Codes for Unknown ISI Channels

The traditional method of sending a training signal to identify a channel, followed by data, may be viewed as a simple code for the unknown channel. Results in blind sequence detection suggest that performance similar to this traditional approach can be obtained without training. However, for short packets and/or time-recursive algorithms, signi cant error oors exist due to the existence of sequences that are indistinguishable without knowledge of the channel. In this paper, we reconsider training signal design in light of recent results in blind sequence detection. We design training codes which combine modulation and training. In order to design these codes, we nd an expression for the pairwise error probability of the joint maximum likelihood (JML) channel and sequence estimator. This expression motivates a pairwise distance for the JML receiver based on principal angles between the range spaces of data matrices. The general code design problem (generalized sphere packing) is formulated as the clique problem associated with an unweighted, undirected graph. We provide optimal and heuristic algorithms for this clique problem. For short packets, we demonstrate that signi cant improvements are possible by jointly considering the design of the training, modulation, and receiver processing.