Wireless Sensor Network, 2009, 1, 458-466
doi:10.4236/wsn.2009.15055 Published Online December 2009 (http://www.scirp.org/journal/wsn).
Copyright © 2009 SciRes. WSN
A Mobile-Agent-Based Adaptive Data Fusion Algorithm
for Multiple Signal Ensembles in
Wireless Sensor Networks
Tianjing WANG1, Zhen YANG2, Guoqing LIU1
1
College of Science, Nanjing University of Technology, Nanjing, China
2Institute of Signal and Information Processing, Nanjing University of Posts&Telecommunications, Nanjing, China
Email: tjingwang@126.com
Received June 8, 2009; revised August 17, 2009; accepted August 18, 2009
Abstract
Distributed Compressed Sensing (DCS) is an emerging field that exploits both intra- and inter-signal correla-
tion structures and enables new distributed coding algorithms for multiple signal ensembles in wireless sen-
sor networks. The DCS theory rests on the joint sparsity of a multi-signal ensemble. In this paper we propose
a new mobile-agent-based Adaptive Data Fusion (ADF) algorithm to determine the minimum number of
measurements each node required for perfectly joint reconstruction of multiple signal ensembles. We theo-
retically show that ADF provides the optimal strategy with as minimum total number of measurements as
possible and hence reduces communication cost and network load. Simulation results indicate that ADF en-
joys better performance than DCS and mobile-agent-based full data fusion algorithm including reconstruc-
tion performance and network energy efficiency.
Keywords: Wireless Sensor Networks, Mobile Agent, Compressed Sensing, Distributed Compressed Sensing,
Joint Sparsity, Joint Reconstruction
1. Introduction
Wireless Sensor Networks (WSN) are an emerging
technology that promises an ability to monitor the
physical world via spatially distributed networks of small
and inexpensive wireless sensor nodes that have the abil-
ity to self-organize into a well-connected network. The
communication tasks consume the limited power avail-
able at such sensor nodes and, therefore, in order to en-
sure their sustained operations, the power consumption
must be kept to minimum. Different from transmission
cost, the computational cost may be negligible for some
applications. For example, WSN monitoring field tem-
perature may use simple functions which essentially are
of insignificant cost. Consequently, a major challenge in
the design of WSN is developing schemes that extract
relevant information about the sensor data with the
minimum energy consumption, especially with transmis-
sion cost. In order to reduce transmission of sensor data,
a new framework for single signal sensing and compres-
sion has developed recently under the rubric of Com-
pressed Sensing (CS) [13]. The implications of CS are
promising for many applications, especially sensing sig-
nals that have a sparse representation in some basis [46].
Based on the intra-signal correlations (between samples
in each signal), CS can perfectly reconstruct a com-
pressible signal from remarkably few linear measure-
ments. In WSN, however, a large number of sensor
nodes presumably observe related phenomena and are
programmed to perform a variety of signals acquisition
tasks. These signals have high correlations and need to
be jointly processed, so the independently encoding and
decoding theory and practice of single signal in a CS
framework can not satisfy such applications. Fortunately,
the ensemble of signals that sensor nodes acquired can be
expected to possess some joint structure, or inter-signal
correlations (between samples across signals), in addition
to the intra-signal correlation of single signal. Most ex-
isting coding algorithms [7,8], however, exploit only
inter-signal correlations and not intra-signal correlations.
A new Distributed Compressed Sensing (DCS) theory
exploits both intra-signal and inter-signal correlation
structures of multiple signal ensembles [911]. In a typi-
cal DCS scenario, each individual node independently
encodes its signal by CS framework and then transmits
*Supported by the Young Teacher Academic Fund of Nanjing Univer-
sity of Technology
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
459
the measurements to a distant Sink node (Sink) by multi-
ple skips. Under the right conditions, Sink can jointly
reconstruct multiple signal ensembles precisely. How-
ever, we note that the network traffic may be extremely
heavy in DCS, resulting in poor performance of the sys-
tem, when the number of sensor nodes is big and the
amount of sensing signals is large. Furthermore, nodes
can not know the suited minimum number of measure-
ments that they need to transmit to Sink under DCS
framework. In order to guarantee perfect reconstruction,
each node has to transmit enough measurements. This
means that DCS only utilizes the inter-signal correlations
in jointly decoding processes, but not in jointly encoding
processes.
In this paper, we design a mobile-agent-based Adap-
tive Data Fusion (ADF) algorithm for multiple signal
ensembles which is inspired by DCS. Instead of passing
large amounts of independently encoding measurements
by DCS in each individual node over the network, Mo-
bile Agent (MA) can fuse multiple signals from node to
node along a shortest path, based on Global Closest First
(GCF) heuristics algorithm [12]. According to the sparse
property of single signal and the joint sparsity of multiple
signal ensembles under the results of data fusion, each
