1. Page 1AES Brief – 25-Mar-03 TDR Spatial Array Digital Beamforming and Filtering L-3 Communications Integrated Systems Garland, Texas 972.205.8411 Timothy.D.Reichard@L-3Com.com Tim D. Reichard, M.S.

2. Page 2AES Brief – 25-Mar-03 TDR Spatial Array Digital Beamforming and Filtering OUTLINE Propagating Plane Waves Overview Processing Domains Types of Arrays and the Co-Array Function Delay and Sum Beamforming – Narrowband – Broadband Spatial Sampling Minimum Variance Beamforming Adaptive Beamforming and Interference Nulling Some System Applications and General Design Considerations Summary

3. Page 3AES Brief – 25-Mar-03 TDR Propagating Plane Waves k Temporal Freq. Spatial Freq. (|k| =  ) s(x o,t) = Ae j(  t - k. x o ) Monochromatic Plane Wave (far-field): k = Wavenumber Vector = direction of propagation x = Sensor position vector where wave is observed x Using Maxwell’s equations on an E-M field in free space, the Wave Equation is defined as:  2 s +  2 s +  2 s = 1.  2 s  x 2  y 2  z 2 c 2  t 2 Governs how signals pass from a radiating source to a sensing array Linear - so many plane waves in differing directions can exist simultaneously => the Superposition Principal Planes of constant phase such that movement of  x over time  t is constant Speed of propagation for a lossless medium is |  x|/  t = c Slowness vector:  = k/  and |  | = 1/c Sensor placed at the origin has only a temporal frequency relation: s([0,0,0], t) = Ae j  t Notation: Lowercase Underline indicates 1-D matrix (k) Uppercase Underline indicates 2-D matrix (R) or H indicates matrix conjugate-transpose

4. Page 4AES Brief – 25-Mar-03 TDR Processing Domains Space-Time s(x, t) = s(t - . x) s(x, t) Space-Freq S(x,  ) e -j  t ejtejt Wavenumber - Frequency S(k,  ) (or beamspace) e -jk. x ejk.xejk.x Wavenumber - Time S(k, t) ejtejt e -j  t e -jk. x ejk.xejk.x

5. Page 5AES Brief – 25-Mar-03 TDR Some Array Types and the Co-Array Function 2-D Array d x Uniform Linear Array (ULA) m= 0 1 2 3 4 5 6 d origin x M = 7 Sparse Linear Array (SLA) m= 0 1 2 3 d x M = 4 Co-Array Function:  C(  ) = w m1 w* m2 where; m1 and m2 are a set of indices for x m2 – x m1 =  - Desire to minimize redundancies and - Choose spacing to prevent aliasing m1,m2 x 0 1d 2d 3d 4d 5d 6d Co-Array # Redundancies 6 2 4 x 0 1d 2d 3d 4d 5d 6d # Redundancies 4 1 2 3 Co-Array“A Perfect Array”

6. Page 6AES Brief – 25-Mar-03 TDR Delay and Sum Beamformer (Narrowband) Delay  0 Delay  1 Delay  M-1 w* 0 w* 1 w* M-1...... y 0 (t) y 1 (t) y M-1 (t)......  z(t) z(t) =  w* m y m (t -  m ) = e j  o t  w* m e -j(  o  m + k o. x m ) = w H y M-1 m=0 Time Domain: M-1 m=0 koko s(x,t) = e j(  o t - k o. x) Freq Domain: Z(  ) =  w* m Y m (  x m ) e -j(  o  m ) =  w* m Y m (  x m ) e j(k o. x m ) = e H WY M-1 m=0 M-1 e is a Mx1 steering vector  -||k o || let  m = (-||k o ||. x m ) / c

