1 radar basic - part ii

RADAR BasicsPart 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 WavesPolarizationEnergy and MomentumThe Electromagnetic Spectrum

Introduction to Radars

Dipole Antenna Radiation

Interaction of Electromagnetic Waves with Material Absorption and Emission Reflection and Refraction at a Boundary Interface DiffractionAtmospheric Effects

RADAR BASICS - I

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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)

RADAR BASICS - I




SOLO Radar Basics

Non-Coherent Pulse Radar

Pulse Radars

Coherent Pulse-Doppler Radar

Range & Doppler Measurements in Pulse-Radar SystemsRange Measurements

Range Measurement Unambiguity

Doppler Frequency Shift

Resolving Doppler Measurement Ambiguity

ResolutionDoppler Resolution

Angle Resolution

Range Resolution

RADAR BASICS - I




SOLO Radar Basics

Pulse Compression WaveformsLinear FM Modulated Pulse (Chirp)

Phase Coding

Poly-Phase Codes

Bi-Phase Codes

Frank Codes

Pseudo-Random Codes

Stepped Frequency Waveform (SFWF)

RADAR BASICS - I




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 - I




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/CirculatorsFerrite 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 fromRadar Basic – Part I

SOLO Radar Basics



SOLOTRANSMITTERS

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SOLO

Electron Tubesfor RF

and Microwaves

MicrowaveTubes

Low Frequency(Gridded Tubes)

Linear BeamTubes

Crossed FieldTubes

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

Filamentleads

Fig. Cutaway view of a Magnetron

pickup loop

a) slot- typeb) vane- typec) rising sun- typed) hole-and-slot- type

Figure 3: forms of the plate of magnetrons

Albert Wallace Hull (1880 – 1966)

Magnetron

Figure 1: the electron path under theinfluence 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-chargewheel 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.

inductivetuningelements

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


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

inputoutput

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 Travelling Wave Tubecollector

inputoutput

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 thehelical 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 Travelling Wave Tube

Characteristics of a TWTThe 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 electrons 1

• 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


SOLO

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

http://www.microwaves101.com/encyclopedia/transmitreceivemodules.cfm

SOLO


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

ATRTube

TRTube

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 GuidesSOLO

SOLO Receiver Equivalent Noise

Boltzman’s constant

Gain = G1Noise Figure = F1

Gain = G2Noise Figure = F2

Gain = GiNoise Figure = Fi

The gain of the receiver is iGGGG 21 ⋅= The noise figure of the receiver is

i

i

GGG

F

GG

F

G

FFF

2121

3

1

21

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

TrRFω=

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

cvTRF /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 strengthc – velocity of light

We assume that from the Antenna emerge radar signal of the Sum S and Difference D

( ) ( )( ) ( ) ( )cRtpFeVD

cRtpeVStj

tj

TRF

TRF

/2

/2

−∆=

−=−

−

ψωω

ωω

Receiver Intermediate Frequency (IF)

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

IFAmp

IFAmp

Band Passat IF

Band Passat IF

S

D'D

'S

( ) tjst IFRFeLO ωω ±1

Mixer

Mixer

First Intemediate Frequaency (1st IF)

( ) ( )( ) ( ) ( )cRtpFeVD

cRtpeVStj

tj

TRF

TRF

/2

/2

−∆=

−=−

−

ψωω

ωω

The Receiver translates the high RF frequency ωRF to a lower frequency to abetter 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

cRtpeVStj

tj

TIF

TIF

/2''

/2''

−∆=

−=−

−

ψωω

ωω

If the mixing frequency is centered at ωRF± ωIF than the output is centered atωIF and at the image 2 ωRF± ωIF .


SOLO

A second mixing frequency is sometimes added to avoid potential problems withimage frequency.

IFAmp

'S''S

( ) tjnd IFIFeLO ωω 22 ±

Mixer

Second Intemediate Frequaency (2nd IF)

IFAmp

'D

''D

Mixer

PhaseShifter

AGC

AGC Band Passat 2nd IF

Band Passat 2nd IF

( ) ( )( ) ( ) ( )cRtpFeVD

cRtpeVStj

tj

TIF

TIF

/2"

/2"2

2

−∆=

−=−

−

ψωω

ωω

The output of the Second Intermediate Frequency (2nd IF)

( ) ( )( ) ( ) ( )cRtpFeVD

cRtpeVStj

tj

TIF

TIF

/2''

/2''

−∆=

−=−

−

ψωω

ωω



SOLO


SOLO

Coherent Pulse-RADAR Block Diagram

Block Diagram of a Simple Coherent Radar

f0Power

Amplifier

SignalGenerator

CoherentOscillator(COHO)

fLO

fRF

fIF

fIF

f0 + fd

fIF + fd fd

f0=fRF + fIF

IF BP &Variable Gain

Amplifier

CYRCULATOR

SIGNAL

PROCESSOR

ANGLETRACKER

DOPPLERTRACKER

RANGETRACKER

SEEKERLOGIC

RADARCENTRAL

PROCESSOR

RADOME

LOW-PASS-FILTER

ANTENNASTABILIZATION

A/D

ANALOG DIGITAL

FREQUENCYSOURCE

RFIF + RECEIVER

ANTENNA

RF Variable

gainLNA

RF Switch

AGC

Stable Local

Oscillator (STALO)

LNA

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Radar EquationRadar 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 mµ εε µ

= × = = r r

The Power Density (Irradiance) intercepted and scatteredby 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 εε 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: [ ]24

i TGT RCVRS A

wattR

σπ

The received Power is given also by:

2/r RCVRS A watt m

24i TGT RCVR

r RCVR

S AS A

R

σπ

=

2lim 4 rTGT R

i

SR

Sσ 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 EquationRadar 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 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

TransmittedMainlobeEnergy

ECM Pod

Ground

A/C Radar

TransmittedSide-lobeEnergy

Missile RADAR Seeker Transmision

fog (gr/cm3)

rain (mm/hr)

air

Target

Direct-pathTarget Return

ECM Pod

Ground

A/C Radar

Target Reflected Energy Return

fog (gr/cm3)

rain (mm/hr)

air

MultipathTarget 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)


Return

Target


ReceivedMainlobe

ClutterEnergy

ECM Pod

Ground

A/C Radar

ReceivedSide-lobeClutteerEnergy

Ground Clutter Return

fog (gr/cm3)

rain (mm/hr)


Return


Target


TransmittedMainlobeEnergy

ECM Pod

Ground

A/C Radar

TransmittedSide-lobeEnergy

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.