node can determine the minimum number of measure-
ments needed to transmit to Sink and effectively reduces
the transmission cost. Extensive experiments in session 4
demonstrate that the energy efficiency and the recon-
struction performance of ADF are more excellent than
DSC and mobile-agent-based Full Data Fusion (FDF)
algorithm.
The organization of this paper is as follows. Section 2
overviews a joint sparsity model in DCS. Single signal
and multiple signals reconstruction methods are dis-
cussed respectively. Section 3 introduces our two mo-
bile-agent-based data fusion algorithms: Full Data Fu-
sion (FDF) and Adaptive Data Fusion (ADF). Detailed
theoretical analyses indicate that FDF and ADF are more
energy efficient than DCS, sufficiently utilizing intra-
signal and inter-signal correlation of multiple signal en-
sembles. Experiments in Section 4 confirm that ADF
indeed reduces large amounts of total transmission cost
and the number of total measurements compared to DCS
and FDF. Section 5 describes conclusions.
2. Distributed Compressed Sensing
2.1. Joint Sparsity Model
The recently introduced theory of DCS enables a new
distributed coding algorithm that exploits both intra- and
inter-signal correlation structures of multiple signals [9].
In this paper, we focus on a Joint Sparsity Models (JSM,
sparse common component +innovations), which can be
improved by mobile-agent-based data fusion. For exam-
ple, a practical situation well-modeled by the JSM is a
group of sensors {Sl
,SJ
} measuring temperatures at a
number of outdoor locations throughout the day. The
temperature readings xl(l{1,2,J}) have both tempo-
ral (intra-signal) and spatial (inter-signal) correlations.
Global factors, such as the sun and prevailing winds,
could have an effect that is both common to all sensors
and structured enough to permit sparse representation in
some basis. More local factors, such as shade, water, or
animals, could contribute localized innovations that are
also structured (and hence sparse in some basis). A simi-
lar scenario could be imagined for a network of sensors
recording light intensities, air pressure, or other phe-
nomena. All of these scenarios correspond to measuring
properties of physical processes that change smoothly in
time and in space and thus are highly correlated.
We adopt language and notation from [9]. Assume that
there exists a known basis
{|,1,,}
m
ii
Rim
ψψΨ=∈=
L
in
which a signal
Tm
lll
xxxmRlJ
=∈∈
LL
can be sparsely represented as
ll
x
θ
, where
((1),,())
T
lll
mθθθ=
L
is a sparse coefficient vector and
0
||||
l
k
θ
=
. Thus, a compressible multiple signal ensem-
ble 1
,,
J
xx
L
shares a common sparse component while
each single signal contains an innovation sparse compo-
nent. That is
ˆ
lCl
θθθ
=+
{1,2,,}
lJ
L
(1)
with
0
,||||
CCCC
xk
θθ
=Ψ=
and
0
ˆ ˆ
ˆ,||||
llll
xk
θθ
=Ψ=
()
lC
kk
< (2)
where kc is a common sparse parameter of θc and kl is an
innovation sparse parameter of θl.
Thus, the signal xc is common to all of
({1,2,,})
l
xlJ
L
and has sparse coefficient vector θc in
basis ψ, and the signal ˆ
({1,2,,})
l
xlJ
L
is the unique
portions of xl and has sparse coefficient vector
ˆ
l
θ
in the
same basis. Denote that Ωc is a tight support set of the
nonzero values in θc and Ωl is a tight support set of the
nonzero values in
ˆ
l
θ
.
To make linear measurements, denote the measure-
ment matrix
()
l
lijnm
ϕ
×
Φ=
({1,2,,})
lJ
L
for the mul-
tisignal ensembles, where a second basis matrix Φl is inco-
herent with Ψ. Thus, a small number of noiseless measure-
ments
((1),
llll
yxy=Φ=
,())({1,2,,})
T
ll
ynlJ
LL
contain sufficient information for approximate reconstru-
ction [1,2]. Mathematically, this can be reduced to a
standard linear algebra problem: we wish to find
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
460
({1,2,,})
l
xlJ
L
which explains the measurements
lllll
yx
θ
=Φ=ΦΨ
({1,2,,})
lJ
L
.
2.2. Joint Reconstruction of Multiple Signal
Ensembles via
1
l
Optimization
To give ourselves a firm footing for analyses, we firstly
consider single signal reconstruction based on the CS
theory, mainly to exploit intra-signal structure at a single
node. CS encodes a workable approximation of the sin-
gle compressible signal
({1,2,,})
l
xlJ
L
with l
nm
=
sampling resources and signal reconstruction can be
achieved by Basis Pursuit (BP) [1,2] or Orthogonal
Matching Pursuit (OMP) [3]. OMP allows faster recon-
struction at the expense of more measurements.
Formally, BP needs to solve the following
1
l
optimi-
zation problem
(P1)
1
min||||
l
θ
subject to
lllll
yx
θ
=Φ=ΦΨ
({1,2,,})
lJ
L
(3)
The simulation results in [2] state that there exists meas-
urements n ck (oversampling factor c 4) are required
to reconstruct
l
x
with high probability, using linear
programming methods e.g. interior point method and
simplex method. That is, an optimal reconstructed signal
**
ll
x
θ
can be achieved by an optimal sparse solution
*
l
θ
of the problem (P1).
In addition to single signal encoding and decoding
methods, the jointly encoding and decoding methods of
multiple signal ensembles are considered in DCS. DCS
expects that 1)
()
Cl
ckk
+
({1,2,,})
lJ
L
measurements
suffice to reconstruct x1, 2)
1
()
J
Cl
l
ckk
=
+
measurements
suffice to reconstruct multiple signal ensembles x1,..., xJ,
for there exists
1
J
Cl
l
kk
=
+
nonzero elements in x1,..., xJ
.
Furthermore, the recovery problem can be formulated
using matrices and vectors as
1
C
J
θ
θ
θ
θ



