Radar Systems Engineering
Lecture 4
The Radar Equation
Dr. Robert M. ODonnell
IEEE New Hampshire Section
Guest Lecturer
IEEE New Hampshire Section
Radar Systems Course 1
Radar Equation 1/1/2010
IEEE AES Society
Block Diagram of Radar System
Transmitter
Propagation
Medium
Target
Radar
Cross
Section
Power
Amplifier
Waveform
Generation
T/R
Switch
Antenna
Receiver
Signal Processor Computer
A/D
Converter
Pulse
Compression
Clutter Rejection
(Doppler Filtering)
User Displays and Radar Control
General Purpose Computer
Tracking
Data
Recording
Photo Image
Courtesy of US Air Force
Used with permission.
Radar Systems Course 2
Radar Equation 1/1/2010
Parameter
Estimation
Thresholding
Detection
The Radar Range Equation Connects:
1. Target Properties - e.g. Target Reflectivity (radar cross section)
2. Radar Characteristics - e.g. Transmitter Power, Antenna Aperture
3. Distance between Target and Radar - e.g. Range
4. Properties of the Medium - e.g. Atmospheric Attenuation.
IEEE New Hampshire Section
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Outline
Radar Systems Course 3
Radar Equation 1/1/2010
Introduction
Introduction to Radar Equation
Surveillance Form of Radar Equation
Radar Equation for Rain Clutter
Radar Losses
Examples
Summary
IEEE New Hampshire Section
IEEE AES Society
Key Radar Functions
Detection
Illuminate selected area with enough energy to detect targets of
interest
Measure target observables
Measure range, Doppler and angular position of detected
targets
Track
Correlate successive target detections as coming from same
object and refine state vector of target
Identification
Determine what target is - Is it a threat ?
Handover
Pass the target on to;
Missile interceptor
Data Collection function
Air Traffic Controller / Operator
Radar Systems Course 4
Radar Equation 1/1/2010
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Radar Range Equation
Power density from
uniformly radiating antenna
transmitting spherical wave
Pt
4 R2
Pt = peak transmitter
power
R = distance from radar
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Used with Permission
Radar Systems Course 5
Radar Equation 1/1/2010
IEEE New Hampshire Section
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Radar Range Equation (continued)
Power density from
isotropic antenna
Power density from
directive antenna
Gain is the radiation intensity of
the antenna in a given direction
over that of an isotropic
(uniformly radiating) source
4 A
Gt = 2
Radar Systems Course 6
Radar Equation 1/1/2010
Pt
4 R2
Pt
Pt G t
4 R2
= peak transmitter
power
= distance from radar
Gt
= transmit gain
.
Courtesy of MIT Lincoln Laboratory
Used with Permission
IEEE New Hampshire Section
IEEE AES Society
Definition of Radar Cross Section (RCS or s)
R
Incident Energy
Radar
Antenna
Target
Reflected Energy
Radar Cross Section (RCS or ) is a measure of the
energy that a radar target intercepts and scatters
back toward the radar
Power of reflected
signal at target
Power density of reflected
signal at the radar
Pt G t
4 R2
Pt G t
4 R2 4 R2
= radar cross section
units (meters)2
Power density of
reflected signal falls
off as (1/R2 )
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Used with Permission
Radar Systems Course 7
Radar Equation 1/1/2010
IEEE New Hampshire Section
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Radar Range Equation (continued)
Power density of reflected
signal at radar
Pt G t
4 R2 4 R2
R
Radar
Antenna
Target
Reflected Energy
The received power = the power density at the radar times the
area of the receiving antenna
Power of reflected
signal from target and
received by radar
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Used with Permission
Radar Systems Course 8
Radar Equation 1/1/2010
Pt G t A e
Pr =
4 R2 4 R2
Pr
= power received
Ae
= effective area of
receiving antenna
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Sources of Noise Received by Radar
The total effect of the
different noise sources is
represented by a single
noise source at the
antenna output terminal.
