Distributed Detection Systems. Hamidreza Ahmadi

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Channel Estimation Error in

Distributed Detection Systems

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Outline

z

Detection Theory

z

Detection Theory

¾

Neyman-Pearson Method

z

Cl

i l Di t ib t d D t ti

z

Classical Distributed Detection

¾

_{Fusion and local sensor rules}

z

Channel Aware Distributed Detection

¾

Perfect and Estimated CSI in the FC

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Detection Theory (NP Method)

y (

)

z

Binary Hypotheses Testing

P x H( | 1) ( | 0)

z

Binary Hypotheses Testing

H0 : x

n

H1

A

=

+

2

n

∼

N(0,

σ

)

( | ) ( | 0) P x H

H1 : x

= +

A n

H1 x (x) H0 x > τ ⎧ γ _{= ⎨} < τ ⎩ ¾

Decision Rule:

P H( 0 |H1) = −1 PD P H( 1 | H0)= PFA FA

P

=

Pr( (x)

γ

=

H1 | H0)

=

Pr(x

> τ

| x

=

n)

H0 x < τ ⎩ ¾

False Alarm Probability:

FA

( ( )

γ

|

)

(

|

)

D

P

=

Pr( (x)

γ

=

H1 | H1)

=

Pr(x

> τ

| x

= +

S

n)

¾

Detection Probability:

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Detection Theory (NP Method)

y (

)

z

Neyman-Pearson Method:

z

Neyman Pearson Method:

Generally Detection probability and False alarm are changing by changing threshold.

Max

P

FA

P

≤ α

Max

P

_D H1 P(x | H1) >

¾

Likelihood Ratio Test (LRT) :

H0 P(x | H1) L(x) P(x | H0) > = τ < ¾

Algorithm:

P

_FA

=

P(

Λ > τ

| H )

0 = α → τ →

P ( )

_D

τ

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Classical Distributed Detection

z

Parallel Topology

z

Parallel Topology

Each Sensor based on its observation ,independently, makes its own decision about the Hypotheses and sends it to the fusion center.

(

)

i i i

u

= γ

(y )

0 0 1 2 N

u

= γ

(u , u , ..., u )

¾Applicable to Power and Bandwidth

limited Networks

¾Performance loss because of accessing to

l ti l i f ti i th t

only partial information in the center as compared with centralized

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Classical Distributed Detection

z

NP Method in DD:

z

NP Method in DD:

For Fixed global false alarm, what are the Optimum local and fusion rules to Maximize global detection probability.

0 D

P

Max Subject to 0 1 N (γ γ, , ...,γ ) 0 F

P

≤ α

1 H

¾

Optimal FC Rule:

LRT

1 1 0 1 1 0 0 H N i i _i H P(u | H ) P(U | H ) (U) P(U | H ) ₌ P(u | H ) > Λ = = τ <

∏

¾

Optimal Local Sensor rules:

More complicated because of distributed nature.

With Conditional independence of sensor observation: LRT

1 0 1 0 H i i i i H P(y | H ) (y ) P(y | H ) > Λ = τ <

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Classical Distributed Detection

z

Person by person optimization (PBPO)

z

Person by person optimization (PBPO)

LRT thresholds at the sensors are coupled with each others and Fusion Center’s.

¾ Each sensor threshold is optimized assuming fixed decision rules at all

¾ Each sensor threshold is optimized assuming fixed decision rules at all other sensors and the FC. This is iteratively done until we reach optima value.

¾ It gives a necessary but not sufficient condition for optimality so several

¾ It gives a necessary but not sufficient condition for optimality, so several Initialization are necessary.

z

Error Exponents:

PBPO is intractable in networks with large number of sensors, So in asymptotic regime, threshold is selected such that gives the best error exponent.( identical thresholds)

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Channel Aware Distributed Detection.

z

Sources of uncertainty:

Noise fading Shadowing

z

Sources of uncertainty:

Noise, fading, Shadowing,

interference in Observation and Transmission channel

z

_{Separation Approach for DD with uncertainty}

:

C

i ti

h

b t

d

t

Communication schemes between sensors and center are

separated from the SP algorithms in decision rules.

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Channel Aware Distributed Detection

z

Channel Aware Fusion rule:

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The ultimate goal is

θ

not recovering Ui

(1)

The ultimate goal is not recovering Ui.

(2)

Î

Optimal detector should consider Channel Conditions in the

1 k ˆ1 ˆkk

I(θ ; y ,..., y ) ≥ θI( ; u ,..., u )

θ

p

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Channel Aware Distributed Detection

z

Channel Aware local sensor rules:

¾ For globally optimal detection, transmission channel is considered in the local sensors. (Energy efficient)

¾ The sensor thresholds are different for different channels. ( Using channel statistics instead of instant CSI)

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Channel Aware Distributed Detection

z

Perfect CSI at the fusion center:

z

Perfect CSI at the fusion center:

¾

Example

:

Independent transmission & observation channels, BPSK (ui= -1 or 1), coherent reception.

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Channel Aware Distributed Detection

z

Suboptimal Fusion Rules:

z

Suboptimal Fusion Rules:

¾

High SNR (Chair-Varshney):

¾

Low SNR (MRC and EGC):

Also can be used when detection indexes of sensors are not known in th FC

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Channel Aware Distributed Detection

z

BPSK with perfect CSI in the FC,

i

l

d

b

i

l

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Channel Aware Distributed Detection

z

Estimated CSI in the Fusion center

T sensor decisions are packed and sent with a training bit.

Assume there is a block fading with coherence time less than the packet length length

.

1

t

(u

=

)

k,1 t k k,1 y = u h +n

¾

LMMSE complex channel estimation:

2 b 1 k k k 1 k 1 E ˆh = E(h | y ) = α y , σ = +(1 )− σ =w2 Ebσerror2 + σ2n k k k,1 k ,1 error 2 n h E(h | y ) α y , σ (1+ ) σ BPSK

Λ

_BPSK K * BPSK MRC k t k 1 _ˆ Re(y h ) K − Λ =

∑

k K j * BPSK EGC k,t k k 1 _ˆ Re(y ), e K ϕ φ φ − Λ =

∑

= BPSK MRC k ,t k k 1 (y ) K

∑

₌ K k 1=

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Channel Aware Distributed Detection

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Channel Aware Distributed Detection

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Channel Aware Distributed Detection

z

OOK modulation and the impact of Number of sensors on

p

Pd.

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Channel Aware Distributed Detection

z

BFSK Sensors with different Pds

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Conclusion

¾ Neyman-Pearson as a detection criterion for maximizing detection probability with the constraint on false alarm rate.

¾ Decentralized Detection, a power and bandwidth efficient detection , p which uses LRT as an optimal rule in the sensors and FC.

¾ Transmission channel aware sensor and fusion rules can improve the

¾ Transmission channel aware sensor and fusion rules can improve the detection performance of DD systems in fading channels.

¾ In high SNR rules using estimated CSI perform like the ones with

¾ In high SNR, rules using estimated CSI perform like the ones with perfect CSI.

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