WO2019200153A1 - System and method of angle-of-arrival estimation, object localization, and target tracking, with received signal magnitude - Google Patents

Abstract

A system and method estimates the angle of arrival of one or more incoming waves and includes measuring a magnitude of a signal received at a first receiver position and measuring a magnitude of a signal received at one or more additional receiver positions, wherein the first receiver position and the one or more additional receiver positions form a receiver array. The angle of arrivals of the signals at the receiver array are estimated based on the measured signal magnitudes.

Description

SYSTEM AND METHOD OF ANGLE-OF-ARRIVAL ESTIMATION, OBJECT LOCALIZATION, AND TARGET TRACKING, WITH RECEIVED SIGNAL

MAGNITUDE

CROSS-REFERENCE TO RELATED APPLICATION(S)

[0001] This application claims the benefit of U.S. Provisional Application No. 62/656,050, filed April 11, 2018 and titled“SYSTEM AND METHOD OF ANGLE-OF-ARRIVAL ESTIMATION, LOCALIZATION, AND TRACKING, WITH RECEIVED SIGNAL MAGNITUDE”, filed April 11, 2018, the contents of which are incorporated by reference in their entirety.

TECHNICAL FIELD

[0002] This invention relates to systems and methods of determining Angle of Arrival (AoA) of signals based on the magnitude of received signals and the tracking of moving objects/entities based on the magnitude of received signals.

BACKGROUND

[0003] In recent years, there has been an increasing interest in using Radio Frequency (RF) signals to obtain information about our surroundings. Imaging, localization, tracking, occupancy estimation, and human activity recognition are some of the many RF sensing applications. Localization and tracking, in particular, are crucial techniques that can be useful in many scenarios such as emergency response, radio navigation, security, surveillance, and smart homes. Angle of Arrival (AoA) estimation, on the other hand, is an important problem that can be used towards localization and tracking. However, most AoA estimation approaches require synchronized phase information, which cannot be obtained on a synthesized array of off-the-shelf RF transceivers.

[0004] Target tracking has been of interest to the research community in the past few years. Some existing tracking work relies on signals with a large bandwidth, while others rely on the availability of a stable absolute phase measurement by using a software defined radio to track, or to estimate the direction of motion. However, large bandwidths or stable absolute phase information are not available in commercial off-the-shelf (COTS) devices. There are a number of works that have demonstrated tracking using COTS devices, albeit with a different approach than ours. However, they require several transceivers, and/or may require transceivers all around the area, and/or require extensive prior calibration experiments in the same environment, and/or are computationally very expensive. .

SUMMARY

[0005] According to one aspect of the disclosure, a method of estimating the angle of arrival of one or more incoming waves includes measuring a magnitude of a signal received at a first receiver position and measuring a magnitude of a signal received at one or more additional receiver positions, wherein the first receiver position and the one or more additional receiver positions form a receiver array. The angle of arrivals of the one or more incoming waves at the receiver array are estimated based on the measured signal magnitudes.

[0006] According to one aspect of the disclosure, a method of estimating a track of one or more moving targets within a region includes measuring a magnitude of a signal received at a first receiver position at a plurality of times (/), wherein the signal received at the first receiver position at each moment in time is comprised of one or more signals transmitted or reflected from one or more moving targets within the region. The track of the one or more targets is estimated based, at least in part, on the received magnitude measurements.

[0007] According to one aspect of the disclosure, a method of estimating tracking information of one or more moving targets within a region includes receiving a plurality of signals at a receiver array comprised of a plurality of receivers, wherein the received signals are transmitted by or reflected off of one or more moving targets located within the region and measuring a magnitude of the plurality of signals received at the plurality of receivers at a plurality of instances in time t. One or more track parameters associated with the one or more moving objects is estimated based on the signal magnitude measurements, wherein the tracking information describes one or more of location, direction, and speed of the one or more moving targets within the region. DESCRIPTION OF THE DRAWINGS

[0008] Figure la is a diagram illustrating a system configured to determine angle of arrival (AoA) associated with a plurality of signals N based only on measured magnitude of the received signals according to some embodiments.

[0009] Figure lb is a flowchart illustrating steps associated with the method of determining the AoA of the plurality of signal N according to some embodiments.

[0010] Figure 2 is a schematic diagram illustrating the selection of possible valid position choices for y2 according to some embodiments.

[0011] Figure 3 is a diagram illustrating a system configured to determine angle of arrival (AoA) associated with a plurality of signals N based only on measured magnitude of the received signals using a pair of receiver arrays or moving receivers according to some embodiments.

[0012] Figures 4a-4c are perspective views illustrating the collection of experimental results utilizing a system configured to determine the AoA associated with signals corresponding with unknown active sources according to some embodiments.

[0013] Figure 5 is a graph illustrating the normalized spectra of the auto -correlation of the magnitude measurements obtained by a moving receiver along two routes with respect to a plurality of unknown active sources according to one embodiment.

[0014] Figure 6a-6c are perspective views illustrating the collection of experimental results utilizing a system configured to determine the AoA associated with signals corresponding with unknown passive sources reflecting signals from a reference source or transmitter according to some embodiments.

[0015] Figure 7 is a graph illustrating the determination of normalized spectra of the auto correlation of the magnitude measurements obtained by a moving receiver along two routes with respect to a plurality of unknown active sources according to one embodiment.

[0016] Figure 8a is a diagram illustrating a system configured to track an active transmitter object using a receiver that measures the magnitude of the received signal over a window of time according to some embodiments. [0017] Figure 8b is a diagram illustrating a system configured to track a passive object using a receiver that measures the magnitude of a signal reflected off of the passive object over a window of time according to some embodiments.

[0018] Figure 8c is a flowchart illustrating steps associated with the method of tracking one or more moving targets/entities according to some embodiments.

[0019] Figure 9a and 9b are diagrams illustrating ambiguity issue that arises in some embodiments and resolution of ambiguity through addition of an additional receiver according to some embodiments.

[0020] Figure lOa is a perspective view illustrating experimental setup and results associated with the tracking of a moving active target according to embodiments.

[0021] Figure lOb is a graph illustrating the accuracy associated with tracking a moving active target in an outdoor setting according to an embodiment.

[0022] Figure lOc is a perspective view illustrating experimental setup and results associated with the tracking of a moving active target in an indoor setting according to embodiments.

[0023] Figure lOd is a graph illustrating the accuracy associated with tracking a moving active target according to an embodiment.

[0024] Figure l la and l lb are perspective views illustrating experimental setup for the tracking of a passive target in both an indoor setting and outdoor setting according to embodiments.

[0025] Figures l2a-l2d are graphs illustrating the tracking of the passive target according to embodiments.

[0026] Figure l3a is a perspective view illustrating the experimental setup for the tracking of a passive object (e.g. person) according to embodiments.

[0027] Figure l3b is a graph illustrating the tracking of the passive target (e.g., person) according to embodiments.

[0028] Figure 14 is a graph illustrating the estimation error associated with tracking of an active object, a passive object and a non-transmitting person according to experimental results. [0029] Figure l5a is a diagram illustrating a system configured to track multiple passive objects using a receiver array that measures the magnitude of the received signal over a window of time according to some embodiments.

[0030] Figure l5b is a flowchart illustrating steps associated with the method of tracking objects utilizing the receiver array measuring signal magnitude over a window of time according to some embodiments

[0031] Figure 16 is a 2D graph illustrating the angle of arrival y^M and motion-induced array parameter y^A according to some embodiments.

[0032] Figure 17 is a flowchart illustrating steps associated with the method of tracking objects according to some embodiments.

[0033] Figure 18 is an equation utilized in some embodiments.

DETAILED DESCRIPTION

[0034] In one aspect of the present disclosure, a system and method is disclosed for estimating the AoA of signal paths arriving at a receiver array using only the magnitude (or equivalently power) of the corresponding received measurements.

[0035] In one embodiment, the receiver array can be a collection of independent receiver antennas (for instance antennas of a number of laptops). In another embodiment, the receiver array can be emulated by a single antenna that is moved to different positions, for instance by using an unmanned vehicle or through other automatic or manual methods. In yet another embodiment, the receiver array could be a collection of a number of receiver antennas that are controlled by the same clock (for instance, a physical antenna array that is manufactured on the same chip).

[0036] According to some embodiments, the disclosed method illustrates how the auto correlation function (and therefore the spectrum content) of the received signal magnitude at the receiver array carries vital information on the AoA. Further, according to some embodiments the system and method disclosed herein provides a framework for the case of angular localization of fixed active transmitters and/or passive objects (including humans) that do not have an active transmitting source onboard. [0037] According to some embodiments, the system and method of AoA estimation provides a framework for tracking moving targets. The framework can track active moving transmitters and/or passive moving objects/humans that do not have a transmitting source onboard. In one embodiment, the proposed framework utilizes only the magnitude of the received signal at a small number of receivers (one or more), estimate the spectral content of signal magnitude and relates it to the track parameters (e.g., location, direction, and speed of the track) and track active and/or passive targets.

[0038] In another embodiment, it further utilizes filters and/or motion dynamics for target tracking, including tracking of both active transmitting targets as well as passive targets.

