Approaches for Angle of Arrival Estimation Wenguang Mao Angle of - - PowerPoint PPT Presentation

Approaches for Angle of Arrival Estimation

Wenguang Mao

Angle of Arrival (AoA)

• Definition: the elevation and azimuth angle of incoming signals
• Also called direction of arrival (DoA)

AoA Estimation

• Applications: localization, tracking, gesture recognition, ……
• Requirements: antenna array
• Approaches:
• Generate a power profile over various incoming angles
• Determine all AoA 𝜄"

𝜾𝟐 𝜾𝟑

Related Concepts

• Synthetic aperture radar (SAR)
• Using a moving antenna to emulate an array
• Alternative way of using physical antenna array
• NOT an estimation approach in the context of AoA
• Most AoA estimation methods can be applied to both physical antenna array

and SAR

• In this presentation, we only focus on antenna array
• May require some modification when applied to SAR

Related Concepts

• Beamforming
• A class of AoA estimation approaches
• MUSIC
• A specific algorithm in subspace-based

approaches

AoA Estimation approaches Beamforming Subspace methods MUSIC

Approaches for AoA Estimation

• Naïve approach
• Beamforming approaches
• Bartlett method
• MVDR
• Linear prediction
• Subspace based approaches
• MUSIC and its variants
• ESPIRIT
• Maximum likelihood estimator
• ……

Key Insights

• Phase changes over antennas are determined by the incoming angle
• Far-field assumption
• Phase of the antenna 1: 𝜚'
• Phase of the antenna 2: 𝜚'
• Then the difference is given by

𝝔𝟑 − 𝝔𝟐 = 𝟑𝝆 𝒆𝒅𝒑𝒕𝜾𝟐 𝝁 + 𝟑𝐥𝝆

Naïve approach

• Determine AoA based on the phase difference of two antenna
• Problems:
• Works for only one incoming signals
• Phase measurement could be noisy
• Ambiguity
• Adopted and improved by RF-IDraw

𝒅𝒑𝒕𝜾𝟐 = (𝚬𝝔 𝟑𝝆 − 𝒍) 𝝁 𝒆

Using Antenna Array

• Received signals at 𝑛-th antenna:

𝒚𝒏 𝒖 = < 𝒕𝒐(𝒖)

𝑶 𝒐?𝟐

𝒇𝒌⋅𝟑𝝆⋅𝝊𝒐⋅(𝒏D𝟐) + 𝒐𝒏(𝒖) 𝑡F(𝑢) : n-th source signals 𝜐F = IJKLMN

O

: phase shift per antenna 𝑂 : the number of sources 𝑁 : the number of antennas 𝑜S(𝑢) : noise terms

Using Antenna Array

• Matrix form:

𝒚𝟐 𝒖 𝑼 𝒚𝟑 𝒖 𝑼 ⋮ 𝒚𝑵 𝒖 𝑼 = 𝒃(𝜾𝟐) 𝒃(𝜾𝟑) ⋯ 𝒃(𝜾𝑶) 𝒕𝟐 𝒖 𝑼 𝒕𝟑 𝒖 𝑼 ⋮ 𝒕𝑶 𝒖 𝑼 + 𝒐𝟐 𝒖 𝑼 𝒐𝟑 𝒖 𝑼 ⋮ 𝒐𝑵 𝒖 𝑼 𝒀 = 𝑩𝑻 + 𝑶 Steering vector: 𝒃 𝜾 = 𝟐 𝒇𝒌𝟑𝝆𝝊 𝜾 𝒇𝒌𝟑𝝆𝝊 𝜾 ⋅𝟑 … 𝒇𝒌𝟑𝝆𝝊 𝜾

𝑵D𝟐 𝑼

Beamforming at the Receiver

• Definition: a method to create certain radiation pattern by combining

signals from different antennas with different weights.

• Will magnify the signals from certain direction while suppressing those

from other directions

Beamforming at the Receiver

• Signals after beamforming using a weight vector 𝒙
• By selecting different 𝑥, the received signal 𝑍 will contain the signal

sources arrived from different direction.

• Beamforming techniques are widely used in wireless communications

𝒁 = 𝒙𝑰𝒀

Beamforming at the Receiver

• Adjust the weight vector to rotate the

radiation pattern to angle 𝜄

• Measure the received signal strength 𝑄(𝜄)
• Repeat this process for any 𝜄 in [0, pi]
• Plot (𝜄, 𝑄(𝜄))
• Peaks in the plot indicates the angle of

arrival

𝜾𝟐 𝜾𝟑

Bartlett Beamforming

• Also called: correlation beamforming, conventional beamforming,

delay-and-sum beamforming, or Fourier beamforming

• Key idea: magnify the signals from certain direction by compensating

the phase shift

• Consider one source signal 𝑡 𝑢 arrived at angle 𝜄d
• Signal at 𝑛-th antenna: xf t = 𝑡(𝑢) ⋅ 𝑓i⋅jk⋅l(Mm)(SD')
• Weight at 𝑛-th antenna: wf = 𝑓i⋅jk⋅l(M)(SD')
• Only when 𝜄 = 𝜄d, the received signal Y = wpX = ∑𝑥S

∗ 𝑦S 𝑢 u is

maximized Phase shift

Bartlett Beamforming

• Weight vector for beamforming angle 𝜄:
• Signal power at angle 𝜄:
• Used by Ubicarse with SAR

𝒙 = 𝒃(𝜾) 𝑸 𝜾 = 𝒁𝒁𝑰 = 𝒙𝑰𝒀 𝒙𝑰𝒀

𝑰 = 𝒙𝑰𝒀𝒀𝑰𝒙 = 𝒙𝑰𝑺𝒀𝒀𝒙 = 𝒃𝑰 𝜾 𝑺𝒀𝒀𝒃(𝜾)

