial fit to the surface peak or parametric search operations in the vicinity of the detected TOA-FOA surface peak. Optionally, a copy-aided parametric estimation method <NUM> is used to further refine TOA and FOA geo-observables and resolve detector ambiguities <NUM>, using known DoF dimension codes <NUM> applied during beacon generation operations. In some aspects, results of the copy-aided parametric estimation method <NUM> is used to further improve estimates of the FOA centroid <NUM>, and optionally the estimate of the timing offset, and the channelized snapshot <NUM> is regenerated <NUM> using the improved FOA centroid <NUM> and optional timing offset estimates, to further improve the geo-observable estimates.

The DoF and data dimensions, and the data dimension codes <NUM> and DoF dimension codes <NUM> are set based on the particular form of redundancy exploited in the FFT/IFFT mechanized resilient least-square (LS) search operation <NUM>, as determined by the stacking operation performed in the data stacking and whitening operation <NUM>. For example, in one aspect where the channelized snapshot is stacked over the inner-code dimension, data dimension Ndata = K<NUM>Nsym, and the data dimension codes <NUM> are the K<NUM>×<NUM> outer subcarrier vector <NUM> phases <MAT>; and DoF dimension MDoF = K<NUM> and the DoF dimension codes <NUM> are the K<NUM>×<NUM> inner subcarrier vector <NUM> phases <MAT>. In a second aspect where the channelized snapshot is stacked over the outer-code dimension, data dimension Ndata = K<NUM>Nsym, and the data dimension codes <NUM> are the K<NUM>×<NUM> inner subcarrier vector <NUM> phases <MAT>; and DoF dimension MDoF = K<NUM> and the DoF dimension codes <NUM> are the K<NUM>×<NUM> outer subcarrier vector <NUM> phases <MAT>.

The key components of this procedure are described in more detail in the next subsections.

<FIG> depicts exemplary operations used to implement the FOA centroid <NUM> and optional timing estimate removal and channelization operations <NUM> in one aspect. The system first performs a FOA centroid and optional timing offset removal operation <NUM>, to remove the estimated FOA centroid <NUM> α̂R and optional estimated timing offset n̂R from the snapshot <NUM>. In some aspects with slotted beacon formats, the prefix and suffix are also discarded prior during the FOA centroid and optional timing offset removal operation <NUM>. The snapshot <NUM> then has duration TR = NrepTsym = NRTADC, where Nrep and NR are integers and TADC = <NUM>/fADC is the ADC sampling period in the receiver's frame of reference.

The snapshot <NUM> is then separated into frequency subcarriers and time symbols <NUM> using a sparse, overlapped, optionally frequency-offset, windowed DFT with overlap time Tsym = NADCTADC, sparsity factor Qsym, DFT length NDFT = QsymNADC, and channelizer window <MAT>, and with frequency offset factor fsym/<NUM> if Ksub is even. The DFT out-put bins corresponding to the active beacon subcarrier frequencies are then selected, and a snapshot equalizer operation <NUM> is applied to those bins. These operations generates a channelized snapshot <NUM> xsub (ksub,nsym) given by <MAT> <MAT> where Nsym = Nrep - Qsym +<NUM> is the number of time symbols in the channelized snapshot and <MAT> is the symbol nsym DFT time-center in the receiver's field of reference, and where GR(ksub,nsym;α̂R) are snapshot channelizer equalizer <NUM> weights that remove effects caused by at least the carrier operation <NUM>, and optionally filtering effects of the reception operations <NUM>. Preferentially, the snapshot equalizer <NUM> weights are given by <MAT> where HR(f) is the aggregate frequency response of the reception operations <NUM>, and where δR(ksub,nsym;α̂R) removes dispersive effects of the FOA centroid <NUM> removal operation <NUM>, <MAT> The aggregate frequency response term is optional, and can be based on modeling of the reception operations <NUM>; or derived from calibration operations performed by the receiver or network, for example, as described in Pattabiraman <NUM>, and can be based on the magnitude or complex value of those reception operations <NUM>.

Table <NUM> lists receiver and channelizer parameters compatible with the beacon generation and transmission parameters shown in Table <NUM>. The receiver assumes a dual-ADC sampling rate of <NUM> million samples per second (Msps) for the LMS Scenario, and <NUM> Msps for the <NUM> Ch. <NUM> Scenario, with sufficient antialiasing filtering to provide a <NUM> and <NUM> protected two-way passband, respectively, covering the active bandwidth of the beacons with ±<NUM> and ±<NUM> of guard band for LO uncertainty, respectively. A mixed-radix DFT with factor-of-four sparsity (Qsym = <NUM>) is assumed in both scenarios, and a separation of <NUM> between successive DFT's. The Table further assumes a <NUM> millisecond snapshot encompassing <NUM> symbol repetitions, <NUM> of which are used in subsequent geo-observable estimation operations, for both scenarios.

