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IEEE New Hampshire SectionRadar Systems Course 1Detection 1/1/2010 IEEE AES Society

Radar Systems Engineering Lecture 6

Detection of Signals in Noise

Dr. Robert M. O’DonnellIEEE New Hampshire Section

Guest Lecturer

Radar Systems Course 2Detection 11/1/2010

IEEE New Hampshire SectionIEEE AES Society

Block Diagram of Radar System

Transmitter

WaveformGeneration

PowerAmplifier

T / RSwitch

Antenna

PropagationMedium

TargetRadarCross

Section

Photo ImageCourtesy of US Air ForceUsed with permission.

PulseCompressionReceiver Clutter Rejection

(Doppler Filtering)A / D

Converter

General Purpose Computer

Tracking

DataRecording

ParameterEstimation Detection

Signal Processor Computer

Thresholding

User Displays and Radar Control

This Lecture



Outline

• Basic concepts– Probabilities of detection and false alarm– Signal-to-noise ratio

• Integration of pulses

• Fluctuating targets

• Constant false alarm rate (CFAR) thresholding

• Summary



Radar Detection –

“The Big Picture”

• Mission –

Detect and track all aircraft within 60 nmi of radar

• S-band λ

~ 10 cm

Example –

Typical Aircraft Surveillance RadarASR-9

Courtesy of MIT Lincoln LaboratoryUsed with permission

RotationRate

12. rpm

Range60 nmi.

TransmitsPulses at~ 1250 Hz



Range-Azimuth-Doppler Cells to Be Thresholded

RotationRate

12.7 rpm

10 Pulses / Half BWProcessed into

10 Doppler Filters

Example –

Typical Aircraft Surveillance RadarASR-9

Range -

Azimuth -

Doppler Cells~1000 Range cells~500 Azimuth cells~8-10 Doppler cells

5,000,000 Range-Az-Doppler Cellsto be threshold every 4.7 sec.

0 50 100

-25

-50

0

Radial Velocity (kts)

Mag

nitu

de (d

B)

Doppler Filter Bank

Az Beamwidth

~1.2 °

As Antenna Rotates~22 pulses / Beamwidth

Is There a Target Present in Each Cell?

RangeResolution

1/16 nmi

RadarTransmitsPulses at~ 1250 Hz

Range60 nmi.



Target Detection in Noise

• Received background noise fluctuates randomly up and down• The target echo also fluctuates…. Both are random variables!• To decide if a target is present, at a given range, we need to set a

threshold (constant or variable)• Detection performance (Probability of Detection) depends of the

strength of the target relative to that of the noise and the threshold setting

– Signal-To Noise Ratio

and Probability of False Alarm

Rec

eive

d B

acks

catte

r Pow

er (d

B)

20

40

30

10Noise

ThresholdFalseAlarm

DetectedStrongTarget

Weak target (not detected)

00 25 50 100 125 150 175

Range (nmi.)



The Radar Detection Problem

RadarReceiver

DetectionProcessing

Measurement Decision

x10 HH or

nx =

Measurement ProbabilityDensity

Target absent hypothesis, Noise onlyTarget present hypothesis, Signal plus noise nax +=

)Hx(p 0

1H )Hx(p 1

For each measurement There are two possibilities:

For each measurement There are four decisions:

0H

0H 1H

1H

Don’t Report

FalseAlarm

DetectionMissedDetection

Truth

Decision

0H



Maximize subject to no greater than specified FAPDP( )α≤FAP

Threshold Test is Optimum

0H

0H 1H

1H

Don’t Report

FalseAlarm

DetectionMissedDetection

Truth

DecisionProbability of Detection:

Probability of False Alarm:

DP

FAP

The probability we choose when is true

The probability we choose when is true

1H 1H

1H 0H

Likelihood Ratio Test

> η=)Hx(p)Hx(p

)x(L0

1

<1H

0H

LikelihoodRatio Threshold

Objective:Neyman-Pearsoncriterion



Basic Target Detection Test

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Voltage

Prob

abili

ty D

ensi

ty Detection

Threshold

Noise Probability DensityNoise OnlyTarget Absent

Probability of False Alarm ( PFA ) PFA

= Prob{ threshold exceeded given target absent }

i.e. the chance that noise is called a (false) target

We want PFA to be very, very low!

