MS_4장 Radar Detection-std

Radar Lecture @ Prof Y Kwag Korea Aerospace Univ. RSP Lab

Chapter 4

Radar Detection

Radar Lecture @ Prof Y Kwag Korea Aerospace Univ.

4.1 Detection in the Presence of Noise

RSP Lab

Detection in the Presence of Noise

Fig 4.1. Simplified block diagram of an envelope detector and threshold receiver

2variancewithnoise,Gaussianwhitemeanzeroadditive:)(

signalechoradar:)(where

)()(signalinput

tn

ts

tnts


RADAR DETECTION

RSP Lab

- Band pass IF filter output signal is

)/tan()(

)1.4()(ofenvelope)(where

)(sin)()(,)(cos)()(

))(cos()(sin)(cos)()( 000

IQ

QI

QI

vvaphaset

tvtr

ttrtvttrtv

tttrttvttvtv

alarmMissVtnts

alarmFalseVtn

DetectionVtnts

T

T

T

)()(

)(

)()(

Radar designers seek to maximize the probability of detection for a given

probability of false alarm


RADAR DETECTION

RSP Lab

- If filter output is composed of

noise alone

)2.4()()(,)()( tntvtntv QQII

noise plus target return signal (sine wave of amplitude A)

)3.4()(sin)()()(

)(cos)()(cos)()()(

ttrtntv

AttnttrtnAtv

QQ

III

Where the noise quadrature components and are uncorrelated zero mean

Low pass Gaussian noise with equal variance, .

)(tnI )(tnQ

2


Joint probability density function

RSP Lab

- The joint probability density function (pdf) of the two random variables is QI nn ,

)4.4(2

)sin()cos(exp

2

1

2exp

2

1),(

2

22

22

22

2

rArnnnnf

QI

QI

- The pdf of the random variable and , represent the modulus and phase

of . The joint pdf for the two random variables is

)(tr )(t

)(tv )(,)( ttr

)8.4(cos

exp2

exp2

),(

)7.4()(casethisin

)6.4(cossin

sincosjacobian:where

)5.4(),(),(

22

22

2

rAArrrf

trJ

r

r

n

r

n

n

r

n

J

Jnnfrf

QQ

II

QI


Rice probability density function

RSP Lab

- The pdf for r alone is obtained by integrating over ),( rf

2

0

22

22

2

2

0

cosexp

2

1

2exp

),()(

drAArr

drfrf

Where the integral inside Eq.(4.9) is known as the modified Bessel function

of zero order.

)10.4(2

1)(

2

0

cos

0

deI

Thus,

2

22

202 2exp)(

ArrAI

rrf

Rice probability density function.

)11.4(

)9.4(


Rayleigh & Gaussian pdf

RSP Lab

)11.4(2

exp)(2

22

202

ArrAI

rrf

- Rice pdf

pdfRayleighpdfRiceA thebecames),alonenoise(0If- 2

.varianceandmeanofathebecames

,largeveryisIf-

2

2

ApdfGaussianpdfRice

A

)12.4(2

exp)(2

2

2

rrrf

)13.4(2

)(exp

2

1)(

2

2

2

Arrf



RSP Lab



RSP Lab

- The density function for the random variable is

* Note that for the case of noise alone (A=0), Eq.(4.15) collapses to

a uniform pdf over the interval {0, 2}.

drrff

r

0

),()(

cos

2

)sin(exp

2

cos

2exp

2

1)(

2

2

22

2 AF

AAAf

x

dexFwhere

22

2

1)(

)14.4(

)15.4(

)16.4(



RSP Lab

- One excellent approximation for the function F(x) is

02

1

51.5339.0661.0

11)( 2

2

2

xe

xxxF x

xofvaluesnegativeforand

)(1)( xFxF

)17.4(

)18.4(


4.2. Probability of False Alarm

RSP Lab

Probability of False Alarm

- Probability of false alarm is define as the probability that a sample

of the signal will exceed the threshold voltage when noise alone.

faP

R )(tr TV

levelthresholdtheinchangessmall

tosensitiveveryisPfa

2

2

2

2

2 2exp

2exp

T

V

fa

Vdr

rrP

T

)19.4( a

fa

TP

V1

ln2 2 )19.4( b


4.2. Probability of False Alarm

RSP Lab

Fig 4.3. Normalized detection threshold versus probability of false alarm


False Alarm Time

RSP Lab

- The false alarm time is faT

- Since the radar operating bandwidth is the inverse of B intt

Maximizing increasing decreased faT TV max)( dR

fa

intfa

P

tT )20.4(

voltaged threshol thepass willdetector envelope theofouput the

thattimeaverageortimenintegratioradarwhere intt

)21.4(

2

2

2exp

1

T

fa

V

BT


MATLAB function que_func.m

Compute -function [PD] = marcumsq(a, b)