Assuming dual-ADC precision of <NUM> bits per I and Q rail, consistent with a low-cost receiver front-end, the size of each snapshot is <NUM> KB for the LMS Scenario, and <NUM> KB for the <NUM> Ch. <NUM> Scenario. Assuming a snapshot is collected <NUM> once per second, backhaul <NUM> of ADC output data to the P/T solution generator requires a snapshot communication bus <NUM> that can support a <NUM> Mbps one-way data-rate for the LMS Scenario, and a <NUM> Mbps one-way data-rate for the <NUM> Ch. <NUM> Scenario, well within capabilities of <NUM> cellular or <NUM> WLAN standards if the P/T solution generator <NUM> is in the NOC <NUM>. Continuous backhaul of snapshots <NUM> to the P/T solution generator <NUM> over the snapshot communication bus <NUM> would require a factor of <NUM> higher data-rate, e.g., <NUM> Mbps and <NUM> Mbps, respectively, for the two scenarios, easily accomplished over Gbps Ethernet if the P/T solution generator <NUM> is on-board the user <NUM>.

The FOA centroid <NUM> and optional timing estimate removal and channelization operations <NUM> can be performed in a number of different manners, for example, using polyphase filtering methods, discrete filter banks centered on each subcarrier frequency, mixtures of radix-<NUM> and non-radix-<NUM> fast Fourier transform (FFT) and inverse-FFT methods, and so on.

<FIG> shows the frequency response of an exemplary DFT window developed for the LMS Scenario using an interpolated Parks-McClellan algorithm. The window passband is set to <NUM>, corresponding ±<NUM> FOA offset expected in this band at the maximum class-<NUM> sUAS airspeed of <NUM> meters/second (<NUM> miles/hour), with ±<NUM> carrier offset margin after FOA centroid <NUM> removal operations. The window stopband is set to <NUM>, sufficient to reject any inter-subcarrier interference due to FOA offset from the receiver center frequency. As shown in this FIG. , the window provides <NUM> dB rejection within its design passband (much less within ±<NUM>), and over <NUM> dB of rejection within its design stopband. The DFT window designed for the <NUM> Ch. <NUM> Scenario has nearly identical performance.

Assuming synchronized beacon transmitters in the network shown in <FIG>, such that tT(ℓ)(tUTC) ≡ tUTC, and defining <MAT> as the actual UTC time (to be estimated in positioning/timing algorithms) at receiver reference time <MAT>, and further assuming short snapshots <NUM> are collected <NUM>, then the channel link gain between the beacon transmitter deployed at network node <NUM>ℓ and a receiver deployed at a user <NUM> is approximated by <MAT>, and the TOA and FOA of the beacon received at that user <NUM> is approximated by <MAT> <MAT> where <MAT> and <MAT> are the TOA and differential TOA (DTOA) of beacon ℓ at time <MAT>, and where pT(ℓ)R(tUTC) = pT(ℓ) - pR(tUTC) is the observed position of network node <NUM>ℓ at the user <NUM>, and uT(ℓ)R(tUTC) = pT(ℓ)(tUTC)/∥pT(ℓ)R(tUTC)∥<NUM> is the observed line-of-bearing (LOB) from the user <NUM> to network node <NUM>ℓ. The TOA, DTOA, and FOA observed at the ADC sampler in the user <NUM> reception operations <NUM> are then given by <MAT> <MAT> <MAT> <MAT> Further assuming accurate equalization of user <NUM> reception operations <NUM>, and ideal suppression of inter-subcarrier interference by the channelizer window used in the FOA centroid <NUM> and optional timing estimate removal and channelization operations <NUM>, channelized snapshot <NUM> xsub(ksub,nsym) is approximated by <MAT> <MAT> where asub(ℓ) is the end-to-end beacon ℓ channelizer output gain, <MAT> <MAT> and where dsub(ksub,nsym;τ,α;ℓ) is the network node <NUM>ℓ beacon ("beacon ℓ") at candidate observed TOA τ and observed FOA α, <MAT> and <MAT> is the analytic discrete Fourier transform of the channelizer window. Assuming additive white Gaussian noise (AWGN) with noise density N<NUM> at the LPF input and ideal LPF equalization, the background interference isub(ksub,nsym) has identical power <MAT> on each channelizer output subcarrier, and beacon ℓ has identical SNR γT(ℓ)R ≈ |asub(ℓ)|<NUM>/Risubisub on each channelizer output subcarrier.

