Writing on Fading Paper and Causal Transmitter CSI

TL;DR

This paper investigates the capacity of wideband fading channels with causal transmitter CSI and no receiver CSI, proposing simple orthogonal coding schemes that achieve capacity limits similar to full CSI scenarios, with applications to discrete channels and capacity per unit cost.

Contribution

It introduces a simple orthogonal coding scheme for channels with causal transmitter CSI that achieves capacity equivalent to full CSI scenarios, extending to discrete channels and capacity per unit cost.

Findings

Orthogonal coding achieves channel capacity with causal CSI.

Capacity with causal CSI matches full CSI in wideband limit.

Proposed coding scheme extends to discrete channels with cost constraints.

Abstract

A wideband fading channel is considered with causal channel state information (CSI) at the transmitter and no receiver CSI. A simple orthogonal code with energy detection rule at the receiver (similar to [6]) is shown to achieve the capacity of this channel in the limit of large bandwidth. This code transmits energy only when the channel gain is large enough. In this limit, this capacity without any receiver CSI is the same as the capacity with full receiver CSI--a phenomenon also true for dirty paper coding. For Rayleigh fading, this capacity (per unit time) is proportional to the logarithm of the bandwidth. Our coding scheme is motivated from the Gel'fand-Pinsker [2,3] coding and dirty paper coding [4]. Nonetheless, for our case, only causal CSI is required at the transmitter in contrast with dirty-paper coding and Gel'fand-Pinsker coding, where non-causal CSI is required. Then we consider a general discrete channel with i.i.d. states. Each input has an associated cost and a zero cost input "0" exists. The channel state is assumed be to be known at the transmitter in a causal manner. Capacity per unit cost is found for this channel and a simple orthogonal code is shown to achieve this capacity. Later, a novel orthogonal coding scheme is proposed for the case of causal transmitter CSI and a condition for equivalence of capacity per unit cost for causal and non-causal transmitter CSI is derived. Finally, some connections are made to the case of non-causal transmitter CSI in [8].