ieee_aes_dbf_and_filtering_25mar03_reichard_revb3
TRANSCRIPT
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Page 1AES Brief 25-Mar-03 TDR
Spatial ArrayDigital Beamforming and Filtering
L-3 Communications Integrated SystemsGarland, Texas
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.