6 radar range-doppler-angular loops
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1
RADARRange – Doppler – Angular
Loops
SOLO HERMELIN
http://www.solohermelin.com
2
SOLO RADAR Range-Doppler-Angular Loops
Table of Contents A/C RADAR Block DiagramA/C - Target KinematicsLOS Vector in Antenna CoordinatesTarget Acceleration ModelSummary of System EquationsDiscretization of a Continuous Linear SystemDiscrete Filter/Predictor Architecture
Compensation for Processing-from-Measurement DelaysKalman Filter - State Estimation in a Linear System (one cycle)
RADAR Range-Doppler LoopsRADAR Angular Loops
References
3
SOLO
A/C RADAR Block Diagram
Block Diagram of a Simple Coherent Radar
f0Power
Amplifier
SignalGenerator
Stable Local
Oscillator (STALO)
CoherentOscillator(COHO)
fLO fRF
fIF
fIF
f0 + fdfIF + fd fd
f0=fRF + fIF
IF AMPLIFIER
CYRCULATOR
SIGNAL
PROCESSOR
ANGLETRACKER
DOPPLERTRACKER
RANGETRACKER
SEEKERLOGIC
A/C RadarCENTRAL
PROCESSOR
RADOME
LOW-PASS-FILTER
ANTENNASTABILIZATION
A/D
ANALOG DIGITAL
FREQUENCYSOURCE
RFIF + RECEIVER
ANTENNA
Return to Table of Content
4
A/C RADAR Range-Doppler-Angular LoopsSOLO
Antenna TrackingLoops
SignalProcessor A/C
Avionics
AngularEstimator
Range and FrequencyEstimators
Estimators
SeekerCentral
Processor
A/C Radar
Antenna Slaving& Mech. Search
Loops
AntennaGimbal
Unit
Slaving& Mech. Search
Commands
AntennaControl
A/CCentral
Processor
Digital
Return to Table of Content
5
At any time t the following vectors define A/C and target kinematics:
SOLO
present A/C position, velocity and acceleration vectorsAAA aVR,,
present target position, velocity and acceleration vectorsTTT aVR,,
A/C - Target Kinematics
A/C
Target
Antenna
6
A/C-Target Kinematics
Define:
SOLO
ATAT VVRRR −=−=
ATMTAT aaVVRRR −=−=−=
→=−= RRRRR AT 1
Target-A/C Antenna (Line-of-Sight) Range Vector
→R1 Line-of-Sight (LOS) Unit Vector
Lω Line-of-Sight (LOS) Angular Velocity Vector
Define:
→→→×= Rtt 11:1 21
1 1 2 21 1 1L R R t tω ω→ → →
= + Λ + Λ Lω Decompose in the LOS and normal to LOS directions
→→→
21 1,1,1 ttR where: are three orthogonal unit vectors defining a right handCartesian System and is its angular velocity vector.
Lω
→→→×= 12 111 tRt
→→→×= 21 111 ttR
A/C
Target
Target
A/C
Target
Antenna
7
A/C-Target KinematicsSOLO
A/C
Target
1 1 2 21 1 1L R R t tω ω→ → →
= + Λ + Λ
−Λ
Λ−
Λ−Λ
=
→
→
→
→
→
→
2
1
1
2
12
2
1
1
1
1
0
0
0
1
1
1
t
t
R
t
t
R
td
d
R
R
ω
ω
8
A/C-Target KinematicsSOLO
1 1 2 21 1 1L R R t tω ω→ → →
= + Λ + Λ
−Λ
Λ−
Λ−Λ
=
→
→
→
→
→
→
2
1
1
2
12
2
1
1
1
1
0
0
0
1
1
1
t
t
R
t
t
R
td
d
R
R
ω
ω
AT VVtRtRRRRRRRR
−=Λ−Λ+=+=→→→→→
2112 11111
( ) ( ) →→→→→→Λ−Λ+Λ−Λ+Λ+Λ++=
2121112122 111111 tRtRRtRtRRRRRRR
( ) ( )
−ΛΛ−Λ+Λ−
+Λ−Λ+Λ+Λ+
Λ−Λ+=
→→→→→→→→→
1112112221222112 111111111 tRRtRRtRRtRRtRtRRRR RR ωω
( )[ ] ( ) ( ) ATRR aatRRRtRRRRRR −=Λ−Λ+Λ−Λ+Λ+Λ+Λ+Λ−=
→→→
2211112222
21 12121 ωω
( )[ ] ( )( ) ATR
R
AT
AT
aatRRR
tRRRRRRR
VVtRtRRRRRRRR
RRRRR
−=Λ−Λ+Λ−
Λ+Λ+Λ+Λ+Λ−=
−=Λ−Λ+=+=
−==
→
→→
→→→→→
→
2211
112222
21
2112
12
121
11111
1
ω
ω
AT RRRRR
−==→1
A/C
Target
9
SOLO RADAR Range-Doppler-Angular Loops
⋅−
⋅=Λ−Λ+Λ
→→
22211 112 tataRRR ATR
ω
We obtain the following kinematic equations that govern the range (R), doppler ( ) andthe angular rates ( ) of the Line-of-Sight (LOS)
R
21,ΛΛ
( )
⋅−
⋅=Λ+Λ−
→→RaRaRR AT 112
221
⋅−
⋅=Λ+Λ+Λ
→→
11122 112 tataRRR ATR
ω
⋅
−
⋅
+
Λ
=
→→RaRa
R
R
R
R
td
dAT 1
1
01
1
0
0
102
( )[ ] ( )( ) ATR
R
AT
AT
aatRRR
tRRRRRRR
VVtRtRRRRRRRR
RRRRR
−=Λ−Λ+Λ−
Λ+Λ+Λ+Λ+Λ−=
−=Λ−Λ+=+=
−==
→
→→
→→→→→
→
2211
112222
21
2112
12
121
11111
1
ω
ω
Range, Doppler & LOS Rate Equations Return to Table of Content
A/C
Target
( ) ( )( ) ( )11122
22211
11
11
2
11
11
2
taR
taRR
R
td
d
taR
taRR
R
td
d
TRA
TRA
⋅+Λ−⋅−Λ−=Λ
⋅+Λ+⋅−Λ−=Λ
ω
ω
10
SOLO RADAR Range-Doppler-Angular Loops
A/C
Target
AntennaAxis
Antenna
The LOS vector deviatesfrom Antenna axis by the two small angles ε1 and ε2 such that rotation matrix from A to L (LOS) is
1Ruu
1Auu
and
The relative angular velocity from A to L (LOS) in Antenna coordinates (A), is:
LOS Vector in A/C Antenna Coordinates
[ ] [ ]
−
−≈
−
−=
−
−==
10
01
1
1
01
1
100
01
01
10
010
01
1
2
12
211
2
12
2
2
1
1
3221
εε
εε
εεεε
εεε
ε
ε
εεεL
AC
Antenna angular velocity21 11121 AAtAAtARA ttAAA
ωωωω ++=
LOS angular velocity 2211 111 ttRRL Λ+Λ+= ωω
( ) ( ) ( )
≈
=
−+
=−=←
2
1
2
1
12
12
2
2
0
0
0
100
01
01
0
0
εε
εε
εεεε
ε
εωωω
AA
AL
AAL
( )
≈
−
Λ
Λ
−
−=←
2
1
2
1
1
2
12 0
10
01
1
2
1
εε
ωωωω
εε
εεω
A
A
At
At
ARR
AAL
02112 =−Λ+Λ− RAR ωεεω
122
211
2
1
εωωε
εωωε
RtA
RtA
A
A
−−Λ=
+−Λ=
11
SOLO RADAR Range-Doppler-Angular Loops
We obtained:
( )( )
1
2
21 1 2 2 1 2 2
22 2 1 1 1 2 1
1
1
A
A
A t A R
A t A R
ε ε ω ω ε ε ε
ε ε ω ω ε ε ε
= Λ + − + − Λ
= Λ + − − − Λ
Therefore:
Since ε1,ε2 << 1 we can use:
1 21 21 1 1A AA A R A t A A t AA t tω ω ω ω= + +
uu uuu uuuAntenna angular velocity
1 1 2 21 1 1L R R t tω ω= + Λ + Λuu uu uu LOS angular velocity
where:
and:
122
211
2
1
εωωε
εωωε
RAt
RAt
t
t
−−Λ=
+−Λ=
2112 Λ−Λ+= εεωω RAR
122
211
2
1
εωωε
εωωε
RAAt
RAAt
t
t
−−Λ≅
+−Λ≅
RAR ωω ≅
Return to Table of Content
LOS Vector in Seeker Coordinates (continue – 1)
A/C
Target
AntennaAxis
Antenna
12
SOLO RADAR Range-Doppler-Angular Loops
Target Acceleration ModelSince the target acceleration vector is not measurable, we assume that it is a random process defined by
