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Research ArticleNonuniformly Spaced Array with the Direct Data DomainMethod for 2D Angle-of-Arrival Measurement in ElectronicSupport Measures Application from 6 to 18GHz

Chen Wu and Janaka Elangage

Defence Research and Development Canada Ottawa Research Centre Ottawa K1A 0Z4 Canada

Correspondence should be addressed to Chen Wu chenwudrdc-rddcgcca

Received 27 May 2020 Revised 24 July 2020 Accepted 3 August 2020 Published 2 September 2020

Guest Editor Jingjing Cai

Copyright copy 2020 Chen Wu and Janaka Elangage )is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

)is paper introduces a 2D angle-of-arrival (AoA) estimator which has a 6ndash18GHz 7-element nonuniformly spaced array (NSA)and a Direct Data Domain- (D3-) based AoA algorithm for a 2D isotropic-element planar array (IEPA) A 2D calibration anddata-transformation method is developed to convert the NSA data to the output of the IEPA so that the NSA-measured data canbe used in the D3 algorithm Using the steering vector (SV) of the IEPA and the results derived from the D3 method a new 2DAoA searching method is also developed which offers frequency-independent performance defined by the probability of AoAestimation when the required estimation accuracy and signal-to-noise ratio (SNR) are given For the applications of electronicsupport measures this paper also presents the use of precalculated SV and data-transformation matrix databases built onpreselected frequency points and a 2D-angle grid that is close to uniformly distributed directions)e simulation results show thatwith good SNR (ge15 dB) the estimator can have 50 probability of AoA estimation with root mean square error (RMSE) less thanor equal to 1deg using just a few samples from the NSA Moreover the study focuses on the applications with low SNR by using moresamples from the NSA Results show that the estimator has 52 and 80 probabilities of AoA estimation with RMSE le1deg and 5degrespectively for phase- or frequency-modulated radar pulses when the SNR is within [minus10 0] dB )e study also shows that theestimator prefers a circular-shaped planar array with a triangular interelement pattern since it presents more symmetricalcharacteristics from different azimuth angles

1 Introduction

)e angle-of-arrival (AoA) of the signal of interest (SOI) isthe most important measurement parameter in an electronicsupport measures (ESM) system to de-interleave interceptedradar signals especially in detecting and classifying lowprobability of intercept signals [1] Traditionally spinningdirection-finding (DF) antenna amplitude-comparisonphase-comparison and interferometry methods are popularAoA measurement methods in ESM systems [2ndash10] Inaddition there are many microwave DF systems developedfor wireless communication applications [11] From a signal-processing perspective there are a number of algorithmsused for signal DF applications Among them multiplesignal classification (MUSIC) [12] and estimation of signal

parameters via rotation invariance techniques (ESPIRT) [13]have been used for many years with different array con-figurations for different applications As examples currentlythese methods are used for 2D AoA estimations for mixedcircular and noncircular incident signals for massive mul-tiple-input multiple-output systems [14] and for uniformrectangular array [15] However to apply these methods astable signal environment is generally required since thesignal covariance matrix needs to be formed A new ap-proach for AoA estimation was introduced using infor-mation geometry (IG) [16] Based on IG Dong et al [17]introduced a simple scaling transform-based informationgeometry (STRIG) method which has more consistentperformance than the original IG method while having ahigh AoA estimation resolution Lonkeng and Zhuang [18]

HindawiInternational Journal of Antennas and PropagationVolume 2020 Article ID 9651650 23 pageshttpsdoiorg10115520209651650

presented a research on 2D DF estimation using arbitraryarrays in MIMO systems and introduced the 2D Fourierdomain line search MUSIC algorithm Krishnaveni et al[19] Devendra and Manjunathachari [20] and Barua et al[21] have surveyed some directions of arrival methodsManyother AoA estimators have also been developed which focuson different array structures and faster processing speedsSome examples are given in [22ndash28]

)ere are also some AoA estimation methods that arebased on compressive sensing (CS) theory [29ndash31] Gurbuzet al [32 33] applied CS to estimate acoustic wave AoA byfocusing on one receiving channel sampling at Nyquist rateand the other channels sampling at CS rate Using the targetbearing as a sparse vector Cevher et al [34] demonstratedmultielement circular acoustic sensor arrays to obtain thebearings of multiple sources by applying an l1-norm min-imization solution called the Dantzig selector [35] Wu andElangage [36] introduced a CS-based ultra-wideband 2DAoA estimation scheme that can estimate AoA of the SOIfrom 2 to 18GHz without any a priori knowledge of theintercepted signals Other examples of treating AoA esti-mation as a sparse recovery problem can be found in[37ndash40] Bayesian CS- (BCS-) based AoA estimation was alsodeveloped in [41ndash43] )e key advantage of the BCS AoAestimator is its ability to estimate signal AoA with fewmeasured data from each element in the array

To estimate the AoA of SOI in a nonstationary envi-ronment and to avoid having to form a signal covariancematrix the Direct Data Domain (D3) method was intro-duced [44ndash47] Using the D3 method and the concept ofsignal cyclostationarity Sarker et al [48] introduced amethod that can handle a number of signals along with theirvarious coherent and noncoherent multipaths and inter-ferences even when the number of signals exceeds thenumber of antenna elements)is was demonstrated using a12-element uniform linear array Some examples of usingthe D3method for AoA estimations can be found in [49ndash54]

Most aforementioned methods and their usages onantenna arrays are suitable for narrow-band applications asthe array elements and their configurations closely relate tothe signal wavelengths Hence they can be used mainly forAoA estimations in communication applications Interfer-ometry-based AoA measurement systems have been widelyused in ESM systems for 2D AoA measurements in ultra-wide frequency ranges [2ndash4] It is desirable to use the longestpossible baseline in the array to achieve low variancemeasurement results However longer baselines requirehigher signal-to-noise ratios (SNRs) in order to resolve theambiguity using the associated trigonometric functions Inaddition even if the wideband antenna elements such ascavity-backed spiral antennas (CBSA) are used in the five-element array in [2] the spacing of the short baseline in thearray is chosen to be less than half of the wavelength of thehighest frequency of interest As a result several differentfive-element interferometers are required to cover a wideoperational frequency range )is results in a large antennaarray footprint Pasala et al [5] introduced a three-elementinterferometer array that used multimodes in the elements

to avoid the short baseline required in the conventional five-element array configuration )e long baseline (larger arrayfootprint) is still required to resolve the ambiguity Howeverthese large-footprint arrays are problematic for small air-borne platforms such as CubeSats (U-class spacecraft) [55]small unmanned aerial vehicles [56 57] and drones

In this paper we propose a novel 2D angle-of-arrival(AoA) estimator for ESM applications from 6 to 18GHz inlow SNR environments with frequency-independent per-formance when the SNR and the required estimation ac-curacy are given )is includes the following contributions

(i) A 2D 7-element compact nonuniformly spacedarray (NSA) is designed with CBSA elements op-erating from 6 to 18GHz which has an arrayfootprint that is smaller than a circle having a414mm radius

(ii) A 2D AoA measurement system applies the D3method on the measured time-domain data(snapshots) from the 7-element NSA )is includesthe following

(a) A 2D data transformation from the 7-elementNSA-measured data to a 2D isotropic-elementplanar array (IEPA) data is introduced in orderto use the NSA data in the D3 method for AoAestimation and six different 2D IEPAs areconsidered

(b) Using 2D IEPA steering vector (SV) and theresults derived from the D3 method a newobjective function is introduced for searchingthe AoA of the SOI solution in both azimuth(Az) and elevation (El) angles )is (1) avoidshaving to solve the ambiguity problem en-countered in interferometric arrays and (2)gives the estimator a frequency-independentperformance

(c) Using a 3D icosphere a 2D-angle grid is in-troduced to give a near-uniform angular dis-tribution in the field of view (FOV) of the 7-element NSA On this grid the data-transfor-mation matrix database and SVs of IEPAs areprecalculated at a list of preselected frequencypoints and stored in a computer prior to AoAestimations Hence the estimator can be usedfor ESM applications without a priori knowl-edge of the SOI

(iii) Since the focus of this development is ESM appli-cation in low SNR environments different SNRlevels are studied with four different commonlyused radar waveforms )ey are the following (1) a13-chip Barker-coded waveform (2) a 20-chip two-valued frequency-coded waveform [58] (3) a 16-chip poly-phase-coded waveform and (4) a ultra-wideband (100MHz chirp) frequency-modulatedcontinuous waveform (FMCW)In addition to the above contributions the findingsof this study are highlighted as follows

2 International Journal of Antennas and Propagation

(i) )e preferred IEPA for this application has a cir-cular-shape which also has the least number ofelements

(ii) )e new AoA estimation scheme shows frequency-independent performance defined by the proba-bility of AoA estimation when the required AoAestimation accuracy is specified for a fixed SNR)isperformance results from using the nonuniformly(nonregularly) spaced CBSA array applying the 2Ddata transformation to convert measured data fromthe NSA to a planar array and then using the new 2DAoA searching method introduced in this paper

(iii) In general the more snapshots used in AoA esti-mation the better the accuracy that can be achievedor the higher the probability of AoA estimation for agiven SNR and desired accuracy

(iv) )e estimator can estimate 2D AoA using fewsnapshots in a high SNR environment

)e paper is organized as follows )e 7-element NSAdesign is presented in Section 2 )e AoA estimation al-gorithm for a planar array with isotropic radiators in freespace is then discussed in Section 3 Section 4 presents a 2Ddata transformation that maps the time-domain sampleddata (or snapshots) from the 7-element NSA to the data of anIEPA so that the D3 method introduced in Section 3 can beapplied Section 5 discusses how to apply the AoA estimationmethod to an ESM system with ultrawide frequency rangewithout a priori knowledge of the frequency of the SOI )esimulation results and discussion are presented in Section 6)e last section gives the conclusions and future work)ereare four appendixes in this paper Appendix A defines the sixIEPAs A description of the 2D-angle grid on the vertices of aunit icosphere is given in Appendix B Four commonly usedradar waveforms with their parameters are given in Ap-pendix C Appendix D explains the reasons why the esti-mator has frequency-independent performance with respectto the probability of AoA estimation mentioned above

