radar systems notes

97.460

RADAR ENGINEERING

NOTES

radarnotes_2006.mif

1. Introduction

- Radar is a(R

- radar operreected ba

- radar can

- radar can

- radar can

- the elemenand receivin

- a portion otions

- the antenn

- the distanctravel to the

- the directi

- also if thersignal (Dopto distingui

-Radar was

- modern ra

- the simplecarrier

- the distancand from th

or R(km)=0

or R(nm)=0RADAR ENGINEERING

n electromagnetic system for the detection and location of objectsAdio Detection And Ranging)

ates by transmitting a particular type of waveform and detecting the nature of the signalsck from objects

not resolve detail or colour as well as the human eye (an optical frequency passive scatterometer)

see in conditions which do not permit the eye to see such as darkness, haze, rain, smoke

also measure the distances to objects

tal radar system consists of a transmitter unit, an antenna for emitting electromagnetic radiationg the echo, an energy detecting receiver and a processor.

f the transmitted signal is intercepted by a reecting object (target) and is reradiated in all direc-

a collects the returned energy in the backscatter direction and delivers it to the receiver

e to the receiver is determined by measuring the time taken for the electromagnetic signal to target and back.

on of the target is determined by the angle of arrival (AOA) of the reected signal.

e is relative motion between the radar and the target, there is a shift in frequency of the reectedpler effect) which is a measure of the radial component of the relative velocity. This can be usedsh between moving targets and stationary ones.

rst developed to warn of the approach of hostile aircraft and for directing anti aircraft weapons.

dars can provide AOA, Doppler, MTI etc.

st radar waveform is a train of narrow (0.1s to 10s) rectangular pulses modulating a sinusoidal

e to the target is determined from the time TR taken by the pulse to travel to the target and returne knowledge that electromagnetic energy travels at the speed of light thus:

.15TR(s)

.081TR(s)

RcTR

2----------= 1/6/06 1

97.460 PART II RADAR

2

- once the pallow echoe

- thus the mtargets are e

- if the PRFThis leads t

- the range

where fP is

- more adva

-e.g. the carpermit the e

- this achievenergy of lo

- this techni

-also CW wfrom the tra

The Radar Rang

- the radar rantenna, tar

- it is used a

-If the trans

R from the

- here the 4

- radars emp

- the gain Gpared to theulse is transmitted by the radar a sufcient length of time must elapse before the next pulse tos from targets at the maximum range to be detected.

aximum rate at which pulses can be transmitted is determined by the maximum range at whichxpected. This rate is called the pulse repetition rate (PRF)

is too high echo signals from some targets may arrive after the transmission of the next pulse.o ambiguous range measurements. Such pulses are called second time around pulses

beyond which second time around pulses occur is called the maximum unambiguous range

the PRF in Hz.

nced signal waveforms then the above are often used

rier may be frequency modulated (FM or chirp) or phase modulated (pseudorandom biphase) toocho signals to be compressed in time after reception.

es high range resolution without the need for short pulses and hence allows the use of the highernger pulses

que is called pulse compression

aveforms can be used by taking advantage of the Doppler shift to separate the received echonsmitted signal. Note: unmodulated CW waveforms do not permit the measurement of range.

e Equation

ange equation relates the range of the radar to the characteristics of the transmitter, receiver,get and the environment.

s a tool to help in specifying radar subsystem specications in the design phase of a program.

mitter delivers PT Watts into an isotropic antenna, then the power density (W/m2) at a distance

radar is

R2 represents the surface area of the sphere at distance R

loy directional antennas to channel the radiated power Pt in a particular direction

of an antenna is the measure of the increased power radiated in the direction of the target, com- power that would have been radiated from an isotropic antenna

RUNAMBIGc

2 f P----------=

PT

4R2

-------------radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Powe

- the target

- the measuis called the

He

Note: the rathe radar.

- the receivicalled the e

- Since the

- Now the mcient receivable signal

Setting Pr=

Note here thby:

As long as tr density from a directional antenna =

intercepts a portion of the incident power and redirects it in various directions

re of the amount of incident power by the target and redirected back in the direction of the radar cross section .

nce the Power density of the echo signal at the radar=

dar cross-section has the units of area. It can be thought of as the size of the target as seen by

ng antenna effectively intercepts the power of the echo signal at the radar over a certain areaffective area Ae

power density (Watts/m2) is intercepted across an area Ae, the power delivered to the receiver is

aximum range Rmax is the distance beyond which the target cannot be detected due to insuf-ed power Pr The minimum power which the receiver can detect is called the minimum detect-Smin.

Smin and rearranging the above equation gives

at we have both the antenna gain on transmit and its effective area on receive. These are related

he radar uses the same antenna for transmission and reception we have

or

PtG

4R2

-------------

PtG

4R2

-------------

4R2

-------------

PrPtG

4R2

-------------

4R2

-------------Ae=

RmaxPtGAe

4( )2Smin-------------------------

14---

=

G4------

Ae

2------=

RmaxPtG

22

4( )3Smin-------------------------

14---

= 1/6/06 3


4

Example:

Use the rad

Prmin =10-13

G=2000=0.23mPRF=524

=2.0m2

Now

From

= 3.1 MW

Note 1: thesassumption

Note 2: thein a given ti

Note 3: Thedue to the e

Rma

PtP

G2

-------=

Pt2(

----=ar range equation to determine the required transmit power for the TRACS radar given

Watts

e three forms of the equation for Rmax vary with different powers of . This results from implicits about the independence of G or Ae from .

introduction of additional constraints (such as the requirement to scan a specic volume of spaceme) can yield other dependence.

observed maximum range is often much smaller than that predicted from the above equationxclusion of factors such as rainfall attenuation, clutter, noise gure etc.

RmaxPtAe

2

42Smin------------------------

14---

=

xc

PRF------------=

r 4( )3R

4

2 PRF( )4------------------------------

1013( ) 4( )3 3 10

8( )2

---------------- 4

000)2 0.23( )2 524( )4 2.0( )---------------------------------------------------------------radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

a) transmitttrain.

- th

- typo

- ty

- ty

b) The wav

c) The receiby the dupl

- d

- d(an

- Ting

-soare

c) The receibetter sensiend has greence

W

PR

D

(LAer may be an oscillator (magnetron) that is pulsed on and off by a modulator to generate the pulse

e magnetron is the most widely used oscillator

pical power required to detect a target at 200 NM is MW peak power and several kW averagewer

pical pulse lengths are several s

pical PRFs are several hundreds of pulses per second

eform travels to the antenna where it is radiated

ver must be protected from damage resulting from the high power of the transmitter. This is doneexer.

uplexer also channels the return echo signals to the receiver and not to the transmitter

uplexer consists of 2 gas discharge tubes called the TR (transmit/receive) and the and an ATRti transmit/receive) cell

he TR protects the receiver during transmission and the ATR directs the echo to the receiver dur- reception.

lid state ferrite circulators and receiver protectors with gas plasma (radioactive keep alive) tubes also used in duplexers

ver is usually a superheterodyne type. The LNA is not always desirable. Although it providestivity, it reduces the dynamic range of operation of the mixer. A receiver with just a mixer frontater dynamic range, is less susceptible to overload and is less vulnerable to electronic interfer-

AVEGUIDE

ESSURIZER

UPLEXERTRANSMITTER

(HPA)

LNA IF STRIP2nd

DETECTORVIDEO AMP

DISPLAY(PROCESSOR)X

~BITE

POWERSUPPLIES

TIMINGELECTRONICS

RADAR BLOCK DIAGRAM AND OPERATION

OW NOISEMPLIFIER) 1/6/06 5


6

.

d) The mixe

- th

- ththe

- thspeati

Noise F

L(GN

Dr and Local Oscillator (LO) convert the RF frequency to the IF frequency.

e IF is typically 300MHz, 140Mz, 60 MHz, 30 MHz with bandwidths of 1 MHz to 10 MHz.

e IF strip should be designed to give a matched lter output. This requires its H(f) to maximize signal to noise power ratio at the output.

is occurs if the |H(f)| (magnitude of the frequency response of the IF strip is equal to the signalctrum of the echo signal |S(f)|, and the ARG(H(f)) (phase of the frequency response) is the neg-

ve of the ARG(S(f)).