Probability of False Alarm ( PFA ) PFA

= Prob{ threshold exceeded given target absent }

i.e. the chance that noise is called a (false) targetWe want PFA to be very, very low!




Basic Target Detection Test

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Voltage

Prob

abili

ty D

ensi

ty Detection

Threshold

Noise

Probability Density

Signal-Plus-Noise

Probability Density

Probability of

False Alarm ( PFA )

Probability of

False Alarm ( PFA )

Signal + NoiseTarget Present

Probability of Detection ( PD ) PD

= Prob{ threshold exceeded given target present }

I.e. the chance that target is correctly detected

We want PD to be near 1 (perfect)!

Probability of Detection ( PD ) PD

= Prob{ threshold exceeded given target present }

I.e. the chance that target is correctly detectedWe want PD to be near 1 (perfect)!

)Hx(p 1




Signal Plus NoiseSNR = 10 dBPD

= 0.61

Signal Plus NoiseSNR = 20 dBPD

~ 1

Detection Examples with Different SNR

• PD

increases with target SNR for a fixed threshold (PFA

)• Raising threshold reduces false alarm rate and increases

SNR required for a specified Probability of Detection

• PD

increases with target SNR for a fixed threshold (PFA

)• Raising threshold reduces false alarm rate and increases

SNR required for a specified Probability of Detection

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Voltage

Prob

abili

ty D

ensi

ty

Noise OnlyDetection ThresholdPFA

= 0.01




Non-Fluctuating Target Distributions

0 2 4 6 800.10.20.30.40.50.60.7

Voltage

Prob

abili

ty

Threshold

Noise pdf

Signal-Plus-Noise pdf

Set threshold rT based

on desired false-alarm probability

Compute detection probability forgiven SNR and false-alarm probability

Is Marcum’s Q-Function(and I0 (x) is a modified Bessel function)

where

Rayleigh Rician

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

2rexprHrp

2

o ( ) ( )RrI2

RrexprHrp o

2

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛ +−= 2

RSNR =

FAeT Plog2r −=

( ) ( )( )FAer

1D Plog2,SNR2QdrHrpPT

−== ∫∞

( ) ( ) drraI2

arexprb,aQ ob

22

∫∞

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=




Probability of Detection vs. SNRPr

obab

ility

of D

etec

tion

(PD)

Signal-to-Noise Ratio (dB)

PFA

=10-4

PFA

=10-6

PFA

=10-10

PFA

=10-12

PFA

=10-8

4 6 8 10 12 14 16 18

0.6

1.0

0.4

0.0

0.2

0.8SNR = 13.2 dB

needed for

PD

= 0.9 and

PFA

= 10-6

Steady Target

Remember

This!



Probability of Detection vs. SNR

SNR = 13.2 dB

needed for

PD

= 0.9 and

PFA

= 10-6

Steady Target

Remember

This!

Prob

abili

ty o

f Det

ectio

n (P

D)

Signal-to-Noise Ratio (dB)4 6 8 10 12 14 16 18 20

0.7

0.9999

0.1

0.3

0.5

0.05

0.9

0.2

0.999

0.99

0.8

0.95

0.98

PFA

=10-4

PFA

=10-6

PFA

=10-10

PFA

=10-12

PFA

=10-8



Tree of Detection Issues

Detection

Non-Fluctuating

Multiple pulses

Swerling I

Fluctuating

Fluctuating

Single Pulse

Square Law Detector

CoherentIntegration

Single PulseDecision

BinaryIntegration

Non-CoherentIntegration

Linear Detector

Non-Fluctuating

Uncorrelated

Swerling IISwerling III Swerling IV

Fully Correlated Partially Correlated



Detection Calculation Methodology

Probability of Detection vs. Probability of False Alarm and Signal-to-Noise Ratio

Determine PDF at Detector Output

• Single pulse• Fixed S/N

Integrate N fixed target pulses

Scan to Scan Fluctuations (Swerling Case I and III)