Solar
Noise
Galactic Noise
Atmospheric
Noise
The noise power at the
receiver is : N = k Bn Ts
Receiver
Ground Noise
Transmitter
(Receiver, waveguide, and duplexer noise)
Noise from Many Sources Competes
with the Target Echo
Radar Systems Course 9
Radar Equation 1/1/2010
Man Made
Interference
Radio Stations, Radars, etc)
k = Boltzmann constant
= 1.38 x 10-23 joules / deg oK
Ts= System Noise
Temperature
Bn = Noise bandwidth of
receiver
IEEE New Hampshire Section
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Radar Range Equation (continued)
Pt G t A e
Pr =
4 R2 4 R2
Signal Power reflected
from target and
received by radar
Average Noise Power
N = k B n Ts
Signal to Noise Ratio
S Pr
=
N N
Courtesy of MIT Lincoln Laboratory
Used with Permission
Assumptions :
G = Gr = Gt
L = Total System
Losses
To= 290o K
Pt G 2 2
S
=
N (4 ) 3 R 4 k Ts B n L
Signal to Noise Ratio (S/N or SNR) is the standard measure of a
radars ability to detect a given target at a given range from the radar
S/N = 13 dB on a 1 m2 target at a range of 1000 km
radar cross section
of target
Radar Systems Course 10
Radar Equation 1/1/2010
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System Noise Temperature
The System Noise Temperature, Ts ,is divided into 3 components :
Ts = Ta + Tr + L r Te
Where:
Ta is the contribution from the antenna
Tr is the contribution from the RF components
between the antenna and the receiver
L r is loss of the input RF components (natural units)
Te is temperature of the receiver
The 3 temperature components can be broken down further :
Ta = (0.88 Tsky 254) / (L a + 290)
T = T (L 1)
and
T = T (F 1)
r
tr
r
e
o
n
Where:
Tsky is the apparent temperature of the sky (from graph)
L a is the dissipative loss within the antenna (natural units)
Ttr is physical temperature of the RF components
Fn is the noise factor of the receiver (natural units)
To is the reference temperature of 290o K
Note that all temperature quantities are in units of oK
Radar Systems Course 11
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Noise Temperature vs. Frequency
Sky Noise Temperature (K)
10,000
1,000
Elevation Angle
5
10
10
1
100
100
1,000
10,000
Frequency (MHz)
100,000
The data on this graph takes into account the following effects:
Galactic noise, cosmic blackbody radiation, solar noise, and
atmospheric noise due to the troposphere
(Adapted from Blake, Reference 5, p 170)
Radar Systems Course 12
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Outline
Radar Systems Course 13
Radar Equation 1/1/2010
Introduction
Introduction to Radar Equation
Surveillance Form of Radar Equation
Radar Equation for Rain Clutter
Radar Losses
Examples
Summary
IEEE New Hampshire Section
IEEE AES Society
Track Radar Equation
Track Radar Equation
Pt G 2 2
S
=
N (4 ) 3 R 4 k Ts B n L
When the location of a
target is known and the
antenna is pointed toward
the target.
Track Example
Courtesy of MIT Lincoln Laboratory
Used with Permission
Radar Systems Course 14
Radar Equation 1/1/2010
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Development of Search Radar Equation
Search Radar Equation
Track Radar Equation
Pav A e t s
S
=
N 4 R 4 k Ts L
Pt G 2 2
S
=
N (4 ) 3 R 4 k Ts B n L
When the location of a
target is known and the
antenna is pointed toward
the target.
When the targets location is
unknown, and the radar has to
search a large angular region
to find it.
Search Volume
Track Example
Where:
Pav = average power
= solid angle searched
t s = scan time for
A e = antenna area
Radar Systems Course 15
Radar Equation 1/1/2010
Search Example
Courtesy of MIT Lincoln Laboratory
Used with Permission
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Search Radar Range Equation
Pav A e t s
S
=
N 4 R 4 k Ts L
Re-write as:
f (design parameters) = g (performance parameters)
Angular coverage
Range coverage
S
4 R
Pav A e
N
=
k Ts L
ts
4
Measurement quality
Time required
Target size
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Used with Permission
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Scaling of Radar Equation
Pav A e t s
S
=
N 4 R 4 k Ts L
4 R 4 k Ts L (S / N )
Pav =
Ae t s
Power required is:
Independent of wavelength
A very strong function of R
A linear function of everything else
Example
Radar Can Perform Search at 1000 km Range
How Might It Be Modified to Work at 2000 km ?
Solutions Increasing R by 3 dB (x 2) Can Be Achieved by:
1.