2 Design of a 7-Element 6ndash18GHz NSA

Figure 1 illustrates the 7-element CBSA array and Figure 2shows nonuniform interelement spacings )e element lo-cations in XY-plane are given in Table 1 )e reasons forchoosing the element locations are to ensure that (1) elementspacings in X and Y directions are as close to half thewavelength of the highest operational frequency as possible(2) the distances between adjacent element centers are biggerthan the diameter of the CBSA element and (3) the overallarray footprint is as small as possible As a result the averageelement spacing in both X and Y directions is about844mm which is a little bigger than half of the free-spacewavelength at 18GHz ie 833mm)e element used in theNSA is a commercial-off-the-shelf 6 to 18GHz CBSA with244mm diameter In addition to ultra-wideband opera-tional frequency band of the array another advantage ofusing the CBSA element is the possibility of neglecting theperformance changes caused by the mutual coupling

between adjacent elements in the array [59] It is mainlybecause the element has about 80deg to 100deg 3 dB beamwidth inits broadside and very low radiation when El-angle ap-proaches 0deg Examples of measured antenna radiation can befound in [60]

X

Y

Z

3

4

5

1

6

7

2El

AZ

Figure 1 A 3D illustration of the 7-element CBSA NSA )ephase-reference center is at the first element center

2672

2538

2581

2565

26722522

2709

2522

2582

2540

2582

2694

Figure 2 )e top view of element centers with interelementspacings (unit is in mm)

Table 1 Antenna element locations

Element pxn (mm) pyn (mm) pzn (mm)

1 00 00 002 2474 679 003 679 2474 004 minus1875 1687 005 minus2590 minus794 006 minus794 minus2590 007 1687 minus1875 00

International Journal of Antennas and Propagation 3

3 AoA Estimation from a Planar Array withIsotropic Elements

Before discussing the AoA estimation algorithm for theNSA this section briefly presents the AoA estimation al-gorithm for a planar array with isotropic elements which isthe 2D version of the development given in [48] Althoughin this study the D3 method was applied to 2D planar arraysgiven in Appendix A the AoA estimation method proposedin this section can also be applicable to other sparse arraystructures such as 2D nested array [22]

31 SOI Representation )e intercepted signal output at thenth antenna element in an IEPA can be expressed as acomplex envelop signal

xn(t) An(t)ej2πfct+ϕ( ) + alefsym(t) +Ω (1)

where A fc and ϕ are the amplitude frequency and initialphase of the waveform of the SOI respectivelyalefsym is the whiteGaussian noise added by the receiver In the followingsimulations it is added to the signal in each receivingchannel based on the signal-to-noise ratio (SNR) defined ineach simulation and the power of the SOIn (1 2 NN) and NN is the total number of elementsin the IEPA Ω represents other unwanted signals such asjamming signals Since signal collision or pulse collision isnot considered in this study Ω is excluded in the latterequations If the received signal in (1) is sampled at a set oftime instants m (1 2 M M + 1) the collected datacan be expressed as

xn(m) An(m)ej2πfcmiddotmmiddotdt+ϕ( ) +alefsym(m) (2)

where dt is the temporal sampling intervalIf the phase reference is at the center element of the

array and it is located at the origin of the coordinate systemthe received signal of the mth snapshot at the nth element is

xn(m) An(m)e(j2πλ) pxn cos(El)cos(Az)+pyn cos(El)sin(Az)+pzn sin(El)[ ]

+ alefsym(m)

(3)

where λ is the SOI free-space wavelength (AzEl) is the SOIincident direction and (pxn pyn pzn) are given in Table 1

32 Cyclostationarity of the SOI )e cyclostationarity is anonlinear transformation operating on a signal and itgenerates finite-strength additive sine-wave componentsthat result in spectral lines [61] A signal x(t) is assumed to

be having a cyclostationary feature with cycle-frequency η ifand only if the product x(t) middot x(t minus τ) for some delay τshows a spectral line at frequency η

)e signals to be detected in our problem are cosine-wave based signals for example

A(t)cos 2πfct( 1113857 (4)

Let us assume the second-order nonlinear transforma-tion which is equivalent to multiplying the signal with itsshifted version or simply to square the SOI the new signalhas a cycle-frequency η that can be

A2(t) middot cos2 2πfct( 1113857

A2(t)

21113888 1113889cos(2πηt) + constant (5)

η 2fc (6)

where it assumes that the delay or shift is zero )e complexenvelope form of (5) is

A2(t)e

j2πηt (7)

In many applications the SOI cyclostationary infor-mation is a priori known information for example thecycle-frequency of the carrier frequency between twocommunication sites However in ESM the interceptedsignal frequency (fc) has to be estimated

33 D3-Based AoAMethod for an IEPA First let us apply asecond-order nonlinear transform to the signal in (2) as wedid in (5) and then regroup the transformed signals into thesignal with cycle-frequency and other frequency compo-nents We have

x2n(m) A

2(m)e

j2πηmiddotmmiddotdt+ Others (8)

where Others includes other frequency components notaround η 2fc and noise Note that in this study the signalcollision is not considered ie the signal to be processedonly has one carrier in the current RF receiver band eventhough the receiver can be tuned from 6 to 18GHz UsingM+ 1 temporal snapshots from each element of anNN-element IEPA the following matrix equation can beconstructed )e detailed derivation and discussion aregiven in Chapter 7 of [48] for a linear array case Here weextend it to the 2D planar array application

Z(NN+1) timesMWMtimes1 E(NN+1)times1 (9)

where

Z

1 middot middot middot ej2π(Mminus1)ηmiddotdt

x21(1) minus x

21(2)e

minusj2πηmiddotdtmiddot middot middot x

21(M) minus x

21(M + 1)e

minusj2πηmiddotdt

⋮ middot middot middot ⋮

x2NN(1) minus x

2NN(2)e


2NN(M) minus x

2NN(M + 1)e

minusj2πηmiddotdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)


W is the unknown weight vector that can be detailed as

W

w1

w2

⋮

wM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)

and E is the excitation vector given as

E

C

0

⋮

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)

)e length of column vector E is NN + 1 and theconstant C can be arbitrarily selected and it also satisfies thefollowing relation with weight vector

C 1113944M

m1wme

j2π(mminus1)ηmiddotdt (13)

However (13) is not a necessary condition since theconstant C gets factored out in the final AoA computationEquation (13) expresses that the sum of the weights producesa gain factor C when they are applied to the SOI at the cycle-frequency )e MoorendashPenrose pseudo-inverse was used tosolve (9) in order to obtain W which has W2 C

M

radic

)e elements from the second row and after in (10) canbe written as x2

n(m) minus x2n(m + 1)eminusj2πηmiddotdt which tells that the

SOI component at the cycle-frequency η 2fc is removedfrom the matrix elements )us the weight vector W ob-tained from (9) minimizes the noise and other frequencycomponents and simultaneously satisfies (13) )is is one ofthe reasons why the algorithm has the ability to measureweak signals from noise and can give a good estimation ofthe AoA of the SOI

)e advantages of (9) are as follows (1) the temporal dataare directly used without being converted to frequencydomain and no covariance matrix is required and (2) thenumber of time-domain snapshots can be very small egM + 1 2 in high SNR cases Hence it can be used indynamic signal environments with good SNR

34 Detecting SOI by Estimating Its AoA Using IEPA For agiven element n in an IEPA when the (n + 1)th row ofmatrixZ in (10) is multiplied with the weight-vector W one has

1113944

M

m1x2n(m) minus x

2n(m + 1) middot e

minusj2πηmiddotdt1113966 1113967wm 0 (14)

From (14) considering wm ne 0 ej2πηmiddotdt ne 0 and (13) thefollowing equation can be derived

1C

1113944

M

m1x2n(m)wm e

j4πλest( ) pxncos(El)cos(Az)+pyncos(El)sin(Az)[ ]

(15)

where it is assumed that pzn 0 λest (cfcest) and c is the

speed of light in free space Equation (15) unveils someimportant results including the following

(i) )e operation 1113936Mm1 x2

n(m)wm means that theweight vector applies a filter to the squared values ofsnapshots

(ii) )e sum of the filtered data in the Item-1 above onlyrelates to the AoA of SOI

(iii) )e weights only can be used to filter the current setof squared values of snapshots used in (9) tominimize the undesired signal at the output Ifsnapshots are changed the weights obtained from(9) will be changed accordingly but the sumwill notbe changed

)e Item-3 in the above discussion reflects the dynamicnature of this method where the processing only deals withthe current set of snapshots fed to (9)

Since (15) uses the nth elementrsquos measured snapshots andhas two unknowns (AzEl) the solution can be calculatedusing any two elements in the array to form two equationsfrom (15) However we are going to introduce a new ap-proach to find AoA in the following sections which uses theIEPA SV and a 2D searching method )e advantage of thenew approach is the ability to avoid having to pick answersfrom an unlimited number of solutions of inverse trigo-nometric functions and the possibility of finding AoA so-lution with no dependency on frequency at a fixed SNR)ereason for the frequency-independent behavior is discussedin Appendix D

)is new AoA searching method is the second importantreason that explains why our scheme can robustly measurethe weak signals and how it can offer a good estimation ofAOA of the SOI)e novelty of this approach is using all theNN elements together for the AoA solution rather thanusing a few elements in the array to find the AoA solutionfrom inverse trigonometric functions

35 IEPA Steering Vector andObjective Function If an IEPAis located at the origin of the Cartesian coordinate system itsSV for an incoming plane wave is

sv

ej1113954kmiddot rrarr1


⋮

ej1113954kmiddot rrarrn

⋮

ej1113954kmiddot rrarrNN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

where 1113954k is the wave vector and rrarrn is the displacement vectorfrom the signal source to the nth element in the array

If the center element of the array is located at the originand its phase is used as the phase reference then the array SVbecomes


sv(az el)

ej(2πλ) px1 cos(el)cos(az)+ py1 cos(el)sin(az)[ ]


⋮

1

⋮

ej(2πλ) pxn cos(el)cos(az)+ pyn cos(el)sin(az)[ ]

⋮

ej(2πλ) pxNN cos(el)cos(az)+ pyNN cos(el)sin(az)[ ]

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(17)

)e SV at the center element is equal to one It is alsoassumed that all the elements have the same locations in zdirection

