Writing on Fading Paper, Dirty Tape With Little Ink: Wideband Limits for Causal Transmitter CSI

Abstract

A wideband Rayleigh 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 pulse position modulation in IEEE Trans. Inf. Theory, vol. 46, no. 4, Apr. 2000 and IEEE Trans. Inf. Theory, vol. 52 no. 5, May 2006) is shown to achieve the capacity of this channel in the wideband limit. This strategy transmits energy only when the channel gain exceeds a threshold, hence only needs causal transmitter CSI. In the wideband limit, this capacity without any receiver CSI is the same as the capacity with full receiver CSI, which is proportional to the logarithm of the bandwidth. Similar threshold-based pulse position modulation is shown to achieve the capacity per unit cost of the dirty-tape channel (dirty paper channel with causal transmitter CSI and no receiver CSI), which equals its capacity per unit cost with full receiver CSI. Then, a general discrete channel with i.i.d. states is considered. Each input has an associated cost and a zero cost input “0” exists. The channel state is assumed 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 noncausal transmitter CSI is derived.