Wireless Sensor Networks based on Compressed Sensing
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Wireless Sensor Networks based on Compressed Sensing Zhuang Xiaoyan
School of Automation Engineering University of Electronic Science and Technology of China Chengdu, China zhuangxyan@
Abstract-The data collected through high densely distributed wireless sensor networks is immense. The asymmetry between the data acquisition and information processing makes a great challenge to the restriction of energy and computation consumption of the sensor nodes, and it limits the application of wireless sensor networks. However, the recent works show that compressed sensing can break through this limitation of asymmetry. Compressed sensing is an emerging theory that is based on the fact that a signal can be recovered through a relatively small number of random projections which contain most of its salient information. In this paper, we introduce the background of compressive sensing, and then applications of compressed sensing in wireless sensor networks are presented. Keywords-wireless sensor networks; data compression; compressed sensing; sparsity
I.
INTRODUCTION
of compressed sensing in wireless sensor networks are presented. Conclusion is made in section IV. II.
COMPRESSIVE SENSING BACKGROUND
Wireless Sensor Networks (WSNs) are widely used in a variety of applications to monitor the physical world via a spatially distributed network of small, inexpensive wireless devices that have the ability to self-organize into a well connected network. A typical wireless sensor network , Which consists of a large number of wireless sensor nodes, the network data transmission are accomplished through multi-hop routing from individual sensor nodes to the data sink. The problem of efficiently transmitting or sharing information from and among a vast number of distributed nodes makes a great challenge to the energy and computation consumption of the sensor nodes. This problem can be resolved by compressive sensing theory. In large-scale wireless sensor network, Compressed Sensing (also known as compressive sampling or CS) is a novel data compression technology to reduce global scale communication cost without introducing intensive computation or complicated transmission control[1]-[3]. This will result in extend the lifetime of the sensor network. Compressing sensor exploits the inherent correlation in some input data set X to compress such data by means of quasi random matrices. If the compression matrix and the original
R M xN
(M
'IIa ,
«
N) . Obviously, the degrees of freedom
ቤተ መጻሕፍቲ ባይዱ
y= Ax = <I>'Pa. (1) M YE R the measurement matrix
978-1-4244-5540-9/10/$26.00 ©2010 IEEE
90
y
s.t. Y = Ax = <1>'1' a \1.1
Wang Houjun, Dai Zhijian
School of Automation Engineering University of Electronic Science and Technology of China Chengdu, China hjwang, daizhijian@
is much smaller than N, then the signal X is
samples, as in Figure 1. For a signal its M linear observations
X=
x ERN , we can obtain
Where
A
E
from its compressed version Y, with high probability, by minimizing a distance metric over a solution space. In this paper, we introduce compressed sensing background in the Section II. In the Section III, applications
data X have certain properties, X can be reconstructed
of y is less than that of X , the equation has no unique solution. However, with the sparsity of signal X, optimization problem. the original signal can be precisely recovered using 11 norm
Compressed sensing is a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition, the theory of compressed sensing extends traditional sensing and sampling systems to a much broader class of signals. The promise of compressed sensing is that a sparse or compressible signal can be recovered from a small salient set of projections. To make this possible, there are two principles [2]: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. Sparsity expresses the idea that the "information rate" of a continuous time signal may be much smaller than suggested by its bandwidth presented in signal, or that a discrete-time signal depends on a number of degrees of freedom which is comparably much smaller than its (finite) length. More precisely, CS exploits the fact that many natural signals are sparse or compressible in the sense that they have concise representations when expressed in the proper basis (such as: Fourier basis, wavelet basis, etc.). The basis can be selected according to the signal's peculiarities. For example, the transfer coefficients of signal
XE RN under basis
'¥ are ai
=
(x, '1IJ(i
=
1,2,,,,
,N)
,
there are no more than K nonzero entries in vector a , where K
called K sparse signal. In contradiction to sparsity, incoherence means that the measurement matrix <I> has dense representation in basis '¥ , and <I> is independent to '¥ . In practice, most of natural signals are sparse or near sparse, and they can be recovered from their compressible
(2)
snapshot of the data field { Xi} ' the projection of the vector X on a projection vector (measurement vector)
School of Automation Engineering University of Electronic Science and Technology of China Chengdu, China zhuangxyan@
Abstract-The data collected through high densely distributed wireless sensor networks is immense. The asymmetry between the data acquisition and information processing makes a great challenge to the restriction of energy and computation consumption of the sensor nodes, and it limits the application of wireless sensor networks. However, the recent works show that compressed sensing can break through this limitation of asymmetry. Compressed sensing is an emerging theory that is based on the fact that a signal can be recovered through a relatively small number of random projections which contain most of its salient information. In this paper, we introduce the background of compressive sensing, and then applications of compressed sensing in wireless sensor networks are presented. Keywords-wireless sensor networks; data compression; compressed sensing; sparsity
I.