Using the channelized snapshot <NUM> model given in (Eq16)-(Eq20), the observed geo-observables <MAT> can in principle be estimated by correlating channelized snapshot <NUM> xsub(ksub,nsym) against <MAT>. Moreover, if fT ≫ <MAT>, then the second dispersive term in (Eq20) can be ignored, and this correlation can be efficiently mechanized using DFT and inverse-DFT (IDFT) methods. However, at high receive SNR this correlation will yield a poor result, or will require a high time-bandwidth product NsymKsub to remove cross-correlation between co-channel beacons. This problem can be overcome by exploiting the spectral or temporal redundancy imposed in this signal at the transmitter, to implement resilient TOA-FOA estimators. This procedure is described below.

<FIG> depicts one aspect employing resilient TOA-FOA estimation, referred to here as the fine least-squares procedure. The channelizer snapshot <NUM> is first stacked along it's inner code dimension <NUM>, generating K<NUM>×<NUM> vector signal x<NUM>(k<NUM>,nsym) = <MAT>, enumerated over K<NUM> outer subcarrier channel indices k<NUM> = <NUM>,. ,K<NUM> - <NUM> and Nsym time symbol indices nsym = <NUM>,. ,Nsym-<NUM>. Using (Eq16)-(Eq20), this signal can be modeled as <MAT> where <MAT> are the K<NUM>×<NUM> inner-code stacked beacon signal vectors, respectively, <MAT> <MAT> and where a<NUM>(τ;ℓ) and d<NUM>(k<NUM>,nsym;τ,α;ℓ) is the K<NUM>×<NUM> beacon ℓ outer-code spectral signature at trial TOA τ and scalar inner-code signal at trial TOA τ and FOA α, respectively, <MAT> <MAT> and the dispersive terms δ<NUM>(nsym;α) and δ<NUM>(k<NUM>,nsym;α) are given by <MAT> <MAT> respectively. If <MAT>, then δ<NUM>(nsym;α̃T(ℓ)R-αR) ≈ <NUM>K<NUM> and s<NUM>(k<NUM>,nsym;ℓ) is nondispersive over the inner-code dimension. Similarly, if fT ≫ <MAT>, then δ<NUM>(k<NUM>,nsym;α) ≈ <NUM> and (Eq23) holds closely. Assuming AWGN background noise and ideal LPF equalization, the K<NUM>×<NUM> background interference vector <MAT> has asymptotic autocorrelation matrix (ACM) Ri<NUM>i<NUM> → RisubisubIK<NUM> on each inner subcarrier channel.

This model is closely analogous to a multi-element antenna array with MDoF degrees-of-freedom (DoF's), where MDoF = K<NUM> is the stacking or "DoF" dimension. Similar to an array, it allows the outer-code signals to be detected and separated with an output (despread) SINR approximated by γT(ℓ)R ~ (MDoF-L+<NUM>)γT(ℓ)R for in presence of strong interference from co-channel beacons (γT(ℓ')R » <NUM>, where ℓ' ≠ ℓ), using well-known, mature linear signal separation methods, e.g., least-squares (LS) algorithms, referred to as code nulling in Agee <NUM>. The method sacrifices one despreader DoF to null each strong signal in the environment, and uses the despreader's remaining DoF's to improve the output SNR of the intended signal. It also admits superresolution geo-observable estimators with accuracy that scales with this output SINR.