X

~G

EFFECT OF LNA ON DYNAMIC RANGE

X

~

1dB compression

Si min

DRSi max

Noise Floor

LSi max

LSi min

DR

1dB compression, mixer

Si min

Si max

Noise FloorGNi+Ne NiSi minG

1dB compression, mixer

Si max GSi max GL

Noise Floorloor

i+Ne) Si minGL

Rradarnotes_2006.mif 1/6/06

radarnotes_2006.mif

i.e

- flt

e) The pulsewhich they

f) The displAngle infor

- thtio

- thsw

- A

- Aan

- s

- Atio

g) The simp

- A

- A

- C

- rotary join

- Mtio

- pof

- man

- m

- brep. H(f) and S(f) should be complex conjugates

or radar with rectangular pulses, a conventional IF lter characteristic approximates a matcheder if its bandwidth B and the pulse width satisfy the relationship

modulation is extracted by the second detector and amplied by video ampliers to levels atcan be displayed (or A to Dd to a digital processor)

ay is usually a CRT; timing signals are applied to the display to provide zero range information.mation is supplied from the pointing direction of the antenna.

e most common type of CRT display is the plan position indicator (PPI) which maps the loca-n of the target in azimuth and range in polar coordinates

e PPI is intensity modulated by the amplitude of the receiver output and the CRT electron beameeps outward from the centre corresponding to range.

lso the beam rotates in angle in synchronization with the antenna pointing angle.

B scope display uses rectangular coordinates to display range vs angle i.e. the x axis is angled the y axis is range.

ince both the PPI and B scopes use intensity modulation the dynamic range is limited

n A scope plots target echo amplitude vs range on rectangular coordinates for some xed direc-n. It is used primarily for tracking radar applications than for surveillance radar.

le diagram has left out many details such as

FC to compensate the receiver automatically for changes in the transmitter

GC

ircuits in the receiver to reduce interference from other radars

ts in the transmission lines to allow for movement of the antenna

TI (moving target indicator) circuits to discriminate between moving targets and unwanted sta-nary targets

ulse compression to achieve the resolution benets of a short pulse but with the energy benetsa long pulse.

onopulse tracking circuits for sensing the angular location of a moving target and allowing thetenna to lock on and track the target automatically

onitoring devices to monitor transmitter pulse shape, power load and receiver sensitivity

uilt in test equipment (BITE) for locating equipment failures so that faulty circuits can belaced quickly

B 1 1/6/06 7


8

h) Instead othen display

- q

- a

- ewh

- m

- d

Th

i) Antennas

- thsou

- th

- parr

Radar Frequ

- m

- s

Thmequf displaying the raw video output directly on the CRT, it might be digitized and processed anded. This consists of:

uantizing the echo level at range-azimuth resolution cells

dding (integrating) the echo level in each cell

stablishing a threshold level that permits only the strong outputs due to target echoes to passile rejecting noise

aintaining the tracks (trajectories) of each target

isplaying the processed information

is process is called automatic tracking and detection (ATD) in a surveillance radar

e most common form of radar antenna is a reector with parabolic shape, fed from a pointrce (horn) at its focus

e beam is scanned in space by mechanically pointing the antenna

hased array antennas are sometimes used. Her the beam is scanned by varying the phase of theay elements electrically

encies

ost radars operate between 220 MHz and 35 Ghz

pecial purpose radars operate out side of this range

Skywave HF-OTH (over the horizon) can operate as low as 4 MHz

Groundwave HF radars operate as low as 2 MHz

millimeter radars operate up to 95 GHz

laser radars (lidars) operate in IR and visible spectrum

e radar frequency letter-band nomenclature is shown in the table. Note that the frequency assign-nt to the latter band radar (e.g. L band radar) is much smaller than the complete range of fre-encies assigned to the letter bandradarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Application

General

- ground-ba

- shipborneis also used

- airborne raand avoidan

- spaceborn

Major Appl

1. Air Traf

- uthe

- hwa

BDesi

HF

VHF

UHF

L

S

Ku

K

Ka

mms of Radar

sed radar is applied chiey to the detection, location and tracking of aircraft of space targets

radar is used as a navigation aid and safety device to locate buoys, shorelines and other ships. it to observe aircraft

dar is used to detect other aircraft, ships and land vehicles. It is also used for mapping of terraince of thunderstorms and terrain.

e radar is used for the remote sensing of terrain and sea, and for rendezvous/docking.

ications

c Control

sed to provide air trafc controllers with position and other information on aircraft ying withinir area of responsibility (airways and in the vicinity of airports)

igh resolution radar is used at large airports to monitor aircraft and ground vehicles on the run-ys, taxiways and ramps.

Table 1:

andgnation

NominalFrequency Range

Specic radar bandsbased on ITU assignments

for region 2

3-30 MHz

30-300 MHz 138-144 MHz216-225 MHz

300-1000 MHz 420-450 MHz890-942 MHz

1000 - 2000 MHz 1215-1400 MHz

2000 - 4000 MHz 2300 - 2500 MHz2700 - 3700 MHz

12 - 18 GHz 13.4 - 14.0 GHz15.7 - 17.7 GHz

18 - 27 GHz 24.05 - 24.25 GHz

40 - 300 GHz 33.4 - 36.0 GHz

40 - 300 GHz 1/6/06 9


10

- Ghigthi

- M

2. Air Navi

- wtur

- te

- r

- d

- g

3. Ship Safe

- th

- d

- arad

- siga

4. Space

- r

- la

- s

5. Remote S

- u

- r

- io

- eag(SyCA (ground controlled approach) or PAR (precision approach radar) provides an operator withh accuracy aircraft position information in both the vertical and horizontal. The operator usess information to guide the aircraft to a landing in bad weather.

LS (microwave landing system) and ATC radar beacon systems are based on radar technology

gation

eather avoidance radar is used on aircraft to detect and display areas of heavy precipitation andbulence.

rrain avoidance and terrain following radar (primarily military)

adio altimeter (FM/CW or pulse)

oppler navigator

round mapping radar of moderate resolution sometimes used for navigation

ty

ese are one of the least expensive, most reliable and largest applications of radar

etecting other craft and buoys to avoid collision

utomatic detection and tracking equipment (also called plot extractors) are available with thesears for collision avoidance

hore based radars of moderate resolution are used from harbour surveillance and as an aid to nav-tion

adars are used for rendezvous and docking and was used for landing on the moon

rge ground based radars are used for detection and tracking of satellites

atellite-borne radars are used for remote sensing (SAR, synthetic aperture radar)

ensing

sed for sensing geophysical objects (the environment)

adar astronomy - to probe the moon and planets

nospheric sounder (used to determine the best frequency to use for HF communications)

arth resources monitoring radars measure and map sea conditions, water resources, ice cover,ricultural land use, forest conditions, geological formations, environmental pollutionnthetic Aperture Radar, SAR and Side Looking Airborne Radar SLAR)radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

6. Law Enfo

- a

- in

7. Military

- s

- n

-

2. The Radar Range

Fro

Al

In ran

Th

It i

Bethe

2.2 Minimu

Ththe

De

If

Rrcement

utomobile speed radars

trusion alarm systems

urveillance

avigation

re control and guidance of weapons

Equation

m page 3 we have

(2.1)

l of the parameters are controllable by the radar designer except for the target cross section .

practice the simple range equation does not predict range performance accurately. The actualge may be only half of that predicted.

is due, in part, to the failure to include various losses

s also due to the statistical nature of several parameters such as Smin, , and propagation losses

cause of the statistical nature of these parameters, the range is described by the probability that radar will detect a certain type of target at a certain distance.

m detectable Signal

e ability of the radar receiver to detect a weak echo is limited by the noise energy that occupies same spectrum as the signal

tection is based on establishing a threshold level at the output of the receiver.

the receiver output exceeds the threshold, a signal is assumed to be present

max

PtAe2

42Smin------------------------

14---

= 1/6/06 11


12

A

- a

Thcro

Thmi

A sig

Thlic

Th

Inso en

At

Thmiing

Nodif

Thab

AlSN

Th

It hvidsample detected envelope is show above

large signal is detected at A

e threshold must be adjusted so that weak signals are detected, but not so low that noise peaksss the threshold and give a false target.

e voltage envelope in the gure is usually from a matched lter receiver. A matched lter maxi-zes the output peak signal to average noise power level

matched lter has a frequency response which is proportional to the complex conjugate of thenal spectrum

e output of a matched lter is the cross correlation between the received waveform and the a rep-a of the transmitted waveform.

e shape of the input waveform to the matched lter is not preserved.

the gure, two signals are present at point B and C. The noise voltage at point B is large enoughthat the combined signal and noise cross the threshold. The presence of noise sometimeshances the detection of weak signals.

point C the noise is not large enough and the signal is lost.

e selection of the proper threshold is a compromise which depends on how important it is if astake is made by (1) failing to recognize a signal (probability of a miss) or by (2) falsely indicat- the presence of a signal (probability of a false alarm)

te: threshold selection can be made by an operator viewing a CRT display. Here the threshold iscult to predict and may not remain xed in time.

e SNR necessary to provide adequate detection must be determined before the minimum detect-le signal Smin can be computed.

though detection decision is done at the video output, it is easier to consider maximizing theR at the output of the IF strip (before detection)

is is because the receiver is linear up to this point

as been shown that maximizing SNR at the output of the IF is equivalent to maximizing theeo output.

