1 radar basic - part ii

RADAR Basics
Part II
SOLO HERMELIN
Updated: 27.01.09Run This
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Table of Content
SOLO
Radar Basics
Basic Radar Concepts
The Physics of Radio Waves
Maxwell’s Equations:
Properties of Electro-Magnetic Waves
Polarization
Energy and Momentum
The Electromagnetic Spectrum
Introduction to Radars
Dipole Antenna Radiation
Interaction of Electromagnetic Waves with Material
Absorption and Emission
Reflection and Refraction at a Boundary Interface
Diffraction
Atmospheric Effects
RADA
BASI

Table of Content (continue – 1)
SOLO
Radar Basics
Basic Radar Measurements
Radar Configurations
Range & Doppler Measurements in RADAR Systems
Waveform Hierarchy
Fourier Transform of a Signal
Continuous Wave Radar (CW Radar)
Basic CW Radar
Frequency Modulated Continuous Wave (FMCW)
Linear Sawtooth Frequency Modulated Continuous Wave
Linear Triangular Frequency Modulated Continuous Wave
Sinusoidal Frequency Modulated Continuous Wave
Multiple Frequency CW Radar (MFCW)
Phase Modulated Continuous Wave (PMCW)
RADA
BASI

SOLO
Radar Basics
Non-Coherent Pulse Radar
Pulse Radars
Coherent Pulse-Doppler Radar
Range & Doppler Measurements in Pulse-Radar Systems
Range Measurements
Range Measurement Unambiguity
Doppler Frequency Shift
Resolving Doppler Measurement Ambiguity
Resolution
Doppler Resolution
Angle Resolution
Range Resolution
RADA
BASI

SOLO
Radar Basics
Pulse Compression Waveforms
Linear FM Modulated Pulse (Chirp)
Phase Coding
Poly-Phase Codes
Bi-Phase Codes
Frank Codes
Pseudo-Random Codes
Stepped Frequency Waveform (SFWF)
RADA
BASI

SOLO
Radar Basics
RF Section of a Generic Radar
Antenna
Antenna Gain, Aperture and Beam Angle
Mechanically/Electrically Scanned Antenna (MSA/ESA)
Mechanically Scanned Antenna (MSA)
Conical Scan Angular Measurement
Monopulse Antenna
Electronically Scanned Array (ESA)
RADAR
BASICS -

SOLO
Radar Basics
Transmitters
Types of Power Sources
Grid Pulsed Tube
Magnetron
Solid-State Oscillators
Crossed-Field amplifiers (CFA)
Traveling-Wave Tubes (TWT)
Klystrons
Microwave Power Modules (MPM)
Transmitter/Receiver (T/R) Modules
Transmitter Summary

SOLO
Radar Basics
Radar Receiver
Isolators/Circulators
Ferrite circulators
Branch- Duplexer
TR-Tubes
Balanced Duplexer
Wave Guides
Receiver Equivalent Noise
Receiver Intermediate Frequency (IF)
Mixer Technology
Coherent Pulse-RADAR Seeker Block Diagram

SOLO
Radar Basics
Radar Equation
Radar Cross Section
Irradiation
Decibels
Clutter
Ground Clutter
Volume Clutter
Multipath Return
Electronic Counter Measures (ECM)

SOLO
Radar Basics
Signal Processing
Binary Detection
Decision/Detection Theory
Radar Technologies & Applications
Radar Operation Modes
References

Continue from
Radar Basic – Part I
SOLO
Radar Basics

SOLO
TRANSMITTERS
Return to Table of Content

SOLO
Electron Tubes
for RF
and Microwaves
Microwave
Tubes
Low Frequency
(Gridded Tubes)
Linear Beam
Tubes
Crossed Field
Tubes
Triode
Pentode
Tetrode
TWT Hybrid
(Twystron)
Klystron Magnetron
CFA
Carcinstron
(MBWD)
Sivan, L., “Microwave Tube Transmitters”, Chapman & Hall, 1994, pg. 4
Transmitters

SOLO

In 1921 Albert Wallace Hull invented the magnetron as a powerful microwawe tube.
resonant cavities anode
catode
Filament
leads
Fig. Cutaway view of a Magnetron
pickup loop
a) slot- type
b) vane- type
c) rising sun- type
d) hole-and-slot- type
Figure 3: forms of the plate of magnetrons
Albert Wallace Hull
(1880 – 1966)
Magnetron

Figure 1: the electron path under the
influence of the varying magnetic field.
1. Phase: Production and acceleration
of an electron beam
2. Phase: velocity-modulation
of the electron beam
Figure 2: The high-frequency
electrical field
3. Phase: Forming of a „Space-Charge Wheel”
Figure 3: Rotating space-charge
wheel in an eight-cavity magnetron
4. Phase: Giving up energy to the ac field
Figure 4: Path of an electron
Magnetron

Magnetron tuning
A tunable magnetron permits the system to be operated at a precise frequency
anywhere within a band of frequencies, as determined by magnetron characteristics.
The resonant frequency of a magnetron may be changed by varying the inductance or
capacitance of the resonant cavities.
inductive
tuning
elements
Tuner frame
anode block
Figure 12: Inductive magnetron tuning

Figure 13: Magnetron M5114B of the ATC-radar ASR-910
Figure 13: Magnetron VMX1090 of the ATC-radar PAR-80
This magnetron is even equipped with the permanent magnets
necessary for the work.
Magnetron

SOLO
The Crossed-Field Amplifier (CFA), is a broadband microwave amplifier that can
also be used as an oscillator (Stabilotron). The CFA is similar in operation to the
magnetron and is capable of providing relatively large amounts of power with high
efficiency. The bandwidth of the cfa, at any given instant, is approximately plus or
minus 5 percent of the rated center frequency. Any incoming signals within this
bandwidth are amplified. Peak power levels of many megawatts and average power
levels of tens of kilowatts average are, with efficiency ratings in excess of 70 percent,
possible with crossed-field amplifiers.
Crossed-Field Amplifier (CFA)
Also other names are used for the Crossed-Field Amplifier
in the literature.
• Platinotron
• Amplitron
• Stabilotron
Figure 2: schematically view of a
Crossed-Field Amplifier
(1) cathode
(2) anode with resonant-cavities
(3) „Space-Charge Wheel”
(4) delaying strapping rings
Figure 1: water-cooled
Crossed-Field Amplifier
L-4756A in its transport case

SOLO
Crossed-Field Amplifier (CFA)
Because of the desirable characteristics of wide bandwidth, high efficiency, and the
ability to handle large amounts of power, the CFA is used in many applications in
microwave electronic systems. When used as the intermediate or final stage in high-
power radar systems, all of the advantages of the CFA are used.
The amplifiers in this type of power-amplifier transmitter must be broad-band
microwave amplifiers that amplify the input signals without frequency distortion.
Typically, the first stage and the second stage are traveling-wave tubes (TWT) and the
final stage is a crossed-field amplifier. Recent technological advances in the field of
solid-state microwave amplifiers have produced solid-state amplifiers with enough
output power to be used as the first stage in some systems. Transmitters with more than
three stages usually use crossed-field amplifiers in the third and any additional stages.
Both traveling-wave tubes and crossed-field amplifiers have a very flat amplification
response over a relatively wide frequency range.
Crossed-field amplifiers have another advantage when used as the final stages of a
transmitter; that is, the design of the crossed-field amplifier allows rf energy to pass
through the tube virtually unaffected when the tube is not pulsed. When no pulse is
present, the tube acts as a section of waveguide. Therefore, if less than maximum output
power is desired, the final and preceding cross-field amplifier stages can be shut off as
needed. This feature also allows a transmitter to operate at reduced power, even when
the final crossed-field amplifier is defective Return to Table of Content

SOLO
Travelling Wave Tube
Travelling wave tubes (TWT) are wideband
amplifiers. They take therefore a special position
under the velocity-modulated tubes. On reason of the
special low-noise characteristic often they are in use
as an active RF amplifier element in receivers
additional. There are two different groups of TWT:
• low-power TWT for receivers
occurs as a highly sensitive, low-noise and wideband amplifier in radar equipments
• high-power twt for transmitters
these are in use as a pre-amplifier for high-power transmitters.
collector
input
output
electron- beam bounching
Amplified Helix Signal
RF-Input
RF induced into Helix
The Travelling Wave Tube (twt) is a
high-gain, low-noise, wide-bandwidth
microwave amplifier. It is capable of
gains greater than 40 dB with
bandwidths exceeding an octave. (A
bandwidth of 1 octave is one in which
the upper frequency is twice the lower
frequency.) Traveling-wave tubes have
been designed for frequencies as low as
300 megahertz and as high as 50
gigahertz. The twt is primarily a voltage
amplifier. The wide-bandwidth and low-
noise characteristics make the twt ideal
for use as an rf amplifier in microwave
equipment.

SOLO
collector
input
output
Figure 5. - electron- beam bounching and a detail-foto of a helix (Measure detail for 20 windings)
The following figure shows the electric fields that are parallel to the electron beam inside the
helical conductor.
The electron- beam bounching already starts at the beginning of the helix and reaches its highest
expression on the end of the helix. If the electrons of the beam were accelerated to travel faster than
the waves traveling on the wire, bunching would occur through the effect of velocity modulation.
Velocity modulation would be caused by the interaction between the traveling-wave fields and the
electron beam. Bunching would cause the electrons to give up energy to the traveling wave if the
fields were of the correct polarity to slow down the bunches. The energy from the bunches would
increase the amplitude of the traveling wave in a progressive action that would take place all along
the length of the TWT.

SOLO
Characteristics of a TWT
The attainable power-amplification are essentially
dependent on the following factors:
• constructive details (e.g. length of the helix)
• electron beam diameter (adjustable by the
density of the focussing magnetic field)
• power input (see figure 6)
• voltage UA2 on the helix
As shown in the figure 6, the gain of the twt has got a linear characteristic of about 26 dB at small
input power. If you increase the input power, the output power doesn't increase for the same gain.
So you can prevent an oversteer of e.g the following mixer stage. The relatively low efficiency of
the twt partially offsets the advantages of high gain and wide bandwidth.
Given that the gain of an TWT effect by the electrons of the beam that interact with the electric
fields on the delay structure, the frequency behaviour of the helix is responsible for the gain.
The bandwidth of commonly used TWT can achieve values of many gigahertzes. The noise
figure of recently used TWT is 3 ... 10 dB.

SOLO
Klystron amplifiers are high power microwave vacuum tubes. Klystrons are velocity-modulated
tubes that are used in some radar equipments as amplifiers. Klystrons make use of the transit-
time effect by varying the velocity of an electron beam. A klystron uses one or more special
cavities, which modulate the electric field around the axis the tube.
Klystron
On reason of the number of the cavities klystrons are divided up in:
• Multicavity Power Klystrons
• Reflex Klystron
Two-Cavity Klystron
A klystron uses special cavities which modulate the electric field around the axis the tube. In the
middle of these cavities, there is a grid allowing the electrons to pass. The first cavity together with
the first coupling device is called a „buncher”, while the second cavity with its coupling device is
called a „catcher”.

SOLO Klystron
• The electron gun produces a flow of electrons1
• The bunching cavities regulate the speed of
the electrons so that they arrive in bunches at the
output cavity.
2
• The bunches of electrons excite microwaves in the
output cavity of the klystron.3
• The microwaves flow into the waveguide , which
transports them to the accelerator.
4
• The electrons are absorbed in the beam stop 5
In a klystron:
http://www2.slac.stanford.edu/vvc/accelerators/klystron.html

SOLO Klystron
Reflex (Repeller) Klystron
Another tube based on velocity modulation, and used to generate
microwave energy, is the reflex klystron (repeller klystron). The reflex
klystron contains a reflector plate, referred to as the repeller, instead
of the output cavity used in other types of klystrons. The electron
beam is modulated as it was in the other types of klystrons by passing
it through an oscillating resonant cavity, but here the similarity ends.
The feedback required to maintain oscillations within the cavity is
obtained by reversing the beam and sending it back through the
cavity. The electrons in the beam are velocity-modulated before the
beam passes through the cavity the second time and will give up the
energy required to maintain oscillations. The electron beam is turned
around by a negatively charged electrode that repels the beam
(„repeller”). This type of klystron oscillator is called a reflex klystron
because of the reflex action of the electron beam.
Three power sources are required for reflex klystron operation:
1. filament power,
2. positive resonator voltage (often referred to as beam voltage) used to
accelerate the electrons through the grid gap of the resonant cavity, and
3. negative repeller voltage used to turn the electron beam around.
The electrons are focused into a beam by the electrostatic fields set up by
the resonator potential (U2) in the body of the tube.The
accompanying graphic shows a circuit diagram with a repeller
klystron using a so called „doghnut”-shaped cavity resonator.

SOLO
Simplified Schematic of the T/R Module
http://www.abacusmicro.com/designs.asp?sub=Links9
http://www.microwaves101.com/encyclopedia/transmitreceivemodules.cfm

SOLO
Radar Receiver
Simplified Radar Receiver (Non-Coherent)
The received RF-signals must transformed in a video-signal to get the wanted
information from the echoes. This transformation is made by a super heterodyne
receiver.
• Circulator
• RF Waveguides
• TR Switches
• Low Noise Amplifier (LNA)
• RF Controllable Gain Amplifier
• Mixer
• IF Band-Pass Filter
• IF Controllable Gain Amplifier

SOLO
Ferrite circulators are often used as a diplexer, generally in modules for active antennae. The
operation of a circulator can be compared to a revolving door with three entrances and one
mandatory rotating sense. This rotation is based on the interaction of the electromagnetic wave
with magnetised ferrite. A microwave signal entering via one specific entrance follows the
prescribed rotating sense and has to leave the circulator via the next exit. Energy from the
transmitter rotates anticlockwise to the antenna port. Virtually all circulators used in radar
applications contain ferrite.
Ferrite circulators
http://www.radartutorial.eu/01.basics/rb01.en.html Return to Table of Content

SOLO
Duplexer with quarter-wave co-axial stubs
ATR
Tube
TR
Tube
A B
C D
During the transmitting pulse, an arc appears across both the tr tube (at the point D) and the
atr tube (at the point C) and causes the tr and atr circuits to act as shorted (closed-end) quarter-
wave stubs. The circuits then reflect an open circuit to the tr (at the point B) and atr (at the point
A) circuit connections to the main transmission line. None of the transmitted energy can pass
through these reflected opens into the atr stub or into the receiver. Therefore, all of the transmitted
energy is directed to the antenna.
„Branch- Duplexer”
During reception the amplitude of the
received echo is not sufficient to cause an
arc across either tube. Under this
condition, the atr circuit now acts as a
half-wave transmission line terminated in
a short-circuit. This is reflected as an open
circuit at the receiver T-junction (at the
point B), three-quarter wavelengths away.
The received echo sees an open circuit in
the direction of the transmitter. However,
the receiver input impedance is matched to
the transmission line impedance so that
the entire received signal will go to the
receiver with a minimum amount of loss.
http://www.radartutorial.eu/01.basics/rb01.en.html

SOLO
keep- alive electrode
main gap
DC ground
ATR-tube for waveguide-stubs with a keep-alive electrode
TR-Tubes
TR tubes are usually conventional spark gaps enclosed in partially evacuated, sealed glass envelopes, as shown in
figure 2. The arc is formed as electrons are conducted through the ionized gas or vapor. You may lower the
magnitude of voltage necessary to break down a gap by reducing the pressure of the gas that surrounds the
electrodes. Optimum pressure achieves the most efficient tr operation. You can reduce the recovery time, or
deionization time, of the gap by introducing water vapor into the tr tube. A tr tube containing water vapor at a
pressure of 1 millimeter of mercury will recover in 0.5 microseconds. It is important for a tr tube to have a short
recovery time to reduce the range at which targets near the radar can be detected. If, for example, echo signals
reflected from nearby objects return to the radar before the tr tube has recovered, those signals will be unable to
enter the receiver.
This TR tube used at microwave frequencies is built to fit into, and become a part of, a wave guide. The
transmitted pulse travels up the guide and moves into the tr tube through a slot. During the transmitting pulse, an
arc appears into the TR tube. One-quarter wavelength away, this action effectively closes the entrance to the
receiver and limits the amount of energy entering the receiver to a small value. The windows of Quartz-glass
(irises) are used to introduce an equivalent parallel-LC circuit across the waveguide for impedance matching.
Tube electron MD 80 S 2 of „Raytheon” Company.

