On the optimal sum capacity for OFDM with on/off power allocation and imperfect channel estimation

We consider a point-to-point frequency-selective orthogonal frequency division multiplexing (OFDM) channel. Based on a linear minimum mean square error (MMSE) channel estimates from pilot signals, a receiver determines a set of 'on' or active subchannels that maximizes a lower bound on a sum capacity. The capacity bound optimization is constrained to a fixed total power, which is a sum of total training power and a sum of equal transmission power on activated subchannels. We show that as number of subchannels N increases, the optimal training power converges to a deterministic constant while the optimal number of active subchannels increases at the rate of (log N -2 log log N)
²
. The associated capacity bound tends to infinity at the rate of log N -2 log log N while that with uniform power allocation of all subchannels converges to a constant.