Comparing (17) with (15) one can find that in order toestimate the AoA (AzEl) a 2D searching method can beused to find (aze ele) by

minazaze

elele

1113944

NN

n1sv2n(az el) minus

1C

1113944

M

m1x2n(m)wm

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(18)

where sv2n is the square of svn at the nth isotropic element in(17) with λ λest which is the same as the right side of (15))e details of the searching method will be given in Section5

4 AoA Estimation Using the 7-Element NSA-Measured Snapshots

)is section discusses how to use snapshots from the 7-element NSA to estimate the AOA of the SOI by takingadvantage of the 2D D3-based AoA method discussed in thelast section

41 2D Transformation Matrix to Convert 7-Element NSA-MeasuredData to IEPAData in Free Space Consider a setupfor the 7-element NSA calibration A far-field source atfrequency fc is placed along the direction (az(p) el(p))and the measured phasor voltages at 7 output ports can beexpressed as yn(p fc) (n 1 2 7) Here p indicates thesource at the pth direction If this source is moved in total P

directions and at each direction one collects the phasorvoltage at each element a measured data matrix Y7timesP(fc)

can be obtained )en replacing the NSA by an NN-ele-ment IEPA and performing the same measurement theoutput phasor voltage matrix from the planar arrayUNNtimesP(fc) can be created In reality instead of makingmeasurement it can be calculated since the isotropic ele-ment in free space does not exist

)e goal is to find the best-fit data-transformationmatrixthat satisfies

TNNtimes7 fc( 1113857 middot Y7timesP fc( 1113857 UNNtimesP fc( 1113857 (19)

)is equation can be solved using the least squaremethod It is achieved by minimizing the following function

minTI

UNNtimesP fc( 1113857 minus TNNtimes7 fc( 1113857 middot Y7timesP fc( 1113857

(20)

where I is the best-fit transformation matrix In order tohave a unique solution of (20) the number of P directionsmust be greater than or equal to the number of antennaelements in the planar array

For frequency fc the transformation matrix (I) has tobe stored in computer memory )en during the operationsnapshots Y7times1(mAzEl fc) from the NSA can be con-verted to snapshots of the NN-element IEPA by

UNNtimes1 mAzEl fc( 1113857 INNtimes7 fc( 1113857Y7times1 mAzEl fc( 1113857

(21)

where the incident direction (Az El) of SOI should bewithin the limits of those directions during the calibrationbut not necessary to be one of the P directions )is is calledthe off-grid case After that the AoA of SOI can be estimatedusing the algorithm discussed in the previous section

Note that different frequencies fc have different trans-formation matrices Obtaining the transformation matricesfor the 6ndash18GHz radar ESM applications without a prioriknowledge of the frequency of SOI is discussed in Section 5

42 Discussion )e calibration method and the data-transformation given in (19) to (21) can be applied to planararrays of any size and any shape Six different IEPAs are usedin this study to demonstrate the approach )ese arrays arepresented in Appendix A

In real applications the data-transformation introducedin this section

(i) eliminates the nonuniformity in the real arraywhich means it transfers a NSA into a regular-spaced IEPA

(ii) converts a 3D real antenna problem into an iso-tropic-element regular-spaced array problem in freespace

(iii) takes into account the mutual coupling between el-ements in a real array when the Ymatrix is measured

(iv) removes the inconsistency among receiver channelswhen the NSA elements are connected to RFreceivers

(v) can remove the electromagnetic interferences by thepresence of near-field scatterers for example thebody of the platform if the calibrationmeasurementis performed with a platform in a big microwaveanechoic chamber

5 Application of the Method to ESM

In the discussion of Sections 3 and 4 although the snapshotsmeasured in time domain are used in (10) and (15) tocalculate the weight vector in (11) the method still requires


the carrier frequency of the intercepted radar signal for thefollowing purposes

(i) To obtain the cycle-frequency(ii) To determine the IEPA SV(iii) To determine the data-transformation matrix

which has to be obtained and stored in computermemory prior to AoA measurements

Unfortunately in radar ESM applications the SOIcarrier frequency has to be estimated (fcest

) )is sectionintroduces an approach to deal with this problem

First let us discuss how to obtain the precalculated SVdatabase and the transformation matrix I database for agiven IEPA based on the 2D-angle grid introduced in Ap-pendix B )en we give the detailed steps of using thesedatabases for AoA estimations

51 Precalculated SV Database and I Database From 6 to18GHz total 2401 equal-spaced frequency points (fci

) arepreselected to develop the databases so that the frequencystep is 5MHz For an IEPA the SV database can be writtenas

squared SV fci z1113872 1113873 sv2n azz(p) elz(p) fci

1113872 1113873 (22)

where n 1 2 NN and i 1 2 2401z 1 2 36 and p is the pth direction in the zth anglezone shown in Figure 17(a) in AppendixB Ptotal 1113936

36z1P(z) 29495 where P(z) is the total di-

rections in the zth angle zone)e I matrix used in (21) is developed in each angle

zone at every frequency (fci) and can be written as

INNtimes7 fci z1113872 1113873 (23)

Basically the number of directions P in (19) is replacedby P(z) in each angle zone Hence for each planar array atevery frequency point and within each angle zone a pair ofprecalculated squared-SV and I matrix is saved in thecomputer memory

52 AoA Measurement without a Priori Knowledge of theCarrier of the SOI During the AoA measurement once thefcest

is obtained from the ESM system the closest frequencypoint (fcI

) in (fci i 1 2 2401) can be found and the

corresponding squared SV(fcI z) and INNtimes7(fcI

z) areused in the AoA estimation To search AoA solution thesecond term in (18) obtained by the D3 method is comparedwith the first term at each direction defined by the 2D-anglegrid in Figure 17(a) in Appendix B through 36 angle zones)en the direction that gives the minimum sum value in(18) is used as the solution

6 Simulation Results and Discussion

In this section the simulation results are first presentedbased on the probability of AoA measurement versus SNRunder the conditions of (1) different RMSE requirements

and (2) using different (M + 1) snapshots during estima-tions )e RMSE of each AoA estimation is defined as

(ΔAz)2 +(ΔEl)2

2

1113971

(24)

where ΔAz and ΔEl are the measurement errors in Az- andEl-angle respectively)en the estimation errors in Az- andEl-angles are discussed Finally the results of the use of asmall number of snapshots for AoA estimation under highSNR conditions are also shown

Simulation cases consist of the data which are processedby the six planar arrays on four different waveforms usingdifferent number of snapshots (M + 1) that varies from 2 to1025 In each case

(i) A section of M + 1 consecutive snapshots is sent toAoA calculation )e starting point of a section ofM + 1 data is randomly selected within a pulse ofdifferent waveforms discussed in Appendix C

(ii) )ere is a total of 100000 AoA estimations(iii) )e incident wave in each AoA estimation is gen-

erated as follows

(a) )e carrier frequency is uniform-randomlypicked in [6 18] GHz

(b) )e SNR of each processed signal is uniform-randomly chosen in [minus20 20] dB It is assumedthat the seven receivers add the same noisepower to the received signals)e noise power iscalculated based on the received signal powerand the noise has white Gaussian distribution

(c) )e incident direction in off-grid case is alsouniform-randomly selected in the FOV

61 Probability of AoA Estimation

611 Comparison with Different IEPAs Tables 2ndash5 providethe SNRs that yield 50 or better probability of AoA esti-mation using 129 257 513 and 1025 snapshots respectivelyfrom the six different arrays (described in Appendix A)under different estimation accuracy requirements In thesetables the values in bold but not italicized show the lowestSNR in a group of 6 SNRs produced by the 6 planar arraysand the bold italic and normal italic values indicate thelargest and the second largest SNRs in a data group re-spectively An IEPA with a lower SNR in a group means thatit has a better capability of measuring AoA than the arrayswith higher SNRs )is is because the array can estimateAoA of weaker signals than the rest of arrays which needstronger signals for the same accuracy

From these tables one can observe that Array-3 has 41out of 64 values given in bold and Array-5 scores 21 )isindicates that Array-3 is the best array among the 6 planararrays to give a good AoA estimation and Array-5 is the nextto Array-3 Note that both of these two arrays are circular-shaped planar arrays with different interelement patterns

On the other hand one can see that Array-6 and Array-1share the most bold-italicized and italic values Array-6 has 50


bold-italicized and 7 italic values and Array-1 has 11 bold-italicized and 38 italic values)ismakes Array-6 to be the leastfavorite array among the 6 IEPAs and Array-1 is the next It isinteresting to note that both Array-6 and Array-1 are rect-angular-shaped planar arrays with different interelementpatterns

)e above observation shows that our AoA estimationmethod prefers the planar array with symmetrical elementdistribution that can be seen from different Az-angles In ad-dition Array-3 has the least number of elements Hence it needsshorter computation time than the other arrays Since Array-3 isthemost favorable planar array out of the 6 arrays the discussionin the rest of this paper uses the data produced by Array-3

612 Probability of AoA Estimation versus SNRFigures 3 and 4 show the results of probabilities of AoAestimation fromArray-3 with estimation accuracies of 1deg and5deg respectively Each subplot of these figures has 4 curvesthat present the results of 4 waveforms From them thefollowing observations can be made

(i) When shorter snapshots are used in AoA calcula-tions ie subplots of in Figures 3(a) and (b) andFigures 4(a) and (b) the 4 curves of the 4 waveformsare very close to each other

(ii) When the number of snapshots is increased themethod still gives very close results for the first threewaveforms but the result of WF-4 starts deviating

from other results as shown in Figures 3(c) and (d)and Figures 4(c) and (d) Eventually when M + 1

1025 in Figures 3(d) and 4(d) the results are worsethan those in Figures 3(andashc) and Figures 4(andashc))is isbecause a longer sequence of FMCW snapshots coversa wider frequency range than those with a shortersequence of snapshots Hence it gives a bigger carrierfrequency estimation error during calculations thatdegrades the probability of AoA estimation Note thatin this study a 100MHz chirp signal has been used

(iii) In general when more snapshots are used in thecalculations better estimation results can be achieved)is can also be clearly observed in Figure 5

