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The Radar Equation

Introduction

2

Introduction• Before target information can be extracted from an echo signal,

that signal must be of sufficient magnitude to overcome the effects of interference.

• The radar equation is used to predict echo power and interfering power to assist in making the determination of whether or not this condition is met. Use of the radar equation accomplishes the following:

– Assists in the design of radar systems to meet the detection specifications set by the users.

– Establishes the relationship between the signal power received and the radar and target parameters

– Describes the power received from interfering sources, including thermal noise, clutter, jamming, and EMI.

– Provides a means for predicting signal-to-interference ratios, and for predicting the maximum range at which targets of a given RCS will produce a specified signal-to- interference ratio.

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• Several parameters affect the signal and interfering power received by the radar system.

– The radar's operating parameters, including• transmitted power, • transmitted energy, • transmitted waveform, • antenna gain and effective aperture, • receiver noise performance• radar system losses, • and the minimum signal-to-interference ratio for detection

– Target parameters, including radar cross-section (RCS), RCS fluctuations, and the Target’s range

– The propagation medium parameters, including RF energy absorption by gasses and the scattering of RF energy by particles in the medium.

• With the above in view, The ability of a radar to detect the presence of a targetis expressed in terms of the Radar Equationwhich is worth deriving because of the insight it gives in the way the radar works.

• We begin with the Transmitter: which has a peak power out put Pt

[ W ].

4

• If this power is radiated isotropically by the antenna, then the power flux ( or, the power density-per unit area ) at a Range R is given by

• Power flux at distance R = Pt / 4πR2 [W/m2 ] (1)

because 4πR2 is the area of a sphere of radius R through which all power must pass.

• If the transmitting antenna is not isotropic, rather it is directional, and concentrates the power towards the target, the equation (1) is modified by introducing the Gain Factor G t.

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• Gain Factor, or Antenna Gain Gt is defined as the relation between the power density of a directional antenna in the middle of the main lobe, and the power density of an isotropic radiator with the same transmitter power.

• The power flux in the direction of the beam is now,

Power Flux at the target = Pt.Gt / 4πR2

[ W/m2 ] (2)

• ERP

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• The power effectively radiated by the Transmitter/antenna combination in the direction of the main beam,is called Effective Radiated Power, ERP.

• It is the product of the power delivered to the transmit antenna, and the gain of the antenna.

• Or ERP =Pt.Gt

• Example: A long range ship and land-based air-search radar can transmit 3MW, with a gain of 4500.– What is its ERP ?– To produce the same power in a receiver at a given range, the

isotropic antenna would be connected to what wattage ?

• Forward propagation

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• Example:– Range of a target from the radar is 300nmi.– Antenna gain is 4500– Total transmitted power, Pt is 3MW

Work out the power density of the Forward Signal ( or Power Per unit area in the beam at the target’s range).

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• The power flux in the direction of the beam is now,

Power Flux at the target = Pt.Gt / 4πR2

[ W/m2 ] (2)

• The Target intercepts a portion of this incident power, and re-radiates it.

• The measure of the incident power intercepted by the Target and radiated back towards the radar is called the Radar Cross-Section,

‘RCS, σ’

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• RCS of the Target :– has units of area.– indicates how large the target appears to be viewed

by the radar– is defined as the power radiated towards the radar

per unit solid angle divided by incident flux/4π steradians.

• depends on – (1) the angle of incidence at which it is viewed, – (2) the radar frequency and,– (3) the polarization used.

• The power re-radiated by the target is now .= Pt.Gt.σ/ 4πR2 [W] (3)

• Scoop

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• Example: If the target of previous example has an RCS of 10m2, how much of illumination power of 0.00348 W/m2 is captured by the ‘scoop’ ?

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• On the return path this power again spreads out over the sphere of area 4πR2.

• Although it does not usually spread out uniformly, the 'gain' of the target is automatically included in the concept of RCS.

• The power density at the radar thus becomes,

Power flux = Pt.Gt.σ/ [4πR2]2 [ W/m2 ] (4)

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• The amount of this returning power that is intercepted by the antenna is determined by its effective area, Ae. The mean power received by the radar, Pr is thus

[W/m2 ] (5)

• This is the radar equation in its fundamental form.

22 ]4/[. RAGPP ettr

13

..(5.b)]..............................4/

44/

4/.4/

: wellas form in this written becan (5)equation

,discussion above thefrom 4/ since and

...(5.a)....................4/

/4.4/

:form in this written becan (5)equation

4/G and /A4

so reception, andission for transm used is areial samebut 4

4.

