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

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

3. 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

4. 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

5. 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 ]

6. 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

7. 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

8. 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

9. 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

10. 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

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

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

13. 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

14. 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

15. 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

16. Estimation Algorithms: MUSIC

17. 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

18. 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

19. 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

20. 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

21. 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

22. Passive Radar for Detection of Ground Moving Objects

23. References http://www.comm.utoronto.ca/~rsadve/Notes/DOA.pdf
tionSecon06.pdf
Combined Use of Various Passive Radar Techniques and Angle of Arrival using MUSIC for the Detection of Ground Moving Objects. Chan et al. p=&arnumber=
Angle-of-Arrival of a Radar Beam in Atmospheric Turbulence. McMillan et al. p=&arnumber=999728

24. Questions?