Channel Aware Distributed Detection in Wireless Network with - PowerPoint PPT Presentation

Channel Aware Distributed Detection in Wireless Network with Correlated Observations Nahal Maleki Department of Electrical and Computer Engineering, University of Rochester

Centralized versus Distributed Detection x 5 Fusion Center x 6 x 1 x 3 y 3 x 4 x 2 fire detection in forest via wireless sensor network Centralized detection: Unlimited energy and bandwidth ⇒ infinite precision for sending observations. Error-free communication channels. Distributed detection: Passing local decisions to the FC. Classical: error-free communication channels. Our model: fading and noise in communication channels. Design of distributed detection system.

The Problem and Our Approach Problem 1(P1) What can be the new architectures for the distributed detection system design in the presence of fading and noise in communication channels?

The Problem and Our Approach Problem 1(P1) What can be the new architectures for the distributed detection system design in the presence of fading and noise in communication channels? Our Approach We propose three new architectures: (i) cooperative fusion architecture with Alamouti’s STC scheme at sensors, (ii) cooperative fusion architecture with signal fusion at sensors, (iii) parallel fusion architecture with local threshold changing at sensors.

The Problem and Our Approach Problem 2(P2) For distributed detection of a Gaussian signal source in noise, what is the optimal transmit power allocation at sensors?

The Problem and Our Approach Problem 2(P2) For distributed detection of a Gaussian signal source in noise, what is the optimal transmit power allocation at sensors? Our Approach For linear fusion rule at the FC and Total or individual transmit power constraints at sensors, Coherent and noncoherent reception mode at the FC, Different communication multiple access channel schemes. We find transmit power allocation at sensors, such that modified deflection coefficient (MDC) at FC is maximized.

Distributed Binary Detection over Fading Channels: Cooperative and Parallel Architectures

Parallel Fusion Architecture Sensing Channel Model H 0 : x k = w k ; H 1 : x k = 1 + w k ; w k ∼ N ( 0 , σ 2 w k ) . S k applies the LRT, u k = 1 f ( x k |H 1 ) π 0 ≷ π 1 . f ( x k |H 0 ) u k = -1 P d k = P ( u k = 1 |H 1 ) and P f k = P ( u k = 1 |H 0 ) . Communication Channel Model y k = u k h k + v k ; h k ∼ CN ( 0 , σ 2 h k ) , v k ∼ CN ( 0 , σ 2 v ) . The FC forms the LRT, U 0 = 1 Λ = f ( y 1 ,..., y K |H 1 ) π 0 ≷ π 1 . f ( y 1 ,..., y K |H 0 ) U 0 = 0 If w k s are uncorrelated, we have Λ = P dk f ( y k | u k = 1 )+( 1 − P dk ) f ( y k | u k = − 1 ) � K P fk f ( y k | u k = 1 )+( 1 − P fk ) f ( y k | u k = − 1 ) . k = 1

Cooperative Fusion Architecture with STC at Sensors Sensing Channel Model S i and S j are cooperative partners. √ S i transmits 1 − α u i , where 0 < α < 1. √ 1 − α u i g ij + η ij , r ij = g ij ∼ CN ( 0 , σ 2 hs ij ) , η ij ∼ CN ( 0 , σ 2 η ) . S j demodulates u i , using the knowledge of g ij , ˆ u i = sgn ( Re ( r ij / g ij )) . � � α α n th slot: S i and S j send 2 u i and 2 u j . � � α α 2 ˆ 2 ˆ ( n + 1 ) th slot: S i and S j send − u j and u i .

Cooperative Fusion Architecture with STC at Sensors Communication Channel Model We have � α � α 2 (ˆ u i h j − ˆ y ij ( n ) = 2 ( u i h i + u j h j ) + v ij ( n ) , y ij ( n + 1 ) = u j h i ) + v ij ( n + 1 ) h i ∼ CN ( 0 , σ 2 h i ) , h j ∼ CN ( 0 , σ 2 h j ) , v ij ( n ) , v ij ( n + 1 ) ∼ CN ( 0 , σ 2 v ) . The FC forms � z i � h ∗ � h ∗ � h j � � y ij ( n ) � h j � � v ij ( n ) � i i = = y ∗ v ∗ h ∗ h ∗ − h i ij ( n + 1 ) − h i ij ( n + 1 ) z j j j � α �� | h i | 2 � � u i � � ˆ h j h ∗ | h j | 2 − h j h ∗ � � �� u i i i + + . h i h ∗ | h j | 2 − h i h ∗ | h i | 2 ˆ u j u j 2 j j U 0 = 1 f ( z i , z j for all pairs |H 1 ) π 0 ≷ Using the h i , h j for all pairs, the FC forms LRT Λ = π 1 . f ( z i , z j for all pairs |H 0 ) U 0 = 0