4

4322

42242

2242

2

422

222

22e

24

2

222

RGP

RGP

GRPP

GA

RAP

ARAPP

AGGG

R

AP

R

APAP

t

t

tr

e

et

eetr

ert

et

eter

as Gt = 4πAe/λ2

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• Defining the maximum range Rmax as the distance where the received signal is equal to the weakest signal Smin, that the receiver can detect.

• This definition gives the following expressions for Rmax:

c).......(6. [m] 4/or

b).......(6. [m] 4/or

(6.a) ....... [m] 4/

4min

322max

4min

22max

4min

2max

SGPR

SAPR

SAGPR

t

et

ett

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• Example: Calculate the maximum range of a radar system which operates at 3 cm with a peak pulse power of 500kW, if its maximum receivable power is 10-13 W, the capture area of its antenna is 5m2 , and the radar crossectional area of the target is 20m2.

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• Finally, the inevitable in-efficiencies in a radar system must now be introduced; which is best done by lumping together as a System Loss factor, Ls, assuming it to be less than unity:

• Power received by the radar, from the target is given by

Pr = [Pt. Gt. Gr. σ. λ2.Ls. ]/[(4π)3.R4] [W] …..(7)• Though this is a complete description of power

received, it is still not useful because it does not indicate whether this power is larger or smaller than the background noise level.

• Unfortunately…

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• Unfortunately noise is always present either as– (a) internal noise from the electronics, or as – (b) external noise from

• (i) the galaxy/s • (ii) the atmosphere • (iii) man made interference, or even • (iv) deliberate jamming signals.

• All these noise are wide-band compared to the radar signal, and one of the functions of the radar receiver is to tailor the bandwidth to accept the received signal, without permitting any unnecessary further noise to enter.

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Average Noise

• Let us say, there is an Average Noise power, N present in the system.

• Let us now compare the Power received from target with the noise power, in what is known as SNR or S/N:

] [

4 43

2

NR

LGGP

N

PSNR srttr

[m]

4

4/1

3

2

max

SNRN

LGGPR srtt

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Example• A short-range surveillance radar operates

at 3GHz and uses a 1m diameter dish for both transmitting and receiving. If the mean transmitting power is 10kW and the noise level is -140dBW, then…

• Calculate the maximum range at which a small aircraft of radar crossection 1m2 could reliably be detected. Assume 5dB losses and an SNR of 13 dB.

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21

The Most Familiar Form of Radar Equation

• Valid for point targets interfered with by thermal noise generated in the radar receiver.

less)(DimensionFactor Noise System F

(Hz)Bandwidth B

K290

)J/ (1.38Constant sBoltzmann' K

LossPath n Propagatio

Loss System The

targetradar to from Range Way One R

(meters)h WavelengtThe

meters) (squre CossectionRadar target The

less)(Dimension

)(

....)4(

0

A

S

43

22

LL

..

T

P

TLLRGP

K

GainAntennaTheG

WattsPowerPeakTransmitThe

Where

EchoSingleessDimensionlFBK

NS

T

oAS

T

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1-Point Targets in Noise• We are familiar with the

equation relating to signal power received from a target >

• If the received power from the interfering sources is known the signal-to-interference ratio is found by dividing signal power by the interfering power >>

RAGP eTTP

42

)4(

takenispower receive system in the

place same at the levelpower ginterferin the

ratio ceinterferen-to-signal The S/I

where

less)(Dimension

P

)4(

I

42

PRAGP

PP

I

eTT

I

RIS

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The simple form of Radar equation…• …expressed the maximum

range Rmax in terms of the key radar parameters, and the target’s radar cross section when the radar sensitivity was limited by receiver noise >>

• Except for the target’s RCS, the parameters of this simple form of radar equation are under the control of the radar designer.

RAGP eTTP

42

)4(

W, detectable

m target, theofn CrossectioRadar

m target, theof aperture effective

,P

where

)4(

min

2

2

t

4

1

min2max

signalMinimumS

AntennaA

gainAntennaG

Wdpowertransmitte

S

GA

e

etPR

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This radar equation states that if long ranges are required:

• the transmitted power must be large,

• the radiated energy must be concentrated into a narrow beam (high transmitting antenna gain),

• the received echo energy must be collected with large antenna aperture,( also synonymous with high gain),

• and the receiver must be sensitive to weak signals.

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• In practice, however, the simple radar equation does not predict the range performance of actual radar equipments to a satisfactory degree of accuracy.

• The predicted values of radar range are usually optimistic. In some cases the actual range might be only half that predicted.

