Detection in Gaussian clutter - unipi.itThe detection algorithm is optimized for a specific Doppler....

Optimum Coherent Detection in Gaussian Disturbance

Maria S. GRECODipartimento di Ingegneria dell’Informazione

University of PisaVia G. Caruso 16, I-56122, Pisa, Italy

[email protected]

Binary Hypothesis Testing Problem

The function of a surveillance radar is to ascertain whether targets are present in the data. Given the observed vector z, the signal processor must make a decision as to which of the two hypotheses is true:

0

1

: Target absent

: Target presentt

H

H

= = +

z d

z s d

The test is implemented on-line for each range cell under test (CUT)

The detection is a binary hypothesis problem whereby we decide either hypothesis H0 is true and only disturbance is present or hypothesis H1 is true and a target signal is present with disturbance.

According to the Neyman-Pearson (NP) criterion (maximize PD while keeping constant PFA), the optimal decision strategy is a likelihood ratio test (LRT):

Neyman-Pearson Criterion

1 1

1 1

0 00 0

1 1

0 0

( ) ( )> >( ) e ln ( ) ln < <( ) ( )

H H

H H

H H

H H

p H p H

p H p Hη η

Λ = ⇒ Λ =

z z

z z

z zz z

z z

( )i iHp Hz z is the probability density function (PDF) of the random vector z

under the hypothesis Hi, i=0,1.

η is the detection threshold (set according to the desired PFA).

The complex multidimensional PDF of Gaussian disturbance is given by:

( )0

10

1( ) ( ) exp H

H Np H p

π−= = −dz z z z R z

R

{ }det , Hermitian matrixH= = →R R R RNx1 data vector

The Optimal Detector

1 1( ) ?Hp H =z zThe complex multidimensional PDF of target + disturbance:

It depends on the target signal model:

If the target vector is deterministic:1 01 0( ) ( )tH Hp H p H= −z zz z s

( )0 ,H ∈z 0 RCN

t= +z s d

{ } ( )1 0

00

11 0

( )

1( ) ( ) exp ( ) ( ) ( )

t t

tH

Ht t t t tH H N

p H

p H E p H p dπ

−

−

= − = − − − ⋅∫

z

s sz z

z s

z z s z s R z s s sR

��

If the target vector is random with known PDF:

1

( )d

N

f

×

p

[ ]1

( ) is the target complex amplitudet dN

fβ β×

=s p

Under some assumptions, we have found for the target signal:

2Doppler frequency target radial velocityt r

d

v Tf

λ= ⇔

The Optimal Detector

2

12 ( 1)

1

( )d

d

j f

d

Nj N f

ef

e

π

π×−

=

p⋮

Temporal Steering Vector

It has the Vandermonde form because the waveform PRF is uniform and target velocity is constant during the CPI.

The detection algorithm is optimized for a specific Doppler.

Since the target velocity is unknown a-priori, p is a known function of

unknown parameters, so the radar receiver should implement multiple

detectors that form a filter bank to cover all potential target Doppler frequencies.

Narrowband Slowly-Fluctuating Target Signal Model

Different models of st have been investigated to take into account different

degrees of a priori knowledge on the target signal:

(1) st perfectly known

(2) st = β p with , i.e., Swerling I model, and p perfectly known;

(3) st = β p with |β| deterministic and ∠β random, uniformly distributed in [0,2π),

i.e. Swerling 0 (or Swerling V) model, and p perfectly known

(4) st = β p with β unknown deterministic and p perfectly known;

2(0, )sβ σ∈CN

Target Signal Models

1

( )t dN

fβ×

=s p

Perfectly Known Target Signal - Case #1

The optimal NP decision strategy is a LRT (or log-LRT):

{ }

1

0

1

0

1 1 1

0

1 1 1 1 1

( )( ) ln ( ) ln ( ) ( )

( )

