Hayward Green Vg

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Direction Finding

for Tracking Radar Using a Simplified,Subspace Based Approach

Dr. Stephen Hayward DERA (Sensors and Processing)

Malvern, Worcs WR14 3PS

[email protected]

Andrew Green DERA (Sensors and Processing)

Malvern, Worcs WR14 3PS

[email protected]

Using the spatial information availablefrom an array of sensors to overcome

target ‘glint’



Multiple, coherent signals

Many pointscatterers on

an aircraft

Sea surfacereflection from asingle scatterer

Tracking radar

Multiple coherent signals can interfere at the receiver

Create non-plane waves at the antenna

Change the response of conventional beams

Can cause monopulse direction finding to fail



Interference effects at the array

Spatial variations in amplitude and phase gradient

Array

Field magnitudein the region of the array

Instantaneous phase gradientin the region of the array

Many pointscatterers on

an aircraft

X

Y

X

Y Regions of

abnormally highphase rate

Regions ofconstructive and

destructiveinterference



Conventional beamforming

Project a manifold vector onto a signal vector

Field magnitudein the region of the

arrayProjection (p) of

manifold vector onsignal subspace

Manifold vector

Signalsubspace is a

vector

Dimension 3

Dimension 2

Dimension 1

θ

Projection (q) ofmanifold vector on

orthogonal subspace

Estimationerror

Scan in azimuth

S c a n

i n

e l e v a t i o

n

Conventional beam scan

M a g n i t u d e o f ‘ p ’



Projection (q) ofsignal vector on

orthogonal subspace

Multi-dimensional beamforming

Project a signal vector onto a manifold subspace

Maximum likelihood approach

Scan in azimuth

S c a n

i n

e l e v a t i o n

Maximum likelihood scan

Projection (p) ofsignal vector on

manifold subspace

Signal vector

Manifoldsubspace is a

plane orvolume

Dimension 3

Dimension 2

Dimension 1

θ

No error




Projection (q) ofmanifold vector on

orthogonal subspace

Multi-dimensional beamforming

Project a manifold vector onto a measured signal subspace

MUSIC approach, using spatial smoothing

Scan in azimuth

S c a n

i n

e l e v a t i o n

MUSIC scan

Projection (p) ofmanifold vector on

signal subspace

Manifold vector

Signal subspace ,found by spatial

smoothing

Dimension 3

Dimension 2

Dimension 1

θ

Low error




Signal model

Single scatterer in far field gives rise to plane waves

Multiple scatterers create composite waves with 3 significant

planar components

Planar array ofsensors in

receiver

Azimuth

Scatterer at arraybroadside

Planarwavefront

Multiplescatterers at

array broadsideElevation

‘Sum’ planar

wavefront

Azimuth ‘differenc

planar wavefront

Elevation ‘difference’

planar wavefront



Results

Combined signal from many scatterers can be approximatelymodelled by a simple linear combination of three waveforms

Multi-dimensional projection approaches such as MUSIC orMaximum Likelihood can reduce or eliminate glint errors subjectto low noise levels

-60 -40 -20 0 20 400

10

20

30

40

50

60

M e a n e s t i m a t i o n e r r o r i n m e t r e s

Mean error in position estimates

-60 -40 -20 0 20 40

0

50

100

150200

250

300

350

400

Noise level in dB

RMS deviation of position estimates

Monopulse performance

MUSIC performance

MUSIC performance

Monopulse performance

Noise level in dB

m e

t r e s

m e

t re s

Low variance Low bias



Bounds on performance

Can use the uniform Cramer-Rao lower bound (CRB) [5] to showlimits on theoretical performance for a given scattererarrangement, signal to noise ratio, and receiver array

Assume that the scatterers are all close together and use aninvertible transform to obtain the mean scatterer position

Generate the uniform CRB on estimates of this mean position

Use targets containing single and multiple scatterers withconstructive and destructive interference at the receiver

