Spatial Array Digital Beamforming and Filtering Tim D

Spatial Array Digital Beamforming and Filtering Tim D. Reichard, M. S. L-3 Communications Integrated Systems Garland, Texas 972. 205. 8411 Timothy. D. Reichard@L-3 Com. com AES Brief – 25 -Mar-03 TDR Page 1

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 AES Brief – 25 -Mar-03 TDR Page 2

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

Processing Domains s(x, t) = s(t - a. x) Space-Time ejk. x e-jwt e-jk. x ejwt S(x, w) S(k, t) Wavenumber Time Space-Freq e-jk. x ejwt (or beamspace) e-jwt Wavenumber Frequency S(k, w) AES Brief – 25 -Mar-03 TDR Page 4

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

Delay and Sum Beamformer (Narrowband) s(x, t) = e j(wot - ko. x) ko y 0(t) y 1(t) w*1 Delay D 1 . . . y. M-1(t) Time Domain: w*0 Delay D 0 . . . S z(t) w*M-1 Delay DM-1 M-1 m=0 z(t) = S w*m ym(t - Dm) = ejwot S w*m e-j(wo. Dm + ko. xm) = w. Hy Freq Domain: M-1 Z(w) = S w*m. Ym(w, xm) m=0 let AES Brief – 25 -Mar-03 TDR e-j(wo. Dm) M-1 = S w*m. Ym(w, xm) ej(ko. xm) = e. HWY Dm = (-||ko||. xm) / c m=0 e is a Mx 1 steering vector Ð -||ko|| Page 6

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

Spatial Sampling M-Sensor ULA Interpolation Beamformer (at location xo): y 0(n) u’ 0(n) Delay D 0 I y 1(n) w 1 Delay D 1 I . . . w 0 y. M-1(n) u’M-1(n) Delay I DM-1 Up-sample z(n) . . . S LPF (p/I) I Down-sample w. M-1 z(n) = S wm S ym(k) * h((n-k)T-Dm) m=0 k • Motivation: Reduce aberrations introduced by delay quantization • Postbeamforming interpolation is illustrated with polyphase filter AES Brief – 25 -Mar-03 TDR Page 8

Minimum Variance (MV) Beamformer • Apply a weight vector w to sensor outputs to emphasize a steered direction (z) while suppressing other directions such that at w = wo: Real {e¢w} = 1 Hence: min E[ |w¢y|2] w yields => wopt = R-1 e / [e¢R-1 e ] Conventional (Delay & Sum Beamformer) Steered Response Power: PCONV(e) = [ e¢WY ] [ Y¢W¢e ] = e¢ R e for unity weights Minimum Variance Steered Response Power: PMV(e) = w¢opt R wopt = [e¢ R-1 e ]-1 • 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¢ AES Brief – 25 -Mar-03 TDR Page 9

ULA Beamformer Comparison ; w = wo PCONV(z) = [e¢(z) R e(z)] PMV(z) = [e¢(z) R-1 e(z)]-1 AES Brief – 25 -Mar-03 TDR Page 10

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

Example Scenario for a Digital Minimum Variance Beamformer Coherent Interference Signal (7 deg away & 5 d. B down from SOI) Signal of Interest (SOI) location PMV(z) = [e¢(z) R-1 e(z)]-1 M-1 Shows Signals Resolvable . x) W(k) = S wmej(k Beam Steered to SOI with 0. 4 degree pointing error m=0 Setup Info used: • N = 500 samples • M = 9 sensors, ULA with d = l/2 spacing • SOI pulse present in samples 100 to 300 • Co-Interference pulse present in samples 250 to 450 AES Brief – 25 -Mar-03 TDR • Aperture Size (D) = 8 d • Array Gain = M for unity wm " m Page 12

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

Adaptive Beamformer Example #2 Robust GSC Architecture Constrained Optimization: min w¢Rw subject to Cw = c and ||B¢wa||2 < b 2 - ||wc||2 where b is constraint placed on adapted weight vector Robust GSC y 0(l) y 1(l) . . . y. M-1(l) Setup: w*c(0) Delay D 0 • For Minimum Variance let C = e¢, c = 1 • e = Array Steering Vector cued to SOI w*c(1) Delay D 1 . . . + S S _ ~ wa w*a, 0(l). . . B • B is Blocking Matrix such that B¢C = 0 • Determine Step Size (m) using Rideal: m = 0. 1*(max l BRideal. B)-1 ~ • wa = B¢wa w*c(M-1) Delay DM-1 z(l) w*a, M-1(l) wa S LMS Algorithm • wc = C¢(CC¢)-1 c Adaptive (Iterative) Portion: • y. B(l) = By(l) ~ • v(l) = wa(l) + mz*(l)B¢ y. B(l) ~ • w (l+1) = v(l), a ||v(l)||2 < b 2 - ||wc||2 (b 2 -||wc||2)1/2 v(l)/||v(l)||, otherwise ~ • z(l) = [wc - wa(l)]¢ y(l) AES Brief – 25 -Mar-03 TDR Page 14

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

Adaptive Beamformer Relative Performance Comparisons RMS Phase Noise = 136 mrad RMS Phase Error = 32 mrad • 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 (m), the faster the adaptation • Additional constraints can be used with these algorithms • min l. PRP is proportional to noise variance => adaptation rate is roughly proportional to SNR AES Brief – 25 -Mar-03 TDR Page 16

Applications to Passive Digital Receiver Systems y 0(t) BPF Digitizer DCM BPF Digitizer y 1(t) . . . y. M-1(t) Steering Vector DCM . . . Adaptive Beamformer Signal Detection and Parameter Encoding • Sparse Array useful for reducing FE hardware while attempting to retain aperture size -> spatial resolution • Aperture Size (D) = 17 d in case with d = l/2 and sensor spacings of {0, d, 3 d, 6 d, 2 d, 5 d} • 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 AES Brief – 25 -Mar-03 TDR Page 17

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 AES Brief – 25 -Mar-03 TDR Page 18

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. AES Brief – 25 -Mar-03 TDR Page 19