>2 <

H

H

H H Ht t

H

H H H H Ht t t t t t t

p Hl

p H

e η

− −

− − − − −

= Λ = = − − −

= + − = ℜ −

z

z

zz z z R z z s R z s

z

s R z z R s s R s s R z s R s

It is the so-called coherent whitening matched filter (CWMF) detector:

( ) ( )1 1 2 1 2 1 2 1 2 1 2 1 2H HH H H Ht t t t t

− − − − − − −= = = =s R z s R R z s R R z R s R z s z

1 2 1 2,t t− −s R s z R z≜ ≜

{ } { } { }1 2 1 2 1 2 1 2 1 2 1 2H H HE E E− − − − − −= = = =z z R zz R R zz R R RR I

matched filteringwhitening transformation

( ) ( )0 10, , , is a real Gaussian r.v.tH H l∈ ∈ ⇒z R z s RCN CN

Performance of the coherent WMF

{ }1

0

1 >( ) 2 <

H

H

Htl e η−= ℜz s R z

{ } { }{ }10 0 02 0H

tm E l H e E H−= = ℜ =s R z

� The fact that d is a complex circular Gaussian random vector implies that [Ch.13, Kay98]:

{ } { } and 0H H TE E= = =dd R R dd

[Kay98] S.M. Kay, Fundamentals of Statistical Signal Processing. Detection Theory. Prentice Hall, 1998.

{ } { }{ }1 11 1 12 2H H

t t tm E l H e E H− −= = ℜ =s R z s R s

{ } { }{ }{ }{ } { }

( )( ){ }{ }

( ) ( )

220 0 0 0

2 21 1 10 0

1 1 1 1

1 1 1 1 1 1 1 1

1 1 * 1 1 1 * 1

var

2

2

H H Ht t t

HH H H Ht t t t

H H H H H H H Ht t t t t t t t

H T H H T Ht t t t t

l H E l E l H H

E e H E H

E

E

E

σ

− − −

− − − −

− − − − − − − −

∗ ∗− − − − − −

= = −

= ℜ = +

= + +

= + + +

= + +

s R z s R z z R s

s R d d R s s R d d R s

s R dd R s s R ds R d d R s d R s d R s s R d

s R dd R s s R dd R s s R d d R s{ }( ){ }

{ } { }

{ } { }{ }

1 1 * 1 1

1 1 * 1 1

1 1 1

22 11 1 1 1

2 2

2 2

2 2

var 2

t

H T H Ht t t t

H T H Ht t t t

H Ht t t t

Ht t

E

E E

l H E l E l H Hσ

∗− − − −

− − − −

− − −

−

= +

= +

= =

= = − =

s R dd R s s R dd R s

s R dd R s s R dd R s

s R RR s s R s

s R s

( ) ( ) { }1 1 * 1 * 0we used the fact that andTH H T T T T

t t t t E∗− − − −= = = =s R d s R d d R s d R s dd


The probability of detection (PD) and probability of false alarm (PFA)

can be calculated using the statistics we have just derived:

{ }

{ }

0

0

0

0

0

1

20

0 0 2200

20

10

11

1 1 11

( )1Pr ( ) exp

22

1exp

22 2

2Pr ( )

2

FA l H

l mx

Hm t t

Ht t

D l H Ht t

l mP l H p l H dl dl

mxdx Q Q

mP l H p l H dl Q Q

η η

σ

ησ

η

ησπσ

η ησπ

ηηησ

+∞ +∞

−= +∞

−−

+∞ −

−

−= > = = −

− = − = =

−− = > = = =

∫ ∫

∫

∫

s R s

s R s

s R s

What is the meaning of ?


