6 radar range-doppler-angular loops

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


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


⋅−

⋅=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


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


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 ≅


LOS Vector in Seeker Coordinates (continue – 1)

A/C

Target

AntennaAxis

Antenna

12


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


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


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



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)


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)




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


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



21

SOLO

Discrete Filter/Predictor Architecture


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)


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)


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.


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


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||


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


( ) ( ) ( ) ( ) ( )+−+−+−+==−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


( )

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



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



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



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


Range Estimator/Predictor using only Range Measurements

The Range Filter Gains KR1, KR2, KR3 are computed using Kalman Filter Method.

35


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


−=

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


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



( ) ( ) ( ) ( ) ( )+−+−+−+==−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


( )

( )

( )

( )

−

−−

−−−

−−

−−

=

−++==

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


( ) ( ) ( )[ ] 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


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


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


( )[ ] ( )[ ]22211211

2221

1211AAAA

AA

AATAAAA QQQQ

QQ

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

QQ

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


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


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



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


Table of Content

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


Table of Content

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


Table of Content

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


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


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


,,,,, 000

58

Random VariablesSOLO Markov Processes



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )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


xxx ==: ( ) ( ) ( ) ( ) ττττ ++=+=+ teteEtxVartV T

xxx :


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




( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )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


,,,,, 000 λλλλλλ


xxx ==:


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


d TT

xxx ++=

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ττττττττ +++++++++=+ tGtQtGtFtVtVtFtVtd

d TT

xxx

60




( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )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 ==:


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



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


d TT

xxx ++= ( ) ( ) QT

tVT

tVtd

dwwww 2

12 +−=

( ) ( )00 ,1

, ttT

tttd

dww φφ −=

where

62




(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




(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



( ) ( ) 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)


(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


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


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


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


( )

( ) ( )

( ) ( )

+

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


( )

( ) ( )

( ) ( )

+

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


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


( ) ( ) TATt eAsIT =−=Φ =

−− 11L

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

−Λ−

+Λ+

−Λ+−Λ

−ΛΛ+

+ΛΛ+

−Λ−−

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

6 radar range-doppler-angular loops

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Transcript of 6 radar range-doppler-angular loops