[0039] In another embodiment, multi dimensions are used for tracking of active moving transmitters and/or passive moving objects/humans that do not have a transmitting source onboard. Receptions across a number of receivers, at each time instant, can represent one dimension. Receptions over time at one receiver antenna position, from a moving transmitter (or transmitters), or from reflections off of a passive moving target (or targets), can provide another dimension. Receptions across frequency slots can yet provide another dimension.

[0040] In one embodiment, multi-dimensional power spectrum of the multi-dimensional received magnitude measurements is estimated and is related to track parameters of one or more active or passive targets.

[0041] In another embodiment, subspace analysis is applied to the multi-dimensional received magnitude measurements to estimate the spectral content. For instance, Multiple Signal Classification (MUSIC) algorithm is applied to the auto-correlation matrix of the multi-dimensional received magnitude measurements in order to calculate the eigen- decomposition of the multi-dimensional received magnitude measurements and further calculate the corresponding pseudo -spectrum signal.

[0042] In yet another embodiment, additional filters such as Particle Filter and/or Joint Probabilistic Data Association Filters are further applied to the output of the estimated multi-dimensional power spectrum or the pseudo-spectrum signal, in order to track targets. [0043] In yet another embodiment, dynamical system models (linear or nonlinear) may additionally be utilized, in conjunction with the estimated spectral content, and with or without utilizing filtering techniques.

[0044] According to one aspect of the disclosure, a robot is utilized to emulate an antenna array and estimate the AoA of active transmitters and passive objects, using only WiFi magnitude measurements. The angular localization has an overall Mean Absolute Error (MAE) of 2.44°, and only takes an average of 0.45 seconds to localize up to four sources/objects.

[0045] According to another aspect, a plurality (e.g., three) of off-the-shelf separate WiFi devices are utilized as receivers to track a plurality of different targets, including an active transmitter, a passive robot that writes the letters of “IPSN” on its path, and a walking passive human. The tracking approach can achieve a margin of error (MAE) of 20 cm for active targets and 26.75 cm for passive ones, and only takes an average of 1.05 seconds to run per 1 m of tracking length.

[0046] According to another aspect, received signal magnitude measurements of a plurality of off-the-shelf separate WiFi devices (e.g., one transmitter and three laptops) on one side of the area are used to track 1 to 3 passive human targets (no device on board), with a mean error of 38 cm in outdoor areas/parking lots, and 55 cm in indoor areas.

[0047] Test results show that AoA can be estimated, with a high accuracy, with only the received signal magnitude measurements, and can be used for efficient angular localization of a plurality of active or passive sources as well as tracking of a plurality of active or passive moving sources.

[0048] In some embodiments, if phase can be more reliably synchronized in a synthesized array of separate COTS receivers in future, or if a synchronized array is already available (for instance a manufactured antenna array), then both magnitude and phase can be used. The approach disclosed herein may then provide an additional sensing mechanism for AoA estimation and/or tracking, and can thus result in a considerably better overall estimation quality using both magnitude and phase.

Angle of Arrival (AoA) ESTIMATION [0049] FIG. la is a diagram illustrating a system 100 configured to determine angle of arrival (AoA) associated with a plurality of signals N transmitted by or reflected off of a plurality of stationary or approximately stationary objects, and FIG. lb is a flowchart that illustrates steps utilized to determine the AoA associated with the plurality of signal N. In this embodiment, the system 100 includes a receiver array comprised of a plurality of individual receivers l02a, l02b, l02c ... 102* (generically referred to as receivers 102), a plurality of signal measurement units l05a, l05b, l05c ... 105* (generically referred to as signal measurement units 105), a collection unit 104, a controller/processor 106 and a reference transmitter signal 108. In some embodiments, the stationary objects themselves include transmitters, and reference transmitter 108 is not required. In some embodiments however, reference transmitter 108 is located at a known angle relative to the receiver antennas (e.g., a small angle relative to the receiver array, approximately 0°). The reference transmitter 108 generates a signal that is measured by the receiver array and reflected off of the plurality of stationary objects. In some embodiments, the reflection of the reference signal from the stationary objects (not shown) constitutes the plurality of signal ai, a₂, ... o IN. In other embodiments, the plurality of signals ai, a₂, ... c IN originate from a plurality of sources (not shown), wherein determining the AoA may be important for tasks such as beam-forming. The plurality of receivers 102 are configured to receive signals transmitted or reflected from one or more (approximately) stationary objects. The term signal refers to any type of electromagnetic (EM) signal, such as a WiFi signal, RF signal, bluetooth, microwave, or other types of well-known signals. Because phase information is not being utilized to determine the AoA, the receivers 102 are not required to operate on the same signal clock or be otherwise synchronized. The signal measurement units 105 measures the magnitude associated with the signal measured by each of the plurality of receivers 102, wherein the measured magnitudes (or equivalently power) are collected by collection unit 104 and provided to controller/processor 106. In response to the measured magnitudes, controller/processor 106 estimates the AoA cpi, cp₂, ... p_N of the plurality of received signals ai, a₂, ... (XN received by each of the receivers 102 according to one or more of the methods disclosed herein. In some embodiments, a signal measurement unit 105 is associated with each of the plurality of receivers 102, while in other embodiments it may be a single component. In other embodiments, a single signal collection unit 104 is utilized to collect magnitudes measurements of signals received from each of the receivers. In other embodiments, one of the receivers may act as the collection unit 105, collecting the magnitude measurements of all the receivers. In some embodiments, controller/processor 106 utilizes AoA estimations to determine location and/or track active or passive objects based only on the measured signal magnitudes. In some embodiments, controller/processor 106 is implemented with one or more of a central processing unit (CPU), graphical processing unit (GPU), application- specific integrated circuit (ASIC), programmable circuit (e.g., field programmable gate array (FPGA)), etc. Processors may include memory for storing instructions, which when executed by the processor implements one or more of the functions described herein.

[0050] In some embodiments, a signal is transmitted from the reference transmitter 108. In some embodiments, no reference transmitter is required if the objects are active objects that include a transmitter. In the case of one or more passive objects, then a reference transmitter is required to generate the signal that is then reflected off of the passive objects. In some embodiments no information regarding the reference transmitter is required. In some embodiments, the location of the reference transmitter is known. In some embodiments, the location of the reference transmitter is selected relative to the receiver array. For example, in one embodiment the reference transmitter 108 is located along the axis utilized to measure the angle of arrival of the one or more incoming signals ai, o_¾, ... o IN. In other embodiments the reference transmitter 108 may be located at other positions.

[0051] In some embodiments, the receivers may form a linear array. In other embodiments, the receivers may be spatially distributed in any manner.

[0052] Consider the N signal paths (labeled ai, a₂, and o IN) arriving at the receiver array comprised of receivers l02a, l02b, l02c, ... 102* at various angles fi, f₂, and f _g, respectively. These signal paths can be caused by active transmitting sources or by passive objects that got illuminated through a transmission in the area and reflected the signal. The term“source” is utilized herein to refer to both active transmitters and passive objects. For purposes of this embodiment the sources (either transmitting or passive) are considered approximately stationary. In subsequent embodiments the tracking of non-stationary objects is provided.

[0053] The AoA of each of the plurality of paths is estimated using only the magnitude of the received signal at each antenna or receiver 102 of the receiver array. Note that AoA estimation results in the angular localization of the objects. The magnitude of the received signal at the array contains information about the AoA of all the signal paths. This is in contrast with typical systems and methods of determining angular localization that require phase measurements associated with the one or more signals.

[0054] Referring to the flowchart shown in Figure lb, at steps 120 and 122, the magnitude of the signal received at each of the plurality of receivers 102 is measured. In some embodiments, each receiver 102 includes hardware for measuring the magnitude of the received signal. In other embodiments, the signal received by each receiver is provided to the plurality of signal measurement units l05a, l05b, l05c, ... 105* to measure the magnitude of the signal received at each of the plurality of receivers. In other embodiments, each receiver l02a, l02b, l02c, ... 102* includes hardware for measuring the magnitude of the received signal. In other embodiments, a plurality of signal measurement units are utilized, each associated with one of the plurality of receivers to measure the magnitude of the received signal. Referring against to Figure lb, at step 120 receiver l02a receives a signal that is representative of incoming signals ai, o_¾, ... an and the reference signal from the transmitter 108 in embodiments utilizing a transmitter and measures the magnitude of the received signal, denoted c(l). At step 122 receiver l02b - for example - receives a signal that is representative of signals ai, o_¾, ... c I_N and the reference signal from the transmitter 108 as measured at the second receiver l02b and measures the magnitude of the received signal denoted c(2). As discussed previously, only the magnitude of the received signals is needed to be measured - not the phase or any other aspect. However, if other aspects of the signal can also be measured, they can also be used in the framework. Thus, the receivers l02a, l02b, l02c, ... 102* do not require a shared clock or synchronicity. In one embodiment, the signal measurement units l05a, l05b, l05c, ... 105* are incorporated as part of each of the plurality of receivers l02a, l02b, l02c, ... 102*, respectively. In other embodiments, a single signal measurement units 105 is utilized to measure signal magnitudes associated with each of the plurality of receivers. Measured magnitudes are collected by collection unit 104 (which could be a separate unit or on one of the receivers) and provided to controller/processor 106, to perform the following steps.