Er

Hr

( ) ( ) ( ) 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/

4mwatt

R

PHES rad

r π=×=

rr

00

00 EH

µε=

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/

4mwattG

R

PHES 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

SG =:

Radar Equation

BϕBϑ

ϕD

ϑD

Antenna

RadiationBeam

Assume for simplicity that the Antenna radiates all the power into the solid angledefined 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

λη

ϑDB

1=ϕϕ

λη

ϕDB

1=

λ - wavelength

ϕϑ DD , - Antenna dimensions in directionsϕϑ,

ϕϑ ηη , - Antenna efficiency in directionsϕϑ,

then ( ) effBB

ADDG22

444

λπηη

λπ

ϕϑπ

ϕϑϕϑ ==⋅

=

where

ϕϑϕϑ ηη DDAeff =:

is the effective area of the Antenna.

2

4

λπ=

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 4AG

R

PASP 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.:

1224AG

R

PP dtransmittereceived π

=therefore

1221 AGAG =or

constA

G

A

G ==2

2

1

1

We already found the constant; i.e.: 2

4

λπ=

A

G

SOLO

Radar Equation

The Power Density (Irradiance) at the target is given by:

TRP

TRG

TRR

EVTarget

Transmitter

[ ]2

Pr

2/

1

4

1mW

LRL

GPS

TGTXMTRopagation

TGTTR

rTransmitte

TR

TRTRr

→

→

=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 EquationThe Power reflected by the target in the receiver direction is:

[ ]WGALRL

GPGASP

TGT

TGTTGT

TGTXMTRopagation

TGTTRTR

rTransmitte

TR

TRTRTGTTGTrTGT

σπ

→

→

==

Pr

24

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

1mW

LRPp

RCVRTGTopagation

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

1mW

LRGA

LRL

GPp

RCVRTGTopagation

RCVRTGTRCVR

TGTTGT

TGTXMTRopagation

TGTTRTR

rTransmitte

TR

TRTRRCVR

TGT

→

→

→

→

=ππ

σ

- Propagation Loss at the Receiver ( > 1 )RCVRL

The Power Density [W/m2] received at the Receiver is

[ ]WL

A

LRGA

LRL

GP

LApP

ceiver

RCVR

RCVR

RCVRTGTopagation

RCVRTGTRCVR

TGTTGT

TGTXMTRopagation

TGTTRTR

rTransmitte

TR

TRTR

RCVRRCVRRCVRCVR

TGT

RePr

2

Pr

2

1

4

11

4

1

/

→

→

→

→

=

=

ππσ

The Power [W/m2] received at the Receiver is

- Effective area in the Receiver Antenna [m2]RCVRA

SOLO

Radar Equation

[ ]WL

G

LRGA

LRL

GPP

ceiver

RCVR

RCVR

RCVRTGTopagation

RCVRTGTRCVR

TGTTGT

TGTXMTRopagation

TGTTRTR

rTransmitte

TR

TRTRRCVR

TGT

Re

2

Pr

2

Pr

2 4

1

4

11

4

1

πλ

ππσ

→

→

→

→

=

the Power [W/m2] received at the Receiver is

πλ

4

2RCVR

RCVR

GA =

( ) [ ]WLLLLRR

GGPP

RCVRRCVRTGTTGTTRTRRCVRTR

TGTRCVRTRTRRCVR

→→

= 223

2

4 πσλ

Using

or

SOLO

Radar Equation

[ ]WL

G

LRGA

LRL

GPP

ceiver

RCVR

RCVRTGTopagation

TGTTRTGTTGT

TGTXMTRopagation

TGTTR

rTransmitte

TR

TRRCVR

TGT

Re

2

Pr

2

Pr

2 4

1

4

11

4

1

πλ

ππσ

→

→

→

→

=

the Power [W/m2] received at the Receiver is

( ) [ ]WLLLR

GPP

RCVRTGTTRTR

TGTTRRCVR 243

22

4 →

=π

σλor

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 −=

5.24

52 =⋅( ) dbdb 134 +=

6.14/5

2 =

( ) dbdb 165 −= 2.34/5

4 =

422 =

( ) dbdb 167 += 54

54 =⋅

( ) dbdb 198 −= 4.64/5

8 =

Decibels GainDecibels = 10 log (Gain)SOLO Decibels

1010

234

111

==

=

db

db

db

db1

4

11+00.1

db5.0

4

5.01+

dbF.0

4

.01

F+

Decibels

Gain

( ) dbdb 9101 −= 4

11

4

11

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

.01

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 performinga certain task.

Clutter

Clutter returns are the vector summation (amplitude and phase) from all of theScattering centers within the radar resolution cell. Thus, the resultant Radar CrossSection (RCS) of the clutter cell is given by:

( )2

1

exp

= ∑

=

scN

kkk j φσσ

where

λππφ kk

Radark

R

c

Rf 4

22 =

= relative phase

Resulta

nt fiel

d for

one p

olariz

ation

1 2

34 5

67

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

jj∑

== 1

• Standard deviation : ( )

11

2

−

−=

∑=

n

xxn

jj

σ


ahMV

pθ

eψ

RψcosR

Ae

Aψ

HorizontalGround

Main LobeBeam

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 angleθp with the horizontal plane.

( )

≤≤−

≤+≤≥

+=

2/2/

0cossin

πψππθ

ψθ p

a

p

a e

hR

e

hR ( )

ψθ

22 cos1cos

R

he a

p −±=+

The doppler frequency shift along the ray R is given by:

( ) ( ) ( )[ ]

ψθψθ

λ

ψθθθθλ

ψλ

cossincoscos

2

cossinsincoscos2

coscos2

22

_

aap

ap

M

ppppMM

clutterd

hR

R

h

R

hV

eeV

eV

Rf

≥

+

−±=

+++==

SOLO Ground Clutter

( )

( ) ( )[ ]

+

−±=

+++=

=

R

h

R

hV

eeV

eV

Rf

ap

ap

M

ppppM

Mclutterd

θψθλ

ψθθθθλ

ψλ

sincoscos2

cossinsincoscos2

coscos2

22

_

( ) pM

aclutterd

VhRf θ

λ

ψsin

20

_

=

==

( ) ψθλ

θ

coscos20

_ pM

e

clutterd

VRf

p =+

=∞→

( ) ψθλ

πθ

coscos2

_ pM

e

clutterd

VRf

p

−=∞→=+

Altitude Line

λψ

θ

M

hR

clutterd

Vef

p

a

20,0

cos

_ =

==

=

clutterdf _

( )RangeR

( )RangeR

Clutter

No Clutter

ClutterPower

ClutterPower

Main LobeClutter(MLC)

AltitudeReturn

λMV2

pMV θ

λcos

2

AAM e

Vcoscos

2 ψλ

pMV θ

λsin

2

pMV θ

λcos

2−

( ) ApA

a

eh

ψθ cossin +

( )

( )

+=

=

ApA

aML

AAM

AAclutterd

e

hR

eV

ef

ψθ

ψλ

ψ

cossin

coscos2

,_

Main Lobe

ahMV

pθ

eψ

RψcosR

Ae

Aψ

HorizontalGround

Main LobeBeam


SOLO Ground Clutter

ah

MV

e

ψ

RψcosR

HorizontalGround


Main LobeBeam

Ae

Aψ

0=pθ( )