Pulse to Pulse Fluctuations (Swerling Case II and IV)

Average over signal fluctuations

Integrate over N pulses

Average over target fluctuations

Integrate from threshold, T, to ∞

PD

vs. PFA

, S/N, &Number of pulses, N



Outline

• Basic concepts




• Summary



Integration of Radar Pulses

Detection performance can be improved by integrating multiple pulses

Calculate

ThresholdCalculate

Coherent Integration Noncoherent Integration

• Adds ‘voltages’

, then square• Phase is preserved• pulse-to-pulse phase coherence required• SNR Improvement = 10 log10

N

• Adds ‘powers’

not voltages• Phase neither preserved nor required• Easier to implement, not as efficient

…

Pulses

TxN1

2N

1nn >∑

=Tx

N1 N

1n

2n >∑

=

Sum CalculateThreshold

Pulses

… 2N

1nnx∑

=

nx

Nx

2x1x 1x

Nx

21x

Calculate22x

Calculate2Nx

2x∑=

N

1n

2nx

Target DetectionDeclared if Target Detection

Declared if



Integration of Pulses

Signal to Noise Ratio per Pulse (dB)0 5 10 15

Prob

abili

ty o

f Det

ectio

n

0.6

1.0

0.4

For Most Cases, Coherent Integration is More Efficientthan Non-Coherent Integration

0.2

0.8SteadyTarget

PFA

=10-6

One Pulse

Ten PulsesCoherent Integration

Ten PulsesNon-Coherent

Integration

CoherentIntegration

Gain

Non-CoherentIntegration

Gain



Different Types of Non-Coherent Integration

• Non-Coherent Integration –

Also called (“video integration”) – Generate magnitude for each of N pulses– Add magnitudes and then threshold

• Binary Integration (M-of-N Detection)– Separately threshold each pulse

1 if signal > threshold; 0 otherwise

– Count number of threshold crossings (the # of 1s)– Threshold this sum of threshold crossings

Simpler to implement than coherent and non-coherent

• Cumulative Detection (1-of-N Detection)– Similar to Binary Integration– Require at least 1 threshold crossing for N pulses



Calculate

Binary (M-of-N) Integration

ThresholdT

CalculateSum

…

Pulse 1

Pulse 2

Pulse

ThresholdT

ThresholdT

2nd

ThresholdM

Individual pulse detectors:

Target present if at least M detections in N pulses

1x1i

Nx

2x

1i,Tx n2

n =≥

∑=

=N

1nnim

N

2nd thresholding:

2i

Ni

0i,Tx n2

n =<Mm ≥Mm <

, target present, target absent

At LeastM of N

Detections

At Least1 of N( ) ( ) kNk

N

MkN/M p1p

!kN!k!NP −

=

−−

= ∑ ( ) NC p11P −−=

21x

Calculate22x

Calculate2Nx

Binary Integration Cumulative Detection



Detection Statistics for Binary Integration

Signal to Noise Ratio per Pulse (dB)

Prob

abili

ty o

f Det

ectio

n

2 4 6 8 10 12 140.01

0.10

0.50

0.90

0.99

0.9991/42/43/4

4/4

“1/4”= 1 Detection

Out of 4 Pulses

Steady(Non-Fluctuating )

TargetPFA

=10-6



Optimum M for Binary Integration

Opt

imum

M

1

10

100

1 10 100Number of Pulses

Steady(Non-Fluctuating )

Target

PD

=0.95

PFA

=10-6

For each binary Integrator, M/N,there exists an optimum M

M (optimum) ≈

0.9 N0.8



Optimum M for Binary Integration

• Optimum M varies somewhat with target fluctuation model, PD and PFA

• Parameters for Estimating MOPT = Na 10b

Adapted from Shnidman in Richards, reference 7

Target Fluctuations

a

b

Range of NNo Fluctuations

0.8 -

0.02 5 –

700Swerling I 0.8 -

0.02 6 –

500Swerling II 0.91 -

0.38 9 –

700Swerling III 0.8 -

0.02 6 –

700Swerling IV 0.873 -

0.27 10 –

700



Detection Statistics for Different Types of Integration

PFA

=10-6

Coherent and Non-Coherent Integration

2 4 6 8 10 12 14 16

0.10

0.50

0.900.95

0.99

0.999

0.01

CoherentIntegration4 Pulses

Non-CoherentIntegration4 Pulses

BinaryIntegration

3 of 4 Pulses

Single Pulse

Signal to Noise Ratio (dB)