Threshold

RMS valueof noise

A B Ctradarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Receiver No

Nona

No

On

No

Nohe

ThTh

Thpli

he

No

Sin

Fnise

ise is unwanted EM energy which interferes with the ability of the receiver to detect wanted sig-ls.

ise may be generated in the receiver or may enter the receiver via the antenna

e component of noise which is generated in the receiver is thermal (or Johnson) noise.

ise power (Watts) = kTBn

where k = Boltzmanns constant =1.38 x 10-23 J/degT = degrees KelvinBn = noise bandwidth

Note: Bn is not the 3 dB bandwidth but is given by:

here f0 is the frequency of maximum response

i.e. Bn is the width of an ideal rectangular lter whose response has the same area as thelter or amplier in question

Note: for many radars Bn is approximately equal to the 3 dB bandwidth (which is easier todetermine)

te: a receiver with a reactive input (e.g. a parametric amplier) need not have any ohmic loss andnce all thermal noise is due to the antenna and transmission line preceding the antenna.

e noise power in a practical receiver is often greater than can be accounted for by thermal noise.is additional noise is created by other mechanisms than thermal agitation.

e total noise can be considered to be equal to thermal noise power from an ideal receiver multi-ed by a factor called the noise gure Fn (sometimes NF)

= Noise out of a practical receiver/Noise out of an ideal receiver at T0

re Ga is the gain of the receiver

te: the receiver bandwidth Bn is that of the IF amplier in most receivers

ce and

Bn

H f( ) 2 fd

H f 0( )

2---------------------------------=

N 0kT 0Bn( )Ga

----------------------------=

GaSoSi-----= Ni kT 0Bn= 1/6/06 13


14

we

rea

Node

he

sub

Probability

Cono

divnu

Th

Th

als

No have

rranging gives:

w Smin is that value of Si corresponding to the minimum output SNR: (So/No) necessary fortection

nce (2.6)

stituting 2.6 into the radar range equation (eqn 2.1) yields

(2.7)

Density Function (PDF)

nsider the variable x as representing a typical measured value of a random process such as aise voltage.

ide the continuous range of values of x into small equal segments of length x, and count thember of times that x falls into each interval

e PDF p(x) is than dened as:

p(x) = lim (No of values in range x at x)x 0 NN

where N is the total number of values

e probability that a particular measured value lies within width dx centred at x is p(x)dx

o the probability that a value lies between x1 and x2 is

te: PDF is always positive by denition

FnSi N iS0 N 0-----------------=

SikT 0BnFnSo

N 0------------------------------=

Smin kT 0BnFnS0N 0-------

min=

Rmax4 PtGAe

4( )2kT 0BnFn S0 N 0( )min--------------------------------------------------------------------=

P x1 x x2<

radarnotes_2006.mif

als

Th

he

als

m1

Nogiv

Va

Va

if x

staco

Uno

e average value of a variable function (x) of a random variable x is:

nce the average value, or mean of x is

o the mean square value is

and m2 are called the rst and second moments of the random variable x.

te: if x represents current, then m1 is the DC component and m2 multiplied by the resistancees the mean power.

riance is dened as

=m2 - m21

riance is also called the second central moment

represents current, 2 multiplied by the resistance gives the mean power of the AC component.

ndard deviation, is dened as the square root of the variance. This is the RMS value of the ACmponent.

iform Probability Density Function K, a < x < a + b

p(x)= 0 x < a, x > a+b

example of a uniform probability distribution is the phase of a random sine wave relativeto a particular origin of time.

p x( ) xd

1=

x( ) ave x( ) p x( ) xd

=

x ave xp x( ) xd

m1= =

x2 ave x2 p x( ) xd

m2= =

2 2

x m1( )2 ave x m1( )2 p x( ) xd

= = = 1/6/06 15


16the constant K is found from the following

hence for the phase of a random sine wave

the average value for a uniform PDF

the mean squared value is

the variance is

the standard deviation is

p x( ) xd

K xda

a b+( )

1 K 1b---= = =

K 12------=

m11b--- x xd

a

a b+( )

a b2---+= =

a a +b

1/b

m1 x0

m21b--- x2 xd

a

a b+( )

a2 ab b23-----+ += =

m2 m12

b2

12------=

b

2 3----------=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Ga

an

we

Ce

Fon

Hefor

Ra

ex(IFan

p(ussian (Normal) PDF)

example of normal PDF is thermal noise

have for the Normal PDF

m1 = x0

m2 = x20 +

2

2 = m2 - m12

ntral Limit Theorem:

The PDF of the sum of a large number of independent, identically distributed randomquantities approaches the Normal PDF regardless of what the individual distribution mightbe, provided that the contribution of any one quantity is not comparable with the resultantof all the others

r the Normal distribution, no matter how large a value of x we may choose, there is always aite probability of nding a greater value

nce if noise at the input to a threshold detector is normally distributed there is always a chance a false alarm.

yleigh PDF

x 0

amples of a Rayleigh PDF are the envelope of noise output from a narrowband band pass lter lter in superheterodyne receiver), also the cross section uctuations of certain types of targets

d also many kinds of clutter and weather echoes.

p x( ) 12

2-----------------

x x0( )2

22

-------------------------

exp=

x0

1/2

x

p(x)

x) xx

2 ave------------------ x

2

2 x2 ave

----------------------

exp= 1/6/06 17


18

he

if x

an

the

thi

he

Th

in

SN

hebil

theat

p(re

2 is replaced by w where w represents power

d ave is replaced by w0 where w0 represents average power

n w 0

s is called the exponential PDF or the Rayleigh Power PDF

re = w0

e Probability Distribution Function

some cases the distribution function is easier to obtain from experiments

R

re we will obtain the SNR at the output of the IF amplier necessary to achieve a specic proba-ity of detection without exceeding a specied probability of false alarm.

output SNR is then substituted into equation 2.6 to obtain Smin, the minimum detectable signalthe receiver input

p(x)

x

m14--- 1=

p w( ) 1w0------ w

w0------ exp=

p(x)

x

x) p x( ) xd

x

=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

he

the

Th

he

Wthe

he

nois:

ThTfa

hepo

Nothe

p(

p(re BV > BIF/2 in order to pass all video modulation

envelope detector may be either a square law or linear detector

e noise entering the IF amplier is Gaussian

re 0 is the variance, the mean value is zero

hen this Gaussian noise is passed through the narrow band IF strip, the PDF of the envelope of noise is Rayleigh PDF

re R is the amplitude of the envelope of the lter output

w the probability that the noise voltage envelope will exceed a voltage threshold VT (false alarm)

(2.24)

e average time interval between crossings of the threshold by noise alone is the false alarm time

re Tk is the time between crossings of the threshold by noise when the slope of the crossing issitive

w the false alarm probability Pfa is also given by the ratio of the time that the envelope is above threshold to the total time

IFAmplier

VideoAmplier

BIF BVseconddetector

x) 120

------------------ v2

20----------

exp=

p R( ) R0------ R

2

20----------

exp=

VT R < < )R0------

VT

R220----------

dRexpVT

2

20----------

exp P fa= = =

T fa1N----

N lim Tk

k 1=

N

= 1/6/06 19


20

W

ba

eq

P f

VT

T f(2.25

here since the average duration of a noise pulse is approximately the reciprocal of the

ndwidth.

uating 2.24 and 2.25

a

tkk 1=

N

Tk

k 1=

N

-----------------

tk aveT k ave

------------------- 1T faBIF------------------= = =

tk tk+1

Tk Tk+1

tk 1BIF---------

a1BIF---------

VT2

20----------

exp=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Example:

for

Noba

None

Ex

Nobeco

WeIF

He

wh

no

No

Th

P f

p(

I0

Pd BIF = 1 MHz and required false alarm rate of 15 minutes, equation 2.25 gives

te: the false alarm probabilities of practical radars are quite small. This is due to their narrowndwidth

te: False alarm time Tfa is very sensitive to variations in the threshold level VT due to the expo-ntial relationship.

ample: for BIF = 1 MHz we have the following:

te: If the receiver is gated off for part of the time (e.g. during transmission interval) the Pfa will increased by the fraction of the time that the receiver is not on. This assumes that Tfa remainsnstant. The effect is usually negligible.

now consider a sine wave signal of amplitude A present along with the noise at the input to thestrip.

re the output of the envelope detector has a Rice PDF which is given by:

2.27

ere I0(Z) is the modied Bessel function of zero order and argument Z

w

for Z large

te: when A = 0 equation 2.27 reduces to the PDF from noise alone

e probability of detection Pd is the probability that the envelope will exceed VT