22

0

_

cos2

coscos2

−±=

=

=

R

hV

eV

Rf

aM

Mclutterd

p

ψλ

ψλ

θ

( ) 00

_

=

==ψ

aclutterd hRf

( ) ψλ

θ

cos20

_M

e

clutterd

VRf

p =+

=∞→

( ) ψλ

πθ

cos2

_M

e

clutterd

VRf

p

−=∞→=+

Altitude Line

λψ

θ

M

hR

clutterd

Vef

p

a

20,0

cos

_ =

==

=

clutterdf _

( )RangeR

( )RangeR

Clutter

No Clutter

ClutterPower

ClutterPower


AltitudeReturn

λMV2

0=pθ

AAM e

Vcoscos

2 ψλ

λMV2

−

AA

a

e

h

ψcossin

( )

=

=

AA

aML

AAM

AAclutterd

e

hR

eV

ef

ψ

ψλ

ψ

cossin

coscos2

,_

Main Lobe

0=pθ

SOLO Ground Clutter

Ground

ahMV

pθ

ψcosR

AeAψ

Main LobeBeam


Cones ofEqui-Range

Rays

R

Projection ofTransmitter& Receiver

on the Ground

Equi-rangePoints

on the Ground

Projection on the Ground

M.L.B.

The Clutter energy froma range R are obtainedfor all points on the groundthat are at the range R fromthe Transmitter/Receiver.

Assuming a flat ground,the points on the grounda a range R > ha are locatedat the intersection of the conical surface with theapex at the Transmitter/Receiver and its altitude lineas the conic axis..

The points on the flatGround having the samerange R from theTransmitter/Receiverare circles.

SOLO Ground Clutter

Ground

ahMV

pθ

ψcosR

AeAψ

Main LobeBeam


Cones ofEqui-doppler

Rays

R

Intersectionof Missile

Velocity Vectorwith the Ground

Ellipsee pθ<

Parabolee pθ=Hyperbolee pθ>

Equi-dopplerPoints

on the Ground


M.L.B.

The points on the groundthat have the same dopplershift are located on raysstarted from Transmitter/Receiver and are at the same angle relative to the MissileVelocity vector VM.

Therefore the points on the Ground that have the same doppler shift are located on the intersection of the conuswith the apex at theTransmitter/Receiver and the conic axis the MissileVelocity vector VM.

EllipseeFor p ⇒<θParaboleeFor p ⇒= θHyperboleeFor p ⇒> θ

SOLO Ground Clutter

Ground

ahMV

pθ

ψcosR

AeAψ

Main LobeBeam


Cones ofEqui-doppler

RaysCones ofEqui-Range

Rays

R

Projection ofTransmitter& Receiver

on the Ground

Intersectionof Missile

Velocity Vectorwith the Ground

Ellipsee pθ<

Parabolee pθ=Hyperbolee pθ>

Equi-dopplerPoints

on the Ground

Equi-rangePoints

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 OFCLUTTER 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 OFCLUTTER OVER THE RANGE AXIS (LOW PRE): (d) CONCENTRATION OFRETURNS ON THE VELOCITY AXIS (HIGH PRF).

SOLO Target in Ground Clutter

SOLOGround 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 LobeBeam


( ) ( )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

SOLOGround Clutter

SOLOClutter

Illuminated Volume Resolution Cell (Pulse Limitted) The Volume Clutter in Range Resolution Cell is give (see Figure) by:

( )2/4

2Clutter τϕϕπ

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 LobeBeam


Choose scatters on the main beam center Groundkk RRuntilkRkR ≥=∆= ,2,1

RADAR

Ik f

cwhere

sVf =⋅= λ

λ12

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 kRk– Range to scatter k

The equivalent Radar Cross Section σClutter of the clutter in the resolution cell of volume VClutter is:

( )2/4

2Clutter τϕϕπ

cRV elaz=

g (0,0) ≈ 1 – antenna patternR – Range to the center of the volume VClutter

See Tildocs # 763310 v1

( ) ( )( )

∑=

+−=Σ

jiN

k

kkk

trver

RcvrXmtr

sc

c

c

RRR

jL

GGPji

,

12

k

kscatter

3

20

2

ClutterVolume 2

22exp

R4,

πσ

πλ

Illuminated Volume Resolution Cell (Pulse Limitted)

∑=

==scN

k k

kscatterClutterClutter

RRV

14

4 σησ ∑

=

=scN

k k

kscatter

Clutter RV

R

14

4 ση

Since the Volume Clutter is on the Main-Beam the effect of it on angle errors is like that of the radar noise.


Main LobeBeam



Target

Ground

A/C RADAR


SOLO

Multipath Return

Target Multipath is the Received Signalfor the mirror reflection the target relativeto 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 RRRrrr

−=

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

( )22ITIT hhXR −+= −

Let compute

Clutter

The Range of the Mirror Target to Seeker is

( ) RhhXR ITITM ≥++= −22

( ) ( ) ( ) ( ) ITITITMMM hhhhhhRRRRRR 42222 =−−+=−=+−

R

hh

RR

hhRR IT

RRR

M

ITM

M 24 2≈+

≈+

=−

..2

GRR

hhRR IT

M ≤≈−If , for all Target scatters k, we cannot distinguish between Target and Target’s Mirror

..2

GRR

hhRR IT

M >≈−If , for some Target scatters, we can distinguish between Target and Target’s Mirror and we willchoose the echoes with the smallest range


Target

Ground

A/C RADAR


SOLO

Multipath Return – Doppler Discrimination

The Target Range-Rate to Seeker is

( )22ITIT

IITTITIT

hhX

hhhhXXR

−+

++=

−

−−

Let compute

Clutter

The Range-Rate of the Mirror of Target to Seeker is

( ) ( ) ( )( )

( )( )

( )( )[ ] ( )[ ] Mi

MiIT

ITITITIT

IITTITITIT

ITIT

IITTITIT

ITIT

IIiTTITITMMM

RR

RRhh

hhXhhX

hhhhXXhh

HHX

HHHHXX

HHX

HHHHXXRRRRRR

44

2222

2

22

2

22

2

22

=++−+

++=

=++

++−

−+++

=−=+−

−−

−−

−

−−

−

−−

024

3

2

>≈+

=−≈+

MIT

RRR

M

M

M

ITMi RR

R

hh

RR

RR

RR

hhRR

M

..2

3GDRR

R

hhRR M

ITM ≤≈− If we cannot distinguish between

Target and Target’s Mirror

..2

3GDRR

R

hhRR M

ITM >≈−

If we can distinguish between Target and Target’s Mirror and we don’t have a Multipath problem.

( )22ITIT

IITTITITM

hhX

hhhhXXR

++

++=

−

−−

Assume that Target & Mirror Target are in the same Range Gate.