-This function uses Parls method to compute PD


4.3. Probability of Detection

RSP Lab

Probability of Detection - Probability of detection is the probability that a sample of will exceed

the threshold voltage in the case of noise plus signal

DP )(trR

)22.4(2

exp2

22

202dr

ArrAI

rP

TV

D

- Assume that the radar signal is a sine waveform with amplitude A, then its power

is . By using (single-pulse SNR) and . 22A 22 2ASNR )1ln()2(

22

faT PV

)24.4()(,

)23.4(1

ln2,2

exp

2/)(

0

2

2

)/1ln(2

2

22

202

22

2

deIQ

P

AQdr

ArrAI

rP

faP

D

fa


Probability of Detection

RSP Lab

- Many approximations for computing Eq.(4.23). Very accurate approximation

presented by North is given by

- Q is called Marcums Q-function. When is small and is large so that the

Threshold is also large, Eq.(4.24) can be approximated by

faP DP

fa

DP

AFP

1ln2

)25.4(

5.0ln5.0 SNRPerfcP faD )26.4(

isfunctionerrorarycomplementthewhere

z

v dvezerfc0

221)(

)27.4(


Table 4.1 Single Pulse SNR(dB)

RSP Lab

dBSNRPandP faD 12.16pulsesinglemin.1099.010


Example 4.1

RSP Lab

dBSNR

FigandTablefromand

BTP

nB

solution:

le pulse.f a singhe SNR o , and t alarm P of falseobability , the prtime t

ration dar integd the ra1GHz. FinBandwidth 0.9 and bection Py of detprobabilit

e 16.67minutrm T false ala: time ofification wing specthe follodar has pulsed raA

fa

fa

faint

D

fa

75.15)(

4.4.1.4

106067.1610

11

sec110

11t

1

12

9

9int


Probability of Detection vs single pulse SNR

RSP Lab

Compute - This program us used to produce Fig. 4.3 with marcumsq.

10-2

10-4

10-6

10-8 10-10 10-12

15.75dB

Assume

Tfa=16.67minutes, PD=0.9,

(SNR)1 ?


4.4. Pulse Integration

Radar pulse integration

: adding radar echoes from many pulses

(1) Coherent (Pre-detection) integration

perform on the quadrature components prior to the envelop detection

(2) Non-coherent (Post-detection) integration

perform after the envelop detector

RSP Lab


Coherent Integration

RSP Lab

Coherent Integration Perfect integrator using Integrating np pulses improve the SNR

The case for non-coherent integration integration loss

- mth pulse

)()( tntsy mm (4.34)

)(ts

)(tnm

: radar return of interest

: white uncorrelated additive noise signal

- Coherent integration of np

(4.35)

ppp n

m

m

p

n

m

m

p

n

m

m

p

tnn

tstntsn

yn

tz111

)(1

)()]()([11

)(



RSP Lab

(4.36)

(4.37)

- Total noise power in z(t) = variance

2

1,

2

21,

*

2

2 11)]()([1

ny

p

n

lm

mlny

p

n

lm

lm

p

nznn

tntnEn

pp

*

11

2 )(1

)(1 pp

n

l

l

p

n

m

m

p

nz tnn

tnn

E

2

ny : single pulse noise power

: 0 (ml) , unity (m=l) ml



- The desired signal power after coherent integration : not change

noise power is reduced by 1/np

RSP Lab

- SNR after coherent integration : Improve np

- Single pulse SNR : (SNR)1 ,

- Integrated (np) pulse SNR : (SNR)np

- SNR after coherent integration : improve np

1)(1

)( SNRn

SNRp

np (4.38)


Non-Coherence Integration


RSP Lab

: Non-Coherence Integration : implemented after the envelope detector

Fig 4.5. Simplified block diagram of a square law detector and non-

coherent integration



RSP Lab

- Square law detector used as an approximation to the optimum receiver

- Define a dimensionless variable (y)

/nn ry

SNRA

Rp 22

2

(4.39)

(4.40)

- The pdf for the new variable

2

)(exp)()()(

2

0

pn

pnn

n

nnn

RyRyIy

dy

drrfyf

(4.41)

2

2

1nn yx

(4.42)



RSP Lab

- The pdf for the variable at the output of the square law detector

pnpnn

nnn RxI

Rx

dx

dyyfxf 2

2exp)()( 0

(4.43)

- Non-coherent integration of np pulses

pn

n

nxz1

(4.44)

- xn are independent, the pdf for variable z

(4.45) )()()()( 21 pnxfxfxfzf

: the modified Bessel function of order

ppnppn

pp

zRnIRnzRn

zzf

p

p

22

1exp

2)( 1

2/)1(

(4.46)

1p

n1pnI



RSP Lab

1)(

)()( 1

pnp

pSNRn

SNRnE

- Non-Coherent integration efficiency

(4.47)

- Integration improvement factor for a specific Pfa

ppp

n

p nnEnSNR

SNRnI

p

)()(

)()( 1

(4.48)

- An empirically derived expressed for improvement factor

(accurate within 0.8dB)

(4.49)

2)log(018310.0)log(140.01

)log(6.46

)/1log(1)235.01(79.6)(

pp

p

fa

DdBp

nn

nP

PnI



RSP Lab

Fig 4.6. Improvement factor VS number of pulses (non-coherent integration).