=



M
,
1
J
x
x
x


=



M
,
1
J
y
y
y


=



M
,
1
0
0
J
Φ


Φ=


Φ

L
%O
L
,
0
0
ΨΨ


Ψ=


ΨΨ

L
%MO(4)
In order to sufficiently utilize inter-signal correlation of
multiple signal ensembles, we assume that
1J
Φ==Φ
L
and then
Φ
%
can be rewritten as
(,,)
diag
Φ=ΦΦ
%
L
. It is possible to let Sink previously
send the same random seed to all sensor nodes in the
interesting field and then the same pseudorandom matrix
Φ can be generated using simple algorithm with seed at
each node.
With sufficient sampling, DCS can reconstruct multi-
ple signal ensembles by solving the following
1
l
opti-
mization problem
(P2)
1
min||||
θ
subject to y
θ
=ΦΨ
%
%
(5)
Due to the optimal spare solution
*
θ
of the problem
(P2), we can get the corresponding reconstructed multi-
signal ensembles
**
x
θ
%
.
3. Mobile-Agent-Based Adaptive Data
Fusion Algorithm
The DCS theory proposes a framework for joint recon-
struction of compressible multi-signal ensembles. How-
ever, each single node independently encodes in DCS,
which does not sufficiently utilize the joint sparsity of
multi-signal ensembles. This operation makes each node
have to transmit a lot of measurements. In this session,
we focus on reducing measurements required to transmit
at each node by mobile-agent-based data fusion in WSN.
A WSN under a DCS framework, as shown in Figure
1, consists of three types of components: Sink, sensor
nodes and communication network. With energy restric-
tion, sensor nodes can not directly communicate with
Sink. For example, the encoding results of a node Sl+1 in
the interesting field are transmitted to Sink by multi skips
routing in Figure 1. Other nodes do the same works. In
Figure 1, we model a WSN as a graph G=(V,E), where
S
d
1
S
2
S
1
J
S
J
S
12
d
1,
JJ
d
2
J
S
3
S
l
S
1
l
S
+
,1
ll
d
+
S
d
Figure 1. A WSN under DCS framework. Circles denote
sensor nodes. The communication distance between node
l
S
and node 1
({1,,1})
l
SlJ
+
∈−
L
is
,1
ll
d
+
and the communica-
tion distance between Sink and
l
S
is approximately as
S
d
by the shortest multiple skips routing. For being simple,
other links representing the communication links among
nodes and Sink are omitted. On the other hand, a route
1
{,,,,,}
SJS
SSSS
LL
represents MA data fusion routing.
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
461
1
{,,,}
SL
VSSS
=
L
and E denotes an edge set represent-
ing the communication links between node-pairs or links
between Sink and nodes. In many applications, a distant
Sink
S
S
retrieves relevant interesting field information
from the sensor nodes. Let
({1,,})
Sl
dlL
L
be the total
communication distance between Sink and a node
({1,,})
l
SlL
L
by the shortest multiple skips routing.
For being simple, Sink is assumed to be far away from
the interesting field so that 1
SSLS
ddd
≈≈
L
and
,1
({1,,1})
Sll
ddlL
+
∈−
?L
, where
,1
ll
d
+
denotes the com-
munication distance between node-pairs
l
S
and
1
l
S
+
[13].
We next describe the communication architecture of
WSN in Figure 1 to motivate the formulation of two data
fusion algorithms for multiple signal ensembles. Assume
that a connected subgraph '''
(,)
GVEG
=⊆
is found,
where
'
V
contains Sink and encode nodes, i.e.,
'1
{,,,}
SJ
VSSSV
=⊂
L
(,)
IHJL
<<
, and
'
E
de-
notes a edge set representing all communication links in
'
V
. For a node
({1,,})