SOLO
„Balanced Duplexer”
Output
• A -3 dB-hybride divides the transmitters power in two parts;
• this part passed the slot of the hybride take a phase-shift of 90°;
• both parts of power cause an arc across both spark gaps
• these arcs short-circuit the waveguide and the power would be reflected;
• the power divides in the -3 dB-hybride once again;
• this part passed the slot of the hybride again take a phase-shift of 90°;
among the parts in the direction of the transmitter occurs a phase-shift of 180° and
these parts of power compensates among each other;
• both parts in the direction of the antenna have the same phase and accumulate to the
full power.
During reception the amplitude of the
received echo is not sufficient to cause an
arc across either spark gap. both parts of
the received echo can pass the spark gaps.
The echoes recur both hybrides and
accumulate their parts in-phase. The loss
of this duplexer is about 0.5 to 1.5 dB.
„Balanced Duplexer” works in accordance with the following principle:

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Wave Guides
SOLO

SOLO
Receiver Equivalent Noise
Boltzman’s constant
Gain = G1
Noise Figure = F1
Gain = G2
Noise Figure = F2
Gain = Gi
Noise Figure = Fi

The gain of the receiver is iGGGG 21 ⋅=
The noise figure of the receiver is
i
i
GGG
F
GG
F
G
F
FF


2121
3
1
2
1
111 −
++
−
+
−
+=
A radar receiver usually has a pre-amplifier (1) characterized by a low noise figure
(F1) and by a high gain (G1) such that the effect of the noise of other amplifiers is
negligible and This is the Low Noise Amplifier (LNA).1FF ≈
The noise energy (white noise) at the Receiver is [ ]jouleFTkEN 0=
where
Kjoulek 
/1038.1 23−
×=
The receiver consists of a number of amplifiers in cascade.
KT 
2900 = room temperature
F receiver noise figure
Receiver Noise Power [ ]wattBFTkN 0=
B - Receiver Bandwidth Return to Table of Content

SOLO
Transmitted RF signal (in phasor form) is ( ) ( )tpetS tj
Tr
RFω
=
p (t) - the pulse train function
At the front-end of the Antenna we receive a shifted and attenuated version of the
transmitted pulse:
( ) ( )
( )cRtpeVtS tj
cv
TRF
/2Re −= −ωω
ωRF - the RF angular velocity
ωT - the target’s Doppler shift
2 R/c time delay between transmission and reception
V – random complex voltage strength
c – velocity of light
We assume that from the Antenna emerge radar signal of the Sum S and
Difference D
( )
( )
( )
( ) ( )cRtpFeVD
cRtpeVS
tj
tj
TRF
TRF
/2
/2
−∆=
−=
−
−
ψωω
ωω

SOLO
The Superheterodyne Receiver translates the high RF frequency ωRF to a lower
frequency for a better processing.
This is done my mixing (nonlinear multiplication) the input frequency ωRF- ωT
with ωRF± ωIF to obtain ωIF - ωT
IF
Amp
IF
Amp
Band Pass
at IF
Band Pass
at IF
S
D
'D
'S
( ) tjst IFRF
eLO ωω ±
1
Mixer
Mixer
First Intemediate Frequaency (1st IF)
( )
( )
( )
( ) ( )cRtpFeVD
cRtpeVS
tj
tj
TRF
TRF
/2
/2
−∆=
−=
−
−
ψωω
ωω
The Receiver translates the high RF frequency ωRF to a lower frequency to a
better processing. This is done my mixing (nonlinear multiplication) the input
frequency ωRF- ωT with ωRF± ωIF to obtain ωIF - ωT .
The IF signal is amplified and bandpass filtered to produce an output at IF frequency
( )
( )
( )
( ) ( )cRtpFeVD
cRtpeVS
tj
tj
TIF
TIF
/2''
/2''
−∆=
−=
−
−
ψωω
ωω
If the mixing frequency is centered at ωRF± ωIF than the output is centered at
ωIF and at the image 2 ωRF± ωIF .

SOLO
A second mixing frequency is sometimes added to avoid potential problems with
image frequency.
IF
Amp
'S
''S
( ) tjnd IFIF
eLO ωω 2
2 ±
Mixer
Second Intemediate Frequaency (2nd IF)
IF
Amp
'D
''D
Mixer
Phase
Shifter
AGC
AGC
Band Pass
at 2nd IF
Band Pass
at 2nd IF
( )
( )
( )
( ) ( )cRtpFeVD
cRtpeVS
tj
tj
TIF
TIF
/2"
/2"
2
2
−∆=
−=
−
−
ψ
ωω
ωω
The output of the Second Intermediate Frequency (2nd
IF)
( )
( )
( )
( ) ( )cRtpFeVD
cRtpeVS
tj
tj
TIF
TIF
/2''
/2''
−∆=
−=
−
−
ψωω
ωω

SOLO
Coherent Pulse-RADAR Block Diagram
Block Diagram of a Simple Coherent Radar
f0
Power
Amplifier
Signal
Generator
Coherent
Oscillator
(COHO)
fLO
fRF
fIF
fIF
f0 + fd
fIF + fd
fd
f0=fRF + fIF
IF BP &
Variable Gain
Amplifier
CYRCULATOR
SIGNAL
PROCESSOR
ANGLE
TRACKER
DOPPLER
TRACKER
RANGE
TRACKER
SEEKER
LOGIC
RADAR
CENTRAL
PROCESSOR
RADOME
LOW-PASS-
FILTER
ANTENNA
STABILIZATION
A/D
ANALOG DIGITAL
FREQUENCY
SOURCE
RFIF +
RECEIVER
ANTENNA
RF
Variable
gain
LNA
RF
Switch
AGC
Stable
Local
Oscillator
(STALO)
LNA
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Radar Equation
Radar Cross Section Definition
SOLO
- Target Radar Cross Section (RCS) [m2
]TGTσ
The incident Power Density (Irradiance) at the target is given by:
2 2 2
/i i i i iS E H H E watt m
µ ε
ε µ
 = × = =  
r r
The Power Density (Irradiance) intercepted and scattered
by the target is given by: [ ]i TGTS wattσ
The received Power Density (Irradiance) is defined as:
2 2 2
/r r r r rS E H H E watt m
µ ε
ε µ
 = × = =  
r r
Power scattered by the target in each steradian: ( ) [ ]/ 4 /i TGTS watt strσ π
Solid angle of receiver as seen from the target: [ ]2
/RCVRA R strΩ=
The received Power is given by:
[ ]2
4
i TGT RCVR
S A
watt
R
σ
π
The received Power is given also
by:
2
/r RCVRS A watt m  
2
4
i TGT RCVR
r RCVR
S A
S A
R
σ
π
=
2
lim 4 r
TGT
R
i
S
R
S
σ π
→∞
=Since RCS is defined in the Far Field:

SOLO
Radar Cross Section σ of a Sphere of Radius r as a Function of the Wavelength λ
Radar Equation

SOLO Radar Equation
Radar Cross Section σ of Different Bodies

Stealth aircraft are practically undetectable by sensors. They exploit the
diagram to minimize scattered and reflected signals, and to focus the
residuals in few directions, different from that of the sensors.
Stealth aircraft are practically undetectable by sensors. They exploit the
diagram to minimize scattered and reflected signals, and to focus the
residuals in few directions, different from that of the sensors.
Contributors to Target RCS
Radar Equation

SOLO
Generic Aircraft Model Scattering Center
Radar Equation

SOLO
rain (mm/hr)
fog (gr/cm3
)
air
Two Way Power Loss (Transmitter -> Target, Target -> Receiver )
Radar Equation

fog (gr/cm3
)
rain (mm/hr)
air
Target
ECM Pod
Ground
A/C Radar
Missile, Target, Environment
fog (gr/cm3
)
rain (mm/hr)
air
Target
Transmitted
Mainlobe
Energy
ECM Pod
Ground
A/C Radar
Transmitted
Side-lobe
Energy
Missile RADAR Seeker Transmision
fog (gr/cm3
)
rain (mm/hr)
air
Target
Direct-path
Target Return
ECM Pod
Ground
A/C Radar
Target Reflected Energy Return
fog (gr/cm3
)
rain (mm/hr)
air
Multipath
Target Return
Target
ECM Pod
Ground
A/C Radar
Target Multipath Return
SOLO
Target Energy Return versus Return from Unwanted Factors
• A/C Radar, Target,
Environment (rain, fog, clutter)
• Radar Seeker Transmission
• Target Energy Return
• Target Multipath Return
• Target ECM Return
• Ground Clutter Return
fog (gr/cm3
)
rain (mm/hr)
air Electronic Counter
Measures (ECM)
Return
Target
ECM Pod
Ground
A/C Radar
Target ECM Return
fog (gr/cm3
)
rain (mm/hr)
Measures (ECM)
Return
Target
Direct-path
Target Return
Received
Mainlobe
Clutter
Energy
ECM Pod
Ground
A/C Radar
Received
Side-lobe
Clutteer
Energy
Ground Clutter Return
fog (gr/cm3
)
rain (mm/hr)
Measures (ECM)
Return
Multipath
Target Return
Target
Direct-path
Target Return
Transmitted
Mainlobe
Energy
ECM Pod
Ground
A/C Radar
Transmitted
Side-lobe
Energy
Target, Multipath, ECM, Clutter Returns
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Radar Equation

Far away from the source of radiation (far field)
the electromagnetic fields and are perpendicular
to each other and to the direction of propagation,
and their amplitudes drop off inversely with the Range R.
E
r
H
r
( ) ( ) ( ) 10202101 constRERRERRER =⇒=
( ) ( ) ( ) 20202101 constRHRRHRRHR =⇒=
That means that the electromagnetic field acts as a spherical wave.
Accordingly the irradiance at a range R from an isotropic radiator (radiating uniformly
in all directions) is:
[ ]2
2
/
4
mwatt
R
P
HES rad
r
π
=×=
rr
0
0
0
0 EH
µ
ε
=
where < > means the time average.
A non-isotropic radiator will radiate more in some direction than in others, and the
maximal irradiation will be:
[ ]2
2
/
4
mwattG
R
P
HES rad
MAXMAXr
π
=×=
rr
where G is the Antenna Gain, a measure of the maximum radiation capability of
the Antenna.
SOLO Radar EquationIrradiation

r
MAXr
S
S
G =:
Radar Equation
Bϕ
Bϑ
ϕD
ϑD
Antenna
Radiation
Beam
Assume for simplicity that the Antenna radiates all the power into the solid angle
defined by the product , where and are the angle from the
boresight at which the power is half the maximum (-3 db).
BB ϕϑ , 2/Bϕ± 2/Bϑ±
ϑϑ
λ
η
ϑ
D
B
1
=
ϕϕ
λ
η
ϕ
D
B
1
=
λ - wavelength
ϕϑ DD , - Antenna dimensions in directionsϕϑ,
ϕϑ ηη , - Antenna efficiency in directionsϕϑ,
then
( ) eff
BB
ADDG 22
444
λ
π
ηη
λ
π
ϕϑ
π
ϕϑϕϑ ==
⋅
=
where
ϕϑϕϑ ηη DDAeff =:
is the effective area of the Antenna.
2
4
λ
π
=
effA
G
SOLO

Radar Equation
Transmitter
IV
Receiver
R
1 2
Let see what is the received power on an
Antenna, with an effective area A2 and range R
from the transmitter, with an Antenna Gain G1
Transmitter
VI
Receiver
R
1 2
2122
4
AG
R
P
ASP dtransmitte
rreceived
π
==
Let change the previous transmitter into a receiver and the receiver into a
transmitter that transmits the same power as previous. The receiver has now an
Antenna with an effective area A1 . The Gain of the transmitter Antenna is now G2.
According to Lorentz Reciprocity Theorem the
same power will be received by the receiver; i.e.:
122
4
AG
R
P
P dtransmitte
received
π
=
therefore
1221 AGAG =
or
const
A
G
A
G
==
2
2
1
1
We already found the constant; i.e.: 2
4
λ
π
=
A
G
SOLO

Radar Equation
The Power Density (Irradiance) at the target is given by:
TRP
TRG
TRR
EV
Target
Transmitter
[ ]2
Pr
2
/
1
4
1
mW
LRL
GP
S
TGTXMTR
opagation
TGTTR
rTransmitte
TR
TRTR
r
  
→
→
=
π
- Transmitter Power [W]TRP
- Transmitter Antenna Gain in the Target directionTRG
- Transmitter Loss (XMTR+Antenna+Radome) ( > 1 )TRL
- Range Transmitter to Target [m2
]TRR
- Propagation Loss from Transmitter to Target ( > 1 )TGTTRL →
SOLO

Radar Equation
The Power reflected by the target in the receiver direction is:
[ ]WGA
LRL
GP
GASP
TGT
TGTTGT
TGTXMTR
opagation
TGTTRTR
rTransmitte
TR
TRTR
TGTTGTrTGT