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 ( )

( )22ITkITk

IITkTkITkITkMk

HHX

hhhhXXR

++

++=

−

−−

integer=+= mRRmR kambiguoussunambiguouMk

RADAR

MkMk f

cwhere

Rf == λ

λ2

integer=+= nffnf kambiguoussunambiguouMk

( )RRIntegi kambiguousk ∆= /

( )ffIntegj kambiguousk ∆= /


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:

TGTXMTRopagation

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

Tkk hhX

hh

hR ++

+=

GNDTGTopagation

GNDTGTk

TGTTGT LRGA

TGT

→

→

Pr

21

1

4

1

πσ

3. Reflected by the ground and reaching the Seeker at the distance ( ) ( )[ ] 2/122_2 TkIIT

TkI

Ik hhX

hh

hR ++

+=

ReceivernPropagatio

22

1

4

1

RCVR

RCVR

GNDTGT

RCVRGNDk

GND L

A

LR

→

→πσ

πλ

4

2RCVR

RCVR

GA =


Target

Ground

A/C RADAR


SOLO

Multipath Return – Signal Power

Therefore the received power from the k scatter mirror is:

Receiver

2

nPropagatio

22

Pr

21

Pr

2

1

4

1

4

11

4

11

4

1

RCVR

RCVRant

GNDTGT

RCVRGNDk

GND

GNDTGTopagation

GNDTGTk

kScatterkScatter

TGTXMTRopagation

TGTTRk

rTransmitte

TR

antXmtrM L

GG

LRLRGA

LRL

GPP

kScatter

k πλ

πσ

ππσ

→

→

→

→

→

→

=

Clutter

( ) ( )[ ] 2/122_1 TkIIT

TkI

Tkk hhX

hh

hR ++

+= ( ) ( )[ ] 2/122

_2 TkIITTkI

Ik hhX

hh

hR ++

+=

( ) ( )( )( ) ( )

+++++++−=Σ ∑

=

Σ

ccj

gG

L

GGPji

jiN

k

ClutterkElkAzproc

trver

RcvrXmtr

k2k1kk2k1kk2k1k,

1 k2k1k

kscatter

proc

Targ

3

20

2

TargetMultipath

RRRRRRRRR

2expRRR

,

L4,

πσσεε

πλ

( )( )

22

21

2

2Targ

3

20

2 ,

4 kkk

ClutterkScatterkElkAz

proc

proc

trver

RcvrXmtrM

RRR

g

L

G

L

GGPP

k

σσεεπ

λ Σ=or:

where: ( )kElkAzant gGG εε ,0 Σ=RCVRRCVRGNDGNDTGTTGTTRTRtrver LLLLLL →→→=

proc

procRcvrRCVR L

GGG

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:

( ) ( )( )( ) ( )

+++++++−=∆ ∑

=

∆

ccj

gG

L

GGPji

jiN

k

ClutterkElkAzElAzproc

trver

RcvrXmtr

k2k1kk2k1kk2k1k,

1 k2k1k

kscatter,

proc

Targ

3

20

2

Az/ElTargetMultipath

RRRRRRRRR

2expRRR

,

L4,

πσσεε

πλ


SOLOElectronic Counter Measures (ECM)

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.


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-DopplerMap is obtained for SignalProcessing

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 alongvarious axes of the data-cube

Space-Time Adaptive Processing:2-D joint adaptive weighting acrossantenna element and pulse number

Beamforming:1-D weighting acrossElectrical Scan Antennaelement number

Pulse Compression:1-D convolution alongthe range axis(“fast time”)

Synthetic Aperture Imaging:2-D matched filtering in slowand fast time

Doppler Processing:1-D filtering or spectralanalysis along the pulse axis(“slow time”)

Run This


Range – Doppler Cells in Σ and ΔAz, ΔEl

SOLOWindowing

• 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 qualityof the data base.

Signal Processing

SOLOWindowing

Rectangular [ ] ≤≤

=otherwise

Mnnw

,0

0,1

Bartlett(triangular) [ ]

≤<−≤≤

=otherwise

MnMMn

MnMn

nw

,0

2/,/22

2/0,/2

Hanning

Hammming

[ ] ( ) ≤≤−

=otherwise

MnMnnw

,0

0,/2cos5.05.0 π

[ ] ( ) ≤≤−

=otherwise

MnMnnw

,0

0,/2cos46.054.0 π

Blackman [ ] ( ) ( ) ≤≤+−

=otherwise

MnMnMnnw

,0

0,/4sin08.0/2cos5.042.0 ππ

Julius Ferdinand von Hann (1839 -1921)

Richard Wesley Hamming (1915 –1998)

Signal Processing

http://upload.wikimedia.org/wikipedia/commons/9/97/Window_function_%28triangular%29.png

http://upload.wikimedia.org/wikipedia/commons/f/ff/Window_function_%28hann%29.png

http://upload.wikimedia.org/wikipedia/commons/7/7c/Window_function_%28hamming%29.png

http://upload.wikimedia.org/wikipedia/commons/c/cb/Window_function_%28blackman%29.png

SOLOWindowing (continue – 1)

cosine

[ ]

≤≤<

−−=

otherwise

MnM

Mnnw

,0

0&5.02/

2/

2

1exp

2

σσ

Lanczos[ ]

≤≤

−

=otherwise

MnM

nnw

,0

0,12

sinc

Gauss

[ ]

≤≤

=

−

=otherwise

MnM

n

M

nnw

,0

0,sin2

cosπππ

[ ]( )

≤≤

−−

=

otherwise

MnI

Mn

I

nw

,0

0,

12

1

0

2

0

α

αKaiser

α=2π

α=3π

Signal Processing

http://upload.wikimedia.org/wikipedia/commons/5/5c/Window_function_%28sine%29.png

http://upload.wikimedia.org/wikipedia/commons/3/3d/Window_function_%28sinc_or_Lanczos%29.png

http://upload.wikimedia.org/wikipedia/commons/f/f2/Window_function_%28gauss%29.png

http://upload.wikimedia.org/wikipedia/commons/b/bf/Window_function_%28Kaiser%3B_alpha_%3D_2_pi%29.png


Bartlett–Hann window

( )

38.0;42,0;62.0

1

2cos

2

1

1

210

210

===

−−−

−−=

aaa

N

na

N

naanw

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

6cos

1

4cos

1

2cos

3210

3210

====

−−

−+

−−=

aaaa

N

na

N

na

N

naanw

πππ

Nuttall window, continuous first derivative; B=2.02

Blackman–Harris window

( )

01168.0;14128.0;48829,0;35875.0

1

6cos

1

4cos

1

2cos

3210

3210

====

−−

−+

−−=

aaaa

N

na

N

na

N

naanw

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

6cos

1

4cos

1

2cos

3210

3210

====

−−

−+

−−=

aaaa

N

na

N

na

N

naanw

πππ

Signal Processing

http://en.wikipedia.org/wiki/Image:Window_function_(bartlett-hann).png

http://upload.wikimedia.org/wikipedia/commons/9/93/Window_function_%28nuttall%29.png

http://upload.wikimedia.org/wikipedia/commons/0/07/Window_function_%28blackman-harris%29.png


