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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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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:
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Detection of Signal in Noise
Threshold Detection :
A-scope presentation(amplitude versus time or range)
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Probability Density Functions (PDF)
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
BTT
tP
faN
kk
N
kk
fa
1
1
1
0
2
2exp
1
T
fa
V
BT
Probability of False Alarm
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Probability of False Alarm
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Integration of Radar Pulses
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Integration of Radar Pulses
4
1
102max
)/(4
)(
NSBFkT
nEnAGPR
n
iet
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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, …
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
Sphere :
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
Aircraft :
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
2
3
2
1
52 Df
Ships:
f frequency (MHz)
D ship displacement (kiloton)
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
Ships:
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
Missiles:
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
The Radar Cross Section
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Radar Cross-Section Fluctuations
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Radar Cross-Section Fluctuations
)2sin()2sin()(1
tfAtfatsN
iiir
target)(1 ts
)(2 ts
)(3 ts)(tsN
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Radar Cross-Section Fluctuations
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Radar Cross-Section Fluctuations
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Radar Cross-Section Fluctuations
Radar Cross-Section Loss in radar equation:
4
1
1
102
max
)/(4
)(
en
fn
iet
LNSBFkT
nEnAGPR
Fluctuations loss : enff LL
1
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Radar Cross-Section Fluctuations
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.
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
PRF (Pulse Repetition Frequency)
Pun f
cTcR 2
unP R
cf
2
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Beam Shape :
Antenna Parameters
),( G
.deg31BB or .deg1,.deg30 BB Pencil Beam : Fan Beam :Typical values Typical values
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
Cosecant Squared Beam:
Antenna Parameters
INTRODUCTION TO RADAR SYSTEMS , Merrill I. Skolnik , Third Edition Chapter 2
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.