7. Page 7AES Brief – 25-Mar-03 TDR Delay and Sum Beamformer (Broadband)  z(n)...... z(n) =   w* m,p y m (n - p) = w H y(n) m=1 J y 1 (n) w* 1,0 z -1 w* 1,1 z -1 w* 1,L-1... y 2 (n) w* 2,0 z -1 w* 2,1 z -1 w* 2,L-1............... y J (n) w* J,0 z -1 w* J,1 z -1 w* J,L-1......... L-1 p=0 J = number of sensor channels L = number of FIR filter tap weights

8. Page 8AES Brief – 25-Mar-03 TDR Spatial Sampling I LPF (  /I) y 0 (n)u’ 0 (n)w0w0 Delay  0  I y 1 (n) z(n) w1w1 Delay  1 I y M-1 (n)u’ M-1 (n)w M-1 Delay  M-1...... I Up-sample Down-sample M-Sensor ULA Interpolation Beamformer (at location x o ) : z(n) =  w m  y m (k) * h((n-k)T-  m ) m=0 M-1 k...... Motivation: Reduce aberrations introduced by delay quantization Postbeamforming interpolation is illustrated with polyphase filter

9. Page 9AES Brief – 25-Mar-03 TDR Minimum Variance (MV) Beamformer Apply a weight vector w to sensor outputs to emphasize a steered direction (  ) while suppressing other directions such that at  =  o : Real {ew} = 1 Hence: min E[ |wy| 2 ] yields => w opt = R -1 e / [eR -1 e ] Conventional (Delay & Sum Beamformer) Steered Response Power: P CONV (e) = [ eWY ] [ YWe ] = e  R e for unity weights Minimum Variance Steered Response Power: P MV (e) = w opt  R w opt = [e  R -1 e ] -1 w MVBF weights adjust as the steering vector changes Beampattern varies according to SNR of incoming signals Sidelobe structure can produce nulls where other signal(s) may be present MVBF provides “excellent” signal resolution wrt steered beam over the Conventional Delay & Sum beamformer MVBF direction estimation accuracy for a given signal increases as SNR increases R = spatial correlation matrix = YY

10. Page 10AES Brief – 25-Mar-03 TDR ULA Beamformer Comparison P MV (  ) = [e(  R -1 e(  )] -1 P CONV (  ) = [e(  R e(  )] ;  =  o

11. Page 11AES Brief – 25-Mar-03 TDR Adaptive Beamformer Example #1 - Frost GSC Architecture For Minimum Variance let C = e, c = 1 e = Array Steering Vector cued to SOI R is Spatial Correlation Matrix = y(l)y(l) R ideal = ss + I  2 = Signal Est. + Noise Est. Determine Step Size (  ) using R ideal :  = 0.1*(3*trace[PR ideal P]) -1 P = I - C(CC) -1 C w c = C(CC) -1 c w(l=0) = w c Constrained Optimization: min wRw subject to Cw = c Setup: z(l) = w(l)y(l) w(l+1) = w c + P[w(l) -  z * (l)y(l)] Adaptive (Iterative) Portion: Non-Adaptive w c Adaptive Algorithm y0(l)y0(l)  y1(l)y1(l) z(l) y M-1 (l)...... Adaptive w w Frost GSC †............ † - General Sidelobe Canceller

12. Page 12AES Brief – 25-Mar-03 TDR Example Scenario for a Digital Minimum Variance Beamformer Signal of Interest (SOI) location Beam Steered to SOI with 0.4 degree pointing error Coherent Interference Signal (7 deg away & 5dB down from SOI) Shows Signals Resolvable N = 500 samples M = 9 sensors, ULA with d = /2 spacing SOI pulse present in samples 100 to 300 Co-Interference pulse present in samples 250 to 450 Setup Info used: Aperture Size (D) = 8d Array Gain = M for unity w m  m W(k) =  w m e j(k. x) m=0 M-1 P MV (  ) = [e(  R -1 e(  )] -1

13. Page 13AES Brief – 25-Mar-03 TDR Example of Frost GSC Adaptive Beamformer Performance Results † † - via Matlab simulation