Prob

abili

ty o

f Det

ectio

n





0.10

0.50

0.900.95

0.99

0.999

0.012 4 6 8 10 12 14 16

PFA

=10-6



BinaryIntegration

3 of 4 Pulses

Single Pulse

Prob

abili

ty o

f Det

ectio

n




Signal to Noise Gain / Loss vs. # of Pulses

Number of Pulses Number of Pulses1 10 100 1 10 100

Sign

al to

Noi

se R

atio

Gai

n (d

B)

Sign

al to

Noi

se R

atio

Los

s (d

B)

0

5

15

10

20

2

4

10

6

0

8

Steady Target

PD

=0.95PFA

=10-6

• Coherent Integration yields the greatest gain• Non-Coherent Integration a small loss• Binary integration has a slightly larger loss than regular

Non-coherent integration

Coherent

Binary(Optimum M)

Binary N1/2

Non-Coherent

Binary N1/2

Binary(Optimum M)

Non-Coherent

RelativeTo CoherentIntegration



Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain

• Non-coherent Integration Can Be Very Inefficient in Correlated Clutter

Equi

vale

nt N

umbe

r of I

ndep

ende

nt P

ulse

s

N=300

5030

2010

3

100

0.001 0.01 0.1 11

5

2

50

10

100

20

rv f/ λσ

IndependentSampling

Region

DependentSampling

Regionvσ

T1fr =

VelocitySpread

OfClutter

PulseRepetition

Rate



Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain

• Non-coherent Integration Can Be Very Inefficient in Correlated Clutter

)T(ρ .99 0.9 0.1 0.02 Eq

uiva

lent

Num

ber o

f Ind

epen

dent

Pul

ses

N=300

5030

2010

3

100

0.001 0.01 0.1 11

5

2

50

10

100

20

rv f/ λσ

IndependentSampling

Region

DependentSampling

Regionvσ

T1fr =

VelocitySpread

OfClutter

PulseRepetition

Rate

Adapted from nathanson, Reference 8



Albersheim Empirical Formula for SNR (Steady Target -

Good Method for Approximate Calculations)

• Single pulse:

– Where:

– Less than .2 dB error for:

– Target assumed to be non-fluctuating• For n independent integrated samples:

– Less than .2 dB error for:

– For more details, see References 1 or 5

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

++=

D

De

FAe P1

PlogBP

62.0logA

B7.1BA12.0A)unitsnatural(SNR

( )B7.1BA12.0Alog44.0n

54.42.6nlog5)dB(SNR 1010n ++⎟⎠

⎞⎜⎝

⎛+

++−=

1.0P9.010P10 D7

FA3 >>>> −−

1.0P9.010P101n8096 D7

FA3 >>>>>> −−

SNRPer

Sample



Outline

• Basic concepts




• Summary



Fluctuating Target Models

• For many types of targets, the received radar backscatter amplitude of the target will vary a lot from pulse-to-pulse:

– Different scattering centers on complex targets can interfere constructively and destructively

– Small aspect angle changes or frequency diversity of the radar’s waveform can cause this effect

• Fluctuating target models are used to more accurately predict detection statistics (PD

vs., PFA, and S/N) in the presence of target amplitude fluctuationsRCS versus Azimuth

B-26, 3 GHz

RCS vs. Azimuth for a Typical Complex Target



Swerling Target Models

=σ Average RCS (m2)

RCSModel

Nature ofScattering Fast Fluctuation

“Pulse-to-Pulse”Slow Fluctuation“Scan-to-Scan”

