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Easy Does It: User Parameter FreeDense and Sparse Methods

for Spectral Estimation

Jian Li

Department of Electrical and Computer Engineering University of FloridaGainesville, Florida

USA

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Spectral Estimation

The goal of spectral estimation is to determine how power distributes over frequency from a finite number of data samples.

Diverse Applications

For example: synthetic aperture radar (SAR) imaging.

Data-Independent ApproachesFFT, Matched Filter, Delay-

and-Sum (DAS)

Poor resolution

High sidelobe levels, especially with missing data.

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FFT; HH

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A SAR imaging example using FFT.

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Data-Adaptive Spectral Estimation

Data-Adaptive Approaches Examples: APES, Capon

Multiple snapshots needed to form reliable sample covariance matrices – fails for single or few snapshots, irregularly sampled data

High computational complexities

High resolution

Low sidelobe levels

Recent Development Iterative Adaptive Approach (IAA)

Applicable to single snapshot scenario

High computational complexities

High resolution

Low sidelobe levels

Dense and accurate

WFFT

IAA

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Iterative Adaptive Approach (IAA)

Each iteration of IAA includes two steps (user parameter

free):

Estimate coefficients:

Update covariance matrix estimate

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IAA-R (IAA with Regularization)

Noise effect taken into account explicitly:

Still user parameter free!

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Active Sensing Example

s

t

y

TnJ s

TnJ s

Clutter ReturnTarget Return

sMatched Filter: z = sTy

Transmitter

Receiver

Active sensing (radar, sonar, etc.) Received signal decomposition:

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sTy

Range-Doppler Imaging

Matched Filter Initialization

Movies Are Nice

Local Quadratic Convergence of IAA Proven.

Radar GMTI Example

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Terrain map

The goal of ground moving target indication (GMTI) is to detect slow moving targets in the stationary background.

yellow or green dots: moving vehicles

STAP

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An

t en

na

Ele

me

nt

s

Pulsesslowtime

1 M

N

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MN samples for fixed range bin

Range

bin

s

fast

time

(J. Ward ’94)

STAP: space-time adaptive processing

Datacube:

Adaptive Processing

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Space-Time Adaptive Processor

(Guerci et al. ’06)

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Angle-Doppler Imaging in STAP

dB

IAADAS

Clutter power distribution over angle-Doppler for a fixed range

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Target angle: 195

A total of 200 targets with constant power

Average SCNR over range: -18.94 dB

Ground truth denoted by x

o

Simulated Ground Truth

Target Detection for Fixed Angle

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Range-Doppler Images

Ideal (total knowledge)

Prior (wrong knowledge)

IAA

dB

GLC (partial knowledge)

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ROC Curves

Median CFAR algorithm applied to target detection

GLC detector

Automatic diagonal loading

Sample Number N = 20

Prior detector

Wrong prior knowledge of the clutter-and-noise covariance matrix

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KASSPER DataSet

Main-beam width: 5 target angles: 190 - 200 (3-D target detection)

A total of 246 targets with varying power

Slow-moving targets and/or weak targets present

o

o o

o195azimuth =

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ROC Curves (KASSPER Data)

Median CFAR algorithm applied for target detection

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Sparse Approaches Related work:

is replaced by to yield a convex optimization problem. LASSO: The least absolute shrinkage and selection operator. BP: Basis pursuit, very similar to LASSO FOCUSS: Focal underdetermined system solution SBL: Sparse Bayesian learning L1-SVD: L1 – singular value decomposition, similar to BP CoSaMP: Compressive Sampling Matching Pursuit

Most existing algorithms require Large computation times User parameters

Hard to decide Performance sensitive to choice of user parameter

Minimize such that is satisfied.

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Kragh et al. Approach

Kragh et al. uses optimization transfer technique to obtain an iterative procedure:

A recent paper on SAR imaging states:

“

’’

This is FOCUSS.

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SLIM

Sparse Learning via Iterative Minimization (SLIM) Solves the User Parameter Problem! (Tan, Roberts, Li, and Stoica, 2010)

SLIM Assumes the Following Hierarchical Bayesian Model:

SLIM is a MAP Approach:

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SLIM Iterations

SLIM Iterates the Following Steps (Starting with DAS):

Given q, SLIM is User Parameter Free– Easy to Use!

Regularized Minimization in SLIM

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Cyclic approach with majorization minimization employed to minimize cost function.

Conjugate gradient + FFT can be used for efficient implementation of SLIM.

For fixed noise variance (i.e., making it a user parameter), SLIM becomes FOCUSS/Kragh et al. Approach.

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FFT for GOTCHA

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SLIM for GOTCHA

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SLIM for GOTCHA

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IAA (Dense) vs. SLIM (Sparse)

IAA is dense; SLIM is sparse.

IAA is more accurate; SLIM tends to bias downward.

IAA has high resolution; SLIM has higher resolution.

IAA’s fast implementation is trickier, especially for non-uniformly sampled data; SLIM is faster and its fast implementation is more straightforward.

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Concluding Remarks

We need to devise dense and sparse methods that are user parameter free – easy to use in practice,

And accurate, And with high resolution, And computationally

efficient.

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Thank you!