Courtesy of MIT Lincoln Laboratory
Used with Permission
Radar Systems Course 17
Radar Equation 1/1/2010
Increasing Pav by 12 dB (x 16)
or 2.
Increasing Diameter by 6 dB ( A e by 12 dB)
or 3.
Increasing t s by 12 dB
or 4.
Decreasing by 12 dB
or 5.
Increasing by 12 dB
or 6.
An Appropriate Combination of the Above
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Search Radar Performance
100 K
R = 3000 km
R = 1000 km
R = 300 km
Average Power (W)
10 K
ASR- 9
Airport Surveillance Radar
ARSR- 4
WSR-88D/NEXRAD
R = 100 km
1K
ASR- 9
TDWR
Search 1 sr
In 10 sec for
1 sq m Target
S/N = 15 dB
Loss = 10 dB
T = 500 deg
100
R = 30 km
10
Courtesy of MIT Lincoln Laboratory.
Used with permission.
ASDE- 3
1
0.1
R = 10 km
Radar Systems Course 18
Radar Equation 1/1/2010
(Equivalent)
10
100
Antenna Diameter (m)
Courtesy of MIT Lincoln Laboratory
Used with Permission
IEEE New Hampshire Section
IEEE AES Society
Search Radar Performance
100 K
R = 300 km
Average Power (W)
10 K
ASDE- 3
Airport Surface Detection
Equipment
R = 3000 km
R = 1000 km
ARSR- 4
WSR-88D/NEXRAD
R = 100 km
1K
ASR- 9
TDWR
Search 1 sr
In 10 sec for
1 sq m Target
S/N = 15 dB
Loss = 10 dB
T = 500 deg
100
R = 30 km
10
Courtesy Target Corporation
ASDE- 3
1
0.1
R = 10 km
Radar Systems Course 19
Radar Equation 1/1/2010
(Equivalent)
10
100
Antenna Diameter (m)
Courtesy of MIT Lincoln Laboratory
Used with Permission
IEEE New Hampshire Section
IEEE AES Society
Search Radar Performance
ARSR- 4
Air Route Surveillance Radar
100 K
R = 3000 km
R = 1000 km
R = 300 km
Average Power (W)
10 K
ARSR- 4
WSR-88D/NEXRAD
R = 100 km
1K
ASR- 9
TDWR
Search 1 sr
In 10 sec for
1 sq m Target
S/N = 15 dB
Loss = 10 dB
T = 500 deg
100
R = 30 km
10
ARSR- 4 Antenna
(without Radome)
ASDE- 3
1
0.1
R = 10 km
(Equivalent)
Courtesy of MIT Lincoln Laboratory
Used with Permission
Radar Systems Course 20
Radar Equation 1/1/2010
10
100
Antenna Diameter (m)
Courtesy of Northrop Grumman.
Used with permission.
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Search Radar Performance
100 K
R = 3000 km
R = 1000 km
R = 300 km
Average Power (W)
10 K
WSR-88D / NEXRAD
ARSR- 4
WSR-88D/NEXRAD
R = 100 km
1K
ASR- 9
TDWR
Search 1 sr
In 10 sec for
1 sq m Target
S/N = 15 dB
Loss = 10 dB
T = 500 deg
100
R = 30 km
10
ASDE- 3
Courtesy of NOAA.
1
0.1
R = 10 km
Radar Systems Course 21
Radar Equation 1/1/2010
(Equivalent)
10
100
Antenna Diameter (m)
Courtesy of MIT Lincoln Laboratory
Used with Permission
IEEE New Hampshire Section
IEEE AES Society
Search Radar Performance
100 K
TDWR
R = 3000 km Terminal Doppler Weather Radar
R = 1000 km
R = 300 km
Average Power (W)
10 K
ARSR- 4
WSR-88D/NEXRAD
R = 100 km
1K
ASR- 9
TDWR
Search 1 sr
In 10 sec for
1 sq m Target
S/N = 15 dB
Loss = 10 dB
T = 500 deg
100
R = 30 km
10
ASDE- 3
1
0.1
R = 10 km
1
(Equivalent)
10
100
Antenna Diameter (m)
Courtesy of Raytheon.