Ta
Using :
−Λ
Λ−
Λ−Λ
=
→
→
→
→
→
→
2
1
1
2
12
2
1
1
1
1
0
0
0
1
1
1
t
t
R
t
t
R
td
d
R
R
ω
ω
waatd
dT
TT
+−=τ1
and τ is the target maneuver time constant
where is a white noise vector with the covariancew ( ) ( ) ( )νδν −= tQwtwE T
( ) ( ) ( ) 2211 111111 ttattaRRaa TTTT ⋅+⋅+⋅=
( ) ( ) ( )[ ]( ) ( ) ( )
( ) ( ) ( ) 2211
2211
2211
111111
111111
111111
ttd
dtat
td
dtaR
td
dRa
ttatd
dtta
td
dRRa
td
d
ttattaRRatd
datd
d
TTT
TTT
TTTT
⋅+⋅+⋅=
⋅+⋅+⋅=
⋅+⋅+⋅=
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) 2112
12111221
1111
11111111
ttaRatatd
d
ttaRatatd
dRtataRa
td
d
TRTT
TRTTTTT
⋅+⋅Λ−⋅+
⋅−⋅Λ+⋅+
⋅Λ−⋅Λ+⋅=
ω
ω
13
SOLO RADAR Range-Doppler-Angular Loops
Target Acceleration Model (continue – 1)
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1 2 2 1
1 2 2 1
2 1 1 2
1 1 1 1
1 1 1 1
1 1 1 1
T T T T
T T R T
T T R T
d da a R a t a t R
d t d t
da t a R a t t
d t
da t a R a t t
d t
ω
ω
= × + Λ × − Λ ×
+ × + Λ × − ×
+ × − Λ × + ×
uu uu uu uu
uu uu uu uu
uu uu uu uu
( ) ( ) ( )νδν −= tqwtwE RTRR
( ) ( ) ( )νδν −= tqwtwE tTtt
waatd
dTT
+−=τ1
We finally obtain, by neglecting terms in , and using :21, ΛΛ RAR ωω ≅
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) tTTRT
TT
tTTRTT
T
RTTTT
T
wRatatatatd
d
wRatatatatd
d
wtataRaRatd
d
+⋅Λ+⋅−⋅−=⋅
+⋅Λ−⋅+⋅−=⋅
+⋅Λ+⋅Λ−⋅−=⋅
1111
1
1111
1
1111
1
1122
1211
1221
ωτ
ωτ
τ
( ) ( )( ) ( ) ( )( ) ( ) ( ) tTRAT
TT
tTRATT
T
RTT
T
wtatatatd
d
wtatatatd
d
wRaRatd
d
+⋅−⋅−=⋅
+⋅+⋅−=⋅
+⋅−=⋅
122
211
111
1
111
1
11
1
ωτ
ωτ
τ
14
SOLO
Singer Target Model
R.A. Singer, “Estimating Optimal Tracking Filter Performance for Manned ManeuveringTarget”, IEEE Trans. Aerospace & Electronic Systems”, Vol. AES-6, July 1970, pp. 437-483
The target acceleration is modeled as a zero-mean random process with exponential autocorrelation ( ) ( ) ( ) TetataER mTT
ττσττ /2 −=+= where σm
2 is the variance of the target acceleration and τT is the time constant of itsautocorrelation (“decorrelation time”).
The target acceleration is assumed to:1. Equal to the maximum acceleration value amax
with probability pM and to – amax
with the same probability.2. Equal to zero with probability p0.3. Uniformly distributed between [-amax, amax]
with the remaining probability 1-2 pM – p0 > 0.
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]max
0maxmax0maxmax 2
210
a
ppaauaauppaaaaap M
M
−−−−+++−++= δδδ
RADAR Range-Doppler-Angular Loops
Target Acceleration Model (continue – 2)
15
SOLO
Singer Target Model (continue 1)
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]max
0maxmax0maxmax 2
210
a
ppaauaauppaaaaap M
M
−−−−+++−++= δδδ
( ) ( ) ( )[ ] ( )
( ) ( )[ ]
( ) ( )[ ] 022
210
2
21
0
max
max
max
max
max
max
max
max
2
max
00maxmax
max
0maxmax
0maxmax
=−−+⋅++−=
−−−−++
+−++==
+
−
−
−−
∫
∫∫
a
a
MM
a
a
M
a
a
M
a
a
a
a
ppppaa
daaa
ppaauaau
daappaaaadaapaaE δδδ
( ) ( ) ( )[ ] ( )
( ) ( )[ ]
( ) ( )[ ]
( )02
max
3
max
02max
2max
2
max
0maxmax
20maxmax
22
413
32
21
2
21
0
max
max
max
max
max
max
max
max
ppa
a
a
pppaa
daaa
ppaauaau
daappaaaadaapaaE
M
a
a
MM
a
a
M
a
a
M
a
a
−+=
−−+−++=
−−−−++
+−++==
+
−
−
−−
∫
∫∫ δδδ
( )02
max
0
222 413
ppa
aEaE Mm −+=−=
σ
Use
( ) ( ) ( )
max0max
00
max
max
aaa
afdaafaaa
a
+≤≤−
=−∫−
δ
RADAR Range-Doppler-Angular LoopsTarget Acceleration Model (continue – 3)
16
SOLO
Target Acceleration Approximation by a Markov Process
w (t) x (t)
( )tF
( )tG ∫x (t)
( ) ( ) ( ) ( ) ( ) ( )twtGtxtFtxtxtd
d +== Given a Continuous Linear System:
Let start with the first order linear system describing Target Acceleration :
( ) ( ) ( )twtata TT
T +−=τ1
( ) ( ) T
T
tta ett τφ /
00, −−=
( ) ( ) [ ] ( ) ( ) [ ] ( )τδττ −=−− tqwEwtwEtwE( ) ( ) [ ] ( ) ( ) [ ] ( )ttRtaEtataEtaE
TT aaTTTT ,τττ +=−+−+
( ) ( ) [ ] ( ) ( ) [ ] ( )τττ +=+−+− ttRtaEtataEtaETT aaTTTT ,
( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) 2,TTTTT aaaaaTTTT ttRtVtaEtataEtaE σ===−−
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tGtQtGtFtVtVtFtVtd
d TT
xxx ++= ( ) ( ) qtVtVtd
dTTTT aa
Taa +−=
τ2
( ) ( )00 ,1
, tttttd
dTT a
Ta φ
τφ −=
where
Target Acceleration Model (continue – 4)
RADAR Range-Doppler-Angular Loops
17
SOLO
( ) ( ) qtVtVtd
dTTTT aa
Taa +−=
τ2
( ) ( )
−+=
−−TT
TTTT
t
T
t
aaaa eq
eVtV ττ τ 22
12
0
( )( ) ( ) ( )
( ) ( ) ( )
<+=+Φ+
>=+Φ=+
−
−
0,
0,,
ττττ
τττ
ττ
ττ
tVetttV
tVetVttttR
TT
T
TTT
TT
T
TTT
TT
aaT
aaa
aaaaa
aa
( )( ) ( ) ( )
( ) ( ) ( )
<+=++Φ
>=+Φ=+
−
−
0,
0,,
ττττ
τττ
ττ
ττ
tVetVtt
tVetttVttR
TT
T
TTT
TT
T
TTT
TT
aaaaa
aaT
aaa
aa
For ( ) ( )2
5 Tstatesteadyaaaaaa
T
qVtVtV
TTTTTT
ττττ ==+≈⇒> −
( ) ( ) ( ) TT
TTTTTTTTe
qeVVttRttR
TT
statesteadyaaaaaaaaττ
ττ τττττ −−
− =≈≈+≈+⇒>2
,,5
Target Acceleration Approximation by a Markov Process (continue – 1)Target Acceleration Model (continue – 5)
RADAR Range-Doppler-Angular Loops
18
SOLO
( ) 2
0 22 T
Taa qde
qdVArea T
TTτττττ τ
τ
=== ∫∫+∞ −+∞
∞−
τT is the correlation time of the noise w (t) and defines in Vaa (τ) the correlation time corresponding to σa
2 /e.One other way to find τT is by tacking the double sides Laplace Transform L 2 on τ of:
( ) ( ) ( ) qdetqtqs sww =−=−=Φ ∫
+∞
∞−
− ττδτδ ττ2L
( ) ( )
( ) ( ) ( )sHqsHs
q
deeq
Vs
T
T
sTssaaaa
T
TTTT
−=−
=
==Φ ∫+∞
∞−
−−−
2
2
/2
1
2
ττ
τττ ττττL
τT defines the ω1/2 of half of the power spectrum
q/2 and τT =1/ ω1/2.
