Partial period autocorrelations of geometric sequences

For a binary pseudorandom sequence {S_i} with period N, the partial period autocorrelation function A_S(τ,k,D) is defined by correlating the portion of the sequence within a window of size D, and start position k, with the portion in another window of the same size but starting τ steps later in the sequence. A distribution of possible partial period autocorrelation values is obtained by allowing the start position K to vary over all possible values O⩽k<N. The expectation value is proportional to the periodic autocorrelation function A_S(τ). In this paper the variance in the partial period autocorrelation values is estimated for a large class of binary pseudorandom sequences, the so-called “geometric sequences.” An estimate is given for the minimum window size D which is needed in order to guarantee (with probability of error less than ε), that a signal has been synchronized, based on measurement of a single partial period autocorrelation value