# Compressive Sensing-Based IoT Applications: A Review
## 3 Compressive Sensing Overview
### Distributed Compressive Sensing  
Conventional CS theory explores only the intra-signal structures (sparsity and compressibility) at a single sensor. However, if the scenario of interest presents an architecture where the data are not only collected by a single sensor but rather with a distributed network of sensors, then exploiting the collaboration of these sensors would lead to an additional gain in the information collected and improve the design of the application. Such scenario arises in several IoT monitoring applications, where a network of wireless sensors is deployed to measure different scalar physical/physiological quantities, such as temperature, light, humidity, human vital signs, etc. If the sensors are densely deployed, their measurements are correlated and they are likely to share certain structures.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Algorithm** & **Minimum Number of Measurements** & **Complexity** \\ \hline BP & \(k\log N\) & \(\mathcal{O}(M^{2}N^{1.5})\) \\ OMP & \(k\log N\) & \(\mathcal{O}(kMN)\) \\ CoSaMP & \(k\log N\) & \(\mathcal{O}(MN)\) \\ SP & \(k\log\left(\frac{N}{k}\right)\) & \(\mathcal{O}(MN\log k)\) \\ Stagewise OMP & \(N\log N\) & \(\mathcal{O}(N\log N)\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Complexity and minimum measurements required for CS reconstruction.  
CS-based multi-sensor architecture with \(J\) sensors can exhibit two scenarios. In the first one, the acquired signals are independent; hence, the reconstruction problem is equivalent to performing \(J\) individual reconstruction operation for each sensor and no collaboration between the sensing nodes can be explored for the reason that each sensor holds no information about the data acquired by the other sensors. However, in the second scenario, which emerges frequently in most IoT applications, the signals acquired from all the sensor are highly correlated, which presents an inter-signal sparsity structure. Therefore, to reconstruct the whole signal ensemble, not only the individual sparsity of each sensor measurements can be exploited, but also the dependency among the samples acquired by different sensors at the same time should be explored to reduce the required number of measurements to assemble the most pertinent information related to the sensed phenomenon. This process,which we refer to as joint measurements recovery, is the motivation for introducing distributed compressive sensing (DCS) concept [56].  
DCS presents a new distributed coding paradigm that exploits both intra- and inter-signal correlation structures. In DCS scenarios, a number of sensors acquire sparse signals which are correlated with each other. Each sensor individually acquires a compressed signal by taking random projections and then transmits it to the fusion node. More importantly, DCS requires no collaboration between the sensors during the acquisition phase. However, the recovery process can exploit the inter-signal correlation between the measurements to reconstruct all the signals simultaneously. The recovery algorithms associated with DCS are derivatives from their CS counter-parts, such as multichannel-BPDN [57], simultaneous OMP (SOMP) [58] and distributed compressive sensing OMP (DCS-SOMP) [56] which is the general form of SOMP.