A Linear Adaptive Algorithm for Data Fusion in Distributed Detection Systems
Rodrigo Pereira David
Department of Telecommunications, INMETRO Rio de Janeiro, Brazil
Raimundo SampaioNeto and César A. Medina
CETUC, Pontifical Catholic University of Rio de Janeiro (PUCRio) Rio de Janeiro, Brazil
Abstract
—In this work we propose an adaptive fusion procedure to perform data detection in distributed detection systems with a decodethenfuse type of receiver. Differently from previously proposed fusion rules, where nodes statistics (i.e. miss detection, false alarm and
a priori
probabilities) and channel transition probabilities are generally needed, the proposed nonassisted adaptive fusion algorithm adapts its coefficients and decision threshold based only on the received signals, so that timevarying or inhomogeneous distributed detection systems are well suited. Computer simulations show that the proposed adaptive fusion strategy delivers a performance very close to the optimal fusion rule.
Keywords— Distributed Detection, Data Fusion, Linear Estimation
I.
I
NTRODUCTION
Distributed detection systems such as wireless sensor networks (WSNs) represent a great technological advance to perform environmental monitoring, surveillance, detection, tracking, etc. with the use of geographically distributed nodes, which are characterized by several constraints such as lowpower and limited computation and communication capabilities. In a typical distributed detection scenario each node collects an observation, computes a local message (either realvalued or discrete), and sends it to a fusion center to perform the decision over the set of possible hypotheses [14]. We assume that nodes do not communicate with each other, and there is no feedback from the fusion center to the nodes. Early works considered the case where the fusion center has knowledge of the local messages (i.e. nonerroneous transmission of local messages between node and fusion center) and, under the conditional independence assumption, optimum fusion rules have been obtained for both binary and
m
ary local nodes output [3,4]. In [5], a channelaware optimal fusion rule was derived for the binary hypothesis case. A suboptimum fusion rule was also proposed in [5] as an approximation for high SNR relieving some of the requirements associated with the
Chair

Varshney
rule [2]. In this suboptimum approach, the local nodes messages are first estimated and then fused using the optimal fusion rule of [3]. Thus, this fusion rule works in a two stage manner (where communication and fusion are treated as independents processes) which is known as DecodethenFuse (DtF) rule [6]. It can be shown that, for high SNR, it has an equivalent performance to the optimal fusion rule [5, 6]. However, the implementation of the DtF rule requires the knowledge of the
a priori
hypothesis probabilities as well as the miss detection and false alarm probabilities, which are unknown in many cases. Some adaptive algorithms were proposed in [79] in order to estimate the miss detection and false alarm probabilities as well as the
a priori
hypothesis probabilities. Nevertheless, these algorithms need long training sequences and these algorithms are computationally expensive. In [10] a simple alternative fusion statistic that approximate ML solution has been proposed in an unknown local sensor detection probability scenario, however the false alarm probability is assumed to be known. In [11], a nonassisted online learning process to adaptively estimate combining weights of the local node messages is proposed. These combining weights account for the differences in nodes reliabilities. This approach considers identical
a priori
hypothesis probabilities and detection threshold equals to zero. In this paper we propose a nonassisted adaptive algorithm that jointly estimates combining weights and decision threshold to perform the optimum fusion rule in the second stage of a DtF rule. Through Monte Carlo simulations, we show that our adaptive fusion has a performance very close to that of the optimum detection algorithm without previous knowledge of the nodes false alarm and missing detection probabilities, channel transition probabilities and
a priori
probabilities. Notation:
Β
old lowercase denotes vectors. The operators
(.)
and
(.)
ℋ
denotes transpose and hermitian transponse respectively,
.
denotes the mean value,
(.)
represents the Gaussian tail distribution,
ℝ(.)
represents the real part and
.
is the sign function. II.
S
YSTEM
M
ODEL
We consider a binary hypothesis testing problem where
K
nodes are connected to a data fusion center in a distributed parallel architecture. Let
and
, denote two hypothesis with
a priori
probabilities
Pr(
)
=
and
Pr(
)
=
, respectively, such that:
: = +1
: = −1
(1)
where
is a random variable representing the binary hypothesis. Each node makes its own independent decision on the hypothesis
and the binary local decisions are mapped to symbols
. Without loss of generality we assume that decision
maps into
= −1 and decision
into
= +1. Finally, symbols are transmitted to the fusion center through a wireless channel. The fusion center follows the DtF rule [6], so that in the first stage the transmitted symbols
,
= 1,,K
, are estimated using a detection technique suitable to the adopted multiple access transmission scheme, generating the estimates
!
,
= 1,,K
. Then, in the second stage, a global decision
"
is made up according to some fusion rule. Considering an ideal transmission channel (
!
=
)
the optimal Bayesian fusion rule was derived in [2] and can be expressed as:
" = # $
+ %$
!
&'
,
(2) where the weights
$
and the threshold
$
are given by [2]:
$
= *
(3)
$
=
= *1−

