Numerical modeling in radar data analyzing

Igor Đurović, Miloš Daković, Vesna Popović

CSS I T

Center for Signals, Systems, and Information TheoryFaculty of Electrical Engineering

University of Montenegro, Podgorica www.tfsa.cg.yu

Abstract

• Radar data recording is often expensive.• It implies expensive equipment and experiments.• Obtained data are usually classified and they are not exchangeable

among different research institutions.• Repeating of experiments requires additional costs.• In order to compare different techniques for radar data analyzing,

development of numerical models to simulate targets in radar systems has an important role.

• The most commonly used are Boeing-727, MiG-25 and F-16 developed by Victor Chen.

• By using mathematical modeling, we have developed a couple of models.

• An UH-1D helicopter model, model of aircraft moving along helicoids path and polarimetric model of ship will be presented.

Radar signal• The two dimensional discrete radar signal is of the form:

• SAR and ISAR are based on the same fundamental principles, the difference is in geometrical configuration.

0

2 2( , ) exp( )exp( 2 )r s

d dq m n j j Bf nT

c c

- reflection coefficient of the corresponding scatterer;d - radar to target distance of the corresponding scatterer.

nradar

flying direction

q

cross-rang rang

LO

S

target

SAR

a)

nt

radar

b)

LOS

target

d

n

nc

ISAR

cross-rang rang

beam width

d

Radar imaging

• The 2D Fourier transform (FT) of the received signal is:

• The periodogram:

represents an SAR image in range/cross-range domain.• In the case of stationary targets in the SAR systems and targets that

are moving with constant velocity in the ISAR systems, resulting radar signal is 2D complex sinusoid and the 2D FT is ideally concentrated on positions proportional to the position of target.

• Radar image can be spread in the 2D FT domain for: fast maneuvering ISAR targets, targets with 3D motion, moving targets in the SAR systems, fast rotation and vibration of the radar target parts - micro-Doppler (m-D) effect.

window function Size of radar image NxM

2( ', ') ( ', ')P m n Q m n

( ', ') ( , ) ( , )exp( 2 ' / 2 ' / )m n

Q m n q m n w m n j m m M j n n N

Reasons for radar image defocusing

• Radar signal can be represented as:

• The 2D FT can be represented in the following form:

• Some more sophisticated techniques for radar imaging are needed

( , ) exp( ( , ))i ii

q m n j m n

(1) (1) ( , )( ', ') ( ' / 2 , ' / 2 ) * '* ' {e }j m ni i iQ m n W m Ma n Nb m n FT

higher order terms in phase function cause defocusing of radar image.

FT of window function, highly concentrated in the 2D FT domain.

Phase function SAR systems ISAR systems (1) (1)

i ia m b n non-moving objects rigid body parts

(2) (1) (1)2 / 2i i ia m a m b n moving objects objects with uniform acceleration

( ) (1)

1 !

pPpi i

p

ma b n

p objects with non-uniform motion

,(1) (1)

1 1 ! !

p kP Kp k

ii ip k

m na m b n d

p k objects with complicated motion patterns

(1) (1) sin( )i i i ii ia m b n c m n objects with vibrations and rotations

UH-1D helicopter model - I

Smaller pulses that can be seen in the right

hand side correspond to the tail rotor flashes.

These flashes correspond to periodic

alignment of the main and tail rotors to maximally reflect the radar signal;

•Signal of a German Air Force Bell UH-1D Helicopter is simulated.•Several effects are emphasized in time frequency representation:

Stationary patterns - rigid body reflection;

Sinusoidal FM signals with a large magnitude in frequency direction - motion of two main blades;

Signals producing lines connecting peaks of the sinusoidal FM signal with time axis - main rotor flashes;

Modulated time tones commonly added to the data tape.

The simplified model of the reflected UH-1D signal can be written as:

UH-1D helicopter model - II

_ _( ) ( ) ( ) ( ) ( )RIG ROT FL M FL Tx t x t x t x t x t

( ) exp( 2 )RIG ix t j tf Signal caused by rigid body

1 2 3 4-10.3 , -2.5 , 2.3 and 2.7 f kHz f kHz f kHz f kHz

( ) [exp( * 2 * sin(2 / )) exp( * 2 * sin(2 / ))]ROT ROT ROT ROT ROT ROTx t j A t T j A t T

Signal caused by rotation of two main blades

175 and 19ROT ROTT ms A kHz

Signal caused by main rotor flashes

_ _ * _( ) ( / 2) ( )FL M FL M ROT t FL Mk

x t t kT h t

_ _ * _( ) ( / 2) ( ) 35.8FL T FL T TAIL t FL T TAILk

x t t kT h t T ms

_

1 2( )

0 elsewhere.ROT

FL M

AH t

Signal caused by tail rotor flashes

Cut-off filters are given in the frequency domain

_

1 2 (7.35 ) 2 (15.7 )( )

0 elsewhere.FL T

kHz kHzH t

Model of aircraft moving along helicoids path - I

• Aircraft is modeled by 17 characteristic point reflectors representing its contours. Coordinates of reflectors are given in range/cross-range domain, with origin in the center of aircraft’s rotation:

• The aircraft is moving in xyz coordinate system

No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

x[m] 0 0 0 0 0 0 0 0 0 -1 1 -3 -2 -1 1 2 3

y[m] 5 4 3 2 1 0 -1 -2 -3 -4 -4 1 2 2 2 2 1

Initial position of the aircraft

helicoids pathcos

sin

h

h

h

x R t

y R t

z V t

30hV m s50hR m srad /8.1

Used parameters are:

Velocity of an object that is moving over this path is:2 2 2 2 2 2 94.8683 /h x h hv y V R m s const

Reflectors rotate, where rotation matrix is given as:cos sin 0

( sin cos 0

0 0 1

)rR

( )r

r r

r

x x

y R y

z z

Coordinates after rotation for angle

Model of aircraft moving along helicoids path - II

• Radar signal reflected from the 17 point scatterers can be obtained by using superposition principle as a sum of individual echoes.

• Radar signal duration is T=0.1024s. Three characteristic time instants t=2s, t=4s and t=6s are used as initial for emitting radar signal.

cross-range

rang

e

cross-range

rang

e

rang

e

cross-range

2D FT (t=2s) 2D FT (t=4s) 2D FT (t=6s)

Simulated SAR Ship Model• The ship is considered to be a rigid body, simulated as a set of points

representing reflectors along the boundaries (contours) of the ships' parts: cabin, stern, left and right side of cutwater (ship’s shell), shipboard, bulwark and three spars.

-20-15

-10-5

05

10

1520

-5

0

5

0

5

10

stern

bulwarksparsshipboard

One side of the ship’s shell

cabin

Solid lines represents contours that are visible for the current radar and ship position. Dashed lines represent contours that are not visible for the current position.

Back Face Culling Algorithm

• The 3D object is represented as the set of connected polygons (triangles). Each triangle is defined by three adjacent points chosen in the counter-clockwise direction.

• For each polygon normal vector is calculated.

(blue and green arrows)• The direction where the radar is facing is

also calculated (pink arrow).• If the vector angle between radar viewing

direction and polygon normal vector is

between -90 degrees 90 degrees,

the polygon is not visible from the

radar position. Otherwise, the polygon

is visible.

AB

C

nThe principle for defining polygons,

and polygon normal vector

Scattering Matrix

• For obtaining polarimetric radar signal the scattering matrix of the scatterer is needed.

• The scattering matrix for each point depends on the geometrical shape to which that point belongs.

• Each point used for the ship simulation is assumed to belong to the one of the five geometrical shapes.

TRIHEDRAL DIHEDRAL

DIPOLE NARROW DIHEDRAL

CYLINDER

1 0

0 1diplaneS

1 0

0 0dipoleS

2 0

0 1narrowdiplaneS

2 0

0 1cylinderS

1 0

0 1trihedralS

Polarimetric (CW) Radar Signal Model

0

2 2( ) ( exp( )) exp( )exp( 2 ( ) )hh hh hv r r

d dq t s s a j j j Bf t mT

c c

20

2 2( ) ( exp( ) )exp( )exp( 2 ( ) )hv hh hv r r

d dq t s a j s a j j Bf t mT

c c

20

2 2( ) ( exp( ) ) exp( )exp( 2 ( ) )vv vh vv r r

d dq t s a j s a j j Bf t mT

c c

0

2 2( ) ( exp( )) exp( )exp( 2 ( ) )vh vh vv r r

d dq t s s a j j j Bf t mT

c c

• H H

• H V

• V H

• V V

By using polarimetric radar, four radar signals are obtained

3D Rotation • Any arbitrary 3D rotation can be represented as combination of rotations

around x, y and z axis. The terms usually used in aviation for these rotations are yaw, pitch and roll.

• If initial position of a point is represented by the vector , its new position, after rotation will be . These two positions are connected by using rotation matrix .

, ,x y z

, ,r r rx y z

x

y

z

yaw

pitch

roll

rR

( )r

r r

r

x x

y R y

z z

- angle of rotation

3D Rotation matrix

cos sin 0

( ) sin cos 0

0 0 1zR

cos 0 sin

0 1 0

sin 0 cosyR

1 0 0

0 cos sin

0 sin cosxR

, , z y zR R R R

yaw pitch roll

Difference between image of the ship visible from the radar position, and image obtained based on

the 2D FT of received radar signal

x1=x

z1 y1

Radarposition

Va

y

z H

Virtual Instrument – Visibility I

Position of the ship

defined by pitch,

yaw, and roll angle.

Position of the radar.

Virtual Instrument – Visibility II

Position of the ship

defined by pitch,

yaw, and roll angle.

Position of the radar.

Radar is at low altitude

After determining which points are visible, they are divided according to the geometrical shape of the part of the ship they belong to.

Virtual Instrument – Polarimetry

Four radar image are obtained by using polarimetric radar

Pitch angle is changed by

sinusoidal law with

frequency 0.5Hz and

amplitude 15 degrees.

Virtual Instrument – Rotation

Although pitch angle is

very small there exist

visible distortions in the

observed SAR image.

This effect is very

important if we want to

estimate ship dimensions.

THANK YOU FOR YOUR ATTENTION