## 1 Introduction
This is usually achieved by coherently accumulating signal samples over an extended period, known as the coherent integration (CI) [6]. However, several critical factors impose constraints on the maximum CI time, including the presence of unknown data bit transitions, residual frequency errors in the received signal, and the existence of phase noise due to receiver oscillator instabilities [3]. Sign reversals within the CI window may occur due to unknown data bit transitions in the received signal, potentially causing partial or complete cancellation of correlation power. The impact of these unknown bit transitions can be mitigated by using pilot signals [6]. A pilot signal is transmitted within the Galileo signals (E1, E5a, E5b, E6) and modernized Global Positioning System (GPS) signals (L1C, L2C, L5) to improve the signal acquisition and tracking [3]. Both the data and the pilot signals are modulated with a unique code to distinguish them from other signals. This code is a pseudo-random noise (PRN) sequence known as a spreading code. Several GNSS signals build a long spreading sequence in a tiered manner whereby a secondary code sequence is used to modify successive repetitions of a primary code. Residual frequency errors impact the received signal after Doppler wipe-off, potentially leading to signal cancellation if the CI exceeds a certain time duration. A smaller residual frequency shift allows for an extended CI before the Signal-to-Noise Ratio (SNR) gain diminishes due to phase wrapping. This phenomenon introduces a trade-off between SNR gain and computational load, as minimizing the residual frequency shift necessitates a finer Doppler search, demanding increased computational operations [3]. The quality of the user receiver clock also limits the CI time. At present, temperature-compensated crystal oscillators (TCXOs) and oven-controlled crystal oscillators (OCXOs) are the most commonly utilized types of clocks. As per the simulations outlined in [7; 11; 12], the TCXO restricts the CI interval to approximately 100 ms, whereas the OCXO enables the utilization of CI times extending up to 1 s. The sensitivity of the receiver is influenced by this fact. Consumer-grade receivers commonly utilize TCXO clocks, while OCXO clocks are typically reserved for professional applications. The higher cost of OCXO clocks, as opposed to TCXO clocks, restricts their widespread adoption in mass-market GNSS receivers [6]. For practical considerations, the CI time cannot be indefinitely extended due to the reasons mentioned above. Consequently, the only recourse is to employ a combination of coherent and non-coherent integration (NCI) to prolong the overall integration time and enhance receiver sensitivity [3; 7]. More specifically, the acquisition begins with coherent correlation, followed by non-coherent accumulation of outputs from multiple coherent correlations through nonlinear operations. Non-coherent accumulation is commonly achieved through the application of Post-Detection Integration (PDI) techniques. These techniques address the constraints associated with coherent accumulation and enable the receiver to capture satellites even in conditions of extremely low \(C/N_{0}\)[7]. However, the NCI time cannot be increased without bounds either, due to limited Doppler estimation accuracy, receiver clock drift instabilities, and the relative movement between the satellites and the receiver [3]. Any errors in determining the Doppler frequency can affect the baseband signal, akin to a mismatch between the chip duration of the received signal and the local code replica. If this mismatch extends over a significant integration interval, it causes blurring in the overall correlation, introducing bias in the final estimation of the code phase [3]. The duration of the NCI also remains limited due to local clock drift instabilities, unless methods to estimate the clock dynamics are implemented [3].