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16-11-2010, 05:06 PM


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Third Semester,M.Tech
Applied Electronics and Instrumentation

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Compressed sensing (CS) is a new method of signal and image measurement and
reconstruction that takes advantage of the fact that many signals observed have sparse
representation under some basis[1]. CS is an emerging model based framework for
signal recovery at a rate significantly below the Nyquist sampling rate. The CS theory
states that a signal having a sparse representation in some bases can be reconstructed
from a small set of random project and implimentationions. Wireless sensor network (WSN) consists of
a large number of spatially distributed signal processing devices (nodes), each with finite
battery lifetime and thus limited computing and communication capabilities [3].
When properly programmed and networked, nodes in a WSN can cooperate to perform
advanced signal processing tasks with unprecedented robustness and versatility,
thus making WSN an attractive low-cost technology for a wide range of remote sensing
and environmental monitoring applications. For applications involving distributed
measurements of some physical phenomenon (e.g. temperature, vibration) that may be
wirelessly transmitted, CS holds promising improvements to these limits. Compressive
measurement of a sensor network is accomplished by prompting each sensor to communicate
its value simultaneously to a central base station. Each sensor value is multiplied
by a random number that changes each measurement and is known by the sensor and
the base station [6].
2.1 Wireless Sensor Networks(WSN)
Sensor networks are the key to gathering the information needed by smart environments,
whether in buildings, utilities, industrial, home, shipboard, transportation
systems automation. Detecting the relevant quantities, monitoring and collecting the
data, assessing and evaluating the information, formulating meaningful user displays,
and performing decision-making and alarm functions [4]
Wireless Sensor Networks have emerged as an important new area in wireless
technology. Sensor network is subject to a unique set of resource constraints such as
finite on-board battery power and limited network communication bandwidth. In a
typical sensor network, each sensor node has a microprocessor and a small amount of
memory for signal processing and task scheduling. Each node is also equipped with
one or more sensing devices such as acoustic microphone arrays, video or still cameras,
infrared (IR), magnetic sensors. Each sensor node communicates wirelessly with a few
other local nodes within its radio communication range. [3,4,5]
Wireless sensor networks (WSN) are gaining increasing research interest for
their emerging potential in both consumer and national security applications. Sensor
networks are envisioned to be used for surveillance, identification, and tracking of targets.
They can also serve as the first line of detection for various types of biological
hazards such as toxic gas attacks. In civilian applications, WSN can be used to monitor
the environment and measure quantities such as temperature and pollution levels.
In most application scenarios, WSN nodes are powered by small batteries,
which are practically nonrechargable, either due to cost limitations or because they
are deployed in hostile environments with high temperature, high pollution levels, or
high nuclear radiation levels. These considerations motivate well energy-saving and
energy-efficient WSN designs. The wireless sensor networks are expected to consist
of thousands of inexpensive nodes, each having sensing capability with limited computational
and communication power and which enable to deploy a large-scale sensor
network [3].
Recent advancement in wireless communications and electronics has enabled
the development of low-cost, low-power, multifunctional miniature devices for use in
remote sensing applications. The combination of these factors has improved the viability
of utilizing a sensor network consisting of a large number of intelligent sensors,
enabling the collection, processing analysis and dissemination of valuable information
gathered in a variety of environments.
A sensor network is composed of a large number of sensor nodes which consist
of sensing, data processing and communication capabilities. Sensor network protocols
and algorithms must possess self-organizing capabilities. Another unique feature of
sensor networks is the cooperative effort of sensor nodes. Sensor nodes are suitable
with an onboard processor. Instead of sending the raw data to the nodes responsible for
the fusion, they use their processing abilities to locally carry out simple computations
and transmit only the required and partially processed data.[3]
Since there are limited bandwidths in wireless sensor networks, it is important
to reduce data bits communicated among sensor nodes to meet the application performance
requirements. It also saves node energy since less bits are communicated
between nodes.Energy limitation is one of the major differences between a WSN and