   σ
π
→
→
==
Pr
2
4
1
- Target Effective area in the Transmitter direction [m2
]TGTA
- Target Gain in the Receiver directionTGTG
- Propagation Loss from Target to Receiver ( > 1 )RCVRTGTL →
The Power Density [W/m2
] received at the Receiver is
[ ]2
Pr
2
/
1
4
1
mW
LR
Pp
RCVRTGT
opagation
RCVRTGTRCVR
TGTRCV
  
→
→
=
π
Target
Transmitter
Receiver
SOLO
- Target Radar Cross Section (RCS) [m2
]TGT TGT TGTA Gσ =

Radar Equation
[ ]2
Pr
2
Pr
2
/
1
4
11
4
1
mW
LR
GA
LRL
GP
p
RCVRTGT
opagation
RCVRTGTRCVR
TGTTGT
TGTXMTR
opagation
TGTTRTR
rTransmitte
TR
TRTR
RCVR
TGT   

  
→
→
→
→
=
ππ σ
- Propagation Loss at the Receiver ( > 1 )RCVRL
The Power Density [W/m2
[ ]W
L
A
LR
GA
LRL
GP
LApP
ceiver
RCVR
RCVR
RCVRTGT
opagation
RCVRTGTRCVR
TGTTGT
TGTXMTR
opagation
TGTTRTR
rTransmitte
TR
TRTR
RCVRRCVRRCVRCVR
TGT   

  
RePr
2
Pr
2
1
4
11
4
1
/
→
→
→
→
=
=
ππ σ
The Power [W/m2
- Effective area in the Receiver Antenna [m2
]RCVRA
SOLO

Radar Equation
[ ]W
L
G
LR
GA
LRL
GP
P
ceiver
RCVR
RCVR
RCVRTGT
opagation
RCVRTGTRCVR
TGTTGT
TGTXMTR
opagation
TGTTRTR
rTransmitte
TR
TRTR
RCVR
TGT   

  
Re
2
Pr
2
Pr
2
4
1
4
11
4
1
π
λ
ππ σ
→
→
→
→
=
the Power [W/m2
π
λ
4
2
RCVR
RCVR
G
A =
( )
[ ]W
LLLLRR
GGP
P
RCVRRCVRTGTTGTTRTRRCVRTR
TGTRCVRTRTR
RCVR
→→
= 223
2
4π
σλ
Using
or
SOLO

Radar Equation
[ ]W
L
G
LR
GA
LRL
GP
P
ceiver
RCVR
RCVRTGT
opagation
TGTTR
TGTTGT
TGTXMTR
opagation
TGTTR
rTransmitte
TR
TR
RCVR
TGT   

  
Re
2
Pr
2
Pr
2
4
1
4
11
4
1
π
λ
ππ σ
→
→
→
→
=
the Power [W/m2
( )
[ ]W
LLLR
GP
P
RCVRTGTTRTR
TGTTR
RCVR 243
22
4 →
=
π
σλ
or
SOLO
,RRR RCVRTR ==Collocated Transmitter & Receiver
with a Common Antenna
RCVRTGTTGTTR LL →→ =
GGG RCVRTR ==

db0
db3
dbdb 326 ⋅=
dbdb 339 ⋅=
db10 10
823
=
( ) dbdb 9101 −=
1
4
5
8
10
=
2
( ) dbdb 132 −=
5.2
4
5
2 =⋅( ) dbdb 134 +=
6.1
4/5
2
=
( ) dbdb 165 −= 2.3
4/5
4
=
422
=
( ) dbdb 167 += 5
4
5
4 =⋅
( ) dbdb 198 −= 4.6
4/5
8
=
Decibels GainDecibels = 10 log (Gain)
SOLO
Decibels

1010
23
4
1
11
=
=
=
db
db
db
db1
4
1
1+00.1
db5.0
4
5.0
1+
dbF.0
4
.0
1
F
+
Decibels
Gain
( ) dbdb 9101 −= 4
1
1
4
1
1
8
10
+==
Decibels GainDecibels = 10 log (Gain)
SOLO
db0 1
db8.0 2.1
db6.0 15.1
db4.0 10.1
dbF.0 4
.0
1
F
+
Decibels

SOLO
Decibels
Radar Parameters Often Expressed in Decibels
• Antenna Gain
• dBi (gain relative to isotropic)
• Power Loss
• dB (power out/power in)
• Power
• dBW (power related to 1 watt)
• dBm (power related to 1 milliwatt)
• Radar Cross Section (RCS)
• dBsm (RCS related to 1 square meter)

SOLO
Clutter is a return or group of returns that is undesirable for the radar performing
a certain task.
Clutter
Clutter returns are the vector summation (amplitude and phase) from all of the
Scattering centers within the radar resolution cell. Thus, the resultant Radar Cross
Section (RCS) of the clutter cell is given by:
( )
2
1
exp 





= ∑=
scN
k
kk j φσσ
where
λ
ππφ kk
Radark
R
c
R
f 4
2
2 =





= relative phase
Resultant field
for
one polarization
1 2
3
4 5
6
7

SOLO
Mathematical Approaches to Characterize Clutter
Clutter
• Clutter Amplitude:
- Statistical quantities: mean, standard deviation
- Statistical distributions: probability amplitude (or power) density or
cumulative probability
• Time Varying Properties:
- Correlation function, power spectral density
• Spatially Varying Properties:
- Spatial distributions, correlations, spectra

SOLO
Characterizing Clutter Using Statistical Quantities
Clutter
• Statistical quantities are useful, but knowing the amplitude distribution is equaly
important
• Mean:
n
x
x
n
j
j∑=
=
1
• Standard deviation :
( )
1
1
2
−
−
=
∑=
n
xx
n
j
j
σ

ah
MV
pθ
e
ψ
R
ψcosR
Ae
Aψ
Horizontal
Ground
Main Lobe
Beam
Transmitter
& Receiver
πθθπθ ≤+≤⇒−≤≤− ppp ee 0
Define a ray R from transmitter
to ground, defined by the angles
e,ψ, relative to Missile velocity
vector.
VM is the Missile (transmitter)
velocity vector, having an angle
θp with the horizontal plane.
( ) 




≤≤−
≤+≤
≥
+
=
2/2/
0
cossin
πψπ
πθ
ψθ
p
a
p
a
e
hR
e
h
R ( )
ψ
θ 22
cos
1cos
R
h
e a
p −±=+
The doppler frequency shift along the ray R is given by:
( ) ( ) ( )[ ]
ψ
θψθ
λ
ψθθθθ
λ
ψ
λ
cos
sincoscos
2
cossinsincoscos
2
coscos
2
2
2
_
aa
p
a
p
M
pppp
MM
clutterd
h
R
R
h
R
hV
ee
V
e
V
Rf
≥








+





−±=
+++==
SOLO
Ground Clutter

( )
( ) ( )[ ]








+





−±=
+++=
=
R
h
R
hV
ee
V
e
V
Rf
a
p
a
p
M
pppp
M
M
clutterd
θψθ
λ
ψθθθθ
λ
ψ
λ
sincoscos
2
cossinsincoscos
2
coscos
2
2
2
_
( ) p
M
aclutterd
V
hRf θ
λ
ψ
sin
20
_
=
==
( ) ψθ
λ
θ
coscos
20
_ p
M
e
clutterd
V
Rf
p =+
=∞→
( ) ψθ
λ
πθ
coscos
2
_ p
M
e
clutterd
V
Rf
p
−=∞→
=+
Altitude Line
λ
ψ
θ
M
h
R
clutterd
V
ef
p
a
2
0,0
cos
_ =












==
=

clutterdf _
( )RangeR
( )RangeR
Clutter
No Clutter
Clutter
Power
Clutter
Power
Main Lobe
Clutter
(MLC)
Altitude
Return
λ
MV2
p
MV
θ
λ
cos
2
AA
M
e
V
coscos
2
ψ
λ
p
MV
θ
λ
sin
2
p
MV
θ
λ
cos
2
−
( ) ApA
a
e
h
ψθ cossin +
( )
( )






+
=
=
ApA
a
ML
AA
M
AAclutterd
e
h
R
e
V
ef
ψθ
ψ
λ
ψ
cossin
coscos
2
,_
Main Lobe
ah
MV
pθ
e
ψ
R
ψcosR
Ae
Aψ
Horizontal
Ground
Main Lobe
Beam
Transmitter
& Receiver
SOLO
Ground Clutter

ah
MV
e
ψ
R
ψcosR
Horizontal
Ground
Transmitter
& Receiver
Main Lobe
Beam
Ae
Aψ
0=pθ
( )
2
2
0
_
cos
2
coscos
2






−±=
=
=
R
hV
e
V
Rf
aM
M
clutterd
p
ψ
λ
ψ
λ
θ
( ) 0
0
_
=
==
ψ
aclutterd hRf
( ) ψ
λ
θ
cos
20
_
M
e
clutterd
V
Rf
p =+
=∞→
( ) ψ
λ
πθ
cos
2
_
M
e
clutterd
V
Rf
p
−=∞→
=+
Altitude Line
λ
ψ
θ
M
h
R
clutterd
V
ef
p
a
2
0,0
cos
_ =












==
=

clutterdf _
( )RangeR
( )RangeR
Clutter
No Clutter
Clutter
Power
Clutter
Power
Main Lobe
Clutter
(MLC)
Altitude
Return
λ
MV2
0=pθ
AA
M
e
V
coscos
2
ψ
λ
λ
MV2
−
AA
a
e
h
ψcossin
( )






=
=
AA
a
ML
AA
M
AAclutterd
e
h
R
e
V
ef
ψ
ψ
λ
ψ
cossin
coscos
2
,_
Main Lobe
0=pθ
SOLO
Ground Clutter

Ground
ah
MV
pθ
ψcosR
Ae
Aψ
Main Lobe
Beam
Transmitter
& Receiver
Cones of
Equi-Range
Rays
R
Projection of
Transmitter
& Receiver
on the Ground
Equi-range
Points
on the Ground
Projection on the Ground
M.L.B.
The Clutter energy from
a range R are obtained
for all points on the ground
that are at the range R from
the Transmitter/Receiver.
Assuming a flat ground,
the points on the ground
a a range R > ha are located
at the intersection of the
conical surface with the
apex at the Transmitter/
Receiver and its altitude line
as the conic axis..
The points on the flat
Ground having the same
range R from the
Transmitter/Receiver
are circles.
SOLO
Ground Clutter

Ground
ah
MV
pθ
ψcosR
Ae
Aψ
Main Lobe
Beam
Transmitter
& Receiver
Cones of
Equi-doppler
Rays
R
Intersection
of Missile
Velocity Vector
with the Ground
Ellipsee pθ<
Parabolee pθ=
Hyperbolee pθ>
Equi-doppler
Points
on the Ground
M.L.B.
The points on the ground
that have the same doppler
shift are located on rays
started from Transmitter/
Receiver and are at the same
angle relative to the Missile
Velocity vector VM.
Therefore the points on the
Ground that have the same
doppler shift are located on
the intersection of the conus
with the apex at the
Transmitter/Receiver and
the conic axis the Missile
Velocity vector VM.
EllipseeFor p ⇒<θ
ParaboleeFor p ⇒=θ
HyperboleeFor p ⇒>θ
SOLO
Ground Clutter

Ground
ah
MV
pθ
ψcosR
Ae
Aψ
Main Lobe
Beam
Transmitter
& Receiver
Cones of
Equi-doppler
RaysCones of
Equi-Range
Rays
R
Projection of
Transmitter
& Receiver
on the Ground
Intersection
of Missile
Velocity Vector
with the Ground
Ellipsee pθ<
Parabolee pθ=
Hyperbolee pθ>
Equi-doppler
Points
on the Ground
Equi-range
Points
on the Ground
M.L.B.
SOLO
Ground Clutter

SIGNALS AND CLUTTER IN A PULSE DOPPLER RADAR, IN THE LOOK-HOTIZON MODESIGNALS AND CLUTTER IN A PULSE DOPPLER RADAR, IN THE LOOK-HOTIZON MODE
SOLO Target in Ground Clutter

SIGNALS AND CLUTTER IN A PULSE DOPPLER RADAR, IN THE LOOK-HOTIZON MODE MODE,
(a) A TYPICAL SITUATION: (b) MAP OF RETURNS ON THE RANGE/VELOCITY PLANE: (c) FOLDING OF
CLUTTER OVER THE RANGE AXIS (LOW PRE):
(d) CONCENTRATION OFRETURNS ON THE VELOCITY AXIS (HIGH PRF).
SIGNALS AND CLUTTER IN A PULSE DOPPLER RADAR, IN THE LOOK-HOTIZON MODE MODE,
(a) A TYPICAL SITUATION: (b) MAP OF RETURNS ON THE RANGE/VELOCITY PLANE: (c) FOLDING OF
CLUTTER OVER THE RANGE AXIS (LOW PRE):
(d) CONCENTRATION OFRETURNS ON THE VELOCITY AXIS (HIGH PRF).
SOLO
Target in Ground Clutter

SOLO
Ground Clutter
Illuminated Ground Area Resolution Cell : Beam Limitted Case
The Main Beam Clutter (Ground) Area in Range Resolution Cell when
is give (see Figure) by:
( ) ( ) pazcRA θϕτ cos/2/tan2/2Clutter =
Ground
Main Lobe
Beam
Transmitter
& Receiver
( ) ( )2//2/tan2tan τϕθ cR elp <
R – range to ground along beam center
φaz – angular beam width in azimuth
φel – angular beam width in elevation
θp –beam grazing angle
τ – pulse width [sec]
c – speed of light 3 108
m/sec
( )Clutter Clutter pAσ σ θ=
σ – ground reflectivity as function of
grazing angle

SOLO
Ground Clutter

SOLO
Clutter
Illuminated Volume Resolution Cell (Pulse Limitted)
The Volume Clutter in Range Resolution Cell is give (see Figure) by:
( )2/
4
2
Clutter τϕϕ
π
cRV elaz=
R – range to ground along beam center
φaz – angular beam width in azimuth
φel – angular beam width in elevation
τ – pulse width [sec]
c – speed of light 3 108
m/sec
Main Lobe
Beam
Transmitter
& Receiver
Choose scatters on the main beam center Groundkk RRuntilkRkR ≥=∆= ,2,1
RADAR
I
k
f
c
where
sV
f =
⋅
= λ
λ
12
r
Their Doppler is given by
According to Range and Doppler of each scatter determine the Range-Doppler cell (i,j) for the
scatter.