Cooperative Fusion Architecture with Signal Fusion at Sensors Sensing Channel Model S j updates its initial decision by fusing r ij and x j and forms u j = 1 ˜ f ( r ij , x j |H 1 ) π 0 ˜ ≷ λ j = π 1 . f ( r ij , x j |H 0 ) u j = -1 ˜ The pair ( S i , S j ) sends √ α ˜ u i , √ α ˜ u j to the FC over two orthogonal channels subject to noise and fading. Communication Channel Model We have y i = √ α ˜ u i h i + v i , y j = √ α ˜ u j h j + v j , h i ∼ CN ( 0 , σ 2 h i ) , h j ∼ CN ( 0 , σ 2 h j ) , v i , v j ∼ CN ( 0 , σ 2 v ) . Using h i , h j for all pairs, the FC forms the LRT U 0 = 1 f ( y i , y j for all pairs |H 1 ) π 0 ≷ Λ = π 1 , to make the final f ( y i , y j for all pairs |H 0 ) U 0 = 0 decision.

Parallel Fusion Architecture with Local Threshold Changing at Sensors Sensing Channel Model In the absence of inter-node communication, S i assumes u j = − u i . S i forms ¯ u i by fusing the assumed decision u j and x i . u i = 1 ¯ f ( x i , u j = − u i |H 1 ) ¯ ≷ λ i = u i = -1 . f ( x i , u j = − u i |H 0 ) ¯ One can verify that u i = 1 if x i > τ ′ u i = − 1 if x i < τ ′ u i = 1 , ¯ i 1 , u i = − 1 , ¯ i 2 , u i = 1 if τ ′ u i = − 1 , ¯ i 2 < x i < τ i , u i = − 1 if τ i < x i < τ ′ u i = 1 , ¯ i 1 where the thresholds τ ′ i 1 , τ ′ i 2 depend on σ 2 w i , ρ i , j and satisfy τ ′ i 2 < τ i < τ ′ i 1 .

Performance Analysis Assumptions Gaussian sensing noises w k are i.i.d. thus P d k = P d , P f k = P f . Sensors are positioned equally distant from the FC and thus h = σ 2 γ 2 ¯ v . h σ 2 Distances between the cooperative partners are assumed equal hs = ( 1 − α ) σ 2 γ 2 across the pairs and therefore ¯ hs . σ 2 η

Parallel Fusion Architecture ¯ � T e 1 P Qn ¯ ( 1 − P f ) K − Qn P e 1 = π 0 f n T e 2 P Qn ( 1 − P d ) K − Qn ¯ � ¯ P e 2 = π 1 d n S 1 { Qn < M } � ¯ � � T e 1 < [ G ( n , n 1 ) D 1 ( n , n 1 )] + 1 { Qn ≥ M } , � 2 | S 1 | dn 1 ∈ S 1 s = 1 S 1 { Qn > M } ( | S 0 | G ( n , n 1 )) t ¯ � � T e 2 < [ min D 2 ( n , n 1 )] + 1 { Qn ≤ M } , | S 0 | t dn 1 ∈ S 0 s = 1 − 1 γ h | a 2 s − 1 − a 2 s − 1  γ h | a 2 s − a 2 s  ¯ n 1 | ¯ | n n 1 n D 1 ( n , n 1 ) =  ( 1 + )( 1 + ) ,  2 2 � − 1 . � ( 1 + 2 ( t 2 − t )¯ | )( 1 + 2 ( t 2 − t )¯ γ h | a 2 s − 1 − a 2 s − 1 γ h | a 2 s − a 2 s D 2 ( n , n 1 ) = n 1 | ) n n 1 n