/
,0 !
= 1
(4)
$
= 2
= *1−
/

,0 !
= −1
(5) where
!
=
and
/
= (
= +13 = −1)
and

=(
= −13 = +1)
are the false alarm and miss detection probabilities of the node
k
decision maker, respectively. Since here we consider also the effect of the transmission channel (
!
4
)
,
/
and

in (4) and (5) are replaced by the false alarm and miss detection probabilities of the equivalent composed decision system formed by the node decision maker system and the wireless transmission channel, which are given by:
/
= 5!
= +13 = −16 =
=
/
7
51−
/
8
6 +51−

7
6
/
8
(6)

= 5!
= −13 = +16 =
=

7
51−

8
6 +51−
/
7
6

8
(7) where
/
8
,

8
the note now are false alarm and miss detection probabilities of the node
k
decision maker,
/
7
= 5!
= +13
= −16
and

7
=
5!
= −13
= +16
are the conditional error probabilities of the node
k
transmitted symbol estimated in the first stage of the fusion center. We also note that, since the coefficients
$
( 4 9)
in (4)(5) depend on the value of
!
(
−1
or +
1
), so the optimal fusion rule as presented in (2)(5) is not a linear combination of the estimated symbols
!
, except for the particular case when

=
/
, for all
k
III.
PROPOSED ALGORITHM
The proposed algorithm relies on a rewritten version of the fusion rule in (2)(5) where the coefficients do not depend on the value of
!
. To do this, we first note that
$
in (4) and (5) can be expressed as:
$
=
1+ !
; <+ 2
1− !
; < =
=
(
− 2
)!
+
(
+2
).
(8) Realizing that
!
= 1
we have:
$
!
=
(
+ 2
)!
+
(
− 2
).
(9) Finally, replacing (9) in (2), the optimal fusion rule can be rewritten as:
" = # >
+ %>
!
&'
,
(10) where the coefficients
>
are now independent of
!
, and are given by:
>
= $
+ %1; (
−2
)
&'
= = *
+ 1;%*51−

6

51−
/
6
/
&'
(11)
>
= 1; (
+ 2
) = = 1;*51−

651−
/
6

/
(12) Note that in the version (1012), the threshold
>
in (11) does not depend only on the
a priori
probabilities as in (3). Based on the structure of the optimal fusion presented in (10)(12), this work proposes a linear adaptive algorithm to joint estimate the weights
>
and the threshold
>
and make the final decision
" .
Let
?(0) = >
(0) >
(0)@>
A
(0)
be the vectors of combining weights at time
i
and
BC (0) = !
(0) !
(0) !
A
(0)
. Then the linear structure in (10) resembles the structure shown in Fig. 1 where the output of a polarity detector device generates the global decision
"(0) = (0)
, where
(0) = ?(0)
BC (0) + >
(0)
. However the optimal coefficients
>
and the threshold
>
in (11) and (12) are dependent of several parameters which are unknown in practice. To overcome this limitation, we propose to estimate
>
and
>
in an adaptive fashion as indicated in Fig. 1. The vector of combining weights
?
and the scalar
>
can be estimated by minimizing a mean squared cost function:
D = EF"
(0) − ?
(0)BC (0) − >
(0)F
G= 3H(0)3
(13) where
"
(0)
I J−1,+1
is an intermediate estimate of
and
H(0) = F"
(0) − ?
(0)BC (0) − >
(0)F
is the output error. Since we assume that a training sequence is not available and the systems parameters (e.g.
/
,