VT2/20 Tfa

12.95 dB 6 min

14.72 dB 10,000 hours

a1

15( ) 60( ) 106 --------------------------------------- 1.11x10

9= =

R) R0------ R

2A

2+

20-------------------

I 0

RA0------- exp=

Z( ) eZ

2Z-------------- 1 1

8Z------- + +

p R( ) RdVT

= 1/6/06 21


22

for

No

In

- w

Thou

No

Pd

p(R

A

-----

V

2----- the conditions RA/0 >> 1 and A >> |R-A|

2.29

te: the area represents the probability of detection

the area represents the probability of false alarm

if Pfa is decreased by moving VT then Pd is also decreased

equation 2.29 we can make the following substitutions:

and

(eqn 2.24)

ith these substitutions, Fig 2.7 is plotted

e performance specication is Pfa and Pd and Fig. 2.7 is used to determine the S/N at the receivertput and the Smin at the receiver input

te: this S/N is for a single radar pulse

12--- 1 er f

V T A

20-----------------( )

VT A( )2

20( )2 2A 0( )

--------------------------------------------

exp 1VT4A-------

1 VT A( )2 0+

8A2( ) 0

---------------------------------------------+

+=

)

VT/0R/0

0

------ 2SN------=

T2

0----- 1

P fa---------ln=radarnotes_2006.mif 1/6/06

radarnotes_2006.m

No

A ate

S/

2.5 Integrat

Fig

- hdete: S/N required is high even for Pd = 0.5. This is due to the requirement for the Pfa to be small.

change in S/N of 3.4 dB can change the Pd from 0.999 to 0.5. When a target cross section uctu-s, the change in S/N is much greater than this

N required for detection is not a sensitive function of false alarm time

ion of Radar pulses

. 2.7 applies for a single pulse only

owever many pulses are usually returned from any particular target and can be used to improvetectionif 1/6/06 23


24

- th

wh

Ex

de

- T

- a

- in

wh

tioe number of pulses nB as the antenna scans is

ere B = antenna beam width (deg)fP = PRF (Hz)

= antenna scan rate (deg/sec)

m = antenna scan rate (rpm)

ample: For a ground based search radar having

B = 1.5 fP = 300 Hz

= 30/s (m = 5 rpm)

termine the number of hits from a point target in each scan

nB = 15

he process of summing radar echoes to improve detection is called integration

ll integration techniques employ a storage device

- the simplest integration method is the CRT display combined with the integrating proper-ties of the eye and brain of the operator.

- for electronic integration, the function can be accomplished in the receiver either beforethe second detector (in the IF) or after the second detector (in the video)

- integration before detection is called predetection or coherent detection

- integration after detection is called postdetection or noncoherent integration

- predetection integration requires the phase of the echo signal to be preserved

- postdetection integration can not preserve RF phase

- for predetection SNRintegrated = n SNRiwhere SNRi is the SNR for a single pulseand n is the number of pulses integrated

- for postdetection, the integrated SNR is less than the above since some of the energy isconverted to noise in the nonlinear second detector

- postdetection integration, however, is easier to implement

tegration efciency is dened as

ere = value of SNR of a single pulse required to produce a given probability of detec-

n

nBB f PS

-------------B f P6m-------------= =

S

S

Ei n( )S N( )1

n S N( )n----------------------=

S N( )1radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

an

wh

No

Ex

- n

- w

thi

- thbe

nf

he

Th

Noof

- ifsea- h

Nosio

- h

No

- Fped is the value of SNR per pulse required to produce the same probability of detection

en n pulses are integrated.

te: for postdetection integration, the integration improvement factor is Ii = nEi(n)

for ideal postdetection, Ei(n) = 1 and hence the integration improvement factor is n

amples of Ii are given in Fig. 2.8a from data by Marcum

ote that Ii is not sensitive to either Pd or Pfa

e can also develop the integration loss as

s is shown in Fig 2.8b

e parameter nf in Fig 2.8 is called the false alarm number which is dened as the average num-r of possible decisions between false alarms

= [no. of range intervals/pulse][no. of pulse periods/sec][false alarm rate]= [TP/][fP][Tfa]

re TP = PRI (pulse repetition interval)fP = PRF

us nf = Tfa /

TfaB

1/Pfa

te: for a radar with pulse width , there are B = 1/ possible decisions per second on the presencea target

n pulses are integrated before a target decision is made, then there are B/n possible decisions/.ence the false alarm probability is n times as great

te: this does not mean that there will be more false alarms since it is the rate of detection-deci-ns is reduced, not the average time between false alarms

ence Tfa is more meaningful than Pfa

te: some authors use a false alarm number nf = nf/n

caution should be used in computations for SNR as a function of Pfa and Pd

ig. 2.8a shows that for a few pulses integrated post detection, there is not much difference from arfect predetection integrator.

S N( )n

Li 101

Ei n( )-------------log= 1/6/06 25


26 radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

- w

- th

hespe

usi

he

somthe

- if

is

he

- fothe

als

No

2.7 Radar C

Crin hen there are many pulses integrated (small S/N per pulse) the difference is pronounced.

e radar equation with n pulses integrated is

2.23

re (S/N)n is the SNR of one of n equal pulses that are integrated to produce the required Pd for acied Pfa

ng equation 2.31 in 2.32

re (S/N)1 is found from Fig. 2.7 and nEi(n) is found from Fig 2.8a.

e postdetection integrators use a weighting of the integrated pulses. These integrators include recirculating delay line, the LPF, the storage tube and some algorithms in digital integration.

an exponential weighting of the integrated pulses is used then the voltage out of the integrator

re Vi is the voltage amplitude of the ith pulse and exp(-) is the attenuation per pulse

r this weighting, an efciency factor can be calculated which is the ratio of the average S/N for exponential integrator to the average S/N for the uniform integrator:

for a dumped integrator

o

for a continuous integrator

te: Maximum efciency for a dumped integrator corresponds to =0

Maximum efciency for a continuous integrator corresponds to n =1.257

ross Section of Targets

oss-section: The ctional area intercepting that amount of power which, when scattered equallyall directions, produces an echo at the radar that is equal to that actually received.

Rmax4 PtGAe

4( )2kT 0BnFn S N( )n----------------------------------------------------------=

Rmax4 PtGAenEi n( )

4( )2kT 0BnFn S N( )1----------------------------------------------------------=

V V i i 1( )[ ]expi 1=

N

=

n2

------ tanhn

2--- tanh

-----------------------=

1 n( )exp[ ]2

n2--- tanh

----------------------------------------= 1/6/06 27


28

=

=

wh

Nothe

- Ee

- E

Noby

- th2.9

-w

dropower reected towards the source/unit solid angle incident power density/4

ere R = rangeEr= reected eld strength at radarEi = incident eld strength at target

te: for most targets such as aircraft. ships and terrain, the does not bear a simple relationship to physical area

M scattered eld: is the difference between the total eld in the presence of an object and theld that would exist if the object were absent

M diffracted eld: is the total eld in the presence of the object

te: for radar backscatter, the two elds are the same (since the transmitted eld has disappeared the time the received eld appears)

e can be calculated using Maxwells equations only for simple targets such as the sphere (Fig.)

hen (the Rayleigh region), the scattering from a sphere can be used for modelling rain-

ps

4R2

R lim

ErEi------

2

2a---------- 1radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

- srad

- thred

-w

NodB

Nofro

- bthe

-he

- hfro

- T

heince varies as -4 in the Rayleigh region, rain and clouds are invisible for long wavelengthars

e usual radar targets are much larger than raindrops and hence the long operation does notuce the target

hen the approaches the optical cross section a2.

te: in the Mie (resonance region) can actually be 5.6 dB greater than the optical value or 5.6 smaller.

te: For a sphere the is not aspect sensitive as it is for all other objects, and hence can be used calibrating a radar system.

ackscatter of a long thin rod (missile) is shown. Where the length is 39 and the diameter /4, material is silver.

re = 0 is the end on view and is small since the projected area is small.

owever at near end on ( 5) waves couple onto the rod, travel the length of the rod and reectm the discontinuity at the far end large .

he Cone Sphere

re the rst derivatives of the cone and sphere contours are the same at the point of joining

2a---------- 1 1/6/06 29


30

Th

Nofro(w

Ex

- Inoe nose-on is shown in Fig. 2.12

te: Fig. 2.12. The for near 0 (-45 to +45) is quite low. This is because scattering occursm discontinuities. Here the discontinuities are small: the tip, the join and the base of the spherehich allows a creeping wave to travel around the sphere)

- when the cone is viewed at perpendicular incidence ( = 90 - , where is the cone halfangle) a large specular return is contained

-from the rear, the is approximately that of a sphere

- the nose on for f above the Rayleigh region and for a wide range of , has a max of

0.42 and a min of 0.012. This gives a very low backscatter(e.g. at = 3 cm, = 10-4 m2.

ample: at S band for 3 targets having the same projected area:

Corner reector: 1000 m2

Sphere 1m2

Cone sphere 10-3 m2

n practice, to achieve a low with a cone sphere, the tip must be sharp, the surface smooth and holes or protuberances allowed.radarnotes_2006.mif 1/6/06

radarnotes_2006.mif 1/6/06 31


32

- a

No

Co

- thing

- thor

- awh

- thtotwi

-A15an

- Tits comparison of nose-on for several cone shaped objects is given in gure 2.13

te: the use of materials such as carbon bre composites can further reduce .