( ) ( ) ( ) TT
TTTTTTTe
qeVttRttR
TT
aaaaaaaττ
ττ τσττττ −−
=≈≈+≈+⇒>2
,,5 2
T
aTqτσ 22
=
Target Acceleration Approximation by a Markov Process (continue – 2)
RADAR Range-Doppler-Angular Loops
Target Acceleration Model (continue – 6)
Return to Table of Content
19
SOLO RADAR Range-Doppler-Angular LoopsSummary of System Equations
RM
TRTTR
wRa
a
R
R
a
R
R
td
d
+
⋅
−+
−Λ=
→
1
0
0
1
0
1
0
/100
10
0102
τ
Range-Doppler Equations
Transversal LOS Equations
( )
( )
+
⋅
⋅
−
−
−
+
Λ
Λ
−−
−−
−
−
−
=
Λ
Λ
t
t
M
At
M
At
Tt
Tt
TRA
RA
RA
RAT
RA
RA
Tt
Tt
w
w
ta
ta
R
R
a
a
RR
R
RR
R
a
a
td
d
0
0
0
0
1
1
0000
1000
0010
0000
001
0
0001
10000
12000
01000
001
00
001
20
00010
2
1
2
2
1
1
2
2
1
1
2
1
2
1
2
1
ω
ω
ε
ε
τω
ω
ω
ωτ
ω
ω
ε
ε
Return to Table of Content
20
SOLO
Discretization of a Continuous Linear System
( ) ( ) ( ) ( )[ ]∫ +−Φ+Φ= −
T
kk dwGuBTxTx0
1 ξξξξ
By changing the integration variable to ξ = λ-(k-1) T we obtain
( ) ( ) ( ) 100 :,:,1, −==−== kk xtxxtxTktTkt
( ) ( ) ( )( ) ( ) ( ) ( )[ ]( )∫−
+−Φ+−Φ=Tk
Tk
dwGuBTkTkxTTkx1
1 λλλλ
or
11111 −−−−− ++= kkkkkk wuDxFxwhere
( ) ( ) ( ) ( )∫∫ −Φ=−Φ=Φ= −−−
T
k
T
kk dwGTwdBTDTF0
1
0
11 ::: ξξξξξ
RADAR Range-Doppler-Angular Loops
Return to Table of Content
21
SOLO
Discrete Filter/Predictor Architecture
RADAR Range-Doppler-Angular Loops
The discrete representation of the sysem is given by
x (k) - system state vector
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )kwkxkHkz
kvkukGkxkFkx
+++=+++=+
111
1
u (k) - system control input
v (k) - system unknown dynamics assumed white gaussian
w (k) - measurement noise assumed white gaussian
k - discrete time counter
22
SOLO
Discrete Filter/Predictor Architecture (continue – 1)
RADAR Range-Doppler-Angular Loops
1. The output of the Filter/Predictor canbe at a higher rate than the input (measurements)
Tmeasurements = m Toutput, m integer
2. Between measurements it will perform State Prediction
( ) ( ) ( ) ( ) ( )( ) ( ) ( )kkxkHkkz
kukGkkxkFkkx
|1ˆ1|1ˆ
|ˆ|1ˆ
++=++=+
3. At measurements it will perform Update State
( ) ( ) ( ) ( )( ) ( ) ( ) ( )11|1ˆ|1ˆ
|1ˆ11
++++=+++−=+
kkKkkxkkx
kkxkHkzk
νν
υ (k) - Innovation
K (k) – Filter Gain
23
SOLO
Discrete Filter/Predictor Architecture (continue – 2)
RADAR Range-Doppler-Angular Loops
The way that the Filter Gain K (k) is definedwill define the Filter properties.
1. K (k) can be choose to satisfy the bandwidth requirements. Since we have Linear Time Constant System a K (k)=constant may be chosen. This is a Luenberger Observer.
2. Since we have a Linear Time Constant System, if we assume White Gaussian System and Measurement Disturbances the Kalman Filter will provide the Optimal Filter/Predictor. An important byproduct is the Error Covariances.
3. The Filter Gain K (k) can be chosen as the steady-state value of the Kalman Filter.
Return to Table of Content
24
RADAR Range-Doppler-Angular LoopsSOLO
From the measurement time kT of Σ, ΔAz, ΔEl channels (A/D processing) until the Digital Signal Processor (DSP) provides the measurements to the Filter a time TG = m T delay passes.
Therefore the Filter will process the measurements y(k-m)T, according to
State vector predictionmkmkmkmkmkmkmk uDxFx −−−−−−+− += ||1 ˆˆ
( )1|1|| ˆˆˆ −−−−−−−−−−− −+= mkmkmkmkmkmkmkmkmk xHyKxx Filtering
Since the Kalman Filter is running the previous estimation is available.1|ˆ −−− mkmkx
Based on the prediction of the measured value can be performedmkmkx −− |ˆky
mkmkmkmkmkmkmk uDxFx −−−−−−+− += ||1 ˆˆ
1|ˆˆ −= kkk xHy
The DSP processes the error and delivers itafter the processing delay TG= mT to the Filter.
kkk yyy ˆ−=∆
11|111|2 ˆˆ +−+−−+−+−+−+− += mkmkmkmkmkmkmk uDxFx
112|111| ˆˆ −−−−−− += kkkkkkk uDxFx
Compensation for Processing-from-Measurement Delays
25
RADAR Range-Doppler-Angular LoopsSOLO
Return to Table of Content
Compensation for Processing-from-Measurement Delays continue – 1)
26
Kalman FilterState Estimation in a Linear System (one cycle)
Sensor DataProcessing andMeasurement
Formation
Observation -to - Track
Association
InputData Track Maintenance
( Initialization,Confirmationand Deletion)
Filtering andPrediction
GatingComputations
Samuel S. Blackman, " Multiple-Target Tracking with Radar Applications", Artech House,1986
Samuel S. Blackman, Robert Popoli, " Design and Analysis of Modern Tracking Systems",Artech House, 1999
SOLO
State at tkx (k)
Evolutionof the system(true state)
Transition to tk+1x (k+1) =F (k) x (k)
+ G (k) u (k)+ v (k)
Measurement at tk+1z (k+1) =
H (k) x (k)+ w (k)
Estimationof the state
StateCovariance
andKalman FilterComputations
Control at tku (k)
State Error Covarianceat tk( )
( ) ( )[ ] ( ) ( )[ ] kkxkxkkxkxE
kkPT /ˆ/ˆ
|
−−
=
Controller
Innovation Covariance
( )( ) ( ) ( )
( )1
1|11
1
+++++
=+
kR
kHkkPkH
kST
Kalman Filter Gain( )( ) ( ) ( )11|1
11 +++
=+− kSkHkkP
kKT
Update StateCovariance at tk+1( )
( ) ( ) ( )111
1|1
+++=++
kWkSkW
kkPT
State PredictionCovariance at tk+1( )
( ) ( ) ( ) ( )kQkFkkPkF
kkPT +
=+|
|1
State Predictionat tk+1( )
( ) ( ) ( ) ( )kukGkkxkF
kkx
+=+
|ˆ
|1ˆ
Measurement Predictionat tk+1
( ) ( ) ( )kkxkHkkz |1ˆ1|1ˆ ++=+
Innovation( ) ( ) ( )kkzkzkv |1ˆ11 +−+=+
Update StateEstimation at tk+1( )( ) ( ) ( )11|1ˆ
1|1ˆ
++++=++
kvkKkkx
kkx
kt
1+kt
StateEstimation
at tk ( )kkx |ˆ
( )kkx |ˆ
( )kx
( )1|1 ++ kkP
( )1| −kkP
( )1|1ˆ ++ kkx
( )1+kx
( )kkP |
( )kkP |1+( )kkx |1ˆ +
( )kt ( )1+kt
Real Trajectory
Estimated Trajectory
Rudolf E. Kalman( 1920 - )
27
Kalman FilterState Estimation in a Linear System (one cycle)
SOLO
State vector prediction111|111| ˆˆ −−−−−− +Φ= kkkkkkk uDxx
Covariance matrix extrapolation111|111| −−−−−− +ΦΦ= kT
kkkkkk QPP
Innovation CovariancekT
kkkkk RHPHS += −1|
Gain matrix computation1
1|−
−= kT
kkkk SHPK
Innovation1|ˆ −−= kkkkk xHyv
Filteringkkkkkk vKxx += −1|| ˆˆ
Covariance matrix updating( ) 1|| −−= kkkkkk PHKIPor
( ) ( ) Tkkk
Tkkkkkkkk KRKHKIPHKIP +−−= −1||
Return to Table of Content
28
