Post on 04-Jun-2018
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Chapter 5cAntennas and Propagation Slide 2
Introduction
Time-domain Signal ProcessingFourier spectral analysis
Identify important frequency-content of signalMatched/Wiener Filter
Optimize signal to noise ratio of output (known signal / noise cov.)
Array Signal ProcessingExploit spatial dimension similar to time-domain SP
This LectureClassical methods: direct extensions of time-domain SPParametric (superresolution) methodsMain Resource: Krim / Viberg paper
Chapter 5cAntennas and Propagation Slide 3
Array Signal Processing
Direction of Arrival EstimationWaves arriving from different directions Induce different phase shifts across arrayFourier-type analysis: Identify different spatial frequencies
Optimal (Linear) BeamformingWiener / Matched-filtering in spatial domain
Limitations of Linear MethodsPerformance limited by size of aperture(regardless of SNR / number of samples)Nonlinear (superresolution) methods
Chapter 5cAntennas and Propagation Slide 4
Signal Model
Narrowband signal:
Signal on the array:
Narrowband AssumptionChanges in s(t)appear simultaneously on array
Signal received at origin
Chapter 5cAntennas and Propagation Slide 5
Signal Model (2)
Restrict attention to xy plane
“Steering Vector”
Collect signals from L antennas
Ant. Coords
Signal
Chapter 5cAntennas and Propagation Slide 6
Signal Model (3)
Steering matrix
Vector of signal waveforms
Steering vector for a ULA
Multiple signals
are baseband waveforms
More compact form
Presence of additive noise
Chapter 5cAntennas and Propagation Slide 7
Assumptions
Exploit spatial dimension: Spatial covariance matrix
Source covariance
Noise covariance
Assuming noise is “white” or uncorrelated from one sensor to the next
Assume P is non-singular matrix (e.g. uncorrelated signals)
Chapter 5cAntennas and Propagation Slide 8
Signal / Noise Subspaces
Suppose that L > M (more antennas than signals)Can partition R according to
Signal Subspace Noise Subspace
Note: Columns of Us span range space of AColumns of Un span its orthogonal complement (null space)
Projection Operators
Chapter 5cAntennas and Propagation Slide 9
Problem Statement
Estimating DOAsFind θm for each of the incoming signalsGiven a finite set of observations {x(t)}Note: In practice have only estimates
Assumption: Know M or how many signals present
Estimating SignalsRecover signals s(t) once DOAs known
Chapter 5cAntennas and Propagation Slide 10
Summary of Estimators
DefinitionsCoherent signals
Signals that are scaled/delayed versions of each other
ConsistencyEstimate converges to true value for infinite data
Statistical efficiencyAsymptotically attains CRB (lower bound on covariance matrix of any
unbiased estimator)
Chapter 5cAntennas and Propagation Slide 12
Spectral-Based vs. Parametric
SpectralForm a function of parameter of interest (DOA)Sweep that function with respect to some parameterIdentify peaksTypically a 1D search. Find DOAs independently
ParametricSimultaneous search of all parametersHigher accuracyIncreased complexity
Chapter 5cAntennas and Propagation Slide 13
Spectral-Based Methods
Beamforming“Steer” a beam and measure output powerPeaks give DOA estimates
Linear beamformer θ0 = steering angle
θ1
θ2
Sources
θ0Po
wer
θ1 θ2
Chapter 5cAntennas and Propagation Slide 14
Bartlett Beamformer
Same as uniform excitation we saw beforeMaximize power collected from look angle θ
For a ULA
Resolution approximately 100º/L
Chapter 5cAntennas and Propagation Slide 15
Bartlett Beamformer (2)
ExampleL=10 Elements, λ/2 spacingResolution of standard ULA approximately 100º/L = 10º(Obtain from HPBW expression)
Chapter 5cAntennas and Propagation Slide 16
Bartlett Beamformer (3)
AdvantagesSimpleRobust
DisadvantagesResolution is limitedInterference of close-by arrivalsStrong side lobes
Chapter 5cAntennas and Propagation Slide 17
Capon’s Beamformer
Revised problem
Minimize total power collectedMaintain gain in “look direction” θ to be 1What does this mean?Like a sharp spatial bandpass filter
Reduce interference from directions other than θ when we are looking in direction θ
Chapter 5cAntennas and Propagation Slide 19
Capon’s Beamformer (3)
AdvantageProvides much narrower main beam. How?Nulls directions that are near look direction
DisadvantagesSacrifice some noise performanceAlso, can be unstable (consider inverse)Resolution still depends on aperture size and SNR
Chapter 5cAntennas and Propagation Slide 20
Subspace-Based Methods
MUSIC (Multiple Signal Classification)Introduced by R. Schmidt in 1980Breakthrough in DOA EstimationExploit structure of signal/noise subspacesResolution no longer depend on array size
Chapter 5cAntennas and Propagation Slide 21
MUSIC
Decompose covariance with EVD
Assume P to be full rank, A (LxM) is “tall” (L>M)Us and A span same (column) subspaceUn spans the orthogonal complement of Us
⇒ Each vector in A is orthogonal to Un
Idea: Sweep θ and see where this goes to 0.