62 AoA Estimation versus Frequency and SNR )e prob-abilities of AoA estimation versus frequency and SNRare shown in Figures 6ndash9 for four different waveformsFrom these figures the following observations can bemade

(i) For a given estimation accuracy ie RMSE atdifferent frequencies the method has about thesame AoA estimation performance against the SNR)e explanation of this performance can be found inAppendix D

(ii) Again the corresponding subplots in first threefigures are basically similar ie the method givessimilar estimation performances for WF-1 to WF-3

Table 2 SNR (dB) values when arrays can produce 50 correct results using 129 snapshots under different estimation accuracies

RMSEle 1deg (Array-1 2 3 4 5 6) RMSEle 2deg (Array-1 2 3 4 5 6) RMSEle 5deg (Array-1 2 3 4 5 6) RMSEle 10deg (Array-1 2 3 4 5 6)WF-1 058minus094minus100minus042minus079087 minus230minus385minus404minus234minus350minus202 minus512minus656minus674minus527minus650minus504 minus690minus792minus779minus703minus803minus694WF-2 071minus089minus095028minus084095 minus216minus383minus391minus221minus365minus181 minus504minus643minus662 minus501minus658minus498 minus689minus802minus787minus687minus810minus671WF-3 063minus090minus100057minus070092 minus219minus374minus382minus224minus363minus196 minus505minus675minus667minus518minus640minus488 minus697minus781minus785minus686minus799minus680WF-4 081minus075minus081048minus065087 minus211minus378minus385minus234minus361minus188 minus506minus669minus666minus507minus670minus492 minus691minus787minus768minus696minus798minus671


RMSEle 1deg (Array-1 2 3 4 5 6) RMSEle 2deg (Array-1 2 3 4 5 6) RMSEle 5deg (Array-1 2 3 4 5 6) RMSEle 10deg (Array-1 2 3 4 5 6)WF-1 minus329minus500minus499minus336minus477minus307 minus568minus759minus766minus582minus747minus569 minus847minus991minus987minus840minus994minus838 minus1020minus1133minus1106minus1011minus1137minus1029WF-2 minus330minus488minus515minus341minus484minus297 minus560minus739minus768minus566minus733minus556 minus837minus999minus1010minus845minus904minus818 minus1024minus1117minus1106minus1028minus1120minus1017WF-3 minus324minus501minus504minus332minus495minus283 minus576minus760minus782minus593minus738minus564 minus841minus988minus999minus855minus990minus853 minus1013minus1112minus1124minus1006minus1128minus994WF-4 minus206minus371minus372minus223minus363minus187 minus454minus636minus652minus481minus619minus424 minus716minus891minus892minus737minus870minus711 minus892minus1006minus1003minus893minus1018minus887




RMSEle 1deg (Array-1 2 3 4 5 6) RMSEle 2deg (Array-1 2 3 4 5 6) RMSEle 5deg (Array-1 2 3 4 5 6) RMSEle 10deg (Array-1 2 3 4 5 6)WF-1 minus498minus672minus699minus518minus673minus478 minus746minus918minus959minus745minus897minus727 minus1006minus1173minus1179minus1025minus1154minus998 minus1187minus1286minus1256minus1175minus1296minus1166WF-2 minus483minus672minus685minus505minus649minus471 minus720minus901minus911minus739minus891minus707 minus988minus1140minus1150minus999minus1148minus998 minus1135minus1257minus1260minus1146minus1275minus1136WF-3 minus488minus666minus691minus521minus648minus492 minus750minus911minus943minus743minus917minus722 minus1002minus1151minus1181minus1010minus1158minus1008 minus1184minus1272minus1273minus1173minus1303minus1169WF-4 minus063minus063minus097071minus054125 minus357minus357minus358minus201minus327minus177 minus592minus592minus606minus466minus634minus448 minus750minus750minus748minus650minus751minus631


(iii) )e subplots in Figure 9 are different from thecorresponding subplots in the other figures and theperformances are not as good as in the other figuresdue to the reason discussed above

63 Estimation Error Distributions in Az and ElFigures 10ndash13 show the histograms of estimation errors ofAz and El when SNR is in different ranges withM + 1 1025 By comparing the left column and the rightcolumn in each figure one can find the following

(i) )e Az-error spreads wider than El-error whichindicates that the AoA estimation method producesbetter estimation results in El than in Az

(ii) )e lower the SNR the wider the spread as expected(iii) Array-3 has almost the same estimation error dis-

tributions when it processes the first threewaveforms

Tables 6ndash9 list the percentages of the absolute value ofestimation errors that are less than or equal to 1deg and 5degwhen the SNR is in different ranges and M + 1 1025 )eresults shown in these tables confirm that the method hasbetter estimation in El-angle than in Az-angle which resultsfrom the definitions of Az- and El-angles

Data in these tables also show the following

(i) When the SNR increases from [0 10] dB to [10 20]dB the method does not improve much of its es-timation accuracy

(ii) From Tables 6 and 7 when the SNR is greater than0dB the estimation performance ismainly determinedby the Az-angle estimation )e results in columns ofldquoBoth Az and Elrdquo and ldquoAzrdquo are very close to each otherand the results in columns of ldquoElrdquo are better than thosein columns of ldquoAzrdquo for both |Er|le 1deg and |Er|le 5deg cases

(iii) From Table 8 (minus10 dBle SNRle 0 dB) although theSNR is less than 0 dB the overall estimation per-formance is slightly less than that of Az-angle es-timation accuracy in both |Er|le 1deg and |Er|le 5degcases )is means that the method starts producingbigger estimation errors in El Nevertheless themethod still can produce better than 52 and 80probabilities of AoA estimation with errors less thanor equal to 1deg and 5deg respectively for the first threewaveforms and better than 33 and 67 proba-bilities of AoA estimation with errors less than andequal to 1deg and 5deg for WF-4 with 513 snapshots

(iv) When the SNR is in [minus20 minus10] dB the method stillhas the capability of making some estimations andproduce better than 20 probability of AoA esti-mation with errors within 5deg for the first threewaveforms

RMSE le 1 (deg)M + 1 = 129

ndash10 10 20ndash20 0SNR (dB)

0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty

ndash10 0 10 20ndash20SNR (dB)

(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)

Figure 3 Probability of AoA measurement of Array-3 when RMSEle 1deg Dashed line is WF-4 result


64 Probability of AoA Estimation Using Few SnapshotsTo study the AoA estimation performance of themethod in agood SNR environment a total of 100000 SNR samples arerandomly picked in [minus20 50] dB and the results are shown inFigure 14 )ere are four groups of curves in each subplot of

the figure )ey represent the use of 2 3 5 and 9 snapshotsin the estimations In each group of curves there are fourcurves for four waveforms Figure 14 shows that the methodcan have good probability of AoA estimation using just a fewsnapshots from the 7-element NSA For example with about

0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)

Figure 5 Probability of AoA measurement of Array-3 on 4 different waveforms with M + 1 17 33 65 129 257 513 and 1025 from rightto left in each subplot Each group of data has 4 curves related to 4 waveforms ooooo lowastlowastlowastlowastlowast and +++++ lines are the results of calculations onWF-4 using 513 1025 and 2049 snapshots respectively

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)

Figure 6 Array-3 probability of AoA estimation versus frequency and SNR (a) RMSEle 1deg and (b) RMSEle 5deg when M + 1 1025 for WF-1

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)

Figure 8 Array-3 probability of AoA estimation versus frequency and SNR (a) RMSEle 1deg and (b) RMSEle5deg when M + 1 1025 for WF-3

1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0Est error (deg) Est error (deg)

5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

Figure 10 )e histograms of Az (left column) and El (right column) estimation errors when 10 dBle SNRle 20 dB with 1025 snapshots (ab) WF-1 (c d) WF-2 (e f ) WF-3 (g h) WF-4

1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

Est error (deg) Est error (deg)

Figure 11)e histograms of Az (left column) and El (right column) estimation errors when 0 dBle SNRle 10 dB with 1025 snapshots (a b)WF-1 (c d) WF-2 (e f ) WF-3 (g h) WF-4


1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)

5Est error (deg) Est error (deg)

Figure 12 )e histograms of Az (left column) and El (right column) estimation errors when minus10 dBle SNRle 0 dB with 1025 snapshots (ab) WF-1 (c d) WF-2 (e f ) WF-3 (g h) WF-4

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

ndash10 ndash5 0(a) (b)

(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10

ndash10 ndash5 0 5 10





ndash10 ndash5 0 5 10Est error (deg) Est error (deg)

Figure 13 )e histograms of Az (left column) and El (right column) estimation errors when minus20 dBle SNRleminus10 dB with 1025 snapshots(a b) WF-1 (c d) WF-2 (e f ) WF-3 (g h) WF-4


Table 6 Array-3 probability of AoA estimation using 1025 snapshots (10 dBle SNRle 20 dB)

Both Az and El |Er|le 1deg |Er|le 5deg Az |Er|le 1deg |Er|le 5deg El |Er|le 1deg |Er|le 5deg

WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669

Table 9 Array-3 probability of AoA estimation using 1025 snapshots (minus20 dBle SNRleminus10 dB and last line using 513 snapshots)


WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045

Table 8 Array-3 probability of AoA estimation using 1025 snapshots (minus10 dBle SNRle 0 dB and last line using 513 snapshots)


WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty

ndash20 0 20 40SNR (dB)

RMSE le 5 (deg)

(b)

Figure 14 Probability of AoA estimation of Array-3 for 4 waveforms with M + 1 2 3 5 and 9 from right to left in each subplot


15 dB SNR and 3 snapshots the method can have 50probability of AoA estimation with RMSEle 1deg

65 Discussion From all the subplots (a) in Figures 3ndash5and Figure 14 one can find that just by increasing SNRit cannot make the method to yield close to 100probability of AoA estimation with RMSE le 1deg )is is

determined by the density of the angles on the 2D-anglegrid (for example Ptotal 29495 in this study) in the 7-element NSA FOV and the number of frequency samples(Ftotal 2401) in [6 18] GHz used to precalculate SV andI databases Increasing these numbers will enhance theperformance of the method However it will slow downthe calculation on a regular desktop computer By usingother means of hardware such as dedicated FPGA with

(a) (b)

(c) (d)

(e) (f )

