nonlinear beamforming peter vouras · distribution statement a : distribution is unlimited ....
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Nonlinear Beamforming
Peter Vouras Naval Research Laboratory
Radar Division, Surveillance Technology Branch, Code 5341 [email protected]
202.404.1859
DISTRIBUTION STATEMENT A: Distribution is unlimited
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Outline
• Introduction – Motivation, Objective, Open Questions
• Overview of Nonlinear Adaptive Processing • Optimal Beamformer Solution • Simulated Results • Summary
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Motivation
• Problem: On receive only, is it possible to improve the performance of DBF for small (or sparse) arrays that operate in dense interference environments?
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• Solution: Nonlinear adaptive algorithms – Nonlinear techniques enable enhanced adaptive degrees of freedom (DOFs)
– e.g., O(N2) vs N-1 DOFs for a linear array – Conventional linear techniques apply spatial filter to complex signal amplitudes. Nonlinear
algorithms apply spatial filter directly to signal power
• Today’s Technology Road Map: Future designs for radars digitize the output of every array element to enable digital beamforming (DBF). In arrays with few elements, DBF yields marginal gains in performance.
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Sparse Digital Arrays Sparse arrays have fewer elements…
Uniform Sparse Arrays – Used extensively on satellites to minimize
antenna size, weight, and power (SWAP) – Sparsity is created by increasing inter-element
spacing – Mainbeam does not scan, so grating lobes can be
set to always point into empty space
Nonuniform Sparse Arrays – Pseudorandom sparse arrays have elevated
average sidelobe levels – Minimum redundancy arrays may have fewest
elements but difficult to determine optimal element placement
– All these approaches have trade-offs – Nested or coprime arrays are a subset of sparse
arrays with highly desirable properties • No grating lobes • Together with nonlinear processing offer
enhanced adaptive DOFs • Computational complexity scales with DOFs
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Grating Lobes
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Definition of Nested Arrays • Nested linear arrays are passive
non-uniform arrays obtained by combining two or more uniform linear arrays with increasing inter-sensor spacing
– Smallest inter-element spacing is λ/2
• Using nonlinear adaptive processing, a nested array with N elements can form O(N2) nulls in the receive pattern
– Conventional linear adaptive processing can create no more than N-1 nulls
– Extra DOFs can also be applied towards sidelobe control or shaping mainbeam
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P. Pal and P. P. Vaidyanathan, “Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom”, IEEE Transactions on Signal Processing, Vol. 58, No. 8, 2010
Uniform Linear Array Nested Linear Array
Length = 6
1
Length = 12
8 12 4 1 6
• Increasing length of array decreases beamwidth • Since array gain is fixed (proportional to N), sidelobe
level must increase
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Recent Theoretical Developments • Nested arrays
– P. Pal and P. P. Vaidyanathan, “Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom”, IEEE Transactions on Signal Processing, Vol. 58, No.8, Aug. 2010
– P. Pal and P. P. Vaidyanathan, “Nested Arrays in Two Dimensions, Part I: Geometrical Considerations,” IEEE Transactions on Signal Processing, Vol. 60, No. 9, Sept. 2012
• Calibration – K. Han, P. Yang, A. Nehorai, “Calibrating Nested Sensor Arrays With Model Errors,”
Proceedings 48th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA., Nov. 2-5, 2014
• Spectrum sensing – D. Cohen and Y. C. Eldar, “Sub-Nyquist Sampling for Power Spectrum Sensing in
Cognitive Radios: A Unified Approach,” IEEE Transactions on Signal Processing, Vol. 62, No. 15, Aug. 2014
• Multiple Input Multiple Output (MIMO) radar – M. Contu and P. Lombardo, “Sidelobe Control for a MIMO Radar Virtual Array,”
Proceedings 2013 IEEE Radar Conference, Ottawa, CA., April 29 – May 3, 2013 6 7/21/2015
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Overview of Nested Array Processing
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Additional adaptive DOFs are embedded in longer weight vector wNL − N2×1 vs N×1
Loss = 0.6 dB Loss = 0.75 dB
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Difference Coarray There is a duality between nonlinear beamforming on the array of physical elements and linear beamforming on a virtual array called the difference coarray
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This weight vector applied to the virtual difference coarray
This weight vector applied to the vectorized covariance matrix……
yields the same beampattern as….
In 2 dimensions………..
Physical array Virtual difference coarray
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Beamformer Objective
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Recall…..
Sample over all angles…..
Unfortunately kernel matrix Q is rank deficient!
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Beamformer Solution -- Lagrangian
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Form Lagrangian....
Set derivatives to zero....
Define complex Lagrange multiplier vector....
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Beamformer Solution -- Decompose w
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Compute SVD of Q....
Decompose w and substitute into previous eqn....
Since....
Set....
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Beamformer Solution -- Final
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Apply constraint equation....
9 nulls at -53°, -40°, -26°, -20°, -10°, 10°, 15°, 33°, 47°
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Iterated Version
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Integrated sidelobe power
Null, mainbeam constraints
Iterated version useful for STAP problem – dimensions (MN)2 × 1
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Evolution of Iterates
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Initial condition must satisfy desired constraints
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Sample Loss Calculation Any adaptation incurs losses which must be carefully considered….
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Adapted Output – 0 Jammers
SNR = 60 dB SNR = 26 dB
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Adapted Output – 1 Jammer
SNR = 60 dB SNR = 26 dB
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Adapted Output – 7 Jammers
SNR = 60 dB SNR = 26 dB
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Summary
• Nonlinear adaptive beamforming techniques on sparse arrays require many data snapshots at low SNRs to achieve desired performance – 1 snapshot suffices at high SNRs
• Potential payoff to radars is enhanced adaptivity in small arrays
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