mplex Targets

e of complex targets (ships, aircraft, terrain) are complicated functions of frequency and view- angle

e can be computed using GTD (Geometric Theory of Diffraction), measured experimentally,found using scale models

complex target can be considered as being composed of a large number of independent objectsich scatter energy in all directions

e relative phases and amplitudes of the echo signals from the individual scatterers determine theal . If the separation between individual scatterers is large compared to the phases will varyth the viewing angle and cause a scintillating echo.

n example of the variation of with aspect angle is shown in Fig. 2.16. The can change bydB for an angular change of 0.33. Broadside gives the max since the projected area is biggerd is relatively at (The B-26 fuselage had a rectangular cross-section)

his data was obtained by mounting the actual aircraft on a turntable above ground and observing with a radar.radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

- a

- athe

- Tco

- thNalis

- aeqtim more economical method is to construct scale models.

n example of a model measurements is given in Fig. 2.17 by the dashed lines. The solid lines are theoretical (computed using GTD) data

he computed data is obtained by breaking up the target into simple geometrical shapes. and thenmputing the contributions of each (accounting for shadowing)

e most realistic method for obtaining data is to measure the actual target in ight. The USval Research Lab has such a facility with L, S, C, and X band radars. The radar track data estab-hes the aspect angle. Data is usually averaged over a 10 x 10 aspect angle interval.

single value cross section is sometimes given for specic aircraft targets for use in the rangeuation. This is sometimes an average value or sometimes a value which is exceeded 99% of thee 1/6/06 33


34

Ex

CoSmSmLaMeLaJum

SmSmCa

PicCaBicMaBir

Insamples of radar cross sections for various targets (in m2))

nventional, unmanned winged missile 0.5all, single engine aircraft 1all ghter, or 4-passenger jet 2

rge ghter 6dium bomber or medium jet airliner 20

rge bomber or large jetliner 40bo jet 100

all open boat 0.02all pleasure boat 2bin cruiser 10

kup truck 200r 100ycle 2n 1d .01

ect 10-5radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

No

e.g

m2

du

Fo

- th

- s

- a

- h

2.8 Cross-S

- th

- vins

- c

- ofra

- th

- aante: even though single values are given there can be large variations in actual for any target

. the AD 4B, a propeller driven aircraft has a of 20 m2 at L band but its at VHF is about 100

This is because at VHF the dimensions of the scattering objects are comparable to and pro-ce a resonance effect

r large ships, an average cross section taken from port, starboard and quarter aspects yields

here is in m2

f is in MHz D is ship displacement in kilotons

is equation applies only to grazing angles i.e. as seen from the same elevation

mall boats 20 ft. to 30 ft. give (X band) approx 5 m2

40 ft. to 50 ft.

10 m2

utomobiles give (X band) of approx 10 m2 to 200 m2

uman being gives as shown:

f (MHz) (m2)

410 0.033 - 2.33 1120 0.098 - 0.997 2890 0.140 - 1.05 4800 0.368 - 1.88 9375 0.495 - 1.22

ection Fluctuations

e echo from a target in motion is almost never constant

ariations are caused by meteorological conditions, lobe structure of the antenna, equipmenttability and the variation in target cross section

ross section of complex targets are sensitive to aspect

ne method of dealing with this is to select a lower bound of that is exceeded some speciedction of the time (0.95 or 0.99)

is procedure results in conservative prediction of range

lternatively, the PDF and the correlation properties with time may be used for a particular targetd type of trajectory

median 52 f D3 2

= 1/6/06 35


36

- th

- thth

- it

- itthe

Sw

Ca

Ca

Ca

Ca

Ca

- Tpra

- Tsme PDF gives the probability of nding any value of between the values of and + d.

e correlation function gives the degree of correlation of with time (i.e. number of pulses)e power spectral density of is also important in tracking radars.

is not usually practical to obtain experimental data for these functions

is more economical to assess the effects of uctuating is to postulate a reasonable model for uctuations and to analyze it mathematically

erling has done this for the detection probabilities of 5 types of target.

se 1

echo pulses received from the target on any one scan are of constant envelope throughoutthe entire scan, but are independent (uncorrelated) scan to scan

This case ignores the effect of antenna beam shape the assumed PDF is:

0

se 2

echo pulses are independent from pulse to pulse instead of from scan to scan

se 3Same as case 1 except that the PDF is

se 4

Same as case 2 except that the PDF is

se 5

Nonuctuating cross section

he PDF assumed in cases 1 and 2 applies to complex targets consisting of many scatterers (inctice 4 or more)

he PDF assumed in cases 3 and 4 applies to targets represented by one large reector with otherall reectors

p ( ) 1ave-----------

ave----------- exp=

p ( ) 1ave-----------

ave----------- exp=

p ( ) 4ave

2----------- 2

ave----------- exp=

p ( ) 4ave

2----------- 2

ave----------- exp=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

- for all cases the value of to be substituted in the radar equation is ave

0 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

PDF for Swerling 1 and 2 targets

average cross section = 20

0 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

PDF for Swerling 3 and 4 targets

average cross section = 20 1/6/06 37


38

- c

- wtha

- N

Thistsam

- a&

- thtuaomparison of the ve cases for a false alarm number nf = 108 is shown in Fig. 2.22

hen detection probability is large, all 4 cases in which is not constant require greater SNRn the constant case (case 5)

ote for Pd =0.95 we have

Case # S/N

1 16.8 dB/pulse

2 6.2 dB/pulse

is increase in S/N corresponds to a reduction in range by a factor of 1.84. Hence if the character-ics of the target are not properly taken into account, the actual performance of the radar (for the

e value of ave) will not measure up to the predicted performance.

lso when Pd > 0.3, larger S/N is required when uctuations are uncorrelated scan to scan (cases 13) than when uctuations are uncorrelated pulse to pulse.

is results since the larger the number of independent pulses integrated, the more likely the uc-tions will average out cases 2 & 4 will approach the nonuctuating case.radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

- Figures 2.23 and 2.24 may be used as corrections for probability of detection (Fig. 2.7) 1/6/06 39


40

Pro

NoNtha

No

(10

Nosqu

Nonu

forfor

No

he

an

Notua

- th

- it

p(cedure:

1) Find S/N from Fig. 2.7 corresponding to desired Pd and Pfa

2) From Fig. 2.23 nd correction factor for either cases 1 and 2 or cases 3 and 4 to beapplied to S/N found in Step 1. The resulting (S/N)1 is that which would apply if detectionwere based on a single pulse

3) If n pulses are integrated, The integration improvement factor Ii(n) is found from Fig.2.24. The parameters (S/N)1 and nEi(n)=Ii(n) are substituted into the radar equation 2.33along with ave.

te: in Fig. 2.24 the integration improvement factor Ii(n) is sometimes greater than n. Here the S/required fro n=1 is larger than for the nonuctuating target. The S/N per pulse will always be lessn that of the ideal predetection integrator.

te: data in Fig. 2.23 and 2.24 are essentially independent of the false alarm number6

radarnotes_2006.mif

- aa

- th

- thtio

- itm.

- thwi

- thby

- H

Nosm

- Tliteircraft ying straight and level t Chi-square distribution with m between 0.9 and 2, and with

ve varying 15 dB from min to max.

e parameters of the tted distribution vary with aspect angle, type of aircraft and frequency

e value of m is near unity for all aspect angles except broadside which give a Rayleigh distribu-n with varying ave

is found that ave has more effect on the calculation of probability of detection than the value of

e Chi-square distribution also describes the cross section of shapes such as cylinders, cylindersth ns (e.g. some satellites). Here m varies between 0.2 and 2 depending on the aspect angle.

e Rice distribution is a better description of the cross section uctuations of a target dominated a single scatterer than the Chi-square distribution with m=2.

ere the Rice distribution is

where s is the ratio of the cross section of the single dominant scatterer to the total crosssection of the smaller scatterers

I0 is a modied Bessel function of zero order

te: when s=1 the results using the Rice distribution approximate the Chi-square with m=2, forall probabilities of detection

he Log Normal distribution has been suggested for describing the cross sections of some satel-s, ships, cylinders, plates, arrays

> 0

where sd = standard deviation of

and m = median of

also the ratio of the mean to median value of is =

p ( ) 1 s+ave----------- s

ave----------- 1 s+( ) I 0 2

ave-----------s 1 s+( ) exp=

p ( ) 12sd

-------------------- 1

2sd2

-------- m------- ln

2exp=

m------- ln

sd2

2-----

exp 1/6/06 41


42

- Cing

Noto-

- thwh

2.9 Transm

Pt

- th

- it

- thomparisons of several distributions models fro false alarm number nf = 106, with all pulses dur-

a scan correlated and pulses in successive scans independent, are shown in Fig. 2.25.

te: The two extreme cases treated are for pulses correlated in any particular scan but with scan-scan independence (slow uctuations), and for complete independence (fast uctuations).