Figure 15 Six planar arrays black dots are the 7-element NSA (see the text for more details)


Z

X

Y

El

Az

Figure 16 An example of a 2D-angle grid formed by an icosphere-based mesh 7-element NSA phase-reference center and the icospherecenter are collocated at the origin of the XYZ-coordinate system

Table 10 Different 2D-angle grids

Nsubdiv Number of angles within FOV (Ptotal) Rough order of angle resolution (deg)

4 462 4325 1841 2166 7371 1087 29495 0548 117927 027)e bold text indicates that Nsubdiv 7 is used in this study

10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)

Figure 17 (a))e 2D-angle grid in the 7-element FOV (Nsubdiv 7) )e grid is subdivided into 36 angle zones that are defined by 12 equaldivisions in Az-angle and 3 divisions in El-angle ie [40deg to 53deg] [53deg to 70deg] and [70deg to 90deg] (b) A zoom-in plot of zone M in (a)


0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)

Figure 18 One pulse of Barker code (a) 13-chip Barker code (b) the Barker code IQ waveform (solid and dashed lines are I and Qrespectively) and (c) phase in each chip of IQ signal

ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)

Figure 19 One pulse of a two-valued frequency-coded waveform (a) a 20-chip two-valued frequency code ie [0 0 0 0 0 0 0 01 0 01 0 01 011 0 1] (b) the IQ of the waveform (solid and dashed lines are I and Q respectively) and (c) the phase in each chip of the signal


ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)

Figure 20 A pulse of a 16-chip P1 poly-phase waveform (a) IQ of the waveform (solid and dashed lines are I and Q respectively) and (b)the phase in each chip of the signal

0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)

Figure 21 One period of a FMCW waveform

Table 11 FMCW signal parameters

Sweep bandwidth (MHz) Sweep direction Sweep time Number of sweeps100 Triangle 500 μs 2

ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )

Figure 22 )e comparison between the values of the first term (blue oooo) and the imaginary part of the second term (red lowastlowastlowastlowast) of (D2)with SNRinfin when the frequency is at 10GHz the incident direction is from (14375deg 8466deg) to Array-3 the waveform is WF-1 and thenumber of snapshots is 512 Blue plus signs are the real part of the second term in (D2) (a) Freq 6GHz (b) Freq 8GHz (c)Freq 10GHz (d) Freq 14GHz (e) Freq 16GHz (f ) Freq 18GHz


ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB

0 20 40 60Element number in array-3

(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)

Element number in array-3

ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )

Figure 23 )e comparison between the values of the first-term (blue oooo) and the imaginary part of the second-term (red lowastlowastlowastlowast) of (D2)with different SNRs when the frequency is at 10GHz the incident direction is from (14375deg 8466deg) to Array-3 the waveform is WF-1 andthe number of snapshots is 512 Blue plus signs are the real part of the second term in (D2) (a) SNR 0 dB (b) SNR 5 dB (c) SNR 10 dB(d) SNR 15 dB (e) SNR 25 dB (f ) SNR 35 dB

6 8 10 12Frequency (GHz)

14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()

lowastlowastlowastlowastlowastlowast SNR = 20dBoooo SNR = 10dBxxxx SNR = 0dB

(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)

Figure 24 Frequency vs probability of AoA estimation for Array-3 and WF-1 with 512 snapshots (a) RMSEle 1deg and (b) RMSEle 5deg


fast access memory the speed of the calculation can beaccelerated

7 Conclusions

)is paper introduces a new radar signal AoA estimator using a7-element 6ndash18GHz CBSA NSA and a 2D D3 method suitablefor ESM applications Unlike the 3- and 5-element CBSA arraysthat are commonly used by interferometry-based AoAmethods our array can be installed on small airborne platforms)e paper also introduces a 2D calibration and data-trans-formation method that allows an IEPA to use the 7-elementNSA-measured data for AoA estimations and converts a full 3Dantenna problem into a 3D free-space problem To calculate theAoA a new 2D AoA search method has been developed bytaking the advantages of the SV of the IEPA and 2D D3 resultswhich avoids having to resolve the ambiguity encountered ininterferometer arrays and also results in frequency-independentperformance )e simulation results show that with high SNR(ge15dB) the estimator can have 50 probability of AoA es-timationwith RMSEle 1deg using just few temporal samplesMoreimportantly with more temporal snapshots the estimator has52 and 80 probabilities of AoA estimation with RMSEle 1degand 5deg respectively for phase- or frequency-modulated radarpulses when the SNR is between minus10 and 0dB Although theperformance can degrade for an ultra-wideband FMCW radarsignal (for example the 100MHz chirp-signal used here) it canstill achieve 33 and 67 probabilities of AoA estimation withRMSEle 1deg and 5deg respectively when SNR is between minus10dBand 0dB Future work is needed to improve the performance ofthe AoA estimation for ultra-wideband FMCW signals in lowSNR conditions)e study also shows that the estimator prefersa circular-shaped planar array with a triangular interelementpattern since this array presents more symmetrical charac-teristics when it is seen from different Az angles In order tofacilitate the practical use of this AoA estimator in ESM ap-plications ie as there is no a priori knowledge of the SOI thispaper also presents the use of precalculated SV andI-databaseswhich are built on a 2D-angle-grid with more uniformly dis-tributed directions with a set of preselected frequency points)e AoA calculation can be further improved by using a denser2D-angle-grid more frequency points for database calculationsand dedicated hardware with fast access memory

Appendix

A 2D Planar Arrays with Isotropic Elements

Six different shapes of planar arrays were used in the study)ese arrays are shown from Figure 15 and they are asfollows

(i) Array-1 (a) uniform-spaced rectangular array with81 elements (white circles) element spacing is 7mmin both x and y directions )e array center is at the41st element

(ii) Array-2 (b) uniform-spaced hexagonal array with61 elements (white circles) the element spacing in

each triangular is 7mm in both x and y directions)e array center is at the 31st element

(iii) Array-3 (c) circular planar array with triangulargrid with 57 elements (white circles) the elementspacing in each triangular grid is 7mm )e arraycenter is at the 31st element

(iv) Array-4 (d) hexagonal planar array with rectan-gular grid with 77 elements (white circles) the el-ement spacing is 7mm in both x and y directions)e array center is at the 39th element

(v) Array-5 (e) circular planar array with rectangulargrid with 69 elements (white circles) the elementspacing is 7mm in both x and y directions)e arraycenter is at the 35th element

(vi) Array-6 (f) rectangular planar array with triangulargrid with 81 elements (white circles) the elementspacing in each triangular grid is 7mm )e arraycenter is at the 41st element

)e following can be found

(i) Arrays have different isotropic-element distributionpatterns

(ii) Since the patterns are different the total number ofisotropic elements (NN) is also different

(iii) )e aperture size of each plane array is about thesame as that of the 7-element NSA

(iv) )e center element in an array is collocated with thefirst element of the 7-element NSA

B Definition of 2D-Angle Grid by Vertices of aUnit Icosphere

It is known that the incident angles defined on a UV-sphere[62] which are frequently used in antenna analysis havemuchmore dense directions near north and south poles thannear the equator In order to obtain more uniform angulardistributions for AoA searching using (18) a 2D-angle griddefined by the unit icosphere [63] is used in this study )e2D-angle grid is formed by subdividing the triangles of aregular icosahedron and the number of subdivisions(Nsubdiv) determines the density of the vertices on the surfaceof an icosphere [64] Figure 16 shows an example of the 2D-angle grid Since the CBSA has about 100deg 3 dB antennabeamwidth in this study the FOV of the 7-element NSAis defined within [Az1 Az2] times [El1 El2] [minus180deg 180deg] times

[40deg 90deg] which are indicated by the asterisks in the exampleshown in Figure 16

Table 10 gives the number of total directions (vertices)within the FOV and their corresponding angular resolutionswith respect to the number of subdivisions of the trianglesof a regular icosahedron Averaged El-angle difference be-tween the vertex on z-axis and surrounding 6 vertices(marked by ldquotimesrdquo in Figure 16) gives a rough order of angleresolution Considering the AoA estimation accuracy andcalculation speed Nsubdiv 7 is used in this study )is 2D-angle grid with angle zones within the FOV is shown inFigure 17


C Four Commonly Used Radar Signals

C1 Barker Code of Length Equal to 13 )e 13-chip Barkercode is shown in Figure 18 which is a binary-phase shift-keying modulation In our simulations the duration ofeach chip (tb) is 200 ns so the total pulse width (PW) is26 μs

C2 Two-Valued Frequency-Coded Waveform Figure 19shows the information of a 20-chip two-valued frequency-coded waveform which is given in Table I of [58] )e two-valued frequency-coded waveforms presented in [58] canyield near-perfect periodic autocorrelation function for radarapplications when the number of chips is a multiple of 4 andthe frequency values are plusmn14tb where tbis the bit duration)e duration of each chip in Figure 19 is 200 ns and the totalPW is 4 μs )e two frequency values are plusmn125MHz )is20-chip two-valued frequency-coded waveform is a kind ofbinary-frequency shift-keying modulation

C3 Poly-Phase Waveform A 16-chip P1 poly-phasewaveform is shown in Figure 20 Each chip width is 1 μs andthe total PW is 16 μs

C4 FMCW )e frequency change of the FMCWwaveformused in this study is shown in Figure 21 Its parameters aregiven in Table 11 Note that it has a very wide linear fre-quency change during chirping

D Explanation of the Frequency-Independent Performance

From Figures 6ndash8 one can observe that for a given AoAestimation accuracy requirement (eg RMSEle 1deg) and theprobability of AoA estimation (eg 50) the required SNRis close to a constant when the frequency is from 6 to18GHz )e following gives the explanation

Taking natural log of both (15) and squared of (17) (18)can be rewritten as

minazaze

elele

1λest

1113888 11138891113944

NN

n1j4π pxn cos(el)cos(az) + pyn cos(el)sin(az)1113858 1113859 minus λest ln

1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)

In this equation the λminus1est term can be removed as it is a

constant )en one has the following equation

minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)

It can be seen that during the solution search the secondterm (after the minus sign in (D2)) has to match or be veryclose to a variable (the first term before minus sign in (D2))which is only determined by the direction and locations ofthe elements in an IEPA Since the first term is frequencyindependent the second term must also be frequency in-dependent )e following simulation results support thisconclusion

