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Optimum Coherent Detection in Gaussian Disturbance
Maria S. GRECODipartimento di Ingegneria dell’Informazione
University of PisaVia G. Caruso 16, I-56122, Pisa, Italy
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
Performance of the coherent WMF
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 ?
Performance of the coherent WMF
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
Performance of the coherent WMF
{ } { } { } { }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
Performance of the coherent WMF
( )
( )( )
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 =
Performance of the coherent WMF
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
Swerling I Target Signal - Case #2
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:
Swerling I Target Signal - Case #2
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
+∞ −
+∞ −
= =
= =
∫
∫
Performance of the non-coherent WMF – SW1
( )( ) ( )( ) ( )
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 +=
Performance of the non-coherent WMF – SW1
PD is a monotonic increasing function of SNRWMF.
If the amplitude is non fluctuating and the phase is uniformly distributed:
is deterministic (nonfluctuating)
is uniformly distributed on [0,2 )
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
Swerling 0 Target Signal – Case #3
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
Performance of the non-coherent WMF – SW0
( ) ( )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
Performance of the non-coherent WMF – SW0
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
≡ =
= − −
∫
∫
Performance of the non-coherent WMF – SW0
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
+ − − ∠= −
+= − ⋅ − ∠
+= −
∫
∫
Performance of the non-coherent WMF – SW0
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
Performance of the non-coherent WMF – SW0
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:
Performance of the non-coherent WMF – SW0
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
Performance of the non-coherent WMF – SW0
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 +
= − ≥
Performance of the non-coherent WMF – SW0
( )
( ) ( )
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
Performance of the non-coherent WMF – SW0
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 =
Performance of the non-coherent WMF – SW0
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)
is uniformly distributed on [0,2 )
je ϕβ β βϕ π
= ⇒
Swerling 0 Target Signal – Case #3
( )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:
Swerling 0 Target Signal – Case #3
( )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
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.
Incoherent pulse train
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−
=
= −
∑