Cooperative Fusion Architecture with STC at Sensors T e 1 P Qn ( 1 − P f ) K − Qn T n , m ¯ � ¯ P e 1 = π 0 f n , m T e 2 P Qn ( 1 − P d ) K − Qn T n , m . ¯ � ¯ P e 2 = π 1 d n , m S 1 { Qn < M } � � � ¯ T e 1 < [ G ( n , m , n 1 , m 1 ) D 1 ( n , m , n 1 , m 1 )] + 1 { Qn ≥ M } � 2 | S 1 | dn 1 , m 1 ∈ S 1 s = 1 S 1 { Qn > M } ( | S 0 | G ( n , m , n 1 , m 1 )) t ¯ � � T e 2 < [ min D 2 ( n , m , n 1 , m 1 )]+ 1 { Qn ≤ M } , | S 0 | t dn 1 , m 1 ∈ S 0 s = 1 � − 1 α 2 ¯ � γ 2 α ¯ γ h ¯ a 1 α ¯ γ h ¯ a 2 h ¯ a 3 D 1 ( n , m , n 1 , m 1 ) = ( 1 + )( 1 + ) − , 8 8 64 α ( t 2 − t )¯ α ( t 2 − t )¯ α 2 ( t 2 − t ) 2 ¯ � − 1 γ 2 � γ h ¯ γ h ¯ h ¯ a 1 a 2 a 3 D 2 ( n , m , n 1 , m 1 ) = ( 1 + )( 1 + ) − 2 2 16 When inter-sensor channels are error-free ¯ a 3 = 0 and when ¯ γ h is high we have , D 2 ( . ) = ( α ( t 2 − t )¯ α ( t 2 − t )¯ D 1 ( . ) = ( α ¯ α ¯ a 1 a 2 a 1 a 2 γ − 2 → diversity gain γ − 2 → diversity gain 8 ) − 1 ¯ ) − 1 ¯ h h 8 2 2

Cooperative Fusion Architecture with Signal Fusion at Sensors ¯ � T e 1 P Qn ¯ ( 1 − P f ) K − Qn P e 1 = π 0 f n T e 2 P Qn ( 1 − P d ) K − Qn ¯ � ¯ P e 2 = π 1 d n S 1 { dn ∈ S 0 } � ¯ � � T e 1 < [ G ( n , n 1 ) D 1 ( n , n 1 )] + 1 { dn ∈ S 1 } → making local decision more reliable , � 2 | S 1 | dn 1 ∈ S 1 s = 1 1 { dn ∈ S 1 } S ( | S 0 | G ( n , n 1 )) t ¯ � � T e 2 < [ min D 2 ( n , n 1 )] + 1 { dn ∈ S 0 } → making local decision more reliable , | S 0 | t dn 1 ∈ S 1 s = 1 − 1 γ h ( a 2 s − 1 − a 2 s − 1  ) 2 γ h ( a 2 s − a 2 s n 1 ) 2  α ¯ α ¯ n n 1 n D 1 ( n , n 1 ) =  ( 1 + )( 1 + ) → no diversity gain ,  4 4 � − 1 → no diversity gain . � ( 1 + α ( t 2 − t )¯ ) 2 )( 1 + α ( t 2 − t )¯ γ h ( a 2 s − 1 − a 2 s − 1 γ h ( a 2 s − a 2 s n 1 ) 2 ) D 2 ( n , n 1 ) = n n 1 n

Parallel Fusion Architecture with Local Threshold Changing at Sensors Qj 4 n 1 , m 1 ¯ � ¯ � P e 1 = π 0 T e 1 P . fj n , m j = 1 Qj 4 n 1 , m 1 ¯ � � ¯ P e 2 = π 1 T e 2 P . dj n , m j = 1 1 { dn , m ∈ S 0 } S � ¯ � � T e 1 < [ G ( n , m , n 1 , m 1 ) D 1 ( n , m , n 1 , m 1 )] + 1 { dn , m ∈ S 1 } , � 2 | S 1 | d ′ s = 1 n 1 , m 1 ∈ S 1 S 1 { dn , m ∈ S 1 } ¯ � ( | S 0 | G ( n , m , n 1 , m 1 )) t � T e 2 < [ min D 2 ( n , m , n 1 , m 1 )] + 1 { dn , m ∈ S 0 } , | S 0 | t s = 1 dn , m ∈ S 0 α 2 ¯ � − 1 γ 2 � γ h ¯ γ h ¯ h ¯ α ¯ a 1 α ¯ a 2 a 3 D 1 ( n , m , n 1 , m 1 ) = ( 1 + )( 1 + ) − , 8 8 64 � − 1 α ( t 2 − t )¯ α ( t 2 − t )¯ α 2 ( t 2 − t ) 2 ¯ γ 2 � γ h ¯ a 1 γ h ¯ a 2 h ¯ a 3 D 2 ( n , m , n 1 , m 1 ) = ( 1 + )( 1 + ) − . 2 2 16