Abstract-In the present work, a discrete-time stationary Rayleigh flat-fading channel with unknown channel state information at the transmitter and the receiver is considered. The law of the channel is assumed to be known at the receiver and the fading process is supposed to be a stationary Gaussian process with an absolutely summable autocorrelation function. The conditional per symbol entropy of the output given the input is shown to converge to a constant for almost every realization of i.i.d. input variables. This implies the existence of the corresponding conditional entropy rate. Moreover, a novel inequality yielding a lower bound for the rate is derived.
I. MOTIVATION AND SETUPWe consider a stationary Rayleigh flat-fading channel where the channel state information is unknown at the transmitter and the receiver. Moreover, the law of the channel is assumed to be known at the receiver. Often, this channel is referred to as noncoherent fading channel. As this scenario corresponds to the basic model of nearly all realistic mobile communication systems it is particularly important. Nevertheless, determining the capacity of this channel turns out to be notoriously difficult and the problem is still open in general.There have been already several attempts to approximate the capacity of noncoherent fading channels by bounds, see, e.g., [1], [2]. For the case of i.i.d. zero-mean proper Gaussian input symbols, which are capacity-achieving in the coherent setup, in [3] bounds on the achievable rate have been derived. One of the hardest problems when studying the capacity or achievable rate of stationary Rayleigh fading channels is the evaluation of the conditional entropy rate of the channel fading process. This is the main topic of the present paper. The proofs and extended material are contained in technical report [4].
A. Channel ModelWe consider an ergodic discrete-time jointly proper Gaussian [5] flat-fading channel, whose output at time k is given by