In this proposal we address the key problems which prevent the efficient and economically viable implementation of Massive MIMO, including the transmitter/receiver sampling complexity, the problem of pilot contamination, and the problem of channel estimation both in TDD and in FDD systems. The key idea of this proposal is that a signi cant dimensionality reduction (and consequently, complexity reduction) in the Massive MIMO frontend processing can be achieved by leveraging the structure of the propagation channels between the base station antenna array and the users. These channels are argued to exhibit sparsity in the angular and delay domain. In short, especially when communication takes place in the mm-waves range, the propagation occurs along discrete multipath components, each of which is characterized by an angle of departure (AoD) and a delay. This inherent sparsity can be leveraged by modern CS algorithms, operating at much better complexity/performance trade off than conventional front-end schemes. Here, sparsity typically means that only a few samples of the signal are actually non-zero, when the signal is represented in a suitable "sparsifying basis". In general, the location of the non-zero components (relative to the signal basis elements) is not known a priori. This new paradigm has been an intriguing topic in mathematics and signal processing in recent years. Note that sparsity-based concepts have been successfully applied in specific communication problems, e.g., the "peak power control problem", the "channel impulse response estimation problem", the "neighbor discovery problem in ad-hoc networks", the "detection of spectral holes in cognitive radio", the "MIMO Radar direction of arrival problem", and several other applications. We will show that leveraging sparsity in communication signals is a viable approach to Massive MIMO implementation with affordable complexity.  The theory and the algorithms developed in this project will therefore lay the foundations for a new generation of air interfaces able to handle a very large number of Tx antennas (in the DL) and Rx antennas (in the UL, or in the channel estimation phase), thus addressing the challenges and the spectral efficiency target performance of 5G networks. As anticipated before in this research proposal we focus on the Massive MIMO scenario, while it is envisioned that in later follow-up phases the C-RAN and DAS architectures (many jointly processed antennas, but physically distributed over the network coverage region) will be also investigated.

The theory of compressed sensing (CS) has shown that a substantial reduction in sampling and storage complexity can be achieved in many relevant linear and non–adaptive estimation problems. Recent theoretical developments have also put the analysis of a whole range of practical non-linear problems within reach. Examples include blind decoding of wireless signals under channel uncertainties, recovery of images from fuzzy snapshots without precise knowledge of the blurring kernel, or more general model uncertainties in conventional CS. The unifying feature of these tasks is that the signal is accessible only through an uncalibrated system, whose description is partially unknown at the time of measurement. Mathematically, one set of parameters (the channel, the kernel, the sensing matrix) is coupled in a multiplicative way to the signal – giving rise to an inherent emph{bilinear structure}. While in conventional CS such model uncertainties inevitably degrade the quality of the recovery, the novel approach is to combine bilinearity and compressibility in order to simultaneously estimate both the signal and the model parameters.The theory of bilinear CS is only at its beginnings, but has recently garnered a significant amount of attention and is developing rapidly. (Because it unites and extends the first two structural assumptions considered in the CS community – sparsity and low-rank – it is sometimes refered to as emph{compressed sensing 3.0}). While we believe that engineering applications are within reach over the duration of the Priority Program, considerable mathematical problems remain to be addressed. Our team of three principal investigators will work towards a comprehensive theory for bilinear CS. On the one hand, we will study, in an abstract context, recovery properties for random subsampling of bilinear maps, as well as bilinear maps of vectors under random subspace conditions. On the other hand, we will work to develop adapted techniques for specific applications, focusing on wireless communication, but also touching on imaging and spectroscopy.