US Patent No. 6,757,323 -- Rapid Signal Acquisition by Spread Spectrum Transceivers
This patent involves some beautiful math related to Kasami sequences. Kasami sequences are useful in communication systems because they have very good cross-correlation properties.
This is another invention developed by the client I call my "genius inventors" because they are so incredibly innovative and prolific. In many ways their inventions are years ahead of their time, and one of the greatest obstacles they face is getting the people they work with (including the Department of Defense) to understand their inventions and its capabilities.
Spread spectrum transceivers communicate using code sequences having low cross-correlations and well-peaked autocorrelations. The initial communications involve broadcasting a beacon signal consisting of a beacon packet repeated at regular intervals (the cycle time). The code sequences may be period-(2n−1) Small Kasami sequences; the beacon packet is a repeated series of (2n/2+1) period-(2n/2−1) progenitor maximal sequences, and behaves like a member of the Kasami family. The acyclic autocorrelation of the beacon packet has regularly-spaced sharp peaks modulated by a pyramidal envelope. The initial communications involve calculating the correlation between the received signal and delayed versions of an internally-generated beacon packet. The length of the initial communications is proportional to the square of the cycle time divided by the width of the acyclic autocorrelation. Synchronization involves locating the peak of the pyramidal envelope, and has a length related to the cycle time times the number of peaks in the acyclic autocorrelation.