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As it becomes increasingly apparent that 4G will not be able to meet the emerging demands of future mobile communication systems, the question what could make up a 5G system, what are the crucial challenges, and what are the key drivers is part of intensive, ongoing discussions. Partly due to the advent of compressive sensing, methods that can optimally exploit sparsity in signals have received tremendous attention in recent years. In this paper, we will describe a variety of scenarios in which signal sparsity arises naturally in 5G wireless systems. Signal sparsity and the associated rich collection of tools and algorithms will thus be a viable source for innovation in 5G wireless system design. We will also describe applications of this sparse signal processing paradigm in Multiple Input Multiple Output random access, cloud radio access networks, compressive channel-source network coding, and embedded …