optimum coherent detection in gaussian disturbance · optimum coherent detection in gaussian...

1

Optimum Coherent Detection in Gaussian Disturbance

Maria S. GRECO and Fulvio GiniDipartimento di Ingegneria dell’Informazione

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

[email protected]

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Introduction+

Array RadarSignal Model

IntroductionIntroduction+ +

Array RadarArray RadarSignal Model Signal Model

Outline of the

Course

Outline Outline of theof the

CourseCourse

Radar Clutter

Modelling andAnalysis

Radar Radar Clutter Clutter

Modelling andModelling andAnalysisAnalysis

OptimumDetection

in Gaussian Disturbance

OptimumOptimumDetectionDetection

in Gaussian in Gaussian DisturbanceDisturbance

OptimumDetection

in non-Gaussian Disturbance

OptimumOptimumDetectionDetection

in nonin non--Gaussian Gaussian DisturbanceDisturbance

Adaptive Detection

in Gaussian Disturbance

Adaptive Adaptive DetectionDetection

in Gaussian in Gaussian DisturbanceDisturbance

ConcludingRemarks

ConcludingConcludingRemarksRemarks

1

2

3 4

5

7

““Coherent Radar Detection in Clutter: Clutter Modelling and Coherent Radar Detection in Clutter: Clutter Modelling and

Analysis + Adaptive Array Radar Signal ProcessingAnalysis + Adaptive Array Radar Signal Processing””

Adaptive Detection

in non-Gaussian Disturbance

Adaptive Adaptive DetectionDetection

in nonin non--Gaussian Gaussian DisturbanceDisturbance

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The binary hypothesis testing problem

The Neyman-Pearson (NP) criterion

The Likelihood Ratio Test (LRT)

The Gaussian detection problem

Target signal models (different degrees of a priori knowledge)

Target model #1 - Perfectly known target signal: the coherent

whitening matched filter (CWMF)

Performance of the CWMF: PFA, PD, and ROC

Outline Section 3.1

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Binary Hypothesis Testing Problem

The function of a surveillance radar is to ascertain whether targets are present in the data. Given a space–time snapshot z (subscript l has been omitted for ease of notation), 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 (decision D0) or hypothesis H1 is true and a target signal is present with disturbance (decision D1).

The associated probabilities are:

{ } { }{ } { }

1 1 0 1

1 0 0 0

Pr , Pr 1 ,

Pr , Pr 1 .

D Miss D

FA No Target FA

P D H P D H P

P D H P D H P

= = = −

= = = −

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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).

Disturbance d encompasses any interference or noise component of the data:

clutter, jamming, and noise:

The three components are usually assumed to be mutually uncorrelated .

=d c + j + n

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A radar receiver performs linear averaging at the antenna (remember that the

antenna pattern integrates everything in its pattern), IF bandpass filters, baseband

anti-aliasing filters, pulse compression filters, etc. The Central-Limit Theorem

(CLT) therefore applies and the output tends to be Gaussian (even when the input

is not Gaussian).

The three disturbance components are assumed zero-mean complex circular

Gaussian , mutually uncorrelated, therefore the disturbance covariance matrix is:

(we omit subscript d for ease of notation).

Optimal processing (2-D space-time or 1-D spatial (angle) or 1-D temporal

(Doppler) processing) refers to the case of a priori known disturbance covariance

matrix R. The resulting performance represents a benchmark for any realistic

(adaptive) approach.

Adaptive processing refers to the case where the disturbance matrix (or in

general the disturbance PDF parameters) is unknown and must be estimated

on-the-fly from the observed data in a CPI.

c j n= + +R R R R

The Gaussian Case

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The complex multidimensional PDF of Gaussian disturbance is given by:

( )0

10

1( ) ( ) exp H

H NMp H p

π−= = −dz z z z R z

R

{ }det , Hermitian matrixH= = →R R R RMNx1 data vector

The Optimal Detector (known R)

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

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{ } ( )1 0

00

11 0

( )

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

t t

tH

Ht t t t tH H NM

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

ν×

v Steering Vector

[ ]1

( ) is the target complex amplitudet d rN

β ν β α×

= ≡s v

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

2Doppler frequency target radial velocityt r

d

v Tνλ

= ⇔

The Optimal Detector (known R)

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2

12 ( 1)

1

( )d

d

j

d

Mj M

e

e

πν

π ν

ν×

−

=

v⋮

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.

The steering vector should be known (calibrated array).

