Optimum Frame Synchronization for Biorthogonally Coded Data

The optimum procedure for locating a sync word periodically inserted in uncoded binary data received over a binary symmetric channel is based on the Hamming or bit distance metric. This concise paper addresses the corresponding frame-sync problem for biorthogonally coded data transmitted over the additive white Gaussian noise (AWGN) channel. For conceptual convenience, the
k
-bit words from the decoder output are treated as "super symbols" from an alphabet of dimension 2
^k
. It is argued that the optimum sync-word search over the decoded data stream is based on a supersymbol distance rule matched to the properties of the biorthogonally coded transmissions over the noisy channel. An optimum frame-sync acquisition algorithm based on this distance rule is formulated, and its performance is investigated. As an example, the performance of this optimum frame-sync algorithm is contrasted analytically with that of a Hamming distance algorithm operating on decoded (32, 6) biorthogonal data, a case of interest to some recent unmanned American space missions.