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The extreme weakness of a global navigation satellite system (GNSS) signal makes it vulnerable to a wide variety of interfering signals, falling within the GNSS frequency bands. One of the main classes of these disturbing signals is represented by continuous wave interference (CWI), which include all signals that can be effectively represented as pure sinusoids. This paper deals with the performance of GNSS signal acquisition in the presence of CWI. An original model describing the impact of CWI on signal acquisition is proposed and the expressions of both false alarm and detection probabilities are mathematically formulated and validated by Monte Carlo simulations. The paper also investigates the role of the interference amplitude and frequency along with the impact of different system parameters on the detection and false alarm probabilities.