# Angle of Arrival (AoA) Calen Carabajal EECS 823.

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Angle of Arrival (AoA) Calen Carabajal EECS 823

Introduction to Angle of Arrival Physics
“Angle between propagation direction of an incident wave and some reference direction” (orientation) Plane wave impinging upon array

Visual Understanding Plane wave impinges on array of antennas with the same orientation and radiation pattern Time delay corresponds to a phase shift between antennas To the right, red lines represent wave front, each with the same relative phase Red dot corresponds incidence of wave front Results in a zero-valued response for this phase

Wave Propagation and Antenna: Specific Case
𝐸 𝑥,𝑦,𝑧 = 𝑒 −𝑗 2𝜋 𝜆 𝑘 𝑥 𝑥+ 𝑘 𝑦 𝑦+ 𝑘 𝑧 𝑧 = 𝑒 −𝑗𝒌∙𝒓 Constant amplitude assumed Received signal for each antenna is 𝑒 −𝑗 2𝜋 𝜆 cos 𝜃 𝑧 𝑖 Summing signals together results in 𝑚=−1 1 𝑒 −𝑗𝑚𝜋 cos 𝜃 Observations 𝑐𝑜𝑠𝜃 – broadside vs boresight Cancellation k*r Frequency dependence

Antenna Arrays, Steering Vector
𝐴𝑟𝑟𝑎𝑦 𝐹𝑎𝑐𝑡𝑜𝑟 𝐴𝐹= 𝑖=1 𝑁 𝑤 𝑖 𝑒 −𝑗𝒌∙ 𝒓 𝑖 = 𝑖=1 𝑁 𝑤 𝑖 𝒔 𝝓 For identical antennas with radiation pattern and orientation 𝑅(𝜃,𝜙), the overall radiation pattern is 𝑌=𝑅 𝜃,𝜙 ∗𝐴𝐹 Wave number 𝑘= 2𝜋 𝜆 = 2𝜋𝑓 𝑐 Steering vector 𝑠(𝜙) can take various forms Specified to arbitrary point of origin Represents relative phases at each antenna 𝒔 𝑻 𝜙 =[ 𝑒 −𝑗𝒌∙ 𝒓 𝟏 𝑒 −𝑗𝒌∙ 𝒓 𝟐 ∙∙∙ 𝑒 −𝑗𝒌∙ 𝒓 𝑵 ]

Applications of Angle of Arrival Estimation: Wireless Sensor Networks
May use antenna array on each sensor node Geodesic location of cell phones Emergency phone calls

Applications of Angle of Arrival: Remote Sensing
AoA estimation provides also allows further characterization of target. Adaptive processes can take advantage of this knowledge Beamsteering/Nullsteering Angle-of-arrival-assisted Radio Interferometry Ground moving objects Coupled with other data (range, Doppler), can extract target location

Limitations of Angle of Arrival Estimation: The Cramer-Rao Bound
CRB provides lower bound on variance in estimations Provides a theoretical limit on ability to discern angle of arrival Considers corrupting noise on the signal 𝒙=𝜶𝒔 𝝓 +𝒏 The Cramer-Rao Bound for AoA estimation is 𝑣𝑎𝑟 𝜙 ≥ 6 𝜎 2 𝛼 𝑁 𝑁 2 −1 𝑘𝑑 2 sin 2 𝜙 𝜎 2 : covariance of noise vector N : number of elements in array d : distance between array elements

Limitations of Angle of Arrival Estimation: Effect of Multipath
Consider either a smooth surface or rough surface Specular surface results in two components—direct component and image component Rough surface results in both the above components as well as diffuse components Fading In extreme case, may result in signal cancellation Approach: Multi-taper Method

Limitations in Angle of Arrival Estimation: Array-based Ambiguities
Ambiguities can introduced to the estimation by the array itself Linear array has infinite ambiguities Planar array has two

Limitations in Angle of Arrival Estimation: Atmospheric Turbulence
Generally small (a few microradians) Can be significant depending on the application Guided missiles

Estimation Algorithms
Correlation Maximum Likelihood Estimation MUSIC: Multiple Signal Classification ESPIRIT: Estimation of Signal Parameters using Rotational Invariance Techniques Matrix Pencil

Estimation Algorithms: Correlation
Non-adaptive estimation 𝑥= 𝑚=1 𝑀 𝛼 𝑚 𝒔 𝜙 +𝒏 Wish to estimate 𝜙 𝑚 The function 𝒔 𝐻 𝜙 𝒔( 𝜙 𝑚 ) has a maximum at 𝜙= 𝜙 𝑚 Optimal for single-user situation Equivalent to DFT of x

