1
Compressed Sensing based Signal and Data
Acquisition in Wireless Sensor Networks and
Internet of Things
Shancang Li, Li Da Xu, and Xinheng Wang
Abstract—The emerging compressed sensing (CS) theory can
significantly reduce the number of sampling points that directly
corresponds to the volume of data collected, which means that
part of the redundant data is never acquired. It makes it possible
to create stand-alone and net-centric applications with fewer
resources required in Internet of things (IoT). CS-based signal
and information acquisition/compression paradigm combines the
nonlinear reconstruction algorithm and random sampling on a
sparse basis that provides a promising approach to compress
signal and data in information systems. This paper investigates
how CS can provide new insights into data sampling and
acquisition in wireless sensor networks and IoT. At first, we
briefly introduce the CS theory in respect of the sampling and
transmission coordination during the network lifetime through
providing a compressed sampling process with low computation
costs. Then, a compressed sensing-based framework is proposed
for IoT, in which the end nodes measure, transmit, and store the
sampled data in the framework. Then, an efficient cluster-sparse
reconstruction algorithm is proposed for in-network compression
aiming at more accurate data reconstruction and lower energy
efficiency. Performance is evaluated with respect to network size
using datasets acquired by a real-life deployment.
Index Terms—Compressed Sensing, Wireless Sensor Networks,
Industrial Informatics, Internet of Things, Information Systems,
Enterprise Systems
I. INTRODUCTION
Researchers found that in information systems, wireless
sensor networks (WSNs), and Internet of Things (IoT), many
types of information has a property called sparseness in trans-
formation process which allows certain number of samples
enabling capturing all required information without loss of
information [1], [2], [3], [4]. IoT has been emerged as a
technological revolution in the information industry [1], [2].
IoT is expected to be a world-wide network of intercon-
nected objects, and its development depends on a number
of new technologies, such as WSNs, cloud computing, and
information sensing [2], [3], [4]. In IoT-based information
systems, a low-cost data acquisition system is necessary to
Manuscript received December 28, 2011;revised October 26, 2012.
Copyright (c) 2009 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
Shancang Li is with College of Engineering, Swansea University,
Swansea SA2 8PP, UK (Phone: +44-1792-602802, Fax: +44-1792-602802,
Email:s.li@swansea.ac.uk).
Li Da Xu is with the Institute of Computing Technology, Chinese Academy
of Sciences, Beijing 100190, China; Old Dominion University, Norfolk, VA
23529, USA
Xinheng Wang is with College of Engineering, Swansea University,
Swansea SA2 8PP, UK (xinheng.wang@swansea.ac.uk).
effectively collect and process the data and information at IoT
end nodes [2], [3], [5], [6]. WSNs have the potential of a
wide range of applications in many industrial systems. WSNs
can be integrated into the IoT, which consists of a number of
interconnected sensor nodes [3], [4], [5].
An IoT can involve thousands of independent components
including computers, sensors, RFID tags, or mobile phones, all
are capable of generating and communicating data, in which
many techniques are involved for data collection, transmission,
and storage [2], [4], [7]. In IoT, a desirable data compression
ratio is very important, which cannot be obtained by current
methods without introducing unacceptable distortions [8], [9].
Furthermore, for most data compression solutions in IoT, three
main problems must be solved: resolution, sensitivity, and
reliability [2], [10], [11], [12].
Recently, an emerging theory named compressed sensing
(CS) has been extensively investigated, with which the data or
signals can be efficiently sampled and accurately reconstructed
with much fewer samples than Nyquist theory [3], [4], [8],
[13]. CS relies on the facts that many types of information
has a property called sparseness in transformation process. The
required information could be obtained from these compress-
edly sampled signals as well as the whole signals sampled
by Nyquist theory [14]. The CS changes the rule of data
acquisition game in information systems by exploiting a priori
data sparsity information [5]. The applications of CS for data
acquisition in WSNs have been studied recently [4], [15], [16],
[17]. In [4], Haupt investigated the compressed sensing for
networked data in WSNs through considering the distributed
data sources and their sampling, transmission, and storage.