Figure 22 shows the comparison between the first termand the imaginary part of the second term in (D2) at eachelement in Array-3 when SNR is equal to infinity One cansee that the imaginary part of the second term in (D2) doesnot change when the frequency changes at a given incidentdirection

Figure 23 shows that the values of the first term and theimaginary part of the second term in (D2) at each element inArray-3 get closer to each other as SNR improves )eperformances of the probability of AoA estimation at dif-ferent frequencies are close to each other at different SNRlevels as shown in Figure 24

Data Availability

)eways to generate the simulation data used to support thefindings of this study are included within the article

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was supported by the Radar Electronic WarfareCapability Development in Radar Electronic Warfare Sectionat Defence Research and Development Canada-Ottawa Re-search Centre Department of National Defence CanadaSpecial thanks to Mr Michael Low from Defence Researchand Development Canada for many good discussion andcomments to improve the quality of the paper


References

[1] P E Pace Detecting and Classifying Low Probability of In-tercept Radar Artech House Norwood MA USA 2ndedition 2009

[2] E Jacobs and E Ralston ldquoAmbiguity resolution in interfer-ometryrdquo IEEE Transactions on Aerospace and ElectronicSystems vol AES-17 no 6 pp 766ndash780 1981

[3] W B Kendall ldquoUnambiguous accuracy of an interferometerangle-measuring systemrdquo IEEE Transactions on Space Elec-tronics and Telemetry vol SET-11 no 2 pp 62ndash70 1965

[4] R C Hansen ldquoMicrowave scanning antennasrdquo in ArraySystems vol 3 Academic Press Cambridge MA USA 1966

[5] K Pasala R Penno and S Schneider ldquoNovel widebandMultimode hybrid interferometer systemrdquo IEEE Transactionson Aerospace and Electronic Systems vol 39 no 4pp 1396ndash1406 2003

[6] E Stephen Lipsky Microwave Passive Direction FindingScitech Publishing Inc Raleigh NC USA 2004

[7] J B Y Tsui Microwave Receivers with Electronic WarfareApplications John Wiley amp Sons Hoboken NJ USA 1986

[8] F Neri Introduction to Electronic Defence Systems ArtechHouse Norwood MA USA 2nd edition 2006

[9] S Chandran Advances in Direction-of-Arrival EstimationArtech House Norwood MA USA 2006

[10] A De Martino Introduction to Modern EW Systems ArtechHouse Norwood MA USA 2012

[11] T E Tuncer and B Friedlander Classical and Modern Di-rection-of-Arrival Estimation Academic Press BurlingtonMA USA 2009

[12] R Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[13] R Roy and T Kailath ldquoESPRIT-estimation of signal pa-rameters via rotational invariance techniquesrdquo IEEE Trans-actions on Acoustics Speech and Signal Processing vol 37no 7 pp 984ndash995 1989

[14] X Wang L Wan M Huang C Shen and K Zhang ldquoPo-larization channel estimation for circular and non-circularsignals in massive MIMO systemsrdquo IEEE Journal of SelectedTopics in Signal Processing vol 13 no 5 pp 1001ndash1016 2019

[15] H Chen C Hou W-P Zhu et al ldquoESPRIT-like two-di-mensional direction finding for mixed circular and strictlynoncircular sources based on joint diagonalizationrdquo SignalProcessing vol 141 pp 48ndash56 2017

[16] M Coutino R Pribic and G Leus ldquoDirection of arrivalestimation based on information geometryrdquo in Proceedings ofthe 2016 IEEE International Conference on Acoustics Speechand Signal Processing (ICASSP) pp 3066ndash3070 ShanghaiChina March 2016

[17] Y-Y Dong C-X Dong W Liu M-M Liu and Z-Z TangldquoScaling transform based information geometry method forDOA estimationrdquo IEEE Transactions on Aerospace andElectronic Systems vol 55 no 6 pp 3640ndash3650 2019

[18] A D Lonkeng and J Zhuang ldquoTwo-dimensional DOA es-timation using arbitrary arrays for massive MIMO systemsrdquoInternational Journal of Antennas and Propagation vol 2017Article ID 6794920 9 pages 2017

[19] V Krishnaveni T Kesavamurthy and B Apama ldquoBeam-forming for direction of arrival (DOA) estimationmdasha surveyrdquoInternational Journal of Computer Applications vol 61 no 112013

[20] M Devendra and K Manjunathachari ldquoLiterature survey onhigh resolution direction of arrival (DOA) algorithmsrdquo

International Advanced Reserch Journal in Science Engi-neering and Technology vol 2 no 2 pp 23ndash31 2015

[21] S Barua S C Lam P Ghosa S Xing and K SandrasegaranldquoA survey of direction of arrival estimation techniques andimplementation of channel estimation based on SCMErdquo inProceedings of the 2015 12th International Conference onElectrical EngineeringElectronics Computer Telecommuni-cations and Information Technology (ECTI-CON) pp 1426ndash1429 Hua Hin )ailand June 2015

[22] Z Zheng and S Mu ldquoTwo-dimensional DOA estimationusing two parallel nested arraysrdquo IEEE CommunicationsLetters vol 24 no 3 pp 568ndash571 2020

[23] H P Shi W Leng ZW Guan and T Z Jin ldquoTwo novel two-stage direction of arrival estimation algorithms for two-di-mensional mixed noncircular and circular sourcesrdquo Sensorsvol 17 no 6 p 1433 2017

[24] Y Wu G S Liao and H C So ldquoA fast algorithm for 2-Ddirectional-of-arrival estimationrdquo Signal Processing vol 83no 8 pp 1927ndash1831 2003

[25] A-J van der Veen ldquoAsymptotic properties of the algebraicconstant modulus algorithmrdquo IEEE Transactions on SignalProcessing vol 49 no 8 pp 1796ndash1807 2001

[26] Q Wang H Yang H Chen Y Y Dong and L H Wang ldquoAlow complexity method for two-dimensional direction-of-arrival estimation using an L-shaped arrayrdquo Sensors (BaselSwitzerland) vol 17 no 12 p 190 2017

[27] A B Gershman M Rubsamen and M Pesavento ldquoOne- andtwo-dimensional direction-of-arrival estimation an overviewof search-free techniquesrdquo Signal Processing vol 90 no 5pp 1338ndash1349 2010

[28] S O Al-Jazzar Z Hamici and S Aldalahmeh ldquoTow-di-mensional AoA estimation based on a constant modulusalgorithmrdquo International Journal of Antennas and Propaga-tion vol 2017 Article ID 3214021 6 pages 2017

[29] E J Candes and T Tao ldquoDecoding by linear programmingrdquoIEEE Transactions on Information Jeory vol 51 no 12pp 4203ndash4215 2005

[30] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation Jeory vol 52 no 4 pp 1289ndash1306 2006

[31] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on InformationJeory vol 52 no 2 pp 489ndash509 2006

[32] A C Gurbuz J H McClellan and V Cevher ldquoA compressivebeamforming methodrdquo in Proceedings of the InternationalConference on Acoustics Speech and Signal Processing LasVegas NV USA March 2008

[33] A C Gurbuz V Cevher and J H Mcclellan ldquoBearing es-timation via spatial sparsity using compressive sensingrdquo IEEETransactions on Aerospace and Electronic Systems vol 48no 2 pp 1358ndash1369 2012

[34] V Cevher A C Gurbuz J H McClellan and R ChellappaldquoCompressive wireless arrays for bearing estimationrdquo inProceedings of the 2008 IEEE International Conference onAcoustics Speech and Signal Processing pp 2497ndash2500 LasVegas NV USA March 2008

[35] E Candes and T Tao ldquo)e dantzig selector statistical esti-mation when p is much larger than nrdquoJeAnnals of Statisticsvol 35 no 6 pp 2313ndash2351 2007

[36] C Wu and J Elangage ldquoMulti-emitter 2-dimensional angle-of-arrival estimator via compressive sensingrdquo IEEE Trans-actions on Aerospace and Electronic Systems vol 56 no 4pp 2884ndash2895 2020


[37] Z-M Liu Z-T Huang Y-Y Zhou and J Liu ldquoDirection-of-arrival estimation of noncircular signals via sparse repre-sentationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 48 no 3 pp 2690ndash2698 2012

[38] Z-M Liu Z-T Huang and Y-Y Zhou ldquoArray signalprocessing via sparsity-inducing representation of the arraycovariance matrixrdquo IEEE Transactions on Aerospace andElectronic Systems vol 49 no 3 pp 1710ndash1724 2013

[39] Z Yang J Li P Stoica and L Xie ldquoSparse methods fordirection-of-arrival estimationrdquo Academic Press Library inSignal Processing Elsevier vol 7 pp 509ndash581 Oxford UK2018

[40] J Cai W Liu R Zong and BWu ldquoSparse array extension fornon-circular signals with subspace and compressive sensingbased DOA estimation methodsrdquo Signal Processing vol 145pp 59ndash67 2018

[41] Z Zhang and B D Rao ldquoSparse signal recovery with tem-porally correlated source vectors using sparse Bayesianlearningrdquo IEEE Journal of Selected Topics in Signal Processingvol 5 no 5 pp 912ndash926 2011

[42] M Carlin P Rocca G Oliveri and A Massa ldquoBayesiancompressive sensing as applied to directions-of-arrival esti-mation in planar arraysrdquo Journal of Electrical and ComputerEngineering vol 2013 Article ID 245867 12 pages 2013

[43] P Rocca M A Hannan M Salucci and A Massa ldquoSingle-snapshot DoA estimation in array antennas with mutualcoupling through a multiscaling BCS strategyrdquo IEEE Trans-actions on Antennas and Propagation vol 65 no 6pp 3203ndash3213 2017

[44] T K Sarkar S Nagaraja and M C Wicks ldquoA deterministicdirect data domain approach to signal estimation utilizingnonuniform and uniform 2-D arraysrdquo Digital Signal Pro-cessing vol 8 no 2 pp 114ndash125 1998

[45] T K Sarkar T K Sarkar H Wang et al ldquoA deterministicleast-squares approach to space-time adaptive processing(STAP)rdquo IEEE Transactions on Antennas and Propagationvol 49 no 1 pp 91ndash103 2001