l
SlJ
L
, its node weight
()
l
wS
denotes the amount of data outgoing from
l
S
. An edge
({1,,1})
l
eElJ∈−
L
is denoted by
1
(,)
lll
eSS
+
=.
The weight of edge
l
e
is equivalent to the weight of
l
S
,
i.e.,
()()
ll
wewS
=. Two metrics,
()
l
te
and
()
l
fe
, are
associated with each edge, describing the transmission
cost and fusion cost on the edge, respectively. In many
WSN applications, however, the fusion cost may be neg-
ligible. For being simple, we do not consider fusion cost
in this paper.
The unit cost of the link for transmitting data from S1
to
1
l
S
+
is abstracted as ,1
() r
lll
ced
βε
+
=+
, where β and r are
tunable parameters based on the radio propagation [14].
Thus the transmission cost
()
l
te
is
()()()
lll
tecewe
=
(6)
Similarly, we approximately define
({1,,})
Sl
elJ
L
is
the communication links between a node Sl and Sink by
the shortest multiple skips routing, and then the trans-
mission cost is approximately as
()()
SlSl
tece
=
()({1,,})
Sl
welJ
L
, where
()
Sl
ce
and
()
Sl
we
are the
unit transmission cost and the weight of
Sl
e
. Obviously,
we can get
()()
Sll
cece
?
, for
,1
Sll
dd
+
?
.
From above definitions, the total network energy con-
sumption in
'''
(,)
GVE
= with different computing
models can be calculated. We firstly consider total net-
work energy consumption of DCS. According to encod-
ing method of a single signal, we assume that individual
node
({1,,})
l
SlJ
L
can gain a sparse coefficient vec-
tor
({1,,})
l
lJ
θ
L
by projecting a sensing signal
({1,,})
l
xlJ
L
into a basis matrix
Ψ
. This operation
may consume some computational energy, but it does not
affect the total network energy consumption and can be
omitted compared to transmission cost.
To conveniently explain the network energy consump-
tion, we assume that Cl
=∅
. Single signal recon-
struction, therefore, needs
l
n
measurements with nl=
c(kc+kl) in a CS framework, and then the total network
energy consumption of DCS can be calculated as follows
11
()()()()
JJ
r
DCSSlSlSCl
ll
Ccewedckk
βε
==
==++
∑∑ (7)
where the number of measurements
()
lCl
nckk
=+
di-
rectly denotes the weight
()
Sl
we
.
We consider the network energy consumption of two
mobile-agent-based data fusion algorithms. Generally
speaking, MA is a special kind of software with small
size, whose transmission cost can be omitted compared
to transmission cost of larger amount of sensing informa-
tion. We will not consider MA transmission cost. Sink
predetermined the MA routing 1
{,,,,,}
SJS
SSSS
LL
by GCF. The detailed mobile-agent-based data fusion
process is presented as follows.
In initialize stage, MA migrates toS1and obtains meas-
urements y1 with
110
||||
nc
θ
=. Subsequently, it migrates
to S2 and finds θc
1
ˆ
θ
and
2
ˆ
θ
by contrasting sparse coef-
ficients θ1 and θ2. Thus, it can calculate the common
measurements yc corresponding to the common sparse
component θ1 of the signal ensembles. This means that
measurements y1 and y2 can be divided into
11
ˆ
C
yyy
=+
and
22
ˆ
C
yyy
=+
, where
1
ˆ
y
and
2
ˆ
y
correspond to the
sparse innovation component
1
ˆ
θ
and
2
ˆ
θ
. MA carrying
measurements
C
y
,
1
ˆ
y
and
2
ˆ
y
with 2
()(
C
weck
=+
12
)
kk
+ migrates to
3
S
. We then repeat above process
on remaining nodes along MA routing until MA returns
to Sink. Such process is a typical mobile-agent-based