The inner-code stacked signal is then formed into K<NUM>Nsym×K<NUM> windowed data matrix <NUM> <MAT>, where tapering windows <NUM> <MAT> and <MAT> satisfy ΣwTOA(k<NUM>) = ΣwFOA(nsym) = <NUM>, and passed through a whitening operation <NUM>, to generate whitened data matrix <NUM> Q<NUM> = <MAT>. The whitening operation <NUM> can be performed, for example, using QR decomposition (QRD) {Q<NUM>,R<NUM>} = QRD(X<NUM>), given by <MAT>, where R<NUM> is an upper-triangular matrix given by <MAT>, and where chol{•} is the Cholesky factor operation. The QRD can be implemented using a number of efficient methods known to those of ordinary skill in the arts, e.g., modified Gram-Schmidt orthogonalization (MGSO). Other whitening operations, e.g., singular value decomposition, can also be used to whiten X<NUM>. The whitened data matrix <NUM> is then used to form SINR-revealing FLS TOA-FOA surface <NUM> γ̂FLS(τ,α;ℓ), by computing intermediate K<NUM>×<NUM> FLS FOA vector <NUM> <MAT> on each inner subcarrier channel using an DFT bank <NUM>; computing K<NUM>×<NUM> whitened FLS linear combiner vector <MAT> for each candidate FOA and data-dimension code <NUM>, for this surface the outer-code, using an IDFT bank; and computing SINR-revealing FLS TOA-FOA surface <NUM> <MAT> Optionally, the FLS FOA vector <NUM> also yields FLS FOA clutter spectrum <NUM> γ̂FLS-clutter(α) = η̂FLS-clutter(α)/(<NUM>-η̂FLS-clutter(α)), where <MAT> which can be used to compute FLS deflection statistic d̂FLS(τ,α;ℓ) = γ̂FLS(τ,α;ℓ)/γ̂FLS-clutter(α), a particularly useful statistic in TOA-FOA spectra containing multiple significant peaks, e.g., due to specular multipath. In some aspects, the clutter statistic is also used to improve the FOA centroid <NUM>, e.g., using formula <MAT> which can be used to regenerate the channelized snapshot <NUM>.

In absence of substantive multipath, the TOA-FOA estimate is then given by<MAT> and the maximal TOA-FOA surface <NUM> value is a metric of the SINR of the FLS combiner output signal at the estimated TOA and FOA, γ̂FLS(ℓ) = γ̂FLS(τ̂T(ℓ)R,α̂T(ℓ)R-α̂R;ℓ). The inner-stacked spectral signature is optionally estimated by â<NUM>(ℓ) = <MAT> where <MAT>. The TOA and FOA error variances are further optionally estimated by <MAT>, where <MAT> <MAT> and where the lower bounds in (Eq34)-(Eq35) are achieved for flat tapering windows <NUM>. Moreover, the background TOA-FOA surface <NUM> values are a factor of <MAT> below γ̂FLS(ℓ), where the lower bound is also achieved for flat tapering windows <NUM>. For this reason, flat tapering windows <NUM> are recommended in absence of channel multipath, and shaped tapering windows <NUM> recommended if multiple TOA-FOA surface <NUM> peaks are expected, e.g., due to strong specular multipath.

The whitening operation <NUM> requires the DoF's of x<NUM>(k<NUM>,nsym), MDoF = K<NUM>, be substantively larger than the number of inner-code stacked signal vectors, Ndata = K<NUM>Nsym. If the tapering windows <NUM> are rectangular, the SINR-revealing metric γ̂FLS(ℓ) can be optionally converted to unbiased SINR estimate <MAT> which holds closely if the interference is i. complex-Gaussian over the outer-code dimension.

Assuming uniform FOA spacing α(kFOA) = (kFOA/KFOA)fsym, (Eq28) can be computed using K<NUM>K<NUM> = Ksub Nsym : KFOA efficient DFT operations. Similarly, assuming uniform TOA spacing τ(nTOA) = (nFOA/NTOA)Tsym/K<NUM>, (Eq29) can be computed using K<NUM>KFOAL K<NUM> : MTOA efficient inverse-DFT (IDFT) operations. These operations are both highly regular and parallelizable, allowing their implementation using efficient FPGA or general-purpose GPU (GPGPU) computation modules.

The FLS TOA-FOA estimates given in (Eq33), and the FLS SINR, are further refined using local search methods in the vicinity of the maximizing FLS TOA-FOA surface value <NUM>. Simple methods for accomplishing include polynomial fit to the surface peak, e.g., using two-dimensional quadratic fit over the nearest neighbors to the maximizing surface grid location. Optionally, parametric search operations that exploit the fully-dispersive form of d<NUM>(k<NUM>,nsym;τ,α;ℓ) given in (Eq25) multiplied by δ<NUM>(k<NUM>,nsym;α) in (Eq27), can be used. In one aspect, Newton and Gauss-Newton recursions are defined over the symbol-normalized TOA-FOA vector <MAT> Defining <NUM>×<NUM> symbol-normalized frequency-time vector <MAT> and K<NUM>Nsym×<NUM> symbol-normalized matrix <MAT>, the recursion is given by <MAT> <MAT> <MAT> <MAT> <MAT> <MAT> <MAT> <MAT> The final TOA and FOA are then given by τ̂T(ℓ)R = (υ<NUM>(ℓ))<NUM>Tsym/K<NUM> and α̂T(ℓ)R = (υ̂<NUM>(ℓ))<NUM>fsym, and the FLS SINR estimate is given by γ̂FLS(ℓ) = <MAT>.