In [15], Fazel proposed a random access compressed sensing
scheme for long-term data gathering in large-scale sensor
networks, which is expected to prolong the life-time of a
sensor network. In [16], Pudlewski et al. applied the CS
theory to the jointly control of the video streaming rates to
reduce the communication cost and increase network capacity.
In [17], Mamaghanian investigated CS for energy-efficient
signals gathering in a wireless body sensor network.
However, for the first time, our work studies information
acquisition in IoT and WSNs with CS from the perspective of
data compressed sampling, robust transmission, and accurate
reconstruction to reduce the energy consumption, computation
costs, data redundancy, and increase the network capacity. A
common task of an IoT end node is to transmit the sensed
data to specific node or fusion center (FC), however, how to
efficiently acquire, store, and transmit among a large number
2
of source nodes remains a challenge [4], [14]. This paper
considers a particular situation that involves with distributed
information sources of data and their acquisition, transmis-
sion, storage, and processing in a large-scale IoT [18]. The
contributions are summarized as follows:
1) We formulate the problem of data acquisition based on
compressed sampling in IoT and WSNs. This is the first
time to apply compressed sampling scheme in IoT with
a theoretical basis.
2) A CS-based information acquisition framework is pro-
posed for IoT, which involves the compressed sampling
at IoT end node, information transmission over IoT, and
accurate data reconstruction at FC. In this framework,
the noise model, communication load, and recovery
accuracy are considered for its industrial applications.
3) By taking the correlation of sensing data over IoT and
WSNs into consideration, an adaptive sparse representa-
tion and corresponding signal reconstruction algorithm
are proposed which offer a higher accuracy and lower
computational complexity compared with pre-existing
group/cluster-sparse reconstruction algorithms.
In the following sections, we will propose a compressed
sensing-based data gathering scheme, in which the compressed
sensing is able to provide a compressed sampling stage with
low computational costs. In Section II, we introduce the main
idea of data compression using compressed sensing. In Section
III, we propose a flexible data acquisition framework for IoT
based on compressed sensing. Experiments of data acquisition
and reconstruction are proposed for IoT to demonstrate the ef-
fectiveness of the proposed approaches in Section IV. Section
V concludes the paper.
II. COMPRESSED SENSING
In a network with n nodes, each node collects or generates
data x
j
, j = 1, . . . , n. For simplicity, we assume that each
sample x
j
is a scalar data (such as temperature, pressure,
etc.) and the collected data is a vector x = [x
1
, . . . , x
n
]
T
,
namely measurements. Thesemeasurements are distributed and
can be shared over the network. IoT may be very large,
and the collection of x at an IoT node might be inefficient
and unreliable. However, compressed sensing theory makes
it possible to accurately reconstruct x based on a highly
compressed decentralized measurement of x [4], [19], [20].
Compressed sensing-based data acquisition is very differ-
ent from the decentralized data acquisition in IoT networks.
Considering a data acquisition model: y = Φx, in which
Φ denotes an m-by-n measurement matrix with m far less
than n. The measurement y is an m × 1 vector, with the
number of elements being far fewer than that of the original
data x. Therefore, y can be transmitted, processed, and stored
with much lower resource requirements than x. In compressed
sensing theory, with a properly designed measurement matrix
Φ, it is possible to recover x from y within a reasonable
accuracy whenever x is compressible.
y = Φx (1)
Compressed sensing is able to measure the data without
requiring any specific prior knowledge [4], [21], [22]. The
required data over the whole network can be reconstructed
based on the measurements as described in Eq.(1), providing
its size m is much smaller than n [23]. In compressed
sensing-based WSNs and IoT, two features can be obtained
for effective data analysis: (1) The compressed sensing-based
method is able to work cooperatively between the nodes,
which means that the collected or generated data by each node
can be distributively processed even without a fusion centre
(FC); (2) The data can be sampled and reconstructed without
prior knowledge. These two features make the compressed
sensing easier to be used for applications where gathering data
is expensive.
A. Conditions for Compressed Sensing
Definition 1: If a signal x = [x
1
, . . . , x
n
]
T
can be repre-
sented over a set of orthonormal n × 1 vector {ψ
i
}
n
i=1
, then
it can be said sparse
x =
n
i=1
θ
i
ψ
i
, or, θ = Ψ
T
x, (2)
in which θ is an n × 1 vector that denotes the weights vector,
and θ
i
=< x, ψ
i
>; Ψ = [ψ
1
, . . . , ψ
n
] is the basis matrix. If
there are k (k ≪ n) nonzero coefficients in θ, then the signal
can be said as k-sparse.