RADAR Range-Doppler LoopsSOLO
The Range-Doppler is defined by the dynamic equation:
( ) ( ) ( ) ( ) ( ) ( )[ ]∫ +Φ+Φ=t
t
RRRRRRR dwGuBttxtttx0
,, 00 ξξξξ
where:
( ) ( ) ( ) ( ) ( ) ( )3132210000 ,,,&,&,, ttttttIttttAtttd
dRRRRRRR Φ=ΦΦ=ΦΦ=Φ
( ) ( ) ( ) ( ) ( ) ( )0
303
202
00 !3!2, 0 tt
ttA
ttAttAIett RRRR
ttAconstA
RR
R
−Φ=+−+−+−+==Φ −=
The general solution of the Linear System is:
RRRRRRR wGuBxAxtd
d ++= or
R
G
u
M
Bx
TR
Ax
TR
wRa
a
R
R
a
R
R
td
d
R
R
RRRR
+
⋅
−+
−Λ=
→
1
0
0
1
0
1
0
/100
10
0102
τ
29
RADAR Range-Doppler LoopsSOLO
( ) ( ) ( ) ( ) ( )+−+−+−+==−Φ −
!3!2
303
202
000
ttA
ttAttAIett RRR
ttAR
R
−Λ=
τ/100
10
0102
RA
−Λ
Λ
=
−Λ
−Λ=
2
2
2
222
/100
/10
10
/100
10
010
/100
10
010
ττ
ττ
RA
−+ΛΛ−Λ
=
−ΛΛ
−Λ=
3
224
2
2
2
2
23
/100
/10
/10
/100
/10
10
/100
10
010
τττ
ττ
τ
RA
( )
Λ+Λ
Λ+
≈
+−+−
+−Λ+Λ+Λ+Λ
−Λ+Λ+
≅Φ<<Λ
<<
<Λ
<
−
1002
1
221
62100
62621
6
62621
222
222
1
3
3
2
2
2
323222342
323222
1
1
32
/
22
TT
T
TT
T
TTT
TTTT
TTT
TTTT
T
TTT
T
e
T
TR
T
ττ
τ
τττ
ττ
τ
( ) ( ) ( ) ( )[ ]∫ +−Φ+Φ= −
T
RRRRRkRRkR dwGuBTxTx0
1 ξξξξ
Discretization of the Continuous Linear System (continue - 1)
30
RADAR Range-Doppler LoopsSOLO
( )
Λ+Λ
Λ+
≈
−
+Λ+Λ+Λ+Λ
−Λ+Λ+
≅Φ<<Λ
<<
<Λ
<
1002
1
221
6100
6621
6
2621
222
222
1
3
3
2
33222342
323222
1
1
3222
TT
T
TT
T
T
TTT
TTT
TTTT
T
TTT
T
T
TR
ττ
τ
τ
τ
Discretization of the Continuous Linear System (continue - 2
111 −−− ++= kRkRRkRRkR wuDxFxwhere
( ) ( ) ( ) ( ) ( ) ( )∫∫ −Φ=−Φ=Φ= −
T
RRkR
T
RRRRR dwGTwdBTTDTTF0
10
::: ξξξξξ
( )
( )
( ) ( ) ( )
( ) ( ) ( ) 1
2
12
10
222
222
11
0
2
0
1
0
1002
1
221
22
1
−
<<Λ
−
≈
−<<−
−
−
≈
−
−−Λ+−Λ
−−−Λ+
≅ ∫−
kR
T
kR
Tutu
TktkTkRR uT
T
udTT
T
TT
T
uDkRR
ττττ
τττ
31
RADAR Range-Doppler LoopsSOLO
Discretization of the Continuous Linear System (continue - 3)
Computation of the covariance matrix
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )∫
∫ ∫
∫ ∫
−Φ−Φ=
−Φ−Φ=
−Φ−Φ==
−
−−−
TTT
T TT
RT
R
q
TRRRR
T T
RT
RRRRRT
kRkRkR
dTGGTq
ddTGwwEGT
ddTGwwGTEwwEQ
0
0 0
0 0111
τττ
λτλλττ
λτλλττ
λτδ
( ) ( )∫ −Φ=−
T
RRRkR dwGTw0
1 : ξξξ ( )
Λ+Λ
Λ+
≈Φ
1002
1
221
222
222
TT
T
TT
T
TR
=
1
0
0
RG
τσ 22 Txq =
( ) ( )11
101
0 +=
+−=−
++
∫ k
T
k
TdT
k
T
kTk ξξξ
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
≅
−−
−−−
−−−
=
−−
−
−
= ∫∫
TTT
TTT
TTT
qd
TT
TTT
TTT
qdTT
T
T
qTT
26
268
6820
12
2
224
12
1
2
23
234
345
02
23
234
0
2
2
ξ
ξξ
ξξξ
ξξξ
ξξξξ
ξ
32
RADAR Range-Doppler LoopsSOLO
Discretization of the Continuous Linear System (continue - 4)
The measurements are:• R ( H = [1 0 0]T ) and ( H = [0 1 0] ) separately R
• R and ( H = [1 1 0]T ) togetherR
Measurements Noise:
The Range and Doppler measurement noise are given by:
( ) ( )
2
222
/289.0
∆++=k
R
RFARkRNS
RkL σσ
( )
2
2
222
2
/
2289.021
∆++
+⋅⋅=
k
Df
fFA
tTtTCR
kfNS
BWkLD
DDσ
λωω
σ
(S/N)k – Signal to Noise ratio at the k batch (from RADAR Equation)
ΔR, ΔBWD – Range and Doppler resolution cells
LR, LCR – linear dimensions of the target in the Range and Cross-range directions
– total noise variations in Range and Doppler22 , kfkR Dσσ
– noise variations in Range and Doppler due to Frequency Agility22 ,
DfFARFA σσ
Rk – Range to the Target at the k batch
33
RADAR Range-Doppler LoopsSOLO
Discretization of the Continuous Linear System (continue - 5)
Summary of RADAR Range-Doppler Dynamics ( ) ( ) ( ) 100 :,:,1, −==−== kk xtxxtxTktTkt
The discrete dynamic system (one step) is given by:
The measurements are:• R ( H = [1 0 0]T ) and ( H = [0 1 0] ) separately R
• R and ( H = [1 1 0]T ) togetherR
== −−−
TTT
TTT
TTT
wwEQ TxTkkkR
26
268
6820
2
23
234
345
2
111 τσ
( ) ( )
2
222
/289.0
∆++=k
R
RFARkRNS
RkL σσ ( )
2
2
222
2
/
2289.021
∆++
+⋅⋅=
k
Df
fFA
tTtTCR
kfNS
BWkLD
DDσ
λωω
σ
( ) 111 −−
⋅= kMkR Rau
11
2
1
222
222
1
0
0
0
2
1002
1
221
−−
−
+
−
−
+
Λ+Λ
Λ+
=
=
kRkR
kTRkTR
kR wuT
T
a
R
R
TT
T
TT
T
a
R
R
x
34
RADAR Range-Doppler LoopsSOLO
Range Estimator/Predictor using only Range Measurements
The Range Filter Gains KR1, KR2, KR3 are computed using Kalman Filter Method.
35
RADAR Range-Doppler LoopsSOLO
Velocity (Doppler) Estimator/Predictor using only Range Measurements
The Doppler Filter Gains KD1, KD2, KD3 are computed using Kalman Filter Method.Return to Table of Content
36
SOLO
Transversal LOS Equations
−=
13
31
ARA
RAA
A
AI
IA
A
ω
ω
−
−=
τ1
00
120
010
1 RR
RAA
Λ
Λ
=
=
2
1
2
2
1
1
2
1
Tt
Tt
a
a
x
x
x
ε
ε
tAAAAAAA wGuBxAxtd
d ++=( )
( )
+
⋅
⋅
−
−
+
Λ
Λ
−−
−−
−
−
−
=
Λ
Λ
t
t
M
At
M
At
Tt
Tt
RA
RA
RA
RA
RA
RA
Tt
Tt
w
w
ta
ta
R
R
a
a
RR
R
RR
R
a
a
td
d
0
0
0
0
1
1
0000
1000
0100
0000
001
0
0001
10000
12000
01000
001
00
001
20
00010
2
1
2
2
1
1
2
2
1
1
2
1
2
1
2
1
ω
ω
ε
ε
τω
ω
ω
ωτ
ω
ω
ε
ε
=
1
0
0
1
0
0
AG
( )
( )
⋅
⋅=
2
1
1
1
2
1
ta
ta
u
M
At
M
At
A
ω
ω
−
−
=
0000
1000
0100
0000
001
0
0001
R
R
BA
RADAR Angular Loops
37
SOLO RADAR Angular Loops
Transversal LOS Equations
Discretization of the Continuous Linear System
AAAAAAA wGuBxAxtd
d ++=
( ) ( ) ( ) 100 :,:,1, −==−== kk xtxxtxTktTkt
( ) ( )111 −−−
++=kAkAAkAAkA wuTDxTFx
where
( ) ( ) ( ) ( ) ( ) ( )∫∫ −Φ=−Φ=Φ=−
T
AAAkA
T
AAAA dwGTwdBTTDTTF0
10
::: ξξξξξ
( ) ( ) ( ) ( ) ( )+−+−+−+==−Φ −
!3!2
303
202
000
ttA
ttAttAIett AAA
ttAA
A
38
SOLO
Transversal LOS Equations
Discretization of the Continuous Linear System (continue - 1)
( ) ( ) ( ) ( ) ( )+−+−+−+==−Φ −
!3!2
303
202
000
ttA
ttAttAIett AAA
ttAA
A
−=
13
31
ARA
RAA
A
AI
IA
A
ω
ω
−
−=
τ1
00
120
010
1 RR
RAA
( ) ( ) 22
2
1: TATAITTF AAAA ++≅Φ=
−−
−
=
322
11
1322
1
2
2
2
IAA
AIA
A
RAAARA
ARARAA
A
ωω
ωω
−−
−
=
2
22
22
1
100
1240
120
τ
τ RR
R
R
R
RR
R
A
( )( )
( )
−++−−
+−++
=
=
2
2
2
322
1132
13
213
2
322
113
2221
1211
TIATAITAIT
TAITT
IATAI
FF
FFTF
RAAAARARA
ARARARAAA
AA
AAA
ωωω
ωωω
RADAR Angular Loops
39
SOLO
Discretization of the Continuous Linear System (continue - 2)
( )
( )
( )
( )
−
−−
−−−
−−
−−
=
−++==
221100
21
21210
21
21
22
2
22
2
322
1132211
TTT
R
TTT
R
RTT
R
RT
R
R
R
TTT
R
RT
TIATAIFF
RA
RA
RA
RAAAAA
ωττ
τω
ω
ω
[ ]
−
−=+≅−=
TTR
R
R
TTT
R
R
TT
TAITFF
RA
RARA
