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 " 𝜾 𝟐 𝜾 𝟑

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 AoA Estimation approaches • MUSIC • A specific algorithm in subspace-based Subspace methods approaches Beamforming 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 𝝔 𝟑 − 𝝔 𝟐 = 𝟑𝝆 𝒆𝒅𝒑𝒕𝜾 𝟐 + 𝟑𝐥𝝆 𝝁

Naïve approach • Determine AoA based on the phase difference of two antenna 𝒅𝒑𝒕𝜾 𝟐 = (𝚬𝝔 𝟑𝝆 − 𝒍) 𝝁 𝒆 • Problems: • Works for only one incoming signals • Phase measurement could be noisy • Ambiguity • Adopted and improved by RF-IDraw

Using Antenna Array • Received signals at 𝑛 -th antenna: 𝑶 𝒇 𝒌⋅𝟑𝝆⋅𝝊 𝒐 ⋅(𝒏D𝟐) + 𝒐 𝒏 (𝒖) 𝒚 𝒏 𝒖 = < 𝒕 𝒐 (𝒖) 𝒐?𝟐 𝑡 F (𝑢) : n-th source signals 𝜐 F = IJKLM N : phase shift per antenna O 𝑂 : the number of sources 𝑁 : the number of antennas 𝑜 S (𝑢) : noise terms

Using Antenna Array • Matrix form: 𝒚 𝟐 𝒖 𝑼 𝒕 𝟐 𝒖 𝑼 𝒐 𝟐 𝒖 𝑼 𝒚 𝟑 𝒖 𝑼 𝒕 𝟑 𝒖 𝑼 𝒐 𝟑 𝒖 𝑼 𝒃(𝜾 𝟑 ) ⋯ = 𝒃(𝜾 𝟐 ) 𝒃(𝜾 𝑶 ) + ⋮ ⋮ ⋮ 𝒚 𝑵 𝒖 𝑼 𝒕 𝑶 𝒖 𝑼 𝒐 𝑵 𝒖 𝑼 𝒀 = 𝑩𝑻 + 𝑶 𝑼 Steering vector: 𝒃 𝜾 = 𝟐 𝒇 𝒌𝟑𝝆𝝊 𝜾 𝒇 𝒌𝟑𝝆𝝊 𝜾 ⋅𝟑 … 𝒇 𝒌𝟑𝝆𝝊 𝜾 𝑵D𝟐

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 𝒁 = 𝒙 𝑰 𝒀 • 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

Beamforming at the Receiver • Adjust the weight vector to rotate the radiation pattern to angle 𝜄 • Measure the received signal strength 𝑄(𝜄) 𝜾 𝟐 • Repeat this process for any 𝜄 in [0, pi] 𝜾 𝟑 • Plot ( 𝜄, 𝑄(𝜄) ) • Peaks in the plot indicates the angle of arrival

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 Phase shift • Consider one source signal 𝑡 𝑢 arrived at angle 𝜄 d • Signal at 𝑛 -th antenna: x f t = 𝑡(𝑢) ⋅ 𝑓 i⋅jk⋅l(M m )(SD') • Weight at 𝑛 -th antenna: w f = 𝑓 i⋅jk⋅l(M)(SD') ∗ 𝑦 S 𝑢 u is • Only when 𝜄 = 𝜄 d , the received signal Y = w p X = ∑𝑥 S maximized

Bartlett Beamforming • Weight vector for beamforming angle 𝜄 : This is why it is called 𝒙 = 𝒃(𝜾) steering vector • Signal power at angle 𝜄 : 𝑰 = 𝒙 𝑰 𝒀𝒀 𝑰 𝒙 = 𝒙 𝑰 𝑺 𝒀𝒀 𝒙 = 𝒃 𝑰 𝜾 𝑺 𝒀𝒀 𝒃(𝜾) 𝑸 𝜾 = 𝒁𝒁 𝑰 = 𝒙 𝑰 𝒀 𝒙 𝑰 𝒀 Covariance matrix • Used by Ubicarse with SAR

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 𝒁𝒁 𝑰 = 𝐧𝐣𝐨 𝒙 𝑰 𝑺 𝒀𝒀 𝒙 𝒙 𝑰 𝒃 𝜾 = 𝟐 s.t. 𝒙 𝑰 (𝒃 𝜾 𝒕 𝒖 𝑼 ) = 𝒕 𝒖 𝑼 Maintain the signals from angle 𝜾

MVDR • Weight vector for beamforming angle 𝜄 : D𝟐 𝒃 𝑰 (𝜾) 𝑺 𝒀𝒀 𝒙 = D𝟐 𝒃 𝑰 (𝜾) 𝒃 𝜾 𝑺 𝒀𝒀 • Signal power at angle 𝜄 : 𝟐 𝑸 𝜾 = 𝒁𝒁 𝑰 = 𝒙 𝑰 𝑺 𝒀𝒀 𝒙 = D𝟐 𝒃 𝑰 (𝜾) 𝒃 𝜾 𝑺 𝒀𝒀

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 𝒓 𝑰 𝒈 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 𝒀 = 𝑩𝑻 + 𝑶 • Covariance matrix of the signals 𝑺 𝒀𝒀 = 𝑭[𝒀𝒀 𝑰 ] = 𝑭[𝑩𝑻𝑻 𝑰 𝑩 𝑰 ] + 𝑭[𝑶𝑶 𝑰 ] 𝑺 𝒀𝒀 = 𝑩𝑭[𝑻𝑻 𝑰 ]𝑩 𝑰 + 𝝉 𝟑 𝑱 𝑺 𝒀𝒀 = 𝑩𝑺 𝑻𝑻 𝑩 𝑰 + 𝝉 𝟑 𝑱 Noise terms Signal terms

MUSIC • Consider the signal term • 𝑆 •• is 𝑂×𝑂 matrix, where 𝑂 is the number of source signals • 𝑆 LL 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(𝜄 j ) ⋯ 𝑏(𝜄 ‘ )] … 𝑏 𝜄 = 0 if 𝜄 = 𝜄 " What we want !!! • Then 𝑟 "

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

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 ‘ ] … 𝑏 𝜄 = 0 for any AoA 𝜄 " • 𝑅 ‘ ' • Plot 𝑞 𝜄 = ‚ ˜(M) and find the peaks ˜ ‚ M ™ š ™ š

Performance Comparison (a) 10 antennas (a) 50 antennas

Performance Comparison (a) SNR 1dB (b) SNR 20dB Music variants Beamforming approaches