data fusion process. In this paper, we term this process a
mobile-agent-based Full Data Fusion (FDF) algorithm
compared to the following adaptive data fusion process.
The total network energy consumption of FDF is shown
as follows
1
1
1
,11
1
1
()()()()
()()
()()
J
FDFllSJSJ
l
Jr
llCl
l
r
SCJ
Ccewecewe
dckkk
dckkk
βε
βε
=
+
=
=+
=++++
+++++
L
L
(8)
We interest in comparing the total energy consumption
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
462
of DCS and FDF. The total transmission cost can be cal-
culated as follows 1
,1
1
()(1)()
J
rr
DCSFDFSClll
l
CCdJckdck
βεβε
+
=
=+−+
1
,1
1
(()())
Jrr
SClll
l
dckdck
βεβε
+
=
=+−+
(9)
Note that
,1
Sll
dd
+
?
and
Cl
kk
>
, it is easy to get
DCSFDF
CC> (10)
As expected, the inequality (10) means that the total
transmission cost of DCS is much larger than FDF. By
analyzing FDF in detail, we find another challenge that
MA will carry more and more amount of data fusion
results along the MA routing. However, data fusion at a
node
l
S
({2,,})
lJ
L
only needs the common meas-
urements
C
y
. If MA still carries all front innovation
measurements
11
ˆ
ˆ
,,
l
yy
L
, it is not in favor of saving
communication cost. This result brings a question
whether we can avoid transmitting an amount of the in-
novation measurements. In this regard, we establish the
following data fusion algorithm named mobile-agent-
based Adaptive Data Fusion (ADF) algorithm. Different
from in FDF, MA only carries the common measure-
ments
C
y
in ADF. Concretely, MA can calculate
ˆ
l
θ
({2,,})
lJ
L
by contrasting
C
θ
and
l
θ
({2,,
l
L
})
J
, when it carrying
C
y
migrates to
l
S
({2,,})
lJ
L
. This allows
l
y
({2,,})
lJ
L
to be
divided into
ˆ
lCl
yyy
=+
in which the number of the
innovation measurement
ˆ
l
y
is ˆ
ll
nck
=. MA carrying
measurements
C
y
with ()
lC
weck
= continuously mi-
grates to
1
l
S
+
. On the other hand, measurements
ˆ
l
y
are
directly transmitted to Sink. We then repeat above proc-
ess on remaining nodes along the same MA routing as
FDF until MA returns to Sink. Then, the total network
energy consumption of ADF is expressed as follows
121
1
,1
2
()()
(()())()()
r
ADFC
J
rrr
llCSlSCJ
l
Cdckk
dckdckdckk
βε
βεβεβε
+
=
=++
+++++++
(11)
We also attend to compare the total energy consumption
of FDF and ADF as follows
1
,1
1
()0
Jr
FDFADFlll
l
CCdckβε
+
=
=+>
(12)
i.e.,
FDFADF
CC> (13)
From (10) and (13), it can be shown that
DCSFDFADF
CCC>> (14)
From (14), we can easily obtain the benefits of ADF.
Firstly, based on DCS, ADF also sufficiently utilizes
both temporal (intra-signal) and spatial (inter-signal)
correlations of multi-signal ensembles to analyze single
signal sparsity structure and multi-signal ensembles
jointly sparsity structure. These sparsity structures make
it possible to perform data fusion of multi-signal ensem-
bles. Second, the innovation measurements are allowed
to directly transmit to Sink, while MA only carries the
common measurements. It benefits reducing transmis-
sion cost. So we can say that ADF provides the optimal
strategy for minimizing total transmission measurements
and transmission cost compared to DCS and FDF.
According to the above discussion, we can reconstruct
multi-signal ensembles by solving the following
1
l
op-
timization problem
(P3)
1
min||||
θ
subject to
ˆ
ˆ
ˆ
y
θ
=ΦΨ
(15)
where
1
C
J
θ
θ
θ
θ



