Acquisition of timing and Doppler-shift in a direct-sequence spread spectrum system

A model for acquisition of a spread-spectrum signal distorted by Doppler shifts is discussed. A joint acquisition of both the time delay and the distorting Doppler velocity, modeled by a Markov chain, is described. The received signal is assumed to undergo a compression/expansion as a function of time. It is shown that the simultaneous presence of the unknown Doppler shift and timing necessitates a modification of conventional serial acquisition strategies. A two-dwell system, using a short-duration correlation to acquire the timing information, with a subsequent longer correlation to acquire the Doppler velocity, is proposed. It is assumed that the system recovers from false alarms, and hence the performance of the system is given by the mean and variance of the synchronization time. Two systems of linear equations that can be solved for the mean and variance are developed. The large sparse matrices are solved numerically. As an example, the performance of such a system for a spread-spectrum signal having a period of 15 chips, a chip duration of t_c=10^-4 s, and operating over a range of 0.0 to 10.0 dB signal-to-noise ratio with v/c equal to 0.0024 is analyzed