# Data Gathering Techniques in WSN: A Cross-Layer View
## 3 Compressed Sensing  
Many data-gathering applications rely on numerous self-powered smart devices to collect real-time data and convey it via a wireless medium to a central entity or entities(e.g., the cloud) to further process it and act upon it. Such devices are expected to perform two basic operations--sensing and wireless connectivity. Two important hindering aspects that derive from these operations influence the performance and need to be considered are: (i) energy consumption associated with these two operations, especially when many of these devices are typically simple with limited computation abilities and battery lifetime (ii) airtime utilization, which can also degrade the performance causing high delays, jitter, battery consumption, etc. Accordingly, one of the main challenges in combating these limitations is reducing the report payload, which affects each report's transmission time and channel utilization. Reducing the payload of the sensed data can be accomplished at different levels; it can be done in the sensing stage by reducing the size of the sensed data, as well as in the report preparing stage by compressing the report size, and in the transmission stage, by selecting which devices need to send reports, thus limiting redundant information. When the reports need to travel multiple hops before reaching their destination, this can be done at the relay stage by pruning, unifying, and compressing reports. In the sequel, we discuss Compressed Sensing (CS). This novel paradigm can reduce report payload at several levels mentioned above, hence lowering the sensing operation's transmission time and energy consumption. Compressed sensing is a signal-processing technique that is most advantageous when the subject signal is sparse in some domain, such that a minimal non-zero vector of coefficients can represent it. The signal sparsity enables a high-quality reconstruction, which is attained by finding the solution to an under-determined linear system of equations with the smallest possible number of non-zero values. Thus, a convex minimization problem needs to be solved to perform the recovery. Note that the CS technique performs non-uniform sampling of the data signal with an average sample rate usually smaller than the minimal rate mandated by the Nyquist-Shannon sampling theorem. A detailed view of the technique can be found in [PERSON] et al. [58] and [PERSON] [59]. Various networking domains can utilize compressed sensing, for example, [PERSON] et al. [60] depict some applications of CS over networks and elucidate the connection between CS and traditional information-theoretic techniques in source coding and channel coding. Particularly, CS is highly suitable for sensed data gathering in wireless sensor networks (e.g., physical phenomena or a scenery), as it can leverage the expected high spatial and temporal correlation between sensing reports sent by neighboring sensors at different times in order to acquire the CS paradigm's desired sparsity. In the following, we review several such data-gathering schemes that utilize CS. [PERSON] et al. [61; 62] consider a densely deployed monitoring sensor network in which reports traverse multiple hops before reaching their destination, a sink. These studies rely on the concept that sensors' readings are spatially correlated; hence, there exists a transform domain in which the sensed signals can be sparsely represented. Both propose a compressive data-gathering (CDG) procedure in which sensors distributively encode their reports by projecting them on a random space basis using random coefficients. These encoded reports can be decoded at the sink using compressive sensing techniques. Specifically, CDG is designed for multi-hop networks where messages need to travel multiple hops before reaching their destination. The sampling process that characterizes the CS compression process is performed individually at each sensor by simple multiplications and additions.