antennas and propagation chapter 5c: array signal ... · chapter 5c: array signal processing and...

Chapter 5c: Array Signal Processingand Parametric Estimation Techniques

Antennas and Propagation

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


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


Signal Model

Narrowband signal:

Signal on the array:

Narrowband AssumptionChanges in s(t)appear simultaneously on array

Signal received at origin


Signal Model (2)

Restrict attention to xy plane

“Steering Vector”

Collect signals from L antennas

Ant. Coords

Signal


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


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)


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


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


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)


Summary of Estimators (2)


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


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


Bartlett Beamformer

Same as uniform excitation we saw beforeMaximize power collected from look angle θ

For a ULA

Resolution approximately 100º/L


Bartlett Beamformer (2)

ExampleL=10 Elements, λ/2 spacingResolution of standard ULA approximately 100º/L = 10º(Obtain from HPBW expression)


Bartlett Beamformer (3)

AdvantagesSimpleRobust

DisadvantagesResolution is limitedInterference of close-by arrivalsStrong side lobes


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


Capon’s Beamformer (2)

Solution


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


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


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.


Comparison: Spectral-based Methods

Parameters:L = 10d = λ/2M = 200 samples


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


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


Forward-Backward Averaging (2)

Including backward signals in our covariance estimate

Consider: pairs of sources are correlatedNew effective source covariance not correlated


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


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


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


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


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


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


ESPRIT

Recall the EVD of R

Steering matrix for ULAVandermonde Matrix


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


ESPRIT (3)

Consider relationship of Ψ and ΦSimilar matricies⇒ Have same eigenvalues


ESPRIT (4)

Tasks

Solve for Ψ

Compute eigenvalues to get Φ ⇒ φ1, φ2, ...

Compute DOAs using

How do we solve this?


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


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

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