Capacity and Degree-of-Freedom of OFDM Channels with Amplitude Constraint

In this paper, we study the capacity and degree-of-freedom (DoF) scaling for the continuous-time amplitude limited AWGN channels in radio frequency (RF) and intensity modulated optical communication (OC) channels. More precisely, we study how the capacity varies in terms of the OFDM block transmission time $T$, bandwidth $W$, amplitude $A$, and the noise spectral density $N_0$. We first find suitable discrete encoding spaces for both cases, and prove that they are convex sets that have a semi-definite programming (SDP) representation. Using tools from convex geometry, we find lower and upper bounds on the volume of these encoding sets, which we exploit to drive pretty sharp lower and upper bounds on the capacity. We also study a practical Tone-Reservation (TR) encoding algorithm and prove that its performance can be characterized by the statistical width of an appropriate convex set. Recently, it has been observed that in high-dimensional estimation problems under constraints such as those arisen in Compressed Sensing (CS) statistical width plays a crucial role. We discuss some of the implications of the resulting statistical width on the performance of the TR. We also provide numerical simulations to validate these observations.