RARA
ARARAAA
ω
ωω
ωω
ωω
2100
210
02
2
2
132112
( )( )
( )
−++−−
+−++
=
=
2
2
2
322
1132
13
213
2
322
113
2221
1211
TIATAITAIT
TAITT
IATAI
FF
FFTF
RAAAARARA
ARARARAAA
AA
AAA
ωωω
ωωω
RADAR Angular Loops
40
SOLO
Discretization of the Continuous Linear System (continue - 3)
( ) ( ) ( )[ ] AA
T
AA
T
AAA BT
ATIdBTAIdBTTD
+=−+≅−Φ= ∫∫ 2
:2
6
00
ξξξξ
=
−
−
=123
231
0
0
0000
1000
0100
0000
001
0
0001
Ax
xA
A B
B
R
R
B
+−
+
=+
22
22
22
133
2
3
22
132
6
TATII
T
ITT
ATI
TATI
ARA
RAA
A
ω
ω
( )
+−
+
=
1
2
131
2
1
2
1
2
13
22
22
AAARA
ARAAA
A
BT
ATIBT
BT
BT
ATI
TD
ω
ω
−
−
=
+
00
10
2
2
2
1
2
13 R
TT
R
R
R
TT
BT
ATI AA
−
=
00
20
02
2
2
2
1
2
R
T
T
BT
RA
RA
ARA ω
ω
ω
−
=
00
10
01
1 RBA
RADAR Angular Loops
41
SOLO
Discretization of the Continuous Linear System (continue - 4)
Computation of the process covariance matrix Qk-1
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( )∫
∫ ∫
∫ ∫
−Φ−Φ=
−Φ−Φ=
−Φ−Φ==
−
−−−
TTT
T TT
AT
A
q
TAAAA
T T
AT
AAAAAT
kAkAkA
dTGGTq
ddTGwwEGT
ddTGwwGTEwwEQ
0
0 0
0 0111
ξξξ
λξλλξξ
λτλλττ
λξδ
( ) ( )∫ −Φ=−
T
AAAkA dwGTw0
1: ξξξ
=
1
0
0
1AG
τσ 22 Ttq =
−
−=
τ1
00
120
010
1 RR
RAA
=
=
1
0
0
1
0
0
1
1 A
A
A G
GG
( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
−−+−+−−−−
−+−−−+−+
=−Φ
2
2
2
322
1132
13
2
13
2
322
113
ξωξξωξω
ξωξωξωξ
ξ
TIATAITAIT
TAITT
IATAI
T
RAAAARARA
ARARARAAA
A
( )( )( ) ( ) ( )
( )( ) ( ) ( )
−−−+−−+
−−++−++
=−Φ
1
2
32
1
2
1313
1
2
32
1
2
1313
2
2
ARAARAARAA
ARAARAARAA
AA
GT
IAATIAI
GT
IAATIAI
GTξωωξω
ξωωξω
ξ
RADAR Angular Loops
42
SOLO
Discretization of the Continuous Linear System (continue - 5)
Computation of the process covariance matrix Qk-1 (continue – 1)
−
−=
τ1
00
120
010
1 RR
RAA
−−
−
=
2
22
22
1
100
1240
120
τ
τ RR
R
R
R
RR
R
AA
( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
−+−−
−
+−−
−
−+−−
−
+−−
−
≈
−
+−+−
−−+
−
+−−
−
−
++−+−
+−+
−
+−−
−
==
2
2
2
2
2
2
2
2
2
2
0
2
22
2
2
2
2
2
22
2
2
2
2
41
2
12
2
1
41
2
12
2
1
22
111
2
12
2
1
22
111
2
12
2
1
τξ
τξ
ξτ
ξ
ξ
τξ
τξ
ξτ
ξ
ξ
ξττ
ωωξω
τ
ξτ
ξ
ξ
ξττ
ωωξω
τ
ξτ
ξ
ξ
ω
TT
T
RR
R
R
T
T
R
TT
T
RR
R
R
T
T
R
TT
T
RR
R
R
T
T
R
TT
T
RR
R
R
T
T
R
RA
RARARA
RARARA
( )( )( ) ( ) ( )
( )( ) ( ) ( )
−−−+−−+
−−++−++
=−Φ
1
2
32
1
2
1313
1
2
32
1
2
1313
2
2
ARAARAARAA
ARAARAARAA
AA
GT
IAATIAI
GT
IAATIAI
GTξωωξω
ξωωξω
ξ
=
1
0
0
1AG
RADAR Angular Loops
43
SOLO
Discretization of the Continuous Linear System (continue - 6)
( )[ ] ( )[ ]22211211
2221
1211AAAA
AA
AATAAAA QQQQ
QQGTGT ===
=−Φ−Φ ττ
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
−+−−−+−−
−
+−−
+++−−−−
++−
+−−
−+−−−−
+−−−
=
4
4
3
3
2
2
1111
2
4
2
3
2
242
2
3
22
2
11
2
4324
22
34
2
11
1622
321
8
12
2
12
2
1
2
12
12
822
1
4
12
24
1
2313
2311
12
13111211
τξ
τξ
τξ
τξ
τξ
ττξ
τττξξξ
τξ
τξ
τξ
τξξξ
τξξ
TTTTQQ
T
RR
RT
RR
R
RR
T
R
TT
RR
R
R
T
RR
R
R
TQ
R
T
R
TT
RR
T
RR
R
R
TT
R
Q
AA
Q
A
A
A
AA
( ) ( )11
101
0 +=
+−=−
++
∫ k
T
k
TdT
k
T
kTk ξξξ
( )
( ) ( )
( )
( )
( ) ( )
+−+−−−
+−
+++−
++
+−−
+−
+−
=−
∫∫
∫∫∫ −
∫ −∫ −
4
5
3
4
2
32
011
011
2
5
2
4
2
3252
2
4
22
3
011
2
5435
22
45
2
011
8082
1
40
12
8
12
2
1
3210
12
4
12
3
4086
1
20
12
820
1
2313
0 2311
12
0 13110 1211
ττττξξξξ
ττττττττξξ
τττ
ξξ
ξξ
ξξξξ
TTTTTdTQdTQ
T
RR
RT
RR
R
RR
T
R
TT
RR
R
R
T
RR
R
R
TdTQ
R
T
R
TT
RR
T
RR
R
R
TT
R
dTQ
T
A
T
A
dTQ
T
A
dTQdTQ
T
A
T
A
T
A
T
A
Computation of the process covariance matrix Qk-1 (continue – 2)
RADAR Angular Loops
44
SOLO
Discretization of the Continuous Linear System (continue - 7)
The measurements in Angular Loops are:• ε1 and ε2 ( H = [1 0 0 1 0 0]T )
Measurements Noise:
The angular measurement noise is given by:
( ) 2,1/
289.02
322
2 =
Θ++
= Θ i
NS
k
R
L
k
idbFA
CRk
i
ii εε σσ
(S/N)k – Signal to Noise ratio at the k batch (from RADAR Equation)
Θ3db i – Antenna Beamwidth in i=1,2 directions
LR, LCR – linear dimensions of the target in the Range and Cross-range directions
– total angular noise variation22
21, kk εε σσ
– noise variations in Range and Doppler due to Frequency Agility22 ,
DfFARFA σσ
• antenna angular rates measured by sensors on the antenna and sometimes alsoon the body
21, AtAt ωω
Rk – Range to the Target at the k batch
RADAR Angular Loops
45
SOLO
Discretization of the Continuous Linear System (continue - 7)Summary of Discretization of RADAR Angular Loops Dynamics
12
1
12
1
12221
1211
12
1
12221
1211
2
1
−−−−−
+
+
=
kkA
A
kAA
AA
kA
A
kAA
AA
kA
A
w
w
u
u
DD
DD
x
x
FF
FF
x
x
( )
( )
( )
−
−−
−−−
−−
−−
==
221100
21
21210
21
21
2
2
22
2211
TTT
R
TTT
R
RTT
R
RT
R
R
R
TTT
R
RT
FF
RA
RA
RA
AA
ωττ
τω
ω
−
−≅−=
TTR
R
R
TTT
R
R
TT
FF
RA
RARA
RARA
AA
ω
ωω
ωω
2100
210
02
2
2112
−
−
==
00
10
2
2
2211 R
TT
R
R
R
TT
DD AA
−
=−=
00
20
02
2
2
2112 R
T
T
DD RA
RA
AA ω
ω
Λ=
1
1
1
1
Tt
A
a
x
ε
Λ=
2
2
2
2
Tt
A
a
x
ε
( )
⋅=
11 1
1
tau
M
At
A
ω
( )
⋅=
22 1
2
tau
M
At
A
ω
If then FA12= -FA21=0 and DA12= -DA21=0 and and are independent.1Ax2Ax0=RAω
RADAR Angular Loops
46
SOLO
Discretization of the Continuous Linear System (continue - 8)Summary of Discretization of RADAR Angular Loops Dynamics (continue – 1)
( ) ( ) ( ) ( )
( )
( ) ( ) ( )
−=−Φ−Φ=
−Φ−Φ==
∫∫
∫ ∫−
−−−
33
33
011
2
0
0 0111
2
II
IIdTQdTGGTq
ddTGwwEGTwwEQ
T
ATt
TT
AT
AAA
T TT
AT
A
q
TAAAA
T
kAkAkA
ξξτσξξξ
λξλλξξλξδ
Process Covariance Matrix Qk-1
Measurement Noise Rk
( ) 2,1/
289.02
322
2 =
Θ++
= Θ i
NS
k
R
L
k
idbFA
CRk
i
ii εε σσ
( )
( ) ( )
( )
( )
( ) ( )
+−+−−−
+−
+++−
++
+−−
+−
+−
=−
∫∫
∫∫∫ −
∫ −∫ −
4
5
3
4
2
32
011
011
2
5
2
4
2
3252
2
4
22
3
011
2
5435
22
45
2
011
8082
1
40
12
8
12
2
1
3210
12
4
12
3
4086
1
20
12
820
1
2313
0 2311
12
0 13110 1211
ττττξξξξ
ττττττττξξ
τττ
ξξ
ξξ
ξξξξ
TTTTTdTQdTQ
T
RR
RT
RR
R
RR
T
R
TT
RR
R
R
T
RR
R
R
TdTQ
R
T
R
TT
RR
T
RR
R
R
TT
R
dTQ
T
A
T
A
dTQ
T
A
dTQdTQ
T
A
T
A
T
A
T
A
RADAR Angular Loops
47
SOLO
During Track Mode the RADAR Seeker performs the following tasks:
• The Angular Tracker uses the Δ Elevation and Δ Azimuth Maps, computes the Radar Errors in the Detected Range-Doppler cells, and controls the gimbals in the Track Mode.