Fluctuation Rate

Swerling I

Swerling IV Swerling III

Swerling II

Similar amplitudes

One scatterer muchLarger than others

Exponential(Chi-Squared DOF=2)

(Chi-Squared DOF=4)

( ) ⎟⎠⎞

⎜⎝⎛

σσ

−σ

=σ exp1p

( ) ⎟⎠⎞

⎜⎝⎛

σσ

−σσ

=σ2exp4p 2




Swerling Target Models

AmplitudeModel

Nature ofScattering Fast Fluctuation

“Pulse-to-Pulse”Slow Fluctuation“Scan-to-Scan”

Fluctuation Rate

Swerling I

Swerling IV Swerling III

Swerling II

Similar amplitudes

One scatterer muchLarger than others

=σ Average RCS (m2)

Central Rayleigh,DOF=4

Rayleigh

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛σ

−σ

=2aexpa2ap

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛σ

−σ

=2

2

3 a2expa8ap




Other Fluctuation Models

• Detection Statistics Calculations– Steady and Swerling 1,2,3,4 Targets in Gaussian Noise– Chi-

Square Targets in Gaussian Noise – Log Normal Targets in Gaussian Noise – Steady Targets in Log Normal Noise– Log Normal Targets in Log Normal Noise– Weibel Targets in Gaussian Noise

• Chi Square, Log Normal and Weibel Distributions have long tails

– One more parameter to specify distribution Mean to median ratio for log normal distribution

• When used– Ground clutter

Weibel– Sea Clutter

Log Normal– HF noise

Log Normal– Birds

Log Normal– Rotating Cylinder

Log Normal



RCS Variability for Different Target Models

0

5

10

15

20

0

5

10

15

20R

CS

(dB

sm)

0 20 40 60 80 1000

5

10

15

20

Sample #

Non-fluctuating Target

Swerling I/II

Swerling III/IV

Constant

HighFluctuation

MediumFluctuation




Fluctuating Target Single Pulse Detection : Rayleigh Amplitude

⎟⎟⎠

⎞⎜⎜⎝

⎛σ

−σ

=2aexpa2)a(p

2Nσσ

=ξ0

1

HTx

HTxz

<

>=

⎟⎟⎠

⎞⎜⎜⎝

⎛σ

−= 2N

2

FATexpP

∫ >= dzda)a(p)a,HTz(pP 1D

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ξ+σ

−=1

1TexpP 2N

2

D

naex:H j1 += φ

Detection Test

Probability ofFalse Alarm(same as fornon-fluctuating)

Average SNR perpulse

Rayleighamplitudemodel

Probability ofDetection Test

⎟⎟⎠

⎞⎜⎜⎝

⎛ξ+= 1

1

FAD PP

nx:H0 =



Fluctuating Target Single Pulse DetectionPr

obab

ility

of D

etec

tion

(PD)

0.6

1.0

0.4

0.0

0.2

0.8

Signal-to-Noise Ratio (dB)0 5 10 15 20 25

Non-Fluctuating

Swerling Case 3,4

Swerling Case 1,2

Fluctuation Loss

For high detection probabilities, more signal-to-noise is required for fluctuating targets.The fluctuation loss depends on the target fluctuations, probability of detection, and probability of false alarm.

PFA

=10-6



Fluctuating Target Multiple Pulse DetectionPr

obab

ility

of D

etec

tion

(PD)

0.6

1.0

0.4

0.0

0.2

0.8

Signal-to-Noise Ratio per Pulse (dB)0 4 8 12 16 20

Steady TargetCoherent IntegrationSwerling 2 Fluctuations Non-Coherent Integration, FrequencyDiversitySwerling 1 FluctuationsCoherent Integration, Single Frequency

PFA

=10-6

N=4 Pulses

• In some fluctuating target cases, non-coherent integration with frequency diversity (pulse to pulse) can outperform coherent integration



Detection Statistics for Different Target Fluctuation Models

Steady TargetSwerling Case ISwerling Case IISwerling Case III

Swerling Case IV

Signal-to-Noise Ratio (SNR) (dB)

Prob

abili

ty o

f Det

ectio

n (P

D)