A signal might be compressible if it can be represented in
terms of a sparse expansion. In fact, compressible signals are
rather ubiquitous which allow compressed sensing in many
far-reaching applications such as data acquisition, data com-
pression, network coding, and others [24]. The sparse signal
or data is measured by taking a smaller number of samples
(m) from the original x using a linear/convex programming
operator Φ, hence Eq.(1) can be rewritten as
y = Φx = ΦΨθ = Aθ (3)
where Φ = [ϕ
1
, ϕ
2
, . . . , ϕ
m
]
T
, A = ΦΨ, k ≤ m ≪ n,
and the n × 1 vector x is compressed into an m × 1 mea-
surement vector y. Eq.(3) admits many to an infinite number
of solutions. In order to find the sparsest solution, it can
be easily solved as an optimization problem by maximizing
“measurement of sparsity” while simultaneously satisfying
Eq.(3). To find a unique sparse solution of Eq.(3), the mea-
surement matrix needs to be successfully designed. Two main
categories of measurement ensembles can be directly applied
in compressed sensing [4], [25], [26], [27]:
Random measurements, Φ is not explicitly used, in which
the measurement y is random linear combinations of the
entries of x. The measurement matrix can be Fourier, Binary
or Gaussian. In compressed sensing framework, the incoherent
measurements can be obtained by random ensembles. Ran-
domness is likely to provide incoherent projections [25], [26].
Incoherent measurements, in which Φ is deterministic that
is assumed to be incoherent with Ψ. The incoherence between
Φ and Ψ can be measured by their mutual coherence [26].
Definition 2: The coherence of two vectors ϕ
i
and ψ
k
can
be defined as
µ = max
i,k
| < ϕ
i
, ψ
k
> | (4)
3
The lower the µ is, the more incoherent Φ and Ψ are. Actually,
in most WSNs or IoT, the network data can be sparsely
represented on a wavelet basis ψ.
In compressed sensing-based framework, signal can be
sampled as y = Φx = ΦΨθ. The backbone of compressed
sensing is two-fold: (1) data is compressible, only a few entries
of θ have a significant amplitude; x is then almost entirely
determined from only a few entries θ; (2) measurements are
incoherent: The measurement matrix A = ΦΨ is incoherent.
In other words, the information carried by a few entries of
θ will spread all over the m entries of y. Each sample y
k
is likely to contain a piece of information of each significant
entry of x.
In order to find the unique sparse solution, it is crucial to
construct a measurement matrix that satisfies conditions such
as null space property, restricted isometry property (RIP), and
some coherence property [25].
Definition 3: For the m-by-n measurement matrix Φ, it is
said to satisfy the RIP (Restricted Isometry Property) or some
coherence property of order K to look for sparse solutions
if there exists a 0 < δ
k
< 1 simultaneously for all k-sparse
signal x ∈ R
n
.
(1 − δ
k
)
m
n
∥x∥
2
2
≤ ∥Φx∥
2
2
≤ (1 + δ
k
)
m
n
∥x∥
2
2
(5)
In practice, we can deterministically construct the measure-
ment matrices using random entries ϕ
ij
as i.i.d realization from
some probability distributions that satisfy RIP.
B. Reconstruction Algorithms
RIP can be one of the sufficient conditions for accurately
reconstructing the compressed signals, which guarantees near
optimal reconstruction of the solution of Eq.(2). RIP requires
that the reconstruction algorithm should be able to find the
sparest vector. Fortunately, this problem can be easily solved.
For Eq.(2), the unknown k-sparse x can be reconstructed
exactly by solving Eq.(6)
min
θ
∥θ∥
p
s.t. y = ΦΨθ (6)
where ∥·∥
p
=
n
i=1
| · |
1/p
denotes the ℓ
p
-norm. Numerous
results have demonstrated that ℓ
p
(0 < p ≤ 1) satisfies RIP
condition. For ℓ
1
, the restricted isometry constants satisfy
δ
k
< 1, which can guarantee the reconstruction conditions.
