ML Estimator for TDOA/FDOA

MLE for
TDOA/FDOA Location
•Overview
• Estimating TDOA/FDOA
• Estimating Geo-Location
1

MULTIPLE-PLATFORM LOCATION
s(
t − t1)
e
jω
s(t)
1
t
Data Link
Emitter to be located
s(
t − t2)
e
jω
s(
t − t3)
e
2
t
Data
Link
jω
3
t
2

ν 21 = ω 2 – ω 1
= constant
FDOA
Frequency-
Difference-
Of-
Arrival
TDOA/FDOA LOCATION
TDOA
Time-
Difference-
Of-
Arrival
ν 23 = ω 2 – ω 3
= constant
s(
t − t1)
e
jω
s(t)
1
t
Data Link
τ 21 = t 2 – t 1
= constant
s(
t − t2)
e
jω
s(
t − t3)
e
2
t
Data
Link
jω
3
t
τ 23 = t 2 –t 3
= constant
3

Estimating TDOA/FDOA
4

SIGNAL MODEL
Will Process Equivalent Lowpass signal, BW = B Hz
– Representing RF signal with RF BW = B Hz
Sampled at Fs > B complex samples/sec
Collection Time T sec
At each receiver:
BPF ADC
cos(ω 1 t)
Make
LPE
Signal
Equalize
-B/2
X RF (f)
X(f)
X f LPE(f)
B/2
f
f
f
5

DOPPLER & DELAY MODEL
s(t) sr (t) = s(t – τ(t))
Tx Rx
R(t)
Propagation Time: τ(t) = R(t)/c
R o
( t)
= R + vt + ( a / 2)
t
2
+ !
Use linear approximation – assumes small
change in velocity over observation interval
For Real BP Signals:
sr o
o
( t)
= s(
t −[
R + vt]
/ c)
= s([
1−
v / c]
t − R / c)
Time
Scaling
Time Delay: τ d
6

~ s
r
DOPPLER & DELAY MODEL (continued)
( t)
=
=
~ s ([ 1
Analytic Signals Model
~ j[ ωct
+ φ(
t)]
s ( t)
=
− v/
c]
t
E(
t)
e
−
E([
1−
v/
c]
t −
τ
τ
Now what? Notice that v

~ s
r
DOPPLER & DELAY MODEL (continued)
( t)
=
=
E(
t
e
Narrowband Analytic Signal Model
− jω
−
c
τ
d
τ
Constant
Phase
Term
α= –ω cτ d
e
d
) e
− jω
c
j{
ω
c
t −ω
( v / c)
t
Doppler
Shift
Term
ω d= ω cv/c
e
c
( v / c)
t −ω
jω
c
t
Carrier
Term
E(
t
c
τ
−
d
+ φ(
t −τ
τ
d
) e
d
)}
jφ(
t −τ
Transmitted Signal’s
LPE Signal
Time-Shifted by τ d
Narrowband Lowpass Equivalent Signal Model
jα
sˆ ( t)
= e e d sˆ
( t − τd
r
− jω
This is the signal that actually gets processed digitally
t
)
d
)
8

CRLB for TDOA
We already showed that the CRLB for the active sensor case is:
1
C(
TDOA)
=
2
8π
N × SNR × B
But here we need to estimate the delay between two noisy
signals rather than between a noisy one and a clean one.
The only difference in the result is: replace SNR by an
effective SNR given by
SNR eff
=
1
SNR
1
1
1 1
+ +
SNR SNR SNR
2
2
rms
where B rms is an effective bandwidth
of the signal computed from the
DFT values S[k].
1
2
B
2
rms
=
1
N
N / 2−1
≈ min{ SNR , SNR
1
∑
k = −N
/ 2
N / 2−1
∑
k
k=
−N
/ 2
2
2
S[
n]
}
S[
k]
2
2
9