Since the target velocity os unknown a-priori, v 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

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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 → coherent whitening matched filter (CWMF)

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

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

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

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

2(0, )sβ σ∈CN

Target Signal Models

1

( )t dN

β ν×

=s v

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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

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The Coherent Whitening Matched Filter (WMF) Detecto r

Hy = w z

The WMF is a linear filter:

w is the optimal weighting vector: 1 1H H Ht t t

− − −= ⇒ = =w s R w R s R s

1

( , )t d sNM

β ν ν×

=s v

{ }1

0

>( ) 2 <

H

H

l e y δ η= ℜ −z

1Ht tδ −s R s≜

1, ,

1 1 1

MN N MH Ht k k n m n m

k n m

w z w z− ∗ ∗

= = =

= = =∑ ∑∑s R z w z

1

1

NM

t×

−=w R s

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To compare the performance of different non-adaptive algorithms with each

other, as well as with adaptive approaches, it is necessary to use some standard

benchmarks.

Important radar performance metrics are the probability of detecting a target

(PD), the probability of declaring a false alarm (PFA), and the accuracy with

which target speed and/or bearing may be measured.

Useful intermediate quantities for the probability of detection are the signal-to-

interference-plus-noise power ratio (SINR) and SINR loss (defined later on).

Equivalently, sometimes we find the signal-to-disturbance power ratio (SDR) .

Minimum Detectable Velocity . The width of the SINR loss notch near

mainlobe clutter determines the lowest velocity detectable by the radar.

Finally, the response of filter w itself is important: it should have a distinct

mainlobe that is as narrow as possible as well as low sidelobes.

Performance Metrics

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( ) ( )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 1 >( ) 2 <

H

H

H Ht t tl e η− −= ℜ −z s R z s R s

{ } { }{ }1 1 10 0 02 H H H

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

� 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, NJ, 1998.

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

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

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

( )( ){ }{ }{ }

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

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

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 *we used the fact that an d 0TH H T T T T

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


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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

120

10

11 1

1

( )1Pr ( ) exp

22

1exp

2 22 2

Pr ( )

FA l H

l mx

Ht t

Hm t t

Ht

D l H

l mP l H p l H dl dl

mx ddx Q Q Q

d

mP l H p l H dl Q Q

η η

σ

ησ

η

ησπσ

η η ησπ

ηηησ

+∞ +∞

−= +∞ −

−−

+∞

−= > = = −

− + = − = = = +

−−= > = = =

∫ ∫

∫

∫

s R s

s R s

s 1

1 22t

Ht t

dQ

d

η−

−

= −

R s

s R s

11 0 1

10

whe2

r 2e 2

HHt tt tH

t t

m md

σ

−−

−

−= =s R s

s R ss R s

≜ What is the meaning of d?

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The filter output is :

The Signal-to-Interference plus Noise ratio (SINR) at the output of w is:

The SINR at the output of the optimal WMF :

( )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 ESINR

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

21

11 1

Ht t H

WMF t tHt t

SINR−

−− −= =

s R ss R s

s R RR s

For perfectly known target signal, we take the real part of the WMF output, therefore the disturbance power is halved:

{ } { } { } { }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

(H1)

if the target signal is deterministic

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

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

( ){ }( ){ }

( )

( )( ){ }

( )

2 21 1

2 21 11

2 21 1

1 2121

2

4 42

22

H Ht t t t

CWMFH HHt tt

H Ht t t t H

t tHH

t tt

SINRE e E

dE

− −

− −−

− −−

−−

= = + ℜ

= = = =

s R s s R s

s R d d R ss R d

s R s s R ss R s

s R ss R d

1therefor e 2 Ht t CWMFd SINR−= =s R s


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( )

( )( )

1

1

2 2

2

FA FA

D D FA

d dP Q d Q P

d

dP Q P Q Q P d

d

η η

η

−

−

= + ⇒ = ⋅ −

= − ⇒ = −

2 12 Ht td −= s R s


where is the SINR at the output of the coherent WMF.

Note that PD is a monotonic increasing function of the SINR at the

output of the optimal filter, therefore to maximize the SINR is equivalent to maximize PD.

PD and PFA completely specify the Neyman-Pearson test performance.

Insight is provided by plotting these probabilities against one another, this is called the Receiver Operating Characteristic (ROC) curve :

In Matlab: see qfunc.m and qfuncinv.m

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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 SINR and the threshold.

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

CWMF_ROC.m


2CWMFSINR dζ = =

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Swerling I Target Signal - Case #2

v perfectly known means: array perfectly calibrated, I & Q channels perfectly matched (i.e. balanced), and we are testing for the presence of a target in a given DOA and Doppler frequency bin.