=



M
, 1
ˆ
ˆ
ˆ
C
J
y
y
y
y



=



M
, 1
00
00
ˆ
00
C
J
Φ


Φ

Φ=


Φ

L
L
MLOM
L
,
0
ˆ
0
Ψ


Ψ=


Ψ

L
MOM
L
.
We can obtain ****
1
ˆ
ˆ
(,,,)
T
CJ
xxxx
=
L
with an optimal
sparse solution ****
1
(,,,)
T
CJ
θθθθ
=
L
, where
**
CC
x
θ
and **
ˆ
ˆ
,({1,,})
ll
xlJ
θ=Ψ∈
L
. Furthermore, multi-signal
ensembles can be reconstructed by
***
ˆ
,
lCl
xxx
=+
({1,,})
lJ
L
.
The above results focus on theoretical analyses of sav-
ing transmission cost in ADF. On the other hand, we are
interested in comparing the joint reconstruction per-
formance of DCS, FDF and ADF. The following simula-
tion results are presented illustrating the better joint re-
construction performance of ADF.
4. Simulation
In our setup, sensor nodes are randomly distributed in a
region of a
5050
mm
×
square. The distance between
Sink and the interesting field is
400
S
dm
=. When con-
sidering transmission cost, we set
2
100//
pJbitm
β=,
2
r
=
and
100/
nJbit
ε= [12] in (6). Furthermore, we
consider a series of example multiple signal ensembles
1
,,
J
xx
K
that satisfy the conditions of joint sparsity
model. The signal components 1
ˆ
ˆ
,,,
CJ
xxx
K
are as-
sumed to be sparse in Discrete Cosine Transform (DCT)
matrix
Ψ
with sparse parameters 1
,,,
CJ
kkk
K
, re-
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
463
46810
2
3
4
5
6
7
x 10
7
Number of nodes J
(a) The total transmission cost
Transmission cost (nJ)
468 10
1
2
3
4
5
Number of nodes J
(b) The relative decreasing rate
The relative decreasing rate (%)
468 10
0
2
4
6
8
10
12 x 10
-9
Number of nodes J
(c) Joint reconstruction error
Joint reconstruction error
468 10
100
200
300
400
500
Number of nodes J
(d) The total number of measurements
Number of measurements
DCS
FDF
ADF
The relative decreasing rate
DCS
ADF&FDF
DCS
ADF&FDF
Figure 2. Effect of the number of nodes on joint reconstruction performance of multiple signal ensembles. We choose signals
with
50
m
=
,
10,
C
k
=
4({1,,})
l
klJ
=∈
L
and
C
n
440,
C
k
==
ˆ
416
ll
nk
==
({1,,})
lJ
L
. (a) The total network transmission
cost, (b) The relative decreasing rate, (c) Joint reconstruction error, (d) The total number of measurements.
20 3040 50 60
0.5
1
1.5
2
2.5
3
3.5
4
x 10
8
Number of nodes J
(a) The total transmission cost
Transmission cost (nJ)
20 3040 50 60
10
12
14
16
18
20
22
24
26
Number of nodes J
(b) The relative decreasing rate
The relative decreasing rate (%)
DCS
FDF
ADF
The relative decreasing rate
Figure 3. Effect of the number of nodes on energy consumption of multiple signal ensembles. We choose signals
with
50
m
=
,
10,
C
k
=
4({1,,})
l
klJ
=∈
L
and
4,
CC
nk
=
ˆ
4({1,,})
ll
nklJ
=∈
L
. (a) The total transmission cost, (b) The relative
decreasing rate.
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
464
spectively. We assign random Gaussian values to the
nonzero coefficients 1
ˆ
ˆ
,,,
CJ
θθθ
K
, and the locations of
nonzero are chosen at random. As a measure of the re-
construction performance, the joint reconstruction error
*
2
1
||||
J
ll
l
exx
=
=−
is designed. The Interior Point (IP)
method in Matlab is used to solve the problem (P2) and
the problem (P3).
Our first experiment chooses signals with the length
50
m
=
and sparse parameters
10,
C
k=
4({1,,})
l
klJ
=∈
L
.
Then the corresponding numbers of measurements are
chosen by
40,
CC
nck==ˆ
16({1,,})
ll
ncklJ
==∈
L
,
where
4
c
=