ere could be partial correlation of pulses within a scan. The results for this case would fall some-ere between the two cases.

itter Power

in the radar equation is the peak power

is is not the instantaneous peak power of the carrier sine wave

is the power averaged over a carrier cycle which occurs at the maximum of a pulse.

e average radar power, Pav is the average transmitter power over the PRI

here = pulse width Tp= PRI fp = PRF

Pav PtT p------ Pt f p= =radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

no

- th

- T

he

- ifof

No

2.10 Pulse R

- PRF is de

- echoes rec

- these can

- consider thbetween Ruw which denes the duty cycle

e typical duty cycle for a surveillance radar is 0.001

hus the range equation in terms of average power is

re (Bn) are grouped together since the product is usually of the order of unity for pulse radars

the transmitted waveform is not a rectangular pulse, we can express the range equation in termsenergy

te: In this form Rmax does not depend explicitly on or fp

epetition Frequency and Range Ambiguities

termined primarily by the maximum range at which targets are expected

eived after an interval exceeding the PRI are called multiple-time-around echoes

result in erroneous range measurements

ree targets A, B and C. here A is within the maximum unambiguous range Runambig, B is

nambig and 2Runambig and C is between 2Runambig and 3Runambig

PavPt

-------- T p-------=

Rmax4 PavGAenEi n( )

4( )2kT 0 Bn( )Fn S N( )1 f p-------------------------------------------------------------------------=

EPav

fp

--------=

Rmax4 EGAenEi n( )

4( )2kT 0 Bn( )Fn S N( )1-----------------------------------------------------------------= 1/6/06 43


44

-one way ofunambiguotargets will

- the numbegets need on

- alternativeamplitude,

- these sche

- one limitaweak multip

- a second l

- the range atheorem

2.11 Antenn

- the gain o

G(,) = po

- G is a func

- from recipdistinguishing multiple time around targets is to operate with a carrying PRF. The echo from anus target will appear at the same place on each sweep, however echoes from multiple time aroundspread out.

r of separate PRFs will depend on the degree of multiple time targets. Second time around tar-ly 2 separate PRFs to be resolved

methods to mark successive pulses to identify multiple time around targets include changingpulse width, frequency, phase or polarization from pulse to pulse

mes are not very successful in practice

tion is the foldover of nearby targets (e.g. nearby strong ground targets, clutter) which can maskle time around targets

imitation is increased processing requirement to resolve ambiguities

mbiguity in a multiple PRF radar can be conveniently decoded by use of the Chinese remainder

a Parameters

f an antenna is

wer radiated per unit solid angle in direction of azimuth and elevation power accepted by antenna from transmitter/4

tion of direction. If it is greater than unity in some directions, it must be less than unity in others

rocity, if an antenna has a larger gain in transmission in a specic direction, then it also has aradarnotes_2006.mif 1/6/06

radarnotes_2006.mif

larger effec

- the most c

- pencil beameasure theweapons co

- to search aon the maxilong at any

- to reduce tsion and wi

- fan beamsbased radar

- even with and the abil

- scan rates

- for long ra

- coverage o

- the elevati

-here

here 0 and

- this patterobserving a

- ldeally m

- csc2 patttrue parabo

- the csc2 having a co

- substitutin

G(

PrPtG

2

4(------------=tive area in that direction

ommon beam shapes fro radar are the pencil beam and the fan beam

ms are axially symmetric with a width of a few degrees. They are used where it is necessary to angular position of a target continuously in azimuth and elevation (e.g. a tracking radar forntrol or missile guidance). These are generated with parabolic reectors.

large sector of sky with a narrow beam is difcult. Operational requirements place restrictionsmum scan time (time for beam to return to the same point) so that the radar can not dwell tooparticular cell.

he number of cells, the pencil beam is replaced with the fan beam which is narrow in one dimen-de in the other.

can be generated with parabolic reectors with a shaped projected area. many long range grounds use fan beams

fan beams, a trade-off exists between the rate at which the target position is updated (scan time)ity to detect weak signals (by use of pulse integration)

are typically from 1 to 60 rpm

nge surveillance, scan rates are typically 5 or 6 rpm

f a simple fan beam is not adequate for targets at high altitudes close to the radar.

on pattern is usually shaped to radiate more energy at high angles as in the csc2 pattern.

for 0 < < m

m are the angular limits of the csc2 t

n is used for airborne search radars observing ground targets as well as ground based radarsircraft. For the airborne case is the depression angle

should be 90 but it is always less

erns can be generated by a distorted section of a parabola or with special multiple horn feed on ala, or with an array such as a slotted waveguide

pattern gives constant echo power Pr independent of range for a target of constant height, h andnstant .

g into the range equation (simple form)

) G 0( )csc 20csc

2-----------------=

0( ) ( )csc22

)3 0( )csc2R

4----------------------------------------- K1

( )cscR

4----------------= 1/6/06 45


46

- now for a

therefore

- hence the

- in practiceterns

Note: the gasame apertu

- the maxim

where is

- this is con

- Note: A

- a typical r

where l is th

2.12 System

- losses in t

- losses whiloss

- losses whiof operator

Note: loss h

Plumbing Lconstant height, h of a target, we have

echo signal is independent of range

Pr varies due to varying with viewing angle, the earth not being at and non perfect csc2 pat-

in of csc2 antennas for ground based radars is about 2 dB less than for a fan beam having there

um gain of any antenna is related to its size by

the antenna efciency which depends on the aperture illumination

trolled by the complexity of the feed design

= Aeff

eector gives a beam width of

e dimension

Losses

he radar reduce the S/N at the receiver output

ch can be calculated include the antenna beam shape loss, the collapsing loss and the plumbing

ch cannot be calculated readily include those due to eld degradation, operator fatigue and lackmotivation

as a value greater than unity - Loss = [Gain]-1

oss

( )csc R h=

PrK1

h4

------- K2= =

G4A

2----------=

deg( ) 65l

---------=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

- lo

- nfre

- a

No

- thgre

Beam Shap

- thsha

- athaad

- less in transmission lines between the transmitter and antenna and between antenna and receiver

ote from the Fig 2.28 that, at low frequencies, the transmission lines introduce little loss. At highquencies the attenuation is signicant

dditional loss occurs at connectors (0.5 dB), bends (0.1dB) and at rotary joints (0.4 dB)

te: if a line is used for both transmission and reception, its loss is added twice

e duplexer typically adds 1.5 dB insertion loss. In general, the greater the isolation required, theater the insertion loss

e Loss

e train of pulses returned from the target to a scanning radar are modulated in amplitude by thepe of the antenna beam

beam shape loss accounts for the fact that the maximum gain is used in the radar equation rathern a gain which changes from pulse to pulse. (this approach is approximate since it does not

dress Pd for each pulse separately)

t the one way power pattern be approximated by a Gaussian shape

here B is the half power beam width

S2 2.782

B2

-------------------exp= 1/6/06 47


48

- nbe

Ex

Nobe

- Ithe

- wbe

- wrec

- aspa

Lim

- li

- th

- liPP

- lithe

- fno

- if

Co

- ifad

- th

- inpla

- c

L(B is the number of pulses received within B and if n is the number of pulses integrated, then theam shape loss (relative to a radar that integrated n pulses with equal gain) is

ample integrating 11 pulses gives L (beam shape) = 1.66 dB

te: the beam shape loss above was for a beam shaped in one plane only (i.e. fan beam or pencilam where the target passes through the centre of the beam)

f the target passe through any other part of the beam the maximum signal will not correspond to signal from the beam centre

hen many pulses are integrated per beamwidth, the scanning loss is taken as 1.6 dB for a fanam scanning in one coordinate, and as 3.2 dB when two coordinate scanning is used

hen the antenna scans so rapidly that the gain on transmission is not the same as the gain oneption, an additional scanning loss is added.

dditional loss for phased array search using a step scanning pencil beam since not all regions ofce are illuminated by the same value of antenna gain.

