Fundamentals of Radar Imaging

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Fundamentals of Radar Imaging

• Allan C. Schell

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The Uses of an Antenna
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Types of Imaging

• Coherent Source Imaging
• When phase is maintained on successive returns
• Incoherent Source Imaging
• When returns and/or sources are uncorrelated

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Angular vs. Spatial Frequency FunctionsAngle
? Fourier Transform ? Spatial
Frequency
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Imaging Properties of an Interferometer
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Multiplicative Arrays

• A type of signal processing antenna consisting of
  two arrays where the outputs are multiplied and
  averaged
• For incoherent sources, the output is
  proportional to the product of the two antenna
  field patterns
• Examples
• Covington and Broten array, 1957
• Mills Cross Antenna, 1953

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Multiplicative Arrays

• The voltage outputs of two antennas G1 and G2
  scanning a source E are multiplied to form

G1G2 ?? F1 (u u) E (u) F2 (u u) E
(u) du du
If the source distribution is incoherent, the
averaged multiplier output is
Output ? F1 (u u) F2 (u u) T (u)
du where T EE
In the spatial frequency domain the antenna
transfer function is
Tau (s) ? f1 (x) f2 (s x) dx
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The Mills Cross Array
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Three Element Green Bank Interferometer16
collinear baselines12 hrs of tracks2700
MHzMax wavelengths24,200
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Imaging Properties of an Interferometer
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The Arsac Array
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Imaging Systems and Error Criteria
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Restoration in the Angular and Spatial Frequency
Domain
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Restoration in the Presence of Noise
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Superresolution Methods
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Imaging Systems and Error Criteria
tR
tE
t tRtE at
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Variational Calculus for Antennas

• Key distinction between time functions and
  radiation patterns
• Time functions obey causality
• Integrated radiation patterns equal ?2 /4p
• Consider an error function E (for example, the
  mean square difference between a desired power
  pattern PD and that of an array P. In the
  spatial frequency domain
• the error is (tD t)2 .)
• Let I be the Integral over the spatial frequency
  range
• Then I ? E du

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Variational Calculus for Antennas (1)
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Variational Calculus for Antennas (2)
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Variational Calculus for Antennas (3)
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Example A Minimum MSE Enhanced Output

• E restored output enhanced output over the
  observed range of spatial frequencies 2
• added spatial frequency components to assure
  positive definiteness of transform 2
• The range of integration is larger than the range
  of received spatial frequencies
• Slide 13

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Outputs of the Array, the Restoration Filter, and
the Enhancement Processor
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Enhancement Illustration - Two Sources
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EnhancementIllustrationTwoEqual Sources
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The Arsac Array
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The Optimum Density Function for S/N 3.5
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Transfer Function of an Optimum Array with 105
Element Spaces and S/N of 3.5
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An Approximation to the Optimum Array for S/N
3.5
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Digital Beamforming
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Digital Beamforming
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Definition
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Digital Beamforming Capabilities
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FFT Beamformer
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Adaptive Nulling
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Adaptive Nulling
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Neural Network Beamformingfor Direction Finding
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A Neural Network Direction Finding System
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Antenna Pattern Correction after Element Failure
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Antenna Pattern Correctionafter Element Failure
(cont.)
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Antenna Pattern Correctionafter Element Failure
(cont.)
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Experimental Results for Two Uncorrelated Sources
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Array Architectures
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An Array Antenna
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Generic Passive Array Architecture
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Generalized Array
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Digital Beamforming Array
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Serial Beamformer
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Parallel (Systolic) Beamformer
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Typical Digital Beamformer Receiver
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A/D Converter Performance
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A Digital Image Processing Array
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Phased Array Trends
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Typical Multilayer Printed Circuit Array
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The End