. Without loss of generality, assume that one
measurement produces 8 bit packet [12]. With increasing
number of nodes
J
, the total number of measurements of
DCS is greatly larger than FDF and ADF in Figure 2(d).
This result causes the transmission cost of DCS also
greatly larger than FDF and ADF in Figure 2(a). To fur-
ther illustrate the advantage of ADF, we consider the rela-
tive decreasing rate calculated as
TC(FDF)-TC(ADF)
TC(FDF)
τ=
in which TC(FDF) and TC(ADF) are the total transmis-
sion cost of FDF and ADF, respectively. Figure 2(b)
clearly shows that the relative decreasing rate linearly
increases with J. This means that the energy efficiency of
ADF is more distinctness with increasing number of
nodes. In Figure 2(c), we emphasize on comparison of
reconstruction performance between DCS and ADF
(FDF). ADF and FDF enjoy less joint reconstruction
errors than DCS, though ADF and FDF use less number
of measurements than DCS. So we can say that ADF per-
forms much better than DCS and FDF.
WSN typically consists of a large number of sensor
nodes, so we need consider energy consumption of much
more nodes in DCS, FDF and ADF to further observe the
advantage of ADF. We repeat the first experiment while
the number of nodes varies from 20 to 60. As expected,
the transmission cost of DCS is further larger than FDF
and ADF as
J
increases in Figure 3(a). Comparing
Figure 3(b) with Figure 2(b), we note that the relative de-
creasing rate scales linearly with
J
. The energy savings
of ADF can be as large as 27%. These results identify that
ADF is an optimal strategy with minimum total number
of measurements and total transmission cost, which con-
sist with the front theoretical conclusions in Section 3.
Experiments in Figure 3 bring another question
whether we can guarantee better joint reconstruction
performance as the number of nodes J increases. In our
joint decoding simulations, we find that computational
time and complexity will greatly increase as J increases.
So measurements in Sink should be grouped according to
applications. This operation consists with the idea in [9].
In the next experiment, we use J=40 nodes and their
measurements are separated in 5 groups. Average recon-
struction error 5
1
()/5
t
t
ee
=
=
is designed to measure the
456789
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
x 10
8
Value of k
C
(a) The total transmission cost
Transmission cost (nJ)
DCS
FDF
ADF
456789
1
2
3
4
5
6
7
x 10
-9
Value of k
C
(b) Average reconstruction error
Average reconstruction error
DCS
ADF&FDF
4 567 8 9
600
800
1000
1200
1400
1600
1800
2000
2200
Value of k
C
(c) The total number of measurements
Number of measurements
DCS
ADF&FDF
Figure 4. Effect of the number of common sparse parameters kc on joint reconstruction of multiple signal ensembles. We
choose
50
m
=
, fix
3({1,,})
l
klJ
=∈
L
, and vary common kc from 4 to 9 and then choose ˆ
4,4({1,,})
CCll
nknklJ
==∈
L
. (a) The
total transmission cost, (b) Average reconstruction errors, (c) The total number of measurements.
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
465
2 4 6
0.5
1
1.5
2
2.5
3
3.5 x 10
8
Value of k
l
(a) The total transmission cost
Transmission cost (nJ)
24 6
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
-7
Value of k
l
(b) Average reconstruction error
Average reconstruction error
246