• Computes the Line-of-Sight (LOS) angular rates for Terminal Guidance.
Because the requirements for gimbals control and those of Terminal Guidancemay be different we can use two Filters, with the same architecture but differentFilter Gains.
RADAR Angular Loops
48
SOLO
Angle Estimator/Predictor
The Angular Filter Gains KA1, KA2, KA3 are computed using Kalman Filter Method.
RADAR Angular Loops
49
Seeker Conceptual Tracking Mode (continue – 1)
Antenna C.G.
A/C
AntennaL.O.S.Apparent
L.O.S.
A/CVelocity
InertialRef. Line
SOLO
Tracking Error
Gr λλε −=
I
G
I
r
Itd
d
td
d
td
d λλε −=
Time Differentiation in an Inertial System.
The Differential Equation of the Estimated Tracking Error is
G
Itd
d λλε −= ˆˆ
where
λ - Estimated LOS inertial angular-rate [rad/sec]
Gλ - Measured (by a rate-gyro on the antenna) gimbal inertial rate [rad/sec]
Kalman Filters are used to estimate both tracking error and L.O.S. inertial angular-rate
λε ˆ,ˆ
TRACK MODE
Return to Table of Content
50
SOLO
References
Y. Bar-Shalom, T.E. Fortmann, “Tracking and Data Association”, Academic Press, 1988
Y. Bar-Shalom, Xiao-Rong Li., “Multitarget-Multisensor Tracking: Principles and Techniques”, YBS Publishing, 1995
S. S. Blackman, “Multiple-Target Tracking with Radar Applications”, Artech House, 1986
Pearson, J.B., Stear, E.B., “Kalman Filter Applications in Airborne Radar Tracking”, IEEE Transactions on Aerospace and Electronic Systems, AES-10, 1974, pp. 319-329
RADAR Range-Doppler-Angular Loops
Return to Table of Content
January 23, 2015 51
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA
52
Random VariablesSOLO
Table of Content
Markov Processes
A Markov Process is defined by:
Andrei AndreevichMarkov
1856 - 1922
( ) ( )( ) ( ) ( )( ) 111 ,|,,,|, tttxtxptxtxp >∀ΩΩ=≤ΩΩ ττ
i.e. the Random Process, the past up to any time t1 is fully defined by the process at t1.
Examples of Markov Processes:
1. Continuous Dynamic System( ) ( )( ) ( )wuxthtz
vuxtftx
,,,
,,,
==
2. Discrete Dynamic System
( ) ( )( ) ( )kkkkk
kkkkk
wuxthtz
vuxtftx
,,,
,,,
1
1
==
+
+
x - state space vector (n x 1)u - input vector (m x 1)v - white input noise vector (n x 1)
- measurement vector (p x 1)z
- white measurement noise vector (p x 1)w
53
Random VariablesSOLO
Table of Content
Markov Processes
Examples of Markov Processes:
3. Continuous Linear Dynamic System( ) ( ) ( )( ) ( )txCtz
tvtxAtx
=+=
Using the Fourier Transform we obtain: ( ) ( )( )
( ) ( ) ( )ωωωωωω
VHVAIjCZH
=−= −
1
Using the Inverse Fourier Transform we obtain:
( ) ( ) ( )∫+∞
∞−
= ξξξ dvthtz ,
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
( ) ( ) ( )( )( )
( ) ( )∫∫ ∫
∫ ∫∫
∞+
∞−
∞+
∞−
−
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
−=−=
−==
ξξξξωξωωπ
ξ
ωωξξωξωπ
ωωωωπ
ξ
ω
dthvddtjHv
dtjdjvHdtjVHtz
th
egrattionoforderchange
V
exp2
1
expexp2
1exp
2
1
int
h (t,τ)v (t) z (t)
54
Random VariablesSOLO
Table of Content
Markov Processes
Examples of Markov Processes:
3. Continuous Linear Dynamic System( ) ( ) ( )( ) ( )txCtz
tvtxAtx
=+=
The Autocorrelation of the output is:
( ) ( ) ( )∫+∞
∞−
= ξξξ dvthtz ,
h (t,τ)v (t) z (t)
( ) ( ) ( )[ ] ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )[ ] ( ) ( ) ( )
( ) ( ) ( ) ( )∫∫
∫ ∫∫ ∫
∫∫
∞+
∞−
−=∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
+=−+−=
−−+−=−+−=
−+−=+=
ζτζζσξξτξσ
ξξξξδξτξσξξξξξτξ
ξξξτξξξττ
ξζ
dhhdthth
ddththddvvEthth
dvthdvthEtztzER
v
t
v
v
zz
2
111
2
212121
2
212111
222111
1
( ) ( ) ( )[ ] ( )τδσττ 2
vvv tvtvER =+=
( ) ( ) ( ) ( ) ( ) 22 expexp vvvvvv djdjRS σττωτδσττωτω =−=−= ∫∫+∞
∞−
+∞
∞−
( ) ( ) ( )( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) 2*2
22
2
expexp
expexpexpexp
expexp
xx
xx
x
RR
zzzz
HHdjhdjh
djdjhhdjdjhh
djdhhdjRSzzzz
σωωχχωχζζωζσ
χχωζζωζχσττζωζζωζτζσ
ττωζτζζσττωτω
χτζ
ττ
=
−=
−=−−−=
−−=−=
∫∫
∫ ∫∫ ∫
∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
=+∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
−=+∞
∞−
( ) ( ) ( ) ( )ωωωω vvzz SHHS *=
55
Random VariablesSOLO
Table of Content
Markov Processes
Examples of Markov Processes:
4. Continuous Linear Dynamic System ( ) ( ) ( )∫+∞
∞−
= ξξξ dvthtz ,
( ) ( ) ( )[ ] ( )τδσττ 2
vvv tvtvER =+= ( ) 2
vvvS σω =
v (t) z (t)( )xj
KH
ωωω
/1+=
( )xj
KH
ωωω
/1+=
The Power Spectral Density of the output is:
( ) ( ) ( ) ( ) ( ) 222
*
/1 x
v
vvzz
KSHHS
ωωσ
ωωωω+
==
( ) ( ) 222
/1 x
vvzz
KS
ωωσω
+=
ω
xω
22
vvK σ
2/22
vvK σ
The Autocorrelation of the output is:( ) ( ) ( )
( ) ( ) ( ) ( )∫∫
∫∞+
∞−
=∞+
∞−
+∞
∞−
−−
=+
=
=
dsss
K
jdj
K
djSR
x
vjs
x
v
zzzz
τωσ
πωτω
ωωσ
π
ωτωωπ
τ
ω
exp/12
1exp
/12
1
exp2
1
2
22
2
22
ωj
xω
R
( ) 0/1 2
22
=−∫
∞→R
s
x
vv dses
K τ
ωσ( ) 0
/1 2
22
=−∫
∞→R
s
x
vv dses
K τ
ωσ
xω−
σ
ωσ js +=
0<τ0>τ
( ) τωσωω xeK
R vvxzz
==2
22
τ
2/22
vvxK σω
( )τωσω
xvxK
−= exp2
22
( ) ( )
( ) ( )
>
+
−−=
−−
<
−
−=
−−
=
∫
∫
→
−→
0exp
Reexp2
1
0exp
Reexp2
1
222
22
222
222
22
222
τω
τσωτ
ωσω
π
τω
τσωτ
ωσω
π
ωω
ωω
x
vx
x
vx
x
vx
x
vx
s
sKsdss
s
K
j
s
sKsdss
s
K
j
x
x
56
Random VariablesSOLO
Markov Processes
Examples of Markov Processes:
5. Continuous Linear Dynamic System with Time Variable Coefficients
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21121&
:&:
tttQteteE
twEtwtetxEtxteT
ww
wx
−=
−=−=
δ
w (t) x (t)
( )tF
( )tG ∫x (t)
( ) ( ) ( ) ( ) ( ) ( )twtGtxtFtxtxtd
d +==
( ) ( ) ( ) ( ) ( )tetGtetFte wxx +=
( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
dwGttxtttx0
,, 00 λλλλ
The solution of the Linear System is:
where:
( ) ( ) ( ) ( ) ( ) ( ) ( )3132210000 ,,,&,&,, ttttttItttttFtttd
d Φ=ΦΦ=ΦΦ=Φ
( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
wxx deGttettte0
,, 00 λλλλ
( ) ( ) ( ) ( ) ( ) twEtGtxEtFtxE +=
57
Random VariablesSOLO
Markov Processes
Examples of Markov Processes:
5. Continuous Linear Dynamic System with Time Variable Coefficients (continue – 1)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21121&
:&:
tttQteteE