This is why it is called steering vector Covariance matrix

Bartlett Beamforming

• Works well when there is only one source signal
• Suffers when there are multiple sources: very low resolution

Minimum Variance Distortionless Response (MVDR)

• Also called Capon’s beamforming
• Key idea: maintain the signal from the desired direction while

minimizing the signals from other direction

• Mathematically, we want to find such weight vector 𝒙 for the

beaming angle 𝜄

𝐧𝐣𝐨 𝒁𝒁𝑰 = 𝐧𝐣𝐨 𝒙𝑰𝑺𝒀𝒀𝒙 s.t. 𝒙𝑰(𝒃 𝜾 𝒕 𝒖 𝑼) = 𝒕 𝒖 𝑼

Maintain the signals from angle 𝜾

𝒙𝑰𝒃 𝜾 = 𝟐

MVDR

• Weight vector for beamforming angle 𝜄:
• Signal power at angle 𝜄:

𝑸 𝜾 = 𝒁𝒁𝑰 = 𝒙𝑰𝑺𝒀𝒀𝒙 = 𝟐 𝒃 𝜾 𝑺𝒀𝒀

D𝟐𝒃𝑰(𝜾)

𝒙 = 𝑺𝒀𝒀

D𝟐𝒃𝑰(𝜾)

𝒃 𝜾 𝑺𝒀𝒀

D𝟐𝒃𝑰(𝜾)

MVDR

• Resolution is significantly enhanced compared to Bartlett method
• But still not good enough
• Better beamforming approaches are developed, e.g., Linear Prediction
• Or resort to subspace based approaches

Subspace Based Approaches

• Beamforming is a way of shaping received signals
• Can be used for estimating AoA
• Can also be used for directional communications
• Subspace based approaches are specially designed for parameter (i.e.,

AoA) estimation using received signals

• Cannot be used for extracting signals arrived from certain direction
• Subspace based approaches decompose the received signals into

“signal subspace” and “noise subspace”

• Leverage special properties of these subspaces for estimating AoA

Multiple Signal Classification (MUSIC)

• Key ideas: we want to find a vector 𝑟 and a vector function 𝑔(𝜄)
• Such that 𝒓𝑰𝒈 𝜾 = 𝟏 if and only if 𝜄 = 𝜄" (i.e., one of AoA)
• Then we can plot 𝒒 𝜾 =

'

• ‚ƒ M

„ =

' ƒ‚ M ••‚ƒ(M)

• The peaks in the plot indicates AoA
• We can expect very sharp peak since 𝑟…𝑔 𝜄 = 0, so the inverse of

its magnitude is infinity

How to find 𝒓 and 𝒈(𝜾)

Multiple Signal Classification (MUSIC)

• MUSIC gives a way to find a pair of 𝑟 and 𝑔(𝜄)
• The signals from antenna array
• Covariance matrix of the signals

𝒀 = 𝑩𝑻 + 𝑶 𝑺𝒀𝒀 = 𝑭[𝒀𝒀𝑰] = 𝑭[𝑩𝑻𝑻𝑰𝑩𝑰] + 𝑭[𝑶𝑶𝑰] 𝑺𝒀𝒀 = 𝑩𝑭[𝑻𝑻𝑰]𝑩𝑰 + 𝝉𝟑𝑱 𝑺𝒀𝒀 = 𝑩𝑺𝑻𝑻𝑩𝑰 + 𝝉𝟑𝑱

Signal terms Noise terms

MUSIC

• Consider the signal term
• 𝑆•• is 𝑂×𝑂 matrix, where 𝑂 is the number of source signals
• 𝑆LL has the rank equal to 𝑂 if source signals are independent
• 𝐵 is 𝑁×𝑂 matrix, where 𝑁 is the number of antenna
• 𝐵 has full column rank
• The signal term is 𝑁×𝑁 matrix, and its rank is 𝑂
• The signal term has 𝑂 positive eigenvalues and 𝑁 − 𝑂 zero eigenvalues, if M>N
• There are 𝑁 − 𝑂 eigenvectors 𝑟" such that 𝐵𝑆••𝐵…𝑟" = 0
• Then 𝐵…𝑟" = 0, where 𝐵 = [𝑏(𝜄')

𝑏(𝜄j) ⋯ 𝑏(𝜄‘)]

• Then 𝑟"

…𝑏 𝜄 = 0 if 𝜄 = 𝜄"

What we want !!!

MUSIC

• 𝑏(𝜄) is the steering function, so it is known
• Needs to determine 𝑟", which needs the eigenvalue decomposition of

the signal term.

• We don’t know the signal term; we only know the sum of the signal

term and the noise term, i.e., 𝑆’’

• All of eigenvectors of the signal term are also ones for 𝑆’’, and

corresponding eigenvalues are added by 𝜏j

• Only need to find the eigenvectors of 𝑆’’ with eigenvalues equal to 𝜏j

MUSIC

• Derive 𝑆’’
• Perform eigenvalue decomposition on 𝑆’’
• Sort eigenvectors according to their eigenvalues in descent order
• Select last 𝑁 − 𝑂 eigenvectors 𝑟"
• Noise space matrix 𝑅‘ = [𝑟•–' 𝑟•–j … 𝑟‘]
• 𝑅‘

…𝑏 𝜄 = 0 for any AoA 𝜄"

• Plot 𝑞 𝜄 =

' ˜‚ M ™š™š

‚˜(M) and find the peaks

Performance Comparison

(a) 10 antennas (a) 50 antennas

Performance Comparison

(a) SNR 1dB (b) SNR 20dB

Beamforming approaches Music variants