( )2( ), 0, , perfectly knownt d sβ ν β σ= ∈s v vCN

( ) ( )20 10, , 0, H

sH H σ∈ ∈ +z R z vv RCN CN

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

sE H E Eβ β β σ= + + = + = +zz v d v d vv R vv 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

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

1

0

1 2 2 1

0

( )ln ln ln ( )

( )H H H H H

s s

H

p H

p Hσ σ −= − + + − +z

z

zR vv R z Rz z vv R z

z

( )

( )

0

1

10

2 11 2

1( ) exp

1( ) exp ( )

HH NM

H HsH NM H

s

p H

p H

π

σπ σ

−

−

= −

= − ++

z

z

z z R zR

z z vv R zvv R

By making use of Woodbury’s identity:

2 1 12 1 1

2 1( )

1

HH s

s Hs

σσσ

− −− −

−+ = −+R vv R

vv R Rv R v

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1

0

1 12 2

2 1>ln ( ) ln ln <1

H

H

H HH

s s Hs

σ σ ησ

− −

−Λ = − + ++

z R vv R zz R vv R

v R v

1

0

21 > <

H

H

H η−v 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 termed the noncoherent WMF

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Coherent vs. noncoherent WMF :


( )22 2 21 1 1 1H H H H disturbanceβ β− − − −= + = +v R z v R v v R d v R v

{ } { } { }{ } { }

{ }

1 1 1

2 1 1

2 1 1

( )H H Ht

H H

H H

e e e

e e

e

β β β

β β

β β

− ∗ − ∗ −

− ∗ −

− ∗ −

ℜ = ℜ = ℜ +

= ℜ + ℜ

= + ℜ

s R z v R z v R v d

v R v v R d

v R v v R d

{ } { } ( ) { }1 1 1 1 1cosH H H H He e eβ β β− − − − −ℜ = ℜ + = ∠ + ℜv R z v R v v R d v R v v R d

We cannot take the real part if we before do not compensate for target phase, in fact if we would do that:

Hence, if we do not know the target phase we have to calculate the modulo (squared) of the WMF output:

t β=s v

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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 (a.k.a. Swerling V) – 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 v RCN

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( )( )

( )

( )( )

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 v R z v

z R z

v R z z R v v R v

v R z v R v

v R z1

,H −v R v

10where Hϕ −∠v R z≜

Swerling 0 Target Signal (a.k.a. Swerling V) – Case #3

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1

0

21 > <

H

H

H η−v 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 e

πβ

β η

β ϕ ϕ ϕπ

β

−

−

−−

−−

Λ = − ⋅

= ⋅

∫v R v

v R v

z v R z

v R z

Swerling 0 Target Signal – Case #3

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Deterministic Unknown Complex Amplitude - Case #4

( ), deterministic unknown, perfectly knownt dβ ν β=s v v

( ) ( )0 10, , ,H H β∈ ∈z R z v RCN CN

This is a composite hypothesis testing problem .

A uniformly most powerful (UMP) test does not exist.

We resort to the generalized likelihood ratio test (GLRT) → the unknown parameters are replaced by their maximum likelihood estimates (MLE).

1

1

00

1

0

( ; ) >( ; ) <( )

H

H

H

H

p He

p Hη

ββΛ = z

z

zz

z

The LRT depends on the unknown complex target amplitude, therefore it cannot be implemented.

11 1

00 0

1 1

0 0

ˆmax ( ; ) ( ; )>( ) max ( ; ) <( ) ( )

H

H

H MLH

GLRT

H H

p H p He

p H p Hβ η

β

β ββΛ = Λ = =

z z

z z

z zz z

z z

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The GLRT is a suboptimal detector. However, it usually produces good detection performance. For large data records it can be shown to be UMP within the class of invariant detectors [Kay98].

1

0

1 1 1

0

( ; )ln ( ; ) ln ( ) ( )

( )H H H

H

p H

p H

ββ β β− −Λ = = − − −z

z

zz z R z z v R z v

z

1 1

1

ˆ argmax ( ) ( )

argmin ( ) ( )

H HML

H

β

β

β β β

β β

− −

−

= − − −

= − −

z R z z v R z v

z v R z v

{ }21 1 1 1

22 111 1

1 1

( ) ( ) 2H H H H

HHH H

H H

eβ β β β

β

− − − −

−−− −

− −

− − = + − ℜ

= + ⋅ − −

z v R z v z R z v R v z R v

v R zv R zz R z v R v

v R v v R vThe minimum is clearly attained when the positive factor containing β is made to vanish.