[46] K Kim T K Sarkar and M S Palma ldquoAdaptive processingusing a single snapshot for a non-uniformly spaced array inthe presence of mutual coupling and near-field scatterersrdquoIEEE Transactions on Antennas and Propagation vol 50no 5 pp 582ndash590 2002

[47] T K Sarkar H Schwarzlander S Seungwon ChoiM S Palma and M C Wicks ldquoStochastic versus deter-ministic models in the analysis of communication systemsrdquoIEEE Antennas and Propagation Magazine vol 44 no 4pp 40ndash50 2002

[48] T K Sarkar M C Wicks M Salazar-Palma andR J Bonneau Smart Antennas IEEE Press A John Wiley ampSon Inc Publication Hoboken NJ USA 2003

[49] K J Kim T K Sarker H Wang and M Salazar-PalmaldquoDirection of arrival estimation based on temporal and spatialprocessing using a direct data domain (D3) approachrdquo IEEETransactions on Antennas and Propagations vol 52 no 2pp 533ndash541 2004

[50] H Chen B X Huang Y LWang and Y Q Hou ldquoDirection-of-arrival based on direct data domain (D3) methodrdquo Journalof Systems Engineering and Electronics vol 20 no 3pp 512ndash518 2009

[51] W Mao G Li X Xie and Q Yu ldquoDOA estimation of co-herent signals based on direct data domain under unknownmutual couplingrdquo IEEE Antennas and Wireless PropagationLetters vol 13 pp 1525ndash1528 2014

[52] P Zhang J Li A Z Zhou H B Xu and J Bi ldquoA robust directdata domain least squares beamforming with sparse con-straintrdquo Progress In Electromagnetics Research C vol 43pp 53ndash65 2013

[53] N Yilmazer and T K Sarker ldquo2D unitary matrix pencilmethod for efficient direction of arrival estimationrdquo DigitalSignal Processing vol 16 pp 767ndash781 2006

[54] N Yilmazer R Fernandez-Recio and T K Sarker ldquoMatrixpencil method for simultaneously estimating azimuth andelevation angles of arrival along with the frequency of theincoming signalsrdquo Digital Signal Processing vol 16pp 796ndash816 2006

[55] C Wu ldquoWideband miniaturized digital receiver technologyfor micronanosat applicationrdquo Scientific Report 2015

[56] E Poliakov and C Wu ldquoCompact PCI computer controlsystem and its control software update for UAV electronicsupport payloadrdquo Technical Memorandum 2009

[57] C Wu and A Young ldquoInitial design considerations of smallUAV EW payloadrdquo Technical Memorandum 2012

[58] N Levanon and U Levanon ldquoTow-valued frequency-codedwaveform with favorable periodic autocorrelationrdquo IEEETransactions on Aerospace and Electronic Systems vol 42no 1 pp pp237ndash248 2006

[59] X Li and M C E Yagoub ldquoWideband cavity-backed spiralantenna design and mutual coupling study in a closely-spacedarrayrdquo in Proceedings of the 2018 18th International Sympo-sium on Antenna Technology and Applied Electromagnetics(ANTEM) Waterloo Canada August 2018

[60] L3HARRISR Antenna System Spiral Antenna 2020 httpswww2l3tcomrandtronantenna_productselements_broadband_spiralshtm

[61] W A Gardner Cyclostationarity in Communications andSignal Processing IEEE Press Piscataway NJ USA 1994

[62] D Lugg ldquoWhat is the difference between a UV sphere and anicosphererdquo 2020 httpsblenderstackexchangecomquestions72what-is-the-difference-between-a-uv-sphere-and-an-icosphere7Etext=A20UV20sphere20has20facescompared20to20the20UV20sphere

[63] Wikipedia the Free Encyclopedia Icosphere 2020 httpsenwikipediaorgwikiIcosphere

[64] W O C Ward ldquoIcosphere generate unit geodesic spherecreated by subdividing a regular icosahedronrdquo 2020 httpswwwmathworkscommatlabcentralfileexchange50105-icospherefocused=3865223amptab=function


presented a research on 2D DF estimation using arbitraryarrays in MIMO systems and introduced the 2D Fourierdomain line search MUSIC algorithm Krishnaveni et al[19] Devendra and Manjunathachari [20] and Barua et al[21] have surveyed some directions of arrival methodsManyother AoA estimators have also been developed which focuson different array structures and faster processing speedsSome examples are given in [22ndash28]

)ere are also some AoA estimation methods that arebased on compressive sensing (CS) theory [29ndash31] Gurbuzet al [32 33] applied CS to estimate acoustic wave AoA byfocusing on one receiving channel sampling at Nyquist rateand the other channels sampling at CS rate Using the targetbearing as a sparse vector Cevher et al [34] demonstratedmultielement circular acoustic sensor arrays to obtain thebearings of multiple sources by applying an l1-norm min-imization solution called the Dantzig selector [35] Wu andElangage [36] introduced a CS-based ultra-wideband 2DAoA estimation scheme that can estimate AoA of the SOIfrom 2 to 18GHz without any a priori knowledge of theintercepted signals Other examples of treating AoA esti-mation as a sparse recovery problem can be found in[37ndash40] Bayesian CS- (BCS-) based AoA estimation was alsodeveloped in [41ndash43] )e key advantage of the BCS AoAestimator is its ability to estimate signal AoA with fewmeasured data from each element in the array

To estimate the AoA of SOI in a nonstationary envi-ronment and to avoid having to form a signal covariancematrix the Direct Data Domain (D3) method was intro-duced [44ndash47] Using the D3 method and the concept ofsignal cyclostationarity Sarker et al [48] introduced amethod that can handle a number of signals along with theirvarious coherent and noncoherent multipaths and inter-ferences even when the number of signals exceeds thenumber of antenna elements)is was demonstrated using a12-element uniform linear array Some examples of usingthe D3method for AoA estimations can be found in [49ndash54]

Most aforementioned methods and their usages onantenna arrays are suitable for narrow-band applications asthe array elements and their configurations closely relate tothe signal wavelengths Hence they can be used mainly forAoA estimations in communication applications Interfer-ometry-based AoA measurement systems have been widelyused in ESM systems for 2D AoA measurements in ultra-wide frequency ranges [2ndash4] It is desirable to use the longestpossible baseline in the array to achieve low variancemeasurement results However longer baselines requirehigher signal-to-noise ratios (SNRs) in order to resolve theambiguity using the associated trigonometric functions Inaddition even if the wideband antenna elements such ascavity-backed spiral antennas (CBSA) are used in the five-element array in [2] the spacing of the short baseline in thearray is chosen to be less than half of the wavelength of thehighest frequency of interest As a result several differentfive-element interferometers are required to cover a wideoperational frequency range )is results in a large antennaarray footprint Pasala et al [5] introduced a three-elementinterferometer array that used multimodes in the elements

to avoid the short baseline required in the conventional five-element array configuration )e long baseline (larger arrayfootprint) is still required to resolve the ambiguity Howeverthese large-footprint arrays are problematic for small air-borne platforms such as CubeSats (U-class spacecraft) [55]small unmanned aerial vehicles [56 57] and drones

In this paper we propose a novel 2D angle-of-arrival(AoA) estimator for ESM applications from 6 to 18GHz inlow SNR environments with frequency-independent per-formance when the SNR and the required estimation ac-curacy are given )is includes the following contributions

(i) A 2D 7-element compact nonuniformly spacedarray (NSA) is designed with CBSA elements op-erating from 6 to 18GHz which has an arrayfootprint that is smaller than a circle having a414mm radius

(ii) A 2D AoA measurement system applies the D3method on the measured time-domain data(snapshots) from the 7-element NSA )is includesthe following

(a) A 2D data transformation from the 7-elementNSA-measured data to a 2D isotropic-elementplanar array (IEPA) data is introduced in orderto use the NSA data in the D3 method for AoAestimation and six different 2D IEPAs areconsidered

(b) Using 2D IEPA steering vector (SV) and theresults derived from the D3 method a newobjective function is introduced for searchingthe AoA of the SOI solution in both azimuth(Az) and elevation (El) angles )is (1) avoidshaving to solve the ambiguity problem en-countered in interferometric arrays and (2)gives the estimator a frequency-independentperformance

(c) Using a 3D icosphere a 2D-angle grid is in-troduced to give a near-uniform angular dis-tribution in the field of view (FOV) of the 7-element NSA On this grid the data-transfor-mation matrix database and SVs of IEPAs areprecalculated at a list of preselected frequencypoints and stored in a computer prior to AoAestimations Hence the estimator can be usedfor ESM applications without a priori knowl-edge of the SOI

(iii) Since the focus of this development is ESM appli-cation in low SNR environments different SNRlevels are studied with four different commonlyused radar waveforms )ey are the following (1) a13-chip Barker-coded waveform (2) a 20-chip two-valued frequency-coded waveform [58] (3) a 16-chip poly-phase-coded waveform and (4) a ultra-wideband (100MHz chirp) frequency-modulatedcontinuous waveform (FMCW)In addition to the above contributions the findingsof this study are highlighted as follows










X

Y

Z

3

4

5

1

6

7

2El

AZ


2672

2538

2581

2565

26722522

2709

2522

2582

2540

2582

2694















+ alefsym(m)

(3)





A(t)cos 2πfct( 1113857 (4)



A2(t)

21113888 1113889cos(2πηt) + constant (5)

η 2fc (6)


A2(t)e

j2πηt (7)



x2n(m) A

2(m)e




where

Z


x21(1) minus x

21(2)e


21(M) minus x

21(M + 1)e

minusj2πηmiddotdt


x2NN(1) minus x

2NN(2)e


2NN(M) minus x

2NN(M + 1)e

minusj2πηmiddotdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)



W

w1

w2

⋮

wM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


E

C

0

⋮

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)


C 1113944M

m1wme



M

radic






1113944

M

m1x2n(m) minus x

2n(m + 1) middot e



1C

1113944

M

m1x2n(m)wm e


(15)











sv



⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)




sv(az el)



⋮

1

⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(17)



minazaze

elele

1113944

NN

n1sv2n(az el) minus

1C

1113944

M

m1x2n(m)wm

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(18)










minTI


(20)