[0055] In one embodiment, at step 124 the spectral content of the measured signal magnitude is estimated using spectral estimation techniques. In some embodiments, spectral estimation techniques include calculating a Fourier transform of the auto correlation of the magnitude measurements. As provided in more detail below, in one embodiment the autocorrelation function A_COrr(A) correlates the magnitude c(l) measured at receiver l02a with the magnitude c(2) measured at receiver l02b, and would likewise correlate the magnitude c(l) measured at receiver l02a with the magnitude c(3) measured at receiver l02c. Further, the autocorrelation function A_COrr(A) correlates the magnitude c(2) measured at receiver l02b with the magnitude c(3) measured at receiver l02c, essentially correlating all measured signal magnitudes with all other measured signal.

[0056] Consider the receiver array of Fig. la, wherein d denotes the distance from the first antenna l02a, as shown in Fig. la. The baseband received signal, due to the N arriving paths, is written as a function of distance d as follows:

wherein a_h is the magnitude of the n^th signal path, l is the wavelength of the signal, f_h is the AoA of the n^th path (measured with respect to the x-axis), m_h is the phase of the n^th signal at the first antenna of the array, and h(ί/) is the receiver noise. Let A_COrr( A) denotes the auto-correlation function of the baseband received signal magnitude, lc(_<i)l, at lag A.

wherein CA is a constant that depends on the total signal power, C_a is a constant that depends on the noise variance

and the signal power, C_{m n} = ^na™“ⁿ y_h =

a_n is the total power of the received signal, and _<5(. ) is the Dirac Delta function. Taking the Fourier transform of A_COrr(A), the following if provided:

Eq. 3 shows that l|c Z(/) | has peaks at the frequencies +(\y_h— ^_ml)/ , for 1 < n < m <N².

₁

For the sake of simplicity, the frequency is normalized with respect to - A, so that the peaks in the spectrum occur at ±| y_h— ip_m\, 1 < n < m < N. It can be seen from Eq. 3 that the locations of the peaks of l|c/Z(/) | contain information about the AoA of the N signal paths. In other embodiments, pseudo-spectrum of the auto-correlation is calculated using sub space analysis techniques such as Multiple Signal Classification (MUSIC) algorithms (more details on this will come later).

[0057] At step 126, these frequency peaks within the frequency spectrum are identified. It should be noted, that while the frequency peaks contain information about the AoA, the frequency peaks do not provide the AoA’ s fi, y2, or y_N directly; rather the frequency peaks provide information related to the difference in the cosine of the various AoA (e.g., cos fi - cos fi). At step 128, the frequency peaks are utilized to determine the differences of the cosines of the AoAs. At step 130 and as discussed in more detail below, the differences in the cosines of the AoA are utilized to create a solution set of all possible AoAs - based solely on the magnitude of the received signals. In some cases the solution set is unique. In other embodiments, the solution set may include ambiguities in which multiple plausible AoAs may be deduced from the differences of the cosines of the AoA. In some embodiments, a second array is utilized to reduce ambiguities (as shown at step 132 and described in more detail below).

AoA ESTIMATION FOR LOCALIZING FIXED SQURCES/QBJECTS

[0058] In some applications, the system and method of determining AoA via measurements of signal magnitude is utilized to locate fixed active or passive objects humans. That is, fixed active signal sources (i.e., with a transmitter on board) or passive signal sources (i.e., no transmitter one board and merely reflecting another transmitted signal) can be located using only the magnitude of the corresponding received signal measurements. The signal measurements can be obtained by using an array of fixed antennas, or by using a single antenna that changes position over time (e.g., antenna mounted on an unmanned vehicle, as shown in Fig. 3), or by a general collection of a number of independent receiver antennas (e.g., antennas of the WiFi cards of two or more laptops).

[0059] In some embodiments, a system and method of localizing fixed sources, either passive or active, is provided.

[0060] Finding all possible arrival angles based on the frequency peaks and their relationship to the arrival angles described at step 130 may be implemented in a number of different ways. In one embodiment, consider the case where there is a dominant incoming signal, for instance a dominant reference source, or the case where one of the incoming signals is considerably stronger than the rest. In this embodiment, the arrival angles can be determined from the peaks identified at step 126 as follows. Consider the case that the signals from the unknown signal sources are of lower transmission power as compared to the reference source at <j>_ref· This case is in particular relevant for applications in which the angles from passive objects is being estimated. In this embodiment, the transmitted reference signal will bounce off of these objects and reach the receiver array with a considerably smaller power than that of the path from our reference transmitting source. In such a case, the AoA estimation problem can be solved as follows. Consider N - 1 unknown sources where the paths arriving from them at the receiver array have a lower power as compared to the reference source. This can happen for both the cases of active and passive sources. In the active case, this can happen when the active transmitters have a lower power as compared to the reference source. On the other hand, the passive case results in a dominant reference source almost all the time. From Eq. 3, the pairwise coefficients C_m,n that correspond to the reference source and an unknown source would be the only significant peaks in the spectrum. More specifically, if the dominant reference source with a higher power is at 0°, and the unknown sources are at angles {fi, . . . ,F_N-I }, then the estimated differences from the spectrum are Q = { l -cos(_{</ /}), . . . , 1— cost^.v-_/) } since the rest of differences will have negligible peaks. Therefore, we can directly estimate the AoAs corresponding to the unknown sources as {cos^_1(l - q) : q E Q).

[0061] In another embodiment, the following algorithm is used to extract all the angles from the peaks in step 130. Consider N signal sources (not shown in Figure la) present on one side of a receiver array, i.e., sources whose AoAs {f_h, 1 £ n < N) satisfy 0° < f_h <180° (see Fig. 1). Let Y = { y_\, y2, . . . , y_N }, where y_h = cos (f_h). Define the function D(U) on a set of real numbers U as the set of all the unique pairwise distances between the elements of U , i.e., D(U) = { I m - Uj \ : , u_j E U, I ¹ j} . Let Q be the set of the absolute values of the pairwise differences of the cosines of AoAs, i.e., Q = D(W). Without loss of generality, Q is assumed to be ordered: Q = {qi, q2, . . ., </ } , qi > qi > > q_M In response, the value of Y is estimated, and the AoAs are obtained by the set of pairwise distances Q determined using Eq. 3.

[0062] The problem of estimating a set of Areal numbers, B, given the multiset of absolute differences (distances) between every pair of numbers, A B, is called the Turnpike problem.. However, it is not possible to obtain a unique solution set using just the set A B. For instance, for a solution B, the sets obtained through translation B + { e} = {b + e : b E B}, mirroring—B = {-b : b E B}, or a combination of both -B + {e}, would also result in the same set of distances A B, for any constant e. Furthermore, when the number of points N > 6, there exist other possible solutions that do not arise from the above construction. [0063] The existing solvers for the Turnpike problem require that the set of distances should contain all the ( ) pairwise distances (or that we know the multiplicity of the non- distinct distances, if any), and they suffer from the translation and mirroring ambiguities as well as other ambiguities. In the AoA estimation problem though, the multiplicity of the possible non-distinct distances is unknown. Furthermore, the aforementioned translation and mirroring ambiguity must be resolved. As a result, the solvers proposed for the Turnpike problem cannot be utilized. In order to overcome the ambiguity arising due to the translation and mirroring of Y, a reference signal source (i.e., a transmitter) is placed at one extreme of the span of angles, say f^ = 0°, so that \|/,_er = 1. This also implies that any valid solution set would only contain \_|/s that are less than or equal to \_|/ref, a condition utilize as shown below.. However, there still exist multiple solutions for a set Q. The proposed method first finds all the valid sets of solutions {y_h: 1 < n < N}, for a given distance set Q. The number of valid solutions is reduced (possibly to a unique solution) by utilizing measurements from two arrays in different configurations.

[0064] In some embodiments, all possible angle solutions corresponding to Y are identified, given the ordered set of distances Q , the AoA corresponding to the reference source at f^ = 0°, and the estimated number of sources (denoted by N). This embodiment relies on the location of the reference source being known. In other embodiments, all possible solutions may be identified within requiring a known location (e.g., f,-_ef = 0°) of the reference source. Estimating the number of sources is described in more detail below. Without loss of generality, the sets Y and Q are taken to include the impact of the new added reference source at f^ (i.e., Y = { i//_rcr, yi, . . . ,y_N-i}· Then, we are interested in estimating the angles of the rest N - 1 unknown sources. The rightmost and leftmost extreme points of the set Y are defined by \|/_ref = 1 and yi = \|/,_er - qi, respectively, as shown in Fig. 2. Consider the positioning of the next point y2, corresponding to cp_- Fig. 2 shows the two possible valid position choices for it. Both these will result in a valid solution set. Similarly, for each of the remaining distances q_t , 3 < I < M, there exist a pair of positions on the line in Fig. 2, whose distance to the two extreme points correspond to that q,. It is easy to confirm that these two positions are the only possible positions given the monotonicity of the set Q. This observation is the base of our proposed approach, which we detail next. Let the set S denote the set of all the sets of valid solutions. Starting with a valid partial solution (VPS), defined as a set S such that D(S) !º Q. All valid solutions are found as follows:

• Initialization: We initialize the set of VPSs with S' ^{1 1} = { { i//,·,._{/ -} qi //_r,f} }, which is the smallest VPS, containing only the two extreme points of Y.