iting Loss

miting in radar can lower the Pd

is is not a desirable effect and is due to a limited dynamic range

miting can be due to pulse compression processing and intensity modulation of CRT (such asI)

miting results in a loss of only a fraction of a dB for large numbers of pulses integrated providing limiting ratio (ratio of video limit level to RMS noise level) is greater than 2

or small SNR in bandpass limiters, the reduction of SNR of a sine wave in narrowband Gaussianise is /4 (approx 1 dB)

the spectrum of the input noise is shaped correctly, this loss can be made negligible

llapsing Loss

the radar integrates additional noise samples along with the wanted signal +noise pulses, theded noise causes degradation called the collapsing g loss

is occurs on displays which collapse range information (C scope which displays El vs Az)

some 3D radars (range, Az, El) that display outputs at all Elevations on one PPI (range, Az) dis-y, the collapsing of the 3D information into 2 D display results in loss

an also occur when the output of a high resolution radar is displayed on a device which is of

beamshape) n

1 2 5.55k2

( ) nB2( )exp

k 1=

n 1( ) 2+

---------------------------------------------------------------------------------=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

co

- Msigno

- thfoe

- r

Ex

the

- c

- F

SN---

Liarser resolution than the radar

arcum has shown that for a square law detector, the integration of m noise pulses, along with nnal + noise pulses with SNR per pulse (S/N)n, is equivalent to the integration of m+n signal _ise pulses each with SNR of

e collapsing loss then is the ratio of the integration loss Li for m+n pulses to the integration loss n pulses

ecall

ample: 10 signal pulses are integrated with 30 noise pulses

Required Pd = 0.9, nf = 108

From Fig 2.8b

Li (40) = 3.5 dBLi (10) = 1.7 dB

refore Li (m,n) = 1.8 dB

ollapsing loss for a linear detector can be much greater than for a square law detector

ig 2.29 shows the comparison of loss for each detector

- m n+( )equivn

m n+------------- S

N )------- n=

m n,( ) Li m n+( )Li n( )

-----------------------=

Li n( )1

Ei n( )-------------= 1/6/06 49


50

No

- trtub

- req

- mof

- thcanin

Op

- a

- th

ity

No

Fie

- wbe

- f

- raaidnideal Equipment

ansmitter power - the power varies from tube to tube (for same type), and with age for a specice. Power is also not uniform over the operating band

- hence Pt may be other than the design value.

- to allow for this, a loss factor of about 2 dB can be used

eceiver noise gure - the NF will vary over the band, hence if the best NF is used in the radaruation, a loss factor must account for its poorer value elsewhere in the band

atched lter - if the receiver is not the exact matched lter fro the transmitted waveform, a lossSNR will occur (typically 1 dB)

reshold level - due to the exponential relationship between Tfa and VT a slight change in VT cause signicant change to Tfa hence, VT is set slightly higher than calculated to give good Tfa

the event of circuit drifts. This is equivalent to a loss

erator Loss

distracted, tired, overloaded, poorly trained operator will perform less efciently

e operator efciency factor (empirical) is where Pd is the single scan probabil-

of detection

te: operator loss is not relevant to systems where automatic detection is done

ld Degradation

hen a radar is operated under eld conditions, the performance deteriorates even more than can accounted for in the above losses.

actors which cause eld degradation are:

- poor training

- weak tubes

- water in the transmission lines

- incorrect mixer crystal current

- deterioration in the receiver NF

- poor TR tube recovery

- loose cable connections

dars should be designed with BIST (built - in system test) and BITE (built - in test equipment) to in performance monitoring

0 0.7 Pd( )2

=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

- a

- B

- w

Ot

- Mof

- inson

- stre

Pro

- th

- th

- th

- fono

- h

- thmabyrad

- acope preventative maintenance plan should be used

ITE parameters to be monitored are

- Pt

- NF of receiver

- Transmitter pulse shape

- recovery time of TR tube

ith no other information available, 3 dB is assumed for eld degradation loss

her Loss Factors

TI radars introduce additional loss. The MTI discrimination technique results in complete losssensitivity for certain target values (blind speeds)

a radar with overlapping range gates, the gates may be wider than optimum for practical rea-s. The additional noise introduced by nonoptimum gate width leads to degradation performance.

traddling loss accounts for loss in SNR for targets not at the centre of a range gate, or at the cen- of a lter in a multiple bank processor

pagation Effects

e radar equation assumes free space propagation

e earths surface and atmosphere have a signicant effect on radar performance

e effects fall into three categories

- attenuation

- refraction by the earths atmosphere

- lobe structure caused by interference between the direct wave and the ground reectedwave

r most microwave radars, attenuation through the normal atmosphere or through precipitation ist signicant

owever reection from rain (clutter) is a limiting factor in radar performance in adverse weather

e deceasing density of atmosphere with altitude results in bending (refraction) of the electro-gnetic wave. This normally increases the line of sight. the refraction can also be accounted for assuming the earth to have a larger radius than actual. A typical earth radius is 4/3 actualius.

t times atmospheric conditions create ducting (or super refraction) and increases the radar rangensiderably. It is not necessarily desirable since it can not be counted on. Also it degrades MTIrformance by extending the range at which ground clutter is seen. 1/6/06 51


52

- thsinpro

2.1

- th

No

NoproFM

Ra

Bl

Cu

Rm4e presence of the earth also breaks the antenna elevation pattern into many lobes. this arisesce the direct and reected waves interfere at the target either destructively or constructively toduce nulls or lobes. This results in non uniform illumination.

4 Other Considerations

e radar equation is now written

te: The following substitutions can be made:

E = Pav/fp = Pt

N0 = N/B (power spectral density of noise)

B 1

T0Fn = Ts

te: The above radar equation was derived for rectangular pulses but applies to other waveformsvided that matched lter detection is used. The equation can be modied to accommodate CW,-CW, pulse doppler MTI or tracking radar.

dar Performance Figure - ratio of pulse power of Transmitter to Smin of receiver

- not often used

ip-Scan ratio - same as single scan Pd

- method used to check performance of ground-based radars

- here an aircraft is own on a radial course and for each scan of the antenna it is recordedwhether or not a target blip is detected. The ration of the number of scans the target wasseen at a particular range to the total number of scans is the blip scan ratio

- head on and tail on aspects are easiest to provide.

mulative Probability of Detection

- if single scan probability of detection id Pd, the probability of detecting a target at leastonce during N scans is the cumulative probability of detection

Pc = 1-(1-Pd)N

Note: the variation of Pd with range might have to be taken into account in computing Pc.

- the variation with range based on the cumulative probability of detection can be the 3rdpower rather than the 4th power which is based on a single scan probability

ax

PavG Aa( )nEi n( )4( )2kT 0 Bn( )Fn S N( )1 f pLs

-------------------------------------------------------------------------------=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Surveillanc

- thpu

- invo

- I

wi

- than

- flarcat

Notha- in practice Pc is not easy to apply. Furthermore radar operators do not usually report adetection the rst time it is observed (which is required by the denition of Pc). Insteadthey report a detection based on threshold crossing on two successive scans, or on two outof three scans

- for track while scan radars, the measure of performance might be the probability of initi-ating a target track rather than just probability of detection.

e Radar Equation

e radar equation which describes the performance of a radar which dwells on the target for nlses is sometimes called the searchlight range equation

a search or surveillance radar, the additional constraint that the radar must search a speciedlume of space in a specied time modies the range equation signicantly.

f represents the angular region to be searched in scan time ts, then we have

where t0 is the time on target = n/fp

0 = solid angle beamwidth

0 AE

where A is the Azimuth beamwidth and E is the Elevation beamwidth

also

th these substitutions the range equation becomes

is indicates that the important parameters for a search radar are the average power and thetenna effective aperture

requency does not appear explicitly, however low frequency is preferred since high power andge aperture are easier to obtain at low frequency and it is easier to build MTI (moving target indi-or) and weather has little effect on performance.

te: the radar equation will be considerably different if clutter or external noise (jamming) rathern receiver noise determine the background for the signal

ts t00-------=

G 40-------=

Rmax4 PavAeEi n( )ts

4kT 0Fn S N( )1Ls------------------------------------------------------= 1/6/06 53


54

Ac

- thacc

- th

- th

MTI and Pulse D

- The dopplvelocity of

- the second

- the MTI raous range (n

- the pulse drange ambig

- MTI is a n

- MTI adds

- practical,

Operation

- a simple Cpower amplcuracy of the Radar Equation

e predicted value of range from the range equation cannot be checked experimentally with anyuracy

e safest means to achieve a specied range performance is to include a safety factor.

is is sometimes difcult to do in competitive bids but results in ne radars

oppler Radar

er shift produced by a moving target may be used in a pulse radar (1) to determine the relativethe target or (2) to separate desired moving targets from undesired stationary clutter

application has been of greater importance

dar usually operates with ambiguous doppler measurements (blind speeds) but with unambigu-o second time around echoes)

oppler radar has a high enough PRF to operate with unambiguous doppler, but at the expense ofuities

ecessity in high quality air surveillance radars that operate in the presence of clutter

cost and complexity and digital signal processing

economical MTI has been available only since the mid 1970s

W radar is shown. In principle the CW radar can be converted into a pulse radar by providing aier and a modulator to turn the amplier on and off.radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Note: The mthe CW pow