Clutter returns are the vector summation (amplitude and phase) from all of the
scattering centers within the radar resolution cell.
SOLO Clutter
The Clutter is obtained by integration (summation) of the signals from the same
range-doppler cells:
where
Nsc – number of scatters in the volume VClutter
σk– Radar Cross Section of scatter k
Rk– Range to scatter k
The equivalent Radar Cross Section σClutter of the
clutter in the resolution cell of volume VClutter is:
( )2/
4
2
Clutter τϕϕ
π
cRV elaz=
g (0,0) ≈ 1 – antenna pattern
R – Range to the center of the volume VClutter
See Tildocs # 763310 v1
( )
( )
( )
∑=












+
−=Σ
jiN
k
k
kk
trver
Rcvr
Xmtr
sc
c
c
R
RR
j
L
GG
Pji
,
1
2
k
kscatter
3
2
0
2
ClutterVolume
2
2
2exp
R4
,

π
σ
π
λ
Illuminated Volume Resolution Cell (Pulse Limitted)
∑=
==
scN
k k
kscatter
ClutterClutter
R
RV
1
4
4
σ
ησ ∑=
=
scN
k k
kscatter
Clutter RV
R
1
4
4
σ
η
Since the Volume Clutter is on the Main-Beam the effect of it on angle errors is
like that of the radar noise.
Main Lobe
Beam
Transmitter
& Receiver

Multipath
Target Return
Target
Ground
A/C
RADAR
SOLO
Multipath Return
Target Multipath is the Received Signal
for the mirror reflection the target relative
to Earth surface.
The vector position of the Target relative
to earth is
LTLTLTT zhyYxXR 111 ++=
r
The vector position of the mirrored
Target relative to earth is
( ) LLTTLTLTLTMT zzRRzhyYxXR 112111_ ⋅−=−+=
rrr
The vector of the signal received by Seeker from the i Target scatter is
IT RRR
rrr
−=
The vector of the signal received by Seeker from the mirrored Target is
( ) ( ) LLTLLTITM zzRRzzRRRR 112112 ⋅−=⋅−−=
rrrrrr
Clutter
hT – altitude above ground surface

SOLO
Multipath Return – Range Discrimination
The Target Range to Seeker is
( )22
ITIT hhXR −+= −
Let compute
Clutter
The Range of the Mirror Target to Seeker is
( ) RhhXR ITITM ≥++= −
22
( )( ) ( ) ( ) ITITITMMM hhhhhhRRRRRR 4
2222
=−−+=−=+−
R
hh
RR
hh
RR
IT
RRR
M
IT
M
M 24 2≈+
≈
+
=−
..
2
GR
R
hh
RR IT
M ≤≈−If , for all Target scatters k, we cannot
distinguish between Target and Target’s Mirror
..
2
GR
R
hh
RR
IT
M >≈−If , for some Target scatters, we can
distinguish between Target and Target’s Mirror and we will
choose the echoes with the smallest range
Multipath
Target Return
Target
Ground
A/C
RADAR

SOLO
Multipath Return – Doppler Discrimination
The Target Range-Rate to Seeker is
( )22
ITIT
IITTITIT
hhX
hhhhXX
R
−+
++
=
−
−−


Let compute
Clutter
The Range-Rate of the Mirror of Target to Seeker is
( )( ) ( )
( )
( )
( )
( )
( )[ ] ( )[ ] Mi
Mi
IT
ITITITIT
IITTITITIT
ITIT
IITTITIT
ITIT
IIiTTITIT
MMM
RR
RR
hh
hhXhhX
hhhhXXhh
HHX
HHHHXX
HHX
HHHHXX
RRRRRR



4
4
2222
2
22
2
22
2
22
=
++−+
++
=
=
++
++
−
−+
++
=−=+−
−−
−−
−
−−
−
−−
0
24
3
2
>≈
+
=−
≈+
M
IT
RRR
M
M
M
IT
Mi RR
R
hh
RR
RR
RR
hh
RR
M



..
2
3
GDRR
R
hh
RR M
IT
M ≤≈− If we cannot distinguish between
Target and Target’s Mirror
..
2
3
GDRR
R
hh
RR M
IT
M >≈− 
If we can distinguish between
Target and Target’s Mirror and we don’t have a
Multipath problem.
( )22
ITIT
IITTITIT
M
hhX
hhhhXX
R
++
++
=
−
−−


Assume that Target & Mirror Target are in the same Range Gate.
Multipath
Target Return
Target
Ground
A/C
RADAR

SOLO
Multipath Return – Angular Discrimination
We found:
( ) ( ) LLTLLTITM zzRRzzRRRR 112112 ⋅−=⋅−−=
rrrrrr
Clutter
The angular separation between Target Scatter k and Target Mirror Scatter k is:
( )
RR
RzzR
RR
RR
M
LLT
M
M
rrrr
×⋅
−=
× 11
2
Multipath Return – Range – Doppler Map
According to Range and Doppler of each scatter mirror:
determine the Range-Doppler cell (i,j) for the scatter mirror.
( ) kITkITMk RhhXR ≥++= −
22 ( )
( )22
ITkITk
IITkTkITkITk
Mk
HHX
hhhhXX
R
++
++
=
−
−−


integer=+= mRRmR kambiguoussunambiguouMk
RADAR
Mk
Mk
f
c
where
R
f == λ
λ
2
integer=+= nffnf kambiguoussunambiguouMk
( )RRIntegi kambiguousk ∆= /
( )ffIntegj kambiguousk ∆= /
Multipath
Target Return
Target
Ground
A/C
RADAR

SOLO
Multipath Return – Signal Power
Assume that The Target and it’s Mirror can be represented each
by Nsc scatters ( k=1,Nsc)
Clutter
The Mirror signal received by the Seeker from scatter k passes
three paths:
  
TGTXMTR
opagation
TGTTRk
rTransmitte
TR
trXmtr
LRL
GP
→
→
Pr
2
1
4
1
π1. Transmitted power from Seeker to Target Scatter k at the distance Rk:
2. Reflected by the target scatter k and reaching the ground at the distance ( ) ( )[ ] 2/122
_1 TkIIT
TkI
Tk
k hhX
hh
h
R ++
+
=
  

GNDTGT
opagation
GNDTGTk
TGTTGT
LR
GA
TGT
→
→
Pr
2
1
1
4
1
πσ
3. Reflected by the ground and reaching the Seeker at the distance ( ) ( )[ ] 2/122
_2 TkIIT
TkI
I
k hhX
hh
h
R ++
+
=
  
ReceivernPropagatio
2
2
1
4
1
RCVR
RCVR
GNDTGT
RCVRGNDk
GND
L
A
LR
→
→π
σ
π
λ
4
2
RCVR
RCVR
G
A =
Multipath
Target Return
Target
Ground
A/C
RADAR

SOLO
Multipath Return – Signal Power
Therefore the received power from the k scatter mirror is:
      
  
  
Receiver
2
nPropagatio
2
2
Pr
2
1
Pr
2
1
4
1
4
11
4
11
4
1
RCVR
RCVRant
GNDTGT
RCVRGNDk
GND
GNDTGT
opagation
GNDTGTk
kScatterkScatter
TGTXMTR
opagation
TGTTRk
rTransmitte
TR
antXmtr
M
L
GG
LRLR
GA
LRL
GP
P
kScatter
k
π
λ
π
σ
ππ
σ
→
→
→
→
→
→
=
Clutter
( ) ( )[ ] 2/122
_1 TkIIT
TkI
Tk
k hhX
hh
h
R ++
+
= ( ) ( )[ ] 2/122
_2 TkIIT
TkI
I
k hhX
hh
h
R ++
+
=
( )
( )
( )( ) ( )









 ++
+++++
−=Σ ∑=
Σ
c
cj
gG
L
GG
Pji
jiN
k
ClutterkElkAzproc
trver
Rcvr
Xmtr
k2k1k
k2k1kk2k1k,
1 k2k1k
kscatter
proc
Targ
3
2
0
2
TargetMultipath
RRR
RRRRRR
2exp
RRR
,
L4
,

π
σσεε
π
λ
( )
( )
2
2
2
1
2
2Targ
3
2
0
2
,
4 kkk
ClutterkScatterkElkAz
proc
proc
trver
Rcvr
XmtrM
RRR
g
L
G
L
GG
PP k
σσεε
π
λ Σ
=
or:
where: ( )kElkAzant gGG εε ,0 Σ=
RCVRRCVRGNDGNDTGTTGTTRTRtrver LLLLLL →→→=
proc
proc
RcvrRCVR
L
G
GG
Targ
=
The Target Multipath received signal is obtained by integration (summation) of the
signals from the same range-doppler cell (i,j):
in the same way:
( )
( )
( )( ) ( )









 ++
+++++
−=∆ ∑=
∆
c
cj
gG
L
GG
Pji
jiN
k
ClutterkElkAzElAzproc
trver
Rcvr
Xmtr
k2k1k
k2k1kk2k1k,
1 k2k1k
kscatter,
proc
Targ
3
2
0
2
Az/ElTargetMultipath
RRR
RRRRRR
2exp
RRR
,
L4
,

π
σσεε
π
λ

SOLO

SOLO Signal Processing

Collecting Pulsed Radar Data: 1 Pulse, Multiple Range-Gates Samples
• when using a coherent receiver, each range sample comprises one “I” sample and
one “Q” sample, forming one complex number I+j Q.
• Each range cells contains an echo from a different range interval.
• Also called Range-Bins, Range-Gates, Fast-Time Samples.

SOLO
Signal Processing
Collecting Pulsed Radar Data: Multiple Pulses
• when using a coherent receiver, each range sample comprises one “I” sample and
one “Q” sample, forming one complex number I+j Q.
• Repeat for multiple pulses in a “coherent processing interval” (CPI) or “dwell”
Sequence of samples for a fixed range bin represents echoes from same range interval
over a period of time.

Perform FFT in Each Range Gate
After FFT a Range-Doppler
Map is obtained for Signal
Processing
FFT
Run This

Perform FFT in Each Range Gate
Data-cube for Signal Processing
Repeat the Operation for each Receiver Channel (Σ,ΔAz,ΔEl,Γ for monopulse antenna
or Σi,j for each element in an Electronic Scanned Antenna)
Range – Doppler Cells in Σ and ΔAz, ΔEl
FFT
FFT
FFT
FFT
Run This

Adaptive algorithms use additional data from the cube for weight estimation.
Datacube for Signal Processing
Standard radar signal processing algorithms correspond to operating in 1- or 2-D along
various axes of the data-cube
Space-Time Adaptive Processing:
2-D joint adaptive weighting across
antenna element and pulse number
Beamforming:
1-D weighting across
Electrical Scan Antenna
element number
Pulse Compression:
1-D convolution along
the range axis
(“fast time”)
Synthetic Aperture Imaging:
2-D matched filtering in slow
and fast time
Doppler Processing:
1-D filtering or spectral
analysis along the pulse axis
(“slow time”)
Run This

SOLO
Signal Processing
Range – Doppler Cells in Σ and ΔAz, ΔEl

SOLO
Windowing
•  Windowing is used for DFT data
to reduce Doppler side lobes
•  Windowing widen main lobe and
this decreases Doppler resolution
•  Windowing reduces the peak of
the DFT producing a processing
loss, PL
•  Windowing causes a modest signal
to noise (S/N) loss, called loss in
peak gain, or LPG.
  Windows are an overlay applied to a given time series to improve the spectral quality
of the data base.
Signal Processing

SOLO
Windowing
Rectangular [ ]


 ≤≤
=
otherwise
Mn
nw
,0
0,1
Bartlett
(triangular) [ ]





≤<−
≤≤
=
otherwise
MnMMn
MnMn
nw
,0
2/,/22
2/0,/2
Hanning
Hammming
[ ]
( )


 ≤≤−
=
otherwise
MnMn
nw
,0
0,/2cos5.05.0 π
[ ]
( )


 ≤≤−
=
otherwise
MnMn
nw
,0
0,/2cos46.054.0 π
Blackman [ ]
( ) ( )


 ≤≤+−
=
otherwise
MnMnMn
nw
,0
0,/4sin08.0/2cos5.042.0 ππ
Julius Ferdinand von Hann (1839 -1921)
Richard Wesley Hamming (1915 –1998)
Signal Processing

SOLO
Windowing (continue – 1)
cosine
[ ]






≤≤<













 −
−
=
otherwise
Mn
M
Mn
nw
,0
0&5.0
2/
2/
2
1
exp
2
σ
σ
Lanczos
[ ]





≤≤





−
=
otherwise
Mn
M
n
nw
,0
0,1
2
sinc
Gauss
[ ]





≤≤





=





−
=
otherwise
Mn
M
n
M
n
nw
,0
0,sin
2
cos
πππ
[ ]
( )







≤≤














−−
=
otherwise
Mn
I
M
n
I
nw
,0
0,
1
2
1
0
2
0
α
α
Kaiser
α=2π
α=3π
Signal Processing

SOLO
Bartlett–Hann window
( )
38.0;42,0;62.0
1
2
cos
2
1
1
210
210
===






−
−−
−
−=
aaa
N
n
a
N
n
aanw
π
Bartlett–Hann window; B=1.46
Low-resolution (high-dynamic-range) windows
Nuttall window, continuous first derivative
( )
012604.0;144232.0;487396,0;355768.0
1
6
cos
1
4
cos
1
2
cos
3210
3210
====






−
−





−
+





−
−=
aaaa
N
n
a
N
n
a
N
n
aanw
πππ
Nuttall window, continuous first
derivative; B=2.02
Blackman–Harris window
( )
01168.0;14128.0;48829,0;35875.0
1
6
cos
1
4
cos
1
2
cos
3210
3210
====






−
−





−
+





−
−=
aaaa
N
n
a
N
n
a
N
n
aanw
πππ
Blackman–Nuttall window
Blackman–Harris window, B=1.98
Blackman–Nuttall window, B=3.77
( )
0106411.0;1365995.0;4891775,0;3635819.0
1
6
cos
1
4
cos
1
2
cos
3210
3210
====






−
−





−
+





−
−=
aaaa
N
n
a
N
n
a
N
n
aanw
πππ
Signal Processing

SOLO
Dolph-Chebyshev window
( ) ( )[ ]
( )
( )[ ]
( ) ( )4,3,2,10cosh
1
cosh
1,,2,1,0,
coshcosh
coscoscos
1
1
1
≈





