On the Convexity of Bit Depth Allocation for Linear MMSE Estimation in Wireless Sensor Networks
Fernando de la Hucha Arce, Panagiotis Patrinos, Marian Verhelst, Alexander Bertrand · IEEE Signal Processing Letters · 2020
Energy efficiency is crucial for a wireless sensor network (WSN) since its nodes are generally powered by energy sources of limited capacity, such as batteries. The bit depth used to quantize the sensor signal samples heavily influences energy consumption, as it strongly impacts the amount of information to be transmitted between the sensor nodes. Bit depth allocation problems seek to assign a certain bit depth to each sensor signal such that energy consumption is minimized while respecting a performance constraint. For multi-channel signal estimation tasks these problems are generally non-convex, and they are often solved through simplifying assumptions or through convex relaxation. However, for linear minimum mean squared error (MMSE) estimation, we show how the matrix inversion lemma allows to transform the MMSE constraint into a convex constraint, which can then be interpreted as a constraint on the excess MMSE due to quantization. As a result, as long as the cost function representing energy consumption is convex, this class of bit depth allocation problems is convex, i.e., if the bit depth variable is relaxed to a real-valued variable. This guarantees global optimality up to discretization of the obtained solution.