Estimation Algorithms: Maximum Likelihood Estimation
Generalize n to an interference vector Vector has property that 𝐸 𝒏 𝒏 𝑯 = 𝑹 𝑛 Both magnitude and AoA are unknown parameters MLE described by 𝜙 , 𝛼 = 𝑚𝑎𝑥 𝛼,𝜙 [ 𝑓 𝑋/𝛼,𝜙 𝒙 ] AoA estimate is where maximum likelihood estimate 𝑃 𝑀𝐿𝐸 𝜙 of spectrum takes maximum 𝑃 𝑀𝐿𝐸 = 𝒔 𝑯 𝑹 𝒏 −𝟏 𝒙 𝟐 𝒔 𝑯 𝑹 𝒏 −𝟏 𝒔 Observations Requires a priori knowledge of interference covariance matrix Highly intensive Impractical algorithm

Estimation Algorithms: MUSIC
Multiple Signal Classification Adaptive technique based on orthogonality of uncorrelated signal covariance matrix 𝒙=𝑺𝜶+𝒏 𝑺 is N x M steering matrix of M steering vectors 𝑺 𝐻 𝒒 𝑚 =0 All eigenvectors 𝒒 𝑚 are orthogonal to the M signal steering vectors Pseudo-spectrum 𝑃 𝑀𝑈𝑆𝐼𝐶 𝜙 = 𝑸 𝑛 𝐻 𝒔 𝝓 𝟐

Estimation Algorithms: MUSIC

Estimation Algorithms: Root-MUSIC
Addresses problem of accuracy in MUSIC due to discretization and need for human interaction Uses a model of the signal--𝑠(𝜙) Algorithm First requires calculation of correlation matrix R 𝑹= 1 𝐾 𝑘=1 𝐾 𝒙 𝒌 𝒙 𝒌 𝐻 provides an estimation of R Decompose R into Q by 𝑹=𝑸𝚲 𝐐 H Partition Q for smallest eigenvalues, 𝑸 𝑛 𝑪= 𝑸 𝒏 𝑸 𝒏 𝐻 , sum diagonals of this matrix provides 𝐶 𝑙 𝑃 𝑀𝑈𝑆𝐼𝐶 −1 𝜙 = 𝑙=𝑁−1 𝑁+1 𝐶 𝑙 𝑧 𝑙 ; 𝑧= 𝑒 𝑗𝑘𝑑𝑐𝑜𝑠𝜙 Roots near unit circle provide 𝜙 𝑚 = cos −1 𝔍 ln 𝑧 𝑚 𝑘𝑑 , for 𝑚=1,2,…,𝑀

Estimation Algorithm: ESPRIT
Estimation of Signal Parameters using Rotational Invariance Techniques Operates based on constant phase shift within S matrix Algorithm First requires calculation of correlation matrix R 𝑹= 1 𝐾 𝑘=1 𝐾 𝒙 𝒌 𝒙 𝒌 𝐻 provides an estimation of R Decompose R into Q by 𝑹=𝑸𝚲 𝐐 H Partition to find 𝑸 𝒔 , M largest eigenvalues of Q. Matrix C defined by 𝑸 𝒔 =𝑺𝑪 Estimate 𝚿= 𝐂 −𝟏 𝚽𝐂 Calculate AoA with 𝜙 𝑚 = cos −1 𝔍 ln 𝑧 𝑚 𝑘𝑑 , for 𝑚=1,2,…,𝑀 𝑧 𝑚 is provided as mth element of diagonal matrix Φ

Estimation Algorithms: Matrix Pencil
Non-statistical technique Time based signal x n = 𝑚=1 𝑀 𝐴 𝑚 𝑧 𝑚 𝑛 + 𝑛 𝑛 Again, 𝑧 𝑚 = e jkdcos 𝜙 m Δ𝑡 Estimate poles using multiple samples of x Use X matrices (as shown) to calculate the system roots 𝑧 𝑚 𝜙 𝑚 = cos −1 𝔍 ln 𝑧 𝑚 𝑘𝑑 , for 𝑚=1,2,…,𝑀

Summary of Methods Discussed
MUSIC/Root-MUSIC Requires assumption of N > M, resolving up to N-1 signals. Large number of signals ESPRIT Requires assumption that N > M as well Pencil Matrix Maximum value N/2 for even N, (N+1)/2 for odd Does not require large number of samples ½ time of Root-MUSIC, less computation If coherent detector is present, same accuracy as Root-MUSIC

Passive Radar for Detection of Ground Moving Objects
Recently developed for border security Utilizes AoA MUSIC technique alongside range-Doppler technique for target location Test operation at 1 GHz using a cell phone antenna emitting a BPSK signal

Passive Radar for Detection of Ground Moving Objects