CRLB for TDOA (cont.)
A more familiar form for this is in terms of the C-T version of
the problem:
σ
TDOA
≥
2π
2
B
rms
1
BT × SNR
eff
“seconds”
2
Brms =
∫ f
∫
2
S(
f )
S(
f )
BT = Time-Bandwidth Product (≈ N, number of samples in DT)
B = Noise Bandwidth of Receiver (Hz)
T = Collection Time (sec)
BT is called “Coherent Processing Gain”
(Same effect as the DFT Processing Gain on a sinusoid)
For a signal with rectangular spectrum of RF width of Bs , then the
bound becomes:
0.
55
σ
TDOA
≥ 2
B
s
BT × SNR
S. Stein, “Algorithms for Ambiguity Function Processing,”
IEEE Trans. on ASSP, June 1981
eff
2
2
df
df
10

CRLB for FDOA
Here we take advantage of the time-frequency duality if the FT:
1
C(
FDOA)
=
2
8π
N × SNR
2
eff × Trms
where T rms is an effective duration
of the signal computed from the
signal samples s[k].
Again… we use the same effective SNR:
SNR eff
=
1
SNR
1
1
1 1
+ +
SNR SNR SNR
2
1
2
T
2
rms
=
1
N
N
/ 2−1
∑
n=
−N
/ 2
N / 2−1
∑
k
n=
−N
/ 2
≈ min{ SNR , SNR
1
2
s[
n]
s[
n]
2
}
2
2
11

CRLB for FDOA (cont.)
A more familiar form for this is in terms of the C-T version of
the problem:
σ
FDOA
≥
2π
2
T
rms
1
BT × SNR
eff
“Hz”
2
Trms For a signal with constant envelope of duration T s , then the bound
becomes:
σ
FDOA
≥ 2
T
s
0.
55
BT × SNR
S. Stein, “Algorithms for Ambiguity
Function Processing,” IEEE Trans. on
ASSP, June 1981
eff
=
∫ t
∫
2
s(
t)
s(
t)
2
2
dt
dt
12

Interpreting CRLBs for TDOA/FDOA
A more familiar form for this is in terms of the C-T version of
the problem:
σ
TDOA
≥
2π
2
B
rms
1
BT × SNR
eff
σ
FDOA
≥
2π
• BT pulls the signal up out of the noise
•Large B rms improves TDOA accuracy
•Large T rms improves FDOA accuracy
Two Examples of Accuracy Bounds:
SNR 1 SNR 2 T = T s B = B s σ TDOA σ FDOA
3 dB 30 dB 1 ms 1 MHz 17.4 ns 17.4 Hz
3 dB 30 dB 100 ms 10 kHz 1.7 µs 0.17 Hz
2
T
rms
1
BT × SNR
eff
13

MLE for TDOA/FDOA
S. Stein, “Differential Delay/Doppler ML
Estimation with Unknown Signals,” IEEE
Trans. on SP, August 1993
We already showed that the ML Estimate of delay for the
active sensor case is the Cross-Correlation of the time signals.
T
∫
C(
τ ) = s ( t)
s ( t + τ )
0
1
2
By the time-frequency duality the ML estimate for doppler
shift should be Cross-Correlation of the FT, which is
mathematically equivalent to
C(
ω)
=
A(
ω,
τ ) =
T
∫
0
T
∫
0
s
s
1
1
( t)
s
( t)
s
2
2
( t)
e
− jωt
( t + τ ) e
dt
dt
− jωt
dt
Find Peak of |C(τ)|
Find Peak of |C(ω)|
The ML estimate of the TDOA/FDOA has been shown to be:
Find Peak of |A(ω,τ)|
14

LPE Rx
Signals
At Two
Receivers
ML Estimator for TDOA/FDOA (cont.)
jα
1( ) t s
2( ) t s
jω
t
= e e d s ( t −τ
1 d
Delay
τ
Called:
• Ambiguity Function
• Complex Ambiguity Function (CAF)
• Cross-Correlation Surface
)
Doppler
ω
ω
ωd “Compare”
Signals
For all
Delays &
Dopplers
Ambiguity Function
τ
Find
Peak
of
|A(ω,τ)|
τ d
15

