Probability distribution of the acquisition time for MAX/TC synchronization with verification

We address code acquisition in spread-spectrum transmissions using the MAX/TC criterion with verification mode. The probability density function (p.d.f.) of the acquisition time is derived by using the direct approach, and the flow-graph approach is used for the moments. These expressions contain previous results on serial and parallel acquisition as particular cases. Two approximations to the acquisition time p.d.f. are given in terms of a Gaussian distribution and a χ² distribution. Their accuracy is evaluated with reference to the case of direct-sequence spread-spectrum multiple-access (DS-SSMA) systems. The χ² approximation is shown to be very accurate, and much better than the Gaussian approximation