in (6)(7) and the
a priori
probabilities
and
) are not known to the receiver, the majority rule was proposed to generate the intermediate estimates
"
(0)
which are more reliable than the individual local decisions [7], so:
"
(0) = #%!
&'
.
(14) The vector
?
and the scalar
>
that minimizes (13) are obtained equating to zero the gradient of the cost function. Solving for
>
we obtain:
>
= L
MN
O
− ?
BC
,
(15) where
L
MN
O
= "
(0)
and
L
BC
= QBC (0)R
. Now, we replace (15) in (13) and minimize for
?
to obtain:
? = S
BCT
U
MV
O
B
,
(16) where
S
BC
= E5BC (0) −
BC
65BC (0) −
BC
6
G
and
U
MV
O
B W
= Q5"
(0) − L
MN
O
65BC (0) −
BC
6R
. We can use different adaptive filtering algorithms to approximate (15)(16). In this paper we use the least mean squares (LMS) and the recursive least squares (RLS) algorithms, as follows.
Fig 1. Proposed Adaptive Fusion Algorithm
A.
ADAPTIVE FUSION BASED ON
LMS
ALGORITHM
The LMS algorithm computes the weights and the threshold in an adaptively fashion by using the steepest descent scheme and instantaneous estimates of the squared error, such that
?(0 + 1)
and
>
(0 +1)
can be computed as:
?(0 +1) = ?(0) −X
?
Y
?
3H(0)3
(17)
>
(0 + 1) = >
(0) − X
7
O
Y
7
O
3H(0)3
(18) where
Y
is the gradient operator,
X
?
and
X
7
O
are updating steps. The recursions in (17)(18) are given by:
?(0 + 1) = ?(0) + X
?
H(0)BC (0)
(19)
>
(0 + 1) = >
(0) + X
7
O
H(0)
(20) A drawback of the LMS algorithm is its sensibility to amplitude variations of the input signal which hampers the suitable sizing of the step
X
?
and
X
7
O
. The normalized LMS algorithm (NMLS) minimizes this problem by normalizing the step
X
?
with respect to the instantaneous power of the observation signal. In the present case this power is constant since
BC
(0)BC (0) = K
. B.
ADAPTIVE FUSION BASED ON
RLS
ALGORITHM
The RLS algorithm recursively searches for the coefficients that minimize the weighted linear least squared cost function:
Z(0) = %[
\T]\]'
H
() = = %[
\T]\]'
3
N
9
(
)
−?
^
B
_(
)
− >
9
3
;
(21) where
H(
)
is the output error defined in the previous section and
9 ` ` [ ` 1
is the forgetting factor. Following the same minimization procedure used to obtain (15) and (16), we arrive at:
>
(0) = LN
MN
O
(0) − ?(0)
N
BC
(0),
(22) where
1
LN
MN
O
(0) = a
1−[ 1−[
0+1
bc
[
0−0=9
"
(),
N
BC
(0) =
a
Td Td
efg
bc [
\T]\]'
B
_(
)
and
?(0) = S_
B hT
(0)UN
MV
O
B
(0),
(23) where,
S_
BC
(0) = [S_
BC
(0 − 1) + 5BC (0) −N
BC
65BC (0) − N
BC
6
(24)
UN
MV
O
B W
(0) = [UN
MV
O
B W
(0 − 1) + 5"
(0) − LN
MN
O
65BC (0) − N
BC
6
(25) and the inverse
S_
B hT
can be computed efficiently using Kalman recursions. IV.
N
UMERICAL
R
ESULTS
This section presents simulations results and evaluate the performance of the proposed LMS and RLS adaptive algorithms where the adaptive fusions (LMS and RLS) are compared to the optimal fusion rule given in (10)(12), (6) and (7) and to the majority fusion rule. In the simulated systems, the node
k
decision is based on the observation:
i
= .j +
= 1,;,,K
(26) where
is defined in (1) with
a priori
probabilities
and
,
j
is a positive constant and
, = 1,;,,K
are mutually independent real Gaussian random variables with zero mean and variance
k
. Node
k
performs a ML detection on
i
to estimate
, thus
= Q i
R
. The resulting probabilities
/
8
and

8
in (6) and (7) are then given by
alj
mk
b
. The node decisions
,
= 1,,K
are BPSK modulated and transmitted to the fusion center through different multipath fading channels, using a modified version of the multiple access scheme proposed in [12] and here referred to as CSCDMA (chipspread code division multiple access) [13]. In the CSCDMA scheme the N chips of the user
k
code sequence
n
o
=
,
. . .
,pT
q
, are all multiplied by the same size M data symbol block
B
. In order to avoid interblock interference in the received signal, it was proposed in [13] that the block
B
be formed by the node
k
message
concatenated with
− 1
zeros, where L is the length of the discrete equivalent channel from node
k
to the fusion center,
s
(0) = t
,
(0) . . .t
,uT
(0)
. Thus the size L block
B
has the vector structure
B
=
v
uT
[13]. The use of the CSCDMA transmission technique preserves the srcinal
_ +
BC (0)
"(0)
>
(0)
?(0)
BC
(0)?(0)
•
H(0)
(0)
"
(0)
Adaptive Alg.
Majority Rule
1
−1
1
An update of the form,
i(0) = c [
\T]\]'
w()
, for
w()
scalar or vector, can be computed recursively as
i(0) = [i(0 −1) +w(0)
.
orthogonality between the users codes in spite of the transmission through a multipath channel, thus allowing the signals transmitted by the nodes to the fusion center to be ideally decoupled in the first stage of the receiver. The decoupled signal
xy
corresponding to node
k
to be processed in the first stage of the fusion center is represented by the size L vector [13]:
xy
=
s
+ z{
= 1,;,,K,
(27) where the noise vector
z{
is complex Gaussian with zero mean and covariance matrix
Qz{
z{
ℋ
R = ;k