(21)






















1113872 1113873 (22)





INNtimes7 fci z1113872 1113873 (23)










(ΔAz)2 +(ΔEl)2

2

1113971

(24)





erated as follows








































RMSE le 1 (deg)M + 1 = 129


0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty


(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)





0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References












































































X

Y

Z

3

4

5

1

6

7

2El

AZ


2672

2538

2581

2565

26722522

2709

2522

2582

2540

2582

2694















+ alefsym(m)

(3)





A(t)cos 2πfct( 1113857 (4)



A2(t)

21113888 1113889cos(2πηt) + constant (5)

η 2fc (6)


A2(t)e

j2πηt (7)



x2n(m) A

2(m)e




where

Z


x21(1) minus x

21(2)e


21(M) minus x

21(M + 1)e

minusj2πηmiddotdt


x2NN(1) minus x

2NN(2)e


2NN(M) minus x

2NN(M + 1)e

minusj2πηmiddotdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)



W

w1

w2

⋮

wM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


E

C

0

⋮

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)


C 1113944M

m1wme



M

radic






1113944

M

m1x2n(m) minus x

2n(m + 1) middot e



1C

1113944

M

m1x2n(m)wm e


(15)











sv



⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)




sv(az el)



⋮

1

⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(17)



minazaze

elele

1113944

NN

n1sv2n(az el) minus

1C

1113944

M

m1x2n(m)wm

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(18)










minTI


(20)




(21)






















1113872 1113873 (22)





INNtimes7 fci z1113872 1113873 (23)










(ΔAz)2 +(ΔEl)2

2

1113971

(24)





erated as follows








































RMSE le 1 (deg)M + 1 = 129


0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty


(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)





0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References













































































+ alefsym(m)

(3)





A(t)cos 2πfct( 1113857 (4)



A2(t)

21113888 1113889cos(2πηt) + constant (5)

η 2fc (6)


A2(t)e

j2πηt (7)



x2n(m) A

2(m)e




where

Z


x21(1) minus x

21(2)e


21(M) minus x

21(M + 1)e

minusj2πηmiddotdt


x2NN(1) minus x

2NN(2)e


2NN(M) minus x

2NN(M + 1)e

minusj2πηmiddotdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)



W

w1

w2

⋮

wM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


E

C

0

⋮

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)


C 1113944M

m1wme



M

radic






1113944

M

m1x2n(m) minus x

2n(m + 1) middot e



1C

1113944

M

m1x2n(m)wm e


(15)











sv



⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)




sv(az el)



⋮

1

⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(17)



minazaze

elele

1113944

NN

n1sv2n(az el) minus

1C

1113944

M

m1x2n(m)wm

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(18)










minTI


(20)




(21)






















1113872 1113873 (22)





INNtimes7 fci z1113872 1113873 (23)










(ΔAz)2 +(ΔEl)2

2

1113971

(24)





erated as follows








































RMSE le 1 (deg)M + 1 = 129


0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty


(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)





0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References





































































W

w1

w2

⋮

wM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)


E

C

0

⋮

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)


C 1113944M

m1wme



M

radic






1113944

M

m1x2n(m) minus x

2n(m + 1) middot e



1C

1113944

M

m1x2n(m)wm e


(15)











sv



⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)




sv(az el)



⋮

1

⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(17)



minazaze

elele

1113944

NN

n1sv2n(az el) minus

1C

1113944

M

m1x2n(m)wm

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(18)










minTI


(20)




(21)






















1113872 1113873 (22)





INNtimes7 fci z1113872 1113873 (23)










(ΔAz)2 +(ΔEl)2

2

1113971

(24)





erated as follows








































RMSE le 1 (deg)M + 1 = 129


0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty


(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)





0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































sv(az el)



⋮

1

⋮


⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(17)



minazaze

elele

1113944

NN

n1sv2n(az el) minus

1C

1113944

M

m1x2n(m)wm

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2

(18)










minTI


(20)




(21)






















1113872 1113873 (22)





INNtimes7 fci z1113872 1113873 (23)










(ΔAz)2 +(ΔEl)2

2

1113971

(24)





erated as follows








































RMSE le 1 (deg)M + 1 = 129


0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty


(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)





0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References













































































1113872 1113873 (22)





INNtimes7 fci z1113872 1113873 (23)










(ΔAz)2 +(ΔEl)2

2

1113971

(24)





erated as follows








































RMSE le 1 (deg)M + 1 = 129


0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty


(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)





0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References



































































































RMSE le 1 (deg)M + 1 = 129


0

05

1

Prob

abili

ty

(a)

RMSE le 1 (deg)M + 1 = 257

0

05

1

Prob

abili

ty


(b)

RMSE le 1 (deg)M + 1 = 513


0

05

1

Prob

abili

ty

(c)

RMSE le 1 (deg)M + 1 = 1025

0

05

1

Prob

abili

ty


(d)





0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References






































































0

05

1

Prob

abili

ty


RMSE le 1 (deg)

(a)

RMSE le 5 (deg)0

05

1

Prob

abili

ty


(b)


0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 129

(a)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 257

(b)


0

05

1

Prob

abili

ty

RMSE le 5 (deg)M + 1 = 513

(c)

0

05

1

Prob

abili

ty


RMSE le 5 (deg)M + 1 = 1025

(d)



1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(a)

1

05

0200

ndash20SNR (dB)

181614121086Freq (GHz)

(b)


1

05

020

0ndash20

SNR (dB)

181614121086Freq (GHz)

(a)

1

05

020

0ndash20

SNR (dB)181614121086

Freq (GHz)

(b)



1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































1500

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5


5

No

of p

oint

s 1000

500

0

1000

500

0

1000

500

0

1000

500

0

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s

1500

1000

500

0No

of p

oint

s


1000

500

0

1000

500

0ndash5 0

(a) (b)

(c) (d)

(e) (f)

(g) (h)

5 ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

ndash5 0 5

No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s

1000

500

0

1000

500

0No

of p

oint

s




1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

1000

500

0No

of p

oint

s 1000

500

0

ndash5 0(a)

5 ndash5 0(b)

5

ndash5 0(d)

5

ndash5 0(f)

5

ndash5 0

(f)

5

ndash5 0(c)

5

ndash5 0(e)

5

ndash5 0

(g)



100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s

100

50

0

100

50

0No

of p

oint

s


(c) (d)

(e)

(g) (h)

(f)

5 10 ndash10 ndash5 0 5 10











WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References






































































WF-1 8576 9642 8584 9650 9981 9987WF-2 8566 9647 8576 9655 9979 9985WF-3 8564 9637 8571 9644 9983 9987WF-4 7591 9119 7664 9132 9397 9669



WF-1 333 2127 714 2351 1746 4582WF-2 318 2074 716 2314 1665 4560WF-3 318 2151 721 2389 1768 4609WF-4 121 910 320 1176 920 3107WF-4 105 1004 332 1273 936 3310



WF-1 8386 9619 8399 9627 9969 9978WF-2 8311 9582 8322 9592 9971 9977WF-3 8475 9614 8381 9620 9977 9985WF-4 5992 8077 6185 8110 8206 9045



WF-1 5363 8148 5660 8154 8320 9135WF-2 5232 8007 5531 8017 8203 9055WF-3 5275 8066 5547 8072 8331 9011WF-4 2630 5105 3001 5222 4932 6931WF-4 3328 6764 3846 6793 6492 8351

ndash200

02

04

06

08

1

Prob

abili

ty

RMSE le 1 (deg)

0 20 40SNR (dB)

(a)

02

04

06

08

0

1

Prob

abili

ty


RMSE le 5 (deg)

(b)






(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References







































































(a) (b)

(c) (d)

(e) (f )



Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































Z

X

Y

El

Az





10908z07

ndash06ndash04

ndash020

0204

06 ndash06ndash04

ndash020

0204

06

x y

M

(a)

z

x y

088086084082

08

0102

03

ndash01

0405 ndash05

ndash04ndash03

ndash02

M

(b)



0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































0

05

1

Bark

er co

de0 2 4 6 8 10 12 14

Time (tb)

(a)

ndash1

0

1

Am

plitu

de

0 2 4 6 8 10 12 14Time (tb)

(b)

0

1

2

3

Phas

e (ra

dian

s)

0 2 4 6 8 10 12 14Time (tb)

(c)


ndash1

0

1

f b lowast

4 t b

0 2 4 6 8 10 12 14 16 18 20Time (tb)

(a)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash1

0

1

Am

plitu

de

(b)

0 2 4 6 8 10 12 14 16 18 20Time (tb)

ndash15

ndash10

ndash5

Phas

e (ra

dian

s)

(c)



ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































ndash1ndash05

005

1A

mpl

itude

0 10 15Time (tb)

5

(a)

0 10 15Time (tb)

ndash2

0

2

Phas

e (ra

dian

s)

5

(b)


0 100 200 300 400 500 600 700 800 900

Time (μ sec)

0

50

100

Freq

uenc

y (M

Hz)




ndash005

0

005

0 20 40 60

(a)

ndash005

0

005

0 20 40 60

(b)

ndash005

0

005

0 20 40 60

(c)

ndash005

0

005

0 20 40 60

(d)

ndash005

0

005

0 20 40 60

(e)

ndash005

0

005

0 20 40 60

(f )



ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































ndash006

ndash004

ndash002

0

002

004

006SNR = 0 dB


(a)

SNR = 5 dB

ndash006

ndash004

ndash002

0

002

004

006


(b)

SNR = 10 dB

ndash006

ndash004

ndash002

0

002

004

006


(c)


ndash006

ndash004

ndash002

0

002

004

006SNR = 15 dB

0 20 40 60

(d)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 25 dB

0 20 40 60

(e)

ndash006

ndash004

ndash002

0

002

004

006


SNR = 35 dB

(f )



14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(a)


14 16 18

1

08

06

04

02

Prob

abili

ty o

f AoA

est

()


(b)




7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References





































































7 Conclusions


Appendix




























minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References












































































minazaze

elele

1λest

1113888 11138891113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D1)



minazaze

elele

1113944

NN


1C

1113944

M

m1x2n(m)wm

⎡⎣ ⎤⎦

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868

2⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (D2)




Data Availability




Acknowledgments



References




































































References




































































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