Equation (Eq33) shows that the FLS TOA-FOA spectrum <NUM> possesses TOA and FOA ambiguity Tsym/K<NUM> and fsym, respectively. In some aspects, this ambiguity is resolved using copy-aided parameter estimation methods <NUM> that exploit the model of a<NUM>(τ;ℓ) given in (Eq24). In one aspect, the copy-aided ambiguity resolution algorithm is given by <MAT> <MAT> <MAT> <MAT> <MAT> where uFLS(ℓ) is given by (Eq40), and where n̂zone(ℓ) and k̂tile(ℓ) are the TOA zone and FOA tile containing the beacon ℓ detection, respectively. The full TOA and FOA geo-observables are then given by <MAT> and α̂T(ℓ)R ← α̂R + α̂T(ℓ)R + k̂tile(ℓ)fsym, respectively. If k̂tile(ℓ) ≠ <NUM> for any detected beacon, then the FOA centroid <NUM> α̂R is optionally recomputed, e.g., using weighted estimate α̂R ← <MAT>, and the channelized snapshot <NUM> is regenerated <NUM> and the subsequent FLS geo-observable estimation operations shown in <FIG> are repeated with the new FOA centroid <NUM> estimate.

The copy-aided ambiguity estimator also provides complex gain estimate <MAT> which can be used to compute the beacon ℓ channelizer output power and phase offset. These parameters provide key inputs for channel calibration operations, e.g., to determine transmit and receive carrier phase, and true channel pathloss, for subsequent network calibration operations.

Table <NUM> lists FLS surface generation parameters usable in the exemplary UTM and IIoT scenarios described here. The degrees of freedom are large enough to separate all of the beacons in the users' FoV's for each scenario, which a roughly factor-of-two excess to account for multipath reflections. In each case, the number of data entries is a large enough multiple of the despreader DoF's to yield a stable QRD and FLS estimate.

Table <NUM> summarizes complexity of the channelization and FLS surface generation operations for the two scenarios, assuming one real multiply-and-add per operation, and assuming the whitening operation <NUM> is performed using a QRD instantiated using Modified Gram-Schmidt Orthogonalization (MGSO). Of these operations, the QRD uses than <NUM>% of the total operations in each scenario. The complexity is well within the capabilities of modern DSP gear, even for the <NUM> Ch. <NUM> Scenario. Moreover, the channelization and DFT/IDFT operations are easily implemented in FPGA (e.g., Xilinx 7K325T or higher devices) or using general-purpose GPU's (GPGPU's).

Table <NUM> summarizes memory requirements of the FLS surface generation procedure for the two scenarios. The memory requirements assuming <NUM>-bit data precision at each stage of processing, and in-place QRD operations. The channelization operation imposes the bulk of memory requirements for each scenario, and is not particularly onerous in any case. The channelization memory is also well within capability of modern DSP, FPGA (e.g., Xilinx 7K325T or higher devices), or GPGPU's.

<FIG>, <FIG> show the FLS TOA-FOA surface <NUM> obtaining for the LMS Scenario and <NUM> Ch. <NUM> Scenario, respectively, and for the surface instantiation parameters given in Table <NUM>. Beacon <NUM> (out of <NUM> beacons searched, and <NUM> beacons detected, i.e., every network node <NUM> in the FoV of User <NUM><NUM>) is chosen for the LMS Scenario, and Beacon <NUM> (out of <NUM> beacons searched, and <NUM> beacons detected, i.e., every network node <NUM> in the FoV of User <NUM><NUM>) is chosen for the <NUM> Ch. <NUM> Scenario, because they each provide a median peak value over all the beacons detected during their respective TOA-FOA searches.

<FIG> shows the exemplary LMS Scenario FLS TOA-FOA surface <NUM> over its full range, generated using the simplified outer-code signal model given in (Eq25). The FLS TOA-FOA surface <NUM> also shows detection of TOA zone <NUM>, using the copy-aided TOA-FOA ambiguity resolution method <NUM> described in (Eq47)-(Eq51). The FOA clutter spectrum <NUM> is clearly visible in the full TOA-FOA surface <NUM>, as well as a single TOA-FOA peak obtained at a <NUM> dB estimated SINR, or <NUM> dB above the background clutter. No other peaks associated with any other network node <NUM> are visible, despite the fact that <NUM> network nodes <NUM> are in the user's <NUM> FoV.

