A new approximation for the distribution of the probability ratio in a sequential probability ratio test (SPRT) using noncoherent integration across a full code period is presented. The new approximation is valid for the carrier-to-noise power ratios typically encountered in GPS acquisition (20？dB-Hz ≤ ≤ 50？dB-Hz), and it allows accurate theoretical performance analysis of the SPRT to be carried out for signals in this range, eliminating the need for lengthy simulations for each scenario under investigation. Thus, the SPRT performance can be readily compared to that of other acquisition strategies for receiver design. Previous approximations in the literature are not valid in the range 20？dB-Hz ≤ ≤ 50？dB-Hz. 1. Introduction Acquisition of direct-sequence spread spectrum (DS/SS) code division multiple access (CDMA) signals, such as the Global Positioning System (GPS) L1 C/A signals [1], involves synchronising the received signal with a locally generated replica signal. Synchronisation is carried out in two dimensions, namely, code phase, , and Doppler uncertainty frequency, . The received signal is correlated with the local replica signal at each code phase and Doppler estimate, and the outputs of the correlator are combined to make a decision that the signal is present and correctly synchronised (an decision) or that the signal is not present or not synchronised (an decision). Each estimate, , is referred to as a “cell” in the search space. The decision making procedure or “search-control strategy” can be based on a fixed or variable number of correlator outputs, and its goal is to make a correct decision in the minimum time. Thus, the main performance metrics are the time required for a decision to be made (the dwell time) in an or an cell, or , respectively, and the probabilities of correct detection, , and false alarm, ？ (the subscript “ ” denotes the “cell level” probabilities). Wald's sequential probability ratio test (SPRT) [2] is known to be the optimal strategy in terms of simultaneously minimising the mean dwell time in both and cells, and , respectively, for a particular performance pair, . The SPRT is widely used throughout the field of communications, with recent applications in many areas, for example, radar detection [3], spectrum sensing for cognitive radio [4], and CDMA acquisition [5–7]. The SPRT is also commonly used as a benchmark for comparing the acquisition performance of CDMA systems, for example, comparing the performances of different code families [8] or the performance of a system under varying fading conditions [9]. Despite
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