=
−=

















=
=
−
−
−
αβ
β
π
β
ω
ω
α
N
Nk
N
N
k
N
W
WIDFTnw
k
k

The α  parameter controls the side-lobe level via the formula:
   Side-Lobe Level in dB = - 20 α
The Dolph-Chebyshev Window (or Dolph window) minimizes the Chebyshev norm of
the side lobes for a given main lobe width 2 ωc:
( ) ( ){ }ωωω WWsidelobes cwwww >=∞= ∑
=
∑
maxmin:min 1,1,
The Chebyshev norm is also called the L - infinity  norm, uniform norm, minimax
norm, or simply the maximum absolute value.
Signal Processing

SOLO
Comparison of Windows
Signal Processing

SOLO
Window
Type
Peak
Sidelobe
Amplitude
(Relative)
Approximate
Width of
Mainlobe
Peak
Approximation
Error
20 log10δ
(dB)
Equivalent
Kaiser
Window
β
Transition
Width
of Equivalent
Kaiser
Window
Rectangular -13 4π/(M+1) -21 0 1.81π/M
Bartlett -25 8π/M -25 1.33 2.37π/M
Hanning -31 8π/M -44 3.86 5.01π/M
Hamming -41 8π/M -53 4.86 6.27π/M
Blackman -57 12π/M -74 7.04 9.19π/M
Signal Processing

SOLO
Signal Processing

SOLO
Effect of Window in the Fourier Transform
• Good Effects
- Reduction of sidelobes
- Reduction of straddle loss
• Bad Effects
- Reduction in peak
- Widening of mainlobe
- Reduction in SNR
No Window
Hamming Window
∑
−
=
1
0
21 N
n
nw
N
21
0
1
0
2
1






∑
∑
−
=
−
=
N
n
n
N
n
n
w
w
N
Signal Processing
Run This

SOLO
Signal Processing
Generation of Σ , ΔAz, ΔEl Range – Doppler Maps
The Parameters defining the Range – Doppler Maps are:
Δ R – Map Range Resolution
Δ f – Map Doppler Resolution
RUnambiguous – Unambiguous Range
fUnambiguous – Unambiguous Doppler
Range – Doppler
Cell
Range – Doppler
Map
f
f
M
R
R
N
sunambiguousunambiguou
∆
=
∆
= &
Range Gates are therefore i = 1, 2, …, N
Number of Range-Doppler Cells = N x M
Doppler Gates are therefore j = 1, 2, …, M
Note: The Map Range & Doppler resolution (Δ R, Δ f) may change as function of
Radar task (Search, Detection, Acquisition, Track). This is done by choosing
the Pulse Repetition Interval (PRI) and the number of pulses in a batch.
resolutionresolution ffRR ≥∆≥∆ &

Generation of Σ , ΔAz, ΔEl Range – Doppler Maps (continue – 1)
The received signal from the scatter k is:
( ) ( )[ ] ( ) ( )ttTktttTkttfCts ddkdk
r
k
r
k ++≤≤++−= τθπ2cos
Ck
r
    – amplitude of received signal
td (t)  – round trip delay time given by ( )
2/c
tRR
tt kk
d
+
=
θk         – relative phase
  The received signal is down-converted to base-band in order to extract the quadrature
components. More precisely sk
r
(t) is mixed with: ( ) [ ] τθπ +≤≤+= TktTktfCty kkk 2cos
  After Low-Pass filtering the quadrature components of Σk, ΔAz k or ΔEl k signals are:
( ) ( )
( ) ( )





=
=
tAtx
tAtx
kkQk
kkIk
ψ
ψ
sin
cos
( ) ( ) 





+−≅−=
c
tR
c
R
fttft kk
kdkk
22
22 ππψ
The quadrature samples are given by:
( ) ( ) 











+−≅=
c
tR
c
R
fjAjAtX kk
kkkkk
22
2expexp πψ
  Ak - amplitude of Σk, ΔAz k or ΔEl k signals
  ψk - phase of Σk, ΔAz k or ΔEl k signals
( ) 











+−











+≅+=
c
tR
c
R
fAj
c
tR
c
R
fAxjxtX kk
kk
kk
kkQkIkk
 22
2sin
22
2cos ππ

Generation of Σ , ΔAz, ΔEl Range – Doppler Maps (continue – 2)
The received signal from the scatter k is:
The energy of the received signal is given by: ( ) ( ) 2
kkkk AtXtXP ==
∗
( ) 











+−











+≅+=
c
tR
c
R
fAj
c
tR
c
R
fAxjxtX kk
kk
kk
kkQkIkk
 22
2sin
22
2cos ππ
where * is the complex conjugate.
Therefore:
kk PA =

Decision/Detection TheorySOLO
Hypotheses
H0 – target is not present
H1 – target is present
Binary Detection
( )0
Hp -  probability that target is not present
( )1
Hp -  probability that target is present
( )zHp |0 -  probability that target is not present and not declared (correct decision)
( )zHp |1 -  probability that target is present and declared (correct decision)
Using Bayes’ rule: ( ) ( ) ( )∫=
Z
dzzpzHpHp |00
( ) ( ) ( )∫=
Z
dzzpzHpHp |11
( )zp -  probability of the event Zz ⊂
Since p (z) > 0 the Decision rules are:
( ) ( )zHpzHp || 01
< -  target is not declared (H0)
( ) ( )zHpzHp || 01
> -  target is declared (H1) ( ) ( )zHpzHp
H
H
|| 01
0
1
<
>

Hypotheses H0 – target is not present H1 – target is present
Binary Detection
( )zHp |0 -  probability that target is not present and not declared (correct decision)
( )zHp |1 -  probability that target is present and declared (correct decision)
( )zp -  probability of the event Zz ⊂
Decision rules are: ( ) ( )zHpzHp
H
H
|| 01
0
1
<
>
Using again Bayes’ rule:
( )
( ) ( )
( )
( )
( ) ( )
( )zp
HpHzp
zHp
zp
HpHzp
zHp
H
H
00
0
11
1
|
|
|
|
0
1
=
<
>
=
( )0
| Hzp -  a priori probability that target is not present (H0)
( )1
| Hzp -  a priori probability that target is present (H1)
Since all probabilities are
non-negative
( )
( )
( )
( )1
0
0
1
0
1
|
|
Hp
Hp
Hzp
Hzp
H
H
<
>

Hypotheses
( )1
| Hzp -  a priori probability density that target is present (likelihood of H1)
( )0
| Hzp -  a priori probability density that target is absent (likelihood of H0)
Detection Probabilities
( ) M
z
D
PdzHzpP
T
−== ∫
∞
1| 1
( )∫
∞
=
Tz
FA
dzHzpP 0
|
( ) D
z
M
PdzHzpP
T
−== ∫∞−
1| 1
PD -  probability of detection = probability that the target is present and declared
PFA -  probability of false alarm = probability that the target is absent but declared
PM -  probability of miss = probability that the target is present but not declared
T -  detection threshold
D
P
FAP
( )1| Hzp( )0
| Hzp
M
P
z
Tz
( )
( )
T
Hzp
Hzp
T
T
=
0
1
|
|
H0 – target is not present H1 – target is present
Binary Detection
( )
( )
( )
( )
T
Hp
Hp
Hzp
Hzp
LR
H
H
=
<
>
=
1
0
0
1
0
1
|
|
:Likelihood Ratio Test (LTR)

Hypotheses
Decision Criteria on Definition of the Threshold T
1. Bayes Criterion
D
P
FAP
( )1
| Hzp( )0
| Hzp
MP
z
T
z
( )
( )
T
Hzp
Hzp
T
T
=
0
1
|
|
Binary Detection
( )
( )
( )
( )
T
Hp
Hp
Hzp
Hzp
LR
H
H
=
<
>
=
1
0
0
1
0
1
|
|
The optimal choice that optimizes the Likelihood Ratio is
( )
( )1
0
Hp
Hp
TBayes
=
This choose assume knowledge of p (H0) and P (H1), that in general are not known a priori.
2. Maximum Likelihood Criterion
Since p (H0) and P (H1) are not known a priori, we choose TML = 1
( )1
| Hzp( )0
| Hzp
M
P z
Tz
( )
( )
1
|
|
0
1
== ML
T
T
T
Hzp
Hzp
D
P
FAP

Hypotheses
Decision Criteria on Definition of the Threshold T (continue)
3. Neyman-Pearson Criterion
DP
γ=FAP
( )1
| Hzp( )0
| Hzp
M
P
z
T
z
( )
( ) PN
T
T
T
Hzp
Hzp
−
=
0
1
|
|
Binary Detection
( )
( )
( )
( )
T
Hp
Hp
Hzp
Hzp
LR
H
H
=
<
>
=
1
0
0
1
0
1
|
|
Neyman and Pearson choose to optimizes the probability of detection PD
keeping the probability of false alarm PFA constant.
Egon Sharpe Pearson
1895 - 1980
Jerzy Neyman
1894 - 1981
( )∫
∞
=
T
TT
z
z
D
z
dzHzpP 1
|maxmax ( ) γ== ∫
∞
Tz
FA
dzHzpP 0
|constrained to
Let use the Lagrange’s multiplier λ to add the constraint
( ) ( )
















−+= ∫∫
∞∞
TT
TT
zz
zz
dzHzpdzHzpG 01 ||maxmax γλ
Maximum is obtained for:
( ) ( ) 0|| 01
=+−=
∂
∂
HzpHzp
z
G
TT
T
λ
( )
( ) PN
T
T
T
Hzp
Hzp
−
==
0
1
|
|
λ
zT is define by requiring that: ( ) γ== ∫
∞
Tz
FA
dzHzpP 0
|

SOLO SEARCH & DETECT MODE
  During Search Mode the RADAR Seeker performs the following tasks:
•   Slaves the Seeker Gimbals to the Designation Target direction (like in Slave Mode).
•   Transmits the RF (by choosing the best waveform).
•   Receives the returning RF.
•   Compute the Σ Range-Doppler Map, chooses the Detection Threshold and policy.
•   Perform Detections Clustering and compute Range and Doppler spread.
Note: Here is important to define the number of Batches that are needed to obtain the
predefined probability of detection, the False Alarm Rate (FAR) and to resolve the different
detections, i.e. the time necessary to perform this task.
•   If a Detection is in the Target Designation (Uncertainty)  Window we go to
    Acquisition Mode.

Target returns are the summation of signals (amplitude and phase)
from all of the scattering centers within the radar resolution cell.
SOLO
Target RCS
where
Nsc – number of scatters in the volume VResol
σk– Radar Cross Section of scatter k
Rk– Range to scatter k
  The equivalent Radar Cross Section σTarget of the target in the resolution cell of volume VResol is:
2N
scatter i4
Target Resol 4
i 1 iR
g
V R
σ
σ η Σ
=
= = ∑
24 N
scatter i
4
i 1Resol iR
gR
V
σ
η Σ
=
= ∑ ( )2/
4
2
Resol τϕϕ
π
cRV elaz=
gΣ (εAz,εEl) – antenna sum pattern ( gΣ(0,0)=1 )
R – Range to the center of the volume VResol
( )
( )
( )( )
∑=
Σ
























+
−=Σ
jiN
k
k
kk
kElkAzproc
trver
Rcvr
Xmtr
sc
c
c
R
RR
j
gG
L
GG
Pji
,
1
2
k
kscatter
proc
Targ
3
2
0
2
Targ
2
2
2exp
R
,
L4
,

π
σεε
π
λ
  In the same way:
gΔ (εAz,εEl) – antenna difference pattern ( gΔ(0,0)=0 )
R  G
A  A
N  T
G  E
E  S
DOPPLER
FILTERS
Range-
Doppler
S cells
Detections
  According to Range and Doppler of each scatter determine the
Range-Doppler cell (i,j) for the scatter.
( )
( )
( )( )
∑=
∆
























+
−=∆
jiN
k
k
kk
kElkAzElAzproc
trver
Rcvr
Xmtr
sc
c
c
R
RR
j
gG
L
GG
Pji
,
1
2
k
kscatter,
proc
Targ
3
2
0
2
Az/ElTarg
2
2
2exp
R
,
L4
,

π
σεε
π
λ

According to the position of Target Uncertainty Window (TUW) versus Clutter chose the
Range – Doppler magnitude (Runambiguous and funambiguous) by defining the Pulse Repetition
Frequency (PRF) and the number of pulses in the batch, and choose resolution Δ R and Δ f.
Improvements
1. Change Range-Doppler cells indexes i,j to
bring the Target Uncertainty Window in
the middle of the Range-Doppler Map
2. Choose on the Range-Doppler Map a
area that includes the Target Uncertainty
Window and perform Ground Clutter
computations only for this area (we may add
Ground Clutter computations in Main Lobe
and Altitude Line: Rk = hI).
Transmits the RF (by choosing the best waveform).
Computation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy

Computation of  the Σ Range-Doppler Map, chooses the Detection Threshold
and policy (continue – 1)
•   Computation of  Noise Threshold in each cell: ( ) ( ) ( ) BFTkjijijiN NoiseNoise 0,,, =Σ⋅Σ=
∗
•   Computation of  Clutter Power in CFAR Window cells
(Cells in area around Target Uncertainty Window):
( ) ( ) ( )∗
Σ⋅Σ= jijijiC CFAR
,,,
•   Computation of  Signal Power in Target
    Uncertainty Window cells:
( ) ( ) ( )∗
Σ⋅Σ= jijijiS ,,,
Window
yUncertaint
Target
•   For each Range-Doppler Cell (i,j) perform the summation of complex signals for all
    the scatters in this cell:
∑∑∑ ===
∆=∆∆=∆Σ=Σ
jijiji N
k
kEljiEl
N
k
kAzjiAz
N
k
kji
,,,
1
,
1
,
1
, ,,

and policy (continue – 2).
DOPPLER
WINDOW
R  W
A  I
N  N
G  D
E  O
     W
R  G
A  A
N  T
G  E
E  S
DOPPLER
FILTERS
S cells
CFAR
Window
R∆
f∆
Target
Uncertainty
Window
( ) ( ) ( )[ ]∑
∗
+ Σ⋅Σ=
n
j Window
CFARNoiseClutter jiji
n
iC ,,
1
Guard
(Gap)
Window
•   Computation of  Clutter + Noise Threshold
•   Coherent Detection:
( ) ( )
( ) ( ) ClutterThjiNiCIf
ClutternoThjiNiCIf
NoiseClutter
NoiseClutter
⇒+>
⇒+≤
+
+
1,
1,
( ) NoiseThNjiS +≥
Window
yUncertaint
Target,
( ) ( ) ( )[ ]∑
∗
+ Σ⋅Σ=
n
j Window
CFARNoiseClutter jiji
n
iC ,,
1
1. If no Clutter declare a Detection in the (i,j) cell of the Target Window if
ThNoise is chosen to assure a predefined
Probability of Detection pd and of False Alarm pFA
( ) NoiseClutterNoiseClutter ThCjiS ++ +≥
Window
yUncertaint
Target,
2. If Clutter declare a Detection in the (i,j) cell of the Target Window if
ThNoise is chosen to assure a predefined
Probability of Detection pd and of False Alarm pFA

•   Coherent Detection (M-out-of-N):
How to Increase Probability of Detection and Reduce Probability of False Alarm:
Suppose that by Coherent Detection using one Range – Doppler Map we have
Probability of Detection pd and Probability of False Alarm pfa.
To Increase Probability of Detection to pD and Reduce Probability of False Alarm
to pFA we use N consecutive batches (at different PRFs) , in each of them performing
the Coherent Detection procedure. We declare a detection in the if we have at least
M Detections for corresponding resolved Range-Doppler cells. In this way:
( )
( )∑=
−
−
−
=
N
Ml
lN
d
l
dD pp
lNl
N
P 1
!!
!
( )
( )∑=
−
−
−
=
N
Ml
lN
fa
l
faFA pp
lNl
N
P 1
!!
!
Example: pd = 0.6, pfa = 10-3
, N = 4, M = 2  gives pD = 0.82, pFA = 6 x10-6
Since we use different PRFs,
to obtain correlation between
Detections we must resolve the
Range-Doppler ambiguities.