Dolph-Chebyshev window

( ) ( )[ ]

( ) ( )[ ]( ) ( )4,3,2,10cosh

1cosh

1,,2,1,0,coshcosh

coscoscos

1

1

1

≈

=

−=

=

=

−

−

−

αβ

β

πβω

ω

α

N

NkN

Nk

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 WWsidelobescwwww >=∞= ∑=∑ 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


Comparison of Windows

Signal Processing

SOLO

Windowing (continue – 3)


WindowType

Peak Sidelobe

Amplitude (Relative)

Approximate Width of Mainlobe

Peak Approximation

Error20 log10δ

(dB)

Equivalent Kaiser

Windowβ

Transition Width

of EquivalentKaiser

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



Signal Processing


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

nnwN

21

0

1

0

2

1

∑

∑−

=

−

=

N

nn

N

nn

w

w

N

Signal Processing

Run This

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 – DopplerCell

Range – DopplerMap

f

fM

R

RN sunambiguousunambiguou

∆=

∆= &

Range Gates are therefore i = 1, 2, …, NNumber 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 choosingthe Pulse Repetition Interval (PRI) and the number of pulses in a batch.

resolutionresolution ffRR ≥∆≥∆ &

SOLO Signal Processing Generation of Σ , ΔAz, ΔEl Range – Doppler Maps (continue – 1)

The received signal from the scatter k is:

( ) ( )[ ] ( ) ( )ttTktttTkttfCts ddkdkrk

rk ++≤≤++−= τθπ2cos

Ckr – amplitude of received signal

td (t) – round trip delay time given by ( )2/c

tRRtt 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

ψψ

sin

cos

( ) ( )

+−≅−=

c

tR

c

Rfttft kkkdkk

2222 ππψ

The quadrature samples are given by:( ) ( )

+−≅=

c

tR

c

RfjAjAtX kkkkkkk

222expexp πψ

Ak - amplitude of Σk, ΔAz k or ΔEl k signals ψk - phase of Σk, ΔAz k or ΔEl k signals

( )

+−

+≅+=

c

tR

c

RfAj

c

tR

c

RfAxjxtX kk

kkkk

kkQkIkk

222sin

222cos ππ

SOLO Signal Processing 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: ( ) ( ) 2kkkk AtXtXP == ∗

( )

+−

+≅+=

c

tR

c

RfAj

c

tR

c

RfAxjxtX kk

kkkk

kkQkIkk

222sin

222cos ππ

where * is the complex conjugate.

Therefore:kk PA =


Decision/Detection TheorySOLO

Hypotheses

H0 – target is not present

H1 – target is present

Binary Detection

( )0Hp - probability that target is not present

( )1Hp - 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) ( ) ( )zHpzHpH

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: ( ) ( )zHpzHpH

H

|| 01

0

1

<>

Using again Bayes’ rule:

( ) ( ) ( )( ) ( ) ( ) ( )

( )zp

HpHzpzHp

zp

HpHzpzHp

H

H

00

011

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 arenon-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 PdzHzpPT

−== ∫∞

1| 1

( )∫∞

=Tz

FA dzHzpP 0|

( ) D

z

M PdzHzpPT

−== ∫∞−

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

DP

FAP

( )1| Hzp( )0| Hzp

MPz

Tz

( )( ) T

Hzp

Hzp

T

T =0

1

|

|

H0 – target is not present H1 – target is present

Binary Detection

( )( )

( )( ) THp

Hp

Hzp

HzpLR

H

H

=<>=

1

0

0

1

0

1

|

|:Likelihood Ratio Test (LTR)


Hypotheses

Decision Criteria on Definition of the Threshold T

1. Bayes Criterion

DP

FAP

( )1| Hzp( )0| Hzp

MPz

Tz

( )( ) T

Hzp

Hzp

T

T =0

1

|

|


Binary Detection

( )( )

( )( ) THp

Hp

Hzp

HzpLR

H

H

=<>=

1

0

0

1

0

1

|


The optimal choice that optimizes the Likelihood Ratio is ( )( )1

0

Hp

HpTBayes =

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

MP z

Tz

( )( ) 1

|

|

0

1 == ML

T

T THzp

Hzp

DP

FAP


Hypotheses

Decision Criteria on Definition of the Threshold T (continue)

3. Neyman-Pearson Criterion

DP

γ=FAP

( )1| Hzp( )0| Hzp

MPz

Tz

( )( ) PN

T

T THzp

Hzp−=

0

1

|

|


Binary Detection

( )( )

( )( ) THp

Hp

Hzp

HzpLR

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 Pearson1895 - 1980

Jerzy Neyman1894 - 1981

( )∫∞

=T

TT

zzDz

dzHzpP 1|maxmax ( ) γ== ∫∞

Tz

FA dzHzpP 0|constrained to

Let use the Lagrange’s multiplier λ to add the constraint

( ) ( )

−+= ∫∫

∞∞

TT

TT

zzzz

dzHzpdzHzpG 01 ||maxmax γλ

Maximum is obtained for:

( ) ( ) 0|| 01 =+−=∂∂

HzpHzpz

GTT

T

λ( )( ) PN

T

T THzp

Hzp−==

0

1

|

|λ

zT is define by requiring that: ( ) γ== ∫∞

Tz

FA dzHzpP 0|

http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Neyman.html

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 differentdetections, 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:

2Nscatter i4

Target Resol 4i 1 iR

gV R

σσ η Σ

=

= = ∑24 N

scatter i

4i 1Resol iR

gR

V

ση Σ

=

= ∑ ( )2/4

2Resol τϕϕπ

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

kkk

kElkAzproc

trver

RcvrXmtr

sc

ccR

RRj

gG

L

GGPji

,

12

k

kscatter

proc

Targ

3

20

2

Targ 22

2expR

,

L4,

πσεε

πλ

In the same way:

gΔ (εAz,εEl) – antenna difference pattern ( gΔ(0,0)=0 )

R G A AN TG EE S

DOPPLERFILTERS

Range-Doppler S cells

Detections

According to Range and Doppler of each scatter determine theRange-Doppler cell (i,j) for the scatter.

( ) ( )( )( )

∑=

∆

+−=∆

jiN

k

kkk

kElkAzElAzproc

trver

RcvrXmtr

sc

ccR

RRj

gG

L

GGPji

,

12

k

kscatter,

proc

Targ

3

20

2

Az/ElTarg 22

2expR

,

L4,

πσεε

πλ

SOLO SEARCH & DETECT MODE

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 tobring the Target Uncertainty Window inthe middle of the Range-Doppler Map

2. Choose on the Range-Doppler Map aarea that includes the Target UncertaintyWindow and perform Ground Cluttercomputations only for this area (we may addGround Clutter computations in Main Lobeand 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

SOLO SEARCH & DETECT MODE

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= jijijiCCFAR

,,,

• Computation of Signal Power in Target Uncertainty Window cells:

( ) ( ) ( )∗Σ⋅Σ= jijijiS ,,,Window

yUncertaintTarget

• For each Range-Doppler Cell (i,j) perform the summation of complex signals for all the scatters in this cell:

∑∑∑===

∆=∆∆=∆Σ=Σjijiji N

kkEljiEl

N

kkAzjiAz

N

kkji

,,,

1,

1,

1, ,,

SOLO SEARCH & DETECT MODEComputation of the Σ Range-Doppler Map, chooses the Detection Threshold and policy (continue – 2).