Courtesy of MIT Lincoln Laboratory
Used with Permission
Radar Systems Course 22
Radar Equation 1/1/2010
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Outline
Radar Systems Course 23
Radar Equation 1/1/2010
Introduction
Introduction to Radar Equation
Surveillance Form of Radar Equation
Radar Equation for Rain Clutter
Radar Losses
Example
Summary
IEEE New Hampshire Section
IEEE AES Society
Radar Equation for Rain Clutter
(and other Volume Distributed Targets)
Pt G 2 2
S
=
N (4 ) 3 R 4 k Ts B n L
Standard radar equation
If the target is a diffuse scatterer (e.g. rain), which
completely fills the range-azimuth-elevation cell of the
radar, then the radar cross section of the target takes the
form:
c 1
(
)(
)
= V and V =
R B R B
4
2 2 ln e 2
And the radar equation becomes:
Pt G c
S
=
N 1024 (ln e 2)R 2 k Ts B n L
2
Note, for
Gaussian
antenna
pattern
2
G
B B
Note, that volume distributed backscatter is a function of 1 / R 2
rather than the usual 1 / R 4
Radar Systems Course 24
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Outline
Radar Systems Course 25
Radar Equation 1/1/2010
Introduction
Introduction to Radar Equation
Surveillance Form of Radar Equation
Radar Equation for Rain Clutter
Radar Losses
Examples
Summary
IEEE New Hampshire Section
IEEE AES Society
System Loss Terms in the Radar Equation
Transmit Losses
Radome
Circulator
Waveguide Feed
Waveguide
Antenna Efficiency
Beam Shape
Low Pass Filters
Rotary Joints
Scanning
Atmospheric
Quantization
Field Degradation
Radar Systems Course 26
Radar Equation 1/1/2010
Receive Losses
Radome
Circulator
Waveguide Feed
Waveguide
Combiner
Receiver Protector
Rotary Joints
Transmit / Receive Switch
Antenna Efficiency
Beam Shape
Scanning
Doppler Straddling
Range Straddling
Weighting
Non-Ideal Filter
CFAR
Quantization
Atmospheric
Field Degradation
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Major Loss Terms in Radar Equation
Beam Shape Loss
Radar return from target with scanning radar is
modulated by shape of antenna beam as it scans
across target. Can be 2 to 4 dB
Scanning Antenna Loss
For phased array antenna, gain of beam less than that
on boresite
Inputs to System Noise Temperature
Noise received by antenna
Local RF noise
Galactic noise
Receiver noise factor
Receive waveguide losses
Antenna ohmic losses
Radar Systems Course 27
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Nature of Beam Shape Loss
Location of Pulses
Radar Equation assumes n
pulses are integrated, all
with gain G.
Antenna
Main
Beam
Except for the pulse at the
center of the beam, the
actual pulses illuminate the
target with a gain less than
the maximum.
Radar Systems Course 28
Radar Equation 1/1/2010
(Adapted from Skolnik, Reference 1, p 82)
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Major Loss Terms in Radar Equation
Waveguide and Microwave Losses
Transmit waveguide losses (including feed, etc)
Rotary joints, circulator, duplexer
Signal Processing Loss
Range and Doppler Weighting
A /D Quantization Losses
Adaptive thresholding (CFAR) Loss
Range straddling Loss
Lens Effect Loss
Refraction in atmosphere causes spreading of beam and
thus degradation in S/N
Atmospheric Attenuation Loss
Attenuation as radar beam travels through atmosphere
(2 way loss)
Radar Systems Course 29
Radar Equation 1/1/2010
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Rectangular Waveguide Attenuation
Frequency
Band
Frequency Range
Attenuation- Lowest to
of Dominant TE10 Mode (GHz) Highest Frequency (dB/100 ft)
UHF
0.35 - 0.53
0.054 - 0.034
L Band
0.96 - 1.44
0.20 - 0.135
S Band
2.6 - 3.95
1.10 - 0.75
C Band
3.95 - 5.85
2.07 - 1.44
X Band
8.2 - 12.40
6.42 - 4.45
Ku Band
12.4 - 18.0
9.58 - 8.04
26.5 - 40.0
21.9 - 15.0
Aluminum
Brass
Ka Band
Silver
Clad
Copper
(Adapted from Volakis, Reference 7, pp 51-40 to 51- 41)
Radar Systems Course 30
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Lens Loss vs. Range
Lens Loss for Point Targets (dB)
The gradient of atmospheric refraction at lower elevation angles,
causes a spreading of the radar beam, and thus a small
diminishment radar power
This lens loss is frequency independent
It is significant only for targets that are at long range.
