ieee_aes_dbf_and_filtering_25mar03_reichard_revb3

8/7/2019 IEEE_AES_DBF_and_Filtering_25Mar03_reichard_RevB3

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Page 1AES Brief 25-Mar-03 TDR

Spatial ArrayDigital Beamforming and Filtering

L-3 Communications Integrated SystemsGarland, Texas

[email protected]

Tim D. Reichard, M.S.


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Spatial Array DigitalBeamforming 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


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Propagating Plane Waves

k

Temporal Freq. Spatial Freq.(|k| = 2 / )

s(x o,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 Maxwells equations on an E-Mfield in free space, the Wave Equation is defined as:

2s + 2s + 2s = 1 . 2s x2 y2 z2 c 2 t2

Governs how signals pass from a radiatingsource to a sensing array

Linear - so many plane waves in differingdirections can exist simultaneously => theSuperposition Principal

Planes of constant phase such thatmovement of x over time t is constant

Speed of propagation for a lossless mediumis | x|/ t = c

Slowness vector: = k/ and | | = 1/c

Sensor placed at the origin has only atemporal frequency relation:

s([0,0,0], t) = Ae j tNotation: Lowercase Underline indicates 1-D matrix (k)Uppercase Underline indicates 2-D matrix (R)

or H indicates matrix conjugate-transpose


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Processing Domains

Space-Time

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

Space-Freq

S(x, )

e -j t

e j t

Wavenumber -Frequency

S(k, )

(or beamspace)

e-j k.x

e jk .x

Wavenumber -Time

S(k, t)

e j t

e -j t

e -j k.x

e jk .x


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Some Array Types andthe 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 ( ) = w m1 w *m2where; 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

x0 1d 2d 3d 4d 5d 6d

Co-Array

# Redundancies

6

2

4

x0 1d 2d 3d 4d 5d 6d

# Redundancies4

123Co-Array A Perfect Array


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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 ym(t - m) = e j o t w*m e -j( o m + ko . xm ) = w Hy

M-1

m=0

Time Domain:M-1

m=0

k o

s(x,t) = e j( o t - k o . x)

Freq Domain:

Z( ) = w*m Ym( , xm) e -j( o m ) = w*m Ym( , xm) e j(k o . xm ) = e HWYM-1

m=0 m=0

M-1

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


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Delay and Sum Beamformer (Broadband)

z(n)

.

.

.

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

J

y 1(n)

w*1,0

z -1

w*1,1

z -1 z -1

w*1,L-1. . .y 2(n)

w*2,0

z -1

w*2,1

z -1 z -1

w*2,L-1

. . .

...

...

y J (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


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Spatial Sampling

I

LPF( /I)

y0(n) u 0(n) w 0Delay

0

I

y1(n)

z(n)

w 1Delay

1

I

y M-1 (n) u M-1 (n) w M-1Delay

M-1

.

.

.

I

Up-sample

Down-sample

M-Sensor ULA Interpolation Beamformer (at location x o):

z(n) = w m 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


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Minimum Variance (MV) Beamformer

Apply a weight vector w to sensor outputs to emphasize a steered direction ( ) whilesuppressing other directions such that at =

o: Real {e w} = 1

Hence: min E [ |w y| 2] yields => w opt = 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 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


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ULA Beamformer Comparison

PMV

( ) =

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

PCONV ( ) =

[e ( ) R e( )]

; = o


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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 w Rw 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-Adaptivew

c AdaptiveAlgorithm

y 0(l )

y 1(l )

z( l )

yM-1 (l )

.

.

.

Adaptive w

w

Frost GSC

.

.

.

.

.

.

- General Sidelobe Canceller


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Example Scenario for a DigitalMinimum Variance Beamformer

Signal of Interest (SOI)location

Beam Steered to SOI with0.4 degree pointing error

Coherent Interference Signal(7 deg away & 5dB down from SOI)

Shows SignalsResolvable

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 me j(k .x)m=0M-1

PMV ( ) =

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

l f d


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Example of Frost GSC AdaptiveBeamformer Performance Results

- via Matlab simulation

Ad i B f E l #2


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Adaptive Beamformer Example #2 -Robust GSC Architecture

Constrained Optimization:

min w Rw subject to Cw = c and ||B w 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 B C = 0

Determine Step Size ( ) using R ideal :

= 0.1*(max BR ideal B)-1

w a = B w a

w c = C (CC )-1 c

~

yB(l ) = By( l )

v( l ) = w a(l ) + z*(l )B yB(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:

~

~

~

LMSAlgorithm

y 0(l )

y 1(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 )

_

w a

w a~

.

..

E l f R b GSC Ad i


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Example of Robust GSC AdaptiveBeamformer Performance Results

- via Matlab simulation

Ad ti B f


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Adaptive Beamformer Relative Performance Comparisons

SOI Pulsewidth retained for both; Robust has better response Robust methods 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 mradRMS Phase Error = 32 mrad

Applications to Passive Digital


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Applications to Passive DigitalReceiver Systems

y0(t)

y1(t)

y M-1 (t)

.

.

.

DCM Digitizer

DCM Digitizer

DCM Digitizer

AdaptiveBeamformer

SignalDetection and

Parameter Encoding

BPF

BPF

BPF SteeringVector

.

.

.

Sparse Array useful for reducing FE hardwarewhile attempting to retain aperture size ->spatial resolution

Aperture Size (D) = 17d in case with d = /2and sensor spacings of {0, d, 3d, 6d, 2d, 5d}

Co-array computation used to verify no spatialaliasing for chosen sensor spacings Tradeoff less HW for slightly lower array gain Further reductions possible with subarrayaveraging at expense of beam-steering

response and resolution performance


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Summary

Digital beamforming provides additional flexibility for spatial filtering andsuppression of unwanted signals, including coherent interferers

Various types of arrays can be used to suit specific applications

Minimum Variance beamforming provides excellent spatial resolutionperformance over conventional BF and adjusts according to SNR of

incoming signals Adaptive algorithms, implemented iteratively can provide moderate to fastmonopulse convergence and provide additional reduction of unwantedsignals 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 bothamplitude and complex phase information

Linear Arrays can be utilized in many ways depending on application andperformance priorities


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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, ArtechHouse, Norwood, MA, 2001.

ieee_aes_dbf_and_filtering_25mar03_reichard_revb3

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