other wireless networks such as wireless local area networks, where energy efficiency
is of a lesser concern. Also, WSNs are often self-configured networks with little or no
pre-established infrastructure as well as a topology that can change dynamically [3]
A wireless sensor network (WSN) generally consists of a basestation (or "gateway")
that can communicate with a number of wireless sensors via a radio link. Data
is collected at the wireless sensor node, compressed, and transmitted to the gateway directly
or, if required, uses other wireless sensor nodes to forward data to the gateway.[5]
Two popular WSN characteristic are:
 Fusion center(FC)
 Adhoc WSNs
 Fusion center(FC):
In Fusion Center network, there is no inter sensor communication; communication
is only between sensors and the FC (Fig 2.1). The FC collects locally
processed data and produces a final estimate;[3]. In the figure s1,s2,s3 indicate
the sensor nodes.
Figure 2.1: A WSN topology with an FC
 Adhoc WSNs:
In Ad hoc WSN,the network itself is responsible for processing the collected
information, and to this end, sensors communicate with each other through the
shared wireless medium (Fig 2.2).
Figure 2.2: Ad hoc Wireless Sensor Network
2.2 Compressive Sensing Background
2.2.1 Consider Sparsity
For large wireless sensor networks, the events are relatively sparse compared
with the number of sources. Compressive sensing is an idea achieve much lower sampling
rate for sparse signals .Consider a real valued, finite length, one dimensional,
discrete time signal X, which view as an N  1 column vector in RN with elements
x[n], n = 1, 2, . . . , N. Here it treats higher-dimensional data by vectorizing it into a
long one diamensional vector.
Any signal in RN can be represented in terms of a basis of N  1 vectors
f igN
i=1 = 1. For simplicity assume that the basis is orthonormal. Forming the N  N
basis matrix := [ 1j 2j:::j n] by stacking the vectors f ig as columns, any signal
X can be expressed as
X = N
i=1Si 1 or X = NNS (2.1)
Where S is the N  1 column vector of weighting coefficients Si =< X; >=
TX. Clearly, X and S are equivalent representations of the same signal,with X in the
time domain and S in the domain.
Focus on signals that have a sparse representation, where X is a linear combination
of just K basis vectors, with K << N. That is, only K of the Si in (1) are
nonzero and (N-K) are zero. Sparsity is motivated by the fact that many natural and
manmade signals are compressible in the sense that there exists a basis where the representation
(1) has just a few large coefficients and many small coefficients. A typical
compression algorithm would simply compute the non-zero coefficients of S and store
their amplitudes and locations
This method of sampling a signal and then compressing it suffers from a few
inherent inefficiencies. First the entire N length signal must be measured which can
be inefficient if N is very large. Second the encoder (compression algorithm) must
compute all of the N transform coefficients even though many of them are small and can
be discarded. Third the positions of each of the transform coefficients must be known
and will therefore require storage [1]. So the use of compressed sensing primarily as
a tool to decrease the number of measurements required to accurately determine the
sensor readings in a wireless sensor network.
2.2.2 Compressive Measurements
Compressive sensing presents an alternative, a more general data acquisition
approach that condenses the signal directly into a compressed representation without
going through the intermediate stage of taking N samples. The remarkable characteristic
of CS is that a K sparse signal can be encoded by multiplying it by a random
matrix,MN,where M is much smaller than N but is larger than K i.e. K << M <<
N. The result of this encoding method is the compressive measurement vector, Y, which
is defined by
Y = MNXN1 (2.2)
Substituting the representation of X from equation (2.1) in to equation (2.2) we get
Y = MN NNS (2.3)
Y = MNS (2.4)
Where  =  is an M  N matrix
Figure 2.3: Compressive sensing measurement process
1a) with (random Gaussian) measurement matrix  and transform matrix .The
coefficient vector S is sparse with K=4. (b) Measurement process in terms of the
matrix product  =  with the four columns corresponding to nonzero Si
highlighted. The measurement vector Y is a linear combination of these four columns.
It is important to discuss how it is possible to reconstruct S from Y and to
ensure that the probability of exact reconstruct can be made close to unity for this measurement
scheme. This is a difficult problem because the locations of the K non-zero
coefficients of X are unknown. The measurement vector Y is just a linear combination
of the columns of which correspond to the non-zero coefficient in S. If the locations
of the non-zero entries of S were known, finding a solution would simply be a matter of
inverting the matrix corresponding to the ordered set of these entries. Here, reconstruction
is possible so long as M  K. A necessary and sufficient condition to show that
the M  Ksystem has a numerically stable inverse is that for any vector V sharing the
same non-zero entries as S we have

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