12 Ht t

−s R s

The filter output can be written as :

The Signal-to-Noise ratio (SNR) at the output is:

Let’s apply this consideration to our case where and

( )H H H Ht ty = = + = +w z w s d w s w d

{ }{ }

{ }{ }

{ }{ }

2 2

2

H H H H H HH Ht t t t t tt tH HH H H HH

E E ESNR

E EE= = = = =

w s w s s w w s s w w sw s s ww Rw w Rww dd w w dd ww d

1t

−=w R s

(H1)

if the target signal is deterministic


{ } { } { } { }1 1 1 1 1H H H H Ht t t t t t te e e e− − − − −ℜ = ℜ + ℜ = + ℜs R z s R s s R d s R s s R d

Therefore, the SINR at the output of the coherent WMF (CWMF) is:

( ){ }( ){ }

( )

( )

2 21 1

2 21 11

21

11

2

42

2

H Ht t t t

CWMFH HHt tt

Ht t H

t tHt t

SNRE e

E

− −

− −−

−−

−

= = + ℜ

= =

s R s s R s

s R d d R ss R d

s R ss R s

s R s


( )

( )( )

1

1

FA CWMF FA

CWMF

CWMFD D FA CWMF

CWMF

P Q SNR Q PSNR

SNRP Q P Q Q P SNR

SNR

η η

η

−

−

= ⇒ = ⋅

−= ⇒ = −

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFA

PD

ζ=-14dB

ζ=0dB

ζ=6dB

ζ=9.5dB

Receiver Operating Characteristic (ROC) curves:

The ROC is a function of only the SNR and the threshold.

Here, the threshold is varied to cover the entire range of interest and the ROC curve is plotted for four different SNR values.

CWMFSNRζ =


Swerling I Target Signal - Case #2

( )2( ), 0, , perfectly knownt d sfβ β σ= ∈s p pCN

( ) ( )20 10, , 0, H

sH H σ∈ ∈ +z R z pp RCN CN

{ } { } { }2 21 ( )( )H H H H

sE H E Eβ β β σ= + + = + = +zz p d p d pp R pp R

Based on these assumptions:

( ) { }22 20, is Rayleigh distributed,

is uniformly distributed on [0,2 )

js se Eϕβ β σ β β σ

ϕ π

= ∈ ⇒ =CN


The log-likelihood ratio (log-LR or LLR) is given by:

1

0

1 2 1 2 1

0

( )ln ln ln ( )

( )H H H H H

s s

H

p H

p Hσ σ− −= − + + − +z

z

zR pp R z R z z pp R z

z

( )

( )

0

1

10

2 11 2

1( ) exp

1( ) exp ( )

HH N

H HsH N H

s

p H

p H

π

σπ σ

−

−

= −

= − ++

z

z

z z R zR

z z pp R zpp R

By making use of Woodbury’s identity:

2 1 12 1 1

2 1( )

1

HH s

s Hs

σσσ

− −− −

−+ = −+R pp R

pp R Rp R p

1

0

1 12 2

2 1>ln ( ) ln ln <1

H

H

H HH

s s Hs

σ σ ησ

− −

−Λ = − + ++

z R pp R zz R pp R

p R p

1

0

21 > <

H

H

H η−p R z

By incorporating the non-essential terms in the threshold, we derive that in this case the optimal NP decision strategy is:


Again, the optimal decision strategy requires calculation of the WMF output, but instead of taking the real part of the output we calculate the modulo of the output: this is due to the fact that the target phase in this case is unknown.

This is sometimes called the noncoherent WMF

Performance of the non-coherent WMF: Signal-to-Noise power ratio (SNR)

Performance of the non-coherent WMF

1

0

2 21 > <

H

H

H H η−=w z p R z1 optimal weightsκ −=w R p

{ }{ }

{ } { }2 2

2 1

2

H Ht t

HWMFH

H

E ESNR SNR E

Eβ −= = ⇒ =

w s w sp R p

w Rww d

( )2 1 2, 0,HWMF s sSNR σ β σ−= ∈p R p CN

Performance of the non-coherent WMF when the target complex amplitude is a zero-mean circular complex Gaussian r.v. (Swerling I).