3.0
0.0
Elevation
Angle
2.0
1.0
1.0
2.0
4.0
0.0
30
Radar Systems Course 31
Radar Equation 1/1/2010
0.5
100
300
Slant Range (nmi)
1000
8.0
3000
(Adapted from Blake, Reference 5, p 192)
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Loss Due to Atmospheric Attenuation
Attenuation vs. Range to Target
Attenuation vs. Frequency
Two way Attenuation to Target (dB)
Two way Attenuation through
Entire Troposphere (dB)
100
10
Elevation
Angle 1
5
10
0.1
100
1,000
10,000
Radar Frequency (MHz)
0,1,5,30 deg
Radar Systems Course 32
Radar Equation 1/1/2010
(X-Band 10 GHz)
100,000
8
Elevation Angle
0.0
6
0.5
4
1.0
2.0
2
5.0
10.0
0
50
100
150
200
250
300
350
Radar to Target Distance (nmi.)
(Adapted from Blake, see Reference 5, p 192)
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Major Loss Terms in Radar Equation
Bandwidth Correction Factor
Receiver not exact matched filter for transmitted pulse
Integration Loss
Non coherent integration of pulses not as efficient as
coherent integration
Fluctuation Loss
Target return fluctuates as aspect angle changes relative
to radar
Margin (Field Degradation) Loss
Characteristics of radar deteriorates over time (~3 dB
not unreasonable to expect over time)
Water in transmission lines
Weak or poorly tuned transmitter tubes
Deterioration in receiver noise figure
Radar Systems Course 33
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Outline
Radar Systems Course 34
Radar Equation 1/1/2010
Introduction
Introduction to Radar Equation
Surveillance Form of Radar Equation
Radar Equation for Rain Clutter
Radar Losses
Examples
Summary
IEEE New Hampshire Section
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Radar Equation - Examples
Airport Surveillance Radar
0 th order
Back of the envelope
Range Instrumentation Radar
A more detailed calculation
Radar Systems Course 35
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Example - Airport Surveillance Radar
Problem : Show that a radar with the parameters listed
below, will get a reasonable S / N on an small aircraft at 60
nmi.
Radar Parameters
Range
Aircraft cross section
Peak Power
Duty Cycle
Pulsewidth
Bandwidth
Frequency
Antenna Rotation Rare
Pulse Repetition Rate
Antenna Size
Azimuth Beamwidth
System Noise Temp.
Radar Systems Course 36
Radar Equation 1/1/2010
60 nmi
1 m2
1.4 Megawatts
0.000525
.6 microseconds
1.67 MHz
2800 MHz
12.7 RPM
1200 Hz
4.9 m wide by
2.7 m high
1.35 o
950 o K
= c/f
4 A
G= 2
= .103 m
= 15670
= 42 dB, (actually 33 dB
with beam shaping losses)
Number of pulses per
beamwidth = 21
Assume Losses = 8dB
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Example - Airport Surveillance Radar
Pt G 2 2
S
=
N (4 ) 3 R 4 k Ts B n L
Pt = 1.4 Megawatts
G = 33 dB = 2000
= .1 m
= 1 m2
k = 1.38 x 10 -23 w / Hz o K
R = 111, 000 m
Ts = 950 o K
B n = 1.67 MHz
L = 8dB = 6.3
(4 ) 3 = 1984
(1.4 x 106 w )(2000)(2000)(.1m)(.1m)(1m2)
(1984 ) (1.11 X 105 m)4 (1.38 x 10 -23 w / Hz o K) (950 o K ) (6.3) (1.67 x 106 Hz)
5.6 x 10+6+3+3-1-1
415 x 10+3+20-23+2+6
5.6 x 10+10
4.15 x 10+2+3+20-23+2+6
5.6 x 10+10
4.15 x 10+10
= 1.35 = 1.3 dB
S / N = 1.3 dB per pulse (21 pulses integrated) => S / N per dwell = 14.5 dB
Courtesy of MIT Lincoln Laboratory
+ 13.2 dB
Used with Permission
Radar Systems Course 37
Radar Equation 1/1/2010
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Example - Airport Surveillance Radar
dB Method
(+)
Peak Power
(Gain) 2
(Wavelength ) 2
Cross section
( 4 ) 3
(Range )4
k
System Temp
Losses
Bandwidth
1.4 MW
33 db
.1 m
