How Good is Phase-Shift Keying for Peak-Limited Fading Channels in the Low-SNR Regime ?

This paper investigates the achievable information rate of phase-shift keying (PSK) over frequency non-selective Rayleigh fading channels without channel st ate information (CSI). The fading process exhibits general temporal correlation characterized by it s spectral density function. We consider both discrete-time and continuous-time channels, and find their asymptotics at low signal-to-noise ratio (SNR). Compared to known capacity upper bounds under peak co nstraints, these asymptotics usually lead to negligible rate loss in the low-SNR regime for slowly time-varying fading channels. We further specialize to case studies of Gauss-Markov and Clar ke’s fading models. I. I NTRODUCTION For Rayleigh fading channels without channel state informat ion (CSI) at low signal-tonoise ratio (SNR), the capacity-achieving input gradually t ends to bursts of “on” intervals sporadically inserted into the “off” background, even unde r vanishing peak power constraints [1]. This highly unbalanced input usually imposes implemen tation challenges. For example, it is difficult to maintain carrier frequency and symbol timi ng during the long “off” periods. Furthermore, the unbalanced input is incompatible with lin ear codes, unless appropriate symbol mapping ( e.g., M -ary orthogonal modulation with appropriately chosen cons tellation sizeM ) is employed to match the input distribution. This paper investigates the achievable information rate of phase-shift keying (PSK). PSK is appealing because it has constant envelope and is amenable t o lin ar codes without additional symbol mappings. Focusing on low signal-to-noise ratio (SN R) asymptotics, we utilize a recursive training scheme to convert the original fading channel without CSI into a series of parallel sub-channels, each with estimated CSI but additional noise t hat remains circular complex white Gaussian. The central results in this paper are as follows. F ir t, for a discrete-time channel whose unit-variance fading process {Hd[k] : −∞ < k < ∞} has a spectral density function SHd(e ) for −π ≤ Ω ≤ π, the achievable rate is (1/2) · [ (1/2π) · ∫ π −π S Hd(ejΩ)dΩ − 1 ] · ρ + o(ρ) nats per symbol, as the average channel SNR ρ → 0. This achievable rate is at most (1/2) · ρ + o(ρ) away from the channel capacity under peak SNR constraint ρ. Second, for a continuous-time channel whose unit-variance fading proc ess{Hc(t) : −∞ < t < ∞} has a spectral density functionSHc(jω) for −∞ < ω < ∞, the achievable rate as the input symbol durationT → 0 is [ 1 − (1/2πP ) · ∫∞ −∞ log (1 + P · SHc(jω)) dω ] ·P nats per unit time, where P > 0 is the envelope power. This achievable rate coincides with t he channel capacity under peak envelopeP . We further apply the above results to specific case studies of Gauss-Markov fading models (both discrete-time and continuous-time) as well as a conti nuous-time Clarke’s fading model. For discrete-time Gauss-Markov fading processes with inno vati n rate 1, the quadratic behavior of the achievable rate becomes dominant only for ρ . Our results, combined with previous results for the high-SNR asymptotics, sugges t that coherent communication can This work has been supported in part by the State of Indiana through the 2 1st Century Research Fund, by the National Science Foundation through contract ECS03-29766, and by the Fellows hip of the Center for Applied Mathematics of the University of Notre Dame. essentially be realized for ≤ ρ ≤ 1/ . For Clarke’s model, we find that the achievable rate scales sub-linearly, but super-quadratically, as O (log(1/P ) · P ) nats per unit time as P → 0. The remainder of this paper is organized as follows. Section II describes the channel model and the recursive training scheme. Section III deals with th e discrete-time channel model, and Section IV the continuous-time channel model. Finally Sect ion V provides some concluding remarks. Throughout the paper, random variables are in capi tal letters while their sample values are in small letters. All logarithms are to base e, and information units are measured in nats. Proofs and further interpretations can be found in the journ al version of the paper [2]. II. CHANNEL MODEL, RECURSIVETRAINING SCHEME, AND EFFECTIVE SNR We consider a scalar time-selective, frequency non-select ive Rayleigh fading channel, written in baseband-equivalent continuous-time form as X(t) = Hc(t) · S(t) + Z(t), for −∞ < t < ∞, (1) whereS(t) ∈ C andX(t) ∈ C denote the channel input and the corresponding output at tim e instantt, respectively. The additive noise {Z(t) : −∞ < t < ∞} is modeled as a zero-mean circular complex Gaussian white noise process with E{Z(s)Z†(t)} = δ(s − t). The fading process{Hc(t) : −∞ < t < ∞} is modeled as a wide-sense stationary and ergodic zero-mean circular complex Gaussian process with unit variance E{Hc(t)H c (t)} = 1 and with spectral density functionSHc(jω) for −∞ < ω < ∞. Additionally, we impose a technical condition that {Hc(t) : −∞ < t < ∞} is mean-square continuous, so that its autocorrelation fun ction KHc(τ) = E{Hc(t + τ)H c (t)} is continuous forτ ∈ (−∞,∞). Throughout the paper, we restrict our attention to PSK over t h continuous-time channel (1). For technical convenience, we let the channel input S(t) have constant envelope P > 0 and piecewise constant phase, i. ., S(t) = S[k] = √ P · e, if kT ≤ t < (k + 1)T, for −∞ < k < ∞.1 The symbol durationT > 0 is determined by the reciprocal of the channel input bandwidth. 2 Applying the above channel input to the continuous-time cha nnel (1), and processing the channel output through a matched filter, 3 we obtain a discrete-time channel X[k] = √ ρ · Hd[k] · S[k] + Z[k], for −∞ < k < ∞. (2) For the discrete-time channel (2) we can verify that • The additive noise{Z[k] : −∞ < k < ∞} is circular complex Gaussian with zero mean and unit variance, i.e., Z[k] ∼ CN (0, 1), and is independent, identically distributed (i.i.d.) for different k. • The fading process{Hd[k] : −∞ < k < ∞} is wide-sense stationary and ergodic zeromean circular complex Gaussian, with Hd[k] being marginallyCN (0, 1). We further notice that {Hd[k] : −∞ < k < ∞} is obtained through sampling the output of the matched filter, hence its spectral density function is SHd(e ) = 1 √ ∫ T 0 ∫ T 0 KHc(s − t)dsdt · ∞ ∑