DOPPLERWINDOW

R W A IN NG DE O W

R G A AN TG EE S

DOPPLERFILTERS

S cells

CFARWindow

R∆

f∆

Target Uncertainty

Window

( ) ( ) ( )[ ]∑ ∗+ Σ⋅Σ=

n

j WindowCFARNoiseClutter jiji

niC ,,

1

Guard(Gap)

Window

• Computation of Clutter + Noise Threshold

• Coherent Detection:

( ) ( )( ) ( ) ClutterThjiNiCIf

ClutternoThjiNiCIf

NoiseClutter

NoiseClutter

⇒+>⇒+≤

+

+

1,

1,

( ) NoiseThNjiS +≥Window

yUncertaintTarget,

( ) ( ) ( )[ ]∑ ∗+ Σ⋅Σ=

n

j WindowCFARNoiseClutter jiji

niC ,,

1

1. If no Clutter declare a Detection in the (i,j) cell of the Target Window if

ThNoise is chosen to assure a predefinedProbability of Detection pd and of False Alarm pFA

( ) NoiseClutterNoiseClutter ThCjiS ++ +≥Window

yUncertaintTarget,

2. If Clutter declare a Detection in the (i,j) cell of the Target Window if

ThNoise is chosen to assure a predefinedProbability 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 haveProbability of Detection pd and Probability of False Alarm pfa.To Increase Probability of Detection to pD and Reduce Probability of False Alarmto 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

lNd

ldD pp

lNl

NP 1

!!

!

( ) ( )∑=

−−−

=N

Ml

lNfa

lfaFA pp

lNl

NP 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 betweenDetections we must resolve theRange-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 thepower of each (i,j) cell, , in each Range-Doppler Map and we add (non-coherently) the powers of each corresponding (i,j) cell to obtaina non-coherent Range-Doppler Map. Now we perform the detection procedureas described before to declare a Detection.

( ) ( ) ( )∗Σ⋅Σ= jijijiS ,,,

SOLO SEARCH & DETECT MODEPerform 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:

( )( )

( )( )∑

∑∑

∑==

il

ill

l

il

ill

l jiS

jiSii

jiS

jiSii

,

,&

,

, 2

2

( )( )

( )( )∑

∑∑

∑==

il

ill

l

il

ill

l jiS

jiSjj

jiS

jiSjj

,

,&

,

, 2

2

Range

Doppler

integer=∆+= mRiRmRlsunambiguoul

RiillRl

∆−= 22σ

integer=∆+= nfifnflsunambiguoul

fjjllf l

∆−= 22σ

If the spread of Target Range/Doppler spread σRl/ σRl are too high, we may remove theTarget detection assumption and declare the group l as Clutter.

lRadar

lf

f

cR

2=

ll fRadar

R f

c σσ2

=

SOLO SEARCH & DETECT MODEPerform 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 choosethe PRFs and the Σ Range-Doppler Map.

clutterdf _

( )RangeR

( )RangeR

Clutter

No Clutter

ClutterPower

ClutterPower


Altitude Return

λMV2

pMV θ

λcos

2

AAM eV

coscos2 ψ

λ

pMV θ

λsin

2

pMV θ

λcos

2−

TargetRange

TargetDoppler

( ) ApA

I

e

h

ψθ cossin +

12

N1 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

yUncertaintTarget

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 toPerform Detection in the Target Uncertainty Window of the Σ Range-Doppler Map asin 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

AzlldbAzAz

jiji

jiji

,,

,,Re

23θ

ε( ) ( )

( ) ( )∑

Σ⋅Σ∆⋅Σ

= ∗

∗

lCluster ll

EllldbElEl

jiji

jiji

,,

,,Re

23θ

ε


SOLO Radar Technologies & Applications

Radar Antenna

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 IslandsRaytheonUHF Phased Array

30 m diameter35,000 elements

25,000 nmi range

http://www.fas.org/spp/military/program/track/cobra_dane.htmRadars for Ballistic Missile Defense

SOLO Anti – Ballistic MissilesRadars 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

Radars for Ballistic Missile Defense


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 (E\F band), 22.5 foot diameter,12,288 elements.


http://upload.wikimedia.org/wikipedia/commons/9/96/Cobra_Judy_Phased_Array.jpg


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, 1987Carpentier, M.H., “Principles of Modern Radar Systems”, Artech House, 1988Le 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

ReferencesRADAR 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, 1991Currie, 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 HouseCook, C.C., Bernfeld, M., “Radar Signals: An Introduction to Theory and Application”,

Artech House, 1993Schleher, 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


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


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

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 –

Stanford University1983 – 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-AMPLITUDEvs FREQUENCYvs FREQUENCY

SOLO

CW semi-active seekerSOLO

Conical scan radarSOLO

Pulse Radar Parameters

Transmitted Signal

Tr

Tp

F0

Tr – Pulse Repetition Interval (PRI)Tp – Transmitted Pulse WidthF0 – Transmitted RF frequencyPp – RF peak power D.C – Duty Cycle = Tp/Tr

Pav – RF average power = Pp*D.C

Pulse Doppler Radar – Clutter

Altitude return

AntennaMain Beam

AntennaSide Lobes

Ground

Target

Power

Frequency

Altitude Side lobe

Main Lobe

Incoming TargetReceding Target

λVc2

Doppler

Range

Radar Altitude

Vradar

Side lobe Clutter

Main Lobe Clutter

Incoming Targets

Receding Targets

CrossingTargets

Noise limitedClutter limited

Pulse Doppler Radar – Clutter

Clutter Cross Section in Doppler Cell

clutterat Gain Antenna )G(

tcoefficien Reflection

widthgate Range R

clutter todirection fligh between Angle

VelocityRadar V

Length WavedTransmitte

thfilter wid FFT B

clutter toRange

)()sin2

(

a

0

g

0

==

======

=

θσ

θ

λ

θσθ

λσ

R

GRV

RB ag2 cos

D

Vf

θλ

=

2 cosD

BB

f V

λθ

=

Detection of Radar SignalsSwerling Models

Case 1The echo received from a target on any look is constant but are independent (uncorrelated) from look to lookThe probability-density function for the RCS:

nsfluctuatio target allover RCS average theis

)exp(1

)(

av

avav

p

σσσ

σσ −=

Case 2The probability-density function for the RCS is same as for Case 1, but the fluctuations are more rapidCases 1 and Case 2 apply to a target consisting of many independent scatterers equal in RCS - Aircrafts

Probability Density Swerling 1

Detection of Radar SignalsSwerling Models

Case 3The 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

pσ

σσ

σσ −=Case 4The probability-density function for the RCS is same as for Case 2, but the fluctuations are more rapidCase 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

Detection of Radar SignalsNoise and False Alarm

The noise is assumed to be Gaussian with probability density function

noise theof valuesquare-mean theis

dv vand v valueebetween th v voltagenoise thefinding ofy probabilit theis )(

)2

exp(2

1)(

0

0

2

0

ψ

ψπψ+

−=

dvvp

dvv

dvvp

The probability that the noise envelope will exceed the voltage threshold VT is Pfa

)2

exp(0

2

ψT

fa

VP −=

Detection of Radar Signals

Integration of pulse trains

The probability of detection for M-out-of-N:

∑=

−−−

=N

Mj

jNd

jdd pp

jNj

NP )1(

)!(!

