All SNs follow two rules throughout acquiring, packing, forward and accu- mulation of data:
• A SN only applies CS measurement accumulation method, when the length of the resulting message is less that when using PF. In other words, each node decides whether to use PF or CS depending on the length of the outgoing message. This way the SNs minimize the energy consumed during radio transmission.
• When the message received by a SN is a CS measurement, the SN is obligated to apply the CS accumulation process. In other words, when the message changes its type from PF to CS, then it continues to be in CS form all the way up through the network till it reaches the sink. These two simple rules and operations help to reduce the overall energy consumption of SNs. However, the energy consumption is not balanced any more. One drawback of the discussed hybrid CS scheme is that there is still the possibility of network partitioning, since the nodes near to the sink may get exhausted and deplete earlier than leaf SNs.
3.4
Summary
In this chapter we have explained the fundamentals of the emerging theory of CS and its applications in WSNs. We have realized that CS provides a very flexible, tunable, resilient and yet efficient sampling method for WSNs. We have listed major advantages of CS over transform coding or other signal compression methods which. CS guarantees a balanced energy consumption by all of the SNs. This important property avoids network partitioning and improves overall resource management of the whole network. It is easily implementable on strict hardware or software platforms of today’s typical SNs. Its resilience against noise and packet loss, makes it very suitable for the harsh operational environment of WSNs.
We have reviewed most important variants of CS which are especially devised for WSNs. In recent years, with the progress of the CS theory in WSNs area, more and more realistic applications of CS are presented. CWS was one of the first variants of CS for WSNs. Though it lacks consideration of hardware implementability, it offers a novel approach which openes a new avenue of research in the area of signal acquisition for WSNs. CDG which is directly based on CWS model, enabled detection of unexpected events. Event detection is a crucial requirement of many critical WSN applications. DCS with its JSM models has categorized a wide range of applications for CS in WSNs. DCS provided an efficient solution to the problem of joint
signal recovery. DCS has many applications in different distributed sampling scenarios, like WSN with star topology, camera arrays, acoustic localization, etc. However, it is not very suitable for multi-hop WSNs. Hybrid CS is a simple and yet effective solution for multi-hop WSNs consisting of very resource-limited SNs. It is one of the most recent improvements to typical CS implementation in WSNs. The only disadvantage of using Hybrid CS is the unbalanced energy consumption that may lead to network partitioning.
Chapter 4
Reordering for Better
Compressibility
In this chapter, we introduce our enhancement to CS in WSN by finding a better labeling (indexing) of the SNs. Our improvement does not affect the basics of CS or its WSN implementation. Consequently, the state of the art techniques can be combined with the methods introduced here. We show that if the signal vector is viewed under a reordering (mapping) function, it is possible to obtain a more compressible signal, i.e., a signal which is sparser in a certain domain such as DCT. Our work offers a new approach to the WSN sampling problem by enhancing the performance of CS.
The major advantages of our proposed technique are:
• A polynomial-time algorithm that finds a permutation of samples of the discrete signal vector f , so that the linear transform of f in frequency domain is sparser than the original ordering of samples.
• A CWS model which is capable of adapting itself to the environment changes. When the state of the environment does not change quickly, our model is capable of reducing spatial sampling rate through con- structing a more compressible view of the spatial signal.
In addition to reconstructing the original signal from compressively sam- pled data, the sink has a second responsibility in our enhanced compressive sampling scheme: Computing sub-optimal reordering of the SNs with sparser DCT representation. We assume the sink to be powerful in processing, i.e., computation and storage.
4.1
Motivation for reordering
In most distributed CS applications, sampling and sparse bases are deter- mined prior to the deployment of the sensor network. Therefore, according to Equation 3.2, µ remains constant because compressive and measurement domains, namely the bases Φ and Ψ do not change after network deploy- ment. Therefore, sparsity factor S is the only parameter that is effective on the minimum number of required measurements m. If we succeed to de- crease S, then we can reconstruct the original signal from fewer samples, saving valuable bandwidth and SNs energy.
4.1.1
Conventional indexing of SNs
In some sampling problems such as WSNs, the ordering of the samples is not dictated by an independent phenomenon. Mostly, we can assign the value sensed by one sensor to the first signal element and the other to the second one, and so on. In this chapter, we focuses on such conditions where we can set the sampling order.
It is important to emphasize that the position and order of SNs are dif- ferent aspects. Reordering only takes place at the sink and all its required computations are done outside the WSN. That’s why our proposed model does not add overhead to the WSN nodes. In simple words, reordering is an alternative view of our WSN signal vector under which we can apply CS more efficiently. It does not require to relocate the SNs or change their positions.
Less measurement requirements after proper reordering
Our enhancement works by improving compressibility, that means increasing sparsity of f under the Ψ-transform, which means decreasing S in Equation 3.2. Then, from Equation 3.2 it implies that a view of the signal that makes it appear to be more compressible, leads to fewer number of compressive mea- surements required for signal reconstruction. Lower compressive sampling rate means more efficient bandwidth usage and decreased energy consump- tion, and hence, prolonging WSN lifetime.
Energy consumption by the SNs is directly related to data transmission rate and number of compressive measurements, but it is inversely related to compressibility of the signal that is sensed by the WSN. Compressibility is increased by finding a mapping under which the WSN signal f is sparser in Ψ-domain.