<FIG> shows results of a <NUM>×<NUM> parametric local search <NUM> centered on the peak of the full FLS TOA-FOA surface <NUM> shown in <FIG>, and generated using the extended outer-code signal model given by (Eq25), multiplied by dispersive term (Eq27). The benefit of the extended outer-code signal model is evident in the maximal peak value of <NUM> dB - a <NUM> dB increase over the full TOA-FOA surface <NUM>, and <NUM> dB above the FOA clutter spectrum <NUM>. The TOA-FOA error for this case is (<NUM> ps, -<NUM>), comparable to the (<NUM> ps, <NUM>) <NUM>th percentile CRB for this case, showing that the method is adhering well to expected performance.

<FIG> shows the exemplary <NUM> Ch. <NUM> FLS TOA-FOA surface <NUM> over its full range, generated using the simplified outer-code signal model given in (Eq25). The FLS TOA-FOA surface <NUM> also shows detection of TOA zone <NUM>, using the copy-aided TOA-FOA ambiguity resolution method <NUM> described in (Eq47)-(Eq51). The FOA clutter spectrum <NUM> is clearly visible in this surface, as well as a single TOA-FOA peak obtained at an <NUM> dB estimated SINR, <NUM> dB above the background clutter. No other peaks are visible, despite the fact that <NUM> network nodes <NUM> are in the user's <NUM> FoV.

<FIG> shows a <NUM>×<NUM> parametric local search <NUM>, centered on the peak of the full FLS TOA-FOA surface <NUM> shown in <FIG>, and generated using the extended outer-code signal model given by (Eq25), multiplied by dispersive term (Eq27). The benefit of the extended outer-code signal model is also evident in the maximal peak value of <NUM> dB - a <NUM> dB increase over the full FLS TOA-FOA surface <NUM>, and <NUM> dB above the FOA clutter spectrum <NUM>. The TOA-FOA error for this case is (-<NUM> ps, <NUM>), comparable to the (<NUM> ps, <NUM>) <NUM>th percentile CRB for this case, showing that the method is also adhering well to expected performance in this Scenario.

Other aspects of the disclosure employ similar operations using alternate stacking methods. These include outer-code stacking, which transforms xsub(ksub,nsym) into K<NUM>×<NUM> outer-code stacked vector <MAT>, and symbol stacking, which transforms xsub(ksub,nsym) into Nsym×<NUM> symbol-stacked vector xsym(ksub,nsym) = <MAT>. Each stacking operation has advantages and disadvantages. Inner-code stacking results in a TOA estimate τ̂T(ℓ)R ≈ τT(ℓ)Rmod(Tsym/K<NUM>), i.e., with range aliased between <NUM> and Tsym/K<NUM>, but with high precision within that range. For this reason it is referred to here as the fine least-squares (FLS) estimator. Conversely, outer-code stacking yields TOA estimate τ̂T(ℓ)R ≈ τT(ℓ)RmodTsym, i.e., with range aliased to full range between <NUM> and Tsym, but with precision that is a factor of K<NUM> coarser. It is referred to here as the coarse least-squares (CLS) estimator. Both estimators provide full range and precision in FOA. The symbol-stacked estimator provides full range and precision in TOA, but no estimate of FOA estimate. In all cases, these issues are resolvable using copy-aided post-processing methods <NUM>.

<FIG> shows the geo-observable based positioning and timing procedure disclosed here, applicable to at least the network shown in <FIG>, and describes the common operations used in each stage. The procedure optimizes maximum-likelihood (ML) objective function <MAT> where FTOA(pR,τR) and FFOA(pR,vR,αR) are the TOA-only and FOA-only ML estimators, respectively, <MAT> <MAT> and where λT = fT/c is the nominal signal-in-space wavelength, <MAT> and <MAT> are given in (Eq34) and (Eq35), respectively. In the aspect described here, the networks nodes <NUM> are assumed to be fixed and have known positions, such that network node <NUM> position pT(ℓ)(tUTC) ≡ pT(ℓ) and velocity vT(ℓ)(tUTC) ≡ <NUM>. If the network nodes' <NUM> system clocks <NUM> are synchronized to UTC, then the user <NUM> system clock <NUM> and LO offsets in the reception operations <NUM> are given by <MAT> and αR = -fTεR/(<NUM>+εR), respectively, allowing the UTC time at known receive reference time <MAT> and the clock rate offset εR to be derived from the observed estimates, and the user <NUM> position and velocity being estimated by the method are given by <MAT> and <MAT>, respectively.

