Revisiting Non-Coherent Detection in Doubly Selective Multipath

We revisit the problem of non-coherent signal detection in multipath scattering by exploring the effect of bandwidth and signaling duration on detection performance in doubly selective rich and sparse multipath. We begin with a general sampled representation of multipath channels that allows us to quantify the independent degrees of freedom (DoF) in non-uniform and sparse channels as a function of the physical propagation environment and the signaling parameters. We then develop the optimal non-coherent detector for such channels. Based on analytical and numerical results, we conjecture that detection performance is optimized in any multipath environment, rich or sparse, if two conditions are met: C1) the number of independent DoF induced by the transmit signal on the channel is approximately SNR/3, a condition empirically observed in earlier work, and C2) the variance of the channel coefficients (each corresponding to a DoF) is identical. We then show how to achieve these conditions given statistical characterization of the channel. In the most general setting of non-uniform power distribution across both delay and Doppler, it may not be possible to meet the two conditions above.