These plots were generated using the empirical approximation in Eq.(4.49).


4.5. Detection of Fluctuating Targets

RSP Lab

Detection Probability Density Function - probability density function for fluctuating targets

0exp1

)(

A

A

A

AAf

avav

02

exp4

)(2

A

A

A

A

AAf

avav

Swerling I & II

Swerling III & IV

(4.50)

(4.51)


Detection of Fluctuating Targets

- Performing the analysis for the general case

RSP Lab

2

2

12

22/)1(

222

2

1exp

)/(

2)/(

AznI

Anz

An

zAzf pnp

n

pp

p

(4.52)

- To obtain f(z) use the relations

)()/(),( AfAzfAzf

- Finally,

(4.53)

(4.54)

dAAzfzf ),()(

(4.55) dAAfAzfzf )()/()(


Detection of Fluctuating Targets

Threshold Selection

RSP Lab

- DiFranco and Rubin general form relating the threshold and Pfa for any

number of pulsed and non-coherent integration

- Incomplete Gamma function

1,1 p

p

Tlfa n

n

VP

dn

en

n

V pT pnV

p

n

p

p

Tl

/

0

11

)!11(1,

1

2

1

)!1(

)2)(1(11

)!1(11,

p

Tp

n

T

p

T

pp

T

p

p

Vn

Tp

p

Tl

V

n

V

nn

V

n

n

eVn

n

V

- Approximated,

(4.56)

(4.57)

(4.58)

RSP Lab Hankuk Aviation Univ.

Threshold vs TV pn

10 100

19.11

Nfa = Marcums false alarm number nfa : pfa

30.05


4.6. Probability of Detection Calculation

- Single pulse SNR is unity (0 dB) range R0 (reference range)

RSP Lab

14

4

00

3

22

0

BFLRkT

GPSNR tR

4003

22

4 BFLRkT

GPSNR t

4

0

0

R

R

SNR

SNR

R

- Single pulse SNR at any range R

- Dividing Eq.(4.66) by Eq.(4.65)

(4.65)

(4.66)

(4.67)

R

RSNR dB

0log40

- SNR at any other range R

(4.68)


Detection of Swerling I targets

Detection of Swerling I Targets

- detection probability for Swerling I type targets (derived by Swerling)

1

1 pSNRV

D ; nePT

11,1

1

111,1

1

1

p

SNRnV

p

p

TI

n

p

pTID ; nen

SNRn

V

SNRnnVP pT

p

(4.76)

(4.77)


2 4 6 8 10 12 14 16 18 20 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR - dB

Pro

babili

ty o

f dete

ction

Swerling V

Swerling I

Probability of detection versus SNR,

single pulse. Pfa= 910

Swerling I and V type fluctuating np=1

Matlab Function pd_swerling1.m


-10 -5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR - dB

Pro

babili

ty o

f dete

ction

np = 1

np = 10

np = 50

np = 100


Swerling I. Pfa= 610



-10 -5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR - dB

Pro

babili

ty o

f dete

ction

np = 1

np = 10

np = 50

np = 100


Swerling I. Pfa= 1210



Detection of Swerling II targets

Detection of Swerling II Targets - detection probability for Swerling II type targets

50,

11

pp

TID ; nn

SNR

VP

- When np>50, Eq.(4.70) is used to compute detection probability

pnC

3

13

SNRnp 1

PnC

4

14

2/2

36 CC

(4.78)

(4.79)

(4.80)

(4.81)


-10 -5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR - dB

Pro

babili

ty o

f dete

ction

np = 1

np = 10

np = 50

np = 100



Detection of Swerling III targets

Detection of Swerling III Targets

- detection probability for Swerling III type targets (derived by Marcum)

- when np=1, 2

op

pp

T

n

pp

TD Kn

SNRnSNRn

V

SNRnSNRn

VP

p

22

2/11

21

2/1exp

2

1,/21

1,1!22/1

0

1

p

p

TIpTI

pp

Vn

TD n

SNRn

VKnV

nSNRn

eVP

Tp

- for np>2

(4.82)

(4.83)



