Tight upper and lower bounds to the information rate of the phase noise channel

Abstract

Abstract-Numerical upper and lower bounds to the information rate transferred through the additive white Gaussian noise channel affected by discrete-time multiplicative autoregressive moving-average (ARMA) phase noise are proposed in the paper. The state space of the ARMA model being multidimensional, the problem cannot be approached by the conventional trellisbased methods that assume a first-order model for phase noise and quantization of the phase space, because the number of state of the trellis would be enormous. The proposed lower and upper bounds are based on particle filtering and Kalman filtering. Simulation results show that the upper and lower bounds are so close to each other that we can claim of having numerically computed the actual information rate of the multiplicative ARMA phase noise channel, at least in the cases studied in the paper. Moreover, the lower bound, which is virtually capacity-achieving, is obtained by demodulation of the incoming signal based on a Kalman filter aided by past data. Thus we can claim of having found the virtually optimal demodulator for the multiplicative phase noise channel, at least for the cases considered in the paper.

Upper and Lower Bounds to the Information Rate Transferred Through First-Order Markov Channels With Free-Running Continuous State

“…The expression of the upper bound is the same as [17], [23], while the lower bound is new. To compute the differential entropy rates appearing in (51) and (52), we need to work out h(R) and h(R|X) by Bayesian filtering, and h(S|R) and h(S|R, X) by Bayesian smoothing.…”

“…Also, the evaluation method of the upper bound is different from [17], where trellis-based techniques are adopted. Both the upper bound and the lower bound are substantially tighter than those of [23] especially when, as it happens with the phase noise spectrum used for deriving the numerical results, inference becomes challenging due to strong phase noise and to the high-dimensional state space.…”

“…The only papers studying the information rate transferred through a channel affected by ARMA phase noise we are aware of are [14], [23], where the method of particle filtering is adopted. Recent investigations on the capacity of the fading channel with hidden Markov state can be found in [24]- [27], the most popular model for the fading spectrum being the first-order ARMA model of [24].…”

“…From these bounds we derive upper and lower bounds to the information rate transferred between the input and the output of communication channels with free-running ARMA continuous hidden state. Specifically, the upper bound is already published in [17], [23], while the lower bound is new. Evaluation of these bounds, which is presented here for the fading channel and for the phase noise channel, is based on the Kalman filter and on the particle filter.…”

“…The novelty compared to [14], where particle filtering techniques are used to compute the information rate, is that we present here upper and lower bounds, while by [14] one can compute only an approximation. Compared to [23], here the evaluation method of the upper bound is new, because one of the terms appearing in the bound is based on a distribution that is allowed here to be multi-modal, while in [23] that distribution is approximated to a Gaussian one. Also, the evaluation method of the upper bound is different from [17], where trellis-based techniques are adopted.…”