[ieee 2007 international conference on computational intelligence and security workshops (cisw 2007)...

The Detection Arithmetic Of Soft Decision Based On Non-ideal Channel In Distributed Detection

Ying-kun Liu Air Force Engineering University

No.1 Feng Hao Road, Xi’an, China [email protected]

Xin-xi Feng Air Force Engineering University

No.1 Feng Hao Road, Xi’an, China [email protected]

Abstract

In this paper we expand the arithmetic of the distributed detection in two aspects. In order to improve the detection performance and use communicated resource rationally, the local decision is extended from hard decision to soft decision. According to non-ideal independent channels, the local decision and reliability information are transmitted to the fusion center. The non-ideal channel is extended from binary symmetry channel to memorial channel. Then we derive the optimal decision form of the local detectors and fusion center by NP(Neyman-pearson) rule. At last stimulation of two detectors shows that using soft decision improves the detection performance and worsen channel depress the detection performance surely.

N

1. Introduction

Distributed detection has received increasing attention in recent years, compared to single detector detection, distributed detection has more strongpoint such as: the bigger overlay area, the higher reliability, the stronger viability and so on. The study of distributed detection has traced back to 1960s. In [1], Tenney and Sandell formulated the distributed detection problem using a Bayesian setting, later the distributed detection developed very quickly, this work was later generalized to multiple sensors by Reibman [2] and by Hoballah [3]. Similarly, under the Neyman-Pearson (NP) criterion, the optimality of the local LRT has been established in [4][5][6]. In the above mentioned papers, all of them have an assumption: the local decisions are transmitted to fusion center with no error. Yet we must face to non-ideal channel during the actual detection process owing to bandwidth、channel fading and channel noise, the non-ideal channel results in the error during the transmission, based on the error

decision received in the fusion center, the performance using the above arithmetic[1-6] is degraded distinctly, so the optimal detection arithmetic must be rectified. A few papers[7][8] have discussed the arithmetic involving of non-ideal channel in distributed detection, but only based on binary symmetry channel, the optimal detection arithmetic of parallel and serial configuration have been studied.

In this paper the channel is memorial Markov channel having finite states. In order to improve the detection performance and use communicated resource rationally, the local detectors apply soft decision. According to NP rule we derive the optimal decision arithmetic. At last stimulation favor our analysis. 2. System model

The distributed detection system comprises N local detectors and independent non-ideal channels and a fusion center, the system figure is shown in Figure 1.

N

Figure 1. The system figure of the distributed detection

In this topology each local detector makes a soft decision based on its own observation gain the

decision and reliability info , then

this decision and reliability info are transmitted through a non-ideal channel to the fusion center, according to the received decision and reliability

information the fusion center makes the final

iy

),1...2,1( NNiui −= iC

ir

iD

2007 International Conference on Computational Intelligence and Security Workshops

0-7695-3073-7/07 $25.00 © 2007 IEEEDOI 10.1109/CIS.Workshops.2007.99

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decision . Meanwhile the channel has finite states.

We assumes that the channel has two states: (1) good state, the error of channel transmission is fewer; (2) bad state, the error of channel transmission is more. Along with the process of detection, the channel state changes between the two states. While transmitting a bit info, the channel state is likely to keep or change, furthermore the shift of channel accords with a series of probability as shown in Figure 2.

0u

Figure 2. Markov shifting figure of channel states

The shift probability of good state and bad state can be

shown by state shift matrix Q ( ), owing to the

stability of the channel, the probability of channel lying the state 1 and 2 are gained respectively by matrix :

⎥⎦⎤

⎢⎣⎡=

2221

1211

aaaa

Q

Q

)1( 1121121 aaa −+=π , )1()1( 1121112 aaa −+−=π . In order to simplify the calculation, the reliability

information 、 is a bit binary data, the input and

output of the non-ideal channel is respectively ( )

and ( ), we assume that the state shifting

probability of each channel is same, when the channel lies in the good state or bad state, the shifting probability of channel input and channel output is respectively shown by matrix and .

iC iD

ii Cu ,

ii Dr ,

1P 2P

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

44434241

34333231

24232221

14131211

1

,,,,,,,,,

,,,

bbbbbbbbbbbb

bbbb

P , .

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

44434241

34333231

24232221

14131211

2

,,,,,,,,,

,,,

cccccccccccc

cccc

P

Where is the probability that when the channel lie in good state the channel input is and the

channel output is , is the probability

that when the channel lie in bad state the channel input is and the channel output is ,

the other elements of matrix and is analogize in turn. The elements of matrix Q ˛ and is estimated by stimulation or testing data, so in the fusion center of matrix Q ˛ and are known.

21b

)1,0( == ii Cu

)0,0( == ii Dr 21c

)1,0( == ii Cu )0,0( == ii Dr

1P 2P

1P 2P

1P 2P

3. The Detection Arithmetic of local detectors and fusion center

In this paper, we derive the optimal decision rule of local detectors and fusion center based on NP rule,

namely under the constraint of , the following

objective function can be obtained the maximum:

α≤0FP

)( 00 αλ −−= FD PPF （1）

Where is the detection probability and the false

alarm probability respectively at the fusion center, and .

