2668 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005
Fusion of Censored Decisions in Wireless Sensor Networks
Ruixiang Jiang and Biao Chen, Member, IEEE
Abstract—Sensor censoring has been introduced for reduced
communication rate in a decentralized detection system where
decisions made at peripheral nodes need to be communicated to a
fusion center. In this letter, the fusion of decisions from censoring
sensors transmitted over wireless fading channels is investigated.
The knowledge of fading channels, either in the form of instanta-
neous channel envelopes or the fading statistics, is integrated in the
optimum and suboptimum fusion rule design. The sensor censor-
ing and the ensuing fusion rule design have two major advantages
compared with the previous work. 1) Communication overhead is
dramatically reduced. 2) It allows incoherent detection, hence, the
phase information of transmission channels is no longer required.
As such, it is particularly suitable for wireless sensor network
applications with severe resource constraints.
Index Terms—Censoring sensor, decision fusion, fading chan-
nels, wireless sensor networks.
I. INTRODUCTION
I
N A WIRELESS sensor network (WSN) tasked with a
distributed detection problem [1], geographically dispersed
sensor nodes are used to make peripheral decisions based on
their own observations. These decisions are transmitted through
wireless channels to a fusion center where a final decision
regarding the state of an event is made. In many WSN ap-
plications involving in situ unattended sensors operating on
irreplaceable power source, severe resource constraints as well
as the time-sensitive nature of many detection problems require
prudent use of power/bandwidth and other resources. The sen-
sor censoring idea, first proposed by Rago et al. in 1996 [2] for
reduced communication rate, is a very suitable candidate for
local sensor signaling.
With censoring sensors, only those sensors with informa-
tive observation, measured by their local likelihood ratio (LR)
values, send the LR to the fusion center. Using the canonical
parallel fusion structure with binary hypothesis and condition-
ally independent sensor observations, it was shown in [2] that
the optimal “no-send” region for any given sensor, defined
on the LR domain, amounts to a single interval for both the
Bayesian and Neyman–Pearson (NP) criteria. This is illustrated
in Fig. 1(a), where [t
1
,t
2
] corresponds to the “no-send” region;
i.e., if the LR falls in-between t
1
and t
2
, the sensor does not
transmit its LR to the fusion center. Furthermore, in the case
of sufficiently small prior probability of the target-present hy-
pothesis and severe communication constraint, the optimal (in
the sense of minimum error probability) lower threshold of the
Manuscript received February 5, 2004; revised September 11, 2004; accepted
November 16, 2004. The editor coordinating the review of this paper and
approving it for publication is W. Liao. This work was presented in part at the
IEEE ICASSP’04, Montreal, QC, Canada, May 2004.
The authors are with Department of Electrical Engineering and Computer
Science, Syracuse University, Syracuse, NY 13244 USA (e-mail: rjiang@ecs.
syr.edu; bichen@ecs.syr.edu).
Digital Object Identifier 10.1109/TWC.2005.858363
Fig. 1. (a) Sensor censoring region. (b) Special case when t
1
=0.
“no-send” region was shown to be 0, i.e., t
1
=0[see Fig. 1(b)].
Similar result was also established later in [3] using the NP
criterion. An intuitive explanation is that when a target is less
likely to be present, the extreme communication constraint
prohibits sending low LR values that happen much more often.
For the case with t
1
=0, the censoring scheme is effectively
an LR test (LRT)-based transmission scheme: Whenever the
local LR exceeds t
2
, the sensor transmits the LR; otherwise,
the sensor remains silent. In this paper, we take the above sensor
censoring to its extreme case—if the local LR exceeds t
2
,the
sensor sends only a single bit,
1
indicating that the LR falls into
the “send” region, instead of the LR value in its entirety. Such
an extreme censoring scheme has also been considered in [3] in
the context of studying locally optimum distributed detection.
Closely related to the present work is the development of
channel-aware decision fusion rules f or WSN where a binary
local sensor signaling is assumed [4]–[6]. Compared with our
previous work, the sensor censoring s cheme enjoys significant
energy efficiency—instead of sending a binary signal at every
time slot, each sensor will stay quiet if its LR falls below t
2
.
