Signal Detection Basics - CFAR

• Types of noise (clutter) and signals (targets)

• Signal separation by comparison (threshold detection)

• Signal Statistics - Parameter estimation

• Threshold determination based on the required Pfa

• CFAR detectors design

• Detection Performance

Vassilis Anastassopoulos, Physics Department

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Noise (clutter) and Signal (Target)

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Detection by comparison with a threshold

Threshold T=amx

Clutter pdf p(x) Signal pdf p(x)

p(x)

x

Pd

Pfa

Mean mx of x

Pd

SCR

Receiver Operating Characteristics

Deferent Detectors

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Signal in background Noise

Targets in Sea Clutter

Targets in Search Radar

Signal Separation

Threshold detection

The Statistics pc(x) of the clutter and

the Target pt(x)

An Estimator z of the statistics of the signal – mean z=mx

An Assumption of the behavior (statistics) of the Target

Determination of Threshold T=az for specific Pfa

Performance assessment PDSCR

Target Declaration

Clutter Statistics Rayleigh Weibull K-pdf

CFAR Detectors CA case OW case

OS case Target Statistics Swerling models

Fundamentals of Radar Signal Theory

Mark Richards

McGraw-Hill

2005

Chapters 6 & 7

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Radar Detection as Hypothesis Testing Hypothesis H0: The measurement is a result of interference only (null Hypothesis)

Hypothesis H1: The measurement is a combined result of interference and echoes from Targets

The signals are described statistically by means of their pdfs and thus we have to employ Statistical Decision Theory.

Actually we need the pdf of the Noise pc(x) and the pdf of the Target pt(x):

Clutter x

Target w

Clutter + Target w+x

Probability of Detection Pd: Probability to correctly declare a target

Probability of False Alarm Pfa: Probability to declare a target when it is not present

Probability of Miss Pm: Probability to miss a target while it is present Pm=1-Pd

Threshold T=amx

Clutter pdf p(x) Signal pdf p(x)

p(x)

x

Pd

Pfa

Mean mx of x

Threshold T=amx

Clutter pdf pc(x)

Signal+Clutter pdf pw+c(x) p(x)

x

Mean mx of x

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Signals and Statistics

0 200 400 600 800 1000 1200-4

-3

-2

-1

0

1

2

3

0 200 400 600 800 1000 1200-4

-3

-2

-1

0

1

2

3

0 200 400 600 800 1000 12000

50

100

150

-4 -3 -2 -1 0 1 2 30

20

40

60

80

100

120

Signal+noise Spectrum and histogram

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Images and Statistics

0 100 200 300 400 500 600 700 8000

50

100

150

200

250

300

0 50 100 150 200 250 3000

50

100

150

200

250

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Decision based on the Likelihood Ratio

Pattern Recognition

Threshold T=amx

Clutter pdf pc(x)

Signal+Clutter pdf pw+c(x) p(x)

x

Mean mx of x

The Neyman-Pearson decision rule:

Choose the threshold T so that Pd is maximized, subject to Pfaa

It is an optimization problem the solution of which leads to the decision rule

T

xd dxHxpP )|( 1

T

xfa dxHxpP )|( 0

0

1

)|(

)|(

0

1

H

H

Hxp

Hxp

x

x

Case of p(x|H0) Gaussian

)1(

1)()|(1

)|(

1

0

0

fa

fa

T

x

T

xfa

PerfT

PTerfdxHxp

dxHxpP

Estimation of the mean

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Clutter and Target Statistics

aa

c

x

c

x

c

axf exp)(

1

x,c,a >0

a

c

xxF exp1)(

Weibull pdf

For a=2 Rayleigh pdf

0 )2/exp()2/(2

1)( 12/

2/

sss

nsp n

n

χ2 distribution 22

2

2

1 nxxxs

0,,, ))(exp()()(

)( 1

bxss

x

s

x

s

bxp bb

Generalized Gamma

0

1

1

)2()(

4)()/()( cxx

m

cdyypyxpxp m

mm

The K-distribution is a compound statistical

model i.e.

The ‘speckle’ x is Rayleigh distributed with mean y,

which is Gamma distributed (b=2).