Music spectrum:
Exhibits peaks when θ is a DOA.
Chapter 5cAntennas and Propagation Slide 22
Comparison: Spectral-based Methods
Parameters:L = 10d = λ/2M = 200 samples
Chapter 5cAntennas and Propagation Slide 23
Coherent Signals
ProblemSignals are correlated with each otherP is no longer full rankMUSIC spectrum will not exhibit peaksExample? Multipath
Techniques to Decorrelate signalsULAForward-backward averagingSpatial smoothing
Chapter 5cAntennas and Propagation Slide 24
Forward-Backward Averaging
Reverse signals in x vector (reverse antennas)
followed by complex conjugate
Introduces a unique phase shift for each steering vector (or source)Can treat as another sample of the same signalBut phase shift introduces decorrelation
Chapter 5cAntennas and Propagation Slide 25
Forward-Backward Averaging (2)
Including backward signals in our covariance estimate
Consider: pairs of sources are correlatedNew effective source covariance not correlated
Chapter 5cAntennas and Propagation Slide 26
Spatial Smoothing
IdeaRelated to FB averagingForm multiple looks of sources by shifting the arrayThis shifts each steering vector (source) by a different phaseRelative phases in each steering vector
are preserved (shift invariance)
Spatial smooth by factor N todecorrelate N sources
Chapter 5cAntennas and Propagation Slide 27
Parametric Methods
Drawback of Spectral MethodsMay be inaccurate (e.g. correlated signals)
Parametric MethodsFully exploit the underlying data modelPowerful, but in general require multi-dimensional searchException: For ULA can exploit model without search
VariantsML (deterministic or stochastic)Subspace fittingRoot MUSICESPRIT
Chapter 5cAntennas and Propagation Slide 28
Deterministic Maximum Likelihood
AssumeZero-Mean, White Gaussian Noisepdf of observed signal (complex Gaussian)
Form Likelihood FunctionIf noise is uncorrelated between samples
Likelihood of observing x(t) = As(t) + n(t) given noise, DOAs, signals
IdeaFind DOAs / signals that make observed x(t) as likely as possible
Chapter 5cAntennas and Propagation Slide 29
Deterministic Maximum Likelihood (2)
(Negative) Log-Likelihood Function
Minima satisfy
Substituting into Log-LikelihoodMinimum:
Make σ as small as possibleInterpretation?When we remove DOAs exactly, resulting power is minimal
Samplecovariance
Projection onto null-space of A
Pseudo-inverse of A
Chapter 5cAntennas and Propagation Slide 30
Deterministic Maximum Likelihood (3)
How do we minimize?
Requires a multidimensional search (numerical)Becomes very complicated for large M
Acceleration methodFind an initial guess with spectral methodFollowed by local optimizer
Chapter 5cAntennas and Propagation Slide 31
Parametric Methods for ULAs
Uniform Linear ArraysSteering matrix has Vandermonde structureCan exploit this strcutureAllows close to ML estimate to be found without searching
ESPRITEstimation of Signal Parameters by Rotation Invariant TechniquesUses the shift-invariance property of A
Chapter 5cAntennas and Propagation Slide 32
ESPRIT
Recall the EVD of R
Steering matrix for ULAVandermonde Matrix
Chapter 5cAntennas and Propagation Slide 33
ESPRIT (2)
Shift property of A
A
Can find a direct method to get Φ
P is full rank, span of Us and A same, which means
For some invertible matrix T
Chapter 5cAntennas and Propagation Slide 34
ESPRIT (3)
Consider relationship of Ψ and ΦSimilar matricies⇒ Have same eigenvalues
Chapter 5cAntennas and Propagation Slide 35
ESPRIT (4)
Tasks
Solve for Ψ
Compute eigenvalues to get Φ ⇒ φ1, φ2, ...
Compute DOAs using
How do we solve this?
Chapter 5cAntennas and Propagation Slide 36
Total Least Squares (TLS)
Want to find A that solves (X,Y tall, A is N x N)Form N dimensional orthogonal basisthat best spans both X and Y
In the N-dimensional subspace, can now equate
Chapter 5cAntennas and Propagation Slide 37
Summary
Array Signal ProcessingLike filtering, but in spatial dimensionCan enhance signalsEstimate locations of sources
Spectral-based MethodsBeamforming (Bartlett, Capon)Subpace-based Method (MUSIC)
Parametric MethodsDirectly exploit underlying signal modelDMLESPRIT