Another important advantage is that the sensor censoring
scheme allows the fusion center to employ fusion rules based
on incoherent detection. Acquiring phase information of trans-
mission channels can be costly as it typically requires training
overhead. This overhead may be substantial for time-selective
fading channels when mobile sensors are involved or the fusion
center is constantly moving [consider, for example, the reach
back channel with the receiver mounted on a unmanned aerial
vehicle (UAV) or a moving vehicle]. Thus, we concentrate on
incoherent-detection-based decision fusion rules in the present
work. Notice that if channel phase information is available, the
fusion rules developed in [4]–[6] can be applied directly—the
1
As usual, this single bit corresponds to a particular waveform that is sent
from the sensor to the fusion center. As is typical in digital communication,
this waveform is represented by a constellation point with appropriately chosen
basis function(s). We assume further that a “1” is sent, indicating that the basis
function is chosen to coincide with the actual waveform.
1536-1276/$20.00 © 2005 IEEE
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005 2669
Fig. 2. Illustration of the ON/OFF signaling for local sensors. Only those
sensors (shaded) whose LR exceeds a certain threshold are alarmed and send
signals to the fusion center.
censoring scheme is equivalent to a binary s cheme with one
constellation point coinciding with the origin.
Specifically, resorting to incoherent detection schemes, we
develop optimum fusion rules for the following two scenarios:
1) when the fading channel envelopes are available at the fusion
center and 2) when only the fading statistics are available.
Under the low signal-to-noise ratio (SNR) regime, we further
reduce the optimal fusion rule into simple test statistics that
are both easy to implement and not subject to prior knowledge
requirement. These test statistics also provide insights into
some simple intuitive test statistics. For example, the censoring
scheme amounts effectively to an
ON/OFF signaling at local
sensors where only alarmed sensors send signals to the fusion
center, as illustrated in Fig. 2. An intuitive detection scheme is
to employ an energy detector (ED); i.e., the fusion center simply
sums up all the signal powers from all the sensors. Indeed,
under certain channel fading models, we show that this simple
scheme is the optimal detector in the low-SNR regime.
We remark here that our emphasis in this work is the de-
velopment of fusion algorithms with
ON/OFF signaling for a
fading environment. Another important issue is the local sensor
decision rule, i.e., how to determine the censoring threshold t
2
.
This is not addressed in this paper. We assume, instead, that
local sensors employ a sensor censoring scheme with known t
2
.
Therefore, the local sensor performance indices (probabilities
of false alarm and detection) can be readily calculated. We
note that censoring threshold design has been addressed in [7]
where detector efficacy is optimized by assuming a simplified
ALOHA protocol for the sensor communications.
The organization of the paper is as follows. In the next
section, we introduce the system model and derive the optimum
LR-based fusion rule with the knowledge of fading channel en-
velope. Two suboptimum fusion statistics are also provided. In
Section III, we derive, under Rayleigh, Ricean, and Nakagami
fading channel models, the optimum fusion rules, assuming
only the knowledge of the fading channel statistics. Numerical
examples are provided in Section IV, followed by conclusions
in Section V.
II. O
PTIMAL FUSION RULE WITH THE KNOWLEDGE
OF
CHANNEL ENVELOPE
The sensor fusion system employing ON/OFF signaling with a
canonical parallel fusion structure is depicted in Fig. 3. The K
sensors collect observations and calculate their respective LR
values. For each sensor, if its LR value exceeds a precalculated
threshold t
2
, it transmits a binary signal (say, u
k
=1)toa
fusion center. Otherwise, if the LR falls below the threshold,
u
k
=0, i.e., the sensor remains silent during this transmission
period. We assume that the observations are independent across
sensors conditioned on any given hypothesis. The probabilities
of false alarm and detection of the kth local sensor node are de-
noted by P
fk
and P
dk
, respectively. They can be computed
easily using the knowledge of the hypotheses under test and the
LR threshold t
2
. The local sensor outputs, u
k
, k =1,...,K,
are transmitted over parallel channels that are assumed to un-
dergo independent flat fading. We denote by h
k
and φ
k
the
fading envelope and phase of the kth channel, respectively. We
further assume a slow fading channel, whereby the channel
remains constant during the transmission of one decision.
The above model yields the channel output for the kth sensor,
given as
y
k
=
n
k
, the kth sensor decides H
0
h
k
e
jφ
k
+ n
k
, the kth sensor decides H
1
(1)
where n
k
is a zero-mean complex Gaussian noise whose real
and imaginary parts are independent of each other and have
equal variance σ
2
, hence, E[|n
k
|
2
]=2σ
2
.