-2 0 2 4 6 8 10 12 14 0.0

0.4

0.6

0.8

1.0

0.2

10 pulses Non-coherently

IntegratedPFA

=10-8

Adapted from Richards, Reference 7



Shnidman Empirical Formulae for SNR (for Steady and Swerling Targets)

• Analytical forms of SNR vs. PD

, PFA

, and Number of pulses are quite complex and not amenable to BOTE* calculations

• Shnidman has developed a set of empirical formulae that are quite accurate for most 1st

order radar systems calculations:

* Back of the Envelope

( )( ) ( ) ( )( )DDDFAFA P1P4ln8.05.0PsignP1P4ln8.0 −−−+−−=η

=α

40N0 ≤

40N41

>

N2,2,NK

,1=

∞ Non-fluctuating target (“Swerling 0 / 5”)

Swerling Case 1

Swerling Case 2

Swerling Case 3

Swerling Case 4

Adapted from Shnidman in Richards, Reference 7



Shnidman Empirical Formulae for SNR (for Steady and Swerling Targets)

( )( )( )

( ) ( )

( ) )SNR(log10dBSNRNXC)unitsnatural(SNR

10CC

8020N2

P10ln7.08.0Pe

K1C

K/525.3P5339.14P4496.18P7006.17C

41

2N2X

10

10C

dB

FA

5

D)14.25P31.27

2

DDD1

dB

D

==

==

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+=

−+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −α++ηη=

∞

−−

∞

99.0P872.0CC

872.0P1.0C

D21

D1

≤≤+

≤≤




Shnidman’s Equation

• Error in SNR < 0.5 dB within these bounds– 0.1 ≤

PD ≤

0.99 10-9

≤

PFA ≤

10-3

1 ≤

N ≤

100

Steady TargetSwerling ISwerling IISwerling IIISwerling IV

SNR

Cal

cula

tion

Erro

r (dB

)

Probability of Detection0.0 0.2 0.4 0.6 0.8 1.0

0.0

-0.5

1.0

0.5

-1.0

SNR Error vs. Probability of Detection

N = 10 pulsesPFA

= 10-6




Outline

• Basic concepts




• Summary



Practical Setting of Thresholds

• Need to develop a methodology to set target detection threshold that will adapt to:

– Temporal and spatial changes in the background noise– Clutter residue from rain, other diffuse wind blown clutter,– Sharp edges due to spatial transitions from one type of

background (e.g. noise) to another (e.g. rain) can suppress targets

– Background estimation distortions due to nearby targets

Display, NEXRAD Radar, of Rain Clouds

Range (nmi)

RainCloud Receiver

Noise

0 1 2

S-Band Data

0

20

40Rain Backscatter Data

Ideal Case-

Little variation in Noise

Pow

erPo

wer

Courtesy of NOAA



Constant False Alarm Rate (CFAR) Thresholding

• Estimate background (noise, etc.) from data– Use range, or range and Doppler filter data– Set threshold as constant times the mean value of background

• Mean Background Estimate =

CFAR Window –

Range Cells CFAR Window –

Range and Doppler Cells

Range

Range cells

Dop

pler

Vel

ocity

cel

ls

Cell to beThresholded

“Guard”Cells

Data Cells Usedto DetermineMean Level ofBackground

∑=

N

1nnx

N1



Effect of Rain on CFAR Thresholding

Cell Under Test

“Guard”

Cells

Data Cells for Mean Level Computation

Window Slides Through Data

2.2 dB9 dB

4.52.6 Slant Range, nmi

Rad

ar B

acks

catte

r (Li

near

Uni

ts)

Rain Cloud

C Band5500 MHz

Receiver NoiseReceiver Noise

Range Cells

Mean Level Threshold CFAR




Effect of Rain on CFAR Thresholding

2.2 dB9 dB

4.52.6 Slant Range, nmi

Rain Cloud

C Band5500 MHz

Range Cells

Cell Under Test

“Guard”

Cells

Data Cells for Mean Level Computation


Receiver NoiseReceiver Noise

Sharp Clutter or Interference Boundaries Can Lead to Excessive False Alarms

Rad

ar B

acks

catte

r (Li

near

Uni

ts)