!

Pd probability of single detection

And the probability of false alarm

∑=

−−−

=N

Mj

jNn

jnn pp

jNj

NP )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 Informationλ

cVPRF 4≥

Extraction of InformationThe Range coverage equals PRI – 2*Pulse_width + recovery timeThe 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

Extraction of InformationTarget range = N PRI + R1

Where R1 = Target Range within PRI

N = number of folds 0…N

)(

1

PRI

RfixN

PRIRR

T

T

=

⊗=

ExampleRT=12,300 m PRI = 1500 m

N = fix(12300/1500) = 8 R1= 12300-1500*8=300 meters

Extraction of Information

PRI1

PRI2

R1

R2

Target

Extraction of InformationResolving 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=PRF2Maximum 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

TargetAmplitude

Range Sampling

Range Gate

CFAR ThresholdE1

E2

E3

R11 R12 R13

∑++=3,2,1

313212111 rangeTarget E

ERERER

Extraction of InformationRange 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 predicatedThe 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 IntTrue Range

Predicted Range

Closing VelocityRange Error

Extraction of InformationDoppler Tracking

TargetAmplitude

Range Sampling

Doppler Gate

CFAR ThresholdE1

E2

E3

D11 D12 D13

∑++=3,2,1

313212111 Doppler Target 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 predicatedThe 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 IntTrue Doppler Predicted DopplerDoppler Error


Extraction of InformationAngular 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:

λαπφ sin

2l=

The outputs from lobes are added and subtracted as vectors

)2

sin(

)2

cos(

φ

φ

∆

Σ

=∆

=Σ

K

KSUM

Difference

Extraction of InformationSum and Difference patterns

Sum

Difference

λαπφ sin

2l=

Output level


Σ∆=Σ∆=

and between angle

))sin(tan(

ϕ

ϕaError

Error output

Target angle

Interference

Multipath

Direct wave

Reflected

wave RadarTarget

Image

Tx Ri Rt

Reflecting Surface

t

HtHr

R

InterferenceMultipath

Phase difference corresponding to the path-length difference

R

HH rtd

22

λπϕ =

The ratio of power incident on the target compared to a target in free space (no multipath)

)2

(sin 4

R

HHKP tr

r λπ=

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

πσ=

Gjr = is the radar antenna gain in the direction of jammerRj = range radar to jammerRt = range radar to targetPj = 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

BWr

=∆

2

81.1 λT

r

Doppler Resolution

( )( )( ) ( ) ( )trtVRtr

ttVRRtr

VVV

Ts

Ts

TsTxA

/

2

:

2

222

222

γ

γ

γ

+=

++=

+−=

Significant range walkwith large BW and T

Significant Doppler walkwith large T c - speed of light

BW – bandwidthλ - wavelengthT - CPI timeVA - aircraft velocityVT - target velocityR - 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

, , , integersk 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 ii

x a e=

= ∑ But also ( )1

modk

i ii

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 11

/

13 3 2 20 1sn N n

r

− + = ÷

( )

2 2 22

/

11 4 3 15 1n s N n

r

− + = ÷

( ) ( )

33

3 3/

5 5 2 12 1N 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 toreturn back to the receiver. Since the electromagnetic waves travel with the speed of lightc (much greater then RADAR andTarget velocities), the received signal is delayed by

c

RRtd

21 +≅

The received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisettttftEtE ddr +−+−= ϕπα 000 2cos

To retrieve the range (and range-rate) information from the received signal thetransmitted signal must be modulated in Amplitude or/and Frequency or/and Phase.

ά < 1 represents the attenuation of the signal


The received signal is:

( ) ( ) ( ) ( )[ ] ( )

( ) ( ) ( )tnoisec

RRtRRtftE

tnoisettttftEtE

fc

ddr

+

+−++−=

+−+−==

2121

0000

/

000

22cos

2cos

00

ϕλππα

ϕπαλ

If we consider only (c = speed of light) then the frequency of the electromagneticwave that reaches the receiver is given by:

ctd

Rd <<

+

−≈

+

+−=

+−+

+−=

c

td

Rd

td

Rd

f

c

tdRd

tdRd

ud

dff

c

RRt

c

RRtf

td

df

21

0

21

0~

00

21210

1

2

1

22

1

ϕπ

ϕππ

λ

+

−=td

RdtdRd

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 isat a distance R2 from the Target, then the frequency of the carrier wave at the Receiver is:

λλ

−

−=td

RdtdRd

ff

21

0

( ) 111111 // RRVVRRR

td

RdET

⋅−=⋅= is the relative velocity between the Target and the Radar source along the line of sight between them

( ) 222222 // RRVVRRR

td

RdMT

⋅−=⋅= 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

EVTarget

Transmitter &Receiver

The transmitted RADAR RF Signal is:

( ) ( ) ( )[ ]ttftEtEt θπ += 00 2cos

( )c

tRtd

02ˆ ≅

Since the received signal preserve the envelope shape of the known transmitted signalwe want to design a Matched Filter that will distinguish the signal from the receiver noise.

the received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisetttffttEtE dDdr +−++−≈ ˆˆ2cosˆ00 θπα

Scaled DownIn Amplitude Two-Way

Delay

Possible Phase ModulationDoppler

Frequency

( ) ( )λ

λ0

/

00 22ˆ 0 tR

fc

tRf

fc

D

−=−≅

=

For R1 = R2 = R we obtain that

SOLO Review of Probability

Exponential Distribution

( ) ( )

<≥−

=00

0exp;

x

xxxp

λλλ

( ) ( )

( )( ) ( )

λλλ

λλ

λλ

1expexp

exp

00exp

0

=−+−−=

−=

∫

∫∞

∞=

−=

∞

dxxxx

dxxxxE

xu

dxxdv

( ) ( ) ( )2

22 1

λ=−= xExExVar

( ) ( )[ ] ( ) ( )