If both h
k
and φ
k
are known, the optimum LR-based decision
fusion rule can be easily derived [4], [6]. Notice that with
censoring, we are replacing {+1, −1} with {1, 0}, hence, with
phase information, the equivalence between the two schemes
(save some scaling factors) is reminiscent to the rotation and
shift invariance principle in digital communications. Thus, we
concentrate now on the incoherent case, i.e., we develop fusion
statistics based on the output envelope, or equivalently, the
output power.
Denote by z
k
the signal power for the kth channel output,
i.e., z
k
= |y
k
|
2
, hence, given h
k
, it is easy to get
p(z
k
|u
k
=0,h
k
)=
1
2σ
2
e
−
z
k
2σ
2
p(z
k
|u
k
=1,h
k
)=
1
2σ
2
I
0
h
k
σ
2
√
z
k
e
−
h
2
k
+z
k
2σ
2
where I
0
(.) is the zeroth-order modified Bessel function of the
first kind.
Using z
k
instead of y
k
in the fusion rule design and assuming
knowledge of the fading channel envelope and the local sensor
performance indices, the logarithmic LR (LLR) can be derived
in a straightforward manner as
Λ = log
p(z
1
,...,z
k
|H
1
)
p(z
1
,...,z
k
|H
0
)
=
k
log
P
dk
p(z
k
|u
k
=1)+(1− P
dk
)p(z
k
|u
k
=0)
P
fk
p(z
k
|u
k
=1)+(1− P
fk
)p(z
k
|u
k
=0)
(2)
=
k
log
P
dk
I
0
h
k
σ
2
√
z
k
e
−
h
2
k
2σ
2
+(1− P
dk
)
P
fk
I
0
h
k
σ
2
√
z
k
e
−
h
2
k
2σ
2
+(1− P
fk
)
. (3)
We consider next the low-SNR approximation for Λ.
2670 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005
Fig. 3. Parallel fusion model in the presence of fading and noisy channel between the local sensors and the fusion center.
Proposition 1: As the channel noise variance σ
2
→∞, i.e.,
SNR → 0, and assuming identical local sensor performance, Λ
in (3) reduces to
Λ
WED
=
k
h
2
k
z
k
. (4)
To verify this, notice from [8] that
I
0
(x)=
∞
i=0
1
4
x
2
i
(i!)
2
. (5)
Applying (5) to I
0
((h
k
σ
2
)
√
z
k
) and keeping only the first two
terms for large σ
2
, we get
I
o
h
k
σ
2
√
z
k
≈ 1+
h
k
σ
2
√
z
k
2
4
. (6)
Plugging this into (3) and using, for small x, e
−x
≈ 1 − x,
we can show that (3) reduces to
Λ ≈
k
log
1+
(P
dk
− P
fk
)
h
2
k
2σ
4
z
k
−
h
2
k
2σ
2
1+P
fk
h
2
k
2σ
4
z
k
−
h
2
k
2σ
2
where we only keep the terms up to the order of 1/σ
4
.Using
the fact that log(1 + x) ≈ x for small x, this can be further
reduced to
Λ ≈
k
(P
dk
− P
fk
)
h
2
k
2σ
4
z
k
−
h
2
k
2σ
2
.
Given that the envelopes h
k
s are known, this test statistic is
equivalent to
k
(P
dk
− P
fk
)h
2
k
z
k
as the term independent
of z
k
can be discarded. Furthermore, if local sensors have
identical performance indices, this statistic is equivalent to (4)
in Proposition 1. This is a weighted sum of the received signal
power from all sensors, hereafter termed as the weighted energy
detector (WED).
III. C
HANNEL-STATIST ICS-BASED FUSION RULES
The LR-based fusion rule developed in the previous section
requires knowledge of the channel fading envelope. Due to
the limited resources, this information may not be available
at the fusion center. Without the knowledge of fading chan-
nel envelope, we derive in this section the channel statistics-
based LRT using the channel output power z
k
. Notice that
obtaining channel fading statistics typically is much less costly
than acquiring instantaneous envelopes. Three popular fading
channel models, namely, the Rayleigh, Ricean, and Nakagami,
are considered in this section.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005 2671
A. Rayleigh Fading Channel
In a purely diffuse scattering environment without a dom-
inant path, the channel is typically modeled as Rayleigh fad-
ing channel. Without loss of generality, we assume that the
Rayleigh fading channel has unit power, i.e., E[h
2
k
]=1. Thus
p(h
k
)=2h
k
e
−h
2
k
.