In some aspects, the network nodes' <NUM> system clocks <NUM> are not fully synchronized to UTC, but network node <NUM> timing offsets <MAT> and rate offsets <MAT> from UTC have been estimated, e.g., using network calibration procedures. In this case, the detected TOA's and FOA's are adjusted to compensate for these offsets. In one aspect, this is performed by setting <MAT> <MAT> prior to computation of the ML estimator. It should be noted that (Eq57) is not exact, as it fails to include division of the second term by <NUM>+εR as shown in (Eq14). However, this effect is minor for user <NUM> system clocks <NUM> with < <NUM> ppm rate offset, and can be removed in subsequent refinements.

Estimating and concentrating τR and αR estimates out of (Eq54)-(Eq55) yields <MAT> <MAT> <MAT> <MAT> Introducing intermediate parameters <MAT> and ũT(ℓ)R = uT(ℓ)R -<IMG> , the velocity is further concentrated out of (Eq61), yielding <MAT> <MAT> The concentrated TOA and FOA objective functions can then be used to find all of the user <NUM> positioning and timing parameters, by conducting a search over position pR alone.

Once the beacon geo-observables, and SINR's have been estimated, and the beacons have been detected or detection failure has been logged <NUM>, a three-stage procedure is used to jointly geolocate the sUAS, and to determine its timing and carrier offset from the beacon network. In the first stage, a coarse areal search is carried out over the entire network geography <NUM>, using the known position of the beacons and optional timing and rate offset estimates <NUM>. Next, a fine areal search is carried out at the optimal search point determined during the coarse search <NUM>. Lastly, a fine altitude search is carried out at the final fine-search location <NUM>.

<FIG> provides more detail on the geo-observable based positioning/timing procedure method performed in one aspect of the disclosure, applicable to each search stage shown in <FIG>. Candidate user <NUM> position coordinates comprising ground and altitude grid coordinates are first set up <NUM>, e.g., using prior information about the search center and ground/altitude grid parameters <NUM>. The search center used in the coarse search stage <NUM> is based on coarse knowledge of the user <NUM> position, e.g., due to wireless communication transceivers <NUM> that the user <NUM> is connected to, or prior user <NUM> position estimates; the search center in the later fine search <NUM> and fine altitude search <NUM> stages are based on results of the prior search stages. Candidate user <NUM> position coordinates are then pruned to a feasible region <NUM> (described in more detail below), based on beacons that have been detected during the geo-observable estimation stage, and beacons that have not been detected during this stage, using network node <NUM> positions, and using beacon detections provided by previous geo-observable estimation operations <NUM>. An initial search is then performed over the pruned candidate user <NUM> position coordinates, using the TOA-only concentrated ML objective function given in (Eq59) <NUM>. At each candidate user <NUM> position coordinate, an additional alternating projections algorithm is used to resolve any additional factor-of-Tamb ambiguity in the TOA estimates, where Tamb = Tsym/K<NUM> if the FLS estimator is used, and the copy-aided ambiguity resolution operation <NUM> is not performed, and Tamb = Tsym otherwise <NUM>. Starting with an initial timing offset estimate τ̂R, an exemplary method performs recursion <MAT> <MAT>
until <IMG> is stable at each candidate user <NUM> position coordinate. The TOA estimates are then updated using τ̂T(ℓ)R ← τ̃T(ℓ)R +Tambn̂zone(ℓ).

In some aspects, the optimal user position then found from the minimum of (Eq59). In other aspects, the FOA-only ML function given in (Eq63) is added to the TOA-only ML function, and the optimal position is found or refined using the combined TOA-FOA ML function. Optionally, the optimal position is further optimized using local search operations <NUM>, e.g., polynomial fit to optimum ML function values or parametric Gauss-Newton method. The velocity and timing offset, and LO offset is then estimated from (Eq62), (Eq58), and (Eq60), respectively <NUM>.

