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Distributed sparse random projections

A second method modifies the randomized gossiping approach by limiting the number of communications each node must perform, in order to reduce overall power consumption  [link] . Each data node takes M projections of its data, passing along information to a small set of L neighbors, and summing the observations; the resulting CS measurements are sparse, since N - L of each row's entries will be zero. Nonetheless, these projections can still be used as CS measurements with quality similar to that of full random projections. Since the CS measurement matrix formed by the data nodes is sparse, a relatively small amount of communication is performed by each encoding node and the overall power required for transmission is reduced.

Centralized algorithms

Decentralized algorithms are used when the sensed data must be routed to a single location; this architecture is common in sensor networks were low power, simple nodes perform sensing and a powerful central location performs data processing.

Compressive wireless sensing

Compressive wireless sensing (CWS) emphasizes the use of synchronous communication to reduce the transmission power of each sensor  [link] . In CWS, each sensor calculates a noisy projection of their data sample. Each sensor then transmits the calculated value by analog modulation and transmission of a communication waveform. The projections are aggregated at the central location by the receiving antenna, with further noise being added. In this way, the fusion center receives the CS measurements, from which it can perform reconstruction using knowledge of the random projections.

A drawback of this method is the required accurate synchronization. Although CWS is constraining the power of each node, it is also relying on constructive interference to increase the power received by the data center. The nodes themselves must be accurately synchronized to know when to transmit their data. In addition, CWS assumes that the nodes are all at approximately equal distances from the fusion center, an assumption that is acceptable only when the receiver is far away from the sensor network. Mobile nodes could also increase the complexity of the transmission protocols. Interference or path issues also would have a large effect on CWS, limiting its applicability.

If these limitations are addressed for a suitable application, CWS does offer great power benefits when very little is known about the data beyond sparsity in a fixed basis. Distortion will be proportional to M - 2 α / ( 2 α + 1 ) , where α is some positive constant based on the network structure. With much more a priori information about the sensed data, other methods will achieve distortions proportional to M - 2 α .

Distributed compressive sensing

Distributed Compressive Sensing (DCS) provides several models for combining neighboring sparse signals, relying on the fact that such sparse signals may be similar to each other, a concept that is termed joint sparsity  [link] . In an example model, each signal has a common component and a local innovation, with the commonality only needing to be encoded once while each innovation can be encoded at a lower measurement rate. Three different joint sparsity models (JSMs) have been developed:

  1. Both common signal and innovations are sparse;
  2. Sparse innovations with shared sparsity structure;
  3. Sparse innovations and dense common signal.

Although JSM 1 would seem preferable due to the relatively limited amount of data, only JSM 2 is computationally feasible for large sensor networks; it has been used in many applications  [link] . JSMs 1 and 3 can be solved using a linear program, which has cubic complexity on the number of sensors in the network.

DCS, however, does not address the communication or networking necessary to transmit the measurements to a central location; it relies on standard communication and networking techniques for measurement transmission, which can be tailored to the specific network topology.

Questions & Answers

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s.
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s. Reply
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Source:  OpenStax, An introduction to compressive sensing. OpenStax CNX. Apr 02, 2011 Download for free at http://legacy.cnx.org/content/col11133/1.5
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