Given p(h
k
), we then calculate the conditional probability
density function (PDF) p(z
k
|u
k
)
p(z
k
|u
k
)=
∞
0
p(z
k
|h
k
,u
k
)p(h
k
)dh
k
.
Straightforward computations yield
p(z
k
|u
k
=0)=
1
2σ
2
e
−
z
k
2σ
2
p(z
k
|u
k
=1)=
1
1+2σ
2
e
−
z
k
1+2σ
2
.
Notice that both of t hem are exponentially distributed with
respective mean values equal to 2σ
2
and 1+2σ
2
. With these
conditional PDF, one can easily construct the LLR as
Λ=
k
log
P
dk
1
1+2σ
2
e
−
z
k
1+2σ
2
+(1− P
dk
)
1
2σ
2
e
−
z
k
2σ
2
P
fk
1
1+2σ
2
e
−
z
k
1+2σ
2
+(1− P
fk
)
1
2σ
2
e
−
z
k
2σ
2
. (7)
Next, we consider low-SNR approximations. We have the fol-
lowing proposition.
Proposition 2: As σ
2
→∞, the LLR in (7) reduces to a form
equivalent to
k
(P
dk
− P
fk
)
z
k
2σ
2
(1 + 2σ
2
)
. (8)
The proof is straightforward by applying first-order Taylor
series expansion for e
−x
and log(1 + x) ≈ x for small x.
With identical local sensors, the above low-SNR approxima-
tion of LLR is equivalent to
Λ
ED
=
k
z
k
(9)
which is termed ED for obvious reasons. This ED comes as
an intuitive detection statistic: From Fig. 2, the more alarmed
sensors, the larger the total received signal power at the fusion
center is.
B. Ricean Fading Channel
If there is a line of sight (LOS) between a local sensor and
the fusion center, the channel is typically modeled as a Ricean
fading channel. The channel gain can be written as
A
k
e
jθ
k
+ w
k
where A
k
e
jθ
k
denotes the “LOS component” and w
k
denotes
the “diffuse component,” assumed to be zero-mean complex
Gaussian with variance σ
2
w
.
Assuming u
k
=1when H
1
is decided, the observation at the
fusion center is
y
k
=
n
k
, the kth sensor decides H
0
A
k
e
jθ
k
+ v
k
, the kth sensor decides H
1
(10)
where v
k
= w
k
+ n
k
is complex Gaussian with zero mean and
variance σ
2
w
+2σ
2
.
Recognizing that (10) is essentially in the same form as (1)
with h
k
, φ
k
, and n
k
replaced with A
k
, θ
k
, and v
k
respectively,
it is straightforward to write out the corresponding LLR in a
similar form as (3). Furthermore, one can show, i n the same
spirit that (4) was derived, that the low-SNR approximation,
assuming identical local sensors, is
Λ
WED2
=
k
A
2
k
z
k
.
This is similar to the WED statistic in (4) except A
k
is the
envelope of LOS, not of the overall channel. Since the LOS
component is typically stationary, A
k
can be easily acquired
through temporal accumulation.
C. Nakagami Fading Channel
Another commonly used flat fading model is the Nakagami
fading channel, which is more general than Rayleigh and
Ricean fading. With unit power assumption, the Nakagami fad-
ing channel has an envelope distribution of the form
P (h
k
)=
2(m)
m
h
2m−1
k
Γ(m)
e
−mh
2
k
where m ≥ 1/2. Therefore
p(z
k
|u
k
=0)=
1
2σ
2
e
−
z
k
2σ
2
P (z
k
|u
k
=1)=
∞
0
p(z
k
|h
k
,u
k
)p(h
k
)dh
k
=
∞
0
1
2σ
2
I
0
h
k
σ
2
√
z
k
× e
−
h
2
k
+z
k
2σ
2
2(m)
m
h
2m−1
k
Γ(m)
e
−mh
2
k
dh
k
=
m
m
Γ(m)
e
−
z
k
2σ
2
2σ
2
×
∞
i=0
z
i
k
(i + m − 1)!(2σ
2
)
m−i
(i!)