twEtwtetxEtxteT
ww
wx
−=
−=−=
δ
w (t) x (t)
( )tF
( )tG ∫x (t)
( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
dwGttxtttx0
,, 00 λλλλ ( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
wxx deGttettte0
,, 00 λλλλ
( ) ( ) ( ) ( ) teteEtxVartV T
xxx ==: ( ) ( ) ( ) ( ) ττττ ++=+=+ teteEtxVartV T
xxx :
( ) ( ) ( ) ( ) ( ) ( ) ττττ +=++=+ teteEttRteteEttR T
xxx
T
xxx :,&:,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ΦΦ+ΦΦ==t
t
TTT
xxx dtGQGttttVttttRtV0
,,,,, 000 λλλλλλ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫+
+Φ+Φ++Φ+Φ=++=+τ
λλτλλλλττττττt
t
TTT
xxx dtGQGttttVttttRtV0
,,,,, 000
58
Random VariablesSOLO Markov Processes
Examples of Markov Processes:
5. Continuous Linear Dynamic System with Time Variable Coefficients (continue – 2)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21121&
:&:
tttQteteE
twEtwtetxEtxteT
ww
wx
−=
−=−=
δ
w (t) x (t)
( )tF
( )tG ∫x (t)
( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
dwGttxtttx0
,, 00 λλλλ ( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
wxx deGttettte0
,, 00 λλλλ
( ) ( ) ( ) ( ) teteEtxVartV T
xxx ==: ( ) ( ) ( ) ( ) ττττ ++=+=+ teteEtxVartV T
xxx :
( ) ( ) ( ) ( ) ( ) ( ) ττττ +=++=+ teteEttRteteEttR T
xxx
T
xxx :,&:,
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
<Φ+Φ+Φ+Φ
>Φ+Φ+Φ+Φ
=+
∫
∫+
0,,,,
0,,,,
,
0
0
000
000
τλλλλλλττ
τλλλλλλττ
ττt
t
TTT
x
t
t
TTT
x
x
dtGQGttttVtt
dtGQGttttVtt
ttR
( )
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
<+Φ+
>Φ+Φ−+Φ+
<Φ+Φ−+Φ
>+Φ
=+
∫
∫
+
+
0,
0,,,
0,,,
0,
,
τττ
τλλλλλλτττ
τλλλλλλττ
ττ
ττ
τ
tttV
dtGQGttttV
or
dtGQGttVtt
tVtt
ttR
T
x
t
t
TTT
x
t
t
TT
x
x
x
59
Random VariablesSOLO Markov Processes
Examples of Markov Processes:
5. Continuous Linear Dynamic System with Time Variable Coefficients (continue – 3)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21121&
:&:
tttQteteE
twEtwtetxEtxteT
ww
wx
−=
−=−=
δ
w (t) x (t)
( )tF
( )tG ∫x (t)
( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
wxx deGttettte0
,, 00 λλλλ
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
<Φ+Φ+Φ+Φ
>Φ+Φ+Φ+Φ
=+
∫
∫+
0,,,,
0,,,,
,
0
0
000
000
τλλλλλλττ
τλλλλλλττ
ττt
t
TTT
x
t
t
TTT
x
x
dtGQGttttVtt
dtGQGttttVtt
ttR
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ΦΦ+ΦΦ==t
t
TTT
xxx dtGQGttttVttttRtV0
,,,,, 000 λλλλλλ
( ) ( ) ( ) ( ) teteEtxVartV T
xxx ==:
( ) ( ) ( ) ( ) ( ) ( ) ττττ +=++=+ teteEttRteteEttR T
xxx
T
xxx :,&:,
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tGtQtGdtFtGQGttFtttVtt
dtGQGttFtttVtttFtVtd
d
T
t
t
TTTTT
x
t
t
TTT
xx
+ΦΦ+ΦΦ+
ΦΦ+ΦΦ=
∫
∫
0
0
,,,,
,,,,
000
000
λλλλλλ
λλλλλλ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tGtQtGtFtVtVtFtVtd
d TT
xxx ++=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ττττττττ +++++++++=+ tGtQtGtFtVtVtFtVtd
d TT
xxx
60
Random VariablesSOLO Markov Processes
Examples of Markov Processes:
5. Continuous Linear Dynamic System with Time Variable Coefficients (continue – 4)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )21121&
:&:
tttQteteE
twEtwtetxEtxteT
ww
wx
−=
−=−=
δ
w (t) x (t)
( )tF
( )tG ∫x (t)
( ) ( ) ( ) ( ) ( ) ( )∫Φ+Φ=t
t
wxx deGttettte0
,, 00 λλλλ ( ) ( ) ( ) ( ) teteEtxVartV T
xxx ==:
( ) ( ) ( ) ( ) ( ) ( ) ττττ +=++=+ teteEttRteteEttR T
xxx
T
xxx :,&:,
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
<Φ+Φ+Φ+Φ
>Φ+Φ+Φ+Φ
=+
∫
∫+
0,,,,
0,,,,
,
0
0
000
000
τλλλλλλττ
τλλλλλλττ
ττt
t
TTT
x
t
t
TTT
x
x
dtGQGttttVtt
dtGQGttttVtt
ttR
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
<+Φ++++++++>+Φ+++++
=+0,,,
0,,,,
τττττττττττττ
τtttGtQtGtFttRttRtF
tGtQtGtttFttRttRtFttR
td
dTTT
xx
TT
xx
x
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
<++++Φ+++++>+Φ+++++
=+0,,,
0,,,,
τττττττττττττ
τtGtQtGtttFttRttRtF
tttGtQtGtFttRttRtFttR
td
dTT
xx
TTT
xx
x
61
Random VariablesSOLO Markov Processes
Examples of Markov Processes:
6. How to Decide if a Input Noise can be Approximated by a White or a Colored Noise
w (t) x (t)
( )tF
( )tG ∫x (t)
( ) ( ) ( ) ( ) ( ) ( )twtGtxtFtxtxtd
d +== Given a Continuous Linear System:
we want to decide if can be approximated by a white noise.( )tw
Let start with a first order linear system with white noise input :( )tw '
( ) ( ) ( )twT
twT
tw '11 +−= w (t)w' (t) ( )
TssH
+=1
1
( ) ( ) Ttt
w ett /
00, −−=φ
( ) ( ) [ ] ( ) ( ) [ ] ( )τδττ −=−− tQwEwtwEtwE ''''
( ) ( ) [ ] ( ) ( ) [ ] ( )ttRtwEtwtwEtwE ww ,τττ +=−+−+
( ) ( ) [ ] ( ) ( ) [ ] ( )τττ +=+−+− ttRtwEtwtwEtwE ww ,
( ) ( ) [ ] ( ) ( ) [ ] ( ) ( )ttRtVwEwtwEtwE wwww ,==−− ττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tGtQtGtFtVtVtFtVtd
d TT
xxx ++= ( ) ( ) QT
tVT
tVtd
dwwww 2
12 +−=
( ) ( )00 ,1
, ttT
tttd
dww φφ −=
where
62
Random VariablesSOLO Markov Processes
Examples of Markov Processes:
6. How to Decide if a Input Noise can be Approximated by a White or a Colored Noise
(continue – 1)
( ) ( ) QT
tVT
tVtd
dwwww 2
12 +−=
( ) ( )
−+=
−−T
t
T
t
wwww eT
QeVtV
22
12
0 t2/T
( ) T
t
ww eV2
0−
−
−T
t
eT
Q 2
12 T
QV statesteadyww 2
=−
( )tVww
( ) ( ) ( ) ( )( ) ( ) ( )
<+=+Φ+
>=+Φ=+
−
−
0,
0,,
ττττ
τττ
τ
τ
tVetttV
tVetVttttR
wwTT
www
wwT
www
ww
( ) ( ) ( ) ( )( ) ( ) ( )
<+=++Φ
>=+Φ=+
−
−
0,
0,,
ττττ
τττ
τ
τ
tVetVtt
tVetttVttR
wwT
www
wwTT
www
ww
For ( ) ( )T
QVtVtV
T statesteadywwwwww 25 ==+≈⇒> −ττ
( ) ( ) ( ) TTstatesteadywwwwwwww e
T
QeVVttRttR
T
ττ
ττττ −−
− =≈≈+≈+⇒>2
,,5
w (t)w' (t) ( )Ts
sH+
=1
1
63
Random VariablesSOLO Markov Processes
Examples of Markov Processes:
6. How to Decide if a Input Noise can be Approximated by a White or a Colored Noise
(continue – 2)
( ) ( ) ( ) TTstatesteadywwwwwwww e
T
QeVVttRttR
T
ττ
ττττ −−
− =≈≈+≈+⇒>2
,,5
( ) T
ww eT
QV /
2ττ ==
τ
T
QV statesteadyww 2
=−
T− T
1−− ⋅ eV statesteadyww ( ) Qde
T
QdVArea T
ww === ∫∫+∞
−+∞
∞− 02
2 ττττ
T is the correlation time of the noise w (t) and can be found from Vww (τ) by tacking the time corresponding to Vww steady-state /e.