500
1000
1500
2000
2500
Value of k
l
(c) The total Number of measurements
Number of measurements
DCS
FDF
ADF
DCS
ADF&FDF
DCS
ADF&FDF
Figure 5. Effect of the number of innovation sparse parameters
l
k
on joint reconstruction of multiple signal ensembles. We
choose
50
m
=
, fix
8
C
k
=
and vary
({1,,})
l
klJ
L
from 1 to 7, and then choose
4,
CC
nk
=
ˆ
4({1,,})
ll
nklJ
=∈
L
. (a) The total
transmission cost, (b) Average reconstruction errors, (c) The total number of measurements.
reconstruction performance, where
(1,,5)
t
et=
L
is the
joint reconstruction error of every group. We perform
experiments with the length of signals
50
m
=
and in-
novation sparse parameters
3
l
k
=
({1,,})
lJ
L
, while
the common sparse parameter kc varies from 4 to 9. The
obtained results in Figure 4 (a)-(c) show the similar con-
clusions in Figure 2 and Figure 3. The number of com-
mon sparse parameters kc greatly affects the total trans-
mission cost of DCS while not FDF and ADF, for DCS
need each node to transmit the common measurements
while FDF and ADF avoid this operation by data fusion.
This sufficiently reveals the advantage of mobile-agent-
based data fusion algorithms. Moreover, the average re-
construction error of DCS, FDF and ADF decrease bene-
fiting from the increasing number of measurements. This
means joint reconstruction performance can be improved
by increasing the total number of mea- surements.
The finally experiments focus on effect of the number
of innovation sparse parameters kl on joint reconstruction
of multiple signal ensembles. We repeat the front ex-
periments with the length of signals m=50 and the com-
mon sparse parameter kc=8, while the innovation sparse
parameter kl (l{1,J}) varies from 1 to 7. As kl (l
{1,J}) increasing, the total transmission cost and the
number of measurements of FDF and ADF fleetly in-
crease compared to Figure 4. Figure 5(a) and Figure 5(c)
reveal that the performance of data fusion is influenced
by the innovation sparse parameters kl, for kl represent
the differences among multi-signal ensembles. At the
expense of more measurements and energy cost, we can
obtain multi-signal ensembles with more details. At the
same time, we gain the better joint reconstruction per-
formance in Figure 5(b), for the total number of meas-
urements increases.
As can be seen, above experiments imply that ADF
sufficiently takes advantage of intra- and inter-signal
correlation of multi-signal ensembles by mobile-agent-
based data fusion. So ADF enjoys better performances
than DCS and FDF.
5. Conclusions
Distributed Compressed Sensing (DCS) extends the the-
ory and practice of Compressed Sensing (CS) to multi-
signal ensembles. A joint sparsity model for multi-signal
ensembles with both intra- and inter-signal correlation
captures the essence of real physical scenarios. This pa-
per provides a new mobile-agent-based Adaptive Data
Fusion (ADF) algorithm. ADF can greatly reduce the
total number of measurements for successful joint recon-
struction compared with DCS. Moreover, ADF can
greatly reduce transmission cost and network load. Ex-
tensive experiments demonstrate that it indeed leads to
T. J. WANG ET AL.
Copyright © 2009 SciRes. WSN
466
better performance than DCS and mobile-agent-based
Full Data Fusion (FDF).
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