E

T

Sketch of Uniform CRB

Bias gradient = E

Variance =T

Unachievableperformance region

Estimator insensitive toparameter being estimated

Estimator isunbiassed



– Uniform CR bound - 1 30dB source

uniform bound

unbiased bound

mean square error

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u

v

-180 -160 -140 -120 -100 -80 -60 -40 -20 0

-70

-60

-50

-40

-30

-20

-10

E

T

Estimation of a single

parameter, from thesignal received on alinear array of sensors

u



– Uniform CR bound - 4 30dB sources

uniform bound

unbiased bound

mean square error

sources in phase

u

v

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-3

-2

-1

0

1

2

3

4

5x 10

-3

-65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15

-60

-40

-20

0

20

40

60

80

E

T

Estimation of a singleparameter, the mean of

from the signal received on alinear array of sensors

u



– Uniform CR bound - 4 30dB sources

uniform bound

unbiased bound

mean square error

sources in anti-phase

-80 -70 -60 -50 -40 -30 -20 -10

-80

-60

-40

-20

0

20

40

60

T

E

-5 -4 -3 -2 -1 0 1 2 3 4 5

x 10-3

-3

-2

-1

0

1

2

3

4

5x 1 0

-3

u

v

Estimation of a singleparameter, the mean of

from the signal received on alinear array of sensors

u



Conclusions

Monopulse estimation does not always achieve performanceequal (or near) to the uniform CRB and better estimators can exist

The close proximity of scatterers within a target allows thereceived signal to be modelled by a subspace of low dimension

Multi-dimensional beamforming approaches such as MUSIC andmaximum likelihood can use this simplified signal model to offerimprovements in bias (compared with monopulse) withoutsignificant increase in variance



Definition

– Let be the natural logarithm of the probability density

of vector random variable , with a vector of nonrandomparameters.

– Then is said to be the log-likelihood function for [1] .

– Let be partitioned as , where is a vector of

parameters of interest and is a vector of unwanted

parameters.

– Then is said to be the profile log-

likelihood for .

Profile Likelihood

& R:

Z

Z R

& R:

Z4

4 4 G I ¯¡ °¢ ±R GI

SUP P& & I

G R: :Z Z@

R

G

[1] - O E Barndorff-Nielson and D R Cox, Inference and Asymptotics, Chapman & Hall, London, 1994.



Direction-of-arrival Estimation

Signal model

– Let there be sources incident on a planar array, with

steering vectors and complexamplitudes

– Let a single vector observation at the array be denoted

where

,

and is a drawn from a zero mean multivariate normal

distribution, with covariance .

N

I I U V I N A I

I N B

B F Z !

N N ! U V U V ¯

¡ °¢ ±A A@ L

4

N B B B ¯¡ °¢ ±

@ L

FT )



Lemma 1

– The profile log-likelihood for

is

Proof

– The observation is parameterised by the vector

, with log-likelihood [2]

(1)

– By definition the minimum norm solution of

is , where is the pseudo-inverse of .

4

N N U U V V G ¯¡ °¢ ±

@ L L

P ( & CONST

T

G

:Z Z ) !! Z

Z4 4 4 R G B ¯

¡ °¢ ±@

( & CONST

T R B B

:Z Z ! Z !

BZ !

B ! Z

! !

[2] - P Stoica and A Nehorai, MUSIC, Maximum Likelihood and the Cramer-Rao Bound , IEEE Trans.

ASSP-37(52), 1989, pp 720-741.



– It follows directly that (1) is maximised byfor all , and substituting into (1) we get

Comments

– When has full column rank ,

and the profile log-likelihood takes the more familiar form

!

B

! Z

!

P (

(

& CONST

CONST

T

T

G

:Z Z !! Z Z !! Z

Z ) !! Z

( ( ! ! ! !

P ( ( ( & CONST T

G

:Z Z ) ! ! ! ! Z



Lemma 2a

Proof

– From the Moore-Penrose conditions [3] we have

– Differentiating both sides wrt to gives

and the result follows directly.

!! ! !

( ( ! ! ! !

( ( (

I I I

G G G

s s s

s s s

! ! ! !! ! !

(

(

I I

) G G

s s s s!! !! !!

I

G

[3] - G H Golub and C F Van Loan, Matrix Computations, 2nd edition, published by the John Hopkins

University Press, 1989.



Lemma 2b

Proof

– Starting with , differentiate both sides by

and , and the result follows directly.

