Upper Bounding the Capacity of Bandlimited AWGN Channels With Peak-Amplitude-Limited Inputs

We derive upper bounds on the capacity of bandlimited linear time-invariant channels with additive white Gaussian noise, where the input signal is subject to a peak-amplitude constraint. It has been shown that when the channel impulse response is square-integrable, unit processes are sufficient to achieve the supremum of the achievable rate for this channel (Ozarow et al., 1988). However, because the characterization of such processes, originally obtained by McMillan (1955) and Shepp (1967), is computationally intractable to evaluate explicitly, the capacity of these channels has not been explicitly computed. To circumvent this difficulty, several works have considered the derivation of lower and upper bounds on the capacity of these channels, expressing such bounds via a scaling factor that multiplies the signal-to-noise ratio (SNR) in the capacity expression for bandlimited Gaussian channels under an average power constraint. While recent works have focused on the lower bound, we focus on deriving upper bounds on the capacity using Gaussian input processes. We utilize a characterization of the autocorrelation function for unit processes derived by Quintanilla (2008), which, following the approach of Shamai and Bar-David (1989), is expressed as spectral constraints on the input power spectral density. The resulting optimization problem is still difficult to solve, so we resort to bounding the SNR scaling factor from above and below. The upper bound on the scaling factor is obtained by restricting the optimization to a finite-dimensional representation and the lower bound is obtained using an explicit construction of a unit process. Combined, these results characterize the upper bound on the SNR scaling factor that can be obtained using Gaussian input processes within a 1.84% relative gap.

M.Sc. student under the supervision of Prof. Shlomo Shamai and Prof. Ron Dabora.