In some aspects, a "pruning" strategy is used to restrict the actual positions searched by the system. <FIG> depicts a strategy for performing this pruning, applicable to each stage in the search strategy shown in <FIG>. Once a set of candidate user positions is set up <NUM>, the slant ranges between the all the beacons and candidate user <NUM> positions are computed, using the known network node <NUM> positions <NUM>. Over-the-horizon (OTH) links are then detected, and candidate user <NUM> positions that connect to more than Lmax-duct OTH network nodes <NUM> associated with detected beacons, or that fail to connect to more than Lmax-fail non-OTH network nodes associated with detected beacons, are then removed from the user location set. These conditions recognize that any candidate location must be inside the FoV of all of the network nodes <NUM> associated with beacons that are detected by the TOA-FOA search procedure, except for ducting events; and must be outside the FoV of all of the network nodes <NUM> associated with beacons that are not detected by the TOA-FOA search, except for blockages, shadowing, etc..

<FIG> illustrates the full positioning search procedure for user <NUM><NUM> in the LMS Scenario. The system determines the horizontal (XY plane) and vertical (Z-axis) position of the user <NUM> both to very high accuracy, and at values comparable to the positioning CRB95. In particular the Z-axis performance exceeds the <NUM> meter FCC E911 mandate, the <NUM> meter (<NUM> foot) WAAS Z-Axis accuracy target, and the <NUM> meter 5GNR XYZ accuracy target. The effect of the "pruning" strategy are also clearly visible in this FIG. , showing the substantive reduction in search region at each stage in the search process.

<FIG> and <FIG> show positioning, velocity, and timing/LO offset estimation performance for all of the users in both scenarios, and Table <NUM> summarizes <NUM>th percentile performance for the exemplary scenarios. Both scenarios demonstrate centimeter-level <NUM>% horizontal (XY plane) positioning accuracy and < <NUM> meter <NUM>% vertical (Z-axis) positioning accuracy - meeting the FCC E911 Z-axis positioning requirement. Both scenarios also demonstrates < <NUM> picosecond <NUM>% clock timing accuracy and < <NUM> ppt <NUM>% clock rate accuracy for all of the scenarios, i.e., near Stratum <NUM> (<NUM> ppt) clock rate accuracy.

The methods described above extend to specular multipath environments in a straightforward fashion. This aspect is also expected to be particularly important in IIoT applications, due to high degrees of multipath expected in warehouse and enterprise environments. However, it will also be important in urban outdoor environments due to reflections from large buildings and structures in the vicinity of users.

Multipath extensions include both multipath mitigation aspects, in which direct and reflection paths are individually identified and used to exclude reflection paths from subsequent positioning and timing solutions, or as part of those solutions; and multipath exploitation aspects, in which direct paths (if available) and reflection paths are identified and used in subsequent positioning and timing solutions. Multipath mitigation aspects usable by those of ordinary skill in the art include:.

Multipath exploitation aspects include multipath fingerprinting methods described in Hilsenrath <NUM> and Wax <NUM>, which exploit the large-scale structure of direct and specular reflections in rich multipath environments.

Additional aspects of the invention are shown Provisional Patent Application <CIT>, entitled "Secure, low-latency, and high-precision interference-resilient navigation and timing using networks of spectrally/temporally redundant beacons," and in the text and drawings disclosed in the paper entitled "Resilient Distributed Positioning Networks: A New Approach to Extreme Low-Latency, High-Precision Positioning and Timing," and the presentation with the same name,
and in Provisional Patent Application <CIT>, entitled "Distributed Resilient Positioning Networks".

The various illustrative blocks and modules described in connection with the disclosure herein may be implemented or performed with a general-purpose processor, a digital signal processor (DSP), an ASIC, a field-programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein.

The functions described herein may be implemented in hardware, software executed by a processor, firmware, or any combination thereof If implemented in software executed by a processor, the functions may be stored on or transmitted over as one or more instructions or code on a computer-readable medium. For example, due to the nature of software, functions described above can be implemented using software executed by a processor, hardware, firmware, hard-wiring, or combinations of any of these.

By way of example, and not limitation, non-transitory computer-readable media may comprise RAM, ROM, electrically erasable programmable read only memory (EEPROM), compact disk (CD) ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other non-transitory medium that can be used to carry or store desired program code means in the form of instructions or data structures and that can be accessed by a general-purpose or special-purpose computer, or a general-purpose or special-purpose processor. For example, if the software is transmitted from a website, server, or other remote source using a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless technologies such as infrared, radio, and microwave, then the coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless technologies such as infrared, radio, and microwave are included in the definition of medium.

Claim 1:
A method, comprising:
synthesizing multitone beacon signals, wherein at least one of subcarrier spacing is selected according to an expected range of frequency-of-arrival for network users or symbol duration is selected according to an expected range of time-of-arrival for the network users;
inducing at least one of spectral or temporal redundancy on the multitone beacon signals; and
transmitting the multitone beacon signals.