The reconstruction of x can be seen as a linear or convex
programming problem and many methods are available to
easily solve this type of problems.
Extensive research efforts have been made to develop var-
ious sparse recovery algorithms, in which there are usually
two groups of methods to perform the sparse recovery. One is
convex relaxing-based recovery algorithms, such as the famous
basis pursuit (BP) that aims at solving the ℓ
1
minimization,
Dantzig Selector, and so on; Another group of commonly used
algorithms are greedy pursuit algorithms based reconstruc-
tion algorithms, such as Matched Pursuit (MP), Orthogonal
Matched Pursuit (OMP), Stagewise OMP (StOMP), Compres-
sive Sampling Matched Pursuit (CoSaMP), Subspace Pursuit
(SP), and so on. Both of the convex programming-based
Infrastructure
Networks
Compressed
Information sampling
Compressed
Distortion-Minimizing
Control
Routing
MAC Protocol
PHY
Routing
MAC Protocol
PHY
Compressed
Information
Reconstruction
Compressed
Distortion-Minimizing
Control
Routing
MAC Protocol
PHY
Infrastructure
Networks
Data Acquisition
Networks
Internet Network
Data Analysis
Networks
Compressed sensing IoT
compressed sensing information encoder compressed information re-constructor
Fig. 1. Compressed sensing scheme over IoT.
and nonparametric greedy-based algorithms have advantages
and disadvantages when applied to different applications. An
advantage of the nonparametric greedy algorithms is that it can
produce a good approximation with a small number of itera-
tions. Meanwhile, the convex programming-based algorithms
have a better reconstruction accuracy. In contrast to BP, basis
pursuit de-noising (BPDN) (also called LASSO) has additional
de-noising performance [27], [28], [29], [30].
C. Noise and Reconstruction Accuracy in Compressed Sensing
In practice, as the data has noise, LASSO is able to
minimize the usual sum of square errors, with a bound on
the sum of the absolute value
min
θ
∥θ∥
1
s.t. Φ
T
(y − ΦΨθ) ≤ λ
1
(7)
Actually, Eq.(7) can also be reformatted as a penalized least
squares estimate problem
arg min
θ
∥y − ΦΨθ∥
2
2
+ λ
2
∥θ∥
1
(8)
By appropriately choosing constants λ
1
and λ
2
, Eq.(8) can
be solved. Therefore, it is possible to accurately reconstruct the
compressed signals without requiring prior knowledge when
the signals are compressible over some domains. One can
accurately reconstruct the sparse vector θ from compressed
y using the reconstruction algorithms mentioned above and
the compression rate can be defined as
ρ =
∥x − ¯x∥
2
2
n
(9)
III. CS-BASED FRAMEWORK IN WSNS AND IOT
In this section, a compressed sensing framework for signal
or data acquisition in WSNs and IoT will be introduced. It
acquires a user-defined continuous packets sequence of data
per interval, and after a compressed sensing-based encoding
procedure the encoded packets are transmitted by wireless
communications. The compressed sensing IoT (CSI) system
simplifies all edge components as IoT nodes, as shown in
4
Fig.1. CSI contains of three phases: (1) The design of com-
pressed sensing information end-node (CSIE), which aims to
reduce the sampling rate and the number of samples without
losing the essential information; (2) The compressed data
delivery scheme, compressed data is delivered to IoT networks
to minimize the received data distortion and communication
burden; and (3) Data reconstruction and analysis at fusion
center(s). The CSI is a flexible architecture to implement a
range of different information acquisition in IoT and WSNs.
A. System Architecture
The essential goal of WSNs and IoT is to accurately acquire
the information about events of interest. The information
acquisition networks usually consist of three core components:
(1) Information sensing system, which can detect and com-
pressively sample the signals of events; (2) Compressed sam-
pling, the systems sample information that are preconditioned
and transmitted over the networks; and (3) Reconstruction
algorithms, the system accurately reconstructs the original
signal from the compressed samples. Inadequate sampling may
cause aliasing in signal reconstruction when the measurement
matrices are not properly selected.