2
(1 + 2σ
2
m)
i+m
where we used (5) in the integration.
In the same spirit as in (2), the LLR under the Nakagami
fading can be constructed given the above conditional proba-
bilities. Unlike the case with the Rayleigh and Ricean fading
2672 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 6, NOVEMBER 2005
Fig. 4. Probability of detection as a function of channel SNR for Rayleigh
fading channels.
channels, the LRT for the Nakagami case involves series with
infinite terms, hence, do not have a closed-form expression. We
show next, however, that at low-channel SNR, i.e., σ
2
→∞,
and with identical local sensors, the LLR again reduces
to an ED.
As σ
2
→∞, the resulting LLR can be derived as
Λ=
k
log
P
dk
P (z
k
|u
k
=1)+(1− P
dk
)P (z
k
|u
k
=0)
P
fk
P (z
k
|u
k
=1)+(1− P
fk
)P (z
k
|u
k
=0)
=
k
log
1+(P
dk
− P
fk
)
m
m
Γ(m)
×
∞
i=1
z
i
k
(i + m − 1)!(2σ
2
)
m−i
(i!)
2
(1 + 2σ
2
m)
m+i
(a)
≈
k
(P
dk
− P
fk
)
m
m
Γ(m)
∞
i=1
z
i
k
(i + m − 1)!(2σ
2
)
m−i
(i!)
2
(1 + 2σ
2
m)
m+i
(b)
≈
k
(P
dk
− P
fk
)
m
m
(m − 1)!
z
k
m!(2σ
2
)
m−1
(1 + 2σ
2
m)
m+1
(c)
≈
k
P
dk
− P
fk
4σ
4
z
k
where all approximations stem from the fact that σ
2
→∞.
Specifically, we used log(1 + x) ≈ x for small x in (a), kept
only the first term in the inner sum in (b), and used (1 +
2σ
2
m)
m+1
≈ (2σ
2
m)
m+1
in (c). Again, with identical sen-
sors, the test statistic reduces to the intuitive ED in (9). An
interesting fact is that this low-SNR approximation is not a
function of the Nakagami s hape parameter m. Furthermore, this
result is the same as that of the Rayleigh fading channel, which
is not surprising, considering that Rayleigh fading is a special
case of Nakagami with m =1.
Fig. 5. Probability of detection as a function of channel SNR for Ricean
fading channels.
IV. PERFORMANCE EVA LUAT I ON
Figs. 4–6 show the simulation results of the detection proba-
bility as a function of channel SNR for various fusion statistics
under the Rayleigh, Ricean, and Nakagami fading channels,
respectively. We also include the coherent LRT assuming the
knowledge of the channel phase information [4]. This coherent
LRT provides uniform performance bound among all detection
statistics. The system false alarm rate at the fusion center is
fixed at P
f0
=0.01. In all examples, the total number of sensor
is 8 with sensor level P
fk
=0.05 and P
dk
=0.5. Some remarks
are in order.
1) In the Ricean fading case, the LOS envelope A
k
is gen-
erated randomly from a uniform distribution U(A − ∆,
A +∆), with A satisfying A
2
/σ
2
w
=1; i.e., the average
Ricean factor is chosen to be 1. Specifically, we choose
A =1, σ
2
w
=1, and ∆=0.2. The variation in A
k
models
the discrepancy of LOS strength for different sensors due
to dispersive geographical locations.
2) In the Nakagami fading case, we choose m =2.
3) From the NP lemma, it is clear that the LR-based fusion
rule provides the best detection performance. Among the
three LRTs, the performance degrades as the prior infor-
mation utilized in each LRT decreases. Thus, coherent
LRT performs better than incoherent LRT using channel
fading envelope, which, in turn, is better than incoherent
LRT using only the fading statistics.
4) As SNR decreases, the two incoherent LRTs approach
their respective l ow-SNR approximations in all cases.
2
5) In all cases, ED and WED suffer performance loss at
high SNR. This is not surprising, given that these alter-
natives are only low-SNR approximations of the optimal
LR-based fusion statistics.
2
We do not include the LRT using fading statistics for the Nakagami fading
in Fig. 6 as it involves infinite sum. While one can truncate the infinite sum, we
notice that the low-SNR approximation itself is already a trundated version.
