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7/30/2019 EECE 522 Notes_09 CRLB Example - Single-Platform Location
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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 +
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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
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7/30/2019 EECE 522 Notes_09 CRLB Example - Single-Platform Location
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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
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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.
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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
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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.
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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
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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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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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Projections Applied to Emitter Location
Shows 2-D ellipses
that result from
projecting 4-D
ellipsoids
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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