1 m2
1984
111 km
1.38 x 10 -23 w / Hz o K
950
8 dB
1.67 MHz
(-)
61.5
66
20
0
33
201.8
228.6
29.8
8
62.2
+ 356.1
- 354.8
+ 1.3 dB
S / N = 1.3 dB per pulse (21 pulses integrated) => S / N per dwell = 14.5 dB
Courtesy of MIT Lincoln Laboratory
( + 13.2 dB)
Used with Permission
Radar Systems Course 38
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Example # 2 Range Instrumentation Radar
Problem : For a C-band pulsed radar with a 6.5 m dish antenna
and 1,000 kW of peak power (0.1% duty cycle), what is the
maximum detection range on a target with 0 dBsm cross
section, a Pd of .9, and Pfa of 10-6 (Assume a Swerling Case 1
target fluctuations and a 1 sec pulse) ?
Radar Parameters
Maximum Detection Range
Probability of Detection
Probability of False Alarm
Target Cross Section
Target Fluctuations
Peak Power
Duty Cycle
Pulsewidth
Frequency
Antenna Size
Number of Pulses Integrated
Radar Systems Course 39
Radar Equation 1/1/2010
?? km
.9
10 -6
0 dBsm ( 1 m2 )
Swerling Case 1
1,000 Kilowatts
0.1 %
1 microsecond
5,500 MHz
6.5 m dish
50
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Detection Statistics for Swerling Case 1
(Probability of Detection = 0.9)
For Coherent Integration
Signal-to-Noise Ratio (dB)
25
S
= 21.2 dB
N
S
S
= N Pulsles
N TOTAL
N PER PULSE
20
S
= 21.2 17.0 = 4.2 dB
N PER PULSE
15
Pfa = 10-6
10
10
20
50
100
200
500 1,000
5,000 10,000
Number of Pulses Non-Coherently Integrated
(Adapted from Blake in Skolnik, see Reference 4, p 192)
Radar Systems Course 40
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Radar Equation Example #2
Radar and Target Parameters Inputs
Peak Power (kilowatts)
Pulse Duration (microseconds)
Noise Bandwidth (MHz)
Transmit Antenna Gain (dB)
Receive Antenna Gain (dB)
Frequency (GHz)
Wavelength (meters)
Single Pulse Signal to Noise Ratio
Target Radar Cross Section (meters)2
k - Boltzmann's Constant 1.38 x 10-23 (w / Hz K )
(4)3
System Noise Temperature ( K )
Total System Losses
Range (kilometers)
Natural Units
1,000
1.0
1.0
5.5
5.45
1.0
598.2
(dB)
60.0
- 60.0
60.0
49.6
49.6
- 25.3
4.2
0.0
- 228.6
33.0
27.8
9.0
519
Antenna
Efficiency
65 %
Diameter (meters) 6
Gain (dB)
49.6
Radar Systems Course 41
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Radar Equation # 2
System Losses
System Losses
(dB)
Bandwidth Correction Factor (dB)
Transmit Loss (dB)
Scanning Antenna Pattern Loss (dB)
Signal Processing Losses (dB)
Atmospheric Attenuation Loss (dB)
Lens Effect Loss (dB)
Integration Loss (dB)
Target Fluctuation Loss (dB)
Margin / Field Degradation Loss (dB)
Total Loss Budget (dB)
0.70
1.30
0.00
1.90
1.80
0.25
0.00
0.00
3.00
8.95
Transmit Losses (dB)
Circulator (dB)
0.40
Switches (dB)
0.40
Transmission Line0.50
1.30
Signal Processing Losses (dB)
Loss Input to System Noise Temperature
Receiver Noise Factor (dB)
Antenna Ohmic Loss (dB)
Receive Transmission Line loss (dB)
Sky Temperature (K)
C-Band at 3
Threshold Loss (dB)
A/D Quantization Loss (dB)
Range Straddling Loss
Weighting Loss
4.00
0.20
0.40
50.00
0.50
0.10
0.20
1.10
1.90
Ts = Ta + Tr + L r Te = 598.2o K
Ta = (0.88 Tsky 254) / (L a + 290)
Tr = Ttr (L r 1)
Radar Systems Course 42
Radar Equation 1/1/2010
and
Te = To (Fn 1)
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Outline
Radar Systems Course 43
Radar Equation 1/1/2010
Introduction
Introduction to Radar Equation
Surveillance Form of Radar Equation
Radar Equation for Rain Clutter
Radar Losses
Examples
Summary
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Cautions in Using the Radar Equation (1)