• Part of this discrepancy is due to the failure of the simple equation to explicitly include the various losses that can occur throughout the system or the loss in performance usually experienced when electronic equipment is operated in the field rather than under laboratory-type conditions.

• Another important factor that must be considered in the radar equation is the statistical or unpredictable nature of several of the parameters.

• The minimum detectable signal Smin and the target cross section are both statistical in nature and must be expressed in statistical terms

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• Other statistical factors which have an effect on the radar performance are the meteorological conditions along the propagation path and the performance of the radar operator, if one is employed.

• In this discussion, the simple radar equation will be extended to include most of the important factors that influence radar range performance.

• If all those factors affecting radar range were known, it would be possible, in principle, to make an accurate prediction of radar performance. But, as is true for most endeavors, the quality of the prediction is a function of the amount of effort employed in determining the quantitative effects of the various parameters.

• Unfortunately, the effort required to specify completely the effects of all radar parameters to the degree of accuracy required for range prediction is usually not economically justified.

• A compromise is always necessary between what one would like to have and what one can actually get with reasonable effort. This will be better appreciated as we proceed through the discussion and note the various factors that must be taken into account.

• A complete and detailed discussion of all those factors that influence the prediction of radar range is beyond the scope of a single session. For this reason many subjects will appear to be treated only lightly. This is deliberate and is necessitated by brevity.

• More detailed information will be found in some of the subsequent chapters or in the references listed at the end of each chapter of your text book

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Minimum Detectable Signal

The ability of a radar receiver to detect a weak echo signal is limited: – by the noise energy that occupies the same portion of

the frequency spectrum as does the signal energy. The weakest signal the receiver can detect is called the minimum detectable signal.

– The specification of the minimum detectable signal is sometimes difficult because of its statistical nature and because the criterion for deciding whether a target is present or not may not be too well defined.

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• Detection is based on establishing a threshold level at the output of the receiver. If the receiver output exceeds the threshold, a signal is assumed to be present. This is called threshold detection.

• Consider the output of a typical radar receiver as a function of time (Fig.).

• This might represent one sweep of the video output displayed on an A-scope. The envelope has a fluctuating appearance caused by the random nature of noise.

• If a large signal is present such as at A in Fig. , it is greater than the surrounding noise peaks and can be recognized on the basis of its amplitude. Thus, if the threshold level were set sufficiently high, the envelope would not generally exceed the threshold if noise alone were present, but would exceed it if a strong signal were present.

• If the signal were small, however, it would be more difficult to recognize its presence.

• The threshold level must be low if weak signals are to be detected, but it cannot be so low that noise peaks cross the threshold and give a false indication of the presence of targets.

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RECEIVER NOISE• Since noise is the chief factor limiting receiver sensitivity, it is necessary to

obtain some means of describing it quantitatively. • Noise is unwanted electromagnetic energy which interferes with the ability

of the receiver to detect the wanted signal. It may originate within the receiver itself or it may enter via the receiving antenna along with the desired signal.

• If the radar were to operate in a perfectly noise-free environment so that no external sources of noise accompanied the desired signal, and if the receiver itself were so perfect that it did not generate any excess noise, there would still exist an unavoidable component of noise generated by the thermal motion of the conduction electrons in the ohmic portions of the receiver input stages. This is called thermal noise, or Johnson noise, and is directly proportional to the temperature of the ohmic portions of the circuit and the receiver bandwidth. The available thermal-noise power generated by a receiver of bandwidth Bn, (in hertz) at a temperature T (degrees Kelvin) is equal to

Available thermal-noise power = kTBn, where k Boltzmann’s constant = 1.38 x 10 - 23 J/deg. If the temperature T is taken to be 290 K, which corresponds approximately to room temperature (62°F), the factor kT is 4 x 10-23 W/Hz of bandwidth.

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• The Total Noise at the output of the receiver may be considered to be equal to Thermal- Noise Power obtained from an “ideal” receiver X by the Noise Figure Fn

K

where

GBkT

NF

ano

on

290T

gain availableG

receiver fromoutput noiseN

T tempstdat receiver ideal ofout noise

receiver practical ofout noise

o

a

o

o

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• The available gain Ga is the ratio of the signal out, So to the signal in, Si and kToBn is the input noise Ni in an ideal receiver. Eq. for Fn may be written as…

out

out

in

in

n

NS

NS

F

32

min2max

4

min

min

i

n

)/(4

S

as expressed bemay signalinput

Ffor eqn. arranging-Re

oonno

et

o

onno

o

onno

NSFBkT

GAPR

N

SFBkTS

N

SFBkT