I I I K K K

I K

)

G G G G G G

G G

s s s s s

s s s s s ss

s s

!! !! ! !! !!

!!!

!! ! ! I

G

K G



Lemma 3 – If is drawn from a m-variate, zero-mean, Normal distribution

with covariance , then is Chi-squared

distributed with degrees of freedom, where

Proof

– The result follows directly from the fact that

is idempotent ( i.e. ) via applicationof Muirhead, theorem 1.4.2 [4].

FT )

( F F!!

R

R RANK RANK !! !

!!

!! !! !!

[4] - R J Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, , 1982



Theorem 1

– The Fisher information matrix, , corresponding to the

profile log-likelihood can be written as

where , ,

and .

P* G

P& G: Z

2E

( ( ( (

U U U VP

( ( ( (

V U V V

* T G

¯

¡ ° ¡ ° ¡ °¢ ±

8 $ ) !! $ 8 8 $ ) !! $ 8

8 $ ) !! $ 8 8 $ ) !! $ 8

DIAG B8 @

N N U

N

U V U V

U U

¯s s

¡ °¡ °s s¢ ±

A A

$ @ L

Fisher Information

N N V

N

U V U V

V V

¯s s¡ °

¡ °s s¢ ±

A A$ @ L



Proof

– According to lemma 1

(2)

; = [

]

P ( ( IK

I K

(

I K

( (

I K

(

I K

*

%

%

%

T G B B

G G

F BG G

B FG G

F FG G

ss s

s s s

s

s s

s

s s

!!! !

!!

!

!!!

!!

@

; =

P

P

IK

I K

&

* %

G

G G G

s

s s

:Z

@



– According to lemma 2, the first term in (2) can be written as

– Since the last term is zero and the first

two terms have the more compact form

( (

I K

(

(

I K

( (

I K

)

)

)

B BG G

B BG G

B BG G

s ss s

s s s s

s

s s

! !!!

! !!!

!

! !!

( ( ! !! !

2E

( ( ( (

U U U V

( ( ( (

V U V V IK

¯ ¡ °¡ °

¡ °¢ ±

8 $ ) !! $ 8 8 $ ) !! $ 8

8 $ ) !! $ 8 8 $ ) !! $ 8



– The second and third terms in (2) are zero because has

zero mean.

– The last term in (3) can be written as

– According to lemma 3 is distributed with mean

and variance which are independent of and

consequently of .

– The last term in (2) is therefore equal to zero, and the

theorem is proved.

F

( (

I I K K

% % F F F FG G G G

s s

s s s s

!!!!

( F F!!

!

G



Comments

– For linear arrays has no dependence on .

– When has full column rank takes the form

which is familiar as the inverse of the bound derived in [2].

! V

! P* G

2E[ ]

( ( ( ( (

U U T 8 $ ) ! ! ! ! $ 8



Bounds on Estimation Accuracy

Cramer-Rao Bounds

– We can use the Fisher information matrix to bound thevariance of an estimator of the scalar function .

– The general form of the bound is as follows [5]

,

where

e T Z T G

[5] - A O Hero, J A Fessler, M Usman, Exploring Estimator Bias-Variance Tradeoffs Using the Uniform CR

Bound , IEEE Trans. Sig. Proc., 44(8), Aug. 1996, pp2026-2041.

4 PT B * T BG G G G G

T

p

e B % T T GZ@



Estimating the mean direction-of-arrival

– For a group of closely spaced point sources it may besufficient to estimate the mean of each coordinate.

– Let

– We can apply an invertible transformation of the parameter

space, .

– Then and .

– To explore the bias-variance tradeoffs involved in

estimating the mean doa, choose , so that

N

I I

T U G

G G 4%

4 T T GG

4

%% P P 4 * *

4 4%

; =4 T G

%% L

4



Uniform Bound

Bias Gradiant norm

– Let , where is a diagonal matrix;

with equal to the beamwidth of the array, and equal

to the dimension of the target.

– Expressions in [5] allow us to compute the uniform CR

bound

4 B BG G

E #% %% %@ #

B

U

U

¯¡ °¡ °

%¡ °¡ °¡ °¡ °¡ °

%¡ °¢ ±

#

L

O M

M O O

L

B U %

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