00, FD PP

)|1(),|1( 000100 HuPPHuPP FD ====

3.1. The soft detection of local detectors

We adopt a person-by-person optimization approach, namely we optimize the decision rule of a local detector fixing decision rules at all other detectors.

The local decision favors the hypothesis

(target existence), favors the hypothesis

(no target existence), is the observation of the ith

detector and the local observations are conditionally independent namely , for all

1=iu

1H 0=iu

0H iy

)|()|()|( kjkikji HypHypHyyp =

1,0, =≠ kji . So the detection probability in (1) can be

given by: 0DP

)|0(),0|1()|1(),1|1()|1(

110

110100HuPHuuP

HuPHuuPHuPP

ii

iiD===+

====== （2）

Where , substituting into

(2), we have:

)|1(1)|0( 11 HuPHuP ii =−==

)|1(),,(),0|1( 1101000 HuPHuuAHuuPP iiiD =+=== （3）

Where ),0|1(),1|1(),,( 101010 HuuPHuuPHuuA iii ==−===

Similarly the false alarm probability is obtained, 0FP

)|1(),,(),0|1( 000000 HuPHuuAHuuPP iiiF =+=== （4）

Substituting (3)(4) into (1), we gain:

iiiii

y iiii

dyHypyuPHuuA

HypyuPHuuACFi

)]|()|1(),,(

)|()|1(),,([

000

110

=∫ −=+=λ

（5）

Where in (5) can be expressed: C

λαλ +==−=== ),0|1(),0|1( 0010 HuuPHuuPC ii

obviously is a constant with regard to the decision rule of local detector i . Paper[8] shows that if of the monotony increasing rule is applied in fusion center then . So in order to maximize

the objective function under the constraint of the optimal decision rule of local detector is :

C

)1,0=(,0),,( 0 > jHuuA ji

α≤0FP i

),,(

),,(

)|(

)|(

10

00

0

1

0

1

HuuA

HuuA

Hyp

Hyp

i

i

u

u

i

i

i

i

λ=

=

<> （6）

Where λ is determined by the false alarm probability α . From (6), we can see the decision form of the local decision is likelihood ratio decision, namely the local likelihood ratio function

)|()|()( 01 HypHypy iii =Λ compare to the local threshold

700700700700696696696696

),,(),,( 1000 HuuAHuuAK iii λ= , when then shows

the target existence, when then shows no

target. When a single decision is transmitted from local detector to fusion center, it can not reflect sufficiently the observation information, so the detection performance is declined. In fact the communication bound is bigger than transmitting a bit decision from local detector to fusion center, so in this paper we apply the soft decision to approve the system performance. In fact how much is higher and

lower than the threshold reflects the reliability of

local decision. We use the difference of and to

quantify to the finite binary bits, how many the binary bits is decided by the communication capacity. We assume that the reliability information is depicted by m bits binary data:

ii Ky ≥Λ )(

ii Ky <Λ )(

)( iyΛ

iK

)( iyΛ iK

While , the local decision ,we assume: ii Ky ≥Λ )( 1=iu

MKAjKK iiiji1, = , （7） mMMj 2,1,...2,1,0 =−=)( −+

If then the reliability of is

expressed:

11,

1, )( +<Λ≤ jiiji KyK 1=iu

jCi =

While , the local decision ,we assume: ii Ky <Λ )( 0=iu

MBKjKK iiiji )(0, −−= , （8） mMMj 2,1,...2,1,0 =−=

If then the reliability of is

expressed:

0,

01, )( jiiji KyK <Λ≤+ 0=iu

jCi =

Where and ( , ) are obtained

by Qiebixuefu inequality, and ˛ are satisfied : iA iB ),[ ∞∈ ii KA ],( ii KB −∞∈

iA iB

0)|)(( 0 ≈>Λ HAyP ii , . 0)|)(( 1 ≈<Λ HByP ii

In addition, we can calculate by the

probability density function of the local likelihood ratio:

)|,(ˆkii HCuP

)1,0,(,)|()|,(ˆ 1, =∫ Λ=== + ktdlHPHjCtuPt

jit

Kikikii

, jiK

∑ ==

)|()|1( 000 HRPRuPP cR cF c∑ ==

（9）

3.2. The detection arithmetic of fusion center

The local decision and reliability information are transmitted to fusion center

through the non-ideal channel, in the fusion center the received decision and reliability information are ,the optimal decision is

derived based on NP rule[4], The detection probability and the false alarm probability can be written as:

),,...,,,( 2211 NNc CuCuCuU =

),,...,,,( 2211 NNc DrDrDrR =

)|()|1( 100 HRPRuPP cR cD c (10)

(11)

Substituting (10) (11) into (1), we have:

[ ] αλλ +−∑ == )|()|()|1( 010 HRPHRPRuPF ccR cc

For a given false alarm probability (12)

α , from (12) the optimum fusion rule that maximizes the detection probability is given by:

⎪⎩

⎪⎨⎧

<Λ=Λ>Λ

==λλγλ

)(,0)(,)(,1

)|1( 0

c

c

c

cRRR

RuP (13)

Where )|()|()( 01 HRPHRPR ccc =Λ , the decision threshold

λ and the randomization constant γ are decided by the false alarm rate α .

