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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
Radar Systems Course 3Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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 Systems Course 4Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 5Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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.
Radar Systems Course 6Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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.)
Radar Systems Course 7Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 8Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 9Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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!
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 10Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 11Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 12Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
∫∞
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−=
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 13Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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!
Radar Systems Course 14Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 15Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 16Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 17Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• Basic concepts
• Integration of pulses
• Fluctuating targets
• Constant false alarm rate (CFAR) thresholding
• Summary
Radar Systems Course 18Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 19Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 20Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 21Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 22Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 23Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 24Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 25Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 26Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
Detection Statistics for Different Types of Integration
Coherent and Non-Coherent Integration
0.10
0.50
0.900.95
0.99
0.999
0.012 4 6 8 10 12 14 16
PFA
=10-6
Non-CoherentIntegration4 Pulses
CoherentIntegration4 Pulses
BinaryIntegration
3 of 4 Pulses
Single Pulse
Prob
abili
ty o
f Det
ectio
n
Signal to Noise Ratio (dB)
Radar Systems Course 27Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
Detection Statistics for Different Types of Integration
Coherent and Non-Coherent Integration
0.10
0.50
0.900.95
0.99
0.999
0.012 4 6 8 10 12 14 16
PFA
=10-6
Non-CoherentIntegration4 Pulses
CoherentIntegration4 Pulses
BinaryIntegration
3 of 4 Pulses
Single Pulse
Prob
abili
ty o
f Det
ectio
n
Signal to Noise Ratio (dB)
Radar Systems Course 28Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 29Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 30Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 31Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 32Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
Outline
• Basic concepts
• Integration of pulses
• Fluctuating targets
• Constant false alarm rate (CFAR) thresholding
• Summary
Radar Systems Course 33Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 34Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 35Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 36Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 37Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 38Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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 =
Radar Systems Course 39Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 40Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 41Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 42Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 43Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
≤≤+
≤≤
Adapted from Shnidman in Richards, Reference 7
Radar Systems Course 44Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Adapted from Shnidman in Richards, Reference 7
Radar Systems Course 45Detection 11/1/2010
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Outline
• Basic concepts
• Integration of pulses
• Fluctuating targets
• Constant false alarm rate (CFAR) thresholding
• Summary
Radar Systems Course 46Detection 11/1/2010
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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
Radar Systems Course 47Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
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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
Courtesy of MIT Lincoln LaboratoryUsed with permission
Radar Systems Course 49Detection 11/1/2010
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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
Window Slides Through Data
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
Courtesy of MIT Lincoln LaboratoryUsed with permission
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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
Window Slides Through Data
Data Cells for Mean Level 2
Use Larger Value
Courtesy of MIT Lincoln LaboratoryUsed with permission
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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
Window Slides Through Data
Data Cells for Mean Level 2
Use Larger Value
“Censored”
Data (Not Usedin Computation of Average)
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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
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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
Radar Systems Course 54Detection 11/1/2010
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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
Radar Systems Course 55Detection 11/1/2010
IEEE New Hampshire SectionIEEE AES Society
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
Radar Systems Course 56Detection 11/1/2010
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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.
Radar Systems Course 57Detection 11/1/2010
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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