• Iteration Update: In iteration /, we place a point at a distance qi from either of the extremes in the existing VPSs. More specifically, for each set S E S(i-J), one test set is generated by adding a point at a distance q_t from the rightmost extreme, and another test set by adding a point at a distance qi from the leftmost extreme. If the pairwise distances of the new sets are a subset of Q , we then add these test sets to S(i-1 ) to generate S(i).

• Algorithm Termination: The algorithm is terminated after M - 1 iterations, which corresponds to exhausting all the elements of Q. A set S e S(M) is a possible solution for Y if the cardinality of S is N and D(S) = Q. All such sets S are utilized to generate S, the final set of all the possible solutions. Algorithm 1 shows the pseudo-code for this algorithm.

Algorithm 1 Finding all possible angle solutions

f

it: end for

72: Cild for

is: S «— fS : S€ S^iM car inalit 's) = J¥,i)(S) = >}

return <$ j = cos

[0065] Remark 1. It can be easily confirmed that the aforementioned algorithm captures all the possible valid solution sets, even when there are distance multiplicities.

[0066] Remark 2. Note that <j>_r,f does not have to be necessarily 0°. As long as it is the smallest possible angle (i.e., all the other angles are greater than it), then the previous algorithm works.

[0067] As discussed above, in some embodiments, Algorithm 1 results in multiple possible solutions. In some embodiments, as shown at step 132 in Figure lb, the ambiguity in solutions is reduced by collecting another set of measurements from a receiver array with a different orientation relative to the first receiver array. As discussed above, because the claimed invention does not rely on phase measurements, the magnitudes measured by the first receiver array and the magnitudes measured by the second receiver array do not need to be synchronous or share a clock. In some embodiments, the new magnitude measurement is obtained by a second fixed receiver array or a collection of a second set of fixed receiver antennas (not shown). In other embodiments, the new magnitude measurement is obtained by an unmanned vehicle that moves along a route with a different orientation. This solution is thus particularly suitable for the case of an unmanned vehicle emulating a receiver array, since traversing two straight routes is a trivial task for an unmanned vehicle.

[0068] Fig. 3 shows an example of such a system 300, in which unmanned vehicle 301 having a signal receiver 305 traverses a first route (labeled“Route 1”) and a second route (Labeled“Route 2”). A reference source 302 generates a reference signal, and one or more active or passive objects 304a and 304b generate a signal or reflect the signal generated by reference source 302 that is received and measured by the receiver located on vehicle 300. A signal measurement unit 308 measures the magnitude of the signals received by the signal receiver 305 at each of the plurality of positions along the first and second routes. A data collection unit 309 collects the measured magnitudes and provides them to controller processor 310 for analysis to determine the AoA cpi and cp₂ of the received signals. Suppose that the AoAs of the signal sources for the first array configuration are {f_h : 1 < n < N} . For the second array that is tilted by an angle W in the clockwise direction, the AoAs are now {f_h +W : 1 < n < N) and the reference source 302 has an angle of arrival W or equivalently, if/_ref= ooc(Ώ). Since cosine is not a linear function of its argument, utilizing two sets of array measurements based on Route 1 and Route 2 results in different sets of pairwise distances Q and Q'. Therefore, all possible angle solutions may be obtained individually for Q and Q' (using Algorithm 1), with the intersection of the two sets providing the valid solution(s). More specifically, let <E>aii,i and <h_aii.2 indicate the AoA solution sets for Q and Q' respectively. The intersection of the two sets <E>_aii,i and <t>_aii.2 - { W } is the final estimated AoAs. In the angles from the two sets <E>_aii,i and <t>_aii.2 (after subtracting W from <I>_aii,2) may never be equal, owing to noise or rounding errors. Therefore, the sets may be compared within a tolerance level. Intuitively, the chance that the two routes have more than one possible common set is considerably small. In some embodiments, if there is more than one solution set in the common set, we can collect measurements along another array route to obtain a unique solution. For example, if an unmanned vehicle is being used, the location of the unmanned vehicle may be modified to emulate a new array. [0069] Criteria for Choosing W: The orientation of the second array W determines the extent of dissimilarity between the sets Q and Q', where a larger W is likely to result in a higher dissimilarity. Therefore, it is preferable to use as large an W as possible. However, we require all the sources in the area to lie on one side of the receiver array (i.e., upper half plane) in the second configuration as well. Therefore, we can use the first set of distances <2, to estimate the largest AoA at the first receiver array as _<¾_max = cos^-1 (1 - qi). This implies that the possible range of values for W is 0 < W < 180° - f_hac· Note that if <j>_r,f for the first route was f_m > 0 instead of 0, where f,,_i,,_i is smaller than all source angles, then the condition for W becomes -</>_min < W < 180° - </>_max , where </>_max = cos^-1 (cos (f p_h ) - qi).

[0070] Choice of Number of Sources: Given a set of unique distances Q and Q', we are interested in estimating the AoAs corresponding to the smallest number of sources that can result in the two sets of distances. For N sources, the maximum number of possible pairwise

distances is

Assuming the cardinality of sets Q and Q' are M and M', the estimated

^ N N

number of sources N should satisfy M < (— ) and M ' < (— ), which translate to the

.. . _jT; l + v 1 + 8 M . _jT; 1+VT+8M, _{T T jT;} . ..

conditions: N > - - - and N > - - - . Hence, we set N _min as the smallest integer satisfying the previous two inequalities.

[0071] Initially, it is assumed that there are N = N_{u n} sources. The AoAs are solved for the sets Q and Q’ separately, using the approach provided in algorithm 1, for example. If the intersection of <E>_aii,i and _aii,2 - { W } is an empty set, we then need to increase N by 1, until we get a non-empty intersection set of solutions. It should be noted in some embodiments it is unlikely that adding an element \_|/new to the true set Y or taking out one element of it will produce the same Q and Q' respectively for both the routes. Hence, it is highly unlikely that using any N other than the true N will produce non-empty intersection set of solutions. In addition, for the case in which finding all the valid solutions with only one measurement array N can be selected, then N can be chosen as iVmin =

, which corresponds to the smallest number of sources that could have resulted in a cardinality of M for Q. If the current N does not result in a valid solution, then N is increased by 1 until we get a non-empty solution set. [0072] Experimental results for the case of fixed sources/objects are provided with respect to Figs. 4-7.

[0073] Referring now to Figures 4a-4c and Figure 5, experimental results for AoA estimation and localization of active transmitting sources are shown according to some embodiments. In Figures 4a-4c, the system utilized includes a robot 400 capable of moving along a route and having a receiver, a reference source 402 (e.g., transmitter) located at a known location, and one or more unknown active transmitters 404a, 404b. In the embodiment shown in Figure 4a the system is installed in an enclosed space (e.g., parking garage), wherein the embodiment shown in Figures 4b and 4c are installed in an outdoor setting. In some embodiments the robot 400 would travel along a first route and measure signal magnitudes at various locations along the route to generate a plurality of magnitude measurements at a plurality of different receiver positions. In this way, the robot 400 acts as essentially a receiver array collecting signal magnitudes at a plurality of locations along the array. In some embodiments the robot 400 may travel along a first route measuring a plurality of signal magnitudes and then along a second route different than the first route (e.g., oriented at an angle W relative to the first route).

[0074] Figure 5 is a graph illustrating the normalized spectra of the auto -correlation of the magnitude measurements obtained by a moving receiver along two routes with respect to a plurality of unknown active sources according to one embodiment. Fine 500 represents the auto-correlation of magnitude measurements associated with the first route and line 502 represents the auto-correlation of magnitude measurements associated with the second route. Both lines are presented in the frequency domain, as illustrated by the x-axis represented as a frequency. The peaks of lines 500 and 502 represent information associated with the angle of arrival (AoA) of the received signals. The dashed lines illustrate the true theoretical peak locations of the magnitude measurements. As shown in Figure 5, the peaks correspond very well with the actual values associated with the AoA in each experiment. The results of these experiments are shown in Table 1, reproduced below. The true AoAs are shown in the first column and the estimated AoAs are shown in the second column. The top part of the table shows results for an embodiment in which two robotic arrays are utilized to measure AoAs. The bottom part of the table shows results for an embodiment in which a dominant reference source and a single array is utilized. The margin of error (MAE) with respect to experiments utilizing two arrays (shown in the top part of the table) was shown to be approximately 1.3°, whereas the MAE in the experiment utilizing a dominant reference source and a single array was shown to be approximately 2.79°. Two

(Bottom)

Dominant

•reference

source &

one array

Table 1

[0075] Referring now to Figures 6a-6c and Figure 7, experimental results for AoA estimation and localization of passive objects are shown according to some embodiments. In Figures 6a-6c, the system utilized includes a robot 600 capable of moving along a route and having a receiver, a reference source 602 (e.g., transmitter) located at a known location, and one or more unknown passive objects 604a, 604b, and stationary humans 606a, 606b.