According to the statistic independence of the local observation and the channels, we have,

1,0,)|,()|( 1 =∏= = jHDrPHRP Ni jiijc (14)

Where,

))|1,1()|0,1()|1,0(

)|0,0(())|1,1()|0,1()|1,0(

)|0,0(()|0,0(

41

3121

11241

3121

111

jii

jiijii

jiijii

jiijii

jiijii

HCuPcHCuPcHCuPc

HCuPcHCuPbHCuPbHCuPb

HCuPbHDrP

==+==+==+

==+==+==+==+

=====

π

π

(15)

Similarly we can obtain , ,

.

)|1,0( jii HDrP == )|0,1( jii HDrP ==

)|1,1( jii HDrP ==

4. Stimulation

We use a two-detector system to stimulate and we assume the observations of the sensors are Gaussian distributed, as follows:

2

)(exp

2

1)|(

2

1|ii

iHysy

Hyfii

−=

π,

2exp

2

1)|(

2

0|i

iHyy

Hyfii π

=

5.221 == ss ,and the observations of the sensors and the non-ideal channels are statistics independent. The state shift matrix of every channel is the same, the elements of the matrix are:

2.0,8.0,1.0,9.0 22211211 ==== aaaa then we can gain 91,98 21 == ππ . Figure 3. shows the receiver output curves (ROC) of three kinds non-ideal channel states: （1） , , 9.044332211 ==== bbbb 023324114 ==== bbbb

05.04334422413123121 ======== bbbbbbbb ;

15.044332211 ==== cccc , , 15.023324114 ==== cccc

35.04334422413123121 ======== cccccccc .

（2） , , 6.044332211 ==== bbbb 023324114 ==== bbbb

; 2.04334422413123121 ======== bbbbbbbb

1.044332211 ==== cccc , , 1.023324114 ==== cccc

4.04334422413123121 ======== cccccccc .

（3）the ideal channels, where , 0,1 21 == ππ

701701701701697697697697

144332211 ==== bbbb , the else elements of the

matrix are zero. 1P

Figure 3. The ROC curves of different channel states

Figure 4. The ROC curves of different reliability bits

From Figure 3. we can see that the channel state better the detection performance better, when the error of channel transmission achieve to some degree the fusion performance is not much better than the local performance.

A bit decision and bits reliability information are transmitted to the fusion center, when the local decision is hard decision, where ,

, , , when

m0=m

9.02211 == bb

1.02112 == bb 25.02211 == cc 75.02112 == cc 1=m , the parameters is the same as the above condition(1). The comparison of the output ROC curves between

and is shown in Figure 4. 0=m 1=m

From Figure.4 we can realize that the more reliability bits the better system performance, but with the increase of the reliability bits the calculation becomes intricacy, commonly . 2≤m

5. Conclusion

In this paper we have studied two extensions, the first, we extend the hard decision to soft decision in local detectors to approve the system performance; the second, we extend binary symmetry channel to memorial non-ideal channels in distributed detection. At last we use a two-detector system to stimulate, the stimulation shows that the performance degrade with the worse channel state and the performance upgrade with more reliability bits, commonly . 2≤m

From this paper, we mention a possible expansion: here we assumed the local soft decision are fixed, is it possible to use globe optimum decision to obtain the local optimal threshold? References [1] Tenney,R.R., and N.R.Sandell,Jr., “Detection With Distributed Sensors”, Journal, IEEE Trans. on AES, 1981, 17(2), pp.501-510. [2] Reibman,A.R., and L.W.Nolte, “Optimal detection and performance of distributed sensor system”,Journal, IEEE Trans. on AES,1987,23(1),pp.24-30. [3] Hoballah,I.Y., and P.K.Varshney, “Distributed Bayesian signal detedion”, Journal, IEEE Trans. on Info.Theory, IT-35,1989,9,pp.995-1000. [4] Ming X., Zhao J W., “On the performance of distributed Neyman-Pearson detection systems”, Journal, IEEE Trans. on Systems Man and Cybernetics.2001,31(1),pp.78-83. [5] Ming Xiang, “On Optimum Distributed Detection and Robustness of System Performance”, Journal, ISIF 2002, pp.156-163. [6] Chen B, Willett P K. “On the optimality of the likelihood -ratio test for local sensor decision rules in the presence of non-ideal channels”, Journal, IEEE Trans Inform Theory, 2005, IT – 51,pp.693 - 699. [7] Xiao-guo, Liang, “Research on fusion algorithm for a kind of two - sensor distributed detection system with non - ideal communication channel”, Journal, Journal of Detection & Control, Vol.28(4),2006,pp.30-33. [8] Ming Xiang, “The distributed detection arithmetic of Neyman-Pearson based on serial configuration”, Journal of Detection & Control, 2002,24(3),pp.5-10.

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