## 1 Introduction  
A Wireless Sensor Network (WSN) is an ad hoc network whose nodes have four basic units: Sensing, processing, communication, and power (often comprising just a battery). Each node operates as long as it has energy [1]. In order to increase sensor nodes autonomy, methods leading to energy saving in WSN have appeared [2]. Transmission bandwidth is tightly associated with energy consumption--the larger the bandwidth the higher the energy consumption [1].  
A signal is said to be sparse if there is a base in which it can be represented with only a few non-zero coefficients. Compressive Sensing (CS) [3] explores sparsity to reduce the number of coefficients required to represent a given signal within some reconstruction fidelity. In CS, few coefficients are obtained through linear measurements (projections), and the original signal can be inferred from them by means of a linear program or greedy algorithm [4]. This works if the quantity of measurements is a bit larger than the signal sparsity (which can be much lower than its _Nyquist_ limit [3]).  
In a WSN, a sink node receives traffic from several sensor nodes, and usually has much more computational power than these other nodes, that must be simple and be low power consuming. Such a scenario fits the CS paradigm: Simple encoding and complex decoding [5]. In this work, the sparsity of collected signals is explored by means of CS to reduce the quantity of transmissions in WSNs, saving sensor nodes energy.  
Notwithstanding CS remarkable signal encoding capability, CS decoding has a phase change behavior: If the quantity of measurements is sufficiently large then a reasonable signal estimative (in terms of mean squared error) is possible; otherwise signal is incorrectly reconstructed. This hampers rate-distortion management by controlling the quantity of coefficients and one must rely mostly on coefficients quantization for rate-distortion control. Therefore, we investigate an incremental quantization scheme for CS coefficients to provide a finner rate-distortion control. In addition, this scheme allows the sensor nodes to first encode signals with a coarse quality and let the sink (the application layer) to ask for extra bits to improve the signal, without requiring significant extra sensor node resources.  
In summary, we investigate CS-coded transmissions in order to save sensor nodes energy. Real-life signals are used in simulations and distinct recovery methods are also investigated. The robustness of CS-coded framework against packets loss is verified, and a novel successive approximation scheme for energy conservation of sensor nodes is proposed.  
After this introduction, Section 2 presents a brief literature review on techniques for energy saving in WSN. This allows to better situate our work within the field and discuss our objectives and contributions. Section 3 brings the Compressive Sensing fundamentals and the reconstruction methods investigated and, within it, Section 3.2 presents the problem model and employed notation. Section 4 discusses the experimental methodology and presents results on the rate-distortion performance and energy consumption of quantized CS measurements encoding in WSN. Section 4.4 presents the impact of packet losses in the rate-distortion performance. The incremental transmissions scheme, using successive approximation, is presented in Section 5. Then, conclusions are presented in Section 6.