RSP Lab

-10 -5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR - dB

Pro

babili

ty o

f dete

ction

np = 1

np = 10

np = 50

np = 100


Detection of Swerling IV targets

Detection of Swerling IV Targets

- detection probability for Swerling IV type targets (derived by Marcum)

- for np



-10 -5 0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

SNR - dB

Pro

babili

ty o

f dete

ction

np = 1

np = 10

np = 50

np = 100


4.7. Cumulative Probability of Detection

RSP Lab


4.7. Cumulative Probability of Detection

- The cumulative probability of detection refers to detecting the target at least

once (range R)

- The target gets closer to the radar, probability of detection increases

- The probability of detection during the n th frame is PDn ,

Then, cumulative probability of detecting during n th frame is

RSP Lab

)1(11

n

i

DC inPP (4.95)


Prob 4.3. A radar detects a closing at R, PD= 0.5 , Pfa=10-7. Compute and

sketch the single look probability of detection as a function of range

(interval 2~20 Km) If successive frame is 1Km, cumulative

probability of detection at 8Km ?

RSP Lab

Example 4.3

RR

SNRSNR R log405210

log40)()( 10

Sol) From table 4.1 the SNR corresponding to PD= 0.5 , Pfa=10-7 is 12dB.

From eq. (4.68) , the SNR at any range R as

-The cumulative probability of detection at 8Km is

9998.0)1)(01.01)(07.01)(25.01)(5.01)(9.01)(999.01(1 39

CP


Example 4.3

< PD versus normalized range>

frame 1 2 3 4 5 6 7 8 9

Range in Km 16 15 14 13 12 11 10 9 8

< range listing for frames 1 through 9 (frame 9 corresponds to R = 8Km) >


4.8 Solving the Radar Equation

RSP Lab

4

1

0

3

2

)()4(

SNRFLkT

GGTfPR

e

rtirt

gain antennareceivingG

gain antennangtransmittiG

tervalindwellT

PRFf

widthpulse

power dtransmittepeakP

r

t

i

r

t

(4.96)

detectionfor requiredSNRminimum(SNR)

figure noisereceiver F

emperaturereceiver tT

constant sBoltzmnn'k

loss systemtotalL

section crosstarget

wavelength

0

e

where,


4.8 Solving the Radar Equation


4.9 Constant False Alarm Rate (CFAR)

RSP Lab

fa

TP

V1

ln2 2

- Relationship between threshold value VT and Pfa

- The process of continuously changing the threshold to maintain a constant

=> Constant False Alarm Rate (CFAR)

Three different types of CFAR

Adaptive threshold CFAR ( assume that interference distribution is known)

Nonparametric CFAR ( unknown interference distribution )

Nonlinear receiver techniques ( normalize root mean square of interference )

(4.97)

powernoisewhere 2,


4.9.1 Cell-Averaging CFAR (single pulse)

< Conventional CA-CAFR >


- The echo return for each pulse is detected by a square law detector

- The Cell Under Test (CUT) is the central cell

- The neighbors of the CUT are excluded from average process due to

spillover

- The output of M reference cells is averaged

- The threshold value is obtained by averaged estimate all reference cell by K0

RSP Lab


ZKY 01

- Cell-averaging CFAR assumes all reference cells contain zero mean

independent Gaussian noise of variance

- In this case, the gamma pdf is,

0;)2/(2

)(2/

)2/(1)2/( 2

zM

ezzf

MM

zM

(4.98)

(4.99)



- The conditional probability of false alarm when y= VT can be written as 22/)( yTfa eyVP

- The unconditional probability of false alarm is

dyyfyVPP Tfafa

0

)()(

where, f(y) is the pdf of threshold,

- Except for the constant K0 is the same as (4.99). Therefore,

0;)()2(

)(2

0

)2/(12

0

yMK

eyyf

M

KyM

Mfa KP

)1(

1

0

- Substituting (4.102) and (4.100) into (4.101) yields

(4.101)

(4.100)

(4.102)

(4.103)


4.9.2 Cell-Averaging CFAR with Non-Coherent Integration

< Conventional CA-CFAR with non-coherent integration>


4.9.2 Cell-Averaging CFAR with Non-Coherent Integration

- The output of each reference cell is the sum of nP squared envelopes

- Total number of summed reference samples is MnP

- The output Y1 is also the sum of nP squared envelopes

- Noise alone is present CUT, Y1 random variable (pdf with 2 nP degrees of

freedom)

- The probability of false alarm is equal to the probability Y1 /Z exceeds threshold

11 /Pr KZYobPfa

- Let conditional probability of false when y= VT be Pfa (VT = y),

unconditional false alarm probability is,

dyyfyVPP Tfafa

0

)()(

where, f(y) is the pdf of the threshold

(4.104)

(4.105)

Documents

MS_4장 Radar Detection-std