- this LO pr

- Coherent m

- let the CW

- therefore t

- the dopple

he

- the referen

Note: equat

Note: for st

- examples

- when fd >

- when fd 40 dB requires < 0.01 rad from pulse to pulse!. This applies to the coho locking orange introduced by the HPA

to I imposed by pulse to pulse instability are:

frequency (f.)-2

o frequency (2fT)-2

phase shift ()-2g ()-2

g 2/(t)22B2/()2B

tude (A/A)2

= interpulse frequency change= bandwidth

pulse width transmission time

= interpulse phase change= timing jitter = pulse width jitter = interpulse amplitude change

gital processor does not experience degradation due to timing jitter of the transmit pulse if thelling the processor timing is started from the detected RF envelope of the transmitted pulse

ctuation of Clutter

stationary clutter - buildings, water towers, mountains, bare hills

A tcos=

A t +( )cos=

t( ) g2 t( ) 2A2

------- t 2-------+ A t2

-------+ sinsinsin=

A2

A( )2------------------ 1

( )2--------------= = 1/6/06 81


82

- dynamic c

- c

- most uctlocated in th

- any motio

- Fig. 4.29 s

- experimen

where Wg(ff0 a =

- this equati

where c is

- or aslutter - trees, vegetation, sea, rain, chaff

an limit the performance of MTI

uating clutter targets can be represented by a model consisting of many independent scattererse resolution cell

n of the scatterers relative to the radar results in a different vector sum from pulse to pulse

hows the power spectral density of clutter for a 1 GHz carrier

tally measured spectra of clutter can be approximated by

(f) = clutter power spectrum) = Fourier transform of the clutter waveform= radar carrier frequency a parameter (given by Fig 4.29)

on can be rewritten as:

(19)

the RMS clutter frequency spread in Hz

W f( ) g f( ) 2 g02

aff 0------

2exp= =

W f( ) W 0f

2

2c2

---------exp=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

where v is

Note:

- the improv

where CA =

- for a singl

where H(f)

- for a singl

h(f

- substitutin

assuming

- if the expo

a

CA

W

0

----------=

CA

W

0

----------=

CA1 ---------= the clutter velocity spread in m/s

ement factor can be written as (20)

Ci/C0 - clutter attenuation averaged over all doppler frequencies

e delay line canceler

(21)

is the frequency response of the canceler

e delay line canceler

) = 1- exp(-j2fT)

= 2jsin(fT)exp(-jfT) (22)

g eqn 19 and 22 into 21 gives

c


84

or

or

Now the av

- hence

- similarly f

- a plot of e

CAf

42

---------=

CAf

16--------=

CAa

2------=erage gain (S0/Si)ave for the single delay line canceler is 2

(25)

or a double canceler, whose average gain is 6

(26)

quation 26 is shown in Fig 4.30 with fp as a parameter. Several representative examples of clut-

p2

c2

---------

p22

2v

2-------------

f p2

2f 0

2------------

I 1cf p

2

22c

2---------------

f p22

82v

2---------------

a f p2

2f 0

2------------= = =

I 2cf p

4

84c

4---------------

f p44

1284v

4----------------------

a2f p

4

24f 0

4----------------= = =radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

ter are indic

- Note: v (

- rain, sea e

- Note: the too great a fa function o

- for exampwhere their

- the genera

Antenna Sc

- a scanning

where nB is

- the receiveated at particular values of v

spectral spread in velocity) is with respect to the mean velocity which for ground clutter is zero.

cho and chaff have non zero mean velocity which must be accounted for

frequency dependence of equations 25 and 26 for the clutter spectrum can not be extended overrequency range since no account is taken of variation of cross section of individual scatterers asf frequency

le the leaves and branches of trees have considerably different reecting properties at Ka band dimensions are comparable to , than at UHF frequencies.

l form for an N pulse canceler with Nl = N-1 delay lines is

anning Modulation

antenna observes a target for time t0

the number of hits received

d pulse train of nite duration t0 has a frequency spectrum which is proportional to 1/t0

I NC2N l

N l!--------

f p2c------------

2N l=

t0nBf p-------

BS------= = 1/6/06 85


86

- hence evennite time o

- if the cluttaffected

- this limita

- to nd the

here WS(f)

and H(f) is

- if the antethe results p

- thus equat(the RMS s

- here the voeld strengt

now

where and

therefore

where Sa (t)

and is tsif the clutter were perfectly stationary, the clutter spectrum would have a nite width due to then target.

er spectrum is too wide due to too short an observation interval, the improvement factor will be

tion is called scanning uctuation or scanning modulation

limitation on Ic we rst nd the clutter attenuation CA

describes the spectrum produced by the nite time on target

the MTI processor frequency response

nna main beam is approximated by a Gaussian shape, the spectrum will also be Gaussian. hencereviously derived for a Gaussian clutter spectrum can be applied.

ions 25 and 26 apply for the antenna scanning uctuations with the correct interpretation of cpread of the spectrum about the mean)

ltage waveform for the clutter is modulated by the antenna power pattern (equal to the two wayh pattern), as it is rotated

B are in degrees

is the modulation of the received signal due to the antenna pattern

he scan rate in /s.

CA

WS f( ) fd0

WS f( ) H f( )

2fd

0

------------------------------------------------=

G ( ) G02.7762B

2----------------------exp=

Sa G0

2.776s-----

2

Bs------ 2------------------------------exp=radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

now

hence

- the spectru

Since this is

- hence

- for the pow

- hence the

- substitutin

s----- = and

m is found by taking the Fourier transform

=

a Gaussian function we must have

Note: this is for the voltage spectrum

er spectrum we have

s due to the antenna scanning is

g this for c in equations 25 and 26 yields

single canceler

and

tBs------ t0=

Sa K2.776t

2

t02

---------------------exp=

Sa K2.776t

2

t02

--------------------- j2ft[ ]expexp td

=

K1

2f

2t02

2.776--------------------exp

2f

2t02

2.776-------------------- f

2

2 f2

----------=

f1.178t0

-------------=

s f

2-------=

s1

3.77t0---------------=

I 1snB

2

1.388-------------=

I 2snB

4

3.853-------------= 1/6/06 87


88

- Note: the waveform istepped scan antenna also is limited in MTI performance by the nite time on target. The times rectangular which gives a different improvement factor.radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

Limiting in

- a limiter ising the disp

- ideally an enough, thesection is la

- this conditfactor I

- if the limimay be a bl

- the limiter

- a nonlinearesults in th

- Figure 4.3Gaussian cl

- here C/L i

Note: the lo15 to 25 dBcanceler. He

- limiting isrequires I at least 60 ddesigned fomodulation MTI Radar

used just before the MTI processor to prevent the residue from large clutter echoes from saturat-lay

MTI radar should reduce clutter to a level comparable to noise. However, when I is not greatclutter residue will appear on the display and prevent the detection of aircraft whose radar crossrger than the clutter residue.

ion can be prevented by setting the limit L relative to the noise N equal to the MTI improvement

t level is set too high, clutter residue obscures part of the display. If the limit is set too low, thereack hole effect

provides a CFAR (Constant False Alarm Rate)

r limiter however causes the spectrum of strong clutter to spread into the canceler pass band ande generation of additional residue which degrades the MTI performance.

2 plots I for 2 pulse and 3 pulse cancelers with various levels of limiting. The abscissa applies toutter spectrum generated by clutter motion (v) or by antenna scanning modulation (nB)

s the ratio of RMS clutter power to the IF limit level

ss of I increases with the complexity of the canceler. Limiting in a 3 pulse canceler will cause a reduction in performance. A 4 pulse canceler with limiting is only 2 dB better than a 3 pulsence adding complexity is not justied in a limiting environment

not needed if processor I is large enough to reduce the largest clutter to the noise level (typically60 dB). This is difcult to achieve since it requires the receiver to have a linear dynamic range ofB, the A/D must have at least 11 bits, the equipment must be stable, the processor must ber I = 60 dB and the number of pulses processed must be sufcient to reduce the antenna scanning

LN---- I= 1/6/06 89


90 radarnotes_2006.mif 1/6/06

radarnotes_2006.mif

5.10 tracking wit

- the track osured from

- the qualityvation and t

- a surveillaradar

- tracks cangrease penc

- a single opond scan ra

- also an op

- these probmatic detec

- the ADT p

- ta- tr- tr- tr- tr- tr

- the autom

- at each rantarget due to

- the integra

- an examplquantized raold

- by locatincalled beam

- if there is required to

- however thto establish

- a computeh Surveillance Radar

f a target can be determined with a surveillance radar from the coordinates of the target mea-scan to scan

of such a track depends on the time between observations, the location accuracy of each obser-he number of extraneous targets present in the vicinity of the tracked target

nce radar which develops tracks on targets it has detected is called a track while scan (TWS)

be obtained by having an operator mark the location of a target on the face of the PPI with ail on each scan

erator however can not handle more than about 6 target tracks when the radar has a twelve sec-te

erators effectiveness in detecting new targets decreases rapidly after 1/2 hour of operation

lems are avoided by automating the target detection and tracking process. This is called auto-tion and tracking (ADT)

erforms the following functions:

rget detectionack initiationack associationack updateack smoothingack termination

atic detection quantizes the range into intervals equal to the range resolution

ge interval the detector integrates n pulses (n is the expected number of hits expected from the the antenna scan rate)

ted pulses are compared with the threshold to determine the presence or absence of a target

e is the moving window de

radar systems notes

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