14. Page 14AES Brief – 25-Mar-03 TDR Adaptive Beamformer Example #2 - Robust GSC Architecture Constrained Optimization: min wRw subject to Cw = c and ||Bw a || 2 <  2 - ||w c || 2 where  is constraint placed on adapted weight vector Setup: For Minimum Variance let C = e, c = 1 e = Array Steering Vector cued to SOI B is Blocking Matrix such that BC = 0 Determine Step Size (  ) using R ideal :  = 0.1*(max  BR ideal B ) -1 w a = Bw a w c = C(CC) -1 c ~ y B (l) = By(l) v(l) = w a (l) +  z * (l)B y B (l) w a (l+1) = v(l), ||v(l)|| 2 <  2 - ||w c || 2 (  2 -||w c || 2 ) 1/2 v(l)/||v(l)||, otherwise z(l) = [w c - w a (l)]  y(l) Adaptive (Iterative) Portion: ~ ~ ~ LMS Algorithm y0(l)y0(l)  y1(l)y1(l) y M-1 (l)...... Robust GSC Delay  0 Delay  1 Delay  M-1 w* c (0) w* c (1) w* c (M-1)  + B......  w* a,M-1 (l) w* a,0 (l) z(l) _ wawa wawa ~......

15. Page 15AES Brief – 25-Mar-03 TDR Example of Robust GSC Adaptive Beamformer Performance Results † † - via Matlab simulation

16. Page 16AES Brief – 25-Mar-03 TDR Adaptive Beamformer Relative Performance Comparisons SOI Pulsewidth retained for both; Robust has better response Robust method’s blocking matrix isolates adaptive weighting to nonsteered response Good phase error response for the filtered beamformer results Amplitude reductions due to contributions from array pattern and adaptive portions The larger the step size (  ), the faster the adaptation Additional constraints can be used with these algorithms min PRP is proportional to noise variance => adaptation rate is roughly proportional to SNR RMS Phase Noise = 136 mrad RMS Phase Error = 32 mrad

17. Page 17AES Brief – 25-Mar-03 TDR Applications to Passive Digital Receiver Systems y 0 (t) y 1 (t) y M-1 (t)...... DCMDigitizer DCMDigitizer DCMDigitizer Adaptive Beamformer Signal Detection and Parameter Encoding BPF Steering Vector...... Sparse Array useful for reducing FE hardware while attempting to retain aperture size -> spatial resolution Aperture Size (D) = 17d in case with d = /2 and sensor spacings of {0, d, 3d, 6d, 2d, 5d} Co-array computation used to verify no spatial aliasing for chosen sensor spacings Tradeoff less HW for slightly lower array gain Further reductions possible with subarray averaging at expense of beam-steering response and resolution performance

18. Page 18AES Brief – 25-Mar-03 TDR Summary Digital beamforming provides additional flexibility for spatial filtering and suppression of unwanted signals, including coherent interferers Various types of arrays can be used to suit specific applications Minimum Variance beamforming provides excellent spatial resolution performance over conventional BF and adjusts according to SNR of incoming signals Adaptive algorithms, implemented iteratively can provide moderate to fast monopulse convergence and provide additional reduction of unwanted signals relative to user defined optimum constraints imposed on the design Adaptive, dynamic beamforming aids in retention of desired signal characteristics for accurate signal parameter measurements using both amplitude and complex phase information Linear Arrays can be utilized in many ways depending on application and performance priorities

19. Page 19AES Brief – 25-Mar-03 TDR References D. Johnson and D. Dudgeon, “Array Signal Processing Concepts and Techniques,” Prentice Hall, Upper Saddle River, NJ, 1993. V. Madisetti and D. Williams, “The Digital Signal Processing Handbook,” CRC Press, Boca Raton, FL, 1998. H.L. Van Trees, “Optimum Array Processing - Part IV of Detection, Estimation and Modulation Theory,” John Wiley & Sons Inc., New York, 2002. J. Tsui, “Digital Techniques for Wideband Receivers - Second Edition,” Artech House, Norwood, MA, 2001.