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Lecture 8

Radar Equation

Dr. Aamer Iqbal Bhatti1

Radar Signal Processing

Dr. Aamer Iqbal Bhatti

Power received from a point target in absence of noise.

If the received power from interfering sources is known,

the signal-to-interference ratio is found by dividing signal

power by interfering power.

2



2

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mWR

GPP T

R

I

T

I

R

PR

GP

P

PSIR

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Radar single antenna is represented by two different but

related parameters: Gain and Affective Aperture.

3



I

ETP

ARR

GPSIR1

4

1

4

122

Radar ERP

Power/Area of radar at target

ERP of target

Power/Area of target power at radar

Power from target captured by radar

SIR

There are three kinds of losses exist in radars:

◦ System Losses: Exists within the system itself.

◦ Propagation Path Losses: Losses in the medium.

◦ Ground Plane Losses: Caused by multiple signal path.

◦ LS :System Losses

◦ LA :Propagation Path Losses

◦ LGP :Ground Plane Losses

GPASI

T

LLLPR

GPSIR

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If more than on hit is processed, the signal to

interference ratio can be increased. A processing gain is

applied to the radar equation.

GP=Processing Gain (usually >>1).

Interfering power is internally-generated noise.

5



GPASI

PT

LLLPR

GGPSIR

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The noise power generated within the radar, taken as a

source at the input to the antenna.

The signal-to-noise ratio for multiple hits processed

together is

The processing gain for noise interference is given by

◦ NL = Number of hits processed together as a look.

◦ Li = The integration Loss.

6



BFkTP oNi

GPASo

PT

LLBFLKTR

GGP

NS

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i

LNP

L

NG )(

It is useful to consider the relationship between pulse

width and bandwidth for gated CW waveform.

Replacing B and GP in equation given on last slide.

The first three factors in numerator are the look energy.

7



1B

iGPASo

LT

LLLFLKTR

GNP

NS

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)(

Signal-to-noise or signal-to noise jamming ratio is directly

proportional to look energy regardless of the waveform

(Discussed later).

PAVG = Average Power

TD = Dwell or look time

8



iGPASo

DAVG

iGPASo

L

iGPASo

LT

LLLFLKTR

GTP

NS

LLLFLKTR

GW

NS

LLLFLKTR

GNP

NS

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)4(

)4(

)4(

)(

SNR in terms of PAVG is useful with pulsed, CW, and

pulse Doppler waveforms.

Energy view of SNR is one of the requirements of Low

Probability of Intercept (LPI) radar.

◦ LPI radar : A radar that “can see but can not be seen”.

LPI is based on the fact that

◦ Target echo detection is done with illumination energy.

◦ Interception is done with peak power.

So LPI requires low peak power but high energy

waveform.

9



A multimodal airborne radar has the following

specifications in one of its modes. (high PRF single target

track).

Find the single hit SNR for a 2 m2 target at 55 nmi.

10



Transmitted Power 10kW

Pulse Width 1.2µs

PRF 250000 pps

Antenna Gain 35dB

Frequency 10.5GHz(λ=0.0286m)

Rx NF 3.5dB

System Loss 1.4dB

Propagation Path Loss 1.6dB

Ground Plane Loss 0 dB

All factors must be in ratio form.

◦ Antenna gain is 3160, the noise factor is 2.24, the system loss is

1.38, the propagation loss is 1.45, the range is 101860 m, and the

ground plane loss is 1.

◦ The SNR for single hit (NL=1) using above equation will be

0.051(-12.9dB).

Example 3-2: If the same radar can coherently integrate

2048 pulses with 1.6dB integration loss Li.

◦ The integrated SNR will be 72 or 18.6dB.

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GPASo

PT

LLBFLKTR

GGP

NS

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)4(

One of the primary uses of the radar equation is the

prediction of the maximum range at which a particular

target can be detected.

Maximum range is found where the SNR is the

minimum.

Example 3-3: Find the maximum range at which the

radar of previous examples can detect a 10m2 target if

the minimum SNR is 16dB. For (a) a single hit, (b) a

2048-hit look.

(a) Maximum range for single hit is 28800m (15.6nmi).

(b) For 2048 hits it is 176800m (95.5nmi).

12



Target echoes for different SNR’s given below the figure.

The number in parentheses is the ratio of the rms signal

voltage to the rms noise voltage and is the square root of

SNR.

13



A factor of the radar equation can be isolated which

represents all system parameters which are inconvenient

to express in dB, called the space gain or space loss.

The space gain is therefore.

The space loss is

14



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GGPSIR

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2

43

2

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G SPACeR

2

2

43)4(m

RL SPACER

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A hypothetical high power early warning radar has the

following specifications.

Transmitted Power 15MW

Type Monostatic

Antenna 94-ft diameter array

Antenna effective aperture 320 m2

Antenna Gain 44700 (46.5dB)

Frequency 1GHz

Rx NF 1.1dB(1.29)

System Loss 1.0dB(1.26)

Propagation Path Loss 1.3dB(1.35)

Processing Gain to Noise 29dB(800)

Minimum SNR 14dB(25.1)

Ground Plane Loss 0 dB(1.0)

Estimate the following.

a) The signal power received from a single hit on a -11.4dBsm

target at a range of 2000km.

b) The noise power present in radar.

c) The SNR for this target for a single hit.

d) The SNR for this target for given processing gain.

e) The maximum range at which this target can be detected.

Solution.

a) 3.62 * 10-15W(-114.4dBm).

b) 5.16 * 10-15W (-112.9dBm).

c) -114.4dBm-(-112.9dBm)=-1.5dB which is not sufficient for

detection.

d) -1.5dB+29dB=27.5dB.

e) Maximum range will be 4350km.

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