In contrast to conventional sensing and sampling systems,
the compressed sensing can extend them to a much broader
class of signals. The compressed sensing-based sampling
process works by taking a small number of samples of a
compressible signal on a sparse basis to reconstruct the origi-
nal signals by using linear/convex optimization methods. The
compressed sensing theory typically requires the projection
matrix to be random, though in practice researchers have often
found that the same idea can be used in other conventional
sampling scenarios [2]. In this section, we summarized the
signals or data that is collected by three models.
1) Node-dependent signal or data acquisition: Each node
acquires i.i.d. signals. In this scenario, the compressed sensing
can be used to effectively reduce the sampling rate without
degenerating the reconstruction performance. A k-sparse sig-
nal x ∈ R
n
can be completely described by the k nonzero
components. x can be sampled with a diversifying matrix and
a measurement vector y can be obtained. The sampling process
can be described as Eq.(10)
y = Ax + ϵ (10)
in which A denotes an m-by-n measurement matrix and ϵ is
noise.
The benefits of this model are: (1) the number of samples
generated by each node can be significantly reduced without
losing the reconstruction accuracy; (2) it may cause the
significant reduction of communications over the networks;
and (3) the computation cost at nodes can be reduced.
2) Cooperative signal or data acquisition between nodes:
In networks (WSN, IoT, etc.), the measurement y can be
represented as
y = [y
1
, · · · , y
m
]
T
=
n
j=1
A
i,j
x
j
(11)
in which y
i
can be easily represented as a linear combination
of the sparsely represented signal x
i
.
Each node is able to compute x
j
by multiplying the corre-
sponding element of matrix A
i,j
, which can be constructed by
choosing the entries as i.i.d realizations from some probability
distribution [2]. Then randomized gossip is used to aggregate
the A
i,j
x
j
on a fusion center. By this way, y is available at
the fusion center.
3) Consensus algorithm-based signal or data acquisition
over networks: In a practical network, most nodes keep a
sleeping mode based on a predefined mechanism; therefore,
the topology changes over time [31]. As a result, it is necessary
to take this situation into consideration for signal acquisition
or data collection.
Consider a network with n nodes at location {p
i
} (i =
1, . . . , n) is monitoring multiple events, assume that N
a
(t)
nodes are in active mode and N
s
(t) nodes are in sleep mode
at time t. Let x
i
denote the source value at p
i
, i ∈ n. Then
measurement y
i
of node i can be represented as
y
i
=
j∈N
A
j,i
x
j
+ ϵ
i
(12)
in which A
i,j
= A
j,i
is the influence of this event on sensor
point p
i
, and ϵ
i
is the random measurement noise of zero
mean. Here x is sparse and A
i,j
can be learned during the
network deployment stage.
Assume that the influence A
ji
= 0, if the distance from
j to i is larger than the communication range. Then the
measurement y
i
becomes y
i
= x
i
+
j∈n
A
j,i
x
j
+ ϵ
i
,
furthermore, for the active nodes in the network, we have
y
a
= ΦAx + ϵ
a
(13)
where A is the n×n matrix with (i, j)-th element being (A
i,j
),
Φ is the m × n measurement matrix that selects the m rows
of A corresponding to the active sensors, and y
a
and ϵ
a
are
the m × 1 measurement vector and noise vector, respectively.
In compressed sensing theory, we aim to recover the n × 1
sparse signal vector x from m measurements. This can be
solved as an optimization problem
min
x
∥Ax − b
a
∥
2
2
+ λ
2
∥x∥
1
s.t. x ≥ 0 (14)
Let N
i
denote the neighboring nodes of i. Assume that each
active sensor i holds the signal x
i
at its own location as well
as the signals x
k
occurring at its inactive neighboring node
∀k ∈ N
a
∪ N
i
. This means node i keeps its measurement x
i
and {x
k
}
k∈N
i
. Then Eq.(14) can be reformulated as
min
i∈N
a
y
i
− x
(i)
i
−
k∈N
s
∪N
i
A
k,i
x
(i)
k
−
j∈N
a
A
k,i
x
(j)
j
2
(15a)
s.t. x
(i)
i
≥ 0, ∀i ∈ N
a
, (15b)
x
(i)
k
≥ 0, ∀k ∈ N
s
∪ N
i
(15c)
Eq.(15) can be reformulated as a separable convex program,
which can be solved with a consensus algorithm by using the
alternating direction method of multipliers.