2( , ), 0,1i i iY H m iσ⇒ ∈ =CN

( ) ( )20 10, , 0, H

sH H σ∈ ∈ +z R z pp RCN CN

{ } { } { }{ } { } { } { }

1 10 0 0

1 1 11 1 1

0

0

H H

H H H

m E Y H E H E

m E Y H E H E Eβ

− −

− − −

= = = =

= = = ⋅ + =

p R z p R d

p R z p R p p R d

1 is a complex circular Gaussian r.v. HiY H−⇒ p R z≜

2

2 2 2 2

2

Exp( ), 0,1

1( ) ( )i

i

R I i i

x

iX Hi

X Y Y Y X H i

p x H e u xσ

σ

σ

−

= = + ⇒ ∈ =

=

Performance of the non-coherent WMF – SW1

0 0( )

1 0

xu x

x

<= ≥

{ } { } { } { }{ }

{ } { } { }{ } { }( ) ( )( )

2 222 1 10 0 0 0

1 1 1

222 11 1 1 1

2 2 21 1 1 1

22 1 1 1 2 1

var

var

1

H H

H H H

H

H H H H

H H H Hs s

Y H E Y H E H E

E

Y H E Y H E H

E E

σ

σ

β β

σ σ

− −

− − −

−

− − − −

− − − −

= = = =

= =

= = =

= + = +

= + = +

p R z p R d

p R dd R p p R p

p R z

p R p p R d p R p p R d

p R p p R p p R p p R p

20

0

21

1

0

1

( )

( )

FA X H

D X H

P p x H dx e

P p x H dx e

ησ

η

ησ

η

+∞ −

+∞ −

= =

= =

∫

∫


( )( ) ( )( ) ( )

20

2 2 20 0 02 2 2 21 1 1 1

20

ln 1ln 1

ln 1

FAFA

FA FA

PP

D FA

P e P

P e e e P

ησ

ση σ σσ σ σ σ

η σ−

− −−

= ⇒ =

= = = =

22 11

20

wh r 1e e 1 Hs WMFSNR

σ σσ

−= + = +p R p

� Receiver Operating Characteristic (ROC) curves:

( )1

1 WMFSNRD FAP P +=


PD is a monotonic increasing function of SNRWMF.

If the amplitude is non fluctuating and the phase is uniformly distributed:

is deterministic (nonfluctuating)


je ϕβ β βϕ π

= ⇒

Swerling 0 Target Signal – Case #3

1 1 11

0 0 0

11

0 0

1 1 1, ,1

0 0 0

2

11 ,,0

0 0

( , ) ( , ) ( )( )

( )( ) ( ) ( )

1( , )( , ) ( )

2

( ) ( )

H H HH

H H H

HH

H H

p H d p H p H dp H

p H p H p H

p H dp H p d

p H p H

ϕ ϕ ϕ

π

ϕ ϕϕ

ϕ ϕ ϕ ϕ ϕ

ϕ ϕϕ ϕ ϕπ

+∞ +∞

−∞ −∞

+∞

−∞

Λ = = =

= =

∫ ∫

∫∫

z zz

z z z

zz

z z

z zz

zz z z

zz

z z

( )1, ,jH e ϕϕ β∈z p RCN

( )( )

( )

( )( )

1

0

2

2

1,0

0

21

01

221 1 1

0

221 1

0

0

21

0

0

1( , )

2( )

( )

1exp ( ) ( )

2

exp

1exp

2

1exp 2 cos( )

2

1exp 2 cos( )

2

H

H

j H j

H

j H j H H

H H

H

p H d

p H

e e d

e e d

d

d e

π

ϕ

πϕ ϕ

πϕ ϕ

π

πβ

ϕ ϕπ

β β ϕπ

β β β ϕπ

β ϕ ϕ β ϕπ

β ϕ ϕ ϕπ

−

−

− − − −

− −

−−

Λ =

− − −=

−

= + −

= − −

= − ⋅

∫

∫

∫

∫

∫

z

z

z

zz

z p R z p

z R z

p R z z R p p R p

p R z p R p

p R z1

,H −p R p

10where Hϕ −∠p R z≜

Swerling 0 Target Signal

1

0

21 > <

H

H

H η−p R z

� Since I0(x) is monotonic in x, the LRT reduces to the noncoherent WMF:

( ) [ ]2

0 0

0

1exp ( )

2I x x d

π

ϕ ϕ ϕπ

−∫≜

where I0(x) is the modified Bessel function of the 1st kind:

( )

( )

2 1

12 1

0

21

0

0

10

1( ) exp 2 cos( )

2

>2 <

H

HH

H

H

H

d e

I e

πβ

β

β ϕ ϕ ϕπ

β λ

−

−

−−

−−

Λ = − ⋅

= ⋅

∫p R p

p R p

z p R z

p R z


1

0

1 > <

H

H

H η−p R z or equivalentely

Performance of the non-coherent WMF when the target amplitude |β| is deterministic, i.e. nonfluctuating, and the target phase is uniformly distributed over [0,2π) (Swerling 0).

The probability of false alarm is the same as before, since it does not depend on the target signal model:

1H

FAP eη

−−= p R p


( ) ( )0 10, , , ,jH H e ϕϕ β∈ ∈z R z p RCN CN

1 1

2

1 1,0

1( ) ( , )

2H Hp H p H dπ

ϕ ϕ ϕπ

= ∫z zz z

1

0

22 2 2 1 > <

H

H

HR IX Y Y Y η−= = + = p R z


The decision rule can be put in the form:

1 11 1( ) ( )D X H Y HP p x H dx p y H dyη η

+∞ +∞

= =∫ ∫

We want to calculate:

PDF of |Y| under the hypothesis H1

1HR IY Y jY −= + = p R z

Let us calculate first the (conditional) PDF of the complex r.v. Y:

{ } { }{ } { }{ } { }{ } { }

1 11 1 1

222 1 11 1 1 1

2 21 11

1 1 1 1 1

, ,

, ,

,

H H j

H H j

H H

H H H H H

m E Y H E H e

E Y m H E e H

E H E

E E

ϕ

ϕ

ϕ ϕ β

σ ϕ β ϕ

ϕ

− −

− −

− −

− − − − −

= = = ⋅

= − = − ⋅

= =

= = =

p R z p R p

p R z p R p

p R d p R d

p R dd R p p R dd R p p R p

11, is a complex Gaussian r.v.HY Hϕ −⇒ = p R z

1 1 1

2

1 1 1, ,0

222

12 21 10

1( , ) ( ) ( , )

2

1 1 1exp

2

R I R IY Y H Y H Y H

j

p y y H p y H p y H d

y e d

π

ϕ

πϕ

ϕ ϕπ

β σ ϕπ πσ σ

≡ =

= − −

∫

∫


21 1 1, ( , )Y H mϕ σ⇒ ∈CN

( )

( )

1

2 2 4 221 1

1, 2 21 10

2 2 4 21

2 21 1 0

2 2 41

02 21 1

2 cos( )1 1( , ) exp

2

1 1exp exp 2 cos( )

2

1exp 2

R I R IY Y H

y y yp y y H d

yy y d

yI y

π

π

β σ β σ ϕϕ

π πσ σ

β σβ ϕ ϕ

πσ σ π

β σβ

πσ σ

+ − − ∠= −

+= − ⋅ − ∠

+= −

∫

∫


Now, to derive the PDF of |Y|, we need to consider the following 2-D transformation of r.v.’s (i.e. from Cartesian to polar coordinates):

( )2 2

arctan

R I

I R

R Y Y Y

Y Y Yϑ

= = +

= ∠ =

From the well-known theorem of r.v. transformations:

Where J is the Jacobian of the transformation:

( ){

1

1 1

1,

1 1, ,cossin

( , ), ( , ) R I

R

I

R IY Y H

Y Y H R Hy ry r

p y y Hp y y H p r Hϑ

ϑϑ

ϑ∠==

∠ ≡ =J


1 r=J

( )

( )

1

1 1

22 41

1 0, 2 21 1

2

1 1,0

22 41

02 21 1

( , ) exp 2 , 0 2 , 0

( ) ( , )

2exp 2 , 0

R H

R H R H

rrp r H I r r

p r H p r H d

rrI r r

ϑ

π

ϑ

β σϑ β ϑ π

πσ σ

ϑ ϑ

β σβ

σ σ

+= − ≤ < ≥

=

+= − ≥

∫

( )1

22 41

1 02 21 1

2( ) exp 2 , 0,R H

rrp r H I r r

β σβ

σ σ +

= − ≥

0

22 1

0 12 21 1

2( ) exp , 0, H

R H

r rp r H r σ

σ σ−

= − ≥ =

p R p

Under the hypothesis H0, the PDF is a Rayleigh function; it can be obtained from the Rice PDF by setting β=0:


2 22 11

HWMFSNR β σ β −= = p R p

The PDF of the envelope R=|Y| under the hypothesis H1 is a Rician function:

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

z

Pro

babi

lity

Den

sity

Fun

ctio

n (P

DF

)

γ=0

γ=1

γ=2

γ=4

γ=6


2 1

1

2, 2 , H

WMF WMF

rz SINR SINRγ β

σ−= p R p≜ ≜

( )1

22 41

1 02 21 1

2( ) exp 2 , 0R H

rrp r H I r r

β σβ

σ σ +

= − ≥


( )

( ) ( )

1

22 41

1 02 21 1

2 2

0

2( ) exp 2

exp ,2

D R H

M

rrP p r H dr I r dr

zz I z dz Q

η η

λ

β σβ

σ σ

γ γ γ λ

+∞ +∞

+∞

+= = −

+= − =

∫ ∫

∫

21 1

1

2where we defined , 2 2 , 2

rz z rγ β σ γ β λ η σ

σ⇒ = =≜ ≜

( ), is the so-called MQ γ λ Marcum Q - function


Summarizing:

( )

1

2 11

exp

22 , 2 , 2ln( )

FA H

HD M M WMF FAH

P

P Q Q SNR P

η

ηβ

−

−−

= −

= = −

p R p

p R pp R p

Note that: 2CWMF

WMF

SNRSNR =


2 2 WMFSINRγ =

( ),MQ γ λ

2 2 0γ =

0 2 4 6 8 10 12 14 16 18

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[ ]WMFSINR dB

DP

coherent WMF (blu)noncoherent WMF (red)

210FAP −=

1010FAP −=

Performance of the WMF

ROC: the noncoherent WMF (SW-0) vs. coherent WMF.

The complex multidimensional PDF of Gaussian disturbance is given by:

( )0 0 22

1( ) ( ) exp

H

H Ndd

p H pσπσ

= = −

dz

z zz z

Nx1 data vector

White Gaussian Noise

( )20 , dH σ∈z 0 ICN

{ } ( )1 0

00

1 0 22

( )

( ) ( )1( ) ( ) exp ( )

t t

tH

Ht t

t t tH H Ndd

p H

p H E p H p dσπσ

−

− −= − = − ⋅

∫

z

s sz z

z s

z s z sz z s s s

��

Perfectly Known Target Signal - Case #1

The optimal NP detection strategy is:

{ }1

0

2

2 >( ) <

H

H

Ht

d

l e ησ

= ℜz s z

( )

( )( )

1

1

FA CWMF FA

CWMF

CWMFD D FA CWMF

CWMF

P Q SNR Q PSNR

SNRP Q P Q Q P SNR

SNR

η η

η

−

−

= ⇒ = ⋅

−= ⇒ = −

2

2 HCWMF t t

d

SINRσ

= s swhere

If the amplitude is non fluctuating and the phase is uniformly distributed:

is deterministic (nonfluctuating)


je ϕβ β βϕ π

= ⇒


( )21, ,j

dH e ϕϕ β σ∈z p ICN

2

2

21

2

0

2

020

0 2

21( ) exp cos( )