[0076] In the embodiment shown in Figure 6c a reference source 602 is located at an angle of cpref = 0° and two humans 606a and 606b are standing at angles cpi = 90° and cp₂ = 110°. As both humans are considered passive objects, the reference source 602 is considered the dominant sources. In this embodiment the robot 600 moves along a single route, sampling magnitude measurements at a plurality of locations to simulate a receiver array. Figure 7 is a graph illustrating the autocorrelation of the measured magnitudes, wherein the peaks associated with the spectrum information represent the angle-of-arrival (AoA) associated with the objects. The dashed lines illustrate the actual angle-of-arrival associated with each object. As shown in Figure 7, the identified peaks are closely aligned with the actual AoA. Table 2, reproduced below, illustrates the results of five different experiments including a number of different objects. As illustrated in the table, the estimated AoA closely aligns with the true AoA in each case, including those having a plurality of objects to be detected. The Margin of Error (MAE) in this experiment was shown to be approximately 2.99°.

Table 2

TRACKING MOVING TARGETS

[0077] Referring now to Figures 8a-8d, a system and method of tracking moving targets is described based only on the measured magnitudes of the received signals. In some embodiments, only a single receiver is utilized, while in other embodiments two or more receivers are utilized to track the one or more moving targets. The steps utilized to track the moving target is shown in Figure 8b, which as described above may be implemented on a processor/controller. In some embodiments, the term active moving target refers to a moving transmitter, such as a moving vehicle that emits a signal as it moves. A passive moving target, in contrast, refers to a moving object that has no transmitter on board, such as a person walking. The system and method described above for utilizing the magnitude of the received signal to estimate Angle of Arrival (AoA) can be utilized to localize and track moving targets. Active target tracking is discussed first with respect to Figures 8a and 8b. Passive target tracking is discussed with respect to Figures 8c and 8d.

[0078] Active Target Tracking: Consider the scenario shown in Fig. 8a, where a known fixed receiver 800 receives wireless signals from a known fixed transmitter 802 and an unknown moving active transmitter 804, which moves with a constant speed, v, along a line that makes an angle 0° with the c-axis, and an angle </>_r with the line connecting active transmitter 804 and receiver 800. In some embodiments, the system further includes a signal measurement unit 801, collection unit 803 and controller/processor 805 for processing the signal magnitudes measured by the receiver 800 and/or signal measurement unit. Although not shown in Figure 8a, in some embodiments, there may be no fixed transmitter for active target tracking while other embodiments may utilize a fixed transmitter for active target tracking, as shown in Figure 8a.

[0079] The total baseband received signal at the receiver at time t, c(t), will then be,

where ai, a₂ are the magnitudes of the paths from fixed transmitter 802 and active transmitter 804, respectively. Parameter mi is the phase of the signal arriving from fixed transmitter 802, m2 is the phase of the signal from active transmitter 804 when the moving target is at its initial position (/ = 0), v is the speed of active transmitter, and

By comparing Eq. 4 and Eq. 1, the equivalence between the two equations is apparent, where Eq. 4 can be considered a special case of Eq. 1 with N = 2 sources, yi = 0 and y2 = y_t. That is, in some embodiments the magnitude-only tracking problem can be considered as the dual of the previous AoA estimation problem, where the active, moving transmitter 804 synthesizes a transmit array, and the quantity /y relates to the angle of departure instead of the angle of arrival. Therefore, we can use our framework from above to estimate \y_t\. In some embodiments, if there are unknown fixed transmitters in the area, they will not affect the tracking quality. This is due to the fact that any fixed transmitter will result in a constant term (such as the first term in Eq. 4) with y = 0. Thus, the non-DC peaks of the spectrum will only correspond to y_t. This is particularly attractive as the signal may bounce off other fixed objects in the area, creating several paths to the receiver.

[0080] In some embodiments, the signal measurement unit 801 is utilized to measure the magnitude of the received signals, collection unit 803 would collect the measured magnitudes from the signal measurement unit 801, and controller/processor 805 would perform the analysis on the measured magnitudes. [0081] Passive Target Tracking: Consider the case in which an unknown passive moving target is tracked. That is, in this embodiment, the target to be tracked (e.g., person 810) does not have a signal source onboard, but rather reflects the incident signal from fixed transmitter 808, as shown in Fig. 8b. The received signal at receiver 806 can then be written as follows:

where the first term is the same as the first term in Eq. 4, g is the reflection coefficient from the moving target 810, m: is the phase of the reflected path when the target is at its initial position ( t = 0), and y_t = cos (<j>_r ) + cos(^r ). Similar to the active case, the duality between Eq. 5 and Eq. 1 can be seen. Thus, the framework described above can be utilized to estimate \y_t I, which, in this case, is equal tol cos(<h ) + cos(^r )l.

[0082] Referring now to Figure 8c, a method of tracking an active and/or passive object is provided. At step 820 the receiver 800 receives and the signal measurement unit 801 measures a signal magnitude associated with the received signal at a first instant in time (e.g., time c(0)). At step 822 the receiver 800 receives and signal measurement unit 801 measures a signal magnitude associated with the received signal at a second instant in time (e.g., time c(t)). As discussed above, because the target (either active or passive) is moving, the target takes on the role of the array, such that a single receiver 800 and associated signal measurement unit 801 may be utilized to measure signal magnitudes. In some embodiments, additional receivers may be utilized to reduce ambiguity in the estimate tracks of objects. At step 824, the controller/processor 805 estimates the spectral content of signal magnitudes using spectral estimation techniques (e.g., Fourier transform of magnitude auto-correlation or MUSIC). At step 826 the frequency peaks are located within the frequency spectrum. At step 828 the frequency peaks are related to the track information (e.g., direction, location, speed) of the moving targets. In some embodiments, at step 830 one or more additional receivers are utilized to reduce ambiguity associated with the tracked objects. This is analogous to the embodiment shown in Figure 3 in which a second array of receivers or second path associated with an unmanned or moving receiver were utilized to reduce ambiguity in detected objects. In this embodiment, however, because the object being tracked is moving and therefore acts as the array, only a single receiver may be required to reduce ambiguity regarding the location, direction and/or speed of the tracked object. At step 832, additional filtering techniques and/or dynamical system modeling may be utilized to improve the track estimation. In general, a filter utilizes a plurality of measurements observed over time, and despite the measurements containing statistical noise and other inaccuracies, produces estimates of unknown variables that tend to be more accurate than those based alone on single measurements. For example, at step 832 a Kalman filter, particle filter, or other well-known filters may further be utilized to improve tracking the position of the objects over time and determine the position, direction and velocity of the active and/or passive objects.

[0083] Resolving the Tracking Ambiguity : In general, the magnitude-based tracking problem is easier than previously described AoA estimation of fixed sources because there is only one angle to be estimated per moving target, and the impact of all non-moving transmitters/objects will not be seen in the power spectrum. However, there are still ambiguities if only one receiver is used, as explained below with respect to Figures 9a and 9b. The tracking problem consists of estimating the location and bearing of Tx_mov 900 at each time instant t. However, using \y_ti\ as the only piece of information would result in ambiguities when solving such a problem. For instance, consider the problem of using one receiver 902 to estimate the bearing of the active target, as shown in Figure 9a. The target has a velocity vector

. It moves in a direction that makes an angle f_Gi with the line connecting the target and the receiver Rxl 902. This results in \y_Ti\ = lcos(r/ _r/)l at the receiver Rxl 902. Then, as shown in Fig. 9a, four different velocity vectors (+v^ and ± can result in the same measurement. Thus, Rxl 902 cannot uniquely estimate the true bearing of the moving target. In addition, the starting point of the target can be any point in the workspace, adding more ambiguity to the solution. In some embodiments, to solve the bearing ambiguity, additional receivers are added. For instance, by adding one more receiver 904, two of those four solutions of Fig. 9a will become invalid, as shown in Fig. 9b. However, in some cases there will still be the ambiguity between the actual velocity vector and the one pointing to the opposite direction. In some embodiments, a dynamical system model (e.g., a nonlinear model) is utilized to represent the motion dynamics of the target. This dynamical system modeling, in conjunction with the receiver measurements, can then improve tracking performance and remove any possible remaining ambiguity of the bearing as well as the location ambiguity, as described below.

Nonlinear Dynamical System Modeling

[0084] Consider the scenario where there are a total of R receivers located at (xrf ,yri ), 1 < i < R, a fixed transmitter T x f _{j x} located at (x_j ,y_j ), and a moving target Txmov located at (x_°(t),y_°(t)) at time t. The state of the target at time t is defined as the 3-dimensional vector xt =

where ()°(l) is the bearing of Txmov at time /, and [.J^ is the transposition operator. A measurement process TT is an R-dimensional vector of measurements from all the receivers: TT = \\i r](t)\, \[\ipr2(i)\, , l[ l rR (/)l] 0 where l[li/>ri(/)l is the measurement obtained at the 0¹ receiver at time t. In case of an active target, this measurement is related to the state of the target as follows:

where w_ri(t ) is the Gaussian measurement noise at receiver n, with variance

On the other hand, in the case of passive target, Yί is related to the state of the target as:

[0085] For the dynamics of x₍, a simple constant-speed motion model, xi_+i = g(x_t), as follows:

wherein W_Xq , w_yo, W_QO are the noise process for the three components of the target state x₀, y₀, and q₀, respectively, and P, is the probability of the target maintaining the same bearing as the previous time instant. In some embodiments, Eq. 8 along with Eq. 6, or Eq. 7, then defines the nonlinear dynamical system of the tracking problem.

