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Nonlinear Beamforming
Peter Vouras Naval Research Laboratory
Radar Division, Surveillance Technology Branch, Code 5341 peter.vouras@nrl.navy.mil
202.404.1859
DISTRIBUTION STATEMENT A: Distribution is unlimited
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.
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
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
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
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
Beamformer Objective
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Recall…..
Sample over all angles…..
Unfortunately kernel matrix Q is rank deficient!
Beamformer Solution -- Lagrangian
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Form Lagrangian....
Set derivatives to zero....
Define complex Lagrange multiplier vector....
Beamformer Solution -- Decompose w
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Compute SVD of Q....
Decompose w and substitute into previous eqn....
Since....
Set....
Beamformer Solution -- Final
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Apply constraint equation....
9 nulls at -53°, -40°, -26°, -20°, -10°, 10°, 15°, 33°, 47°
Iterated Version
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Integrated sidelobe power
Null, mainbeam constraints
Iterated version useful for STAP problem – dimensions (MN)2 × 1
Evolution of Iterates
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Initial condition must satisfy desired constraints
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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