2

2 > <

d

H

d

H

NH

d

NH

d

d e

I e

βπσ

βσ

βϕ ϕ ϕ

π σ

βλ

σ

−

−

Λ = − ⋅

= ⋅

∫z p z

p z

H N=p pwhere

1

0

> <

H

H

H ηp z

Since I0(x) is monotonic in x, the LRT reduces to:


( )2

exp 2 , 2ln( )dFA D M WMF FAP P Q SNR P

N

ησ = − = −

2

2where: WMF

d

NSNR

βσ

=

ROC:

1

0

2 > <

H

H

H ηp zor equivalentely

The integration in this case is said coherent, because all the pulse samples are first weigthed by p and then coherently summed up. The modulo is taken after summation. The gain in the SNR for coherent integration is equal to N.

Swerling 0 vs Swerling 1

Incoherent pulse train - Incoherent integration

( ) ( )( ) ( )

0( )

1

: Target absent

: Target presentj n

z n d n H

z n Ae d n Hφ

= = +

( )

11

0 0

1

1,1

0 0

1

1 1 10

10

00

( , )( )

( )( ) ( )

( ( ) , ) ( )

( )( ( ) )

HH

H H

N

H Nn

Nn

n

p H dp H

p H p H

p z n H p H d

z np z n H

ϕ

ϕ

ϕ ϕ

ϕ ϕ ϕ

+∞

−∞

+∞−

−= −∞

−=

=

Λ = =

= = Λ

∫

∏ ∫∏

∏

zz

z z

zzz

z z

2

2

0 2

2 ( )( ( )) d

A

d

A z nz n I e σ

σ

− Λ = ⋅

Applying the logarithm: ( )1

0

ln ( ) ln ( )N

n

z n−

=

Λ = Λ∑z

1

0

1

0 20

2 ( ) >( ) ln <

H

H

N

n d

A z nl I η

σ

−

=

=

∑z

The calculation of I0 is not easy,

so we have to resort to some

approximation

( )2

0ln 14

xI x for x≅ ≪

( )0ln 1I x x for x≅ ≫

Incoherent pulse train


Then for x<1 (small SNR) the detector becomes:

1

0

12

0

>( ) <

H

H

N

n

z n η−

=∑

For x>>1 (large SNR) the detector becomes:

1

0

1

0

>( ) <

H

H

N

n

z n η−

=∑

Both detectors perform an incoherent integration, since they first

calculate the modulus of each sample, then sum up the results.

The linear approximation fits the ln I0 very well for x>10dB, the square law detector is an excellent fit for x<5dB.


The integration gain for coherent processing is N

For the incoherent integration, the calculation is not so easy because

in general it depends on N, PD and PFA.

For very large values of N, the gain of the quadratic detector can be

approximated (in dB) with . As a rule of thumbs we use .

For the linear detector

Albersheim’s equation is used.

1010log 5.5N −N

Binary integration

� Any coherent or incoherent detection is given by comparing the

detection statistic with a threshold, so the output of the detector is binary in the sense that it takes one of only two possible outcomes.

� If the entire detection process is repeated N times for a given

range, Doppler, or angle cell, N binary detections will be available.

� Each decision of “target present” will have some probability PD of

being correct, and prob. PFA of being incorrect. To improve the reliability of the detection decision, the decision rule can require that

a target is detected on some number M out of N decisions before it is

finally accepted as a valid target detection.

� This process is called binary integration with a “M of N” rule.

� With the binary integration both overall PFA and PD are given by

( )( , ) 1N

N kk

k M

NP M N p p

k−

=

= −

∑

Detection in Gaussian clutter - unipi.itThe detection algorithm is optimized for a specific Doppler....

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