Signal detection in sparse multipath channels

In this paper, we revisit the problem of signal detection in multipath environments. Existing results implicitly assume a rich multipath environment. Our work is motivated by physical arguments and recent experimental results that suggest physical channels encountered in practice exhibit a sparse structure, especially at high signal space dimension (i.e., large time-bandwidth product). We first present a model for sparse channels that quantifies the independent degrees of freedom (DoF) in the channel as a function of the physical propagation environment and signal space dimension. The number of DoF represents the delay-Doppler diversity afforded by the channel and, thus, critically impacts detection performance. Our focus is on two types of non-coherent detectors: the energy detector (ED) and the optimal non-coherent detector (ONCD) that assumes full knowledge of channel statistics. Results show, for a uniform distribution of paths in delay and Doppler, the channel exhibits a rich structure at low signal space dimension and then gets progressively sparser as this dimension is increased. Consequently, the performance of the detectors is identical in the rich regime. As the signal space dimension is increased and the channel becomes sparser, the ED suffers significant degradation in performance relative to the ONCD. Finally, our results show the existence of an optimal signal space dimension - one that yields the best detection performance - as a function of the physical channel characteristics and the operating signal to noise ratio (SNR).