[0086] To estimate the state of the moving target, the conditional probability of the target having a state x given all the measurements up to time t, p(xt IYΐ: _/ ) is computed. In the filtering literature, this probability is referred to as the filtering Probability Density Function (PDF). Then, the mean of the computed PDF, E{xt IY1:_/ }is utilized as estimate for the target state at time t . In some embodiments, because the dynamical system is nonlinear, a particle filter is utilized to compute the filtered PDF. A Particle Filter (PF) is a probability distribution represented by a set of random samples, called particles, drawn from that distribution. Such a representation is desirable because it can easily model nonlinear transformations of random variables, which makes it particularly suitable for the problem at hand. The basic idea of a PF is that, at each time instant, samples (or particles ) are drawn from a proposal distribution

is the total number of particles. Those particles are then given importance weights, w_t^, that describe how well they fit the current measurement Y^. That set of weighted particles represent the filtering PDF p(xt I Yΐ:_ί) at time t. Afterwards, a resampling step is performed to neglect particles with very low weights (very low probability of producing the current measurement) to allow focus on the particles with high weights. Specifically, a new set of I particles are drawn from the distribution defined by w_t ^[l] over the values of x '¹.

[0087] In some embodiments, the particle filter as described with respect to Algorithm 2, shown below, is utilized. For example, at line 2 of the algorithm shown below, particles for an initial state are drawn from an initial distribution z_\ (xi), which can depend on any prior information obtained about the initial state of the target (or is taken to be uniform when no prior information is available). At line 3, the PF algorithm calculates the importance weight of each particle as the probability of getting the measurement Yi, given that the state of the target is this particle. This probability can be easily calculated using the measurement model in Eq. 6 or 7 (depending on whether an active or passive target is being tracked). At line 6, the particles are resampled, which results on the discarding of low weight particles. At line 7, motion dynamics are enforced by evolving the resampled particles according to a motion model (for example, the motion model described with respect to Eq. 8). After the tracking period T is over, the estimated track of the object E{x_t I Yΐ:ί} 11 i:T is smoothed by passing it through a spatial moving average filter.

Algorithm 2 Parti de filler for motion tracking

Input: Total fracking time T_> Number of particles /, Measurements

Output: Estimate of the target states xi_{: j\}

i. initialise

2 Sample

¾ Compute the importance weights

and normalise n

4; Estimate the initial target state * ~ E{ i |Yi } ~

$: for 2 < f < T do

6; Sample x^ _r for Ϊ— f fro the distribution defined bypixt-x ~ x¾^{) ~} 4_~r

7; Sample xjp^ ~ pi ^_} )

Compute tire importance weights

and normalise

9; Estimate the target state x_(:

ift; end for [0088] A benefit of this framework is that the computation time is short, does not require any prior calibration in the same environment, and has a good tracking and localization quality.

[0089] The AoA estimation framework provides a dual setting for tracking a moving target, wherein the magnitude of the received signal at one or more receivers are used for tracking. Additional filtering and motion dynamics can also be utilized to further improve tracking performance. The approach can track active transmitting targets as well as passive moving objects or humans as shown below in the various experiments provided in Figures l0a-l4.

[0090] Figure lOa and lOc are perspective views illustrating experimental setup and results associated with the tracking of a moving active target according to embodiments of the present invention. In the embodiment shown in Figures lOa and lOb, an active object (transmitter 1000) is positioned in a region to be monitored along with a plurality of stationary receivers l002a, l002b, and l002c. The embodiment shown in Figure lOa is provided in an outdoor setting whereas the embodiment shown in Figure lOb is implemented in an indoor setting. Figures lOb and lOd illustrate the ability to track the active object 1000 as it moves through the respective regions. In Figure lOb, the true path of the active object 1000 is shown by line 1004 and the tracked path estimated by the plurality or receivers l002a, l002b, and l002c is shown by the dashed line 1006. Similarly, in Figure lOd the true path of the active object 1000 is shown by line 1008 and the tracked path estimated by the plurality of receivers l002a, l002b, and l002c is shown by the dashed line 1010.

[0091] Figure l la and l lb are perspective views illustrating experimental setup for the tracking of a passive target according to embodiments of the present invention. In both embodiments, the system includes a reference source 1100 (e.g., fixed transmitter), a plurality of receivers H02a, H02b, and H02c, and a moving passive target 1104. Tracking of the passive target 1104 is shown in the graphs of Figures l2a-l2d. In Figure l2a the passive target 1104 outlines the letter“I”, in Figure l2b the passive target 1104 outlines the letter“P”, in Figure l2c the passive target 1104 outlines the letter“S”, and in Figure l2d the passive target 1104 outlines the letter“N”. In each embodiments, the actual path of the passive target is shown by line 1108, while the estimated or tracked path of the passive target 1104 is shown by dashed line 1110. In each case, the results indicate a good ability to track the location of a passive object, with a mean average error of approximately 23.27 centimeters.

[0092] Figure l3a is a perspective view illustrating the experimental setup for the tracking of a passive object 1304 (e.g. person) according to embodiments of the present invention, and Figure l3b is a graph illustrating the tracking of the passive target (e.g., person) according to embodiments of the present invention. In this embodiment, the system relies on a reference transmitter 1300 and a plurality of receivers l302a, l302b, and l302c. Rather than a robot, in this embodiment a person acts as the object (passive object) and detected signals indicate a reflection of the transmitter reference signal off of the person. Figure 13b is a graph illustrating the correlation between the actual path taken by the person as shown by solid line 1308 and the estimated path of the person as indicated by dashed line 1310. Once again, the proposed framework provides accurate tracking of the object. In one embodiment the margin of error between the actual path and estimated path was approximately 30.26 centimeters.

[0093] Figure 14 is a graph illustrating the cumulative distribution function (CDF) curve of the absolute tracking estimation error for both active and passive examples. For example, line 1400 indicates the tracking error associated with tracking of an active object, line 1402 shows the tracking error associated with tracking a passive object, and line 1404 shows the tracking error associated with tracking a person (also passive). As shown in Figure 14, the proposed framework illustrates low estimation errors for each of the different types of objects tracked.

TRACKING MOVING TARGETS USING MULTIPLE DIMENSIONS

[0094] Figure l5a illustrates a system 1500 configured to determine angle of arrival (AoA) associated with a plurality of signals N transmitted by or reflected off of a plurality of moving targets and to utilize the determined AoA to track the moving objects. Figure l5b is a flowchart illustrating a plurality of steps utilized to track the objects according to some embodiments. These embodiments illustrate the ability to utilize multi-dimensional analysis and extend the tracking approach of Fig. 8c to track a plurality of targets. The embodiment shown in Figure 15 utilizes a plurality of receivers to measure signal magnitudes at a plurality of different distances d. In addition, the plurality of receivers measure signal magnitudes at a plurality of different instances in time t, Then, all the signal magnitude measurements across both dimensions (e.g., time and receiver positions) are utilized to tracking moving targets. As compared with previous embodiments disclosed with respect to Figure la, 3, and 8a, for example, the embodiment shown in Figure l5a relies on analysis is more than one dimension to detect the position, direction, and speed of the one or more moving objects.

[0095] In some embodiments, more than two dimensions are used. In some embodiments of this work, other dimensions can include different frequency slots. A benefit of this multi-dimensional framework is that it can improve the tracking performance.

[0096] In some embodiments, system 1500 includes a transmitter 1502, a plurality of receivers l504a, l504b, ... 1504*, a plurality of signal measurement units l507a, l507b, ... 1507*, a collection unit 1506, and a controller processor 1508. In the embodiment shown in Figure 15, system 1500 is configured to track the movements of a plurality of objects (both passive and active). The plurality of receivers l504a, 1504, .. 1504* are configured to receive signals transmitted from and/or reflected from the plurality of objects l5l0a, l5l0b, l5l0c located within the region to be monitored. The plurality of signal measurement units l507a, l507b, ... 1507* measure signal magnitudes associated with the received signals, which are collected by the collection unit 1506. Although the receivers l504a, 1504, ... 1504* do not need to operate on the same clock or be otherwise synchronized sufficiently to detect and measure phase information, the measured signal magnitudes are measured both in the spatial reference frame as well as the temporal reference frame. In some embodiments, measurements in the spatial reference frame are referenced to the location of the first receiver l504a in the array, with the location of the second receiver l504b being given by a distance di from the first receiver and the location of subsequent receivers 1504* being given by a distance d_* from the first receiver. In some embodiments, the location of transmitter 1502 is known and positioned at a relatively low angle with respect to the orientation of the receiver array. In other embodiments, the location of transmitter 1502 is not required to be known.