The radar equation is simple enough, that just about
anyone can learn to use and understand
Unfortunately, the radar equation is complicated enough
that anyone can mess it up, particularly if you are not
careful
See next viewgraph for relevant advice
Radar Systems Course 44
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IEEE New Hampshire Section
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Cautions in Using the Radar Equation (2)
The Sanity Check
Take a Candidate Radar Equation
Check it Dimensionally
R and are distance
is distance squared
Pt is energy / time
S / N , G , and L are dimensionless
k Ts is energy
B n is (time)-1
Check if Dependencies Make Sense
Pt G 2 2
S
=
N (4 ) 3 R 4 k Ts B n L
Increasing Range and S/N make requirements tougher
Increasing Power and Antenna Gain make radar more capable
Decreasing Wavelength and Radar Cross Section make the
radar less capable
Radar Systems Course 45
Radar Equation 1/1/2010
IEEE New Hampshire Section
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Radar Equation and the Detection Process
Target Fluctuation Statistics
Swerling Model 1, 2, 3, or 4
Other
Radar Parameters
Transmitter Power
Antenna Gain
Frequency
Pulse Width
Waveform
Type of Detection
Linear
Square Law
Range
Radar to Target
Target Characteristics
Cross Section vs.
Angle and Frequency
Radar
Equation
Properties of
Propagation Medium
Attenuation vs. Frequency
Rain Requirements
Radar Systems Course 46
Radar Equation 1/1/2010
PD
Probability
Of
Detecting Target
Signal to Noise
Ratio (S/N)
Detection
Probability
Of
False Alarm
(Detecting Noise)
Detection Threshold
Constant
Adaptive
Noise Statistics
Gaussian
Other
PFA
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Summary
The radar equation relates:
Radar performance parameters - Detection range, S/N, etc.
and
Radar design parameters - Transmitter power, antenna size,
etc.
There are different forms of the radar equations for different
radar functions
Search, Track
Scaling of the radar equation allows the radar designer to
understand how the radar parameters may change to
accommodate changing requirements
Be careful, if the radar equation leads to unexpected results
Do a sanity check !
Look for hidden variables or constraints
Compare parameters with those of a real radar
Radar Systems Course 47
Radar Equation 1/1/2010
IEEE New Hampshire Section
IEEE AES Society
References
1. Skolnik, M., Introduction to Radar Systems, McGraw-Hill,
New York, 3rd Ed., 2001
2. Barton, D. K., Modern Radar System Analysis, Norwood,
Mass., Artech House, 1988
3. Skolnik, M., Editor in Chief, Radar Handbook, New York,
McGraw-Hill, 3rd Ed., 2008
4. Skolnik, M., Editor in Chief, Radar Handbook, New York,
McGraw-Hill, 2nd Ed., 1990
5. Blake, L. M., Radar Range Performance Analysis, Silver
Spring, Maryland, Munro, 1991
6. Nathanson, F. E., Radar Design Principles, New York,
McGraw-Hill, 1st Ed., 1991
7. Volakis, J. L., Antenna Engineering Handbook, McGraw-Hill,
New York, 4th Ed., 2007
Radar Systems Course 48
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Contributors
Dr Stephen D. Weiner
Dr. Claude F. Noiseux
Radar Systems Course 49
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Homework Problems
From Reference 1, Skolnik, M., Introduction to Radar
Systems, 3rd Edition, 2001
Radar Systems Course 50
Radar Equation 1/1/2010
Problem 1-5
Problem 1-6
Problem 2-22
Problem 2-24
Problem 2-25
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