ML Estimator for TDOA/FDOA (cont.)
How well do we expect the Cross-Correlation Processing to
perform?
Well… it is the ML estimator so it is not necessarily optimum.
But… we know that an ML estimate is asymptotically
• Unbiased & Efficient (that means it achieves the CRLB)
• Gaussian
ˆ
−1
θML
~ N ( θ,
I ( θ))
Those are some VERY nice properties that we can make use of
in our location accuracy analysis!!!
16

Properties of the CAF
• Consider when τ = τ T
d [ ] 2
∫
jω
t
A(
ω,
τd
) = s(
t)
e
|A(ω,τ d )|
ω d
width ∼ 1/T
• Consider when ω = ω d
|A(ω d ,τ)|
τ d
width ∼ 1/BW
ω
τ
A(
ω
d
0
d
e
− jωt
dt
like windowed FT of sinusoid
where window is |s(t)| 2
, τ)
=
T
∫
0
s(
t)
s(
t
−
τ
d
correlation
+ τ)
dt
17

TDOA ACCURACY REVISITED
TDOA Accuracy depends on:
» Effective SNR: SNR eff
» RMS Widths: B rms = RMS Bandwidth
XCorr
Function
XCorr
Function
~1/B rms
Narrow B rms Case
Poor Accuracy
TDOA
Wide B rms Case
Good Accuracy
TDOA
2
Brms =
∫
f
∫
2
S(
f )
S(
f )
2
2
df
df
Low Effective SNR
Causes Spurious Peaks
On Xcorr Function
Narrow Xcorr Function
Less Susceptible to
Spurious Peaks
18

FDOA ACCURACY REVISITED
FDOA Accuracy depends on:
» Effective SNR: SNR eff
» RMS Widths: D rms = RMS Duration
XCorr
Function
XCorr
Function
~1/D rms
Narrow D rms Case
Poor Accuracy
FDOA
Wide D rms Case
Good Accuracy
FDOA
2
Brms =
∫
f
∫
2
S(
f )
S(
f )
2
2
df
df
Low Effective SNR
Causes Spurious Peaks
On Xcorr Function
Narrow Xcorr Function
Less Susceptible to
Spurious Peaks
19

COMPUTING THE AMBIGUITY FUNCTION
Direct computation based on the equation for the ambiguity
function leads to computationally inefficient methods.
In EECE 521 notes we showed how to use decimation to
efficiently compute the ambiguity function
20

Estimating Geo-Location
21

TDOA/FDOA LOCATION
Data Link
Data
Link
Centralized Network of P
•“P-Choose-2” Pairs
“P-Choose-2” TDOA Measurements
“P-Choose-2” FDOA Measurements
• Warning: Watch out for Correlation
Effect Due to Signal-Data-In-Common
Data
Link
22

TDOA/FDOA LOCATION
Pair-Wise Network of P
• P/2 Pairs
P/2 TDOA Measurements
P/2 FDOA Measurements
• Many ways to select P/2 pairs
• Warning: Not all pairings are equally
good!!! The Dashed Pairs are Better
23

TDOA/FDOA Measurement Model
Given N TDOA/FDOA measurements with corresponding 2×2 Cov. Matrices
( ˆ τ1, ˆ ν1),
( ˆ τ 2,
ˆ ν 2 ), … , ( ˆ τ N , ˆ ν N ) Assume pair-wise network, so…
C C , … , C
TDOA/FDOA pairs are uncorrelated
For notational purposes… define the 2N measurements r(n) n = 1, 2, …, 2N
r
2n−1
r
2n
= ˆ τ
= ˆ ν
n
n
,
,
1,
2
n
n
=
=
1,
2,
1,
2,
…,
…,
N
N
N
r
=
[ 1 r2
! r2
N
r ]
Data Vector
Now, those are the TDOA/FDOA estimates… so the true values are notated as:
s
2n−1
s
2n
= τ
= ν
n
n
,
,
n
n
( τ1, ν1),
( τ 2,
ν 2 ), … , ( τ N , ν N
=
=
1,
2,
1,
2,
…,
…,
N
N
)
s
=
[ 1 s2
! s2N
s ]
“Signal” Vector
T
T
24