Mean Level Threshold CFAR




Greatest-of Mean Level CFAR

• Find mean value of N/2 cells before and after test cell separately

• Use larger noise estimate to determine threshold

• Helps reduce false alarms near sharp clutter or interference boundaries

• Nearby targets still raise threshold and suppress detection

Cell Under Test

“Guard”

CellsData Cells for Mean Level 1


Data Cells for Mean Level 2

Use Larger Value




Censored Greatest-of CFAR

• Compute and use noise estimates as in Greatest-of, but remove the largest M samples before computing each average

• Up to M nearby targets can be in each window without affecting threshold

• Ordering the samples from each window is computationally expensive

Cell Under Test

“Guard”

CellsData Cells for Mean Level 1


Data Cells for Mean Level 2

Use Larger Value

“Censored”

Data (Not Usedin Computation of Average)



Mean Level CFAR Performance

PFA

=10-6

1 PulseMatched FilterMean Level CFAR –

10 SamplesMean Level CFAR –

50 SamplesMean Level CFAR –

100 Samples

CFARLoss

Signal-to-Noise Ratio per Pulse (dB)

Prob

abili

ty o

f Det

ectio

n (P

D)

4 6 8 10 12 14 16 18 0.01

0.999

0.99

0.90

0.50

0.10

0.9999



CFAR Loss vs. Number of Reference Cells

PFA

=10-8

10-6

10-4

Number of Reference Cells0 10 20 30 40 50

CFA

R L

oss

(dB

)

2

1

0

3

4

5

6

PD

= 0.9

The greater the number of reference cells in the CFAR, the better is the estimate of clutter or noise and the less will be the loss in detectability. (Signal to Noise Ratio)

Adapted from Richards, Reference 7



CFAR Loss vs Number of Reference Cells

For Single Pulse Detection Approximation

CFAR Loss (dB) = -

(5/N) log PFA

Dotted Curve PFA = 10-4

Dashed Curve PFA = 10-6

Solid Curve PFA = 10-8

N = 15 to 20 (typically)

Since a finite number of cells are used, the estimate of the clutter or noise is not precise.

Number of Reference Cells, N

CFA

R L

oss,

dB

N=100

N=3

N=10

N=30

N=1

PFA

10-810-610-4

2 5 7 10 20 50 70 100

4

0.4

1

0.5

0.2

0.3

0.7

3

2

5

Adapted from Skolnik, Reference 1



Summary

• Both target properties and radar design features affect the ability to detect signals in noise

– Fluctuating targets vs. non-fluctuating targets– Allowable false alarm rate and integration scheme (if any)

• Integration of multiple pulses improves target detection– Coherent integration is best when phase information is available– Noncoherent integration and frequency diversity can improve

detection performance, but usually not as efficient• An adaptive detection threshold scheme is needed in real

environments

– Many different CFAR (Constant False Alarm Rate) algorithms exist

to solve various problems

– All CFARs algorithms introduce some loss and additional processing



References

1. Skolnik, M., Introduction to Radar Systems, McGraw-Hill, New York,3rd

Ed., 2001.2. Skolnik, M., Radar Handbook, McGraw-Hill, New York, 3rd

Ed., 2008.3. DiFranco, J. V. and Rubin, W. L., Radar Detection, Artech

House, Norwood, MA, 1994.4.

Whalen, A. D. and McDonough, R. N., Detection of Signals in Noise, Academic Press, New York, 1995.

5. Levanon, N., Radar Principles, Wiley, New York, 19886. Van Trees, H., Detection, Estimation, and Modulation

Theory, Vols. I and III, Wiley, New York, 20017. Richards, M., Fundamentals of Radar Signal Processing,

McGraw-Hill, New York, 20058. Nathanson, F., Radar Design Principles, McGraw-Hill, New

York, 2rd

Ed., 1999.



Homework Problems

• From Skolnik, Reference 1– Problems 2.5, 2.6, 2.15, 2.17, 2.18, 2.28, and 2.29– Problems 5.13 , 5.14, and 5-18

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