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

7/21/2015 2

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?

7/21/2015 3

• 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

4 7/21/2015

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

5 7/21/2015

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

7/21/2015

Additional adaptive DOFs are embedded in longer weight vector wNL − N2×1 vs N×1

Loss = 0.6 dB Loss = 0.75 dB

7

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

8

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

9 7/21/2015

Recall…..

Sample over all angles…..

Unfortunately kernel matrix Q is rank deficient!

Beamformer Solution -- Lagrangian

10 7/21/2015

Form Lagrangian....

Set derivatives to zero....

Define complex Lagrange multiplier vector....

Beamformer Solution -- Decompose w

11 7/21/2015

Compute SVD of Q....

Decompose w and substitute into previous eqn....

Since....

Set....

Beamformer Solution -- Final

12 7/21/2015

Apply constraint equation....

9 nulls at -53°, -40°, -26°, -20°, -10°, 10°, 15°, 33°, 47°

Iterated Version

13 7/21/2015

Integrated sidelobe power

Null, mainbeam constraints

Iterated version useful for STAP problem – dimensions (MN)2 × 1

Evolution of Iterates

14 7/21/2015

Initial condition must satisfy desired constraints

Sample Loss Calculation Any adaptation incurs losses which must be carefully considered….

15

16

Adapted Output – 0 Jammers

SNR = 60 dB SNR = 26 dB

7/21/2015 16

17

Adapted Output – 1 Jammer

SNR = 60 dB SNR = 26 dB

7/21/2015 17

18

Adapted Output – 7 Jammers

SNR = 60 dB SNR = 26 dB

7/21/2015

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

19