INTRODUCTION TO RADAR SYSTEMS, Merrill I. Skolnik, Third EditionChapter 2 INTRODUCTION TO RADAR...

INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2

INTRODUCTION TO RADAR SYSTEMS

Chapter 2 :

The Radar Equation

Third Edition By : Merrill I. Skolnik


The Simple form of Radar Equation

),,,,(

4min

4

1

min2max SAGPfS

AGPR et

et

is radar cross section

G is gain of antenna

eA is effective aperture of antenna

tP is peak power

minS Minimum of detectable signal by receiver

Under control of Radar designer Target parameter


The failure of Simple Form

1. Statistical nature of Smin (determined by reciever noise).

2. Fluctuation of radar cross section.3. Losses.4. Propagation effects (earth‘s surface, weather and atmosphere)

),,,,,,( minmax fadet PPSAGPfR

The probability of detection : PdTherefore:

The probability of false alarm : Pfa must be considered. and

This mean:


Detection of Signal in Noise

Threshold Detection :

A-scope presentation(amplitude versus time or range)


Receiver Noise

NBTkThermal Noise Power:

out

out

in

in

an

outn

N

SNS

GBTk

N

Tatreceiveridealofoutnoise

receiverpracticalofoutnoiseF

00

out

outnnin N

SFBTkS 0

min

0min

out

outn N

SBFTkS

min02

4max

/4 NSBFTk

AGPR

n

et

out

out

in

in

n

NSN

S

F


Probability Density Functions (PDF)


Envelope Detector:

Probability of False Alarm

Probability of Noise Voltage in input of IF stage is Gaussian PDF :

0

2

0 2exp

2

1)(

n

n

vvp 2

0 nv

Mr. Rice has shown that the noise in output of IF is Rayleigh :

0

2

0 2exp)(

RR

Rp

0

2

0

2

0 2exp

2exp)(

T

VT

VdR

RRRVp

T

Probability of False Alarm :

0

2

2exp

T

fa

VP


BTT

tP

faN

kk

N

kk

fa

1

1

1

0

2

2exp

1

T

fa

V

BT



Probability of Detection

Rice Probability density function )(2

exp)(0

00

22

0 RA

IARR

Rps

)(

00 RA

I Zero-order Modified Bessel Function

TVTd dR

RAI

ARRRVpP )(

2exp)(

00

0

22

0


Integration of Radar Pulses

B

r

x

?

360sec60

r is Rotate Per Minute (rpm) of Antenna

pf is Pulse Repetition frequency (PRF) of Radar

B is Half Power Beam Width (HPBW) of Antenna

n is number of pulse per scan ( hit per scan )

r

B

r

Bx

6360

60

sec

x is time on target

?6

sec1

n

f

r

B

p

P

r

B fn6



Ni NSn

NSnE

)/(

)/()( 1Integration Efficiency :

)()( nEnnI ii Integration Improvement Factor :

Types of Integration:

1. Pre-detection integration (Coherent integration ).2. Post-detection integration (Non-coherent integration ).

1. Pre-detection integration needs to phase of pulses.2. Post-detection integration don’t need to phase of pulses

)(

1log10)(

nEnL

ii Integration Loss: J. I. Marcum, 1954, Rand Corporation report



4

1

102max

)/(4

)(

NSBFkT

nEnAGPR

n

iet


The Radar Cross Section

2

2

244densitypower Incident

angle solidunit source rdpower towa Reflected

i

r

E

ER

Types of Targets:

• Simple targets such as: Sphere, Cylinder, Flat plate, Rod, Ogive and cone• Complex targets such as: Aircraft, Ship, Building, …



Sphere :



Aircraft :



2

3

2

1

52 Df

Ships:

f frequency (MHz)

D ship displacement (kiloton)



Ships:



Missiles:


Radar Cross-Section Fluctuations



)2sin()2sin()(1

tfAtfatsN

iiir

target)(1 ts

)(2 ts

)(3 ts)(tsN



Swerling Target Models:

Case 1:Scan to Scan Fluctuation (Rayleigh scattering ) or slow Fluctuation with PDF:

0)exp(1

)(

avav

p

Case 2: Pulse to Pulse Fluctuation or Fast Fluctuation with same PDF of case 1:

Case 3: Scan to Scan Fluctuation with PDF:

Case 4: Pulse to Pulse Fluctuation or Fast Fluctuation with same PDF of case 3:

Case 0: No-Fluctuation in radar cross section occurs.

Swerling assume that target is very big and contain many small targets

0)2

exp(4

)(2

avav

p



Radar Cross-Section Loss in radar equation:

4

1

1

102

max

)/(4

)(

en

fn

iet

LNSBFkT

nEnAGPR

Fluctuations loss : enff LL

1



Decorrelation by frequency Diversity & Agility :

Frequency Diversity:

Frequency Agility:

Multiple TX/RX in different frequency is used. Example is air traffic control radar for reliability of detection.

Pulse to Pulse change in radar frequency by a wide band TX.It don’t used for MTI radars.


PRF (Pulse Repetition Frequency)

Pun f

cTcR 2

unP R

cf

2


Antenna Parameters

AAG ae

22

44Gain :

BBdB

BB

GG

26000log10

26000

vB D

65h

B D

65

or

is vertical beam width is horizontal beam width

4

),(antennabyacceptedpower

anglesolidunitdirectionandinradiatedpower

G

),(),( DG A

D is directivity of antenna

4

),(antennabyradiatedPower

anglesolidunitdirectionandinradiatedPower

D


Beam Shape :

Antenna Parameters

),( G

.deg31BB or .deg1,.deg30 BB Pencil Beam : Fan Beam :Typical values Typical values


Cosecant Squared Beam:

Antenna Parameters

20 csc)( GG

4

4

143

22 csc

4 Rk

R

GPP tr

h

Rcsc41

h

kPr

2arg1arg etTretTr PP

1T 2T

h h

1R 2R

Antenna Pattern

Range

Height

TX/RX


Cosecant Squared Beam:

Antenna Parameters


Antenna Parameters

Revisit Time : Scan rime of antenna

Rotate Per Minute (RPM) r

r 1. Number of pulse for suitable signal to noise.2. Target speed.

is depending to :

rPractical value for is :

• 5-6 rpm for long range air traffic control radars.

• 10-15 rpm for long range military radars.

• 30-60 rpm for high speed targets.

INTRODUCTION TO RADAR SYSTEMS, Merrill I. Skolnik, Third EditionChapter 2 INTRODUCTION TO RADAR...

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