eece 522 notes_09 crlb example - single-platform location

7/30/2019 EECE 522 Notes_09 CRLB Example - Single-Platform Location

1/12

1

CRLB Example:

Single-Rx Emitter Location via Doppler

],[);( 111 Ttttfts +

Received Signal Parameters Depend on Location

EstimateRx SignalFrequencies: f1, f2, f3, ,fN

Then Use Measured Frequencies to Estimate Location

(X, Y, Z, fo)

],[);( 222 Ttttfts +

],[);( 333 Ttttfts +


2/12

2

Problem Background

Radar to be Located: at Unknown Location (X,Y,Z)

Transmits Radar Signal at Unknown Carrier Frequencyfo

Signal is intercepted by airborne receiver

Known (Navigation Data):

Antenna Positions: (Xp(t), Yp(t),Zp(t))

Antenna Velocities: (Vx(t), Vy(t), Vz(t))

Goal: Estimate Parameter Vector x = [X Y Z fo]T


3/12


4/12

4

Estimation Problem Statement

Given:Data Vector:

Navigation Info:

[ ]TNtftftf ),(~

),(~

),(~

)(~

21 xxxxf !=

)(,),(),(

)(,),(),()(,),(),(

)(,),(),(

)(,),(),(

)(,),(),(

21

21

21

21

21

21

Nzzz

Nyyy

Nxxx

Nppp

Nppp

Nppp

tVtVtV

tVtVtVtVtVtV

tZtZtZ

tYtYtY

tXtXtX

!

!

!

!

!

!

Estimate:Parameter Vector: [X Y Z fo]

T

Right now only want to consider the CRLB

Vector-Valued function of a Vector


5/12

5

The CRLBNote that this is a signal plus noise scenario:

The signal is the noise-free frequency values The noise is the error made in measuring frequency

Assume zero-mean Gaussian noise with covariance matrix C: Can use the General Gaussian Case of the CRLB

Of course validity of this depends on how closely the errors

of the frequency estimator really do follow this

Our data vector is distributed according to: )),(()(~

Cxf~xf

Only Mean Shows

Dependence on

parameterx!

Only need the first term in the CRLB equation:

[ ]

=

j

T

iij

xx

)()( 1 xfC

xfJ

I use J for the FIM instead ofI to avoid confusion with the identity matrix.


6/12

6

Convenient Form for FIM

To put this into an easier form to look at Define a matrix H:

Called The Jacobian off(x)

[ ]4321

valuetrue

|||),( hhhhxtx

H

x

=

=

=

f

where

xx

x

x

x

h

"

#

=

=

),(

),(

),(

2

1

N

j

j

j

j

tf

x

tfx

tfx

Then it is east to verify that the FIM becomes:

HCHJ 1= T


7/12

7

CRLB Matrix

The Cramer-Rao bound covariance matrix then is:

[ ]1

1

1)(

=

=

HCH

JxC

T

CRB

A closed-form expression for the partial derivatives needed forH can be

computed in terms of an arbitrary set of navigation data see Reading Version

of these notes.

Thus, for a given emitter-platform geometry it is possible to compute the

matrix H and then use it to compute the CRLB covariance matrix in (5), from

which an eigen-analysis can be done to determine the 4-D error ellipsoid.

Cant really plot a 4-D ellipsoid!!!

But it ispossible to project this 4-D ellipsoid down into 3-D (or even 2-D) so

that you can see the effect of geometry.


8/12

8

Projection of Error Ellipsoids

A zero-mean Gaussian vector of two vectors x & y:

The the PDF is: [ ]

TTTyx =

{ }

C

C

1

2

1

exp)(det2

1

)( 21= T

p

= yyx

xyx

CC

CC

C

The quadratic form in the exponential defines an ellipse:

kT = C 1 Can choose kto make size of

ellipsoid such that falls

inside the ellipsoid with a

desired probability

Q: If we are given the covariance C how is x alone is distributed?

A: Extract the sub-matrix Cx out ofC

See also Slice Of Error Ellipsoids


9/12

9

Projections of 3-D Ellipsoids onto 2-D Space

2-D Projections Show

Expected Variation

in 2-D Space

xy

z

Projection ExampleTzyx ][= Tyx ][=xFull Vector: Sub-Vector:

We want to project the 3-D ellipsoid for

down into a 2-D ellipse forx

2-D ellipse still shows

the full range of

variations ofx andy


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10

Finding Projections

To find the projection of the CRLB ellipse:

1. Invert the FIM to get CCRB2. Select the submatrix CCRB,sub from CCRB3. Invert CCRB,sub to get Jproj

4. Compute the ellipse for the quadratic form ofJproj

Mathematically:

T

TCRBsubCRB

PPJ

PPCC

1

,

=

=( ) 11 = Tproj PPJJ

P is a matrix formed from the identity matrix:

keep only the rows of the variables projecting onto

For this example, frequency-based emitter location: [X Y Z fo]T

To project this 4-D error ellipsoid onto the X-Y plane:

=

00100001P


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11

Projections Applied to Emitter Location

Shows 2-D ellipses

that result from

projecting 4-D

ellipsoids


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12

Slices of Error Ellipsoids

Q: What happens if one parameter were perfectly known.

Capture by setting that parameters error to zero slice through the error ellipsoid.

Slices Show

Impact of Knowledge

of a Parameter

xy

z

Impact:

slice = projection when ellipsoid not tilted

slice < projection when ellipsoid is tilted.

Recall: Correlation causes tilt

eece 522 notes_09 crlb example - single-platform location

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