Short Codes with Mismatched Channel State Information: A Case Study

The rising interest in applications requiring the transmission of small amounts of data has recently lead to the development of accurate performance bounds and of powerful channel codes for the transmission of short-data packets over the AWGN channel. Much less is known about the interaction between error control coding and channel estimation at short blocks when transmitting over channels with states (e.g., fading channels, phase-noise channels, etc...) for the setup where no a priori channel state information (CSI) is available at the transmitter and the receiver. In this paper, we use the mismatched-decoding framework to characterize the fundamental tradeoff occurring in the transmission of short data packet over an AWGN channel with unknown gain that stays constant over the packet. Our analysis for this simplified setup aims at showing the potential of mismatched decoding as a tool to design and analyze transmission strategies for short blocks. We focus on a pragmatic approach where the transmission frame contains a codeword as well as a preamble that is used to estimate the channel (the codeword symbols are not used for channel estimation). Achievability and converse bounds on the block error probability achievable by this approach are provided and compared with simulation results for schemes employing short low-density parity-check codes. Our bounds turn out to predict accurately the optimal trade-off between the preamble length and the redundancy introduced by the channel code.