[0097] With reference to Figure l5b, at step 1520, the plurality of receivers l504a, l504b, ... 1504* measure the received signal transmitted and/or reflected from the plurality of objects. In some embodiments, the multi-dimensional received signal c(t, d), is a function of time t and distance d along the array, expressed as follows:

wherein ao and fo are the complex magnitude and angle of arrival corresponding to the direct signal path from the transmitter 1502 to the receiver array, a_n and cp_n are the complex magnitude and angle of arrival corresponding to the n^th target at the receiver array, h(cI, t ) is the receiver noise, parameter i//^M represents the motion-based array parameter created by the movement of the object being tracked. The parameter /^A represents the angle-of-arrival of the signals from the n^th targets appears jointly with the parameter i//^M in the two- dimensional spectrum generated from I c(t, d) I², which is expressed as follows:

wherein d(...) is the 2D Dirac delta function, and z,_{, d} if_t, f_d) represents the modeling error term in the 2D spectrum. The locations of the peaks in the 2D spectrum (as shown in Figure 16, for example) give the corresponding pairs of y^A and i//^M values for each of the moving targets. With reference to Figure l5b, at step 1522 the 2D spectral content of signal magnitude is obtained. At step 1524 the location of the peaks within the 2D spectrum are determined. And at step 1526 the frequency peaks are utilized to determine the angle of arrival and motion induced array parameters

associated with the measured signal magnitudes.

[0098] At step 1528, additional filtering techniques and/or dynamical system modeling may be employed to improve track estimation. As discussed in more detail below, this may include a non-linear filter such as a particle filter (PF) with a Joint Probabilistic Data Association Filter (JPDAF). In other embodiments, other frameworks may be utilized to reduce ambiguity and therefore improve track estimation. For example, two targets may have the same AoA values y^A, but they could be distinguishable from one another based on the motion-induced parameter values y^, or vice versa. As described in more detail below with respect to Figure 17, various frameworks and filters may be utilized to track the objects and reduce ambiguity in solutions sets. In some embodiments, the 2D spectrum defined by Equation 10 provides two peaks corresponding to each target in the area. For example, the nth target generates peaks in the spectrum at

, Y_h ) (^— Y_{h >} - Y_h )· In some embodiments, this ambiguity is eliminated by selecting the location of the reference transmitter 1502. For example, in some embodiments, the reference transmitter 1502 is placed at one extreme of the angle-of-arrival space of the receiver array (fo = 0° or fo = 180°). Assuming placement of the transmitter at fo = 0° as shown in Fig. l5a, then

= 1 - cos f_h, which is a quantity that lies in the interval [0, 2]. This implies that -

lies in [-2, 0]. Because these two intervals are disjointed, we can restrict the search space of y^A in the spectrum to the [0, 2] interval. Then, the n^th target generates only one peak in the limited spectrum at

, y_h \ thereby eliminating the ambiguity in the sign of both the y parameters. A similar analysis can be derived for the case when the transmitter is located such that fo = 180°. The proposed joint framework therefore eliminates the ambiguity in the peaks and provides a larger search space for multiple targets in the spectrum. Figure 16 shows an example of a 2D spectrum with three peaks corresponding to three targets in the region -2 <

< 2, and 0 < y^A < 2. The locations of the peaks in the

Y_h ) space are (1.2, 1.4), (1.2, 0.6), and (-1.2, 0.6). In the 1D analysis for

, all the peaks would not be resolvable since they have the same absolute value of 1.2. Similarly, the 1D analysis for the y_h dimension includes two peaks also not resolvable due to a shared absolute value of 0.6. However, in the joint 2D spectrum all three peaks are resolvable.

[0099] Figure 17 is a flowchart illustrating multi-dimensional framework for estimating a 2D spectrum from the raw spatio-temporal magnitude- squared measurements I c(t, d) I² according to some embodiments. The embodiment shown in Figure 17 utilizes the Multiple Signal Classification (MUSIC) algorithm, although as described above other spectral analysis techniques may be utilized. In some embodiments, a benefit of using MUSIC for the joint estimation of parameters is that the resolvability of paths in each dimension depends on the length of the arrays in both dimensions. That is, while a longer time window better resolves paths in the dimension of time, it can also help resolve paths in the dimension of space, i.e. paths that have the same ⁴ but different y .

[00100] At step 1700, the measured signal is organized into a matrix C that represents the 2-dimensional magnitude measurements. For example, in some embodiments the matrix C is a MA X M_T matrix of magnitude- squared measurements in the spatio-temporal window. In some embodiments, the plurality of receiver antennas l504a, l504b, .... 1504* sample the received signal at a rate of l/T_s samples/sec for a duration of T_Window The sample rate l/T_s and duration T_Wmdow may be programmed or selected dynamically based on the application. For example, in one embodiment the sample duration T _mdow is selected to be approximately 0.5 seconds. Selecting a relatively small duration T _mdow allows objects to be tracked without requiring that they move in straight lines. The sample rate l/T_s is selected based on the maximum frequency content for fi and f

[00101] The number of samples in space and time are thus M_A and Mr = \T_Wmdow/T_s

J, respectively. The matrix C can thus be expressed as follows:

(ID wherein a = c ((i - 1) d_a , (j - 1) T_s ) is the measured 2D received signal described in Equation 9.

[00102] In some embodiments, the 2D spectral content is estimated by defining a steering vector s(i _n , ip^) based on the equation shown in Figure 18, which includes components describes the array measurements at time t = 0, array measurements at time t = T_s, and array measurements at time t = (Mr- 1 )T_S. A vectorized form of C can be written in terms of the steering vectors of the paths arriving at the Rx array as follows:

Where ( ) denotes the vectorized form of a matrix, S is an MAMTXN matrix whose H^l|1 column i

[00103] At step 1702 the matrix C is utilized to generate a 2D spectrum P using Fourier analysis, or a pseudo-spectrum P of the measurement matrix C using subspace- based analysis techniques (e.g., MUSIC). For example, the MUSIC algorithm is utilized in some embodiments to calculate the eigen-decomposition of the correlation matrix R_c of the measurement vector (C), using the following equation:

Where R_A = E{AA^H}, R = E | hh" } , and E{ } is the expectation operator. The eigenvectors of R_c can be divided into bases of a signal subspace, whose dimension is equal to the rank of R_A, and bases of a noise subspace, which is orthogonal to all the steering vectors corresponding to the N signal paths arriving at the receiver array. A pseudospectrum P(y/^M, y ) as follows:

Wherein EN is a matrix whose columns constitute the bases for the noise subspace. The pseudo spectrum P(y Z¹, y ) has peaks at the locations

y , n = 1, ..., N, because the steering vectors corresponding to these locations are orthogonal to the noise subspace EN- Extracting the locations of the peaks of P(y/^M, y/^A) provides the required

pairs needed for tracking the N targets. In some embodiments, the MUSIC algorithm assumes that all different N signals are uncorrelated. This assumption is not valid in some scenarios in which scattering and multipath propagation are present. In some embodiments, spatial smoothing is utilized to uncorrelate the signals. For example, in one embodiments, the matrix R_c is calculated by averaging the correlation matrices of different subsets of the antenna array, given that each of the subsets is a set of contiguous antennas. To address the correlation of signals in the 2D framework, spatial smoothing is extended to spatio- temporal smoothing MUSIC. The matrix C is divided into overlapping sub-matrices C^sub of size M^^ub x M ^Ub each. The correlation matrix R, is then calculated as the average of the correlation matrices R ^ub of the sub-matrices C^sub

[00104] At step 1704, the locations of the peaks within the 2D spectrum P of the data left, d) 1² in a time window of duration T_Wmdow· In one embodiment, the pseudo spectrum R(y^M , I//' ) is provided by the following equation as:

where J is the number of detected peaks int eh pseudospectrum. This information is subsequently utilized to estimate the tracks of the A targets.

[00105] In some embodiments, additional dynamical system modeling and/or filtering may be utilized. At step 1706 a dynamical system is utilized to represent the motion of the one or more targets, along with the relation of the 2D spectrum locations of the detected peaks within the pseudospectrum are utilized to track multiple targets. In some embodiments, extraction of information regarding the location and heading of the targets at time t relies on application of the 2D spatio-temporal smoothing MUSIC algorithm on the data left, d) I² in a time window of duration T_Wmdow starting at time t, to extract the set of peaks y₍ at time t. To avoid issues associated with ambiguity and associate the measurements associated with the motion of the targets are modeled as a nonlinear dynamical system. In addition, a non-linear filter such as a particle filter (PF) with a Joint Probabilistic Data Association Filter (JPDAF) is utilized to solve the dynamical system and determine an estimate regarding the location and movement of each object. [00106] For example, in one embodiment a transmitter (Tx) is presumed located at

(. ct ,yr) and a receiver array (Rx) array is presumed centered at (X_R, \_R) such that its array axis is parallel to the x-axis, as shown in Figure 15. The state of the n^th target at time t is defined as a 4-dimensional vector x? = [x_n(t), y_n(t), q_h(ί), v, ///] ' , where x„( l ), y_n(t) define the location of the n^th target at time t, ()„( l ) is the direction of the target with respect to the A-axis at time t , and v„(l) is the speed of the target at time t. The measurement process y_h( l ) is defined as the pair ( y_h (ί ), y_h (ί ), which is related to the target’s state as follows:

and

where hci and h_A are measurement noise processes with variances

and s^_A, respectively. On the other hand, a simple motions dynamics model is utilized in which a target maintains the same direction of motion with probability P_c, and occasionally changes that direction with probability 1 - P_c. More specifically the state of the n^th target evolves with time according to the model

follows:

wherein h_Ch, h_gh, h_qgi, and h_ngi are all dynamics noises processes with variances

, q_hq , and q_hnh, respectively and U(0, 2p) is the uniform distribution in the interval [0, 2 p]. For the estimation of the state of the n^th target x? at time t, the filtering Probability Density Function (PDF) p(x \ yi_:( ) of the n^th target’s state at time t given all the measurements up to time t. The mean of this PDF is the estimate of the target’s state x? = E{Xt \l _l:t}·

[00107] At step 1708, the dynamical system is solved to estimate the location, direction and/or speed of each of the plurality of targets. For example, in some embodiments, a Particle Filter (PF) is utilized to compute the filtering PDF of the n^th target. The Particle Filter (PF) approximates probability distribution using samples (or particles) drawn from that distribution. For example, in one embodiment the Particle Filter (PF) described in Algorithm 3, reproduced below.

