Page 1AES Brief – 25-Mar-03 TDR Spatial Array Digital Beamforming and Filtering L-3 Communications...

AES Brief – 25-Mar-03 TDR

Spatial Array Digital Beamforming and Filtering

L-3 Communications Integrated SystemsGarland, Texas972.205.8411

[email protected]

Tim D. Reichard, M.S.


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


Propagating Plane Waves

k

Temporal Freq. Spatial Freq. (|k| = )

s(xo,t) = Ae j(t - k . xo)

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:

2s + 2s + 2s = 1 . 2sx2 y2 z2 c2 t2

• 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 jt

Notation: Lowercase Underline indicates 1-D matrix (k) Uppercase Underline indicates 2-D matrix (R) or H indicates matrix conjugate-transpose


Processing Domains

Space-Time

s(x, t) = s(t - . x)s(x, t)

Space-Freq

S(x, )

e-jt

ejt

Wavenumber -Frequency

S(k, )

(or beamspace)

e-jk.x

ejk.x

Wavenumber -Time

S(k, t)

ejt

e-jt

e-jk.x

ejk.x


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

dorigin

x

M = 7

Sparse Linear Array (SLA)

m= 0 1 2 3

d x

M = 4

Co-Array Function:

C() = wm1w*m2

where; m1 and m2 are a set of indices for xm2 – xm1 =

- 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

# Redundancies4

1

2

3Co-Array “A Perfect Array”


Delay and Sum Beamformer (Narrowband)

Delay0

Delay1

DelayM-1

w*0

w*1

w*M-1

.

.

.

y0(t)

y1(t)

yM-1(t)

.

.

.

z(t)

z(t) = w*m ym(t - m) = ejot w*m e-j(om + ko . xm) = wHy

M-1

m=0

Time Domain: M-1

m=0

ko

s(x,t) = e j(ot - ko . x)

Freq Domain:

Z() = w*mYm(xm) e-j(om) = w*mYm(xm) ej(ko . xm) = eHWY

M-1

m=0 m=0

M-1

e is a Mx1 steering vector -||ko||let m = (-||ko|| . xm) / c


Delay and Sum Beamformer (Broadband)

z(n)

.

.

.

z(n) = w*m,p ym(n - p) = wHy(n) m=1

J

y1(n)

w*1,0

z-1

w*1,1

z-1 z-1

w*1,L-1. . .y2(n)

w*2,0

z-1

w*2,1

z-1 z-1

w*2,L-1. . .

...

...

yJ(n)

w*J,0

z-1

w*J,1

z-1 z-1

w*J,L-1. . .

...

L-1

p=0

J = number of sensor channels

L = number of FIR filter tap weights


Spatial Sampling

I

LPF(/I)

y0(n) u’0(n) w0

Delay0

I

y1(n)z(n)

w1

Delay1

I

yM-1(n) u’M-1(n) wM-1

DelayM-1

.

.

.I

Up-sample

Down-sample

M-Sensor ULA Interpolation Beamformer (at location xo):

z(n) = wm ym(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


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 => wopt = R-1 e / [eR-1e ]

Conventional (Delay & Sum Beamformer) Steered Response Power:

PCONV(e) = [ eWY ] [ YWe ] = eR e for unity weights

Minimum Variance Steered Response Power:

PMV(e) = woptR wopt = [eR-1e ]-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


ULA Beamformer Comparison

PMV() =

[e(R-1 e()]-1

PCONV() =

[e(R e()]

; = o


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)

• Rideal= ss + I2 = Signal Est. + Noise Est.

• Determine Step Size () using Rideal:

= 0.1*(3*trace[PRidealP])-1

• P = I - C(CC)-1C

• wc = C(CC)-1c

• w(l=0) = wc

Constrained Optimization:

min wRw subject to Cw = c

Setup:

• z(l) = w(l)y(l)

• w(l+1) = wc + P[w(l) - z*(l)y(l)]

Adaptive (Iterative) Portion:

No

n-A

dap

tive

w

c AdaptiveAlgorithm

y0(l)

y1(l)

z(l)

yM-1(l)

.

.

.

Ad

apti

ve

w

w

Frost GSC†

.

.

.

.

.

.

†- General Sidelobe Canceller


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 wm m

W(k) = wmej(k.x)m=0

M-1

PMV() =

[e(R-1e()]-1


Example of Frost GSC Adaptive Beamformer Performance Results†

†- via Matlab simulation


Adaptive Beamformer Example #2 - Robust GSC Architecture

Constrained Optimization:

min wRw subject to Cw = c and ||Bwa||2 < 2 - ||wc||2

where is constraint placed on adapted weight vectorSetup:

• 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 Rideal:

= 0.1*(max BRidealB)-1

• wa = Bwa

• wc = C(CC)-1c

~

• yB(l) = By(l)

• v(l) = wa(l) + z*(l)B yB(l)

• wa(l+1) = v(l), ||v(l)||2 < 2 - ||wc||2

(2-||wc||2)1/2 v(l)/||v(l)||, otherwise

• z(l) = [wc - wa(l)]y(l)

Adaptive (Iterative) Portion:

~

~

~

LMSAlgorithm

y0(l)

y1(l)

yM-1(l)

.

.

.

Robust GSC

Delay0

Delay1

DelayM-1

w*c(0)

w*c(1)

w*c(M-1)

+

B...

w*a,M-1(l)

w*a,0(l)

z(l)

_

wa

wa~

.

.

.


Example of Robust GSC Adaptive Beamformer Performance Results†

†- via Matlab simulation


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


Applications to Passive Digital Receiver Systems

y0(t)

y1(t)

yM-1(t)

.

.

.

DCM Digitizer

DCM Digitizer

DCM Digitizer

Adaptive Beamformer

Signal Detection and

ParameterEncoding

BPF

BPF

BPF

S

tee

rin

g V

ec

tor

.

.

.

• 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


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


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.

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