How to Increase Probability of Detection and Reduce Probability of False Alarm:
•   Non-Coherent Detection:
To Increase Probability of Detection we use N consecutive batches, we compute the
power of each (i,j) cell,                                       , in each Range-Doppler Map and we
add (non-coherently) the powers of each corresponding (i,j) cell to obtain
a non-coherent Range-Doppler Map. Now we perform the detection procedure
as described before to declare a Detection.
( ) ( ) ( )∗
Σ⋅Σ= jijijiS ,,,

Perform Detections Clustering and compute Range and Doppler spread.
• Clustering
The Target signal may be spread in more then one
Σ Range-Doppler cell. Clustering Process is to group
the detections in the Σ Range-Doppler Map.
Group l parameters are mean and spread:
( )
( )
( )
( )∑
∑
∑
∑
==
i
l
i
ll
l
i
l
i
ll
l
jiS
jiSi
i
jiS
jiSi
i
,
,
&
,
,
2
2
( )
( )
( )
( )∑
∑
∑
∑
==
i
l
i
ll
l
i
l
i
ll
l
jiS
jiSj
j
jiS
jiSj
j
,
,
&
,
,
2
2
Range
Doppler
integer=∆+= mRiRmR lsunambiguoul
Rii llRl
∆−=
22
σ
integer=∆+= nfifnf lsunambiguoul
fjj llfl
∆−=
22
σ
If the spread of Target Range/Doppler spread σRl/ σRl are too high, we may remove the
Target detection assumption and declare the group l as Clutter.
l
Radar
l
f
f
c
R
2
=
ll
f
Radar
R
f
c
σσ
2
=

Perform Detections Clustering and compute Range and Doppler spread.
•  Altitude Line and Main Lobe Clutter
The Interceptor altitude above ground hI is unknown.
Therefore is necessary to search for Altitude Line
and the Main Lobe Clutter in order to properly choose
the PRFs and the Σ Range-Doppler Map.
clutterdf _
( )RangeR
( )RangeR
Clutter
No Clutter
Clutter
Power
Clutter
Power
Main Lobe
Clutter
(MLC)
Altitude
Return
λ
MV2
p
MV
θ
λ
cos
2
AA
M
e
V
coscos
2
ψ
λ
p
MV
θ
λ
sin
2
p
MV
θ
λ
cos
2
−
Target
Range
Target
Doppler
( ) ApA
I
e
h
ψθ cossin +
1
2
N
1 2 M

Range-Doppler Map
•  Check that the detection are from returns in
    the Main Lobe by comparing the signal power
    with the antenna Γuard power.
( ) ( ) ( ) ∗∗
Γ⋅Γ>Σ⋅Σ= jijijiS ,,,
Window
yUncertaint
Target
    If true the received signal is in the Main Lobe
    If not the received signal is in the Side Lobe and
    therefore rejected.

SOLO ACQUISITION MODE
  During Acquisition Mode the RADAR Seeker performs the following tasks:
•   Slaves the Seeker Gimbals to the Designated Target direction.
•   The Angular Tracker is initialized.
•   Confirms that the Detection is steady and in the Designated Zone by solving the
     ambiguities in Range and Doppler by using a number of Batches with different
     PRFs (Pulse Repetition Frequency).
•   The Angular Tracker uses the Δ Elevation and Δ Azimuth Maps, computes the
     Radar Errors in the Detected Range-Doppler cells, and  controls the Antenna Beam
     in the Track Mode, by closing the track loops.
•   Compute the Σ and Δ Range-Doppler Maps.

SOLO ACQUISITION MODE
In the Acquisition Mode the RADAR Seeker Signal Processor continue to
Perform Detection in the Target Uncertainty Window of the Σ Range-Doppler Map as
in Detection Mode, performing Detection cells Clustering.
The Δ Elevation and Δ Azimuth Maps, are used to compute the Angular Radar Errors
in the Detected Range-Doppler cells. For a cluster of l cells:
( ) ( )
( ) ( )∑ 







Σ⋅Σ
∆⋅Σ
= ∗
∗
lCluster ll
AzlldbAz
Az
jiji
jiji
,,
,,
Re
2
3θ
ε
( ) ( )
( ) ( )∑ 







Σ⋅Σ
∆⋅Σ
= ∗
∗
lCluster ll
EllldbEl
El
jiji
jiji
,,
,,
Re
2
3θ
ε

SOLO
Radar Technologies & Applications

SOLO Anti – Ballistic Missiles
AN/FPS – 108 Cobra Dana
Calibration Fixture
First deployed in 1977, the AN/FPS-108 radar
operates in the 1215-1400 MHz band using a 29m
phased array antenna. The primary mission is to
track and collect data on foreign intercontinental
ballistic missile (ICBM) and submarine launched
ballistic missile (SLBM) test launches to the
Kamchatka impact area and the broad ocean impact
areas in the Pacific Ocean. The metric and signature
data collected support START 2 and INF treaty
monitoring, and scientific and technical intelligence
efforts.
Aleutian Islands
Raytheon
UHF Phased Array
30 m diameter
35,000 elements
25,000 nmi range
http://www.fas.org/spp/military/program/track/cobra_dane.htm
Radars for Ballistic Missile Defense

AN/FPS-115 PAVE PAWS Radar
PAVE PAWS reached initial operating
capability 4 April 1980 at Otis AFB in
Massachusetts, and 15 August at Beale AFB,
California
PAVE is an Air Force program name, that,
contrary to some reports, does not have an
expansion, while PAWS stands for Phased
Array Warning System. The radar is used
primarily to detect and track sea-launched
and intercontinental ballistic missiles. The
system also has a secondary mission of Earth-
orbiting satellite detection and tracking.
Information received from the PAVE PAWS
radar systems pertaining to SLBM/ICBM and
satellite detection is forwarded to the United
States Space Command's Missile Warning
and Space Control Centers at Cheyenne
Mountain Air Force Base, Colo. Data is also
sent to the National Military Command
Center and the US Strategic Command.
http://www.fas.org/spp/military/program/track/pavepaws.htm
•UHF Phased Array
•1792 elements
•22.1 meter diameter
•3,000 nmi

AN/FPS-115 PAVE PAWS Radar
Peak Power 1,792 active elements at 325
watts = 582.4 kilowatts (kW)
Duty Factor 25% (11% search, 14%
track)
Average Power 145.6 kW
Effective Transmit
Gain
37.92 dB
Active Radar Diameter 22.1 m
Frequency 420 MHz – 450 MHz
Radar Detection Range 5,556 km (3,000 nmi)
Wavelength 0.69 m at 435 MHz
Sidelobs -20 dB (1st
), -30 dB (2nd
)
-- 38 dB (root mean square)
Face Tilt 20 degrees
Number of Faces 2
3 db Beam Width 2.2 degrees
Specifications
http://www.fas.org/spp/military/program/track/pavepaws.htm

Cobra Judy Ballistic Missile Tracking Radar AN/SPQ - 11
http://en.wikipedia.org/wiki/AN/SPQ-11
Close up view of the front of Cobra Judy radar,
1983
Passive electronically scanned array 2900-3100 MHz (EF band), 22.5 foot diameter,
12,288 elements.

ACTIVE PHASED ARRAY RADAR (APAR)
http://www.thales-systems.ca/projects/apar/apar.pdf
During live missile firing tests held by the Royal Netherlands Navy (RNLN) in March 2005, the
APAR radar system successfully guided two Evolved SeaSparrow Missiles (ESSM) and two
Standard Missiles (SM2) simultaneously to various targets, destroying them all.
APAR, Thales' Active Phased Array
Radar, is the world's most sophisticated
multi-function radar. Its non-rotating
antenna houses four faces that together
cover the full 360 degrees. Each face
consists of more than 3000 very small
radar transmitter/receiver (T/R)
elements, giving the radar its unique
capabilities and high operational
availability. The inherent agility of
APAR guarantees a high performance in
the most adverse conditions, under
severe electronic protection measures.
APAR makes use of Interrupted
Continuous Wave Illuminations (ICWI)
technology, a concept that has been
developed in the international Tri-lateral
Frigate Cooperation formed by the
Netherlands, Germany and Canada.
http://www.thales-nederland.nl/nl/news/archive/2005/april26-2005.shtml
http://www.netherlands-embassy.org/tromp/prapar.htm

AN/TPS-59 (V)3  Tactical Missile Defense Radar
  Developed for the United States Ballistic Missile Defense Organization
(BMDO) and the United States Marine Corps, the TPS-59 (V)3 is designed to
operate with HAWK and Patriot.
When integrated with HAWK, the TPS-59 (V)3/HAWK system is the most cost
effective TMD system currently in production with successfully validated
performance against ballistic missiles as well as air breathing threats.
The radar has been designed to be rapidly transported by truck, helicopter, or C-
130 cargo plane.
Performance
    Frequency 1215 – 1400 Hz
    Transmitter Power 46 kW
Tactical Ballistic Missiles
   Range 400 nmi (740 km) with continuous
coverage to 106
ft (305 km)
   Elevation Beam Steering -5º to 60º
   Azimuth Sector Coverage 360º
Launch/Impact Point prediction 3-5 km circular probability for 50 – 750
km range TBMs
Surveillance Volume 95 x 10 nmi3
(603
x 106 km3
)
Air Breathing Targets
    Range 300 nmi (555 km) with continuous
coverage to 105
ft (30.5 km)
    Elevation Beam Steering -2º to 20º
    Azimuth Sector Coverage 360º
Reliability
    MTBF             2,000 hours
    Availability     0.9947
Lockheed MartinRadars for Ballistic Missile Defense

Sea-Based X-Band Radar
Sea-Based X-Band Radar is a floating, self-propelled,
mobile radar station designed to operate in high winds
and heavy seas. It is part of the United States
Government's Ballistic Missile Defense System.
The Sea-Based X-Band Radar is mounted on a 5th
generation Norwegian-designed, Russian-built CS-50
semi-submersible twin-hulled oil-drilling platform.
Conversion of the platform was carried out at the
AMFELS yard in Brownsville, Texas; the radar mount
was built and mounted on the platform at the Kiewit
yard in Ingleside, Texas, near Corpus Christi. It will be
based at Adak Island in Alaska but can roam over the
Pacific Ocean to detect incoming ballistic missiles.
ST. LOUIS, Jan. 10, 2006 -- Boeing [NYSE: BA]
announced today the arrival in Hawaii of the Sea-
Based X-Band Radar (SBX) built for the U.S. Missile
Defense Agency. This marks an interim stop in the
vessel's transport operation, originating in the Gulf
of Mexico and maneuvering through the Straits of
Magellan, ultimately destined for Adak, Alaska.
http://cryptome.sabotage.org/sbx1-birdseye.htm

Skolnik, M.I., “Introduction to RADAR Systems”,
3th
Ed., 2003
Mahafza, B.R.,“Radar Systems Analysis and Design Using MATLAB”,
Chapman & Hall, 2000
Skolnik, M.I., “RADAR Handbook”, McGraw Hill, 2nd
Ed.,
Stimson, G.W., “Introduction to Airborne RADAR”,
References RADAR Basics

Baton, D.K., “Radar System Analysis And Modeling”,
Long, M.W.,“Radar Reflectivity of Land and Sea”, Artech House,
Baton, D.K., “Modern Radar System Analysis”,
Lacomme, P., Hardange, J.-P., Marchais, J.-C., Normant, E.,
“Air and Spaceborne Radar Systems: An Introduction”, SciTech Publishing, 2001

Knott, E.F., Schaeffer, J.F., Tuley, M.T., “Radar Cross Section”, 2nd
Ed.,
Knott, E.F., “Radar Cross Measurements”
Kolosov, A., “Over-the-Horizon Radar”, Artech House, 1987
Carpentier, M.H., “Principles of Modern Radar Systems”, Artech House, 1988
Le Chevalier, F., “Principles of Radar and Sonar Signal Processing”, Artech House, 2002
References

Blackman, S., Popoli, R.,“Design and Analysis of Modern Tracking Systems”,
Artech House, 2nd
Ed, 1999
Blackman, S.,“Multiple Trget Tracking with Radar Applications”,
Artech House, 1986
Bar Shalom, Y., Li, X.R., Kirubarajan, T.,“Estimation with Applications to Tracking
and Navigation”,
Bar Shalom, Y.,“Multitarget-Multisensor Tracking :Applications and Advances”, Vol. 2,
Artech House, 1992
References
RADAR Basics

Wehner, D.R., “High-Resolution Radar ”, Artech House, 2nd
Ed., 1995
Carrara, W.G/. Goodman, R.S., Majewski, R.M.,“Spotlight Synthetic Aperture Radar:
Signal Processing Algorithms”, Artech House, 1995
Rihaczek, A.,“Principles of High Resolution Radar”, Artech House, 1996
Soumekh, M, “Synthetic Aperture Radar Signal Processing with MATLAB Algorithms “,
John Wiley & Sons, 1999
References
RADAR Basics

Balanis, C.A., “Antenna Theory: Analysis and Design ”, 2nd
Ed., John Wiley, 2005
Tsui, J.B., “Microwave Receivers with Electronic Warfare Applications”, John Wiley,
2nd Ed., 2005
Nathanson, F.E.,”Radar Design Principles: Signal Processing and the Environment”,
McGraw Hill, 1st
Ed., 1969,2nd
Ed.,1991
Macfadzean, R.H.M.,” Surface-Based Air Defense Systems Analysis”, Artech House,
1992
References
RADAR Basics