Algorithm 3: Particle Filter for Motion Tracking

Input: Total tracking time T. Number of particles I, Number of

moving people N, Measurements Y _cg

Output: Estimate of the target states ”_T, « ~ 1, 2, . . . , N

5: Compute the importance w the jPDAF m

Algorithm 2, and normalize

a. Estimate the initial state of the ^i} target as it” -

1

11: end for

12: Compute the importance weights

using the JPDAF in

Algorithm 2. and normalize

1.3: Estimate the state of the

target as x?

i* end for

[00108] The weights w]^t,n^ assigned by the Particle Filter (PF) represent how well they fit the current set of measurements Yi (step four in the Particle Filter Algorithm). However, the Particle Filter lacks the knowledge of which of the measurements in Yi is generated by which target (referred to as association problem). In some embodiments, this problem is overcome through the use of a Joint Probabilistic Data Association Filter (JPDAF) to calculate the importance weights (see the discussion of Algorithm 4, below). Having calculated the important weights using the JPDAF, the particle filter resamples the particles to retain the most heavily weighted particles. The resampled particles evolve according to the motion model and the process is repeated for consecutive time instants.

[00109] As discussed above, the motion modeling utilized introduces errors, such that some of the measurements turn out to be false alarms not associated with any target. The probability of such alarms is denoted as a probability P_FA The opposite problem is associated with missing measurements from the solution set Y*. The probability of a target miss is denoted as a probability 1 -P_D, where P_D is the detection probability. The JPDAF algorithm calculates the probabilities of all possible association profiles given the current set of measurements and particles. An association profile co matches each target to one of the J_t measurements. That is, an association profile co is a set of A pairs (k, l ) where / = 1, 2, ..., N, k e{0, 1 , . . . , }, and a pair (k, l ) represents assigning the measurements y_/,. to the I^th target. The probability of the n^th target generating the measurement »/// can be computed by summing the probabilities of all the association probiles which assign the measurement »/// to the n^th target. The set of all such association profiles by W,,,. i_.c.. W,,,. = { co; (j, n) e co } . The algorithm for the Joint Probabilistic Data Association Filter for Particle Weight Calculation is reproduced below.

Algorithm 4: Joint Probabilistic Data Association Filter for Particle Weight Calculation

Input Aii current particles :

Output: 1 he particles·' importance weights w^{| J}'^fS*

l: Calculate the

2: Calculate g ^ f |x ***1), which denotes the probability of

the easurement being generated b the n target having

a state

according Eq 12 and Eq. 13

3; Generate all possible association profiles «, where e> ~

i k ϊk ¾ 10, L , _rJ l ~ { L . . N | } and (1% I) is a pair

assign ng the measurement ¾ to the / ^s ta get

; Calculate the probability of each association profile as

where &½ is a subset of with targets net being assigned to

any of the measurements_» Ie„ & _™ | ( E, I j€ <¾& ¾]·

S: Calcniat the probability that a measurement is caused by

the rfo target b_ίh by summing over all assodaiioa profiles

making such

6; Calculate the importance wei hts

[00110] Experimental results associated with one or more embodiments described with respect to Figures l5a-l7 are described in more detail in the following journal article, C. Karanam, B. Korany, Y Mostofi“Tracking from One Side - Multi-Person Passive Tracking with WiFi Magnitude Measurements”, 18th ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN), April 2019, incorporated by reference herein. [00111] While the disclosed embodiments focused on the harder problem of AoA estimation, object localization, and tracking based on magnitude measurements, if additional phase measurement is available, it can be easily incorporated in the framework.

[00112] While the invention has been described with reference to an exemplary embodiment(s), it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment(s) disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.

Claims

CLAIMS:

1. A method of estimating the angle of arrival of one or more incoming waves, the method comprising:

measuring a magnitude of a signal received at a first receiver position,

measuring a magnitude of a signal received at one or more additional receiver positions, wherein the first receiver position and the one or more additional receiver positions form a receiver array,

estimating the angle of arrivals based on the measured signal magnitudes.

2. The method of claim 1, where the signal magnitude measurements are related to the angle of arrival through spectral estimation.

3. The method of claim 2, where spectral estimation involves auto-correlation function analysis, Fourier analysis, or subspace analysis, relating the signal magnitude measurements to the angle of arrivals.

4. The method of claim 1, wherein estimating the angle of arrivals based on the measured signal magnitudes further includes:

estimating spectral content of measured signal magnitudes using spectral estimation techniques;

locating frequency peaks within the estimated spectral content; and

iteratively analyzing information obtained from the located frequency peaks to estimate the angle of arrivals of the one or more incoming waves.

5. The method of claim 4, wherein iteratively analyzing information obtained from the located peaks includes:

determining, based on the located frequency peaks, values representing a difference between the cosines of the angles of arrival; initializing a solution set bounded based on information known about the reference source;

utilize the difference between the cosines of the angles of arrival to identify possible solution within the bounded solution set; and

iteratively evaluate the possible solutions to determine validity of possible angles of arrival.

6. The method of claim 1, where the receiver array is comprised of a plurality of receivers, each receiver located at a different receiver position.

7. The method of claim 1, wherein the receiver array is an emulated array comprised of a single receiver that moves through a plurality of positions.

8. The method of claim 1, further including:

measuring a magnitude of a signal received at one or more additional receiver positions that form a second receiver array and utilizing the measured magnitudes at both first and second arrays to jointly estimate AoA of incoming waves.

9. The method of claim 1, further including estimating an angular location of one or more objects located within a region, wherein one or more of the incoming signals are transmitted by the one or more objects or represent reflections off of the one or more objects, wherein angular localization of the one or more objects is based on the estimated angle of arrival from transmission by or reflection from the one or more objects.

10. The method of claim 9, further including:

transmitting a reference signal from a reference source, wherein the reference signal is included in the signal measured at the plurality of receiver positions and wherein a location of the reference source relative to the receiver array is known.

11. A method of estimating a track of one or more moving targets within a region, the method comprising:

measuring a magnitude of a signal received at a first receiver position at a

plurality of times (/), wherein the signal received at the first receiver position at each moment in time is comprised of one or more signals transmitted by or reflected from one or more moving targets within the region; and

estimating the track of one or more targets based, at least in part, on the received magnitude measurements.

12. The method of claim 11, further including,

estimating the spectral content of the measured signal magnitudes measured at the plurality of times /;

relating the spectral content to parameters of the track of the moving targets; estimating the track of one or more moving targets based on this relationship, wherein the track estimates include one or more of location, direction, and speed of each moving target within the region.

13. The method of claim 11, wherein relating the spectral content to parameters of the track includes locating frequency peaks within the estimated spectral content and wherein track estimates are related to the frequency peaks located within the estimated spectral content.

14. The method of claim 11, further including:

measuring a magnitude of a signal received at a second receiver position at a plurality of time (/), wherein the signal received at the second receiver position at each moment in time is comprised of one or more signals transmitted by or reflected from one or more objects within the region, and utilizing the magnitude of signals received at the second receiver position to

jointly estimate the track of the one or more targets.

15. The method of claim 11, further including:

transmitting a reference signal from a reference source, wherein the reference signal is included in the signal magnitudes measured at the plurality of times t and wherein a location of the reference source relative to the receiver array is known.

16. A method of estimating tracking information of one or more moving targets within a region, the method comprising:

receiving a plurality of signals at a receiver array comprised of a plurality of

receivers, wherein the received signals are transmitted by or reflected off of one or more moving targets located within the region;

measuring a magnitude of the plurality of signals received at the plurality of

receivers at a plurality of instances in time t;

estimating track parameters associated with one or more moving targets based on the signal magnitude measurements, wherein the tracking information describes one or more of location, direction, and speed of the one or more moving targets within the region.

17. The method of claim 16, further comprising:

estimating the 2D spectral content of the measured signal magnitudes utilizing one or more of auto-correlation function analysis, Fourier analysis, subspace analysis, or Multiple Signal Classification (MUSIC);

relating the 2D spectral content to the parameters of the track of the moving

targets; estimating track parameters associated with one or more moving targets based on this relationship, wherein the tracking information describes one or more of location, direction, and speed of the one or more objects within the region.

18. The method of claim 17, further comprising:

applying one or more filters to the spectral content to improve the estimated track parameters.

19. The method of claim 18, wherein applying one or more filters includes applying a motion model to the movement of objects within the region.

20. The method of claim 19, wherein applying one or more filters includes applying a particle filter with a Joint Probabilistic Data Association Filter (JPDA) to the tracking information.