TDOA/FDOA Measurement Model (cont.)
Each of these measurements r(n) has an error ε(n) associated with it, so…
r = s +
Because these measurements were estimated using an ML estimator (with
sufficiently large number of signal samples) we know that error vector ε is a
zero-mean Gaussian vector with cov. matrix C given by:
⎡C1
0 0 ⎤
⎢
⎥
C = diag{ C1
, C2,
… , CN
} = ⎢ 0 # 0 ⎥
⎢
⎥
⎢
⎣ 0 0 CN
⎥
⎦
The true TDOA/FDOA values depend on:
Emitter Parms: (xe , ye , ze ) and transmit frequency fe xe = [ xe ye ze fe ] T
Assumes that
TDOA/FDOA
pairs are
uncorrelated!!!
Receivers’ Nav Data (positions & velocities): The totality of it called xr r = s(
xe
; xr
) +
ε
ε
• Deterministic “Signal” + Gaussian Noise
• “Signal” is nonlinearly related to parms
To complete the model… we need to know how s(x e;x r) depends on x e and x r.
Thus we need to find TDOA & FDOA as functions of x e and x r
25

TDOA/FDOA Measurement Model (cont.)
Here we’ll simplify to the x-y plane… extension is straight-forward.
Two Receivers with: (x 1, y 1, Vx 1, Vy 1) and (x 2, y 2, Vx 2, Vy 2)
Emitter with: (x e, y e)
(Let R i be the range between Receiver i and the emitter; c is the speed of light.)
The TDOA and FDOA are given by:
s ( x , y ) = τ
1
e
e
=
=
s ( x , y , f ) = ν
2
e
e
e
1
c
12
f
c
e
12
⎛
⎜
⎝
=
⎡
⎢
⎢
⎢⎣
=
R
1
− R
c
2
2
2
2
2
( x1
− xe
) + ( y1
− ye
) − ( x2
− xe
) + ( y2
− ye
) ⎟
⎠
f
c
e
d
dt
( R − R )
( x1
− xe
) Vx1
+ ( y1
− ye
)
( x − x
2 ) + ( y − y )
1
1
e
2
1
e
Vy
1
2
−
( ) ( )
( ) ( ) ⎥ ⎥⎥
x2
− xe
Vx2
+ y2
− ye
Vy2
2
2
x − x + y − y
2
e
2
⎞
e
⎤
⎦
26

CRLB for Geo-Location via TDOA/FDOA
Recall: For the General Gaussian Data case the CRLB depends on a FIM that
has structure like this:
[ J(
θ)
]
nm
T
⎡∂µ
x ( θ)
⎤ −1
⎡∂µ
x ( θ)
⎤
= ⎢ ⎥ Cx
( θ)
⎢ ⎥
⎣ ∂θn
'$
$ $ ⎦$
$ &$
$ ⎣ ∂θm
$ $ $ % ⎦
variability
of mean w.r.t. parms
+
1 ⎡
1 ( ) 1 ( ) ⎤
− ∂Cx
θ − ∂Cx
θ
tr⎢Cx
( θ)
Cx
( θ)
⎥
2 ⎢
⎥
'$
⎣
∂θn
∂θm
$ $ $ $ $ &$
$ $ $ $ $ $ % ⎦
variability
of cov. w.r.t. parms
Here we have a deterministic “signal” plus Gaussian noise so we only have the
1 st term… Using the notation introduced here gives…
C
CRLB
( x
e
)
⎡ T
∂s
( x
= ⎢
⎢⎣
∂xe
H T
e
)
C
−1
∂s(
x
∂x
e
e
) ⎤
⎥
⎥⎦
−1
Called the “Jacobian” … for the 3-D location with
TDOA/FDOA will be a 2N × 4 matrix whose columns
are derivatives of s w.r.t. each of the 4 parameters.
H
()
27

