Radar Imaging and Waveform Design

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Radar Imaging and Waveform Design

• Birsen Yazici
• Electrical, Computer and Systems Engineering
• Rensselaer Polytechnic Institute
• May 26, 2005

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Outline

• Range-Doppler imaging and adaptive diversity
  waveform design B. Yazici and G. Xie
• Synthetic aperture inversion for arbitrary flight
  path in the presence of noise and interference
  B. Yazici and M. Cheney
• Conclusions

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Filtered Backprojection Method

• Range-Doppler imaging and waveform design based
  on simplified forward model that takes advantage
  of underlying invariant structure
• SAR imaging based on physics based forward model
  that takes into antenna beam pattern, array
  configuration, transmitted pulse, geometric
  spreading factors and flight path

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Range-Doppler Imaging

• Ideal point target model
• Ideal extended target model

Doppler stretch
Time delay
Target reflectivity function
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Clutter/Interference and Noise Models

• Target Object of interest
• Clutter All unwanted reflections and other
  scatterers
• Noise Thermal noise at the receiver

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Problem Statement

• How do we recover target reflectivity function
  embedded in noise and clutter?
• What waveform(s) shall we transmit?
• To suppress noise clutter/interference
• For simultaneous estimation of target velocity
  and range

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Approach

• View received echo from target as the Fourier
  transform of the target with respect to affine
  group evaluated at the transmitted waveform
• Fourier transform of functions defined on range
  and Doppler plans is not equal to the standard
  Fourier transforms with respect to range and
  Doppler
• Fourier transform of functions defined on
  range-Doppler plane is a matrix valued function

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Advantages

• Address receiver and waveform design problems
  simultaneously
• Bypasses the concept of ambiguity function
• Waveform design for simultaneous estimation of
  target range and velocity
• Provides design of clutter rejecting waveforms
• Method of data fusion to synthesize high
  resolution range-Doppler images from multiple
  narrowband radars operating coherently

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How to Estimate Target

• Send sufficient number of pulses to cover the
  spectrum of the target
• Use Fourier inversion formula for reconstruction
  Filtered back projection

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How to Suppress Clutter/Interference

• Perform minimum mean square error filtering based
  on the spectra of target and clutter/interference
• Optimal MMSE filter spectrum
• Optimal filtering

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Range-Doppler Image

• Implement inverse Fourier transform in different
  ways to obtain different receiver and waveform
  design algorithms
• Only need the diagonal terms for trace sum
• Alternatively use matrix elements of each
  transform

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Algorithm 1
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Algorithm 2
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Numerical Simulations

• Transmit waveforms derived from Laguerre
  polynomials

Target
Clutter
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Numerical Simulations
Wiener Filter
Difference between waveforms used in Algorithm 2
and deterministic algorithm
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Numerical Simulations
Algorithm 2
Deterministic algorithm
Algorithm 1
Comparison of Algorithm 1, Algorithm 2 and
deterministic algorithm
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Conclusions

• Clutter rejection can be performed either in
  transmission or reception
• Clutter rejection in transmission reduces
  receiver complexity
• In adaptive transmission, matching is NOT
  performed with respect to transmitted waveforms
• No limitation in the choice of orthogonal basis,
  but should be in the range space of the Wiener
  filter to avoid redundancy
• Algorithms applicable to MIMO scenario
• Narrowband case
• Discrete affine group

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SAR inversion for arbitrary flight path in the
presence of noise and interference
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Forward Model
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Problem Statement

• Received data model
• C unwanted ground reflectivity function
• n additive noise.
• Problem determine from .

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Noise and Interference Model

• Additive noise model Stationary in fast time,
  uncorrelated in slow time

• Target and interference model Not necessarily
  stationary

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Image Formation

• Form image by filtered back projection

• Determined by minimizing the mean square
  error

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How to solve for the filter?

• Apply change of variables so that when the
  forward and inverse map are composed the
  resulting operator is a pseudo differential
  operator.
• Perform stationary phase calculations to
  determine the leading order contributions to mean
  square error
• Perform variational calculations with respect to Q

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How to solve for the filter?

• In general the filter is the solution of an
  integral equation
• If we assume that the target and interference are
  stationary then,
• Jacobian coming from the change of variables

• When filter becomes

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Numerical Simulations
Target
Target and interference
Deterministic FBP reconstruction
Statistical FBP reconstruction
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Numerical Simulations
Target
Statistical FBP reconstruction
Details of statistical FBP
Deterministic FBP reconstruction
Details of deterministic FBP
Target and interference
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Numerical Simulation
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Conclusion

• Microlocal reconstruction extended to a
  statistical setting
• Bistatic and multistatic mode of operation
• Design of flight trajectories for multistatic
  UAVs
• Design of waveforms

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Inversion Methods

• Solving PDEs (Bellini et al. 1979)
• Filtered backprojection (Tretiak et al. 1980,
  Metz et al. 1995, Kuchment et al. 1994)
• Fourier Relation (Tretiak et al., Innouye et al.
  1989, Metz Pan, Kuchment et al.)
• Circular harmonic decomposition (Hawkins et al.
  1988)

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Fourier Transform of the Affine Group

• Irreducible unitary representations of the affine
  group
• Fourier transform
• Fourier inversion
• Discrepancy operator