The Subsequence Weight Distribution of Summed Maximum Length Digital Sequences

The statistics of the weight distribution of subsequences of digital sequences called hybrid-sum sequences, formed from the modulo-two sum of several maximum length sequences, are analyzed. The results indicate that a relation exists between the statistics of the weight distribution of the subsequences and the characteristic polynomials of the component maximum length sequences. An analysis procedure is developed for identifying a large group of sequences with good statistical properties for applications requiring the generation of analog pseudorandom noise by filtering digital sequences. Using the analysis approach, the filtering process is approximated by the convolution of the sequence with a sum of unit-step functions. The first five moments of the resulting weighttuples are used to characterize the statistics of the filtered sequence. The analysis reveals that the statistics of the signals generated with the hybrid-sum generator are potentially superior for many applications to the statistics of signals generated with single maximum length sequence generators. Furthermore, fewer calculations are required to evaluate the statistics of a large group of hybrid-sum generators than are required to evaluate the statistics of the same size group of approximately equivalent maximum length sequences. Efficient algorithms that may be used to evaluate the statistics of hybrid-sum sequences are indicated, and example calculations are given.