CRLB for Geo-Loc. via TDOA/FDOA (cont.)
TDOA/FDOA Jacobian:
H
= ∆
∂s(
x
∂x
e
e
)
⎡ ∂s1(
xe
)
⎢
∂x
⎢ e
⎢ ∂s2
( xe
)
= ⎢ ∂xe
⎢
(
⎢
⎢∂s2
N ( xe
)
⎢
⎣ ∂xe
∂
∂xe
∂s1(
xe
)
∂ye
∂s2
( xe
)
∂ye
(
∂s2N
( xe
)
∂y
e
∂
∂ye
∂s1(
xe
)
∂ze
∂s2
( xe
)
∂ze
(
∂s2N
( xe
)
∂z
e
∂
∂ze
∂s1(
xe
) ⎤
∂f
⎥
e ⎥
∂s2
( xe
) ⎥
∂f
⎥
e
(
⎥
⎥
∂s2N
( xe
) ⎥
∂f
⎥
e ⎦
Jacobian can be computed for any desired Rx-Emitter Scenario
Then… plug it into () to compute the CRLB for that scenario:
C
CRLB
( x
e
)
[ ] 1
T −1
−
H C
= H
∂
∂fe
28

CRLB Studies
The Location CRLB can be used to study various aspects of the emitter location
problem. It can be used to study the effect of
• Rx-Emitter Geometry and/or Platform Velocity
• TDOA accuracy vs. FDOA accuracy
• Number of Platforms
• Platform Pairings
• Etc., Etc., Etc…
Once you have computed the CRLB Covariance C CRLB you can use it to
compute and plot error ellipsoids.
Assumes Geo-Location Error
is Gaussian…
Usually reasonably valid
Emitter
Sensor 1
k 1 σ TDOA
V V
Faster than doing Monte
Carlo Simulation Runs!!!
Sensor 2
k 2 σ FDOA
29

CRLB Studies (cont.)
Error Ellipsoids: If our C CRLB is 4×4… how do we get 2-D ellipses to plot???
Projections!
z
Projections of 3-D Ellipsoids onto 2-D Space
y
x
2-D Projections Show
Expected Variation
in 2-D Space
30

CRLB Studies (cont.)
Another useful thing that can be computed from the CRLB is the CEP for the
location problem. For the 2-D case:
within 10%
1
2
2
1
CEP ≈ 0.
75 λ + λ = 0.
75 σ + σ
Eigen-Values of the
2-D Cov. Matrix
Down Range
2
2
Diagonal elements of
the 2-D Cov. Matrix
CEP = 80
CEP = 40
CEP = 20
CEP = 10
Cross Range
“Circular Error Probable”
= radius of a circle that when
centered at the estimate’s mean,
contains 50% of the estimates
CEP Contour Plots are
Good Ways to Assess
Location Performance
31

CRLB Studies (cont.)
Sensor
Geometry and TDOA vs. FDOA Trade-Offs
Both Important FDOA Important TDOA Important
Target
TDOA Only
FDOA Only
TDOA/FDOA
Target
TDOA Only
Target
FDOA
Only
25
32

Estimator for Geo-Location via TDOA/FDOA
Because we have used the ML estimator to get the TDOA/FDOA
estimates the ML’s asymptotic properties tell us that we have
Gaussian TDOA/FDOA measurements
Because the TDOA/FDOA measurement model is nonlinear it is
unlikely that we can find a truly optimal estimate… so we again
resort to the ML. For the ML of a Nonlinear Signal in Gaussian we
generally have to proceed numerically.
One way to do Numerical MLE is ML Newton-Raphson (need vector
version):
θˆ
k+
1
= θˆ
k
⎧
⎪⎡∂
− ⎨⎢
⎪⎢
⎩⎣
2
Hessian: p×p matrix
ln p(
x;
θ)
⎤
T ⎥
∂θ∂θ
⎥⎦
−1
⎫
∂ln
p(
x;
θ)
⎪
⎬
∂θ
⎪
⎭
However, the “Hessian” requires a second derivative…
This can add complexity in practice… Alternative:
Gauss-Newton Nonlinear Least Squares based on linearizing the model.
θ=
θˆ
k
Gradient: p×1 vector
33