One other way to find T is by tacking the double sides Laplace Transform L 2 on τ of:
( ) ( ) ( ) QdetQtQs s
ww =−=−=Φ ∫+∞
∞−
− ττδτδ ττ2'' L
( ) ( )
( ) ( ) ( )sHQsHsT
Q
deeT
QVs sT
sswwww
−==
=
==Φ ∫+∞
∞−
−−−
2
/
2
1
2ττ ττ
τL( ) ( ) 22/1/1 ωωω
+= Q
Qww
ω
T/12/1 =ω
Q
2/Q
T/12/1 −=−ω
T can be found by tacking ω1/2 of half of the power spectrum Q/2 and T=1/ ω1/2.
64
Random VariablesSOLO Markov Processes
Examples of Markov Processes:
( ) ( ) 22/1/1 ωωω
+= Q
Qww
ω
T/12/1 =ω
Q
2/Q
T/12/1 −=−ω
Let return to the original system: ( ) ( ) ( ) ( ) ( ) ( )twtGtxtFtxtxtd
d +==
w (t) x (t)
( )tF
( )tG ∫x (t)
6. How to Decide if a Input Noise can be Approximated by a White or a Colored Noise
(continue – 3)
Compute the power spectrum ofand define Q and T.
( )ωjsww =Φ ( )tw
then can be approximated by the white noise with( )tw ( )tw '
( ) ( ) [ ] ( ) ( ) [ ] ( )τδττ −=−− tQwEwtwEtwE ''''
then can be approximated by a colored noise that can be obtained by passingthe predefined white noise through a filter
( )tw( )tw ' ( )
sTsH
+=1
1
If F of eigenvalue maximum
1F of constant time minimumT =<51
If F of eigenvalue maximum
1F of constant time minimumT =>52
65
Random VariablesSOLO Markov Processes
Examples of Markov Processes: 7. Digital Simulation of a Contimuos Process
Let start with a first order linear system with white noise input :( )tw '
( ) ( ) ( )twT
twT
tw '11 +−= w (t)w' (t) ( )
TssH
+=1
1
( ) ( ) Ttt
w ett /
00, −−=φ ( ) ( )00 ,
1, tt
Ttt
td
dww φφ −=
( ) ( ) ( ) ( ) ( )∫ −−− +=t
t
TtTtt dwT
etwetw0
0 '1/
0
/ τττ
Let choose t = (k+1) ΔT and t0 = k ΔT
( ) ( ) [ ] ( ) ( ) [ ] ( )τδττ −=−− tQwEwtwEtwE ''''where
( )[ ] ( ) ( )[ ] ( )( )
( )
Tkw
Tk
Tk
TTkTT dwT
eTkweTkw
∆
∆+
∆
∆+−∆− ∫+∆=∆+
1'
1
/1/ '1
1 τττ
66
Random VariablesSOLO Markov Processes
Examples of Markov Processes: 7. Digital Simulation of a Contimuos Process (continue – 1)
Define: TTe /: ∆−=ρ
( ) ( ) [ ] ( ) ( ) [ ] ( )[ ] ( )[ ] ( ) ( ) [ ] ( ) ( ) [ ]
( )
( )( )
( )[ ]( )
( )[ ] ( ) ( ) ( )2/21/12
2
1
12
/12
2
1 1
122112
/1/1
1111
12
122
1
''''1
''''
11
21
21
ρτ
ττττττ
ττ
ττδ
ττ
−=−===
−−=
∆−∆∆−∆
∆−∆+
∆
∆+−
∆+
∆
∆+−
∆+
∆
∆+
∆ −
∆+−∆+−
∫
∫ ∫
T
Qe
T
Qe
T
T
QdQ
Te
ddwEwwEwET
ee
TkwETkwTkwETkwE
TTTk
Tk
TTk
Tk
Tk
TTk
Tk
Tk
Tk
Tk Q
TTkTTk
( )[ ] ( ) ( )[ ] ( )( )
( )
Tkw
Tk
Tk
TTkTT dwT
eTkweTkw
∆
∆+
∆
∆+−∆− ∫+∆=∆+
1'
1
/1/ '1
1 τττ
Define w’ (k) such that:
( ) ( ) [ ] ( ) ( ) [ ] T
QkwEkwkwEkwE
2:'''' =−−
( ) ( )2
1
1
':'
ρ−= kw
kw
Therefore:( )[ ] ( ) ( )kwTkwTkw '11 2ρρ −+∆=∆+
67
RADAR Range-Doppler LoopsSOLO
Start with:
( ) ( ) ( ) ( ) ( ) ( )3132210000 ,,,&,&,, ttttttIttttAtttd
d Φ=ΦΦ=ΦΦ=Φ
Using the Laplace’s Transform we obtain
( ) ( ) ( )sAttssI
Φ=Φ−Φ ~,
~00 ( ) ( ) 1~ −−=Φ AIss
−Λ=
τ/100
10
0102A
+−Λ−
−=−
τ/100
1
012
s
s
s
AsI
( ) ( ) ( )( ) ( )
( ) ( )( )
( ) ( )
( ) ( )
+
Λ−+Λ−Λ−Λ
Λ−+Λ−Λ−
=
Λ−++Λ++
Λ−+=− −
τ
τ
τ
ττττ
τ
/1
100
/1
/1
11
00
/1/1
1/1/1
/1
1222222
2
222222
22
222
1
s
ss
s
s
s
s
ssss
s
s
ssss
sss
ssAsI
68
RADAR Range-Doppler LoopsSOLO
( )
( ) ( )
( ) ( )
+
Λ−+Λ−Λ−Λ
Λ−+Λ−Λ−
=− −
τ
τ
τ
/1
100
/1
/1
11
222222
2
222222
1
s
ss
s
s
s
s
ssss
s
AsI
( ) ( )
( ) ( )
+
Λ+−Λ−
Λ−+Λ+
+−Λ
Λ++
Λ−Λ+
Λ
−Λ−
Λ
Λ+−ΛΛ+
Λ−+ΛΛ+
+−Λ−
Λ+Λ−
Λ−Λ
Λ++
Λ−
=
τ
ττττ
τ
ττττ
/1
100
/121
/121
/1/1/1
21
21
22
/121
/121
/1/11
21
21
21
21
22
22
s
sssssss
sssssss
69
RADAR Range-Doppler LoopsSOLO
( )
( ) ( )
( ) ( )
+
Λ+−Λ−
Λ−+Λ+
+−Λ
Λ++
Λ−Λ+
Λ
−Λ−
ΛΛ+−ΛΛ+
Λ−+ΛΛ+
+−Λ−
Λ+Λ−
Λ−Λ
Λ++
Λ−
=− −
τ
ττττ
τ
ττττ
/1
100
/121
/121
/1/1/1
21
21
22
/121
/121
/1/11
21
21
21
21
22
22
1
s
sssssss
sssssss
AsI
( ) ( ) TATt eAsIT =−=Φ =
−− 11L
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
−Λ−
+Λ+
−Λ+−Λ
−ΛΛ+
+ΛΛ+
−Λ−−
Λ+
=
−
Λ−Λ−Λ−ΛΛ−Λ
Λ−Λ−Λ−ΛΛ−Λ
τ
τ
τ
ττττ
τττ
/
22
/
22
/
00
/12/12/12
1
2
/12/12/12
1
2
1
T
TTTTTTT
TTTTTTT
e
eeeeeee
eeeeeee
70
RADAR Range-Doppler LoopsSOLO
( ) ( ) TATt eAsIT =−=Φ =
−− 11L
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
−Λ−
+Λ+
−Λ+−Λ
−ΛΛ+
+ΛΛ+
−Λ−−
Λ+
=
−
Λ−Λ−Λ−ΛΛ−Λ
Λ−Λ−Λ−ΛΛ−Λ
τ
τ
τ
ττττ
τττ
/
22
/
22
/
00
/12/12/12
1
2
/12/12/12
1
2
1
T
TTTTTTT
TTTTTTT
e
eeeeeee
eeeeeee
( )
Λ+Λ
Λ+
≈
+−+−
+−Λ+Λ+Λ+Λ
−Λ+Λ+
≅Φ<<Λ
<<
<Λ
<
−
1002
1
221
62100
62621
6
62621
222
222
1
3
3
2
2
2
323222342
323222
1
1
32
/
22
TT
T
TT
T
TTT
TTTT
TTT
TTTT
T
TTT
T
e
T
T
T
ττ
τ
τττ
ττ
τ
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