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1

1ˆ

H

ML Hβ

−

−= v R zv R v

1 1

1 1

estimation error

( )ˆH H

ML H H

ββ β− −

− −

+≡ = +v R v d v R dv R v v R v��

{ } { }

{ } { }

{ }

11

1 1

2121 1

21 1

1 1

2 11

ˆ 0

1ˆ

1

HH

ML H H

HH H

ML H H

H H

HH

EE E

E E E

E

β β

β β

−−

− −

−− −

− −

− −

−−

− = − = − =

− = =

= =

v R dv R dv R v v R v

v R dv R dd R v

v R v v R v

v R dd R v

v R vv R v

Complex Amplitude ML Estimate

The MLE of β is unbiased and efficient: the MSE on the estimates of amplitude and phase coincide with their Cramér-Rao lower bounds (CLRBs) .

31

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1

1

00

211

10

ˆ( ; )>ln ( ) ln <( )

H

H

HMLH

GLRT HH

p H

p H

βη

−

−Λ = =z

z

z v R zz

z v R v

21

1 11

ˆ ˆ( ) ( )H

H HML ML H

β β−

− −−− − = −

v R zz v R z v z R z

v R v

As a consequence we find the GLRT to be:

1

0

21 > <

H

H

H η−v R z

Incorporating the denominator in the threshold, again we find that the decision strategy is to compare the modulo-squared WMF output to a threshold:


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Performance of the non-coherent WMF: Signal-to-Interference plus Noise power ratio (SINR)

Performance of the non-coherent WMF

1

0

2 21 > <

H

H

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

{ }{ }

{ } { }2 2

2 1

2

H Ht t

HWMFH

H

E ESINR SINR E

Eβ −= = ⇒ =

w s w sv R v

w Rww d

( )2 1 2

2 1

, 0,

, deterministic

Hs s

WMFH

SINRσ β σ

β β

−

−

∈=

v R v

v R v

CN

33

33/191

(a) 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 vv 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

− −

− − −

= = = =

= = = ⋅ + =

v R z v R d

v R z v R v v R d

1 is a complex circular Gaussian r.v. HiY H−⇒ v 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

<= ≥

34

34/191

{ } { } { } { }{ }

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

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

− −

− − −

−

− − − −

− − − −

= = = =

= =

= = =

= + = +

= + = +

v R z v R d

v R dd R v v R v

v R z

v R v v R d v R v v R d

v R v v R v v R v v R v

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

+∞ −

+∞ −

= =

= =

∫

∫


35

35/191

( )( ) ( )( ) ( )

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

w e 1h re 1 Hs WMFSINR

σ σσ

−= + = +v R v

� Receiver Operating Characteristic (ROC) curve s:

( )1

1 WMFSINRD FAP P +=


PD is a monotonic increasing function of SINRWMF.

36

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(b) 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/Swerling V).

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

1H

FAP eη

−−= v R v


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

1 1

2

1 1,0

1( ) ( , )

2H Hp H p H dπ

ϕ ϕ ϕπ

= ∫z zz z

37

37/191

1

0

22 2 2 1 > <

H

H

HR IX Y Y Y η−= = + = v 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 −= + = v R z

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

38

38/191

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

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

− −

− −

− −

− − − − −

= = = ⋅

= − = − ⋅

= =

= = =

v R z v R v

v R z v R v

v R d v R d

v R dd R v v R dd R v v R v

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

39

39/191

( )

( )

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

= = +

= ∠ =

40

40/191

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

+= − ≤ < ≥

=

+= − ≥

∫

41

41/191

( )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−

= − ≥ =

v R v

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

HWMFSINR β σ β −= = v R v

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

42

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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−= v R v≜ ≜

� Rice_PDF.m

43

43/191

( )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 In Matlab: marcumq.m

44

44/191

10

2 1

1,!

where (.) is the incomplete gamma function

and is the WMF output SINR:

n

D Hn

HWMF

eP n

n

SINR

ξξ η

ξ ξ β

−+∞

−=

−

= ⋅Γ +

Γ

= =

∑ v R v

v R v


Summarizing:

( )

1

2 11

exp

22 , 2 , 2ln(1 )

FA H

HD M M WMF FAH

P

P Q Q SINR P

η

ηβ

−

−−

= −

= =

v R v

v R vv R v

Sometimes, PD is instead expressed as follows:

Note that: 2

CWMFWMF

SINRSINR =

45

45/191


2 2 WMFSINRγ =

( ),MQ γ λ

2 2 0γ =

� Q_Marcum.m

46

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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

� CHvsNCH.m

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

47

47/191

Performance of the WMF

ROC for narrowband detector: noncoherent WMF (SW -0) vs. coherent WMF.

[ ]WMFSINR dB

2 12 HWMF CWMFSINR SINR β −= = v R v

� CHvsNCH.m

coherent WMF (blu)noncoherent WMF (red)

optimum coherent detection in gaussian disturbance · optimum coherent detection in gaussian...

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