DiFranco, J.V., Rubin, W.L., “Radar Detection”, Artech House, 1981
Barkat, M.,“Signal Detection And Estimation”, Artech House, 1991
Schleher, D.,C., Ed.,“Automatic Detection and Radar Data Processing”, Artech House,
1980
Minkoff, J.R.,“Signals, Noise and Active Sensors: Radar, Sonar, Laser Radar ” ,
John Wiley & Sons, 1992
References
RADAR Basics

Barton, D.K., “Radars Volume 4: Radar Resolution and Multipath Effects ”,
Artech House, 1975
Barton, D.K., “Radars Volume 2: Radar Equation”, Artech House, 1974
Barton, D.K., “Radars Volume 1: Monopulse Radar”, Artech House, 1977
Barton, D.K., “Radars Volume 3: Pulse Compression”, Artech House, 1974
References
RADAR Basics

Barton, D.K., “Radars Volume 7: CW and Doppler”, Artech House, 1978
Barton, D.K., “Radars Volume 5: Radar Clutter”, Artech House, 1974
Barton, D.K., “Radars Volume 6: Freqency Agility and Diversity”, Artech House, 1974
References
RADAR Basics

Morris, G., “Airborne Pulsed Doppler Radar”, Artech House, 1996
Scheer, J.A., Kurtz, J.L.,Ed., “Coherent Radar Performance Estimation” , Artech House,
1993
Jenn, D., “Radar and Laser Cross Section Engineering: Lessons Learned from the
Aviation Industry”, American Institute of Aeronautics & Astronautics, 2005
Nitzberg, R.,”Adaptive Signal Processing for Radar”, Artech House, 1991
Currie, N.C.,Ed., “Techniques of Radar Reflectivity Measurement, Atech House, 1984
References
RADAR Basics

Hovanessian, S.A.,“Radar Detection and Tracking Systems” , Artech House,1973
Hovanessian, S.A.,“Radar System Design and Analysis” , Artech House,1984
Levanon, N.,“Radar Principles” , John Wiley & Sons, 1988
Peebbles, P.Z., “Radar Principles “, John Wiley & Sons, 1998
References
RADAR Basics

Brookner, E.,“Tracking and the Kalman Filter Made Easy” , John Wiley & Sons, 1998
Brookner, E., Ed., “Radar Technology” , Artech House
Cook, C.C., Bernfeld, M., “Radar Signals: An Introduction to Theory and Application”,
Artech House, 1993
Schleher, D.C.,“MTI and Pulsed Doppler Radar”, Artech House, 1991
References
RADAR Basics

Galati, G., Ed.,“Advanced Radar Techniques and Systems”,
IEE Radar, Sonar, Navigation and Avionics Series 4, Peter Peregrinus Ltd., 1993
Sabatini, S., Tarantino, M., “Multifunction Array Radar: System Design and Analysis”,
Artech House, 1994
Ulaby, F.T., Fung, A.K., Moore, R.K., “Microwave Remote Sensing, Active and Passive:
Radar Remote Sensing and Surface Scattering and Emission Theory”, Vol. 2,
Artech House, 1982
Farina, A., “Antenna-Based Signal Processing Techniques for Radar Systems”,
Artech House, 1992
References
RADAR Basics

Barton, D.K., Leonov, A.I., Leonov, S.A., Morozov, A.I., Hamilton, P.C.,
“Radar Technology Encyclopedia ” , Artech House, 1997
Jelalian, A.V.,“Laser Radar Systems”, Artech House, 1991
Edde, B., “Fundamentals of Radar: Self Study Course”, IEEE, 1999
Blake, L.V., “Radar Range Performance Analysis”, Artech House, 1986

Zmuda, H., Touglian, E.N., “Photonic Aspects of Modern Radar ” , Artech House, 1994
Neri, F., “Introduction to Electronic Defense Systems”, SciTech Publishing, Incorporated,
2006
References
RADAR Basics

SOLO
References
RADAR Basics
1. S. Hermelin, “My RADAR Reference Books”,
2. S. Hermelin, “Electromagnetic Waves & Photons”,
3. S. Hermelin“Short History of Radar Beginnings”,
4. S. Hermelin “Airborne Radars1”,
. “Airborne Radars2”,
. “Airborne Radars Examples2”,
. “Airborne Radars Examples1”,
5. S .Hermelin, “Pulse Radar Doppler Seeker”,

SOLO
References
S. Hermelin, “Range & Doppler Measurements in RADAR Systems”,
RADAR Basics
S. Hermelin, “Clutter Models”,
S. Hermelin, “Radar Signal Processing”,
S. Hermelin , “Fourier Transforms in Radar”,
S. Hermelin, “Matched Filters and Ambiguity Functions for Radar Signals”,
S. Hermelin, “Pulse Compression Waveforms”,
S. Hermelin, “Detection Decisions”,
Georgia Tech Lectures in RADAR

January 15, 2015 215
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 –
Stanford University
1983 – 1986 PhD AA

http://www.radartutorial.eu

SOLO
Formation of Standing Waves
Standing wave in a string (both ends clamped).
Formation of standing wave through reflection
of a sinusoidal wave at a fixed end.

SOLO
“Introduction to Airborne Radar”,
George W. Stimson, 2nd Ed.
Scitech Publishing

CW semi-active seeker
MISSILE SIGNALS-AMPLITUDEMISSILE SIGNALS-AMPLITUDE
vs FREQUENCYvs FREQUENCY
SOLO

Pulse Radar Parameters
Transmitted Signal
Tr
Tp
F0
Tr – Pulse Repetition Interval (PRI)
Tp – Transmitted Pulse Width
F0 – Transmitted RF frequency
Pp – RF peak power
D.C – Duty Cycle = Tp/Tr
Pav – RF average power = Pp*D.C

Pulse Doppler Radar – Clutter
Altitude
return
Antenna
Main Beam
Antenna
Side Lobes
Ground
Target
Power
Frequency
Altitude Side lobe
Main Lobe
Incoming TargetReceding Target
λ
Vc2
Doppler
Range
Radar Altitude
Vradar
Side lobe
Clutter
Main Lobe
Clutter
Incoming
Targets
Receding
Targets
Crossing
Targets
Noise limited
Clutter limited

Pulse Doppler Radar – Clutter
Clutter Cross Section in Doppler Cell
clutteratGainAntenna)G(
tcoefficienReflection
widthgateRangeR
cluttertodirectionflighbetweenAngle
VelocityRadarV
LengthWavedTransmitte
thfilter widFFTB
cluttertoRange
)()
sin2
(
a
0
g
0
=
=
=
=
=
=
=
=
=
θ
σ
θ
λ
θσ
θ
λ
σ
R
GR
V
RB ag
2 cos
D
V
f
θ
λ
=
2 cosD
B
B
f V
λ
θ
=

Detection of Radar Signals
Swerling Models
Case 1
The echo received from a target on any look is constant but are
independent (uncorrelated) from look to look
The probability-density function for the RCS:
nsfluctuatiotargetalloverRCSaveragetheis
)exp(
1
)(
av
avav
p
σ
σ
σ
σ
σ −=
Case 2
The probability-density function for the RCS is same as for Case 1, but
the fluctuations are more rapid
Cases 1 and Case 2 apply to a target consisting of many independent
scatterers equal in RCS - Aircrafts

Probability Density Swerling 1

Swerling Models
Case 3
The fluctuation is assumed to independent from look to look as in Case
1, but the probability density function is given by
)
2
exp(
4
)( 2
avav
p
σ
σ
σ
σ
σ −=
Case 4
The probability-density function for the RCS is same as for Case 2, but
the fluctuations are more rapid
Case 3 and Case 4 apply to a target that can be represented as one large
reflector together with other small reflectors - Ships
In all the above cases the RCS value in the radar equation is the
average RCS.
The probability of detecting a given RCS can be calculated

Probability Density Swerling 3

Noise and False Alarm
The noise is assumed to be Gaussian with probability density function
noisetheofvaluesquare-meantheis
dvvandvvalueebetween thvvoltagenoisethefindingofyprobabilittheis)(
)
2
exp(
2
1
)(
0
0
2
0
ψ
ψπψ
+
−=
dvvp
dv
v
dvvp
The probability that the noise envelope will exceed the voltage threshold VT is Pfa
)
2
exp(
0
2
ψ
T
fa
V
P −=

Integration of pulse trains
The probability of detection for M-out-of-N:
∑=
−
−
−
=
N
Mj
jN
d
j
dd pp
jNj
N
P )1(
)!(!
!
Pd probability of single detection
And the probability of false alarm
∑=
−
−
−
=
N
Mj
jN
n
j
nn pp
jNj
N
P )1(
)!(!
!
Pn probability of false alarm in single detection

•Extraction of Information
• PRF selection guide lines:
– Incoming targets – High PRF
• No Doppler ambiguity
• Range ambiguity
– Receding targets – Medium PRF
• Range and Doppler ambiguity
– Crossing targets – low PRF given target and
main lobe clutter are not co-range – other wise
no detection
• No range ambiguity
• Doppler ambiguity

Extraction of Information
λ
cV
PRF 4≥

The Range coverage equals PRI – 2*Pulse_width + recovery time
The pulse width is defined by PRI and maximum available duty cycle
Example:
PRF = 100 KHz; Duty cycle 10%
PRI = 1/PRF = 10 µsec= 1500 meters
Maximum pulse width = 150 meters
Maximum range coverage 1500 – 300 = 1200 meters.
In this example if range uncertainty is smaller than 1200 meters, one PRF is suffice.
Otherwise a set of PRF’s must be selected to cover the uncertainty.
All targets ranges above PRI are folded within PRI – range ambiguity

Target range = N PRI + R1
Where R1 = Target Range within PRI
N = number of folds 0…N
)(
1
PRI
R
fixN
PRIRR
T
T
=
⊗=
Example
RT=12,300 m PRI = 1500 m
N = fix(12300/1500) = 8 R1= 12300-1500*8=300 meters

PRI1
PRI2
R1
R2
Target

Resolving range ambiguity
Use at least 2 PRF’s so
RT = N PRI1+R1 = M PRI2 + R2
PRF1 and PRF2 are derived from basic clock so K1*clock= PRF1 and K2*clock=PRF2
Maximum unambiguous range is 1/ K1*K2*clock.
Where K1 and K2 are prime numbers.
Example
Clock = 20 KHz K1=6 K2=7
PRF1 = 120 KHz PRF2 = 140 KHz
PRI1= 1250 m PRI2~ 1070 m
R1= 950 m R2= 70 m
True range = 7*1250+950= 9*1070+70=9700 m
N= 7 M=9
Same applies for resolving the Doppler ambiguity.

Extraction of InformationRange Tracking
Target
Amplitude
Range Sampling
Range Gate
CFAR Threshold
E1
E2
E3
R11 R12 R13
∑
++
=
3,2,1
313212111
rangeTarget
E
ERERER

Range Tracking
The goal is to predict where the target range will be on following
detection
Based on current range using a tracking algorithm (KALMAN) the
following target range is predicated
The error between real position and actual position is calculated and the
tracking parameters are updated
Tracking algorithms implementing 2 integration will extract the range
and range rate for the target on line of sight.
Comp Int Int
True Range
Predicted Range
Closing Velocity
Range Error

Doppler Tracking
Target
Amplitude
Range Sampling
Doppler Gate
CFAR Threshold
E1
E2
E3
D11 D12 D13
∑
++
=
3,2,1
313212111
DopplerTarget
E
EDEDED

Doppler Tracking
The goal is to predict where the target Doppler will be on following
detection
Based on current Doppler using a tracking algorithm (KALMAN) the
following target Doppler is predicated
The error between real Doppler and actual Doppler is calculated and the
tracking parameters are updated
Tracking algorithms implementing 1 integration will extract the Doppler
on line of sight.
Comp Int
True Doppler Predicted DopplerDoppler Error

Angular Information
The Monopulse concept:
L
θ
θ
L sinθ
The path difference between the signals as received at each lobe is L sin θ
The phase difference φ for a wavelength λ:
λ
α
πφ
sin
2
l
=
The outputs from lobes are added and subtracted as vectors
)
2
sin(
)
2
cos(
φ
φ
∆
Σ
=∆
=Σ
K
KSUM
Difference

Sum and Difference patterns
Sum
Difference
λ
α
πφ
sin
2
l
=
Output level

Σ∆=
Σ
∆
=
andbetweenangle
))sin(tan(
ϕ
ϕaError
Error output
Target angle

Interference
Multipath
Direct wave
Reflected
wave Radar
Target
Image
TxRi Rt
Reflecting Surface
t
Ht
Hr
R

Interference
Multipath
Phase difference corresponding to the path-length difference
R
HH rt
d
22
λ
π
ϕ =
The ratio of power incident on the target compared to a target in free space (no multipath)
)
2
(sin4
R
HH
KP tr
r
λ
π
=
So the power at a specific ranges can be decrease to zero – fading effect.
Additional effect is an angular tracking error due to the presence of image

Interference
ECM
The Signal to Jammer ration is Prt/Prj
4
2
4 t
j
jrjj
rtt
rj
rt
R
R
GGP
GGP
P
P
π
σ
=
Gjr = is the radar antenna gain in the direction of jammer
Rj = range radar to jammer
Rt = range radar to target
Pj = is the jammer power within radar bandwith

Interference
• Jamming techniques
– ON-OFF
– RGPO
– VGPO
– Towed
– Expandable
– Chaff
– Cross-eye
– Inverse gain
– others
ECM

SOLO
Figure: estimating of the angular position
Monopulse Concept
Monopulse radars find their origin in tracking systems.
Since the late 1970s, the principle of monopulse has been
adapted to suit PSR and SSR systems and is in common
operational use world-wide today
A target will be seen by a radar from the moment it enters the main antenna beam or
from the moment it is illuminated by the transmitted radar antenna beam. A search
radar always makes an error in the determination of the direction of the target
because it makes the assumption that the target is situated in the direction of the axis
of the main beam of the antenna. This error is of the order of the beam width of the
main antenna beam.

http://atcsl.tripod.com/radar_theory.htm

AV
TxV
R
Airborne Radar Vehicle
TsV
TV
Ground Moving Target
r∆
r∆
Range Resolution






=∆
2
81.1 c
BW
r 





=∆
2
81.1 λ
T
r
Doppler Resolution
( )
( )
( ) ( ) ( )trtVRtr
ttVRRtr
VVV
Ts
Ts
TsTxA
/
2
:
2
222
222
γ
γ
γ
+=
++=
+−=