INTRODUCTION
of compressed sensing in wireless sensor networks are presented. Conclusion is made in section IV. II.
COMPRESSIVE SENSING BACKGROUND
Wireless Sensor Networks (WSNs) are widely used in a variety of applications to monitor the physical world via a spatially distributed network of small, inexpensive wireless devices that have the ability to self-organize into a well connected network. A typical wireless sensor network , Which consists of a large number of wireless sensor nodes, the network data transmission are accomplished through multi-hop routing from individual sensor nodes to the data sink. The problem of efficiently transmitting or sharing information from and among a vast number of distributed nodes makes a great challenge to the energy and computation consumption of the sensor nodes. This problem can be resolved by compressive sensing theory. In large-scale wireless sensor network, Compressed Sensing (also known as compressive sampling or CS) is a novel data compression technology to reduce global scale communication cost without introducing intensive computation or complicated transmission control[1]-[3]. This will result in extend the lifetime of the sensor network. Compressing sensor exploits the inherent correlation in some input data set X to compress such data by means of quasi random matrices. If the compression matrix and the original
R M xN
(M
'IIa ,
«
N) . Obviously, the degrees of freedom
ቤተ መጻሕፍቲ ባይዱ
y= Ax = <I>'Pa. (1) M YE R the measurement matrix
978-1-4244-5540-9/10/$26.00 ©2010 IEEE
90
y
s.t. Y = Ax = <1>'1' a \1.1
Wang Houjun, Dai Zhijian
School of Automation Engineering University of Electronic Science and Technology of China Chengdu, China hjwang, daizhijian@
is much smaller than N, then the signal X is
samples, as in Figure 1. For a signal its M linear observations
X=
x ERN , we can obtain
Where
A
E
from its compressed version Y, with high probability, by minimizing a distance metric over a solution space. In this paper, we introduce compressed sensing background in the Section II. In the Section III, applications
data X have certain properties, X can be reconstructed
of y is less than that of X , the equation has no unique solution. However, with the sparsity of signal X, optimization problem. the original signal can be precisely recovered using 11 norm
Compressed sensing is a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition, the theory of compressed sensing extends traditional sensing and sampling systems to a much broader class of signals. The promise of compressed sensing is that a sparse or compressible signal can be recovered from a small salient set of projections. To make this possible, there are two principles [2]: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. Sparsity expresses the idea that the "information rate" of a continuous time signal may be much smaller than suggested by its bandwidth presented in signal, or that a discrete-time signal depends on a number of degrees of freedom which is comparably much smaller than its (finite) length. More precisely, CS exploits the fact that many natural signals are sparse or compressible in the sense that they have concise representations when expressed in the proper basis (such as: Fourier basis, wavelet basis, etc.). The basis can be selected according to the signal's peculiarities. For example, the transfer coefficients of signal
XE RN under basis
'¥ are ai
=
(x, '1IJ(i
=
1,2,,,,
,N)
,
there are no more than K nonzero entries in vector a , where K
called K sparse signal. In contradiction to sparsity, incoherence means that the measurement matrix <I> has dense representation in basis '¥ , and <I> is independent to '¥ . In practice, most of natural signals are sparse or near sparse, and they can be recovered from their compressible
(2)
snapshot of the data field { Xi} ' the projection of the vector X on a projection vector (measurement vector)