}
u
.
The first stage of the fusion center performs a ML detection on
xy
using a filter matched to the channel impulse response
s
to generate the estimate
!
= Qℝ5s
ℋ
xy
6R
. Thus the resulting probabilities
/
7
e

7
in (6) and (7) are given by
~• €s
€
mk

[13]. The channels
s
are mutually independent and modeled by a time invariant random FIR filter with L = 4 taps, with the coefficients of node k channel given by
t
,]
= ‚
]
ƒ
,]
, where
ƒ
,]
,
= 9,1,, − 1
, are Gaussian complex mutually independent random variables, with zero mean and
EFƒ
,]
F
2
G =
1. The values of
ƒ
,]
are randomly drawn and kept fixed throughout each simulation run. The coefficients
‚
]
, satisfies
c 3
„T\'
‚
\
3
=1 with
‚
=
0.8671,
‚
=9.…†…‡
,
‚
=
0.2178 and
‚
ˆ
=
0.1092. Results are averaged over 10000 runs, with 100 node messages transmitted in each run. The LMS algorithm uses updating steps
X
?
=
X
7
O
= 0.0005 (normalized by K) and the RLS algorithm uses a forgetting factor
[
= 0.999. Knowledge of the channels
s
by the receivers is also assumed. We compare the proposed LMS and RLS adaptive fusion with the optimal fusion rule (1012) and with the majority fusion rule (14), which is also the rule applied to obtain the reference signal for the proposed adaptive fusions. Figures 2 and 3 show the decision error rate (DER) for each decision rule and for different values of signal to noise ratio at the nodes (local node SNR =
j
mk
). We deploy a WSN with K = 7 nodes, with
= 0.1 and
= 0.9, where the average SNR at the receiver (fusion center) are the same for all nodes (i.e., average channel SNR =
€s
€
mk

).
Fig 2. Decision error rate vs local node SNR for average channel SNR = 8 dB , K=7 nodes,
= 9.1
and
= 9.‰
Fig. 2 is for an average channel SNR of 8 dB and Fig. 3 is for a reduced channel SNR of 2 dB. Note that for the majority fusion rule, whose output is used as reference signal in the proposed algorithms, there is a strong degradation when the channel SNR of 2 dB is simulated, however, the LMS and RLS adaptive fusions maintain essentially the same performance of the optimal fusion rule for both scenarios, thus, they exhibit robustness to low channel SNR.
Fig 3. Decision error rate vs node SNR for average channel SNR = 2 dB, K=7 local nodes,
= 9.1
and
= 9.‰
In the next experiment, we deploy a WSN with K = 7 nodes with different configurations, for nodes 1 to 3 we set the local SNR =
Š‹Œ
Ž
, for node 4 local SNR =
9.‡ Š‹Œ
Ž
and for nodes 5 to 7 we set local SNR =
1.‘ Š‹Œ
Ž
. For the average channel SNR we set for nodes 1 and 2 a channel SNR = 8 dB, nodes 3 to 5 a channel SNR = 2 dB and nodes 6 to 7 channel SNR = 7 dB. In Fig. 4, the DER vs
Š‹Œ
Ž
is depicted for different fusion rules. In this inhomogeneous scenario, as expected, the
Fig 4. Decision error rate vs
Š‹Œ
Ž
, K= 7 local nodes, average channel SNR: nodes 1 and 2 = 8 dB, nodes 3 to 5 = 2 dB , nodes 6 and 7 = 7 dB,
= 9.1
and
= 9.‰
performance of majority rule degrades considerably, while the adaptive fusion algorithms still maintain only a slight loss of performance when compared to the optimal fusion rule, evidencing that the proposed algorithms are robust to inhomogeneous WSNs. C.
C
ONCLUSION
In this paper we proposed nonassisted LMS and RLS based adaptive algorithms to implement the fusion procedure in inhomogeneous WSNs. The proposed algorithms use majority rule estimates obtained locally at the fusion center as a reference signal. It was verified by simulations that the proposed schemes exhibit a performance very close to that of the optimal fusion rule.
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R
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