Km(x) is the modified bessel function of the third kind

and order m, while c is the scale parameter.

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High Resolution Clutter

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GC-pdf modeling

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Estimation of the mean

Threshold T=amx

Clutter pdf pc(x)

Signal+Clutter pdf pw+c(x) p(x)

x

Mean mx of x

T

xfa dxHxpP )|( 0

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

16

18

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mean estimation using the averager:

mx=[x(1)+x(2)+…+x(n)]/n

(is it an optimal estimator?) n=10

Uniform distribution with mean 0.5

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Estimation of the mean

T

xfa dxHxpP )|( 0

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.580

5

10

15

Threshold T=amx

Clutter pdf pc(x)

Signal+Clutter pdf pw+c(x) p(x)

x

Mean mx of x

Mean estimation using the averager:

mx=[x(1)+x(2)+…+x(n)]/n

Optimal estimator:

For the same number of variables the

smallest variance in the estimation of

the required parameter. n=50

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Estimation of the mean and Pfa evaluation

Since the threshold T=amx=az is not

fixed, but its is actually a random

variable which follows a new pdf

(?),

We have to determine (steps)

1. A way to find an optimal

estimator.

2. To find the pdf of the estimate

3. To modify the expression for Pfa

and Pd.

azT

xfa dxHxpazxPP )|(][ 00

0

0

00

)(][

)|(][

dxxpaxxP

dxHxpazxPP

z

azT

xfa

Threshold T=amx

Clutter pdf pc(x)

Signal+Clutter pdf pw+c(x) p(x)

x

Mean mx of x

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Case of Weibull - statistics

aa

c

x

c

x

c

axf exp)(

1

a

c

xxF exp1)(

1

2][

a

rcxE rr

Fits well to sea statistics.

Mathematically it is well described.

With statistical fit tests or using

moment matching techniques we

can certify (or not) the validity of

the distribution.

However, estimation of the parameters

at each point of the signal space

we need an optimal estimator.

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Parameter estimation with statistical fit-tests

Histogram with blue (experimental pdf) for a squared Sea

region on a VV-SAR image with scale parameter c=492,1 and

και shape parameter a=2,16. Parameter estimation for this

region (many points) was carried out by the MATLAB

command “Wblfit” and confidence interval 99%.

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Maximum Likelihood Estimation

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Maximum Likelihood Estimation

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Maximum Likelihood Estimation

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Maximum Likelihood Estimation

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ML-Parameter Estimation – Weibull pdf (1/3)

2/1)2/(

exp2

)(

cc

xxcxf

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ML-Parameter Estimation – Weibull pdf (2/3)

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ML-Parameter Estimation – Weibull pdf (3/3)

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The pdf of the estimator statistic (1/3)

2/1)2/(

exp2

)(

cc

xxcxf

Weibull pdf

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The pdf of the estimator statistic (2/3)

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The pdf of the estimator statistic (3/3)

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OW-CFAR Performance

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Basic steps for CFAR detection

• Knowledge of the statistics of the clutter.

• Optimal statistic for mean estimation.

• Pdf of this statistic.

• Evaluation of Pfa.

• Threshold evaluation.

• Pd evaluation for various SCRs (Performance assessment - ROC)

0

0

00

)(][

)|(][

dxxpaxxP

dxHxpazxPP

z

azT

xfa Nc

fa

N

TP

1)(

1

2/

0

1

11

)(][

)|(][

dxxpaxxP

dxHxpazxPP

z

azT

xd • Assessment using Simulation

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CFAR performance

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Conclusions (1/2)

•For robust detectors we employ Order Statistic Estimators.

•Performance Assessment contains performance on clutter

edges, and target masking.

•Clutter maps constitute a-priori knowledge for the clutter

statistics of a regions. Accelerate detection.

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• We examined CFAR detection of point

targets in specific clutter statistics and

described performance assessment.

• Many other detection topics remain:

- Distributed targets

- Coherent detection

- Detection using multi-channel signals

• What is detection and its quality; How this

can be used in decision fusion;

Conclusions (2/2)