5
B. Sparse Representation
The CSIE samples the original information based on com-
pressed sensing theory and then deliver the samples through
CSI. The sampling rate (or the number of samples) is de-
termined in this process, while the measurement matrix is
pre-selected and shared between the sender and receiver as
described in Section II.A.
CSIE has two advantages: (1) it runs on low complexity
and can be used over thin-node of IoT; and (2) it takes the
advantage of the temporal correlation between continuous data
matrices. CSIE is able to effectively sample the compressible
signal x on a certain basis. However, a few promising schemes
are available for the design of the CS-based information en-
coder to sparsify/compress bases for information or data. Since
an information system (e.g., wireless sensor network) consists
of many data sources that are able to monitor the information
related to a certain spatially varying phenomenon, if the
nodes are deployed in a random manner such that they cover
uniformly a given surface, then sparsifying transformation
may be readily borrowed from traditional signal processing.
In this case, many well-developed tools such as discrete-
cosine-transform (DCT), discrete-Fourier-transform (DFT) or
discrete-wavelet-transform (DWT) may be used to de-correlate
and sparsify the sensor data [4].
Fig.2 illustrates an example of compressed sensing-based
spatially correlated data acquisition network, where DWT is
used for sparsification. Actually, the non-sparse raw data in
Fig.2(a) can easily be sparsely represented over a wavelet basis
as shown in Fig.2(b).
In IoT, remote data collection involves specific collections
that often provide redundant data which cannot be accounted
for by a standard compression technique. In a general frame-
work, let us consider that N observations of the same moni-
toring area are available: {y
i
}(i = 1, · · · , N) such that
y
i
= A
Λ
i
x + n
i
(16)
where {A
Λ
i
}(i = 1, · · · , N) are N independent random sub-
matrices of Φ with Card(Λ
i
) = M. It is clear that x
can be reconstructed from the N compressed observations
{y
i
}
i=1,...,N
. According to Eq.(5), we propose a substitution
decomposition solution with the following
min
θ
∥θ∥
1
s.t.
N
i=1
∥y
i
− A
i,j
θ∥
2
2
≤ ϵ (17)
It can be further recast in the following Lagrangian form
arg min
θ
N
i=1
∥y
i
− A
i,j
θ∥
2
2
+ λ
2
∥θ∥
1
(18)
For a sensor network with changing topology that the data
is made of N readings {x
i
}
i=1,··· ,N
such that each reading
x
i
is a noiseless observation of the same sensing area x, each
observation is compressed using compressed sensing such that
ρ = M/t. Compression is made by solving the problem in
min
θ
N
i=1
∥y
i
− ϕ
i
Ψθ∥
2
2
+ λ
2
∥θ∥
1
(19)
(a) Monitoring scenario
0 500 1000 1500 2000 2500
−2000
0
2000
4000
6000
8000
10000
12000
(b) Sparse representation of monitoring data
Fig. 2. Illustration of the compressibility of network.
in which λ
2
is a regularization parameter, which is a trade-off
between the sparse representation of signals and reconstruction
accuracy.
C. Noise Model, Communication Load, and Recovery Accu-
racy
This subsection will discuss the noise model in compressed
sensing. For compressed problem the measurement noise can
be modeled as
y = Ax + ϵ (20)
where y ∈ R
n
is the measurement, A ∈ R
m×n
is the
measurement matrix, and ϵ is assumed to be a Gaussian
random vector with i.i.d. components. Let I
m
is an identity
matrix of size m, for the normalized measurement matrix A
= [A
i,j
], each component here is assumed to be distributed as
A
i,j
∼ N (0, 1/m), i = 1, . . . , m, and j = 1, 2, . . . , n.
Similarly, the input noise model can be given by
y = A(x + ϵ) (21)
where ϵ ∼ N (0, I
n
) is a Gaussian random vector with i.i.d
components.
In multihop networks (such as WSNs, IoT, and so on),
the random projection of sensing data can be computed and
delivered to every subset of nodes using a gossip or consensus
scheme, or they might be delivered to FC(s) using clustering
and aggregation techniques [4]. In some ad hoc networks,
explicit routing information is difficult to be obtained and