Significant range walk
with large BW and T
Significant Doppler walk
with large T c - speed of light
BW – bandwidth
λ - wavelength
T - CPI time
VA - aircraft velocity
VT - target velocity
R - stand-off Range
Long CPI can lead to target doppler walk or smearing.
The degree of smearing is a function of λ2
.
Mechanism for Moving Target Smearing

Range & Doppler Measurements in RADAR SystemsSOLO
Chinese Remainder Theorem
The original form of the theorem, contained
in a third-century AD book by Chinese
mathematician Sun Tzu and later republished
in a 1247 book by Qin Jiushao.
Suppose n1, n2, …, nk are integers which are
pairwise coprime. Then, for any given
integers a1,a2, …, ak, there exists an integer x
solving the system
1 1 1 1 1
2 2 2 2 2
1 2
0
0
0
, , , integers
k k k k k
k
x n t a n a
x n t a n a
x n t a n a
t t t are
≡ + > >
≡ + > >
≡ + > >
L L L L L
L
or in modern notation
( )mod 1,2, ,i ix a n i k≡ = L ai is the reminder of x : ni
x

Chinese Remainder Theorem (continue – 1)
A Constructive Solution to Find x
( )mod 1,2, ,i ix a n i k≡ = L
x
Define 1 2: kN n n n= L
For each i, ni and N/ni are coprime.
Using the extended Eulerian
algorithm we can therefore find
integers ri and si such that
( )/ 1i i i irn s N n+ =
Define
Therefore ei divided by ni has the remainder 1 and divided by nj (j≠i) has the remainder 0,
because of the definition of N.
( ): / 1i i i i ie s N n rn= = −
( ) ( )1 mod 0 modi i i je n and e n i j= = ∀ ≠
Because of this the solution is of the form
1
k
i i
i
x a e
=
= ∑ But also ( )
1
mod
k
i i
i
a e x N
=
=∑

Chinese Remainder Theorem (continue – 2)
A Constructive Solution to Find x (Example)
( )mod 1,2, ,i ix a n i k≡ = L
1 2 3: 60N n n n= × =
( )
( )
( )
2 mod 3 ,
3 mod 4 ,
1 mod 5 .
x
x
x
≡
≡
≡
1 2 33, 4, 5n n n= = =
1 2 3/ 20, / 15, / 12N n N n N n= = =
( ){ { { {
11 1
1
/
13 3 2 20 1
sn N n
r
 
− + = ÷
 
( ){ { { {
2 2 2
2
/
11 4 3 15 1
n s N n
r
 
− + = ÷
 
( ){ { ( ){ {
33
3 3
/
5 5 2 12 1
N nn
r s
 
+ − = ÷
 
( ): /i i ie s N n= ( )1 : 2 20 40e = = ( )2 : 3 15 45e = = ( )3 : 2 12 24e = − = −
1 2 32, 3, 1a a a= = =
( )1 1 2 2 3 3 2 40 3 45 1 24 191x a e a e a e= + + = × + × + × − =
Check:
191 63 3 2 47 4 3 38 5 1= × + = × + = × +
( )/ 1i i i irn s N n+ =Find ri and si such that:
Compute:
Therefore:
and ( )11 191 11 mod 60x N= ¬ = =
11 3 3 2 2 4 3 2 5 1= × + = × + = × +

The transmitted RADAR RF
Signal is:
( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=
E0 – amplitude of the signal
f0 – RF frequency of the signal
φ0 –phase of the signal (possible modulated)
The returned signal is delayed by the time that takes to signal to reach the target and to
return back to the receiver. Since the electromagnetic waves travel with the speed of light
c (much greater then RADAR and
Target velocities), the received signal
is delayed by
c
RR
td
21 +
≅
The received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisettttftEtE ddr +−+−= ϕπα 000 2cos
To retrieve the range (and range-rate) information from the received signal the
transmitted signal must be modulated in Amplitude or/and Frequency or/and Phase.
ά < 1 represents the attenuation of the signal

The received signal is:
( ) ( ) ( ) ( )[ ] ( )
( ) ( ) ( )tnoise
c
RR
tRRtftE
tnoisettttftEtE
fc
ddr
+










 +
−++−=
+−+−=
=
21
21
0
000
/
000
2
2cos
2cos
00
ϕ
λ
π
πα
ϕπα
λ
If we consider only (c = speed of light) then the frequency of the electromagnetic
wave that reaches the receiver is given by:
c
td
Rd
<<




















+
−≈






+












+−=











 +
−+




 +
−=
c
td
Rd
td
Rd
f
c
td
Rd
td
Rd
ud
d
ff
c
RR
t
c
RR
tf
td
d
f
21
0
21
0~
00
2121
0
1
2
1
2
2
1

ϕ
π
ϕπ
π
λ






+
−=
td
Rd
td
Rd
fd
21
is the doppler frequency shift at the receiver
Christian Johann Doppler first observed the effect in acoustics.

TV
1R
If the Radar Receiver is at a distance R1 from the Target and the Receiver is
at a distance R2 from the Target, then the frequency of the carrier wave at the
Receiver is:
λλ






−






−=
td
Rd
td
Rd
ff
21
0
( ) 11111
1
// RRVVRRR
td
Rd
ET

⋅−=⋅= is the relative velocity between the Target and the
Radar source along the line of sight between them
( ) 22222
2
// RRVVRRR
td
Rd
MT

⋅−=⋅= is the relative velocity between the Target and the
Receiver along the line of sight between them
SOLO
Doppler Frequency Shift

Matched Filters in RADAR SystemsSOLO
α MV
R
EV
Target
Transmitter&
Receiver
The transmitted RADAR RF
Signal is:
( ) ( ) ( )[ ]ttftEtEt θπ += 00 2cos
( )
c
tR
td
02ˆ ≅
Since the received signal preserve the envelope shape of the known transmitted signal
we want to design a Matched Filter that will distinguish the signal from the receiver noise.
the received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisetttffttEtE dDdr +−++−≈ ˆˆ2cosˆ 00 θπα
Scaled Down
In Amplitude
Two-Way
Delay
Possible
Phase ModulationDoppler
Frequency
( ) ( )
λ
λ
0
/
0
0 22ˆ
0 tR
f
c
tR
f
fc
D

−=−≅
=
For R1 = R2 = R we obtain that

SOLO Review of Probability
Exponential Distribution
( )
( )



<
≥−
=
00
0exp
;
x
xx
xp
λλ
λ
( ) ( )
( )
( ) ( )
λ
λλ
λλ
λλ
1
expexp
exp
0
0exp
0
=−+−−=
−=
∫
∫
∞
∞
=
−=
∞
dxxxx
dxxxxE
xu
dxxdv
( ) ( ) ( ) 2
22 1
λ
=−= xExExVar
( ) ( )[ ] ( ) ( )
( )[ ]
1
0
0
1exp
expexpexp
−∞
∞






−=−
−
=
−==Φ ∫
λ
ω
λω
λω
λ
λλωωω
j
xj
j
dxxxjxjEX
Probability Density Functions
Cumulative Distribution Function
Mean Value
Variance
Moment Generating Function
( ) ( )
( )



<
≥−−
=−= ∫∞−
00
0exp1
exp;
x
xx
dxxxP
x
λ
λλλ
( ) ( ) 2
0
2
2
22 2
λω ω
=
Φ
=
=
d
d
jxE X
Distributions
examples
Table of Content

Chi-square Distribution
( )
( )
( )
( )
( )





<
≥−
Γ=
−
00
02/exp
2/
2/1
;
2/2
2/
x
xxx
kkxp
k
k
( ) kxE =
( ) kxVar 2=
( ) ( )[ ]
( ) 2/
21
exp
k
X
j
xjE
−
−=
=Φ
ω
ωω
Mean Value
Variance
( )
( )
( )





<
≥
Γ=
00
0
2/
2/,2/
;
x
x
k
xk
kxP
γ
Γ is the gamma function ( ) ( )∫
∞
−
−=Γ
0
1
exp dttta a
( ) ( )∫ −= −
x
a
dtttxa
0
1
exp,γγ is the incomplete gamma function
Distributions
examples

Student’s t-Distribution
( ) ( )[ ]
( ) ( )( ) 2/12
/12/
2/1
; +
+Γ
+Γ
= ν
ννπν
ν
ν
x
xp
( )



=
>
=
1
10
ν
ν
undefined
xE
( )
( )



∞
>−
=
otherwise
xVar
22/ ννν
Mean Value
Variance
Moment Generating Function not defined
( ) ( )[ ]
( )
∑
∞
=






−











 +






Γ
+Γ
+=
0
2
!
2
3
2
1
2
1
2/
2/1
2
1
;
n
n
n
nn
n
x
x
xP
ν
ν
ννπ
ν
ν
∞
−
−=Γ
0
1
exp dttta a
( ) ( ) ( ) ( )121: −+++= naaaaa n L
It get his name after W.S. Gosset that wrote
under pseudonym “Student”
William Sealey
Gosset
1876 - 1937
Distributions
examples
Table of Content

Uniform Distribution (Continuous)
( )





>>
≤≤
−=
bxxa
bxa
abbaxp
0
1
,;
( )
2
ba
xE
+
=
( ) ( )
12
2
ab
xVar
−
=
( ) ( )[ ]
( ) ( )
( )abj
ajbj
xjE
−
−
=
=Φ
ω
ωω
ωω
expexp
exp
Mean Value
Variance
( )







>
≤≤
−
−
>
=
bx
bxa
ab
ax
xa
baxP
1
0
,;
Distributions
examples
Moments
Table of Content

Rayleigh Distribution
( ) 2
2
2
2
exp
;
σ
σ
σ






−
=
x
x
xp
( )
2
π
σ=xE
( ) 2
2
4
σ
π−
=xVar
Mean Value
Variance
( ) 





−−= 2
2
2
exp1;
σ
σ
x
xP
( ) ( )








−





−−=Φ jerfi
22
2/exp1 22 σωπ
σωσωω
John William Strutt
Lord Rayleigh
(1842-1919)
Distributions
examples
Moments
Rayleigh Distribution is the chi-distribution with k=2( ) ( ) ( )
( )
( )k
k
k
k
k
k
k U
k
p k
χ
σ
χ
σ
χ
χ 







−
Γ
=
−−−
Χ 2
212/2
2
exp
2/
2/1

Given X and Y, two independent gaussian random variables, with zero means and the
same variances σ2
Example of Rayleigh Distribution
( ) 




 +
−= 2
22
2
2
exp
2
1
,
σσπ
yx
yxpXY
find the distributions of R and Θ given by: ( )XYYXR /tan& 122 −
=Θ+=
( ) ( )
( ) ( ) θθ
σπ
θ
σ
σπσ
θθ
dprdrp
drdrr
ydxdyx
ydxdyxpdrdrp
r
XYR
Θ
Θ
=





−=





 +
−==
22
2
22
22
22
exp
22
exp,,
where:
( ) πθ
π
θ 20
2
1
≤≤=Θp
( ) 0
2
exp 2
2
2
≥





−= r
rr
rpr
σσ
Uniform Distribution
Solution
Table of Content
x
y
r
θ

Rice Distribution
( ) 










 +
−
= 202
2
22
2
exp
,;
σσ
σ
σ
vx
I
vx
x
vxp
( )
2
π
σ=xE
( ) 2
2
4
σ
π−
=xVar
Mean Value
Variance
( ) 





−−= 2
2
2
exp1;
σ
σ
x
xP
( ) ( )








−





−−=Φ jerfi
22
2/exp1 22 σωπ
σωσωω
Stuart Arthur Rice
1889 - 1969
Distributions
examples
where:
( ) ( )∫ 





−=




 π
ϕ
σ
ϕ
πσ
2
0
220
'
2
'cos
exp
2
1
d
vxvx
I

Rice Distribution
The Rice Distribution applies to the statistics of the envelope of the output of a bandpass
filter consisting of signal plus noise.
Example of Rice Distribution
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )[ ] ( ) ( )[ ] ( )tAtntAtn
ttnttntAtnts
SC
SC
00
000
sinsincoscos
sincoscos
ωϕωϕ
ωωϕω
−++=
+++=+
X = nC (t) and Y = nS (t) are gaussian random variables, with zero mean and the same
variances σ2
and φ is the unknown but constant signal phase.
Define the output envelope R and phase Θ:
( )[ ] ( )[ ]
( )[ ] ( )[ ]{ }ϕϕ
ϕϕ
cos/sintan
sincos
1
22
AtnAtn
AtnAtnR
CS
SC
+−=Θ
−++=
−
( ) ( ) ( ) ( )
( )
222
22
22
2
2
2
22
cos
exp
2
exp
22
sin
exp
2
cos
exp,,
σπ
θ
σ
θϕ
σ
σπσ
ϕ
σ
ϕ
θθ
drdrrAAr
ydxdAyAx
ydxdyxpdrdrp XYR





 +
−




 +
−=





 −
−




 +
−==Θ
Solution
( ) ( ) ( ) ( )∫∫ +




 +
−




 +
−== Θ
ππ
θϕ
σ
θϕ
σπσ
θθ
2
0
222
222
0 2
cos
exp
22
exp, d
rArAr
drprp RR

Rice Distribution
Example of Rice Distribution (continue – 1)
( ) ( ) ( ) ( )∫∫ 





−




 +
−== Θ
ππ
ϕ
σ
ϕ
πσσ
θθ
2
0
22
22
2
2
0
'
2
'cos
exp
2
1
2
exp, d
rAArr
drprp RR
where:
( ) ( )∫ 





−=




 π
ϕ
σ
ϕ
πσ
2
0
220 '
2
'cos
exp
2
1
d
rAAr
I
is the zero-order modified Bessel function of the first kind
( ) 










 +
−= 202
22
2
2
exp,;
σσσ
σ
Ar
I
Arr
ArpR Rice Distribution
Since I0 (0) = 1, if in the Rice Distribution we take A = 0 we obtain:
Rayleigh Distribution( ) 





−== 2
2
2
2
exp,0;
σσ
σ
rr
ArpR
Table of Content

Weibull Distribution
( )





<
>≥













 −
−




 −
=
−
00
0,,exp
,,;
1
x
x
xx
xp
αγµ
α
µ
α
µ
α
γ
αµγ
γγ
( ) ( )













 −
−−== ∫∞−
γ
α
µ
αµγαµγ
x
dxxpxP
x
exp1,,;,,;
( ) 





+Γ=
γ
α
1
1xE
∞
−
−=Γ
0
1
exp dttta a
Ernst Hjalmar
Waloddi Weibull
1887 - 1979
Mean Value
Variance( ) ( )22 2
1 xExVar −





+Γ=
γ
α
Distributions
examples
Table of Content

1 radar basic - part ii

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