( )[ ]1

0

0

1exp

expexpexp

−∞

∞

−=−

−=

−==Φ ∫

λωλω

λωλ

λλωωω

jxj

j

dxxxjxjEX

Probability Density Functions

Cumulative Distribution Function

Mean Value

Variance

Moment Generating Function

( ) ( ) ( )

<≥−−

=−= ∫∞− 00

0exp1exp;

x

xxdxxxP

x λλλλ

( ) ( )2

0

2

222 2

λω ω

=Φ==d

djxE X

Distributionsexamples

Table of Content


Chi-square Distribution

( )( )

( )( ) ( )

<

≥−Γ=

−

00

02/exp2/

2/1;

2/2

2/

x

xxxkkxp

k

k

( ) kxE =

( ) kxVar 2=

( ) ( )[ ]( ) 2/21

expk

X

j

xjE−−=

=Φ

ω

ωω



Mean Value

Variance Moment Generating Function

( )( )

( )

<

≥Γ=

00

02/

2/,2/

;

x

xk

xk

kxP

γ

Γ is the gamma function ( ) ( )∫∞

− −=Γ0

1 exp dttta a

( ) ( )∫ −= −x

a dtttxa0

1 exp,γγ is the incomplete gamma function


SOLO Review of ProbabilityStudent’s t-Distribution

( ) ( )[ ]( ) ( ) ( ) 2/12 /12/

2/1; ++Γ

+Γ= νννπννν

xxp

( )

=>

=1

10

νν

undefinedxE

( ) ( )∞

>−=

otherwisexVar

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

xxP

νν

ννπνν


− −=Γ0

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


Table of Content


Uniform Distribution (Continuous)

( )

>>

≤≤−=

bxxa

bxaabbaxp

0

1,;

( )2

baxE

+=

( ) ( )12

2abxVar

−=

( ) ( )[ ]( ) ( )

( )abj

ajbj

xjE

−−=

=Φ

ωωω

ωωexpexp

exp



Mean Value

Variance


( )

>

≤≤−−

>

=

bx

bxaab

ax

xa

baxP

1

0

,;


Moments

Table of Content


Rayleigh Distribution

( )2

2

2

2exp

;σ

σσ

−

=

xx

xp

( )2

πσ=xE

( ) 2

2

4 σπ−=xVar



Mean Value

Variance


( )

−−=

2

2

2exp1;

σσ x

xP

( ) ( )

−

−−=Φ jerfi

222/exp1 22 σωπσωσωω

John William Strutt

Lord Rayleigh

(1842-1919)


Moments

Rayleigh Distribution is the chi-distribution with k=2( ) ( ) ( )

( ) ( )kk

k

k

k

k

k Uk

pk

χσ

χσ

χχ

−

Γ=

−−−

Χ 2

212/2

2exp

2/

2/1



Given X and Y, two independent gaussian random variables, with zero means and thesame variances σ2

Example of Rayleigh Distribution

( )

+−=2

22

2 2exp

2

1,

σσπyx

yxpXY

find the distributions of R and Θ given by: ( )XYYXR /tan& 122 −=Θ+=

( ) ( )

( ) ( ) θθσπ

θσ

σπσθθ

dprdrpdrdrr

ydxdyxydxdyxpdrdrp

r

XYR

Θ

Θ

=

−=

+−==

22

2

22

22

22exp

22exp,,

where:( ) πθ

πθ 20

2

1 ≤≤=Θp

( ) 02

exp2

2

2≥

−= r

rrrpr σσ

Uniform Distribution


Solution

Table of Content

x

y

rθ


Rice Distribution

( )

+−=

202

2

22

2exp

,;σσ

σσ vx

I

vxx

vxp

( )2

πσ=xE

( ) 2

2

4 σπ−=xVar



Mean Value

Variance Moment Generating Function

( )

−−=

2

2

2exp1;

σσ x

xP

( ) ( )

−

−−=Φ jerfi

222/exp1 22 σωπσωσωω

Stuart Arthur Rice1889 - 1969


where:

( ) ( )∫

−=

π

ϕσ

ϕπσ

2

0220 '

2

'cosexp

2

1d

vxvxI


Rice Distribution

The Rice Distribution applies to the statistics of the envelope of the output of a bandpassfilter consisting of signal plus noise.

Example of Rice Distribution

( ) ( ) ( ) ( ) ( ) ( ) ( )( )[ ] ( ) ( )[ ] ( )tAtntAtn

ttnttntAtnts

SC

SC

00

000

sinsincoscos

sincoscos

ωϕωϕωωϕω

−++=+++=+

X = nC (t) and Y = nS (t) are gaussian random variables, with zero mean and the samevariances σ2 and φ is the unknown but constant signal phase.

Define the output envelope R and phase Θ:

( )[ ] ( )[ ]( )[ ] ( )[ ] ϕϕ

ϕϕ

cos/sintan

sincos1

22

AtnAtn

AtnAtnR

CS

SC

+−=Θ

−++=−

( ) ( ) ( ) ( )

( )222

22

22

2

2

2

22

cosexp

2exp

22

sinexp

2

cosexp,,

σπθ

σθϕ

σ

σπσϕ

σϕθθ

drdrrAAr

ydxdAyAxydxdyxpdrdrp XYR

+−

+−=

−−

+−==Θ

Solution

( ) ( ) ( ) ( )∫∫ +

+−

+−== Θ

ππ

θϕσ

θϕσπσ

θθ2

0222

222

0 2

cosexp

22exp, d

rArArdrprp RR


Rice Distribution

Example of Rice Distribution (continue – 1)

( ) ( ) ( ) ( )∫∫

−

+−== Θ

ππ

ϕσ

ϕπσσ

θθ2

022

22

2

2

0

'2

'cosexp

2

1

2exp, d

rAArrdrprp RR

where:

( ) ( )∫

−=

π

ϕσ

ϕπσ

2

0220 '

2

'cosexp

2

1d

rAArI

is the zero-order modified Bessel function of the first kind

( )

+−=202

22

2 2exp,;

σσσσ Ar

IArr

ArpR Rice Distribution

Since I0 (0) = 1, if in the Rice Distribution we take A = 0 we obtain:

Rayleigh Distribution( )

−==

2

2

2 2exp,0;

σσσ rr

ArpR

Table of Content


Weibull Distribution

( )

<

>≥

−−

−

=

−

00

0,,exp,,;

1

x

xxx

xpαγµ

αµ

αµ

αγ

αµγ

γγ

( ) ( )

−−−== ∫

∞−

γ

αµαµγαµγ x

dxxpxPx

exp1,,;,,;

( )

+Γ=

γα 1

1xE


− −=Γ0

1 exp dttta a

Ernst HjalmarWaloddi Weibull

1887 - 1979



Mean Value

Variance( ) ( ) 22 21 xExVar −

+Γ=

γα


Table of Content

1 radar basic - part ii

Science

solo radar basicsrf

s of content

array esaradar basics

antenna msaesamechanically

material absorption

emission reflection

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